Kabus.eu

Pigreads: The Python-integrated GPU-enabled reaction-diffusion solver using OpenCL for cardiac electrophysiology and other applications

Mon, 23 Feb 2026 00:00:00 +0000

This article was previously published in Computer Physics Communications, 110088 (Kabus, et al., 2026). More about Pigreads can be found online at its code repository on GitLab, its documentation. A PDF of the article can be downloaded at the website of Computer Physics Communications 110088 (DOI: 10.1016/j.cpc.2026.110088), or directly here.

Authors:

Desmond Kabus^1,2 (ORCID: 0000-0002-6965-5211)
Hans Dierckx^1,2 (ORCID: 0000-0003-0899-8082)
Tim De Coster^1,2 (ORCID: 0000-0002-4942-9866)

Institutions:

Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Albinusdreef 2, 2333 ZA Leiden, The Netherlands
Mathematical Institute, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands

DOI: 10.1016/j.cpc.2026.110088

Keywords:

reaction-diffusion
finite differences
Python
OpenCL
GPU
electrophysiology

Highlights:

GPU-accelerated reaction-diffusion solver in up to three dimensions.
Minimal NumPy-friendly API with OpenCL kernels.
Includes commonly used reaction terms for cardiac electrophysiology.
Open-source, tested, documented, with examples and tutorials.

Abstract:

Pigreads is a streamlined Python module for efficient numerical solution of reaction-diffusion systems on graphics cards (GPU), with CPU fallback. It exposes a simple and straight-forward NumPy-compatible API. Users may employ built-in models – including electrophysiology examples – or supply custom reaction terms. Supported features include 0D-3D uniform Cartesian grids, no-flux and periodic boundary conditions, anisotropic diffusion, spatially varying diffusion and reaction, and localised source terms. The project is open-source, tested, documented, and distributed with examples and tutorials.

Introduction

Reaction-diffusion systems appear across physics, chemistry, biology, and ecology; they are used to study pattern formation and excitable waves, for instance in electrophysiology (Cannon et al., 2014; Clayton et al., 2011; Kapral & Showalter, 1995; Lechleiter et al., 1991; Murray, 1976; Rotermund et al., 1990). Their numerical solution can be computationally demanding, so efficient implementations are essential, ranging from ad-hoc research codes to large general software packages (Niederer et al., 2011). However, modern applications such as inverse problems, parameter estimation, uncertainty quantification, and machine learning, require a compact code with minimal installation requirements that is portable across platforms, runs fast, and can easily be integrated into other software or frameworks.

Pigreads is a Python module (van Rossum et al., 1995) that provides a minimal, concise, and NumPy-friendly API (Harris et al., 2020) to define and run reaction-diffusion simulations while performing the costly computations efficiently on accelerators: Computational kernels implemented in OpenCL (Stone et al., 2010) run on both GPUs and CPUs as a fallback via PyOpenCL (Klöckner et al., 2012). The Pigreads module includes several pre-defined models relevant to electrophysiology; users can also supply custom reaction terms.

Representative examples are illustrated in Fig. 1: simulations of the conduction in a 1D cable of excitable cells, 2D pieces of cardiac tissue like monolayers in culture wells, and numerical electrophysiology experiments of whole organs in 3D like the ventricles or atria of the human heart. The 0D case, i.e., without diffusion, is also supported to perform single cell simulations.

Figure 1: Pigreads can be used to solve reaction-diffusion systems in up to three dimensions. A. Cardiac action potentials for the model by Courtemanche et al. (1998) in the 0D case, i.e., without diffusion. B. Three travelling pulses in a cable (1D) followed by a fourth blocked pulse in the same model. C. A spiral wave in a circular 2D domain in the model by Marcotte & Grigoriev (2017) that was stimulated by increasing the main variable on a line between high and low values of the recovery variable. D. Pseudo-electrograms for a travelling wave in a 2D piece of tissue with periodic boundary conditions on the top and bottom and different cell types for mid-wall, endo-, and epi-cardial cells following the model by ten Tusscher & Panfilov (2006) with 15% fibrosis, stimulated three pulses at the locations and times labelled S0–2. E. Three-dimensional cardiac electrophysiology simulations in human bi-ventricular (Hren, 1996, 1998; Keldermann et al., 2009) and human bi-atrial (De Coster et al., 2018; Dossel et al., 2011) geometries with faster diffusion along muscle fibres. A single pulse is sent out from the apex or the sino-atrial node, respectively. We use the models by ten Tusscher & Panfilov (2006) for the ventricles and by Courtemanche et al. (1998) for the atria.

The module is available on GitLab, where it is properly tested using continuous integration, and is well-documented on Read the Docs, with examples and tutorials. Pigreads can be installed from the Python Package Index PyPI via:

$ pip install pigreads

Consider also installing the optional dependencies for the command line interface (CLI) and creating plots and videos via:

$ pip install 'pigreads[all]'

The CLI can be used to run and visualise Pigreads simulations defined in a YAML file (Net, 2023) adhering to a Pydantic scheme as described in the documentation.

Mathematical problem definition

We generally define reaction-diffusion systems as: $$ \partial_t {{\underline{{u}}}}(t, {{\bm{{x}}}}) = \underbrace{ {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}}({{\bm{{x}}}}) \nabla {{\underline{{u}}}}(t, {{\bm{{x}}}}) }_{\text{diffusion}} + \underbrace{ {{\underline{{r}}}}({{\underline{{u}}}}, {{\bm{{x}}}}) }_{\text{reaction}} + \underbrace{ {{\underline{{s}}}}(t, {{\bm{{x}}}}) }_{\text{source}} \qquad{(1)}$$ for time $t\in[0, T]$ and space ${{\bm{{x}}}} = {{{{\left[ x, y, z \right]}}}^\mathrm{T}} \in\heartsuit\subset\mathbb R^3$ in the domain $\heartsuit$, with initial conditions ${{\underline{{u}}}}(0, {{\bm{{x}}}})$ and no-flux boundary conditions $0 = {{\bm{{n}}}}\cdot {{\bm{{D}}}} \nabla u$ for diffused variables on ${{\bm{{x}}}} \in \partial\heartsuit$. At each point in time and space, the states vector ${{\underline{{u}}}}(t, {{\bm{{x}}}}) \in \mathbb R^{\mathtt{Nv}}$ consists of Nv elements, the so-called states or variables, $u_0$, $u_1$, …, $u_{\mathtt{Nv}-1}$. For the diffusion term, we define the diffusivity matrix ${{\bm{{D}}}}({{\bm{{x}}}}) \in \mathbb R^{3\times 3}$ and the selection matrix ${{\underline{{P}}}} \in \mathbb R^{\mathtt{Nv}\times\mathtt{Nv}}$ to select which variables to diffuse and how strongly. The selection matrix often takes the sparse form ${{\underline{{P}}}} = \operatorname{diag}(P_{0}, ..., P_{\mathtt{Nv}-1})$ with mostly zeros. The reaction term ${{\underline{{r}}}}({{\underline{{u}}}}, {{\bm{{x}}}}) \in \mathbb R^{\mathtt{Nv}}$ describes the local dynamics of the system and may vary in space, for instance in parameters or even in the model equations themselves. We refer to a specific local choice of ${{\underline{{r}}}}_{\mathtt{imodel}}({{\underline{{u}}}})$ with fixed parameters as a model; ${{\underline{{r}}}}$ refers to the reaction term, i.e., all models. The source term ${{\underline{{s}}}}(t, {{\bm{{x}}}}) \in \mathbb R^{\mathtt{Nv}}$ can be used to add external influences to the system, for instance to stimulate the system at specific times and locations.

For no source term, ${{\underline{{s}}}} = 0$, homogeneous and isotropic diffusion, ${{\bm{{D}}}}({{\bm{{x}}}}) = D = \text{const.}$, and only two variables, $u = u_0$ and $v = u_1$, with only $u$ diffusing, $P_0 = 1$, $P_1 = 0$, the system reduces to: $$ \begin{align} \partial_t u(t, {{\bm{{x}}}}) &= D \nabla^2 u(t, {{\bm{{x}}}}) + r_u(u, v) \\ \partial_t v(t, {{\bm{{x}}}}) &= r_v(u, v) \end{align} \qquad{(2)}$$

Technical overview

Pigreads organises simulation components around three objects: an instance of the class Models that defines local reaction terms, an array inhom that defines the domain and which model to use where, and an array states holding all state variables on a uniform Cartesian grid; see also the flowchart in Fig. 2.

Figure 2: Main components of a simulation to solve a reaction-diffusion problem using Pigreads. The state variables at each point in the grid at the frames are stored in the variable states. The main computations are done in the method models.run, which performs Nt forward Euler steps with time step dt, outputting the subsequent frame after these Nt steps.

Numerical choices are deliberately simple: finite-difference stencils for spatial derivatives and explicit forward-Euler time stepping. Supported capabilities include 0D-3D grids, no-flux and periodic boundary conditions, spatially varying anisotropic diffusivity and reaction terms, and source terms.

Diffusion and reaction kernels run in OpenCL on the selected GPU/CPU device; Python performs setup, parameter changes, and in- and output while results remain accessible as NumPy arrays for analysis and plotting, for instance using Matplotlib (Hunter, 2007) or PyVista (Sullivan & Kaszynski, 2019).

Note that execution of the kernels requires a compatible OpenCL runtime environment. For example, for Nvidia graphics cards, the runtime is provided through the CUDA toolkit and corresponding device drivers. For AMD devices, it is available via the AMD ROCm platform, and for Intel devices, through the Intel oneAPI toolkit. Alternatively, the PoCL implementation can be used for CPU-based or generic OpenCL execution.

We validated our code by using it to successfully compute the correct solution to the problem proposed by Niederer et al. (2011): simulation of the model by ten Tusscher & Panfilov (2006) for 100 ms in a slab of 3 × 7 × 20 mm. As a benchmark, on an NVIDIA GeForce RTX 2080 Super GPU with Max-Q Design, the computations take 0.025 s at the coarsest resolution of 0.5 mm and 0.05 ms; and 41.9 s at the finest resolution of 0.1 mm and 0.005 ms.

Unit testing is used to ensure all components of Pigreads work as expected: Code changes are automatically tested using continuous integration where the API is exercised and numerically validated against known results. The tests cover 100% of the code.

Hands-on example

In this section, we define and run a simple 2D simulation of a spiral wave in a two-variable model that breaks up to spiral chaos in a circular domain with no-flux boundary conditions. With this example, it is shown in which steps a Pigreads simulation is usually defined.

First, define the coordinates of the geometry to be used. In this example, we use a 2D plane with 200 points in both $x$ and $y$:

import pigreads as pig
import numpy as np
R = 10
z, y, x = np.mgrid[0:1, -R:R:200j, -R:R:200j]
Nz, Ny, Nx = x.shape
dz, dy, dx = pig.deltas(z, y, x)

Pigreads is optimised for three-dimensional space. For lower-dimensional simulations, set the number of points in additional dimensions to one, as done above for the $z$-dimension. Note that np.mgrid is used to define a dense multi-dimensional grid where the coordinates, in this case, are defined as the range of integers from 0 up to but excluding 1 in z, i.e., $z \in \{0\}$, and -R to R divided into 200 equal steps including both end points in y and x. See the documentation of NumPy for details. The function pig.deltas returns the grid spacing dz, dy, dx that is used in the arrays z, y, x, which each have a shape of (Nz, Ny, Nx). Space in Pigreads is periodic by default, such that the points at index ix = 0 are neighbours with the points at ix = 1 and ix = Nx - 1, and likewise for the other dimensions.

The grid spacing dz, dy, dx must be chosen small enough to resolve the dynamics of the system, but large enough to keep the computational cost manageable. For instance, at insufficient spatial resolution, the conduction is slower along grid axes than diagonal to them. One may use this effect to determine an as large as possible grid spacing dx such that the difference between the fastest and slowest stimulation at some distance is less than a given threshold, for instance 5%.

The integer field inhom defines which points are outside (inhom == 0) and inside the domain $\heartsuit$ (inhom > 0). Pigreads implements no-flux boundary conditions on $\partial\heartsuit$. In our example, we define the domain $\heartsuit$ as a disk of radius $R$:

inhom = np.ones(x.shape, dtype=int)
r = np.linalg.norm((x, y, z), axis=0)
inhom[r >= R] = 0

Values of inhom larger than zero can be used to select one or multiple models, i.e., reaction terms ${{\underline{{r}}}}$. For an inhom value of 1, models[0] is used; and models[1] for a value of 2, etc. One or more of the available models can be selected using an instance of the Models class:

models = pig.Models()
models.add("marcotte2017dynamical", beta=1.389)
Nv = models.Nv

The key of a model – an identifying string – is here used to choose a model and keyword arguments are used to set model parameters to different values than the defaults; the function add may be called multiple times to add more models. When using multiple models, the largest number of variables Nv is used.

Various models are pre-defined in Pigreads, and defining another model is straight-forward, see section 5. A list of the available models can also be obtained programmatically using:

for key in pig.Models.available.keys():
 print(key)
# aliev1996simple
# barkley1991model
# beeler1977reconstruction
# ...

For some models, there are multiple sets of parameters available, they may be accessed via the additional meta data of a model definition:

key = "tentusscher2006alternans"
model_def = pig.Models.available[key]
endo = model_def.meta["parameter sets"]["endo"]
for parameter, value in endo.items():
 print(f"{parameter} = {value}")
# g_Ks = 0.392
# g_to = 0.073
# s_offset = 28.0
# s_variant = 1.0
models_ = pig.Models()
models_.add(key, **endo)

In the next step, memory needs to be allocated to store the state variables of the model. For this, we typically use a 5D array of shape (Nfr, Nz, Ny, Nx, Nv) called states in Pigreads. It consists of the values of all Nv variables at each point in space on Nfr so-called frames in time.

One way to initialise a states array is using the function models.resting_states: It creates an array of the correct shape and fills its first frame with the appropriate resting values for each model according to the value of inhom:

Nfr = 100
states = models.resting_states(inhom, Nframes=Nfr)
assert states.shape == (Nfr, Nz, Ny, Nx, Nv)

Note that this allocates a potentially large amount of memory, as states is a 5D array of shape (Nfr, Nz, Ny, Nx, Nv). For large simulations, it may be better to only store a few frames and overwrite older frames, or to save frames to disk and not keep them in memory, for instance via memory mapping:

states = np.lib.format.open_memmap("states.npy",
 mode='w+', dtype=np.float32,
 shape=(Nfr, Nz, Ny, Nx, Nv),
)
states[:1] = models.resting_states(inhom, Nframes=1)

If the full history over time does not need to be kept, it is also possible to create a states array of only a few frames, continually overwriting them in an infinite loop. We use this scheme in some of the interactive examples that can be found in the code repository.

Initial conditions can then be set in the frame with ifr = 0:

states[0, x < -8, 0] = 1
states[0, y < 0, 1] = 2

Or equivalently expressed in mathematical notation: $$ \begin{align} u_0(t = 0, {{\bm{{x}}}}) &= \begin{cases} 1 & \text{if}\, x < -8 \\ u_{0,\,\text{rest}} & \text{else} \end{cases} \\ u_1(t = 0, {{\bm{{x}}}}) &= \begin{cases} 2 & \text{if}\, y < 0 \\ u_{1,\,\text{rest}} & \text{else} \end{cases} \end{align} \qquad{(3)}$$

The calculation of the diffusion term ${{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}}$ is implemented as a weighted sum of neighbouring points. The weights can be calculated using the function weights, which also requires the diffusivity ${{\bm{{D}}}}$ as input:

diffusivity = pig.diffusivity_matrix(Df=0.03)
weights = models.weights(dz, dy, dx, inhom, diffusivity)

Calculating the weights is an expensive operation. Once they are calculated for a given geometry, it is possible to store and re-use them. Together with inhom, they fully encode all the necessary information about geometry needed for the numerical scheme in models.run below.

Finally, the simulation can be started using run to advance the simulation from one frame to the next, which is done in a loop over the number of frames Nfr. The function run does Nt forward Euler steps and only returns the final states after those steps:

Nt = 200
dt = 0.025
for ifr in range(Nfr - 1):
 states[ifr + 1] = models.run(
 inhom, weights, states[ifr],
 Nt=Nt, dt=dt,
 )

Note that for numerical stability, the time step dt needs to be chosen small enough to at least meet the Courant-Friedrichs-Lewy (CFL) condition (Courant et al., 1928), which informally can be summarised as: “The wave must travel less than one grid length per time step.” A smaller time step will lead to higher accuracy at higher computational cost. We usually choose dt as large as possible while still obtaining a solution that does not change significantly for smaller dt.

Additional arguments to run may be used to add stimulus currents at specific times and locations. See the documentation for more details.

Now that the simulation is done, the 5D array states containing the result can be analysed and visualised, for instance with Matplotlib (Hunter, 2007):

import matplotlib.pyplot as plt
plt.imshow(states[-1, 0, :, :, 0])
plt.show()

Or as a movie using FFmpeg:

from pigreads.plot import movie
movie("example.mp4", states[:, 0, :, :, 0])

Additional more in-depth examples are available in the comprehensive API documentation and in the Git repository of this project.

Reaction terms

The so-called models define the reaction term ${{\underline{{r}}}}$ of the reaction-diffusion equation (Eq. 1). While Pigreads comes with a variety of pre-defined models, it is also easily possible to define a model.

A model can be defined by adding it to the dictionary of available models using just a few lines of OpenCL code, for instance the model by FitzHugh (1961) and Nagumo et al. (1962):

import pigreads as pig
from pigreads.schema.model import ModelDefinition
pig.Models.available["fitzhugh1961impulses"] = \
ModelDefinition(
 name="FitzHugh 1961 & Nagumo 1962",
 description=
 "A 2D simplification of the Hodgkin-Huxley model.",
 dois=[
 "https://doi.org/10.1016/S0006-3495(61)86902-6",
 "https://doi.org/10.1109/JRPROC.1962.288235",
 ],
 variables={"u": 1.2, "v": -0.625},
 diffusivity={"u": 1.0},
 parameters={"a": 0.7, "b": 0.8, "c": 3.0, "z": 0.0},
 code="""
 *_new_u = u + dt * (v + u - u*u*u/3 + z + _diffuse_u);
 *_new_v = v + dt * (-(u - a + b*v)/c);
 """,
)

To import a model from a Myokit (Clerx et al., 2016) or CellML (Clerx et al., 2020) file, a converter is provided in the repository.

While the main application of Pigreads is cardiac electrophysiology, there are various more general pre-defined models: The trivial model defines diffusion of a single variable without a reaction term, the model by Gray & Scott (1983) is used in pattern formation studies, and the models by Barkley (1991) and Hodgkin & Huxley (1952) are some of the simplest and earliest models for electrophysiology in general. Simple example simulations for these models are shown in Fig. 3.

Figure 3: Pigreads includes various pre-defined models. The panels each contain a 2D simulation of a different model and a time trace at the marked reference point in the domain for manually chosen simulation parameters. See also the model overview in Table ¿tbl:models?. A. The trivial model defines diffusion of a single variable. B. The model by Gray & Scott (1983) describes the formation of various patterns depending on chosen parameters. C. One of the simplest models with excitation and recovery enabling spiral waves is the model by Barkley (1991). D. The model by Hodgkin & Huxley (1952) of the electrical conduction in giant nerve fibre of squids is one of the first mathematical models of electrophysiology. Here, instead of a time trace, we show a simulation of a single cell.

An overview of the pre-defined models for cardiac electrophysiology is given in Fig. 4. The simulations shown in this figure have been performed using resolutions, timings and stimuli that, for each model, are found using an automatic procedure, as they depend on the model’s dynamics: time step dt, grid spacing dx, and the stimulus amplitude $\Delta u$ to be applied in a region by instantaneously increasing the first variable by the given value. In the remainder of this section, we outline this automatic procedure to find these parameters for a given cardiac electrophysiology model. Python scripts for this procedure are also published as a separate, second repository.

Figure 4: Models for cardiac electrophysiology that are pre-defined in Pigreads. For each model, we present a the transmembrane voltage for a simulation of a single cell of three pulses with different intervals between the pulses and a 2D simulation of a spiral wave initiated with an S0-S1-S2 protocol. See also the model overview in Table ¿tbl:models?. A.–E. Phenomenological models. F.–J. Models for atrial cells. K.–O. Models for ventricular cells. Note: The spiral wave is not necessarily stable in all models and may break up into spiral chaos.

First we must determine the time step dt and the stimulus $\Delta u$. We consider a large space of possible values $(\mathtt{dt}, \Delta u) \in S \subset [{10}^{-8}, 0) \times [{10}^{-5}, {10}^{5}]$, in which we run single-cell simulations (${{\bm{{D}}}} = 0$). We then narrow down this search space $S$ to the valid regime $S' \subset S$ of a single action potential. We do so by checking these criteria:
- Are all variables finite at all points in time?
- Does the first variable $u$ increase and decrease?
- Does $u$ oscillate less than five times?
- Does $u$ move away from the resting value?
- Does $u$ excite past the stimulated value?
- Does $u$ only change smoothly, i.e., without large jumps?
- Does $u$ cover a large fraction of the observed range of it across simulations?
We then pick dt at the 80th percentile of valid simulations $S'$, followed by $\Delta u$ at the 30th percentile of valid simulations at the chosen dt.
We then run an additional single-cell simulation to determine the action potential duration (APD), i.e., the time that $u$ is above a 30% threshold.
With a 1D simulation at a very fine grid spacing dx obtained via the CFL condition (Courant et al., 1928), we can then determine the conduction velocity (CV). Here, we for now set the diffusivity ${{\bm{{D}}}} = 1$ without loss of generality, as we can rescale it to any desired CV, as outlined below.
We refine the values of CV and APD by measuring them for the third stimulus in 1D simulations with critical timings of stimulation.
In a series of 2D simulations at ${{\bm{{D}}}} = 1$, we determine the largest grid spacing dx such that the difference between the fastest and slowest stimulation at some distance along the axes and diagonal to them is less than 5%.
If necessary for numerical stability in 2D simulations, i.e., the CFL condition, we reduce dt.

The result of this automatic procedure is given in a tabular overview of all pre-defined models in Pigreads in Table ¿tbl:models?. This table includes references to the original publications of the models with their number of variables Nv, as well as the determined parameters for discretisation, dt and dx, the stimulus strength $\Delta u$, and resulting APD and CV at diffusivity ${{\bm{{D}}}} = 1$ in the units of the model. Note that these parameters can be scaled to a desired velocity via: $$ D \propto \mathrm{CV}^2, \quad \mathtt{dx} \propto \mathrm{CV}, \quad \mathtt{dt} = \text{const}. \qquad{(4)}$$ which follows for Eq. 1 from keeping the velocity and diffusivity the same in grid units, i.e., $D \cdot \mathtt{dt} / \mathtt{dx}^2$ and $\mathrm{CV} \cdot \mathtt{dt} / \mathtt{dx}$ are both constant.

In cardiac electrophysiology, stimuli in the form of instantaneous changes in variables, like the provided $\Delta u$, are considered less physically realistic than current-based stimuli which take the form of a source term ${{\underline{{s}}}}$ in Eq. 1. Usually, a suitable amplitude of a current-based stimulus can be obtained by spreading out the stimulus $\Delta u$ over some time $T$: $$ s_0 = \frac{\Delta u}{T} \qquad{(5)}$$

While the provided stimulus amplitude $\Delta u$ and resolutions in time and space work well in our tests, they may not be optimal for all applications. The user may want to adjust them for their specific use case.

Positioning across alternatives

Most solvers for cardiac electrophysiology have no or only partial GPU support; these codes utilise parallelisation on large numbers of CPU cores, for instance alphabetically: Alya Red (Vázquez et al., 2016), BeatBox (Antonioletti et al., 2017), CEPS (Leguèbe et al., 2025), cbcbeat (E. Rognes et al., 2017), Chaste (Pitt-Francis et al., 2009), Continuity (Gonzales et al., 2016), fenicsx-beat (H. Finsberg, 2025), FiniteWave (Nezlobinsky et al., 2026), Ithildin (Kabus et al., 2024), lifex-ep (Africa et al., 2023), openCARP (Plank et al., 2021), OpenCMISS (Bradley et al., 2011), simcardems (H. N. T. Finsberg et al., 2023), and svMultiPhysics (Updegrove et al., 2016).

The solvers that use GPUs vary in terms of the numerical scheme, backend, and language used for the API to set up numerical simulations: Abubu.js is a finite-differences code using WebGL which is accessed via JavaScript in a web browser (Kaboudian et al., 2019); CardioMat uses finite differences in the Matlab Parallel Computing Toolbox (Biasi et al., 2025); and MonoAlg3D is a finite-volumes solver using MPI and CUDA that is interfaced with using custom INI files containing parameters (Berg et al., 2025).

Python’s robust packaging system, scientific libraries for numerical computing, data visualisation, and machine learning, as well as its ease of use, make it an attractive choice for developing scientific software. There is another GPU-based cardiac electrophysiology solver developed around the same time as Pigreads that also integrates well into the Python ecosystem: TorchCor uses finite elements with the PyTorch deep learning library (Zhou et al., 2026). Pigreads fundamentally differs from this as it uses finite differences and OpenCL as the backend. Due to the simpler numerical scheme and use of optimised low-level OpenCL code, Pigreads is usually faster than TorchCor for the same problem sizes. Pigreads solves the benchmark problem proposed by Niederer et al. (2011) around ten times faster than TorchCor: On the same hardware, TorchCor computes the solution in 9.2 s for the coarsest resolution, and 264 s for the finest resolution. For Pigreads’ performance, see section 3.

Outlook

While we here have focused on cardiac electrophysiology, Pigreads can be used for reaction-diffusion systems in general. Due to its simple API, it is easy to use and integrate into existing Python code, enabling researchers to build on a solid foundation. We envision that Pigreads will be used in the future for parameter estimation, inverse problems, uncertainty quantification, sensitivity analysis, and model reduction. It could also be integrated into clinical workflows for personalised medicine.

Its nature as a Python module also facilitates building, for instance, an interactive graphical user interface or a web application on top of it. This would make it even more accessible to a wider audience, for instance, to researchers without programming experience. We are including a proof-of-concept interactive demo in the Python module allowing to attempt to terminate atrial fibrillation, i.e., spiral chaos, by tapping a touch screen.

As new models can easily be defined, another use case is designing, fitting, and testing new models. The module could also be used for educational purposes, for instance in courses on mathematical biology, dynamical systems, or scientific computing.

While Pigreads could be extended in various ways – for instance more advanced time stepping methods, adaptive mesh refinement, support for more complex geometries, or other kinds of governing equations – we have chosen to keep the module minimal to keep it as stable as possible. While we are open to receive pull requests of minor tweaks, bugfixes, and additional model definitions and geometry files, no additional features should be added unless they are absolutely necessary. There is always the possibility to fork the project and add additional features if needed, for instance, on a per-project basis.

We hope that Pigreads will be a useful tool for researchers across various fields of science and that it will enable scientific progress.

Data availability

Pigreads can be installed from PyPI. The source code is available on GitLab, and its documentation on Read the Docs.

Scripts to generate the figures and data presented in this manuscript are available in a separate Git repository.

Addenda

Acknowledgments:

DK would like to thank Martina Chirilus-Bruckner and Daniël Pijnappels for inspiring discussions.

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process. During the preparation of this work the authors used ChatGPT and GitHub Copilot in order to help with writing the manuscript and minor parts of the code. After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

Funding:

This work was supported by the Netherlands Organisation for Scientific Research (NWO Open Mind grant 2025/TTW/02025375 to TDC). DK was supported by KU Leuven grant GPUL/20/012, and the Society, Artificial Intelligence and Life Sciences (SAILS) initiative of Leiden University.

Competing interests:

The authors have declared that no competing interests exist.

Author contributions:

DK: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualization. HD: Writing – Review & Editing, Supervision. TDC: Conceptualization, Writing – Review & Editing, Supervision, Project administration, Funding acquisition.

References

Africa, P. C., Piersanti, R., Regazzoni, F., Bucelli, M., Salvador, M., Fedele, M., Pagani, S., Dede’, L., & Quarteroni, A. (2023). lifex-ep: A robust and efficient software for cardiac electrophysiology simulations. BMC Bioinformatics, 24(1). https://doi.org/10.1186/s12859-023-05513-8

Antonioletti, M., Biktashev, V. N., Jackson, A., Kharche, S. R., Stary, T., & Biktasheva, I. V. (2017). BeatBox–HPC simulation environment for biophysically and anatomically realistic cardiac electrophysiology. PLOS ONE, 12(5), e0172292. https://doi.org/10.1371/journal.pone.0172292

Barkley, D. (1991). A model for fast computer simulation of waves in excitable media. Physica D: Nonlinear Phenomena, 49(1–2), 61–70. https://doi.org/10.1016/0167-2789(91)90194-e

Berg, L. A., Oliveira, R. S., Camps, J., Wang, Z. J., Doste, R., Bueno-Orovio, A., Santos, R. W. dos, & Rodriguez, B. (2025). MonoAlg3D: Enabling cardiac electrophysiology digital twins with an efficient open source scalable solver on GPU clusters. https://doi.org/10.1101/2025.04.09.647733

Biasi, N., Seghetti, P., Parollo, M., Zucchelli, G., & Tognetti, A. (2025). A Matlab toolbox for cardiac electrophysiology simulations on patient-specific geometries. Computers in Biology and Medicine, 185, 109529. https://doi.org/10.1016/j.compbiomed.2024.109529

Bradley, C., Bowery, A., Britten, R., Budelmann, V., Camara, O., Christie, R., Cookson, A., Frangi, A. F., Gamage, T. B., Heidlauf, T., Krittian, S., Ladd, D., Little, C., Mithraratne, K., Nash, M., Nickerson, D., Nielsen, P., Nordbø, Ø., Omholt, S., Pashaei, A., Paterson, D., Rajagopal, V., Reeve, A., Röhrle, O., Safaei, S., Sebastián, R., Steghöfer, M., Wu, T., Yu, T., Zhang, H., & Hunter, P. (2011). OpenCMISS: A multi-physics & multi-scale computational infrastructure for the VPH/Physiome project. Progress in Biophysics and Molecular Biology, 107(1), 32–47. https://doi.org/10.1016/j.pbiomolbio.2011.06.015

Cannon, J., McCarthy, M. M., Lee, S., Lee, J., Börgers, C., Whittington, M. A., & Kopell, N. (2014). Neurosystems: Brain rhythms and cognitive processing. European Journal of Neuroscience, 39(5), 705–719. https://doi.org/10.1111/ejn.12453

Clayton, R., Bernus, O., Cherry, E. M., Dierckx, H., Fenton, F. H., Mirabella, L., Panfilov, A. V., Sachse, F. B., Seemann, G., & Zhang, H. (2011). Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Progress in Biophysics and Molecular Biology, 104(1-3), 22–48. https://doi.org/10.1016/j.pbiomolbio.2010.05.008

Clerx, M., Collins, P., De Lange, E., & Volders, P. G. (2016). Myokit: A simple interface to cardiac cellular electrophysiology. Progress in Biophysics and Molecular Biology, 120(1-3), 100–114. https://doi.org/10.1016/j.pbiomolbio.2015.12.008

Clerx, M., Cooling, M. T., Cooper, J., Garny, A., Moyle, K., Nickerson, D. P., Nielsen, P. M., & Sorby, H. (2020). CellML 2.0. Journal of Integrative Bioinformatics, 17(2-3), 20200021. https://doi.org/10.1515/jib-2020-0021

Courant, R., Friedrichs, K., & Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100(1), 32–74. https://doi.org/10.1007/BF01448839

Courtemanche, M., Ramirez, R. J., & Nattel, S. (1998). Ionic mechanisms underlying human atrial action potential properties: Insights from a mathematical model. American Journal of Physiology-Heart and Circulatory Physiology, 275(1), H301–H321. https://doi.org/10.1152/ajpheart.1998.275.1.h301

De Coster, T., Claus, P., Seemann, G., Willems, R., Sipido, K. R., & Panfilov, A. V. (2018). Myocyte remodeling due to fibro-fatty infiltrations influences arrhythmogenicity. Frontiers in Physiology, 9. https://doi.org/10.3389/fphys.2018.01381

Dossel, O., Krueger, M. W., Weber, F. M., Schilling, C., Schulze, W. H. W., & Seemann, G. (2011). A framework for personalization of computational models of the human atria. 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 4324–4328. https://doi.org/10.1109/iembs.2011.6091073

E. Rognes, M., E. Farrell, P., W. Funke, S., E. Hake, J., & M. C. Maleckar, M. (2017). cbcbeat: An adjoint-enabled framework for computational cardiac electrophysiology. The Journal of Open Source Software, 2(13), 224. https://doi.org/10.21105/joss.00224

Finsberg, H. (2025). fenicsx-beat - an open source simulation framework for cardiac electrophysiology. Journal of Open Source Software, 10(114), 8416. https://doi.org/10.21105/joss.08416

Finsberg, H. N. T., Herck, I. G. M. van, Daversin-Catty, C., Arevalo, H., & Wall, S. (2023). simcardems: A FEniCS-based cardiac electro-mechanics solver. Journal of Open Source Software, 8(81), 4753. https://doi.org/10.21105/joss.04753

FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466. https://doi.org/10.1016/s0006-3495(61)86902-6

Gonzales, M., Guthals, A., Kerckhoffs, R., Krishnamurthy, A., Lionetti, F., Van Dorn, J., Villongco, C., Vincent, K., & McCulloch, A. (2016). Continuity: A problem-solving environment for multiscale biological modeling [Computer software]. Cardiac Mechanics Research Group; University of California, San Diego. https://continuity.ucsd.edu

Gray, P., & Scott, S. K. (1983). Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Chemical Engineering Science, 38(1), 29–43. https://doi.org/10.1016/0009-2509(83)80132-8

Harris, C. R., Millman, K. J., Walt, S. J. van der, Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N. J., Kern, R., Picus, M., Hoyer, S., Kerkwijk, M. H. van, Brett, M., Haldane, A., Río, J. F. del, Wiebe, M., Peterson, P., Gérard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C., & Oliphant, T. E. (2020). Array programming with NumPy. Nature, 585(7825), 357–362. https://doi.org/10.1038/s41586-020-2649-2

Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500–544. https://doi.org/10.1113/jphysiol.1952.sp004764

Hren, R. (1996). A realistic model of the human ventricular myocardium: Application to the study of ectopic activation. [PhD thesis, Dalhousie University]. https://dalspace.library.dal.ca//handle/10222/55139

Hren, R. (1998). Value of epicardial potential maps in localizing pre-excitation sites for radiofrequency ablation. A simulation study. Physics in Medicine and Biology, 43(6), 1449–1468. https://doi.org/10.1088/0031-9155/43/6/006

Hunter, J. D. (2007). Matplotlib: A 2D graphics environment. Computing in Science & Engineering, 9(3), 90–95. https://doi.org/10.1109/MCSE.2007.55

Kaboudian, A., Cherry, E. M., & Fenton, F. H. (2019). Real-time interactive simulations of large-scale systems on personal computers and cell phones: Toward patient-specific heart modeling and other applications. Science Advances, 5(3). https://doi.org/10.1126/sciadv.aav6019

Kabus, D., Cloet, M., Zemlin, C., Bernus, O., & Dierckx, H. (2024). The Ithildin library for efficient numerical solution of anisotropic reaction-diffusion problems in excitable media. PLOS ONE, 19(9), 1–26. https://doi.org/10.1371/journal.pone.0303674

Kapral, R., & Showalter, R. (1995). Chemical waves and patterns. Kluwer.

Keldermann, R. H., ten Tusscher, K. H. W. J., Nash, M. P., Bradley, C. P., Hren, R., Taggart, P., & Panfilov, A. V. (2009). A computational study of mother rotor VF in the human ventricles. American Journal of Physiology-Heart and Circulatory Physiology, 296(2), H370–H379. https://doi.org/10.1152/ajpheart.00952.2008

Klöckner, A., Pinto, N., Lee, Y., Catanzaro, B., Ivanov, P., & Fasih, A. (2012). PyCUDA and PyOpenCL: A scripting-based approach to GPU run-time code generation. Parallel Computing, 38(3), 157–174. https://doi.org/10.1016/j.parco.2011.09.001

Lechleiter, J., Girard, S., Peralta, E., & Clapham, D. (1991). Spiral calcium wave propagation and annihilation in xenopus laevis oocytes. Science, 252(5002), 123–126. https://doi.org/10.1126/science.2011747

Leguèbe, M., Coudière, Y., Calvez, L., Pannetier, V., Douanla-Lontsi, C., Davidović, A., Bécue, P.-E., Juhoor, M., & Zemzemi, N. (2025). CEPS: A cardiac electrophysiology exploration tool for new mathematical models and methods. Journal of Open Source Software, 10(114), 8910. https://doi.org/10.21105/joss.08910

Marcotte, C. D., & Grigoriev, R. O. (2017). Dynamical mechanism of atrial fibrillation: A topological approach. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(9). https://doi.org/10.1063/1.5003259

Murray, J. (1976). On travelling wave solutions in a model for the belousov-zhabotinskii reaction. Journal of Theoretical Biology, 56(2), 329–353. https://doi.org/10.1016/S0022-5193(76)80078-1

Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50(10), 2061–2070. https://doi.org/10.1109/jrproc.1962.288235

Net, T. A. A. döt, Ingy AND Müller. (2023). YAML ain’t markup language, revision 1.2.2. YAML Language Development Team. https://yaml.org/spec/1.2.2/

Nezlobinsky, T., Okenov, A., Vandersickel, N., & Panfilov, A. V. (2026). Finitewave: A lightweight and accessible framework for cardiac electrophysiology simulations [Computer software]. Ghent University, Belgium. https://github.com/finitewave/Finitewave

Niederer, S. A., Kerfoot, E., Benson, A. P., Bernabeu, M. O., Bernus, O., Bradley, C., Cherry, E. M., Clayton, R., Fenton, F. H., Garny, A., Heidenreich, E., Land, S., Maleckar, M., Pathmanathan, P., Plank, G., Rodríguez, J. F., Roy, I., Sachse, F. B., Seemann, G., Skavhaug, O., & Smith, N. P. (2011). Verification of cardiac tissue electrophysiology simulators using an N-version benchmark. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369(1954), 4331–4351. https://doi.org/10.1098/rsta.2011.0139

Pitt-Francis, J., Pathmanathan, P., Bernabeu, M. O., Bordas, R., Cooper, J., Fletcher, A. G., Mirams, G. R., Murray, P., Osborne, J. M., Walter, A., Chapman, S. J., Garny, A., Leeuwen, I. M. M. van, Maini, P. K., Rodríguez, B., Waters, S. L., Whiteley, J. P., Byrne, H. M., & Gavaghan, D. J. (2009). Chaste: A test-driven approach to software development for biological modelling. Computer Physics Communications, 180(12), 2452–2471. https://doi.org/10.1016/j.cpc.2009.07.019

Plank, G., Loewe, A., Neic, A., Augustin, C., Huang, Y.-L., Gsell, M. A. F., Karabelas, E., Nothstein, M., Prassl, A. J., Sánchez, J., Seemann, G., & Vigmond, E. J. (2021). The openCARP simulation environment for cardiac electrophysiology. Computer Methods and Programs in Biomedicine, 208, 106223. https://doi.org/10.1016/j.cmpb.2021.106223

Rotermund, H.-H., Engel, W., Kordesch, M., & Ertl, G. (1990). Imaging of spatio-temporal pattern evolution during carbon monoxide oxidation on platinum. Nature, 343(6256), 355–357. https://doi.org/10.1038/343355a0

Stone, J. E., Gohara, D., & Shi, G. (2010). OpenCL: A parallel programming standard for heterogeneous computing systems. Computing in Science & Engineering, 12(3), 66–73. https://doi.org/10.1109/MCSE.2010.69

Sullivan, B., & Kaszynski, A. (2019). PyVista: 3D plotting and mesh analysis through a streamlined interface for the Visualization Toolkit (VTK). Journal of Open Source Software, 4(37), 1450. https://doi.org/10.21105/joss.01450

ten Tusscher, K. H. W. J., & Panfilov, A. V. (2006). Alternans and spiral breakup in a human ventricular tissue model. American Journal of Physiology-Heart and Circulatory Physiology, 291(3), H1088–H1100. https://doi.org/10.1152/ajpheart.00109.2006

Updegrove, A., Wilson, N. M., Merkow, J., Lan, H., Marsden, A. L., & Shadden, S. C. (2016). SimVascular: An open source pipeline for cardiovascular simulation. Annals of Biomedical Engineering, 45(3), 525–541. https://doi.org/10.1007/s10439-016-1762-8

van Rossum, G., Drake, F. L., et al. (1995). The Python programming language. Centrum voor Wiskunde en Informatica Amsterdam. https://www.python.org

Vázquez, M., Houzeaux, G., Koric, S., Artigues, A., Aguado-Sierra, J., Arís, R., Mira, D., Calmet, H., Cucchietti, F., Owen, H., Taha, A., Burness, E. D., Cela, J. M., & Valero, M. (2016). Alya: Multiphysics engineering simulation toward exascale. Journal of Computational Science, 14, 15–27. https://doi.org/10.1016/j.jocs.2015.12.007

Zhou, B., Balmus, M., Corrado, C., Cicci, L., Qian, S., & Niederer, S. A. (2026). TorchCor: High-performance cardiac electrophysiology simulations with the finite element method on GPUs. SoftwareX, 33, 102521. https://doi.org/10.1016/j.softx.2027.102521

Calendar of travel photos 2025

Fri, 26 Dec 2025 20:00:00 +0100

Cover photo: Hapynionbrug, Leiden, Nederland

PhD Defence of Desmond Kabus

Tue, 10 Jun 2025 10:00:00 +0200

My PhD defence has taken place on 10 June 2025, 10:00 at the Arenberg Castle in Leuven, Belgium.

The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Please also see the acknowledgements in which I would like to thank you all for your continuing support.

Venue: Arenberg Castle

As it is the oldest university in the low countries, the campus of KU Leuven consists of buildings from many centuries. I had the privilege to invite you to the Castle of Arenberg built in the 16th century, located just a few kilometres outside Leuven.

10:00: Entrance

10:30: Presentation

11:15: Questions from the jury

12:15: Congratulations

12:30: Reception

Catering

Gifts

Group photos

14:30: End of the reception

The day continued with an early dinner only for close family members and concluded with a pub crawl in the Oude Markt in the city centre of Leuven.

Towards data-driven generation of individualised human heart models from scalable heart muscle sheets

Tue, 10 Jun 2025 09:45:00 +0200

A PDF version of the full thesis is available online, but you can also read the full contents on this website at the links below.

Dissertation presented in partial fulfillment of the requirements for the degrees of:

Doctor of Science (PhD): Mathematics
Doctor of Philosophy (PhD): Medicine

Institutions:

KU Leuven, Arenberg Doctoral School, Faculty of Science
Leiden University Medical Center (LUMC), LUMC Graduate School, Faculty of Medicine

Supervisors:

Dr. Hans Dierckx (LUMC)
Prof. Dr. Daniël A. Pijnappels (LUMC)
Dr. Antoine A.F. de Vries (LUMC)
Prof. Dr. Tom Van Doorsselaere (KU Leuven)

Members of the examination committee:

Prof. Dr. Stefan Van Aelst (chair) (KU Leuven)
Prof. Dr. Piet Claus (KU Leuven)
Prof. Dr. Jasmina Magdalenić Zhukov (KU Leuven)
Prof. Dr. Roeland Merks (Leiden University)
Dr. Vincent Portero (LUMC)
Prof. Dr. Maxime Sermesant (Inria, IHU Liryc, 3IA Côte d’Azur)

Abstract

Heart rhythm disorders like atrial and ventricular tachycardia and fibrillation can be treated in various ways. These arrhythmias are characterised by abnormal electrical activity in the heart such as re-entrant circuits—for instance spiral waves. Clinicians can choose from methods ranging from medication to surgical interventions like ablation or implantation of a pacemaker or defibrillator. These choices depend on a large variety of factors that, in the end, all come down to tweaking the electrical patterns to restore a healthy heart rhythm.

It is not always clear which treatment is the best for a particular patient and what outcome can be expected. The diagnosis and treatment of the cause of an arrhythmia, for instance by localisation of re-entrant circuits, is a complex process as it should also take into account the patient’s unique anatomy and physiology. This is why personalised computational models of the heart—so-called cardiac digital twins—hold great promise for the future of cardiology.

Zooming in to the level of individual heart muscle cells, creating a computational model of their electrical activity remains challenging: The response of individual cells to stimuli must be measured and combined into a model that should then be able to predict behaviour on the tissue and organ level—which can be a big leap requiring a lot of validation.

In this dissertation, we explore computational methods for personalised modelling of cardiac electrophysiology. We create software packages to numerically simulate the reaction-diffusion equations for the electrical activity of the heart. We also develop methods to detect and study re-entrant circuits in the heart as phase defects. We describe arrhythmia formation via quasiparticles in Feynman-like diagrams. Lastly, we create data-driven models for cardiac electrophysiology directly from optical voltage mapping data on monolayers—videos of the excitation waves in two-dimensional tissue samples.

While the reaction-diffusion based software package can be used for highly detailed simulations of the electrical patterns in the heart, the novel data-driven approach allows streamlined creation of models specific to individual tissue samples at much lower computational cost. Spiral wave dynamics can be predicted from just focal wave data. The phase defect approach offers a new way to study re-entrant circuits in the heart: With the quasiparticle view, we uncovered deeper insights into the mechanisms of arrhythmia formation.

The presented methods are a few steps towards the creation of fully personalised cardiac digital twins. In the future, the data-driven model creation pipeline could be used to refine a general model of the cardiac excitation waves to a patient-specific one. In conjunction with the phase defect approach, this could lead to improved diagnosis and treatment strategies for heart rhythm disorders that are truly personal.

Chapters

Acknowledgments from dissertation

Tue, 10 Jun 2025 09:30:00 +0200

This text is part of my dissertation. The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Besides all of the knowledge I gained about computational modelling of the heart, I can say that in my “global PhD”, I certainly learned a thing or two about the logistics of interdisciplinary and international collaboration: countless moves between the Netherlands and Belgium, negotiating the bureaucracy of two universities, and the challenges of building bridges between mathematics and physics on the one hand, and biology and medicine on the other. Over these four years, I have become a cosmopolitan European—and I am grateful to all the people I have met along the way who helped shape the person that I am now. From my birth place of Bochum to Kortrijk, Gent, Leuven, and of course my new home of Leiden, I have been welcomed with open arms. Truly, thank you all.

The term “Doktorvater” (doctoral father) is a fitting description for the role of a PhD supervisor, though I have been lucky enough to have a whole doctoral family to support me. I would like to thank my supervisors Hans Dierckx, Daniël Pijnappels, and Antoine de Vries for pushing me to become a better scientist day by day. They gave me the freedom to explore the ideas that interested me at any moment, while also providing the guidance to keep me on track. Hans, thank you for the brainstorming sessions drawing diagrams on the blackboard distributing colourful chalk dust all over the room. Daniël, thank you for building this truly interdisciplinary lab, a place where groundbreaking research is done every day. Twan, thank you for perfecting the art of scientific writing and for always being available for an interesting chat.

To my extended team of supervisors of Alexander Panfilov, Katja Zeppenfeld, and Tom Van Doorsselaere, as well as my supervisory committee, thank you for your support. Sasha, it still makes me smile when we talked about being star-struck when meeting your scientific idols.

HeartKOR! What a rollercoaster ride it has been. From the first video calls after my job interview during the pandemic with Louise Arno and Lore Leenknegt where we talked as if I was already part of the team, to the group growing with Marie Cloet and Nathan Dermul, the summer school in Bordeaux, and the many productive meetings in Kortrijk and online. I am grateful for both: friendship and science. Louise, thank you for being my carpool buddy; Lore, thanks for letting me store all my moving boxes in your garage while I went on a trip to teach sailing; Marie, thank you for listening to me, tea in hand, whenever I needed to vent; Nathan, big thanks for giving me the grand tour of Roeselare. And Aaron Gobeyn, thank you for showing me how smoothly a master’s project can be done.

And of course, gratitude to the Laboratory of Experimental Cardiology with all its members, too many to list. It is an honour and a privilege to be part of this group. Just to name a few: thanks to Tim De Coster for deep mathematical and not-so-mathematical discussions, to Niels Harlaar for introducing me to the optical voltage mapping data, and the technicians Cindy Bart, Juan Zhang, and Sven Dekker for preparing cells for my experiments, etc. Minka Bax, thank you for welcoming me in Leiden while I had to quarantine immediately after moving. Balázs Ördög, thank you for kayaking, sailing, and driving me and my boxes on my first move to Belgium. I am always happy to help in our office gardening sessions, Vincent Portero. Nogmaals hartelijk dank aan iedereen in het lab!

Also thank you, my co-authors Christian Zemlin and Olivier Bernus, for the collaboration on the Ithildin paper. I look forward to many more opportunities to work together. Also I would like to thank Jürgen Dreher, who originally introduced me to the field of cardiac modelling for my bachelor’s and master’s theses.

Talking about sailing, which has played a big part in my free time to balance out the academic work, I would like to thank everyone in the SIRUB sailing club, especially Michael Hornig, for countless adventures on different waters. Sei es das Mittelmeer, die Ostsee, das IJsselmeer, oder die Seen in Friesland, wir segeln weiter! Mögen alle Winschen richtig belegt werden. Also a big thank you to Merian for the weekends of sailing shenanigans in Scheveningen. Digging out a boat from the sand after a storm is not something you forget easily.

Then, my dear friends all around the world, some of whom I have known for decades and others I have only met in these last four years probably while travelling. Thank you for being there for me, when we are once again following all ScotRail lines on the map to just find some remote station on GeoGuessr (Schweinstein’s gang: Lisa, Julia, Basti, Pascal), when we are hiking over to Hattingen on the weekend (Miguel, Niels, Niklas, Robin² ), or when we finally update each other about our lives (Kira, Racha, Adriana, Lena). Thank you for all the memories we have made together and the ones we will still make (Dina, Dennis, Zoë, Emil, Veronica).

Meine Großeltern, Doris und Ralf, ich freue mich immer auf die Zeit, die wir zusammen verbringen, und hoffentlich werden es noch viele Jahre sein. Danke an meine Eltern, Iris und Diethelm, die in mir die wissenschaftliche Neugier geweckt haben und mich immer unterstützen, egal wo ich gerade bin. Wer hätte gedacht, dass wir beide letztendlich in der Medizin landen würden, als ich Physik studiert habe und du Biologie, danke Muriel! Ich bin immer für dich da, liebe Schwester, auch wenn München und Leiden so weit auseinander liegen.

And if I forgot to mention you and yet you are reading this, know that I am grateful for you too.

Some people think of life as a book with chapters to be closed and opened. Instead, I believe life is more like a river, with tributaries and confluences and meanders. I am not a believer in strict beginnings and endings, not a believer in strict categories to fit in. So let’s see what things will remain and what things will change.

I am looking forward to the journey ahead.

Desmond Kabus
Leiden, 2024-12-05

Steps towards true cardiac digital twins

Tue, 10 Jun 2025 09:15:00 +0200

This chapter is part of my dissertation. The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Discussion

Ranging from microscopic quasiparticle interactions and detailed simulations of individual ion channels, to macroscopic tissue-first models which may be used in cardiac digital twins: in this dissertation, we have presented mathematical tools for cardiac electrophysiology research. On the one hand, there are analysis methods for the electrical excitation patterns of the heart—mainly centred around the concept of phase defects, their detection, see chapter 3 (Kabus et al., 2022), and their study via Feynman-like diagrams, see chapter 4 (Arno et al., 2024a). On the other hand, we have presented numerical methods for the simulation of these patterns, following both the well-established reaction-diffusion approach in chapter 2 (Kabus, Cloet, et al., 2024) and a novel data-driven model for the dynamics of cardiac excitation waves as observed in optical voltage mapping data in chapter 5 (Kabus, De Coster, et al., 2024).

Simulation of cardiac excitation patterns using the reaction-diffusion based approach has been thoroughly explored and proven over the decades. Our software follows this approach and reaps the benefits of the established methods and the vast variety of in-silico tissue models that have been designed and published for the bidomain and monodomain description. This enables us to build on the existing knowledge of cardiac electrophysiology and to explore phenomena on state-of-the-art level with our software framework.

With the reaction-diffusion based software packages Ithildin and Pigreads, it is possible to perform in-silico versions of tissue-level experiments, allowing for the study of complex wave patterns for instance in monolayers. The influence of tissue geometry, anisotropy, and heterogeneity as well as the differences between different types of cells can be investigated, as well as the impact of drugs that affect the electrical behaviour of the cells, and many more phenomena. As with all software developed and articles written in the course of this PhD project, they are freely accessible¹ to the public following the philosophy of open source and open science.

We have also collected and implemented useful methods for the analysis of excitation patterns as Python modules. Both in-silico simulation data and in-vitro optical mapping data are supported by the Ithildin and Sappho Python modules, respectively. Signal processing methods such as smoothing, filtering, and data transformations, as well as robust algorithms for the computation of quantities of interest for cardiology such as maps of LAT, APD, and CV are included.

During arrhythmias, the heart’s electrical activity is governed by re-entrant waves, which can either be anatomical or functional re-entry. In both of these cases, the spiral wave core drives the electrical activity of the entire wave. Instead of using phase singularities, we have proposed to describe the core as phase defects. In two-dimensional excitable media, these phase defect lines show the extent of conduction blocks which can either be the functional rotor core or anatomical obstacles. With the detection of phase defects in chapter 3, we offer a more robust way to detect the driving core of spiral waves, a tool which may become useful in the context of ablation procedures.

New insights were gained into the formation of arrhythmias following the quasiparticle viewpoint. Three elementary cardions were identified as the intersections of wave fronts with phase defects, wave backs with phase defects, and the end points of phase defects—heads, tails, and pivots, respectively. They are always born in particle-antiparticle pairs that can annihilate and are associated with two different kinds of charges, the $Q$-charge in phase and the $P$-charge in space. Compound cardions like the growth, shrink, filament, and core particles were identified and their role in arrhythmogenesis was discussed in chapter 4. There is a strong analogy between the cardions and the elementary particles in the Standard Model of particle physics. One of the main results of chapter 4 is to describe wave dynamics, particularly formation of figure-of-eight re-entry, using Feynman-like diagrams. Both in-silico and in-vitro data were analysed using these methods which can be seen as an extension of the phase singularity description of spiral waves. The use of these diagrams may provide new, deeper understanding of the mechanisms of arrhythmogenesis.

A full cardiac digital twin of a patient’s heart requires modelling the electrophysiology on the whole-heart scale, i.e., simulations in three spatial dimensions for long durations of minutes to hours in length and a large computational domain in space. The reaction-diffusion based approach is, due to the CFL condition, still too computationally expensive, preventing their effective use in most clinical settings. Distephym, the data-driven model for excitation waves we presented in chapter 5, is a promising alternative: Due to the different way of encoding wave dynamics in the state space, it is possible to run at much coarser temporal resolutions while remaining stable. The model equations are encapsulated in the polynomial and the updating equations for the memory-encoding moving averages and gradient-approximating standard deviations—all of which are simple and computationally cheap to evaluate. We can fit the model effectively just with data that can be observed in mapping experiments. The model can generalise to unseen activation patterns: despite being trained on focal waves, it can predict the dynamics of spiral waves. We attribute this to the use of stochastic burst pacing, which “stress-tests” the tissue during data collection to cover as much of the possible states the tissue can be in as possible.

While we have so far only used the data-driven model on optical voltage mapping data of cardiac in-vitro monolayers, the method is general and can be applied to other data sets, different scales in time and space, and even different excitable media. The dynamics are encoded in the state space, such that the model can learn given enough data. However, the model is, of course, limited to the spatial and temporal scales that are encoded in the memory terms and gradient approximations. Using multiple EMAs and SDs can be used to encode and learn more complex dynamics, for instance precise changes to action potential morphology, cell memory effects or cell aging. While currently, we have only trained one model for the entire tissue, it is possible to train multiple models for different regions of the tissue, or even individual cells, to better capture the heterogeneity of the tissue. The model also must not necessarily stay stationary in time, but can be updated as new data come in. This online-learning approach is a promising extension of the method which may be used for real-time prediction of excitation patterns in the future. Another possibility is to use the data-driven model as a correction term to an existing ionic model to improve its predictive power for a given tissue sample. Instead of a polynomial, more complex function approximators such as neural networks should also be explored to model the dynamics. Finally, the use of the advection-based description of the wave propagation in combination with a general function approximator for the dynamics of the tissue is a new approach of modelling these excitation patterns. The limitations and advantages of this method are still to be explored in more detail.

Outlook

The methods presented in this dissertation are a few steps towards the creation of fully personalised cardiac digital twins. The data-driven model creation pipeline could be extended to learn the intricate dynamics seen in individual tissue samples, building up a library of models for different types of cells. Multiple parameter sets at a fixed model complexity for different regions of the tissue could then be assembled to create a model of the entire heart. The parameters could then—with the online-learning approach—be tweaked to better fit the observed patterns, tweaked to better represent the patient’s heart. Tissue parameters are encoded in the model and could be compared with parameters for other patients and cell types in the library to transfer the knowledge gained from one patient to another. The model could then be used to predict the outcome of different treatment strategies, for instance, how an ablation in a certain region would impact the wave propagation in the heart.

As the model is only trained on the observed excitation patterns, it just needs to “watch” a patient’s heart for an extended amount of time while inferring a model of the behaviour of the cells. One could imagine a future where a patient undergoes a mapping procedure, and the model learns from the incoming data in real time. One big hurdle to overcome is how to obtain these data. Optical voltage mapping is generally speaking not possible in a clinical setting, only less invasive methods like ECGs or EGMs are available, possibly from intra-cardiac catheters. It remains to be seen how well the data-driven model can be trained on these data. In the future, novel methods may become available to observe the electrical activity of the heart in more detail, which would greatly improve the quality of the data-driven models. In contrast to the classical approach to model the ionic currents in the heart which are manually tuned to fit the limited available data, data-driven models get better the more data are available.

Due to the resurgence of interest in machine learning in the last decade, novel methods for in-silico modelling are taking shape, re-thinking the way we model the electrical activity of the heart. These methods can not only be used for the creation of digital twins, but also to gain deeper insights into the mechanisms of arrhythmia formation via in-silico experiments or the particle-based viewpoint of arrhythmogenesis. Machine learning is also promising for the solution of inverse problems in cardiac electrophysiology. While, in this work, we advocate the creation of digital twins, the presented methods may be used in any of these other directions to develop better clinical diagnostics and treatments.

In view of the associated savings in computational cost and the ability to learn directly from observed data, the data-driven approach presented in this dissertation is a promising alternative to the classical reaction-diffusion based approach. This jump in computational efficiency lets us get closer to full real-time simulations of the entire heart which would allow for personalised testing of treatment strategies. These models would make only minimal assumptions about the underlying dynamics reducing bias and become better and better at predicting the waves the more data are available. Combining these new approaches, to build upon and learn from the existing methods, will be another key step towards the creation of better in-silico models of the heart.

The next years will be a pivotal time for the field of cardiac electrophysiology. We have shown the potential of data-driven models to predict the cardiac excitation patterns in a tissue sample. The next steps will be to extend and grow the method, to learn from more data, to move to larger scales. Due to continued efforts of the scientific community, we keep getting closer to the goal of creating true cardiac digital twins for the unique electrical behaviour of each patient’s heart.

References

Arno, L., Kabus, D., & Dierckx, H. (2024a). Analysis of complex excitation patterns using Feynman-like diagrams. Scientific Reports, 14(1), 28962. https://doi.org/10.1038/s41598-024-73544-z

Kabus, D., Arno, L., Leenknegt, L., Panfilov, A. V., & Dierckx, H. (2022). Numerical methods for the detection of phase defect structures in excitable media. PLOS ONE, 17(7), 1–31. https://doi.org/10.1371/journal.pone.0271351

Kabus, D., De Coster, T., de Vries, A. A. F., Pijnappels, D. A., & Dierckx, H. (2024). Fast creation of data-driven low-order predictive cardiac tissue excitation models from recorded activation patterns. Computers in Biology and Medicine, 169, 107949. https://doi.org/10.1016/j.compbiomed.2024.107949

The code repositories can be found in the data availability statements of the articles, or on https://gitlab.com/heartkor, https://gitlab.com/dkabus, and https://kabus.eu. ↩︎

Introduction to computational modelling of cardiac excitation waves

Tue, 10 Jun 2025 09:00:00 +0200

This chapter is part of my dissertation. The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Basics of cardiac electrophysiology

The vital mechanical contraction of the heart is driven by electrical waves that travel through the heart muscle tissue—the myocardium. In sinus rhythm—the normal heart rhythm—the heart activates in a well-choreographed sequence to pump blood effectively to the organs: The sinoatrial node in the right atrium acts as the pacemaker of the heart emitting an electrical pulse. The pulse then travels to the left atrium via Bachmann’s bundle, such that both atria contract at the same time. After the pulse reaches the atrioventricular node, it travels through the bundle of His and the Purkinje fibres to and through the ventricular wall, causing the ventricles to contract after the atria (Schiebler, 2005; Zipes & Jalife, 2014). Photographs of an anatomical model of the human heart can be found in Fig. 1.

Figure 1: Anterior view of the human heart in an anatomical model. The organ has four chambers: the smaller right (I) and left atria (II) superior of the larger right (III) and left ventricles (IV). Electrical waves coordinate its contraction pumping blood to the organs. Blood-flow is regulated by the opening and closing of valves between the chambers.

Arrhythmias are deviations from this normal heart rhythm. An abnormally low heart rate is called bradycardia¹, while an abnormally high heart rate is called tachycardia². Ventricular tachycardia (VT) is associated with a high risk of sudden cardiac death and is caused by re-entrant circuits in the ventricles, i.e., electrical waves that circulate in the ventricles repeatedly exciting them too early. It can evolve into ventricular fibrillation (VF), which is chaotic electrical activity in the ventricles leading to severely reduced pumping capacity of the heart and sudden cardiac death if not treated immediately with a defibrillator. Atrial fibrillation (AF) is the most common arrhythmia and is not immediately life-threatening, but it can lead to stroke and heart failure if not treated properly (Zipes & Jalife, 2014).

The field of cardiac electrophysiology studies how electrical waves propagate through the heart and how they cause arrhythmias. It is a field that operates on a variety of different levels: from the patient, to the organ, tissue, cell, and subcellular level. The field is highly interdisciplinary, combining knowledge from medicine, biology, chemistry, physics, mathematics, computer science, and engineering. The goal is to not just understand and diagnose the electrical activation patterns, but also to control them to ensure the proper working of each patient’s heart.

In recent years, efforts have been made to create personalised computational models of the heart, so-called cardiac digital twins, which could be used to test and optimise treatment strategies for individual patients (Bhagirath et al., 2024; Costabal et al., 2020; Gillette et al., 2021; Koopsen et al., 2024; Martin et al., 2022; Niederer et al., 2019; Shahi et al., 2022; Trayanova et al., 2020). Machine learning based approaches, for instance, utilising the data from wearable sensors (Chen et al., 2025) and other biomedical applications such as optoelectronics for cardioversion (Portero et al., 2024) show promise to impove the diagnosis and treatment of cardiac arrhythmias and their causes in the coming years. I have contributed to the concept of cardiac digital twins in a variety of ways through scientific articles, some of which are included in this dissertation:

the development of numerical software to simulate the electrical activity of the heart, see chapter 2 (Kabus, Cloet, et al., 2024),
the detection of re-entrant circuits in the heart as phase defects, see chapter 3 (Kabus et al., 2022),
the description of arrhythmia formation via quasiparticles in Feynman-like diagrams, see chapter 4 (Arno et al., 2024a), and
the creation of data-driven models for cardiac electrophysiology directly from optical voltage mapping data, see chapter 5 (Kabus, De Coster, et al., 2024).

The remainder of this chapter provides background knowledge that is helpful to understand the work presented in the published articles. It begins with a brief overview of cardiac action potentials and central quantities used to describe them. We then discuss the cell cultures and experiments that were conducted to obtain the data used for the creation of the computational models, and how these data are processed. Next, an overview of conventional in-silico models of cardiac electrophysiology is given, followed by an introduction into phase mapping. The introductory chapter concludes with the basics of machine learning on which the data-driven models are based.

Propagation of action potential waves

Excitation waves propagate by neighbouring cells activating each other. Each heart muscle cell—cardiac myocyte—activates and deactivates in a repeating cycle: the action potential. The activation, also known as depolarisation, is much quicker than the deactivation, the repolarisation. The cell’s transmembrane voltage $u$—in units of —is the difference in electrical potential between the inside and the outside of the cell; it is one of the central quantities in the field of cardiac electrophysiology. During the action potential of a single cell, the transmembrane voltage first increases from a resting potential of about to a peak of about exact values vary between cell types. This is called the upstroke of the action potential. After the peak, the voltage remains roughly constant for to the plateau phase, before it decreases back to the resting voltage (Zipes & Jalife, 2014).

We call the time at which $u$ rises above a threshold, the local activation time (LAT); and when it falls below threshold, the local deactivation time (LDT). The time between a cell’s activation and deactivation is called the action potential duration (APD): $$ \text{APD} = \text{LDT} - \text{LAT} \qquad{(1)}$$ Analogously, we call the time between deactivation and activation the diastolic interval (DI): $$ \text{DI} = \text{LAT} - \text{LDT} \qquad{(2)}$$ The cycle length (CL) is the time between two subsequent activations of a cell: $$ \text{CL} = \text{LAT}_2 - \text{LAT}_1 \qquad{(3)}$$ Note that these quantities depend on the chosen threshold. For example, when measured at of the upstroke, the APD is more precisely referred to as APD₃₀ .

Another central quantity is the conduction velocity (CV), which is the speed at which the excitation wave travels through the tissue. It is usually measured as the average speed of the sharp wave front between two points in space: $$ \text{CV} = \frac{\Delta x}{\Delta t} \qquad{(4)}$$ where $\Delta x$ is the distance between the two points and $\Delta t$ is the time it takes for the wave front to travel this distance. This definition is prone to errors when the points are not well-aligned with the direction of wave propagation. Also, as it is averaged, it does not capture local variations in CV. More rigorous definitions exist, for instance, by taking the infinitesimal limit of the above equation and the length of the velocity vector ${{\bm{{v}}}} \in \mathbb R^{D}$, where $D$ is the number of spatial dimensions: $$ \text{CV} = {{\left\lVert {{\bm{{v}}}} \right\rVert}} \qquad{(5)}$$

In Fig. 2, typical action potentials of an atrial model are shown for four pulses in a cable simulation as functions of space and time. In the space-time plot in panel C, the CV is measured as the slope of the wave front. Depending on the timing of the pulses, the APD and CV varies. The fourth pulse is blocked, i.e., it does not propagate through the cable.

Figure 2: Atrial action potentials in a cable simulation. These data were obtained using a numerical simulation of the model by Courtemanche et al. (1998), as will be introduced in section 2.3. A. Subsequent action potentials have different APD₃₀ depending on the previous DI, i.e., timing. B. The excitation wave travels through the cable. C. In a space-time plot of the activation pattern, the CV can be measured as the slope $\Delta x / \Delta t$ of the wave front. In this view, it can also easily be seen that the fourth pulse is blocked.

APD and CV depend on the so-called restitution characteristics which vary between different cell types. Restitution curves are plots of APD or CV against the DI or CL, an example of these curves is shown in Fig. 5.7.

As it has such a big impact on the action potential, the timing of the stimuli is another important aspect of cardiac electrophysiology. The so-called stimulus protocol is the sequence of the timings and locations of the stimuli, usually applied using an electrode. Common protocols are S1S2 where the tissue is re-excited just behind a part of the wave back of a first wave, or burst pacing where several stimuli are applied at the same location in quick succession. Due to inhomogeneities in the tissue, this might lead to re-entry—repeated excitation, which will be discussed below. In stochastic burst pacing (SBP), stimuli at the same electrode location are applied at random intervals to record the response of the tissue to a large variety of timings to get a more complete picture of the cells’ behaviour in unusual settings (Kabus, De Coster, et al., 2024).

A model cell line of human atrial myocytes

To study the electrical properties of cardiomyocytes, it is common to use model cells that can be relatively easily obtained, cultured, and manipulated in the laboratory. There is a wide variety of cells in use, for example neonatal rat ventricular myocytes (NRVMs) (Majumder et al., 2016), or pluripotent stem cell-derived cardiomyocytes (PSC-CMs) (Karakikes et al., 2015).

At Leiden University Medical Center, a new cell line was recently developed that was derived from human fetal atrial tissue which was conditionally immortalised using a lentiviral vector-based system. These so-called human immortalised atrial myocytes (hiAMs) are more representative of human atrial myocytes than other cells enabling their use in a wide range of experiments with cells (Harlaar et al., 2022):

hiAMs are monoclonal, meaning that they are derived from a single cell and are hence genetically identical. This leads to a high consistency in phenotypic properties of the cells, such as their CV and APD. They also posses superior electrophysiological properties compered to human embryonic stem cell-derived atrial myocytes (hESC-AMs) or human induced pluripotent stem cell-derived atrial myocytes (hiPSC-AMs). Observed CVs of hiAMs are around with APD₈₀ of compared to at similar APDs for hESC-AMs (Harlaar et al., 2022).

hiAMs can be expanded almost indefinitely as de-differentiated cells in the presence of the inducing agent doxycycline, and upon its removal, they re-differentiate into functional cardiomyocytes. This allows for the creation of large quantities of cells for experiments, such as the creation of monolayers—one-cell-thick layers grown on a flat surface. As the cells are randomly oriented in the monolayer, the resulting two-dimensional and approximately homogeneous and isotropic tissue can be used to study action potential propagation and re-entry in a controlled way. One use of these monolayers is to study the effects of pro- or anti-arrhythmic drugs on the electrical properties (Harlaar et al., 2022).

A visual outline of the creation of hiAMs is given in figure 1 of the article introducing the cell line (Harlaar et al., 2022): After the cells are isolated from a human fetus, they are transduced with a lentiviral vector encoding the recombinant simian virus 40 large T (SV40LT) antigen under the control of a doxycycline-inducible promoter. The effect of SV40LT on the cells is that they continuously divide. Cell clones are filtered based on selection criteria, such as the ability to proliferate in a doxycycline-dependant manner and to undergo cardiomyogenic differentiation into excitable and contractile cells. The selected cells now have the ability to proliferate in the presence of doxycycline and to differentiate into functional atrial myocytes upon its removal, i.e., without SV40LT expression (Harlaar et al., 2022).

The hiAM cell lines are a promising new tool for studying the electrical properties of the human atria. In chapter 5, optical voltage mapping of focal waves in hiAM monolayers was used as an in-vitro data set to train a data-driven in-silico model. The spiral waves predicted by that model were compared to spiral waves in hiAM monolayers (Kabus, De Coster, et al., 2024). In the following section, we will present some experiments that are commonly performed to measure the electrophysiological activity of cardiomyocytes, such as patch-clamping and optical mapping, which are relevant for the work in this project.

Common experiments in cardiac electrophysiology

Experiments in the discipline of medicine are often graded on a scale from most-invasive to least-invasive as follows, see also Fig. 3:

In-vivo experiments are performed on living organisms, usually laboratory animals. This is the most realistic setting, but also the most invasive, expensive, and ethically challenging.
Ex-vivo experiments are performed on tissue samples or organs extracted from organisms that is artificially kept alive outside of the animal. This still requires animal sacrifice.
In-vitro experiments are performed “in the glass”—i.e., on isolated cells or tissues that are kept alive in a culture dish. Cell lines are cultured to study the behaviour of (populations of) individual cells.
In-silico experiments are performed on computers. The term refers to the use of silicon in computer chips. This is naturally the least invasive method, but also the most abstract. In-silico models must be created using in-vivo, ex-vivo, or in-vitro data and validated against them.

Figure 3: Scale of the invasiveness of medical experiments. The scale ranges from most invasive on the left to least invasive on the right.

In this dissertation, we focus on in-silico experiments motivated by in-vivo, ex-vivo, and in-vitro experiments. In the following sections, we will briefly introduce the types of experiments that are the most relevant for the research presented in this dissertation.

Measurement of single-cell behaviour using patch clamping

Patch clamping is an in-vitro or ex-vivo technique to measure the electrical activity of a single cell. Under a microscope, a glass pipette is carefully attached to the cell membrane. Depending on the quantity that should be measured, the pipette can either leave the cell membrane intact, for instance, to measure the activity of individual ion channels, or penetrate the cell membrane to measure the activity of the entire cell. As the pipette is filled with a solution that conducts electricity, the electrical response of the cell to an applied voltage can be measured (Hill & Stephens, 2021).

Patch clamping was used to measure electrical properties of hiAMs, such as various ionic currents to compare their electrophysiology with that of other cell types. As these cells are an in-vitro model of human atrial myocytes, they should behave similarly to the cells they aim to model. Patch-clamping was therefore mainly used as a tool to validate the hiAMs as a new in-vitro model in the original publication introducing them by Harlaar et al. (2022). This validation justifies the use of hiAMs for future electrophysiological experiments.

Fig. 4 shows how the steady-state current is obtained for a hiAM. A voltage step protocol is applied to stimulate the cell, and the current over time is measured. After an initial transient phase, the current stabilises to the steady-state current. The result of this experiment is a current-voltage curve describing the specific current across the membrane as a function of the applied voltage.

Figure 4: Measurement of the steady-state current of hiAMs using patch clamping. A. The voltage is varied in steps from to and held for . B. The specific current across the membrane is measured for each voltage step. C. Steady-state current as a function of voltage. Experimental data from Harlaar et al. (2022), recorded at Amsterdam Medical Center.

Since the publication of Harlaar et al. (2022), additional patch-clamp experiments have been performed to measure a greater variety of ionic currents in hiAMs. These data are currently being analysed and will be published in a follow-up paper which will introduce a new in-silico electrophysiological model of hiAMs (De Coster et al., in preparation). The in-silico model will follow the classical approach to design a mathematical model of the cells’ electrophysiology to work in the context of the bi- and monodomain description which will be introduced in section 2.

Mapping techniques for tissue-level electrical excitation waves

To study the propagation of excitation waves in cardiac tissue, its electrical activity must be recorded simultaneously at multiple locations. As the electrical activity over space is recorded, this is called mapping. While the transmembrane voltage can not easily be measured directly for mapping, we instead measure by proxy.

On the one hand, electrodes can be used to record the electrical potential in so-called electrograms (EGMs) at different locations, see also section 2.1. The electrodes may be placed on the surface of the tissue or inserted into the tissue. This can be done in-vitro, ex-vivo, or even in-vivo. The electrodes are often arranged in a grid or linear patterns. Catheters with multiple electrodes can be inserted into the heart, such as the PentaRay catheter by Biosense Webster with a 5-star-shaped array of 20 electrodes or the HD Grid catheter by Abbott with a 4x4 grid of electrodes (Berte et al., 2020). Recordings from multiple beats can be combined to create phase maps of recurring patterns of activation, see also section 3.

On the other hand, optical mapping techniques usually record the fluorescence of voltage- or Ca²⁺ -sensitive dyes (Entcheva & Kay, 2021). A commonly used potentiometric dye is di-4-ANEPPS, which, when it absorbs light in blue to green wavelengths, emits red light whose intensity changes in a way that is approximately proportional to the transmembrane potential (Loew, 1996).

The setup for optical mapping typically consists of the following components: A light source, such as a halogen lamp, whose light may need to be filtered to the desired wavelengths. A dichroic mirror reflects the excitation light to the sample, while allowing the emission light to pass through. An emission filter also blocks the excitation light and only allows the desired, emitted wavelengths to pass through to the camera. The camera and a stimulation mechanism such as an electrode are controlled by a computer. A photograph of a simple optical mapping setup is shown in Fig. 5.

Figure 5: Optical mapping setup with a camera out of frame at the top, green light from the top, lenses for focusing and magnification, and the sample—a 24-well plate. As a simple mechanism for stimulating the tissue, a bipolar electrode is placed in the well. The camera and electrodes are controlled by a computer. To keep the tissue at body temperature, the sample is placed on a heating plate.

Both high temporal and spatial resolution are required for optically mapping the electrical activity of cardiac tissue, fast enough to capture the rapid changes over time and fine enough to capture the details of wave propagation through the tissue. The amount of noise is higher at higher resolutions due to less photons being collected per pixel and frame. Frame rates used for the data collected in this work are in the range of 1 to per frame. The spatial resolution is typically around 150 to per pixel, depending on the choice of lenses and camera. For example, the MiCAM05-Ultima camera by SciMedia has a resolution of 100x100 pixels onto which the imaging plane is projected (Harlaar et al., 2022).

As an example of the data that can be recorded with optical mapping, we show in Fig. 6 a sequence of snapshots of the electrical activity in a hiAM monolayer in a 6-well dish. This experiment shows the formation of a figure-of-eight spiral wave under burst pacing. During subsequent pulses, the gap of blocked conduction between the two sides of a wave front builds up to the point where there is enough space for the front to turn in on itself, creating a pair of spirals.

Figure 6: Figure-of-eight spiral formation under burst pacing in hiAM monolayers in a 6-well dish. A. The stimulation protocol in gray consists of three pulses at labelled S1-3, followed by a burst of 20 pulses at , B1-20. These stimuli trigger focal waves emanating from the electrode at the top left in the frame. The response of the central pixel is shown in purple. B.-I. Snapshots of the activity at the times indicated by the thin black lines in A. Frames are chosen at times when an excitation wave crosses a region of interest to show the build-up to figure-of-eight spiral wave formation from pulse to pulse. E. At the top left, a spiral wave forms in sync with the pacing, labelled R1. F.-I. The gap between the two parts of the wave front is so wide, that in the next frames, the wave front can turn around and merge with the next wave front, forming a figure-of-eight spiral wave. H. The two spirals in the figure-of-eight spiral wave are labelled R2 and R3.

In contrast to single cell measurements, optical mapping can capture the complex patterns of excitation waves in tissue. The data are particularly rich as each pixel may have a different time series of excitation and average cell properties, providing a large variability in the data. This variability can be used in the development of data-driven methods to model the electrical activity of cardiac tissue, a main focus of this work.

Processing and analysis of mapping data

Especially at higher resolutions, optical mapping recordings can be so noisy and obscured by other artefacts making it difficult to extract useful information from them. To focus on the relevant parts of the data, there are a variety of processing steps that can be applied. It is important to balance the removal of noise with preservation of the signal without distortion.

One of the most used steps in data processing is smoothing which gets its name from making noisy, jagged data smoother. This is typically done by applying an averaging filter to the data: The smoothed value at a given point is the average of the values of the data points in a small neighbourhood around it. A larger neighbourhood will result in smoother data, but the signal will be less clear, i.e., blurred. If the neighbourhood is too small, the noise will not be removed effectively. Smoothing and blurring are, mathematically speaking, the same operation, which is described by the diffusion equation, also known as the heat equation (Eq. 27). Smoothing can also be done in the frequency domain by applying a low-pass filter, for instance via the Fourier transform. Optical mapping data can be smoothed in time and the two spatial dimensions, note however that smoothing in time violates the causality of the data, meaning that the smoothed data at a given time point will depend on future data points.

A way to smoothen data without violating causality is to use an exponential moving average (EMA). The EMA is a recursive filter that gives more weight to recent data points. The EMA of a time series $u(t)$ at time $t$ is defined as (Heckert et al., 2002): $$ s(t) = \alpha \, u(t) + [1-\alpha] \, s(t-\Delta t) \qquad{(6)}$$ where $s(t)$ is the smoothed data, $\alpha$ is the smoothing factor, and $\Delta t$ is the time step. The EMA gets its name from the fact that the weight of the contribution of each data point to the smoothed value decreases exponentially over time. This can be seen if Eq. 6 is rewritten as: $$ \dot s(t) = \tilde\alpha \, u(t) - \tilde\alpha \, s(t) \qquad{(7)}$$ defining $\tilde\alpha = \frac{\alpha}{\Delta t}$. Multiplyling by ${{\mathrm{e}^{\tilde\alpha t}}}$ and integrating yields: $$ s(t) = {{\mathrm{e}^{-\tilde\alpha t}}} \, u(0) + \tilde\alpha \int_0^t {{\mathrm{e}^{\tilde\alpha \, (t' - t)}}} \, u(t') \;\mathrm{d}t' \qquad{(8)}$$ Hence, the EMA really is a weighted average of the previous data points, with exponentially decaying weights. As it can be efficiently implemented using the recursive formula in Eq. 6, the EMA is a popular choice for smoothing data in real-time applications, such as optical mapping.

Statistical data transformation techniques can also be used to improve the quality of the data. It is common to remove the mean along each dimension in the data and rescale to unit variance. This is called whitening (Abu-Mostafa et al., 2012). For instance, in optical mapping data, the recorded intensity at points that are further away from the camera will be lower than those closer to it. After whitening each pixel by dividing by the standard deviation of its signal over time, the data will have the same variance at each point. This can also be useful for comparing data from different experiments.

To be able to compare optical mapping recordings with electrode-based EGM maps, we can calculate approximate monopolar pseudo-EGMs via a convolution: $$ u_\text{EGM}({{\bm{{x}}}}, t) \propto \int_{\heartsuit} \mathrm{d}{{\bm{{x}}}}' \; \frac{ \nabla \cdot {{\bm{{D}}}} \nabla u(t, {{\bm{{x}}}}') }{ {{\left\lVert {{\bm{{x}}}} - {{\bm{{x}}}}' \right\rVert}} } \qquad{(9)}$$ where ${{\bm{{x}}}}$ is the position of the electrode, $\heartsuit$ is the tissue domain, ${{\bm{{D}}}}$ is the diffusion matrix, and $u$ is the signal representing the transmembrane voltage. These monopolar pseudo-EGMs may be converted to bipolar EGMs by subtracting a reference EGM from each pseudo-EGM. The reference EGM can for instance be any one of the EGMs or the average of all pseudo-EGMs. For more details and additional context, see section 2.1 and chapter 2, (Kabus, Cloet, et al., 2024).

Many quantities of interest in cardiac electrophysiology can be simply obtained via thresholding and differences, such as LAT, LDT, APD, DI, and CL (section 1.1). The CV (Eq. 5) is a quantity that is highly sensitive to noise and artefacts in the data. An elegant algorithm to estimate the velocity vector from LAT maps—a scalar field $T({{\bm{{x}}}})$ in space ${{\bm{{x}}}} \in \mathbb R^{D}$—is the method by Bayly et al. (1998). The gradient $\nabla T$ of this field is the “slowness”—a vector pointing in the same direction as ${{\bm{{v}}}}$, but with the inverse value. We can then obtain the velocity vector via the relation: $$ \begin{aligned} {{\bm{{v}}}} &= \frac{\nabla T}{{{\left\lVert \nabla T \right\rVert}}^2} \end{aligned} \qquad{(10)}$$ However, as LAT increases in discrete steps such that neighbouring grid points either have the same LAT or differ by one or more steps, the gradient is not well-defined. To overcome this, the data can either be smoothed or interpolated for instance by fitting a linear function to the data in a small neighbourhood around each grid point, which is the approach by Bayly et al. (1998).

In Fig. 7, the CV is estimated from LAT and LDT maps of a pulse in the same optical voltage mapping recording of a hiAM monolayer as in Fig. 6. Linear functions—planes—are fitted at each grid point to the neighbouring LAT and LDT values. We use this to get a smooth version of the LAT and LDT maps, as well as the CV via the slope of these planes. It can be seen that in the region of interest where the figure-of-eight reentry forms, the CV of both the wave front and back is lower than in the rest of the tissue, while the APD is prolonged.

Figure 7: Estimation of CV for noisy optical mapping data for the same recording as in Fig. 6. A. Parts of three subsequent excitation waves are passing through the tissue, seen here in the optical signal $u$. B. The smoothed LAT map of pulse B5. C. The smoothed LDT map of pulse B5. D. APD of pulse B5, i.e., the difference of LDT and LAT, is prolonged in the circled region of interest. E. CV of the wave front of pulse B5 is lower in the region of interest. F. The same is true for the CV of the wave back at higher variance.

In the course of this PhD project, besides the main contributions of models of the excitation waves—the reaction-diffusion solvers Pigreads and Ithildin (Kabus, Cloet, et al., 2024) and the data-driven model creation pipeline Distephym (Kabus, De Coster, et al., 2024), to be introduced in the following sections—we have developed various companion software packages to process and analyse the data obtained from the aforementioned experiments: mainly patch clamping, optical mapping, and in-silico simulations. These packages, which contain the methods introduced in this section and more, are (1) the Python module for Ithildin (Kabus et al., 2022; Kabus, De Coster, et al., 2024; Kabus, Cloet, et al., 2024), a Python package to process and analyse excitation wave data, originally designed for data from the reaction-diffusion solver Ithildin and (2) Sappho (Kabus, De Coster, et al., 2024), another Python package to process and analyse optical mapping data. Both packages are compatible with each other. Just as the main contributions, these packages are open-source and available on GitLab.

In-silico models of cardiac electrophysiology

A common way to model the electrical activation patterns that control the mechanical contraction of the heart is the reaction-diffusion equation in the monodomain description. It arises from a number of assumptions about the nature of the electric potentials and ion concentrations in the heart muscle tissue. In this section, we will give a derivation of the monodomain equations from first principles. We will also illustrate how to numerically solve them and present typical solutions to the monodomain equations.

Bidomain description—the cell as an electrical circuit

The first assumption we usually make to describe the electrical activity in the heart’s cells is that the ion concentrations and arising potentials are continuous across the tissue, i.e., instead of looking at the microscopic structure of each cell, we instead zoom out to the mesoscopic tissue scale. The motivation for this is that the activating cells locally show similar behaviour overall—neighbouring cells usually fire together (Tung, 1978). With this assumption, we can now define the electric potentials $u_\text{i}(t, {{\bm{{x}}}})$ on the interior of the cell continuum and $u_\text{e}(t, {{\bm{{x}}}})$ on the exterior, in units of ${\mathrm{{m}{V}}}$ (Clayton et al., 2011; Neu & Krassowska, 1993). These potentials are also commonly referred to as the intra-cellular and extra-cellular potentials, respectively.

The sketch in Fig. 8 shows how the interior and exterior are separated by the cell membrane which consists of a lipid bilayer—modelled as a capacitor, and protein structures which regulate the flow of ions across the cell membrane, i.e., the total ion current $I_\text{ion}$ per unit volume in ${\mathrm{{A}{/}{m}^3}}$. The membrane has a capacitance per unit volume of $C=\beta_\text{m} C_\text{m}$ in ${\mathrm{{F}{/}{m}^3}}$, obtained from the product of the specific cell membrane capacitance $C_\text{m}$ in ${\mathrm{{F}{/}{m}{^2}}}$ and the ratio $\beta_\text{m}$ in ${\mathrm{\frac{1}{m}}}$ of the cells’ surface area to their volume (Clayton et al., 2011). The behaviour of the ion channels in the membrane is quite complex and depends not just on the potentials and ion concentrations, but also on whether or not the structures act like an open or closed “gate”, which is encoded in so-called gating variables. The collective behaviour of the ion channels is described by a cell model for cardiac electrophysiology, which was pioneered by Hodgkin & Huxley (1952) and iterated over in the following decades. In section 2.4.3 and section 2.6, we will provide more detail about a variety of different cell models. The total current across the cell membrane is $C\partial_t u + I_\text{ion}$.

Figure 8: Sketch of the main quantities in the bidomain description. On the left, a close-up view of the cell membrane is depicted which, on the right, is represented by an electical circuit. The membrane separates the cells’ interior from the exterior such that we consider different ion concentrations in each. The lipid bilayer acts like a capacitor $C$ and the protein structures control the ionic currents $I_\text{ion}$, which are here depicted as an adaptive resistor $R$. The difference between the interior and exterior potentials, $u_\text{i}$ and $u_\text{e}$ respectively, is called the transmembrane voltage $u = u_\text{i} - u_\text{e}$. The total current across the cell membrane is $C\partial_t u + I_\text{ion}$. Adapted from Hodgkin & Huxley (1952) and Alberts et al. (2002).

To each of the electric fields on the interior $-\nabla u_\text{i}$ and exterior $-\nabla u_\text{e}$ in units of ${\mathrm{{V}{/}{m}}}$, we can apply Ohm’s law to obtain the current densities $-{{\bm{{\sigma}}}}_\text{i} \nabla u_\text{i}$ and $-{{\bm{{\sigma}}}}_\text{e} \nabla u_\text{e}$ in units of ${\mathrm{{A}{/}{m}{^2}}}$. We denote the conductivity tensors as ${{\bm{{\sigma}}}}_\text{i}$ and ${{\bm{{\sigma}}}}_\text{e}$, for the interior and exterior respectively, in units of ${\mathrm{{S}{/}{m}}} = {\mathrm{\frac{1}{{\Omega}{/}{m}}}}$ (Clayton et al., 2011). These general conductivity tensors enable anisotropic conduction, i.e., current can spread more quickly in some directions, such as along a fibre direction versus across them. More details on the influence of this tensor on the solutions ${{\underline{{u}}}}$ will be provided in section 2.5.

As the current and charge are conserved, the total current across the cell membrane $C\partial_t u + I_\text{ion}$ takes the role of a source term for current densities on the exterior and a sink term on the interior. Therefore, for the divergence of the current densities, we obtain: $$ \nabla\cdot{{\bm{{\sigma}}}}_\text{i} \nabla u_\text{i} = -\nabla\cdot{{\bm{{\sigma}}}}_\text{e} \nabla u_\text{e} = C\partial_t [u_\text{i} - u_\text{e}] + I_\text{ion} \qquad{(11)}$$ By expressing Eq. 11 in terms of the transmembrane voltage $u$ and the extra-cellular potential $u_\text{e}$, we obtain on the domain $\heartsuit\subset\mathbb R^3$ and for duration $T$ in ${\mathrm{{m}{s}}}$: $$ \begin{aligned} C\partial_t u &= \nabla\cdot{{\bm{{\sigma}}}}_\text{i}\nabla[u + u_\text{e}] - I_\text{ion} & \text{on}\,\heartsuit\times[0,T] & \;\;\;\text{(A)} \\ -\nabla\cdot[{{\bm{{\sigma}}}}_\text{i} + {{\bm{{\sigma}}}}_\text{e}] \nabla u_\text{e} &= \nabla\cdot{{\bm{{\sigma}}}}_\text{i} \nabla u & \text{on}\,\heartsuit\times[0,T] & \;\;\;\text{(B)} \\ 0 &= {{\bm{{n}}}} \cdot {{\bm{{\sigma}}}}_\text{i}\nabla[u + u_\text{e}] & \text{on}\,\partial\heartsuit\times[0,T] & \;\;\;\text{(C)} \\ 0 &= {{\bm{{n}}}} \cdot {{\bm{{\sigma}}}}_\text{e}\nabla u_\text{e} & \text{on}\,\partial\heartsuit\times[0,T] & \;\;\;\text{(D)} \\ u &= u_\text{init} & \text{on}\,\heartsuit\times\{0\} & \;\;\;\text{(E)} \\ u_\text{e} &= u_\text{e,init} & \text{on}\,\heartsuit\times\{0\} & \;\;\;\text{(F)} \end{aligned} \qquad{(12)}$$ where, using the normal vector ${{\bm{{n}}}}$ on $\partial\heartsuit$, we have added the boundary condition that no currents may leave the domains, and initial conditions. This is called the bidomain continuum description of activating tissue, due to the assumption of continuous quantities across the two domains outside and inside the cells (Clayton et al., 2011). The boundary conditions for the extra-cellular potential (Eq. 12 D), can further be generalised to couple to the electrical potential around the organ, in the so-called bath (Tung, 1978).

Note that the two main variables in the bidomain equations, the transmembrane voltage $u$ and the extra-cellular potential $u_\text{e}$, are strongly coupled to each other in Eq. 12 B. Their dynamics only differ due to the discrepancy in anisotropy in ${{\bm{{\sigma}}}}_\text{i}$ and ${{\bm{{\sigma}}}}_\text{e}$.

The bidomain equations can be solved with a variety of different numerical methods (Clayton et al., 2011). For instance, a temporal step $t \rightarrow t + \Delta t$ in a naive forward Euler scheme could consist of these tasks:

Calculate the diffusion term: $$ I_\text{i} = \nabla\cdot{{\bm{{\sigma}}}}_\text{i}\nabla[u + u_\text{e}] \qquad{(13)}$$
Update $u$ based on Eq. 12 A: $$ u \leftarrow u + \Delta t [I_\text{i} + I_\text{ion}] \qquad{(14)}$$ respecting the no-flux boundary condition Eq. 12 C.
Calculate the diffusion term: $$ I_u = \nabla\cdot{{\bm{{\sigma}}}}_\text{i} \nabla u \qquad{(15)}$$
Update $u_\text{e}$ by solving the generalised Poisson equation based on Eq. 12 B: $$ \nabla\cdot{{\bm{{\sigma}}}} \nabla u_\text{e} = -I_u \qquad{(16)}$$ with ${{\bm{{\sigma}}}} = {{\bm{{\sigma}}}}_\text{i} + {{\bm{{\sigma}}}}_\text{e}$, respecting the no-flux boundary condition Eq. 12 C.

As it is an elliptic partial differential equation, solving the Poisson equation usually involves implicit solvers which are computationally much more expensive than the solution of the reaction-diffusion equation in the first step, Eq. 12 A.

Monodomain description—a reaction-diffusion system

Another assumption can be made to get to a simpler description of the activation patterns. Assume that the anisotropy of the conduction on the interior and exterior is the same, i.e., let the conductivities be proportional to each other, ${{\bm{{\sigma}}}}_\text{e} := \lambda {{\bm{{\sigma}}}}_\text{i}$ for a dimensionless constant $\lambda > 0$ (Clayton et al., 2011). From Eq. 12, we then obtain: $$ \begin{aligned} &\nabla\cdot{{\bm{{\sigma}}}}_\text{i} \nabla u = -\nabla\cdot[{{\bm{{\sigma}}}}_\text{i} + \lambda {{\bm{{\sigma}}}}_\text{i}] \nabla u_\text{e} = -[1 + \lambda]\nabla\cdot{{\bm{{\sigma}}}}_\text{i} \nabla u_\text{e} \\ \Rightarrow &C\partial_t u = \nabla\cdot{{\bm{{\sigma}}}}_\text{i}\nabla u + \nabla\cdot{{\bm{{\sigma}}}}_\text{i}\nabla u_\text{e} - I_\text{ion} = \nabla\cdot{{\bm{{\sigma}}}}_\text{i}\nabla u - \frac{1}{1+\lambda} \nabla\cdot{{\bm{{\sigma}}}}_\text{i} \nabla u - I_\text{ion} \\ \Rightarrow &\partial_t u = \frac{1}{C} \frac{\lambda}{1+\lambda} \nabla\cdot{{\bm{{\sigma}}}}_\text{i} \nabla u - \frac{I_\text{ion}}{C} \end{aligned} \qquad{(17)}$$

From this, the main monodomain equation follows: $$ \partial_t u = \nabla\cdot{{\bm{{D}}}} \nabla u - \frac{I_\text{ion}}{C} \qquad{(18)}$$ where we have defined the diffusivity tensor $$ {{\bm{{D}}}} := \frac{1}{C} \frac{\lambda}{1+\lambda} {{\bm{{\sigma}}}}_\text{i} \qquad{(19)}$$

The full monodomain description can be obtained by taking the other quantities that influence the ionic currents $I_\text{ion}$ into account. These quantities describe the state of the cells, for instance via ion concentrations and gating variables. These variables are collected in the state variable vector ${{\underline{{u}}}} (t, {{\bm{{x}}}})$ whose first component usually is the transmembrane voltage $u$. Let ${{\underline{{r}}}} ({{\underline{{u}}}})$ be the reaction term whose first component is $r({{\underline{{u}}}}) = - I_\text{ion}/C$ in ${\mathrm{{V}{/}{s}}}$. We can then rewrite Eq. 18 as the reaction-diffusion equation: $$ \partial_t {{\underline{{u}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}} ({{\underline{{u}}}}) \qquad{(20)}$$ where the unitless projection matrix ${{\underline{{P}}}}$ encodes whether a variable is to be subjected to the diffusion operator. For most cell models, only the transmembrane voltage is diffused, ${{\underline{{P}}}} = \operatorname{diag}(1, 0, ..., 0)$.

A side note on notation: We use different notation to distinguish vectors and matrices in physical space and state space. Capital letters are used for matrices, ${{\bm{{D}}}}$, ${{\underline{{P}}}}$; while lowercase letters are used for vectors, ${{\bm{{x}}}}$, ${{\underline{{u}}}}$. Boldface symbols are vectors or matrices in physical space, ${{\bm{{x}}}}$, ${{\bm{{D}}}}$; while underlined quantities are vectors or matrices with respect to the state variables, ${{\underline{{u}}}}$, ${{\underline{{P}}}}$. To illustrate this notation, Eq. 20 would read as follows in Einstein sum notation (Kabus, Cloet, et al., 2024): $$ \partial_t u_m(t,{{\bm{{x}}}}) = P_{mm'} \; \partial_{n} D_{nn'}({{\bm{{x}}}}) \; \partial_{n'} u_{m'}(t, {{\bm{{x}}}}) + r_{m}({{\underline{{u}}}};{{\bm{{x}}}}) \qquad{(21)}$$ where $m, m' \in \{1, ..., M\}$ for $M$ state variables and $n, n' \in \{1, 2, 3\}$ the spatial dimensions with $\partial_n = \partial_{x_n}$.

With no-flux boundary and initial conditions, the full monodomain description takes the form of this system of partial differential equations: $$ \begin{aligned} \partial_t {{\underline{{u}}}} &= {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}} ({{\underline{{u}}}}) & \text{on}\,\heartsuit\times[0,T] & \;\;\;\text{(A)} \\ 0 &= {{\bm{{n}}}} \cdot \nabla {{\underline{{u}}}} & \text{on}\,\partial\heartsuit\times[0,T] & \;\;\;\text{(B)} \\ {{\underline{{u}}}} &= {{\underline{{u}}}}_\text{init} & \text{on}\,\heartsuit\times\{0\} & \;\;\;\text{(C)} \end{aligned} \qquad{(22)}$$

In a more general context, the monodomain equations are an example of a reaction-diffusion system, which can be used to describe a wide variety of phenomomena. All kinds of excitation waves can be modelled as reaction-diffusion systems, which, besides muscle tissue and other tissues in the body like coupled neurons (Cannon et al., 2014), includes waves in chemistry (Kapral & Showalter, 1995; Rotermund et al., 1990), or, more abstractly, the spread of epidemics (Arno et al., 2024a; Lechleiter et al., 1991). Even the propagation of a mexican wave in a sports stadium, or the spread of forest fires with subsequent recovery (Fig. 9) can be seen as excitable media.

Figure 9: Forest fires as an excitable system can also be modelled by the reaction-diffusion equations (Eq. 22). In the depicted non-scientific simulation, the spread of a forest fire is modelled on a planet in the shape of a human heart to illustrate the similarities between the two phenomena of wildfire spread and cardiac electrical excitation waves. CC-BY 2022 Judith Verdonck, Desmond Kabus et al.; Team HeartKOR at KU Leuven and Digital Arts and Entertainment at Howest.

Numerical solvers of reaction-diffusion problems

To solve the monodomain equations (Eq. 22) numerically, in contrast to the bidomain equations, one now does not need to solve Poisson’s equation. A forward Euler scheme for this is:

Calculate the diffusion term: $$ {{\underline{{d}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} \qquad{(23)}$$
Update ${{\underline{{u}}}}$ based on Eq. 22 A: $$ {{\underline{{u}}}} \leftarrow {{\underline{{u}}}} + \Delta t [{{\underline{{d}}}} + {{\underline{{r}}}}] \qquad{(24)}$$ respecting the no-flux boundary condition Eq. 22 B.

While this simpler numerical scheme is an advantage of the monodomain over the bidomain description, the monodomain description has the disadvantage that it is at a higher level of abstraction from the physical reality. Some level of detail necessarily gets lost during abstraction. For instance, one can not exactly recover the intra- and extra-cellular potentials, $u_\text{i}$ and $u_\text{e}$, from the transmembrane voltage $u$.

A variety of finite-differences or -elements software has been published to solve Eq. 22, some of which also have support for solving the bidomain description, Eq. 12, (Africa et al., 2023; Antonioletti et al., 2017; Arens et al., 2018; CARMEN, 2024; Cooper et al., 2020; Finsberg et al., 2023; Niederer et al., 2011; Plank et al., 2021; Rognes et al., 2017). To have control of every aspect of the numerical solution, we have also developed our own reaction-diffusion solvers. By designing every component that goes into the software ourselves, we can optimise the code perfectly for our use cases, such as including tools geared towards cardiac electrophysiology and omitting features that are irrelevant to us. Additionally, we can tweak the code to ideally match available hardware.

The first of our reaction-diffusion solvers is Ithildin (Kabus, Cloet, et al., 2024): It was written in the C++ programming language (ISO, 2023) and utilises parallelisation with OpenMPI (Graham et al., 2006), an open implementation of the message passing interface (MPI), for efficient simulations in 2D and 3D on central processing units (CPUs). It also offers a wide variety of methods for recording additional data during the simulation, such as pseudo-EGMs. Beginning with work by C. Zemlin and O. Bernus and continued development by H. Dierckx from 2009 onwards, in this PhD project, this code was polished, thoroughly improving its code in regards to the numerical methods and user-friendliness. Ithildin was published as a scientific research paper for this PhD project which is reproduced in chapter 2.

A more minimal alternative is the Python-integrated GPU-enabled reaction diffusion solver, or in short Pigreads, which we wrote to be published along with this dissertation. Instead of on CPUs, Pigreads solves the reaction-diffusion equation on graphical processing units (GPUs). While GPUs are, as the name implies, designed for calculations related to computer graphics—such as rendering 3D objects, shaders, or image processing, all of their computational power is opened up in general-purpose computing on GPUs (GPGPU). This paradigm is particularly useful for highly-parallel numerical problems, for instance, where at each of a large number of points, the same equations need to be solved independently from one another. While there are multiple solutions to utilise GPGPU capabilities, for Pigreads, we use the Open Computing Language (OpenCL) (Stone et al., 2010) to enable cross-platform GPU-parallelisation. For instance, on Nvidia GPUs, OpenCL capabilities are provided by the proprietary but highly-optimised Compute Unified Device Architecture (CUDA) (Du et al., 2012; Nickolls et al., 2008). Similar implementations exist for other GPU architectures; there even is a CPU fallback making it possible to run Pigreads on machines without a dedicated GPU. Pigreads is designed to have as little overhead as possible, only solving the reaction-diffusion equations (Eq. 22), without any bells and whistles for maximal performance. Instead, it is relying on a simple interface with the Python programming language via Pybind11 (Jakob et al., 2017; van Rossum et al., 1995). This way, any desired custom feature can easily be implemented in a Python script with numerical and scientific Python (Harris et al., 2020; Virtanen et al., 2020) which then calls the Pigreads module.

Both of our solvers, Ithildin and Pigreads, use central finite differences to implement the spatial derivatives in Eq. 22, such as the diffusion term $\nabla \cdot {{\bm{{D}}}} \nabla u$. In the finite differences method, derivatives are approximated by calculating a weighted sum in a neighbourhood around each point on a grid—the discretisation of the domain $\heartsuit$. For instance, the one-dimensional Laplacian $\partial_x^2 u$ can be calculated as: $$ \partial^2_x u(x) \approx \frac{u(x + \Delta x) - 2 u(x) + u(x - \Delta x)}{\Delta x^2} \qquad{(25)}$$ which is an approximation that is accurate to the second order of the grid spacing $\Delta x > 0$. The weights in this case are $\frac{1}{\Delta x^2}$ for the previous and following grid point, and $\frac{-2}{\Delta x^2}$ for the grid point $x$ to calculate the Laplacian at. Note that these weights need to be adjusted if any of the neighbouring points are outside of the domain. Finite-difference weights numerically discretising the diffusion term $\nabla \cdot {{\bm{{D}}}} \nabla u$ and respecting the no-flux boundary conditions (Eq. 22) also depend on the varying value of ${{\bm{{D}}}}({{\bm{{x}}}})$ over space. The diffusion matrix ${{\bm{{D}}}}$ is usually left constant in time though, such that the weights must be calculated only once at the beginning of the numerical simulation (Clayton et al., 2011).

A final issue to address in the design of finite differences solvers of Eq. 22 using an forward Euler time stepping scheme is the Courant-Friedrichs-Lewy (CFL) numerical stability criterion (Courant et al., 1928): If the time step $\Delta t$ is too large, critical details of the wave propagation in the reaction-diffusion equation might be neglegted leading to an inaccurate or even unstable solution, i.e., $u \rightarrow \infty$. The CFL condition ensures that $\Delta t$ is small enough such that the waves do not travel further than one spatial grid point during each time step. Ithildin automatically sets the time step low enough to satisfy the criterion as follows (Kabus, Cloet, et al., 2024; Li et al., 1994): $$ \Delta t < {{\left[ 2\max{({{\underline{{P}}}})} \sum_{n=1}^{N} \frac{D_{nn}}{\Delta x_n^2} \right]}}^{-1} \qquad{(26)}$$ Therefore, the maximum time step is linked to the chosen grid spacing.

Common solutions of reaction-diffusion problems

The heat equation

The two terms on the right-hand side of Eq. 22 A, encode the two mechanisms enabling the propagation of excitation waves. The diffusion term ${{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}}$ spatially couples the medium and leads to the spreading of ${{\underline{{u}}}}$. Considering only the diffusion term, i.e., for the trivial model ${{\underline{{r}}}} = 0$, Eq. 22 is reduced to the heat equation: $$ \begin{aligned} \partial_t u &= \nabla \cdot {{\bm{{D}}}} \nabla u & \text{on}\,\heartsuit\times[0,T] & \;\;\;\text{(A)} \\ 0 &= {{\bm{{n}}}} \cdot \nabla u & \text{on}\,\partial\heartsuit\times[0,T] & \;\;\;\text{(B)} \\ u &= u_\text{init} & \text{on}\,\heartsuit\times\{0\} & \;\;\;\text{(C)} \end{aligned} \qquad{(27)}$$ An example of a typical solution $u(t, {{\bm{{x}}}})$ of the heat equation is obtained via numerical simulation using Pigreads and shown in Fig. 10: The sharp right angles of the square-shaped heat distribution $u_\text{init}(x)$ at $t=0$, gradually get smoothed out such that the temperature will eventually equalise in the entire domain for $t\to\infty$. These equations also describe how initially uneven concentrations of chemicals eventually reach a homogeneous steady-state through diffusion, which is why Eq. 27 A is sometimes also called the diffusion equation. This is also the origin of the names of the diffusion term, diffusion operator, etc.

Figure 10: A solution of the heat equation. The heat equation (Eq. 27) describes that an initially uneven distribution of heat gets diffused until a steady-state of homogeneous temperature is reached.

The model by P. Gray & Scott (1983)

The reaction term ${{\underline{{r}}}}$ describes the local reaction at each point of the medium $\heartsuit$. Let there be two variables, ${{\underline{{u}}}} = {{{{\left[ a, b \right]}}}^\mathrm{T}}$, scaled to the interval $[0,1]$, which each describe the concentrations of one of two chemicals $A$ and $B$. One of the models by P. Gray & Scott (1983) describes the chemical reactions: $$ \begin{aligned} A + 2B &\to 3B & \;\;\;\text{(A)} \\ B &\to C & \;\;\;\text{(B)} \end{aligned} \qquad{(28)}$$ for the product $C$. Reactant $A$ is fed into the system at rate $f$, a model parameter; and the parameter $k$ determines how quickly reactant $B$ is removed. $A$ takes the role of an activator whose presence increases the reaction rate, while $B$ inhibits, i.e., slows down the reaction. The model equations then take the form: $$ \begin{aligned} \partial_t a &= D_a \nabla^2 a - a b^2 + f [1 - a] & \;\;\;\text{(A)} \\ \partial_t b &= D_b \nabla^2 b + a b^2 - [f + k] b & \;\;\;\text{(B)} \end{aligned} \qquad{(29)}$$ with diffusivities $D_a = 1$ and $D_b = 0.5$. For various parameters $k$ and $f$, the solutions of Eq. 29 are leopard-like dotted patterns or zebra-like line patterns, or even spiral waves. For example, Fig. 11 illustrates a Pigreads simulation of the model with $k = 0.062$ and $f = 0.055$ starting at a triangular initial distribution of $B$ and constant maximal concentration $a=1$ of reactant $A$. We used a grid spacing of $\Delta x = \Delta y = 1$ and a time step of $\Delta t = 0.1$ in arbitrary spatial and temporal units.

Figure 11: A solution of the model by P. Gray & Scott (1983). Starting from a simple initial condition in chemical $B$, a complex shape develops for parameters $k = 0.062$ and $f = 0.055$, cf. Eq. 29. Arbitrary units in time and space are used, the concentrations $a$ and $b$ are scaled to [0,1].

This example illustrates how the interplay of diffusion and reaction can lead to complex and chaotic behaviour: Slight variations in initial conditions can lead to vastly different outcomes.

Excitation waves in cardiac muscle tissue

Nowadays, there is a large variety of electrophysiological models that describe the excitation waves that can be found in various cells. One of the first and most influential was the model of the giant squid axon by Hodgkin & Huxley (1952), describing the neurons as an electrical circuit, as described in section 2.1, gaining them the 1963 Nobel Prize in Physiology or Medicine (Nobel Prize, 1963).

Most models fit into two categories: phenomenological and detailed ionic models. The detailed models (Courtemanche et al., 1998; Majumder et al., 2016; Paci et al., 2013; ten Tusscher & Panfilov, 2006) aim to describe as many electrophysiological processes inside each cell as possible to provide a full description of their behaviour starting on the single cell level, zooming out to the tissue level. The goal of the phenomenological models is to describe the overall wave dynamics on the tissue level while simplifying microscopic details within the cells. This reduces their computational cost, such that larger scale simulations are possible than with more detailed models. The model by FitzHugh (1961) and Nagumo et al. (1962) can be seen as a two-variable simplification of the four-variable Hodgkin & Huxley (1952) model. The model by Barkley (1991) follows similar design goals, however it was specifically designed to simulate the spiral waves arising in the Belousov-Zhabotinsky reaction. Other early phenomenological models specifically designed for cardiac excitation patterns are the models by Karma (1993); Aliev & Panfilov (1996), Fenton & Karma (1998), and C. Mitchell (2003). The minimal four-variable model by Bueno-Orovio et al. (2008) can be used to emulate other models, such as the ones by ten Tusscher et al. (2004) or Priebe & Beuckelmann (1998).

Using the smoothed version of the Karma model (Byrne et al., 2015; Karma, 1993, 1994; Marcotte & Grigoriev, 2017), we have run a Pigreads simulation on a monolayer, i.e., two-dimensional tissue, approximately in 6-well size ($R = {11~\mathrm{{m}{m}}}$) at a resolution of 200x200 pixels, with homogeneous and isotropic diffusivity $D = {0.03~\mathrm{{mm}^2{/}{m}{s}}}$, with selection matrix $P_u = 1$ and $P_v = 0.05$ for the state variables ${{\underline{{u}}}} = {{{{\left[ u, v \right]}}}^\mathrm{T}}$, using a time step of $\Delta t = 0.025$. In this simulation in Fig. 12, some of the typical electrophysiological behaviour of cardiac myocytes can be observed: The state variable $u$ which represents the transmembrane voltage in arbitrary units, smoothly switches between excited and resting state in waves of excitation that propagate through the medium. The depolarisation at the wave front, the up-stroke, occurs on a quicker time-scale than the repolarisation at the wave back. Between excitation and recovery, the transmembrane voltage stays roughly constant in the so-called plateau phase. This cycle from polarised resting state, over upstroke, plateau, and recovery, back to the resting state is called an action potential. The restitution variable $v$ describes the internal state of the cells, most crucially how recently the waves have been excited. At a high value of this variable $v$, excitation is inhibited, as can be seen at $t={0~\mathrm{{m}{s}}}$ in Fig. 12 A.

Figure 12: Simulation of the smoothed Karma model in a 6-well monolayer. Due to the initial distribution of the restitution variable, a rotor forms.

In this simulation (Fig. 12), we have chosen initial conditions such that a rotor forms: Due to the high value of $v=1$ in the lower half of the monolayer at $t={0~\mathrm{{m}{s}}}$, the initial stimulus in $u$ only triggers an action potential in the upper half, a conduction block line between the halves forms. Once the tissue recovers, it is again possible to excite the lower half. The wave turns around and follows the conduction block line creating a spiral wave rotor. These re-entrant waves have been identified as one of the mechanisms of tachycardia (R. A. Gray et al., 1995).

Influence of tissue heterogeneity and anisotropy on electrical excitation waves in the heart

An anisotropic diffusivity tensor ${{\bm{{D}}}}$, i.e., one that is not the same in every direction, encodes at which time scale diffusion—and equivalently conduction—takes place in the different directions. Cardiac muscle tissue is structured in fibres that may be further aligned in sheets. The diffusion along the fibres, encoded in diffusivity $D_{\text{f}}$, and within the sheets, $D_{\text{s}}$, is faster than in the normal direction, $D_{\text{n}}$ (Clayton et al., 2011): $$ {{\bm{{D}}}} = D_{\text{f}} {{\bm{{e}}}}_{\text{f}} {{{{\bm{{e}}}}}^\mathrm{T}}_{\text{f}} + D_{\text{s}} {{\bm{{e}}}}_{\text{s}} {{{{\bm{{e}}}}}^\mathrm{T}}_{\text{s}} + D_{\text{n}} {{\bm{{e}}}}_{\text{n}} {{{{\bm{{e}}}}}^\mathrm{T}}_{\text{n}} \qquad{(30)}$$ for the orthonormal basis vectors ${{\bm{{e}}}}_\text{f,s,n}$ along the fibres, within the sheet, and normal to the sheets, and ${{\bm{{a}}}} {{{{\bm{{b}}}}}^\mathrm{T}}$ denotes the outer product of the column vectors ${{\bm{{a}}}}$ and ${{\bm{{b}}}}$. Note that additionally the diffusivity ${{\bm{{D}}}}({{\bm{{x}}}})$ and the reaction term ${{\underline{{r}}}}({{\bm{{x}}}}; {{\underline{{u}}}})$ in Eq. 22 may depend on location, encoding spatial inhomogeneities.

An example of the fibre direction influencing the wave propagation can be seen in another Pigreads simulation of the Karma model in Fig. 13. All parameters were left the same as in Fig. 12, except the diffusivity tensor ${{\bm{{D}}}}$: For a diffusivity along the fibres $D_{\text{f}} = {0.03~\mathrm{{mm}^2{/}{m}{s}}}$ and in the normal direction in the sheet $D_{\text{s}} = \frac{1}{5} D_{\text{f}}$, we consider fibres whose direction vary at a constant rate along the horizontal axis, as can be seen Fig. 13 A. Particularly comparing panels D of Fig. 12 and Fig. 13, it can be seen how the slower conduction normal to the fibres has changed the shape of the wave front.

Figure 13: Simulation showing the influence of fibre direction on the wave propagation in a 6-well monolayer of the Karma model.

One way to numerically model fibrosis, i.e., regions of cells that can not be excited (Kazbanov et al., 2016), is to treat vertices in the finite-differences grid as exterior points. This was randomly done to of the vertices in the Pigreads simulation shown in Fig. 14. Here, the wave is slightly disturbed but still shows roughly the same behaviour as in the unmodified simulation (Fig. 12).

Figure 14: Simulation showing the effect of fibrosis on the wave propagation in a 6-well monolayer of the Karma model.

The deterministic dynamics of Eq. 22 are chaotic though, so small differences grow over time; the solutions diverge (A. Mitchell & Bruch Jr, 1985). Consider another simulation where a different subset of of vertices are modelled as fibrosis. While at first, the two solutions in Fig. 14 and Fig. 15 evolve in quite similar ways, the final frames show very different states.

Figure 15: Simulation showing chaotic behaviour in the wave propagation in a 6-well monolayer of the Karma model. Initially small differences grow until divergence from Fig. 14.

Different types of cells, the variation between cells of the same type over space, or diseased cells may be described by changing model parameters. In the Pigreads simulation in Fig. 16, we consider an inhomogeneity at the centre of the domain, where the model parameter $\epsilon$ of the Karma model as formulated in Marcotte & Grigoriev (2017) is increased by . This leads to a quicker recovery in that region, such that the rotor is sped up. This, in turn, drives the formation of more rotors across the domain. The break up of this single spiral into multiple rotors can be interpreted as a model of tachycardia evolving into fibrillation (section 1).

Figure 16: Simulation with inhomogeneous restitution parameters influencing the wave propagation in a 6-well monolayer of the Karma model.

Another useful inhomogeneity is to use different, but compatible cell models for different types of cells. For instance, the model by ten Tusscher & Panfilov (2006) shows slightly different behaviour for cells in the ventricular epicardium—the outside surface of the ventricles, endocardium—the inside surface, and in the middle of the ventricular wall. A sketch-like simulation that emphasises these slight differences in the excitation along with pronounced stronger diffusion along fibres in the horizontal axis can be found in Fig. 17. Due to the differences between the epi- and endocardial variants of the model, the symmetry of the wave front is broken. Models that differ may also have numerically unstable behaviour at the interfaces between them. As these models differ only slightly, they are still compatible.

Figure 17: Simulation of a 2D domain representing a cross-section of the ventricular wall with different variants of the model by ten Tusscher & Panfilov (2006). It can be seen that the excitation is different between variants.

Detailed, ionic models of cardiac electrophysiology

Where the phenomenological models (section 2.4.3) strive for simplicity and faithfulness at the tissue level, this often comes at the cost of accuracy at the single cell level. This is in contrast to the design goals of ionic models (Courtemanche et al., 1998; Majumder et al., 2016; Paci et al., 2013; ten Tusscher & Panfilov, 2006), where it ideally includes all ionic currents and other mechanisms that effect the electrophysiological behaviour of cardiac myocytes and other excitable cells.

To create such a model, most authors adapt an existing model by optionally modifying the model equations, such that more mechanisms in the cells at hand are captured, followed by tweaking the model parameters with data from a variety of sources: Patch-clamping is usually used to record current-voltage curves that capture how a current across the cell membrane is affected by the transmembrane voltage $u$. Also obtained from patch-clamping, activation/inactivation curves can provide insight into when and how quickly the ion channel gates open in a model. Action potential duration and conduction velocity restitution curves are also used to fit such an ionic model, which may be obtained from optical mapping. Details on these methods can be found in section 1.3.

The main ion current $I_\text{ion}(t, {{\bm{{x}}}})$ in the domain ${{\bm{{x}}}}\in\heartsuit$ over time $t\in[0,T]$ typically consists of a sum of individual currents similar to the following (Courtemanche et al., 1998; Hodgkin & Huxley, 1952): $$ I_\text{ion} = I_\text{Na} + I_\text{K} + I_\text{Ca} + ... \qquad{(31)}$$ plugging into the bi- or monodomain equations (Eq. 12 or Eq. 20). The currents $I_\star(t, {{\bm{{x}}}})$ are often defined as the product of scalar conductances $G_\star$, unit-less gating variables $g_{\star, a}(t, {{\bm{{x}}}})$, and the displacement of the transmembrane voltage $u$ from the equilibrium potentials $u_{\star}(t, {{\bm{{x}}}})$ (Courtemanche et al., 1998; Hodgkin & Huxley, 1952): $$ I_\star = G_\star {{\left[ u - u_{\star} \right]}} \prod_a g_{\star a} \qquad{(32)}$$

The equilibrium potential $u_{\star}(t, {{\bm{{x}}}})$ for an ion species, e.g. Na⁺ , K⁺ , Ca²⁺ , etc., can be calculated as (Courtemanche et al., 1998): $$ u_{\star} = \frac{R T}{Z F} \log\frac{{{\left[ \star \right]}}_\text{e}}{{{\left[ \star \right]}}_\text{i}} \qquad{(33)}$$ with the extra- and intracellular ion concentrations ${{\left[ \star \right]}}_\text{e, i}(t, {{\bm{{x}}}})$, the valence $Z$ of the ion, temperature $T$ in ${\mathrm{K}}$, the gas constant $R = {8.3143~\mathrm{{J}{/}{K}{/}{mol}}}$, and Faraday constant $F = {96486.7~\mathrm{{C}{/}{mol}}}$.

Ion gates $g_{\star a}(t, {{\bm{{x}}}})$ can be modelled as (Courtemanche et al., 1998): $$ \partial_t g_{\star a} = \frac{ g_{\star a \infty} - g_{\star a} }{ \tau_{\star a} } \qquad{(34)}$$ with the steady state $g_{\star a \infty}$ and time scale $\tau_{\star a}$.

For gating variables following Eq. 34, instead of using an forward Euler scheme Eq. 24, one can use the Rush-Larssen time stepping scheme (Courtemanche et al., 1998; Rush & Larsen, 1978): $$ g_{\star a} \leftarrow g_{\star a \infty} + [ g_{\star a} - g_{\star a \infty} ] {{\mathrm{e}^{-\frac{\Delta t}{\tau_{\star a}}}}} \qquad{(35)}$$

It is common in the ionic models to, besides the transmembrane voltage $u$, have gating variables $g_{\star, a}$ and ion concentrations ${{\left[ \star \right]}}_\text{e, i}$ as state variables ${{\underline{{u}}}}(t, {{\bm{{x}}}})$. To keep the number of state variables as low as possible, the other quantities in Eqs. 31-34 are expressed in terms of ${{\underline{{u}}}}$.

Ionic models aim to capture the behaviour of a specific type of cell or cell line. For example, the 21-variable model by Courtemanche et al. (1998) is designed for human atrial myocytes, while the 18-variable atrial model by Paci et al. (2013) describes human induced pluripotent stem cell derived cardiomyocytes (hiPSC-CMs) in an atrial-like phenotype. Tissue-level simulations for both of these are presented in Figs. 18, 19. To stimulate a rotor in these two simulations, an S1S2 protocol was used: When the wave back of an initial stimulus (S1) is sensed at a sensor location, a second stimulus is placed, such that the new excitation wave closely follows the wave back, creating a rotor. A visualisation of this protocol can also be found in Fig. 2.5. We here use current-based stimuli, essentially an additional source term on the right-hand-side of Eq. 22 A. For the simulation of the model by Paci et al. (2013) in Fig. 19, we use an additional stimulus (S0) before the protocol as, due to the restitution properties, the action potential duration of the first pulse is much longer than that of the second pulse. In the resulting wave dynamics of the two models, it can be seen how different two models for different cell lines behave even when both of them are designed as models for the same organ, in this case the atria.

Figure 18: Simulation of the model by Courtemanche et al. (1998) in a large culture dish stimulated twice following an S1S2 protocol. A meandering rotor is formed by the second stimulus at the wave back.

Figure 19: Simulation of the atrial model by Paci et al. (2013) in a 6-well culture dish stimulated three times following an S0S1S2 protocol. The well is just big enough to fit one rotor.

Limitations and alternatives

The in-silico models in the monodomain description presented in this section are powerful tools for studying the complex wave dynamics in heart muscle tissue. While here, we have shown simulations mostly on homogeneous and isotropic 2D discs to model monolayers, higher-dimensional simulations with anisotropy and heterogeneities can be performed. By choosing appropriate weights for the spatial derivatives, it is also possible to support curved surfaces like for instance the thin atrial wall (Davydov et al., 2000; Dierckx et al., 2013; Kabus, Cloet, et al., 2024).

Newer ionic models capture a lot of detail on the microscopic scale, while producing realistic electrical activation patterns on the tissue scale. Using them on the whole-organ scale, would make them a useful tool in the clinical context for personalised medicine: Procedures could be tested on a cardiac digital twin of the patient before they are actually implemented to test their efficacy and their impact on the patient. However, as the available computational power is usually still not big enough to tackle such large-scale whole-heart simulations, they are currently not feasible for patient care. This is mainly due to the resolution needed in space and time, i.e., $\Delta x$ and $\Delta t$, to accurately resolve the wave front, cf. Eq. 26. Also, the number of variables in detailed models—and hence the number of model equations, negatively impacts computational costs. With a higher number of model parameters, models also require more quite diverse and extensive measurements. Measuring every current in a cell line via patch-clamping is typically not feasible to create a full ionic model, so data from previously published models are imported that are assumed to also fit well to the cell line at hand. The variation between individual cells of a cell line is also often not taken into account. Finally, the currents are often re-balanced by scaling their conductances $G_\star$ so the action potential shape, and restitution characteristics match with observations. However, as these conventional ionic models are designed on the single-cell scale, they model single cells quite well.

One of the goals of this project was to explore alternatives or modifications to these conventional in-silico models of cardiac electrophysiology. High-resolution optical voltage mapping recordings of electrical activation patterns on the tissue scale (section 1.3.2) hold promise to enable fully-automatic, data-driven generation of electrophysiological models directly on the tissue-level. In the research paper presented in chapter 5, we explore one such method that can be used to quickly generate data-driven tissue excitation models from optical mapping data using a simple low-order polynomial model.

This method and other alternative models use machine learning techniques some of which will be presented in section 4. In section 3, we introduce methods that are useful to study the excitation patterns that can be observed in-vitro—in optical mapping data, as well as in-silico—the models that have been introduced in this section. These tools are used in the method presented in chapter 5.

Phase analysis of electrical activation patterns

Consider a model ${{\underline{{r}}}}({{\underline{{u}}}})$ in the single cell context, i.e., Eq. 22 with no spatial coupling ${{\bm{{D}}}} = 0$. Most of the models we consider in the context of cardiac electrophysiology capture both depolarisation and repolarisation—i.e., excitation and recovery. For a deterministic, continuous model function ${{\underline{{r}}}}({{\underline{{u}}}})$ to describe this behaviour, it is typically encoded in at least two variables ${{\underline{{u}}}} = {{{{\left[ u, v \right]}}}^\mathrm{T}}$. The variables of a model span the so-called state space. In state space, the model function is a vector field that describes the reaction at each possible state of the cells.

Phase analysis is a useful tool to understand the behaviour of a model by describing its state with a simple clock-like periodic variable—a so-called phase or phase angle ${\varphi}$. This section will introduce the main concepts of phase, phase singularities, phase defects, and their relation to conduction blocks. Phase defects are the main topic of two more scientific articles included in this dissertation, Kabus et al. (2022) in chapter 3 and Arno et al. (2024a) in chapter 4.

State space and phase

As an example, consider the reaction term ${{\underline{{r}}}}({{\underline{{u}}}}) = {{{{\left[ r_u, r_v \right]}}}^\mathrm{T}}$ of the simple two-variable model by Aliev & Panfilov (1996): $$ \begin{aligned} r_u &= - k u {{\left[ u - a \right]}} {{\left[ u - 1 \right]}} - u v & \;\;\;\text{(A)} \\ r_v &= - \epsilon v - \epsilon k u {{\left[ u - a - 1 \right]}} & \;\;\;\text{(B)} \\ \epsilon &= \epsilon_0 + \frac{\mu_1 v}{u + \mu_2} & \;\;\;\text{(C)} \end{aligned} \qquad{(36)}$$ with constant scalar parameters $k$, $a$, $\epsilon_0$, $\mu_1$, and $\mu_2$.

At each point ${{\underline{{u}}}} = {{{{\left[ u, v \right]}}}^\mathrm{T}}$ of the state space, this defines a unique reaction of the cells. This reaction is visualised as a vector function plot in Fig. 5.2. This figure is also coloured by the component of the reaction term in $u$-direction, $r_u$, or in the slightly different notation of that article $R_u$. For a starting state of $u=0.15$ and $v=0$, following the reaction ${{\underline{{r}}}}$ yields the trajectory ${{\underline{{u}}}}(t)$ drawn in black—the path following ${{\underline{{r}}}}$ through state space. It describes an arc around the state space, returning to the stable resting point, an attractor at $u=0$ and $v=0$. Small perturbations from this point lead to a restoring force towards the point. Diffusion and stimulus currents can be seen as additional source terms for $u$, see also Eq. 22. These offsets can raise $u$ above the critical threshold $a$ such that the system follows a similar trajectory to the aforementioned drawn one.

Therefore, under excitation, for instance from stimuli or in a spiral wave as in Fig. 12 or Fig. 2.5, the system’s state ${{\underline{{u}}}}$ evolves along a loop-like trajectory. This so-called inertial manifold (Temam, 1990) or dynamical attractor is visualised in Fig. 5.3 by colouring each point in the state space by how often it was visited in a two-dimensional monodomain simulation of the model by Aliev & Panfilov (1996). States near the dynamical extractor evolve towards and follow it (Cross & Hohenberg, 1993; Keener, 2004; Mikhailov et al., 1994). Encircled by the loop-like inertial manifold is a region—a “hole”—which contains significantly less likely states, i.e., ${{\underline{{u}}}}$ that the system in a normal excitation pattern does not evolve towards. We call this region of unlikely states the forbidden region in the state space (Arno et al., 2021).

Under repeated excitation, the system keeps evolving around in this loop in state space. So, with these cyclic trajectories in state space, it is natural to describe the system using an angle. This so-called phase describes the position on the cycle in state space. For a given state space evolution ${{\underline{{u}}}}(t, {{\bm{{x}}}})$ over time $t \in [0, T]$ for each point ${{\bm{{x}}}} \in \heartsuit$ in the medium, we get a function ${\varphi}(t, {{\bm{{x}}}})$. We choose ${\varphi}= 0$ as the phase at the resting state, and from there, it increases monotonously going around the loop, returning to ${\varphi}= 2\pi = 0$ (mod $2\pi$). The phase can be thought of as a “clock” for the state of the cells. A small phase value means that the system just excited, a large value closer to $2\pi$ means that the system is recovering or recovered. The details of this depend on the chosen definition of phase.

One such definition for phase is the classical phase or activation phase as an angle in state space (Bray & Wikswo, 2002; Clayton et al., 2006; R. A. Gray et al., 1998): $$ {\varphi}_\text{act} = \operatorname{arctan2}{{\left( R - R_*, V - V_* \right)}} + c \qquad{(37)}$$ where $V$ is a variable encoding the excitation of the system—typically the transmembrane voltage $u$, and $R$ is the restitution variable encoding the recovery of the system. $R_*$ and $V_*$ are constant threshold values inside the cycle’s forbidden zone, and $c$ is an offset which is typically chosen such that ${\varphi}= 0$ corresponds to the resting state.

For the definition of the restitution variable $R$, there are a few options, which all come with their advantages and disadvantages in describing the state of the system: One option is to use another state variable from ${{\underline{{u}}}}$ as the restitution variable, $R = v$. This is a common choice, particularly for two-variable models such as the models by Aliev & Panfilov (1996) or C. Mitchell (2003). If there are more than two state variables, the angle in any 2D plane in that model’s state space may be used. In Table 3.1, it can be seen that for the model by Bueno-Orovio et al. (2008), the authors choose to use the state variables $w$ and $s$ as $V$ and $R$ respectively, despite both of these variables having a less intuitive interpretation than the transmembrane voltage $u$ (Kabus et al., 2022). Another option is to extract $R$ from the temporal evolution of $V$, e.g., $R$ as a time delayed version of $V$ (R. A. Gray et al., 1995). This has the advantage to work when only a single state variable $V$ of the system can be observed. Alternatively, the Hilbert transform of $V$ may be used as $R$ (Bray & Wikswo, 2002), or the exponential moving average, see Eq. 6, (Kabus, De Coster, et al., 2024). This has the advantage to be more robust to noise.

Another way to define the phase is via the LAT, see section 1.1. The LAT at each point ${{\bm{{x}}}} \in \heartsuit$ over time $t \in [0, T]$ can be defined as the last time $t$, at which $V$ rose over a threshold $V_*$: $$ t_\text{LAT} ({{\bm{{x}}}}, t) = \operatorname{argmax}_{t' \le t} {{\left\{V ({{\bm{{x}}}}, t') = V_* \wedge \partial_t V ({{\bm{{x}}}}, t') > 0 \right\}}} \qquad{(38)}$$ LAT maps can, in the clinical context, be measured from catheter recordings in the heart (Cantwell et al., 2015), see also section 1.3.2. These maps are then used to identify conduction block lines (CBL), i.e., lines on the surface of the heart muscle that the conduction wave does not cross. CBLs are linked to arrhythmogenesis (Ciaccio et al., 2016), i.e., the formation of heart rhythm disorders, such as tachycardia and fibrillation (section 1).

An LAT-based phase can be defined as (Arno et al., 2021): $$ {\varphi}_\text{LAT} ({{\bm{{x}}}}, t) = f(t - t_\text{LAT} ({{\bm{{x}}}}, t)) \qquad{(39)}$$ with a scaling function $f: [0,\infty) \rightarrow [0,2\pi]$, such that: $$ \begin{aligned} f(0) &= 0 \\ \lim_{\tau \to \infty} f(\tau) &= 2 \pi \\ \forall \tau: f'(\tau) &\ge 0 \end{aligned} \qquad{(40)}$$ These properties ensure that the LAT-based phase has the properties we expect from a phase to be monotonously increasing from the resting state at ${\varphi}= 0$. As the time since last excitation increases, we expect ${\varphi}$ to approach a value close to $2\pi$. For a characteristic time $\tau_0$ of recovery, choices for this scaling function $f$ are $f(\tau) = 2\pi \tanh {{\left( \frac{\tau}{\tau_0} \right)}}$ (Arno et al., 2021; Kabus et al., 2022), or $f(\tau) = \frac{2\pi\tau}{\tau_0}$, clipped to the range $[0, 2\pi]$ (Arno et al., 2024a).

In Fig. 3.3, the different phase definitions are visually compared. For the state space phase ${\varphi}_\text{act}$ (Eq. 37), points where all phases meet can be found at at the centres of spiral waves; while the LAT-based phase ${\varphi}_\text{LAT}$ (Eq. 39) exhibits discontinuous jumps, e.g. at conduction block lines. So, the different phase definitions give us different points of view on the nature of spiral wave cores.

Phase singularities and phase defects

There are two different but related ways of describing the centres of spiral waves—also referred to as rotor cores: Phase singularities (PSs) are located where all phases meet (R. A. Gray et al., 1998), while phase defects (PDs) are those points where the phase exhibits a discontinuous jump (Arno et al., 2021; Tomii et al., 2021). PSs have co-dimension 2, meaning that in two spatial dimensions, they are points, and in three dimensions, they are lines which are called filaments. PDs have co-dimension 1, so in 2D, they are phase defect lines (PDLs) and in 3D, they are ribbon-like phase defect surfaces (PDSs) (Arno et al., 2021, 2024b).

In Fig. 20, the relation between PSs and PDs is illustrated for a Pigreads simulation of the Karma model. In this example, the phase was calculated for the two phase definitions (Eqs. 37, 39), then the cosine method (Tomii et al. (2021), Kabus et al. (2022)), which will be introduced in chapter 3 (Eq. 3.12), is used to find jumps in the phase. In the state space phase ${\varphi}_\text{act}$, PSs can be seen, in contrast to the extended PDLs in ${\varphi}_\text{LAT}$.

Figure 20: Phase singularities and defects in a simulation of the Karma model. Different phase definitions were used to convert the recordings in the variables $u$ (panel A.) and $v$, to the state space phase ${\varphi}_\text{act}$ (Eq. 37, panel B.), or the LAT phase ${\varphi}_\text{LAT}$ (Eq. 39, panel C.). The cosine method was used to find discontinuities in the phases (Eq. 3.12, panels E and F, respectively). For the state space phase, point-like PSs can be seen, while for the LAT phase, extended PDLs can be seen (panel D.). Note that the two PSs on the right lie on the same PDL, a conduction block line. The region considered excited is coloured yellow and labelled “E” in panel D, the unexcited region is labelled “U”.

In Fig. 2.10, a similar view is provided in 3D for the model by Bueno-Orovio et al. (2008). The PDSs are ribbon-like surfaces, here identified with a method that ascribes them with finite thickness and filament lines.

Re-entrant waves can form around an inhomogeneity in the medium—like an obstacle—which is called anatomical re-entry, but they can also exist in homogeneous tissue, where the rotor is not anchored to an obstacle in so-called functional re-entry (Fenton et al., 2000). The overall behaviour of a rotor is mainly driven by its core; if the core is undisturbed, the rotor will persist. Describing re-entry as PSs or PDs can offer deeper insights into the interactions of rotors with one another, how re-entrant circuits multiply and how they can be removed. For instance, re-entry that keeps forming around the same obstacle can be prevented via ablation: Locate the in-excitable obstacle, for instance as PDLs in LAT mapping recordings, and make more tissue in-excitable, for instance by burning or freeze-burning, such that afterwards, no more anatomical re-entry is possible around the new obstacle (Deisenhofer et al., 2010; Haissaguerre et al., 1998).

The detection of PDs is the topic of chapter 3, which was also published as a scientific research paper (Kabus et al., 2022). In chapter 4, another research paper published as part of this dissertation (Arno et al., 2024a), the interactions of PDLs, wave fronts and backs are for the first time studied as quasiparticles with Feynman diagrams, a sketch of particle interactions from the field of particle physics. The arising quasiparticles are further studied in up-coming research by Gobeyn et al. (in preparation).

Machine learning techniques for creating data-driven excitation models

The long-term goal of computational medicine is to create personalised digital twins—a high-detail, high-fidelity, high-resolution representation of entire organs or even the entire body of a patient (Niederer et al., 2019; Trayanova et al., 2020). A digital twin would allow testing treatments on it first to evaluate their effectiveness and impact before actually performing them on the patient. An ideal, full digital twin for cardiac electrophysiology would enable faster-than-real-time simulations of the electrical activation patterns based on all available measurements of a patient’s heart, such as its geometry, or electrocardiograph (ECG), EGM, and mapping recordings. An ideal digital twin would be fully data-driven, i.e., a model of the heart is created from minimal assumptions extracting all relevant information about the dynamics from the data.

There are two fundamental challenges in creating such data-driven digital twins:

Making use of as much of the available data as possible.
Designing numerical methods that are sufficiently fast for clinical applications.

Regarding the first point of inclusion of more data, note that most in-silico models of cardiac electrophysiology are fitted using fairly little data—mainly patch clamping and restitution curves, as we have discussed in section 2.7. The model fitting procedure also typically requires human input and is far from automated. This makes these conventional models a suboptimal starting point in the design of digital twins: They are both not data-driven and can not easily be created automatically.

Regarding the second point of computational cost, in section 2.3, we have seen that the numerical solution of a reaction-diffusion system—a system of partial differential equations (PDEs)—requires adequate resolution in space and time for numerical stability (Courant et al., 1928). In the context of cardiac electrophysiology, the computational effort to resolve those systems of equations is therefore quite high making them too slow for clinical applications with the currently available computers: For larger tissue volumes, resolutions are required to be roughly in the order of $\Delta x \approx {0.1~\mathrm{{m}{m}}}$ in three spatial dimensions and $\Delta t \approx {0.01~\mathrm{{m}{s}}}$ in time (Clayton et al., 2011). As an order-of-magnitude estimate, at this resolution, a reaction-diffusion simulation of of an organ in size would require calculations of the reaction term and the Laplacian on 10¹⁴ nodes in space and time, i.e., 100 trillion nodes. A possible workaround to reduce the computational cost is to use a surrogate model, i.e., an algorithm producing good approximations of the original model while, ideally, being much more computationally efficient (Raissi et al., 2019).

With the tools from machine learning, many options for the design of surrogate models and digital twins are opened up. In this section, we present some of these possible options after we introduce some of the basic methods which they build upon.

The learning problem

As a general optimisation problem consider an unknown function ${{\bm{{y}}}} = \hat f({{\bm{{x}}}})$—the target function—and a data set of $N$ pairs of input and output vectors: the inputs ${{\bm{{x}}}}_n \in \mathcal X$, indexed by $n \in {{\left\{1, ..., N \right\}}}$ and corresponding outputs ${{\bm{{y}}}}_n \in \mathcal Y$ (Abu-Mostafa et al., 2012). We now are looking for the optimal function $f \in \mathcal F$ which fits these data ${{\bm{{y}}}}_n \approx f({{\bm{{x}}}}_n)$ within a set of possible candidate functions $\mathcal F$, the hypothesis set. This process is called fitting the function $f$, optimising the function, or even “learning” the function. Except for the simplest cases, this is usually done with computers, hence the field of function optimisation is often called machine learning.³

More specifically, the problem of finding a function $f$ given annotated data, points ${{\bm{{x}}}}_n$ and labels ${{\bm{{y}}}}_n$, such that ${{\bm{{y}}}}_n \approx f({{\bm{{x}}}}_n)$, is called supervised machine learning. This is in contrast to unsupervised machine learning, in which patterns in unlabelled data ${{\bm{{x}}}}_n$ should be discovered. Unsupervised machine learning tasks are for instance finding clusters in data, measuring the similarity of data, or finding trends in the data. Supervised and unsupervised machine learning are often used together, for instance, one can use principal component analysis to transform high-dimensional data onto a space with fewer dimensions in which the data have the most variance. In this lower-dimensional space, the set of function candidates to be searched for an optimal solution is then also smaller.

A common way to state the so-called learning problem of supervised machine learning is in terms of a loss function $\mathcal L$ which is defined such that it is minimal for the optimal choice of $f \in \mathcal F$, for instance: $$ \mathcal L(f) = \frac {1}{N} \sum_{n=1}^N l{{\left( {{\bm{{y}}}}_n, f({{\bm{{x}}}}_n) \right)}} \qquad{(41)}$$ using a distance function $l : \mathcal Y \times \mathcal Y \to \mathbb R$ in the output space $\mathcal Y$, e.g.: $$ l({{\bm{{y}}}}_a, {{\bm{{y}}}}_b) = {{\left\lVert {{\bm{{y}}}}_a - {{\bm{{y}}}}_b \right\rVert}} \qquad{(42)}$$

Many optimisation problems in supervised machine learning then take the following form:

Given $N$ data ${{\bm{{x}}}}_n \in \mathcal X$ and ${{\bm{{y}}}}_n \in \mathcal Y$, find the function $f\in\mathcal F$ in the hypothesis set which minimises the loss function $\mathcal L$ (Eq. 41).

While the learning problem can be stated in more general forms, this version suffices for the methods introduced in the following.

Least-squares method

As a more concrete example of an optimisation problem, consider $N$ data points ${{\bm{{x}}}}_n \in \mathcal X \subseteq \mathbb C^{D + 1}$ in $D$ dimensions, indexed by $n \in {{\left\{1, ..., N \right\}}}$ and $d \in {{\left\{0, 1, ..., D \right\}}}$, and corresponding observed scalar function values $y_n \in \mathcal Y \subseteq \mathbb C$, sampled from an unknown target function $y_n = \hat f({{\bm{{x}}}}_n)$. For simpler notation, we define the zero-th element of all vectors ${{\bm{{x}}}}_n$ to be equal to one, i.e., $\forall n: x_{n, 0} = 1$.

One of the simplest classes of functions in this form are linear functions, i.e., the linear regression model, using a vector of weights ${{\bm{{w}}}} \in \mathcal W \subseteq \mathbb C^{D+1}$: $$ y_n \approx f({{\bm{{x}}}}_n) = {{{{\bm{{w}}}}}^\mathrm{T}}{{\bm{{x}}}}_n \qquad{(43)}$$ Due to the convention of defining $x_{n,0} = 1$, the zero-th element in the weight vector $w_0$ takes the function of a so-called bias; the entire function $f$ is shifted by this offset. For this hypothesis set, each point in the parameter space ${{\bm{{w}}}} \in \mathcal W$ maps to a function $f \in \mathcal F$.

For the hypothesis in Eq. 43 and Euclidean distance in $\mathcal Y$, the loss function Eq. 41 then takes this form (Abu-Mostafa et al., 2012): $$ \mathcal L({{\bm{{w}}}}) = \frac {1}{N} \sum_{n=1}^N {{\left\lVert y_n - {{{{\bm{{x}}}}}^\mathrm{T}}_n{{\bm{{w}}}} \right\rVert}}^2 = \frac {1}{N} {{\left\lVert {{\bm{{y}}}} - {{\bm{{X}}}} {{\bm{{w}}}} \right\rVert}}^2 \qquad{(44)}$$ where we have re-written the right-hand side using the matrix of all data points ${{\bm{{X}}}} \in \mathbb C^{N\times(D+1)}$, again with the convention that $\forall n: x_{n,0} = 1$.

To find the optimal choice of the weights ${{\bm{{w}}}}$, the parameters of the hypotheses functions, we minimise the loss by finding the weights for which the gradient of the loss is zero (Abu-Mostafa et al., 2012): $$ \begin{aligned} {{\bm{{0}}}} &= N \nabla_{{{\bm{{w}}}}} \mathcal L({{\bm{{w}}}}) = \nabla_{{{\bm{{w}}}}} {{\left\lVert {{\bm{{y}}}} - {{\bm{{X}}}} {{\bm{{w}}}} \right\rVert}}^2 = {{{{\bm{{X}}}}}^\mathrm{T}} {{\left( {{\bm{{y}}}} - {{\bm{{X}}}} {{\bm{{w}}}} \right)}} \\ \Rightarrow {{\bm{{w}}}} &= {{\left[ {{{{\bm{{X}}}}}^\mathrm{T}} {{\bm{{X}}}} \right]}}^{-1} {{{{\bm{{X}}}}}^\mathrm{T}} {{\bm{{y}}}} \end{aligned} \qquad{(45)}$$

Therefore, to fit a linear regression with the least-squares method (Eq. 45), only matrix multiplication and matrix inversion operations are needed; this is usually realised using Gaussian elimination. The least-squares method gets its name from the minimisation of a loss function that is the sum of squared distances, as sketched in Fig. 21. Least-squares is a very robust method due to the straight-forward use of common matrix operations and its lack of super-parameters. It is used as a component of many other optimisation methods when a linear function needs to be fitted.

Figure 21: Sketch of linear and parabolic least-squares fits. This optimisation method finds the linear function (solid line) that minimises the sum of the squares of the distances between predicted and true output values $y$. The data have been sampled from the target function (dotted) with random Gaussian additive noise. Despite being sampled from a linear function, the fitted parabola (dashed) has a lower loss function value.

Polynomial fitting

The least-squares method can, for instance, be used to not just fit linear functions to data ${{\bm{{x}}}} \in \mathcal X \subseteq \mathbb C^{D + 1}$ and $y \in \mathcal Y \subseteq \mathbb C$, but any polynomial of finite degree, such as: $$ \begin{aligned} f({{\bm{{x}}}}) = f(x_1, x_2) = w + w_1 x_1 + w_2 x_2 + w_{11} x_1^2 + w_{12} x_1 x_2 + w_{22} x_2^2 \\ f({{\bm{{z}}}}) = w z_0 + w_1 z_1 + w_2 z_2 + w_{11} z_{11} + w_{12} z_{12} + w_{22} z_{22} \end{aligned} \qquad{(46)}$$ Note that this polynomial is just a linear function with respect to the polynomial expansion ${{\bm{{z}}}}({{\bm{{x}}}})$ of the data, which, in this case, is defined as: $$ {{\bm{{z}}}}({{\bm{{x}}}}) = {{{{\left[ z_0, z_1, z_2, z_{11}, z_{12}, z_{22} \right]}}}^\mathrm{T}} = {{{{\left[ 1, x_1, x_2, x_1^2, x_1 x_2, x_2^2 \right]}}}^\mathrm{T}} \qquad{(47)}$$

Fitting a polynomial of a given degree can therefore be done by applying the corresponding polynomial transform to the input data ${{\bm{{x}}}}$ followed by applying the least-squares method to find a linear function to the transformed data ${{\bm{{z}}}}$, $y$. In Fig. 21, a parabola is also fitted to the data using polynomial expansion and the least-squares method.

Neural networks

The central idea behind neural networks is to approximate complex functions by combining many simple functions. This is captured in the universal approximation theorem, which states that a sufficiently complex neural network can approximate any continuous function given enough data (Hornik et al., 1989). Neural networks are inspired by the brain: Neurons are arranged in a network and, when stimulation by its neighbouring input neurons is strong enough, a neuron can fire which may, in turn, stimulate other neurons.

As sketched in Fig. 22, a neural network ${{\bm{{y}}}} = f({{\bm{{x}}}})$ consists of $L$ layers of neurons, i.e., nodes, indexed by $\ell \in \{0, ..., L\}$ and $d \in \{0, ..., D_\ell\}$. The value of the zeroth layer is set to the input, ${{\bm{{x}}}}_{0} = {{\bm{{x}}}} \in \mathbb C^{D_x+1}$; while the output of the $L$th layer is the output of the neural network, ${{\bm{{y}}}} = {{\bm{{x}}}}_L \in \mathbb C^{D_y+1}$. The value $x_{\ell, d} \in \mathbb C$ of each neuron in all layers except the zeroth is calculated based on the values ${{\bm{{x}}}}_{\ell - 1} \in \mathbb C^{D_{\ell-1}+1}$ of the previous layer using a linear term with weights ${{\bm{{w}}}}_{\ell, d} \in \mathbb C^{D_{\ell-1}+1}$ and a so-called activation function $\vartheta_{\ell, d} : \mathbb C \to \mathbb C$ (Abu-Mostafa et al., 2012): $$ x_{\ell, d} = \vartheta_{\ell, d}{{\left( {{{{\bm{{w}}}}}^\mathrm{T}}_{\ell, d}{{\bm{{x}}}}_{\ell-1} \right)}} \qquad{(48)}$$ Note that again $\forall \ell: x_{\ell, 0} = 1$, such that $w_{\ell, d, 0}$ is the bias in the linear term.

Figure 22: Sketch of a fully connected feed-forward neural network. Each layer contains nodes, also known as neurons, whose value is computed based on the values of the nodes in the previous layer. Each layer contains a constant node with $x_{\ell,0} = 1$.

The activation function $\vartheta_{\ell, d}$ plays a crucial role in neural networks as a scaling function which is applied to the linear combination of inputs from the previous layer, introducing non-linearities that facilitate learning complex patterns. The choice of activation function significantly impacts a neural network’s performance, influencing both its expressiveness and the ease of training. Two widely used activation functions are the sigmoid function $\vartheta_{\text{sig}}$ and the rectified linear unit (ReLU) $\vartheta_{\text{ReLU}}$.

The sigmoid function introduces smooth, differentiable non-linearity and is defined as: $$ \vartheta_{\text{sig}}(x) = \frac{1}{1 + {{\mathrm{e}^{-x}}}} \qquad{(49)}$$ However, it can suffer from the vanishing gradient problem, which makes training deep networks challenging (Hochreiter, 1991; Hochreiter et al., 2001).

The rectified linear unit (ReLU), on the other hand, is defined as: $$ \vartheta_{\text{ReLU}}(x) = \max(0, x) \qquad{(50)}$$ While not differentiable at $x=0$, ReLU is more computationally efficient than the sigmoid function.

The weights $w_{\ell d b}$ are usually stored in a large vector ${{\bm{{W}}}}$, which is trained using an optimisation algorithm, such as stochastic gradient descent (SGD) or one of its variants. The optimisation algorithm changes the weights in the direction that minimises a loss function (Eqs. 41, 44) that quantifies the discrepancy between the output of the neural network and the target output.

To find the gradient of the loss function with respect to the weights, the back-propagation algorithm is used. It gets its name from the fact that, in contrast to forward propagation, which computes the output of the network given the input, it computes the gradient of the loss function by propagating the error backwards through the network to find how much each node has contributed to it.

At each node or neuron, the back-propagation algorithm follows the chain rule to compute the gradient of the loss with respect to the weights. To sketch out the back-propagation algorithm, recall this general form of the loss function and main equation for the output of a node (Eq. 48), rewritten using the linear terms $s_{\ell d}$: $$ \begin{aligned} \mathcal L({{\bm{{W}}}}) &= \sum_{d=0}^{D_L} c(x_{L d}, y_d) \\ x_{\ell d} &= \vartheta_{\ell d}(s_{\ell d}) \\ s_{\ell d} &= {{{{\bm{{w}}}}}^\mathrm{T}}_{\ell d} {{\bm{{x}}}}_{\ell-1} \end{aligned} \qquad{(51)}$$ For target values ${{\bm{{y}}}}$ and the predicted values ${{\bm{{x}}}}_L$, the gradient of the loss function with respect to the weights for back-propagation is given by: $$ \begin{aligned} \frac{\partial \mathcal L}{\partial w_{\ell d b}} &= \sum_{q=0}^{D_{L}} c'(x_{L q}, y_{q}) {\frac{\partial x_{L q}}{\partial w_{\ell d b}}} \\ { \frac{\partial x_{\ell d}}{\partial w_{k b q}} } &= \begin{cases} \vartheta'_{\ell d}(s_{\ell d}) \; {\frac{\partial s_{\ell d}}{\partial w_{k b q}}} & \text{if } \ell > 0 \\ 0 & \text{otherwise} \end{cases} \\ { \frac{\partial s_{\ell d}}{\partial w_{k b q}} } &= \sum_{p=0}^{D_{\ell-1}} {{\left[ w_{\ell d p} {\frac{\partial x_{\ell-1, p}}{\partial w_{k b q}}} + x_{\ell-1, p} {\frac{\partial w_{\ell d p}}{\partial w_{k b q}}} \right]}} \\ {\frac{\partial w_{\ell d b}}{\partial w_{k q p}}} &= \begin{cases} 1 & \text{if } \ell = k \text{ and } d = q \text{ and } b = p \\ 0 & \text{otherwise} \end{cases} \end{aligned} \qquad{(52)}$$ where $\ell, k$ are indices of the layers and $d, b, q, p$ are indices of the neurons in the layers. The gradient can then be calculated by first computing the values $x_{\ell d}$ and $s_{\ell d}$ for all neurons in the network using forward propagation (Eq. 51), followed by calculating the weighted sum of products in the network for each weight (Eq. 52). Only the edge with $\ell = k$ and $d = q$ and $b = p$ has a non-zero derivative with respect to the weight $w_{k q p}$, so only its and the weights of the edges “down-stream” of this node contribute to its component of the gradient.

As they are general function approximators, neural networks have a wide variety of use cases. However, it can be difficult to train them, as the number of parameters needed for many tasks can be too great to be learned from the available data, the so-called curse of dimensionality. There are several ways to address this issue, such as regularisation or using algorithms that make better use of their available parameters, such as convolutional neural networks (CNNs) for input data ${{\bm{{x}}}}$ with a grid-like structure, i.e., images or videos.

To regularise the training of neural networks, to prevent so-called overfitting, one can, for example, randomly set a percentage of the node’s values to zero during training, which is called dropout. This way, the network can not rely too heavily on any individual node. Alternatively, in so-called weight decay, one may add a term to the loss function that “penalises” large weights, e.g. the L1 or L2 norm of the vector of all weights ${{\bm{{W}}}}$.

CNNs are well-suited for images as inputs, as they can capture spatial patterns in the data using convolutional layers. These layers apply a stencil to the data that combines the values in a neighbourhood of the input data, which can extract useful features from the data. Usually, this is implemented as weighted sums of the input data. The dense layers of a CNN can then combine these local features to predict the output. CNNs have been successfully applied to a variety of computer vision tasks, such as image classification, object detection, and image segmentation. One such application is cell counting, where CNNs have been used to count the number of cells and nuclei in images of tissue samples (Van Valen et al., 2016; Weisrock et al., 2024). Due to the variability in the shapes and sizes of cells, this task is challenging using traditional image processing techniques, while CNNs can achieve performance similar to a human expert.

In summary, neural networks and their extensions are useful tools for general function approximation and have been applied to a wide range of tasks.

Physics-informed neural networks

Another promising use case of neural networks is to compute solutions $u(t, {{\bm{{x}}}})$ to PDEs of the following form, which is coined by Raissi et al. (2019) as physics-informed neural networks (PINNs): $$ \partial_t u + \mathcal D(u) = f = 0 \qquad{(53)}$$ with a nonlinear differential operator $\mathcal D(u)$ in space, to be amended with boundary and initial conditions in $u$. We define the left hand side as $f(t, {{\bm{{x}}}})$, the residual of the PDE. The solution $u(t, {{\bm{{x}}}})$ is then described using a neural network. It is the solution to this particular instance of the problem; for different initial and boundary conditions, another solution must be found and the neural network re-trained from scratch. PDEs that fulfil Eq. 53 are for instance the reaction-diffusion equation (Eq. 22) or the one-dimensional Burgers’ equation: $$ \partial_t u + u \partial_x u - c \partial_x^2 u = f = 0 \qquad{(54)}$$

As the term “physics-informed” implies, the physical knowledge of the PDE governing the system is included as regularisation terms in the to-be-minimised loss function which penalises unrealistic solutions: $$ \mathcal L = \frac {1}{N} \sum_{n=1}^N {{\left\lVert u_n - u(t_n, {{\bm{{x}}}}_n) \right\rVert}}^2 + \frac {1}{C} \sum_{c=1}^C {{\left\lVert f(t_c, x_c) \right\rVert}}^2 \qquad{(55)}$$ We here use $N$ labelled training data points $t_n, {{\bm{{x}}}}_n, u_n$; and $C$ collocation points $t_c, {{\bm{{x}}}}_c$, at which to check the PDE (Eq. 53). Additional terms can be added to enforce, for instance, boundary conditions or initial conditions.

This physics-based regularisation has the additional benefit to make this method quite robust to overfitting to the data. PINNs can also work without discretisation of domain and time. The neural network $u$ is trained as usual, for example, using gradient descent, or a quasi-Newton method, enabling the use of the robust machine learning libraries developed in the last decades. The derivatives of $u$ in the loss function can be found using automatic differentiation, i.e., the exact derivatives of the neural network can be found analytically based on the chosen network architecture. More structure can also be imposed on the neural network, for instance a forward Euler or Runge-Kutta time-stepping scheme of high order, resulting in a discrete time model (Raissi et al., 2019).

PINNs have been successfully used to solve various problems, e.g., starting with the Burgers’ equation, and Schrödinger’s equation (Raissi et al., 2019), to fluid mechanics (Cai et al., 2021), meteorology, chemical engineering, and biophysics (Toscano et al., 2024). PINNs have also been used in the context of cardiology: for the reconstruction of LAT maps from sparse recordings (Costabal et al., 2020), the reconstruction of excitation waves from mechanical deformation (Dermul & Dierckx, 2024), or as a surrogate model for reaction-diffusion systems for the model by Aliev & Panfilov (1996; Martin et al., 2022).

Another use case of PINNs is data-driven discovery of PDEs: Given data from a solution to a system of PDEs, it is possible to recover not only the solution to this system but also system parameters (Raissi et al., 2019). The data-driven discovery of PDEs via PINNs could also be applied to find cardiac electrophysiology model parameters.

Creation of data-driven cardiac excitation models

Besides the aforementioned PINN-based approaches, a variety of other machine learning techniques have been tried to create cardiac excitation models to various degrees of success. In simple terms, the goal is to approximate the dynamics described by the bi- or monodomain descriptions using more versatile and faster models that can be trained on data.

One simple approach to this task is to approximate the reaction term in the reaction-diffusion equation (Eq. 22) with a neural network: $$ \partial_t {{\underline{{u}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}}_\text{NN} ({{\underline{{u}}}}) \qquad{(56)}$$ However, this approach is futile because, in most cases, only some of the elements of the reaction term can be directly measured in experiments. One could however use a neural network in this way to approximate an existing model, yielding a surrogate model that, depending on the complexity of the original model, could be faster to evaluate. The numerical solution of the system remains the same though: The reaction-diffusion equation still needs to be solved and the usual constraints of the CFL condition (Eq. 26) apply. Meaning, the speed-up would only be marginal.

Another naive approach is to use a dense neural network to approximate the entire right-hand side of the reaction-diffusion equation (Eq. 22) at lower resolution in space and time: $$ \partial_t {{\underline{{u}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}} ({{\underline{{u}}}}) \approx \dot{{{\underline{{u}}}}}_\text{NN} {{\left( {{\left\{{{\underline{{u}}}}_n \right\}}}_{n=1}^N \right)}} \qquad{(57)}$$ where the state vectors ${{\underline{{u}}}}_n$ at $N$ points in the neighbourhood of ${{\underline{{u}}}}$ are used as input of the neural network. This approach also has some notable drawbacks: The dimension of the network’s input space is quite large as it grows with the number of points $N$ and the dimension of the state vector ${{\underline{{u}}}}$. So, the network also needs to be quite large and hence has a lot of parameters to train, requiring more data and more time for training. Also, no information about the spatial structure is used, so the network has to learn the spatial structure from scratch. There is also no information about the underlying physics, further increasing the complexity of the task.

One could remedy the latter two drawbacks by using a CNN instead of a dense network. A selection of spatial features could be extracted by convolutional layers $\hat u_\text{conv}$ and subsequently used by a dense neural network $\dot{{{\underline{{u}}}}}_\text{dense}$: $$ \partial_t {{\underline{{u}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}} ({{\underline{{u}}}}) \approx \dot{{{\underline{{u}}}}}_\text{dense} {{\left( \hat u_\text{conv}{{\left( {{\left\{{{\underline{{u}}}}_n \right\}}}_{n=1}^N \right)}} \right)}} \qquad{(58)}$$ This approach has the advantage that the network can be trained to learn spatial features reducing the number of parameters compared to Eq. 57. However, the network still has to learn the underlying physics from scratch. We found that the spatial features extracted can be enough to describe the diffusion term, and hence the spread of the excitation wave, but not the reaction term. The restitution characteristics of the tissue involve temporal features that are not included in the spatial features extracted by the convolutional layers. We therefore would also need to include points from previous time steps in the input of the network, increasing the dimensional complexity of the task yet again. The dense network would get both spatial and temporal features as input.

Now, we propose to only use features that are known to be relevant for excitation waves in cardiac electrophysiology. For the purposes of this project, this is the approach that gave the best results. In a first prototype with code name Melange, we used a dense neural network to approximate the reaction-diffusion equation (Eq. 22) with a reduced set of features extracted from optical voltage mapping data:

the transmembrane voltage $u$, proportional to the change in observed light intensity,
approximate monopolar EGMs $u_\text{EGM}$ (Eq. 9),
a moving average $\tilde u$ of the transmembrane potential $u$ over the last few time steps (Eq. 6),
the absolute value ${{\left\lVert \nabla u \right\rVert}}$ of the gradient of the transmembrane voltage,
the local activation time phase ${\varphi}_\text{LAT}$ (Eq. 39),
the APD of the most recent pulse (Eq. 1),
and DI between the two most recent activations (Eq. 2).

Or, expressed as a PDE: $$ \partial_t {{\underline{{u}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}} ({{\underline{{u}}}}) \approx \dot{{{\underline{{u}}}}}_\text{Melange} {{\left( u, u_\text{EGM}, \tilde u, {{\left\lVert \nabla u \right\rVert}}, {\varphi}_\text{LAT}, t_\text{APD}, t_\text{DI} \right)}} \qquad{(59)}$$

In further research, we streamlined this model to only use the transmembrane potential $u$ and the moving average $\tilde u$ as input features, as well as an approximation $g$ of the gradient ${{\left\lVert \nabla u \right\rVert}}$ calculated using the standard deviation of $u$ in a small neighbourhood around each point. This model—code named Distephym, short for data-driven in-silico tissue-based electrophysiology model—is the subject of chapter 5 and has been published as a standalone paper (Kabus, De Coster, et al., 2024): $$ \partial_t {{\underline{{u}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}} ({{\underline{{u}}}}) \approx \dot{{{\underline{{u}}}}}_\text{Distephym} {{\left( u, \tilde u, g \right)}} \qquad{(60)}$$ Instead of a neural network, a low-order polynomial $\dot{{{\underline{{u}}}}}_\text{Distephym}$ is fitted to the data. The method is used to both fit a surrogate model of an existing in-silico tissue model, i.e., the model by Aliev & Panfilov (1996), and the method is used to create a model based on in-vitro data—optical voltage mapping data of human atrial myocytes (Harlaar et al., 2022). Even when the model is only trained on data from focal excitation waves, it is able to predict the dynamics of spiral waves. The optimisation method of the Distephym models is automated to such an extent that only minimal input is needed to create a model from the spatio-temporal recordings of excitation waves—i.e., optical voltage mapping data. This model which can run at much lower resolutions in time and space, will be discussed in detail in chapter 5.

The fast and automatic creation of data-driven models is a crucial step forward in the continued study of heart rhythm disorders. The speed-up holds promise to improve the use of computer models to predict the outcome of surgery or therapy, advancing towards a true cardiac digital twin.

References

Abu-Mostafa, Y. S., Magdon-Ismail, M., & Lin, H.-T. (2012). Learning from data (Vol. 4). AMLBook New York.

Africa, P. C., Piersanti, R., Regazzoni, F., Bucelli, M., Salvador, M., Fedele, M., Pagani, S., Dede’, L., & Quarteroni, A. (2023). Lifex-Ep: A robust and efficient software for cardiac electrophysiology simulations. BMC Bioinformatics, 24(1), 389. https://doi.org/10.1186/s12859-023-05513-8

Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., & Walter, P. (2002). Molecular biology of the cell (4th ed.). Garland Science.

Aliev, R. R., & Panfilov, A. V. (1996). A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals, 7(3), 293–301. https://doi.org/10.1016/0960-0779(95)00089-5

Antonioletti, M., Biktashev, V. N., Jackson, A., Kharche, S. R., Stary, T., & Biktasheva, I. V. (2017). BeatBox—HPC simulation environment for biophysically and anatomically realistic cardiac electrophysiology. PLOS ONE, 12(5), e0172292. https://doi.org/10.1371/journal.pone.0172292

Arens, S., Dierckx, H., & Panfilov, A. V. (2018). GEMS: A fully integrated PETSc-based solver for coupled cardiac electromechanics and bidomain simulations. Frontiers in Physiology, 9, 1431. https://doi.org/10.3389/fphys.2018.01431

Arno, L., Kabus, D., & Dierckx, H. (2024a). Analysis of complex excitation patterns using Feynman-like diagrams. Scientific Reports, 14(1), 28962. https://doi.org/10.1038/s41598-024-73544-z

Arno, L., Kabus, D., & Dierckx, H. (2024b). Strings, branes and twistons: Topological analysis of phase defects in excitable media such as the heart. In arXiv preprint arXiv:2401.02571. https://doi.org/10.48550/arXiv.2401.02571

Arno, L., Quan, J., Nguyen, N. T., Vanmarcke, M., Tolkacheva, E. G., & Dierckx, H. (2021). A phase defect framework for the analysis of cardiac arrhythmia patterns. Frontiers in Physiology, 12. https://doi.org/10.3389/fphys.2021.690453

Barkley, D. (1991). A model for fast computer simulation of waves in excitable media. Physica D: Nonlinear Phenomena, 49(1-2), 61–70. https://doi.org/10.1016/0167-2789(91)90194-e

Bayly, P. V., KenKnight, B. H., Rogers, J. M., Hillsley, R. E., Ideker, R. E., & Smith, W. M. (1998). Estimation of conduction velocity vector fields from epicardial mapping data. IEEE Transactions on Biomedical Engineering, 45(5), 563–571. https://doi.org/10.1109/10.668746

Berte, B., Zeppenfeld, K., & Tung, R. (2020). Impact of micro-, mini-and multi-electrode mapping on ventricular substrate characterisation. Arrhythmia & Electrophysiology Review, 9(3), 128. https://doi.org/10.15420/aer.2020.24

Bhagirath, P., Strocchi, M., Bishop, M. J., Boyle, P. M., & Plank, G. (2024). From bits to bedside: Entering the age of digital twins in cardiac electrophysiology. Europace, 26(12), euae295. https://doi.org/10.1093/europace/euae295

Bray, M.-A., & Wikswo, J. P. (2002). Considerations in phase plane analysis for nonstationary reentrant cardiac behavior. Physical Review E, 65(5), 051902. https://doi.org/10.1103/PhysRevE.65.051902

Bueno-Orovio, A., Cherry, E. M., & Fenton, F. H. (2008). Minimal model for human ventricular action potentials in tissue. Journal of Theoretical Biology, 253(3), 544–560. https://doi.org/10.1016/j.jtbi.2008.03.029

Byrne, G., Marcotte, C. D., & Grigoriev, R. O. (2015). Exact coherent structures and chaotic dynamics in a model of cardiac tissue. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(3), 033108. https://doi.org/10.1063/1.4915143

Cai, S., Mao, Z., Wang, Z., Yin, M., & Karniadakis, G. E. (2021). Physics-informed neural networks (PINNs) for fluid mechanics: A review. Acta Mechanica Sinica, 37(12), 1727–1738. https://doi.org/10.1007/s10409-021-01148-1

Cantwell, C. D., Roney, C. H., Ng, F. S., Siggers, J. H., Sherwin, S. J., & Peters, N. S. (2015). Techniques for automated local activation time annotation and conduction velocity estimation in cardiac mapping. Computers in Biology and Medicine, 65, 229–242. https://doi.org/10.1016/j.compbiomed.2015.04.027

CARMEN. (2024). Cardiac ElectroPhysiology Simulator (CEPS). IHU Liryc, Inria. https://carmen.gitlabpages.inria.fr/ceps/

Chen, S., Fan, S., Qiao, Z., Wu, Z., Lin, B., Li, Z., Riegler, M. A., Wong, M. Y. H., Opheim, A., Korostynska, O., et al. (2025). Transforming healthcare: Intelligent wearable sensors empowered by smart materials and artificial intelligence. Advanced Materials, 2500412. https://doi.org/10.1002/adma.202500412

Ciaccio, E. J., Coromilas, J., Wit, A. L., Peters, N. S., & Garan, H. (2016). Formation of functional conduction block during the onset of reentrant ventricular tachycardia. Circulation: Arrhythmia and Electrophysiology, 9(12), e004462. https://doi.org/10.1161/CIRCEP.116.004462

Clayton, R., Zhuchkova, E., & Panfilov, A. (2006). Phase singularities and filaments: Simplifying complexity in computational models of ventricular fibrillation. Progress in Biophysics and Molecular Biology, 90(1-3), 378–398. https://doi.org/10.1016/j.pbiomolbio.2005.06.011

Cooper, F. R., Baker, R. E., Bernabeu, M. O., Bordas, R., Bowler, L., Bueno-Orovio, A., Byrne, H. M., Carapella, V., Cardone-Noott, L., Cooper, J., Dutta, S., Evans, B. D., Fletcher, A. G., Grogan, J. A., Guo, W., Harvey, D. G., Hendrix, M., Kay, D., Kursawe, J., Maini, P. K., McMillan, B., Mirams, G. R., Osborne, J. M., Pathmanathan, P., Pitt-Francis, J. M., Robinson, M., Rodriguez, B., Spiteri, R. J., & Gavaghan, D. J. (2020). Chaste: Cancer, Heart and Soft Tissue Environment. Journal of Open Source Software, 5(47), 1848. https://doi.org/10.21105/joss.01848

Costabal, F. S., Yang, Y., Perdikaris, P., Hurtado, D. E., & Kuhl, E. (2020). Physics-informed neural networks for cardiac activation mapping. Frontiers in Physics, 8. https://doi.org/10.3389/fphy.2020.00042

Courant, R., Friedrichs, K., & Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100(1), 32–74. https://doi.org/10.1007/BF01448839

Cross, M. C., & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Rev. Modern Phys., 65(3), 851. https://doi.org/10.1103/RevModPhys.65.851

Davydov, V., Zykov, V., & Yamaguchi, T. (2000). Drift of spiral waves on nonuniformly curved surfaces. Macromolecular Symposia, 160, 99–106. https://doi.org/10.1002/1521-3900(200010)160:1<99::AID-MASY99>3.0.CO;2-Y

De Coster, T., Kabus, D., Verkerk, A. O., Veldkamp, M. W., Harlaar, N., Dekker, S. O., Vries, A. A. F. de, Pijnappels, D. A., & Panfilov, A. V. (in preparation). Ionic mechanisms underlying human immortalised atrial action potential properties: Insights from a mathematical model.

Deisenhofer, I., Zrenner, B., Yin, Y., Pitschner, H.-F., Kuniss, M., Großmann, G., Stiller, S., Luik, A., Veltmann, C., Frank, J., et al. (2010). Cryoablation versus radiofrequency energy for the ablation of atrioventricular nodal reentrant tachycardia (the CYRANO study) results from a large multicenter prospective randomized trial. Circulation, 122(22), 2239–2245. https://doi.org/10.1161/CIRCULATIONAHA.110.970350

Dermul, N., & Dierckx, H. (2024). Reconstruction of excitation waves from mechanical deformation using physics-informed neural networks. Scientific Reports, 14(1), 16975. https://doi.org/10.1038/s41598-024-67597-3

Dierckx, H., Brisard, E., Verschelde, H., & Panfilov, A. V. (2013). Drift laws for spiral waves on curved anisotropic surfaces. Physical Review E, 88, 012908. https://doi.org/10.1103/PhysRevE.88.012908

Du, P., Weber, R., Luszczek, P., Tomov, S., Peterson, G., & Dongarra, J. (2012). From CUDA to OpenCL: Towards a performance-portable solution for multi-platform GPU programming. Parallel Computing, 38(8), 391–407. https://doi.org/10.1016/j.parco.2011.10.002

Entcheva, E., & Kay, M. W. (2021). Cardiac optogenetics: A decade of enlightenment. Nature Reviews Cardiology, 18(5), 349–367. https://doi.org/10.1038/s41569-020-00478-0

Fenton, F. H., & Karma, A. (1998). Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 8(1), 20–47. https://doi.org/10.1063/1.166311

Fenton, F. H., Karma, A., Hastings, H. M., & Evans, S. J. (2000). Transition from ventricular tachycardia to ventricular fibrillation as function of tissue characteristics computer model. Proceedings of the 22nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Cat. No. 00CH37143), 1, 391–394. https://doi.org/10.1109/IEMBS.2000.900757

Finsberg, H. N. T., van Herck, I. G. M., Daversin-Catty, C., Arevalo, H., & Wall, S. (2023). simcardems: A FEniCS-based cardiac electro-mechanics solver. Journal of Open Source Software, 8(81), 4753. https://doi.org/10.21105/joss.04753

FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466. https://doi.org/10.1016/s0006-3495(61)86902-6

Gillette, K., Gsell, M. A. F., Prassl, A. J., Karabelas, E., Reiter, U., Reiter, G., Grandits, T., Payer, C., Štern, D., Urschler, M., Bayer, J. D., Augustin, C. M., Neic, A., Pock, T., Vigmond, E. J., & Plank, G. (2021). A framework for the generation of digital twins of cardiac electrophysiology from clinical 12-leads ECGs. Medical Image Analysis, 71, 102080. https://doi.org/10.1016/j.media.2021.102080

Gobeyn, A., Kabus, D., & Dierckx, H. (in preparation). ZEUS method for the numerical detection of topological quasiparticles arising in the context of excitable media.

Graham, R. L., Shipman, G. M., Barrett, B. W., Castain, R. H., Bosilca, G., & Lumsdaine, A. (2006). Open MPI: A high-performance, heterogeneous MPI. 2006 IEEE Int. Conf. On Cluster Computing, 1–9. https://doi.org/10.1109/CLUSTR.2006.311904

Gray, P., & Scott, S. (1983). Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability. Chemical Engineering Science, 38(1), 29–43. https://doi.org/10.1016/0009-2509(83)80132-8

Gray, R. A., Jalife, J., Panfilov, A., Baxter, W. T., Cabo, C., Davidenko, J. M., & Pertsov, A. M. (1995). Nonstationary vortexlike reentrant activity as a mechanism of polymorphic ventricular tachycardia in the isolated rabbit heart. Circulation, 91(9), 2454–2469. https://doi.org/10.1161/01.CIR.91.9.2454

Gray, R. A., Pertsov, A. M., & Jalife, J. (1998). Spatial and temporal organization during cardiac fibrillation. Nature, 392(6671), 75–78. https://doi.org/10.1038/32164

Haissaguerre, M., Jais, P., Shah, D., Takahashi, A., Hocini, M., Quiniou, G., Garrigue, S., Le Mouroux, A., P, L. M., & J, C. (1998). Spontaneous initiation of atrial fibrillation by ectopic beats originating in the pulmonary veins. New England Journal of Medicine, 339, 659–666. https://doi.org/10.1056/nejm199809033391003

Harlaar, N., Dekker, S. O., Zhang, J., Snabel, R. R., Veldkamp, M. W., Verkerk, A. O., Fabres, C. C., Schwach, V., Lerink, L. J. S., Rivaud, M. R., Mulder, A. A., Corver, W. E., Goumans, M. J. T. H., Dobrev, D., Klautz, R. J. M., Schalij, M. J., Veenstra, G. J. C., Passier, R., Brakel, T. J. van, Pijnappels, D. A., & Vries, A. A. F. de. (2022). Conditional immortalization of human atrial myocytes for the generation of in vitro models of atrial fibrillation. Nature Biomedical Engineering, 6(4), 389–402. https://doi.org/10.1038/s41551-021-00827-5

Heckert, N. A., Filliben, J. J., Croarkin, C. M., Hembree, B., Guthrie, W. F., Tobias, P., & Prinz, J. (2002). Handbook 151: NIST/SEMATECH e-handbook of statistical methods. https://doi.org/10.18434/M32189

Hill, C. L., & Stephens, G. J. (2021). An introduction to patch clamp recording. Patch Clamp Electrophysiology: Methods and Protocols, 1–19. https://doi.org/10.1007/978-1-0716-0818-0_1

Hochreiter, S. (1991). Untersuchungen zu dynamischen neuronalen Netzen. Diploma, Technische Universität München, 91(1), 31.

Hochreiter, S., Bengio, Y., Frasconi, P., Schmidhuber, J., et al. (2001). Gradient flow in recurrent nets: The difficulty of learning long-term dependencies. A field guide to dynamical recurrent neural networks. IEEE Press In.

Hornik, K., Stinchcombe, M., & White, H. (1989). Multilayer feedforward networks are universal approximators. Neural Networks, 2(5), 359–366. https://doi.org/10.1016/0893-6080(89)90020-8

ISO. (2023). ISO/IEC 14882:2023: Programming languages — C++ (Seventh, p. 2101). International Organization for Standardization.

Jakob, W., Rhinelander, J., & Moldovan, D. (2017). pybind11 – seamless operability between C++11 and Python. https://github.com/pybind/pybind11

Kapral, R., & Showalter, R. (1995). Chemical waves and patterns. Kluwer.

Karakikes, I., Ameen, M., Termglinchan, V., & Wu, J. C. (2015). Human induced pluripotent stem cell–derived cardiomyocytes: Insights into molecular, cellular, and functional phenotypes. Circulation Research, 117(1), 80–88. https://doi.org/10.1161/circresaha.117.305365

Karma, A. (1993). Spiral breakup in model equations of action potential propagation in cardiac tissue. Physical Review Letters, 71(7), 1103. https://doi.org/10.1103/PhysRevLett.71.1103

Karma, A. (1994). Electrical alternans and spiral wave breakup in cardiac tissue. Chaos: An Interdisciplinary Journal of Nonlinear Science, 4(3), 461–472. https://doi.org/10.1063/1.166024

Kazbanov, I. V., Ten Tusscher, K. H., & Panfilov, A. V. (2016). Effects of heterogeneous diffuse fibrosis on arrhythmia dynamics and mechanism. Scientific Reports, 6(1), 20835. https://doi.org/10.1038/srep20835

Keener, J. P. (2004). The topology of defibrillation. Journal of Theoretical Biology, 230(4), 459–473. https://doi.org/10.1016/j.jtbi.2003.11.033

Koopsen, T., Gerrits, W., van Osta, N., van Loon, T., Wouters, P., Prinzen, F. W., Vernooy, K., Delhaas, T., Teske, A. J., Meine, M., Cramer, M. J., & Lumens, J. (2024). Virtual pacing of a patient’s digital twin to predict left ventricular reverse remodelling after cardiac resynchronization therapy. Europace, 26(1), euae009. https://doi.org/10.1093/europace/euae009

Li, N., Steiner, J., & Tang, S. (1994). Convergence and stability analysis of an explicit finite difference method for 2-dimensional reaction-diffusion equations. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 36(2), 234–241. https://doi.org/10.1017/S0334270000010377

Loew, L. M. (1996). Potentiometric dyes: Imaging electrical activity of cell membranes. Pure and Applied Chemistry, 68(7), 1405–1409. https://doi.org/10.1351/pac199668071405

Majumder, R., Jangsangthong, W., Feola, I., Ypey, D. L., Pijnappels, D. A., & Panfilov, A. V. (2016). A mathematical model of neonatal rat atrial monolayers with constitutively active acetylcholine-mediated k+ current. PLoS Computational Biology, 12(6), e1004946. https://doi.org/10.1371/journal.pcbi.1004946

Martin, C. H., Oved, A., Chowdhury, R. A., Ullmann, E., Peters, N. S., Bharath, A. A., & Varela, M. (2022). EP-PINNs: Cardiac electrophysiology characterisation using physics-informed neural networks. Frontiers in Cardiovascular Medicine, 8. https://doi.org/10.3389/fcvm.2021.768419

Mikhailov, A. S., Davydov, V., & Zykov, V. (1994). Complex dynamics of spiral waves and motion of curves. Physica D: Nonlinear Phenomena, 70(1-2), 1–39. https://doi.org/10.1016/0167-2789(94)90054-X

Mitchell, A., & Bruch Jr, J. (1985). A numerical study of chaos in a reaction-diffusion equation. Numerical Methods for Partial Differential Equations, 1(1), 13–23. https://doi.org/10.1002/num.1690010104

Mitchell, C. (2003). A two-current model for the dynamics of cardiac membrane. Bulletin of Mathematical Biology, 65(5), 767–793. https://doi.org/10.1016/S0092-8240(03)00041-7

Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50(10), 2061–2070. https://doi.org/10.1109/jrproc.1962.288235

Neu, J. C., & Krassowska, W. (1993). Homogenization of syncytial tissues. Critical Reviews in Biomedical Engineering, 21(2), 137–199.

Nickolls, J., Buck, I., Garland, M., & Skadron, K. (2008). Scalable parallel programming with cuda: Is cuda the parallel programming model that application developers have been waiting for? Queue, 6(2), 40–53. https://doi.org/10.1145/1365490.1365500

Niederer, S. A., Lumens, J., & Trayanova, N. A. (2019). Computational models in cardiology. Nature Reviews Cardiology, 16(2), 100–111. https://doi.org/10.1038/s41569-018-0104-y

Nobel Prize. (1963). The Nobel prize in physiology or medicine 1963. https://www.nobelprize.org/prizes/medicine/1963

Paci, M., Hyttinen, J., Aalto-Setälä, K., & Severi, S. (2013). Computational models of ventricular-and atrial-like human induced pluripotent stem cell derived cardiomyocytes. Annals of Biomedical Engineering, 41(11), 2334–2348. https://doi.org/10.1007/s10439-013-0833-3

Portero, V., Deng, S., Boink, G. J. J., Zhang, G. Q., de Vries, A. A. F., & Pijnappels, D. A. (2024). Optoelectronic control of cardiac rhythm: Toward shock-free ambulatory cardioversion of atrial fibrillation. Journal of Internal Medicine, 295(2), 126–145. https://doi.org/10.1111/joim.13744

Priebe, L., & Beuckelmann, D. J. (1998). Simulation study of cellular electric properties in heart failure. Circulation Research, 82(11), 1206–1223. https://doi.org/10.1161/01.res.82.11.1206

Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045

Rognes, M. E., Farrell, P. E., Funke, S. W., Hake, J. E., & Maleckar, M. M. C. (2017). Cbcbeat: An adjoint-enabled framework for computational cardiac electrophysiology. The Journal of Open Source Software, 2(13), 224. https://doi.org/10.21105/joss.00224

Rush, S., & Larsen, H. (1978). A practical algorithm for solving dynamic membrane equations. IEEE Transactions on Biomedical Engineering, 4, 389–392. https://doi.org/10.1109/TBME.1978.326270

Schiebler, T. H. (2005). Anatomie: Histologie, entwicklungsgeschichte, makroskopische und mikroskopische anatomie, topographie. Springer.

Shahi, S., Fenton, F. H., & Cherry, E. M. (2022). A machine-learning approach for long-term prediction of experimental cardiac action potential time series using an autoencoder and echo state networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(6). https://doi.org/10.1063/5.0087812

Temam, R. (1990). Inertial manifolds. The Mathematical Intelligencer, 12(4), 68–74. https://doi.org/10.1007/BF03024036

ten Tusscher, K. H. W. J., Noble, D., Noble, P.-J., & Panfilov, A. V. (2004). A model for human ventricular tissue. American Journal of Physiology-Heart and Circulatory Physiology, 286(4), H1573–H1589. https://doi.org/10.1152/ajpheart.00794.2003

Tomii, N., Yamazaki, M., Ashihara, T., Nakazawa, K., Shibata, N., Honjo, H., & Sakuma, I. (2021). Spatial phase discontinuity at the center of moving cardiac spiral waves. Computers in Biology and Medicine, 130, 104217. https://doi.org/10.1016/j.compbiomed.2021.104217

Toscano, J. D., Oommen, V., Varghese, A. J., Zou, Z., Daryakenari, N. A., Wu, C., & Karniadakis, G. E. (2024). From PINNs to PIKANs: Recent advances in physics-informed machine learning. arXiv Preprint arXiv:2410.13228. https://doi.org/10.48550/arXiv.2410.13228

Trayanova, N. A., Doshi, A. N., & Prakosa, A. (2020). How personalized heart modeling can help treatment of lethal arrhythmias: A focus on ventricular tachycardia ablation strategies in post‐infarction patients. WIREs Systems Biology and Medicine, 12(3). https://doi.org/10.1002/wsbm.1477

Tung, L. (1978). A bi-domain model for describing ischemic myocardial DC currents [PhD thesis]. MIT, Cambridge, MA.

van Rossum, G., Drake, F. L., et al. (1995). The Python programming language. Centrum voor Wiskunde en Informatica Amsterdam. https://www.python.org

Van Valen, D. A., Kudo, T., Lane, K. M., Macklin, D. N., Quach, N. T., DeFelice, M. M., Maayan, I., Tanouchi, Y., Ashley, E. A., & Covert, M. W. (2016). Deep learning automates the quantitative analysis of individual cells in live-cell imaging experiments. PLoS Computational Biology, 12(11), e1005177. https://doi.org/10.1371/journal.pcbi.1005177

Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S. J., Brett, M., Wilson, J., Millman, K. J., Mayorov, N., Nelson, A. R. J., Jones, E., Kern, R., Larson, E., Carey, C. J., Polat, İlhan, Feng, Y., Moore, E. W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E. A., Harris, C. R., Archibald, A. M., Ribeiro, A. H., Pedregosa, F., van Mulbregt, P., & SciPy 1.0 Contributors. (2020). SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17, 261–272. https://doi.org/10.1038/s41592-019-0686-2

Weisrock, A., Wüst, R., Olenic, M., Lecomte-Grosbras, P., & Thorrez, L. (2024). MyoFInDer: An AI-based tool for myotube fusion index determination. Tissue Engineering, ja. https://doi.org/10.1089/ten.tea.2024.0049

Zipes, D. P., & Jalife, J. (2014). Cardiac electrophysiology: From cell to bedside. Elsevier.

from the Greek word βραδύς (bradýs) for “slow” ↩︎
from the Greek word ταχύς (tachýs) for “fast” ↩︎
The term artificial intelligence (AI) is much more vague. Not just machine-learned functions, but, in the most broad sense, any computer algorithm that seems in some sense “intelligent” can be called AI. For instance, allegedly, the bot of Reddit user u/versaceblues won their university’s AI poker competition with the algorithm if isMyTurn: goAllIn(), to which all other bots folded. In recent years, AI has become an overused buzzword that is used essentially interchangeably with the word “algorithm”, “program”, or even “mathematics”. We refrain from using the term AI, as it is too vague, and instead prefer to use more precise terms, such as machine learning, where applicable. ↩︎

Calendar of travel photos 2024

Wed, 04 Dec 2024 20:48:09 +0100

Cover photo: 東京スカイツリー, 日本国 (Sky Tree, Tokyo, Japan)

Analysis of complex excitation patterns using Feynman-like diagrams

Fri, 22 Nov 2024 00:00:00 +0000

This article was previously published in Scientific Reports 14, 28962 (2024). (Arno et al., 2024a) and is a chapter of my dissertation. The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Authors:

Louise Arno^1,3 (ORCID: 0000-0001-9384-519X)
Desmond Kabus^1,2,3 (ORCID: 0000-0002-6965-5211)
Hans Dierckx¹ (ORCID: 0000-0003-0899-8082)

Institutions:

Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Etienne Sabbelaan 53, 8500 Kortrijk, Belgium
Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Albinusdreef 2, 2333 ZA Leiden, The Netherlands
iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Oude Markt 13, 3000 Leuven, Belgium

Correspondence: dr.h.dierckx@gmail.com

DOI: 10.1038/s41598-024-73544-z

Abstract:

Many extended chemical and biological systems self-organise into complex patterns that drive the medium behaviour in a non-linear fashion. An important class of such systems are excitable media, including neural and cardiac tissues. In extended excitable media, wave breaks can form rotating patterns and turbulence. However, the onset, sustaining and elimination of such complex patterns is currently incompletely understood. The classical theory of phase singularities in excitable media was recently challenged, as extended lines of conduction block were identified as phase discontinuities. Here, we provide a theoretical framework that captures the rich dynamics in excitable systems in terms of three quasiparticles: heads, tails, and pivots. We propose to call these quasiparticles ‘cardions’. In simulations and experiments, we show that these basic building blocks combine into at least four different bound states. By representing their interactions similarly to Feynman diagrams in physics, the creation and annihilation of vortex pairs are shown to be sequences of dynamical creation, annihilation, and recombination of the identified quasiparticles. We draw such diagrams for numerical simulations, as well as optical voltage mapping experiments performed on cultured human atrial myocytes (hiAMs). Our results provide a new, unified language for a more detailed theory, analysis, and mechanistic insights of dynamical transitions in excitation patterns.

Introduction

Some of the most intriguing waves and patterns in nature arise from the spatial coupling between excitable elements. Examples include the emergence of intelligence in neural tissue (Cannon et al., 2014), epidemic spread across a population (Smith et al., 2002), intracellular waves (Lechleiter et al., 1991), chemical oxidation waves (Kapral & Showalter, 1995; Rotermund et al., 1990) and the emergence of cardiac arrhythmias in the heart muscle (Allessie et al., 1973; Gray et al., 1998). Our work was motivated by the case of heart rhythm disorders, that had a global mortality rate of in 2019, with increasing numbers every year. Hence, cardiac arrhythmias remain one of the largest causes of death worldwide (WHO, 2020).

While the local excitation dynamics can be measured and modelled, a firm theoretical description at the mesoscale is still lacking. Hence, much effort is currently invested in numerical modelling (Clayton et al., 2011) and machine learning approaches (Trayanova et al., 2021). Our aim is to complement such approaches with semi-analytical methods that give more insight in why patterns develop over time in a certain manner.

In this paper, we revise the classical mesoscale theory of excitation that has been used for cardiac arrhythmia analysis over the past decades (Christoph et al., 2018; Clayton et al., 2006; Clayton et al., 2011; Gray et al., 1998). Similar to historical developments in particle physics, we coin that the classical phase singularities in excitation are formed of more elementary quasiparticles. While this finer structure remains hidden in simple dynamical regimes, we show below in simulation and experiment that it becomes relevant during the processes of wave block, vortex initiation and recombination. Such insight is relevant for excitation patterns, since precisely those events need to be thoroughly understood if one wants to predict or control the emergent dynamics in those systems. A graphical outline of the analysis method presented in this paper can be found in Fig. 1.

Figure 1: Conceptual framework for the analysis of excitation patterns using quasiparticles and Feynman-like diagrams. A. A linear-core spiral wave (Bueno-Orovio et al., 2008) in a numerical simulation on a human biventricular geometry, see the Methods section below. B. Numerical simulation (Bueno-Orovio et al., 2008) in a two-dimensional domain, which could represent a pattern on the outer surface of the heart or in a cell culture. C. The linear phase based on local activation time (LAT) Eq. 3. D. Spatial analysis of the observed pattern, showing excited ($\mathrm{E}$, yellow) and unexcited ($\mathrm{U}$, white) regions. They are separated by the wave fronts ($\mathrm{F}$, cyan), wave backs ($\mathrm{B}$, magenta), and phase defect lines ($\mathrm{Z}$, gray). The end points of these three different curves are topologically preserved as quasiparticles: heads (h, blue), tails (t, red), and pivots (p, green). E. Top: Depending on the orientation, we assign positive or negative charges $Q=\pm \frac{1}{2}$ to heads and tails, and charges $P = \pm \frac{1}{2}$ to pivots. Bottom: Several combinations of head, tail, and pivot are seen to travel together through the medium, akin to bound states in particle physics: the classical tip ($\mathrm{f} = \mathrm{h}+\mathrm{t}$), the spiral core ($\mathrm{c} = \mathrm{h}+\mathrm{t}+2\mathrm{p}$), and the newly identified phase defect growth sites ($\mathrm{g}=\mathrm{h}+\mathrm{t}+\mathrm{p}$), and shrinking sites ($\mathrm{s}=\mathrm{t}+\mathrm{p}$) seen during arrhythmogenesis. For an overview of all quantities, see Table 1. F. We propose to keep track of the recombinations of building blocks over time using Feynman-like diagrams.

particle	notation	simplified notation	charge $Q$	charge $P$	composition
head	$\mathrm{h}^{\pm}$	$\mathrm{h}^{\pm}$	$\pm\frac{1}{2}$	0
tail	$\mathrm{t}^{\pm}$	$\mathrm{t}^{\pm}$	$\pm\frac{1}{2}$	0
pivot	$\mathrm{p}_{\pm}$	$\mathrm{p}^{\pm}$	$0$	$\pm\frac{1}{2}$
filament; tip	$\mathrm{f}^{\pm\pm}$	$\mathrm{f}^{\pm}$	$\pm 1$	$0$	h$^\pm$ + t$^\pm$
core; phase singularity	$\mathrm{c}^{\pm\pm}_{\pm\pm}$	$\mathrm{c}^{\pm}$	$\pm 1$	$\pm 1$	h$^\pm$ + t$^\pm$ + 2p$_\pm$
shrink site	$\mathrm{s}^{\pm}_{\pm}$	$\mathrm{s}^{\pm}$	$\pm\frac{1}{2}$	$\pm\frac{1}{2}$	p$_\pm$ + t$^\pm$
growth site	$\mathrm{g}^{\pm\pm}_{\pm}$	$\mathrm{g}^{\pm}$	$\pm 1$	$\pm\frac{1}{2}$	h$^\pm$ + t$^\pm$ + p$_\pm$

Table 1: Overview of quasiparticles (cardions) in excitable media, together with their charges, notation, composition, and colour code. In the extended notation, superscript denotes the head/tail charge $Q$ in units of $\frac{1}{2}$ and subscripts denote the pivot charge $P$. As only quasiparticles of the same charge bind, the simplified notation indicates this with a single superscript $\pm$. The list is non-exhaustive and reflects our current knowledge.

As we illustrate our general theoretical framework below with an analysis of experiments performed on cultured human heart cells (Harlaar et al., 2022), we now describe the current state-of-art and associated challenges in the description and understanding of cardiac excitation patterns.

The mechanical contraction of each myocyte in the heart is triggered by electrical depolarisation of the cells, which first propagates through a dedicated conduction system, but is thereafter passed on from cardiomyocyte to cardiomyocyte. As a result, the activation wave can re-excite itself, or a local focal source can start extra excitations. Such abnormal emerging behaviour manifests as a cardiac arrhythmia.

A simple arrhythmia in which the excitation wave follows a fixed path in time, or which originates from a region of ectopic impulse formation can be treated by ablation therapy, in which the trigger or path is physically destroyed (Haissaguerre et al., 1998). Even though efficiency has improved significantly throughout the years, the success rate is rather low for arrhythmias with incompletely understood spatiotemporal organisation (Cronin et al., 2019; Mujović et al., 2017). Such arrhythmias include ventricular fibrillation (Gray et al., 1998; Jalife et al., 1998), which is the most lethal one, and atrial fibrillation, which affects approximately of the world population, and is responsible for one third of stroke cases (Samol et al., 2016). In both atrial and ventricular fibrillation, as well as in ventricular tachycardia, rotating vortices of activation have been observed (Allessie et al., 1973), named spiral waves, scroll waves, or cardiac rotors. While computer simulations of the heart (Gray et al., 1995) and initial observations (Gray et al., 1998) often yield long-lived stable rotors, complex patterns detected in real hearts commonly show a more complex regime that exhibits conduction blocks and transient rotors which do not even complete a full turn. There is a consensus that interacting wave fragments, also called wavelets (Arno et al., 2021; Lee et al., 2020; Moe & Abildskov, 1959) play a role in atrial and ventricular fibrillation. However, the precise mechanisms of onset, sustainment, and termination of complex arrhythmias remain to be further elucidated (Aras et al., 2017; Shibata et al., 2022).

We now review the concepts of phase and phase singularities that are used to describe excitable media. Winfree (1973) already noted that many biological processes are cyclic in nature, and therefore could be described naturally by a ‘phase’. Near the centre of a rotating spiral wave, phase singularities may occur (Winfree & Strogatz, 1983), which form filament curves in three dimensions. Given two observed variables $V$ and $R$ that depend on the position ${{\bm{{r}}}}$ and time $t$ in a spatiotemporal pattern, it is common to call the polar angle in the state space spanned by $V$ and $R$ the ‘phase of activation’: $$\begin{aligned} \varphi({{\bm{{r}}}},t) &= \operatorname{atan2}( R({{\bm{{r}}}},t) - R_*, V({{\bm{{r}}}},t) - V_*) \end{aligned}\qquad{(1)}$$ Here, $\operatorname{atan2}$ is the two-argument arc-tangent function, delivering values in $(-\pi,\pi]$. In a seminal work, Gray et al. (1998) analysed ventricular fibrillation patterns in this manner (Gray et al., 1998), with $V({{\bm{{r}}}},t)$ the optical intensity representing normalised transmembrane potential, and $R({{\bm{{r}}}},t) = V({{\bm{{r}}}},t+\tau_0)$ its time-delayed version. The main idea here is that points that differ in phase by an integer multiple of $2\pi$ are effectively in the same state. After spatial filtering of the signal, distinct points occurred where all phases meet, called phase singularities (Winfree & Strogatz, 1983). Since then, analysis tools (Bray & Wikswo, 2003; Gray et al., 1998), ablation strategies (Narayan et al., 2012), and mathematical theory (Clayton et al., 2006; Panfilov & Dierckx, 2017) have been developed based on the assumed existence of phase singularities during arrhythmias. A phase singularity can also be regarded as the intersection of two contour lines of two different state variables, which also allows to locate them numerically (Fenton & Karma, 1998). An extension of this topological approach that focuses on the dynamical transitions between the contour intersections was described by Marcotte & Grigoriev (2017).

Characterising excitable media in terms of phase singularities was pioneered based on observations in oscillating chemical reactions (Winfree, 1973; Zhabotinsky & Zaikin, 1973). When a phase singularity is tracked in computational models of cardiac excitation (Fenton & Cherry, 2008), however, it is often seen to follow a star-like or zig-zag pattern (Krinsky et al., 1992). Such a ’linear rotor core’ also occurs at the interface between regions with different action potential duration, where it was historically called a ‘reverbator’ (Krinsky, 1966, 1968). Linear core rotors have for decades escaped a thorough theoretical description, in contrast to the better-behaved ‘rigidly rotating rotors’. The latter are now well understood in terms of response function theory (Biktashev et al., 1994; Henry & Hakim, 2000; Keener, 1988; Verschelde et al., 2007), which enables to forecast their motion if they are long-lived and weakly interacting with other rotors, boundaries or inhomogeneities. However, in arrhythmia management, the key processes to control are the transient processes causing the ‘birth’ of rotors from wave breaks during arrhythmogenesis, and the elimination of rotors under pharmacological treatment, scarification of the medium, or by mutual collisions. Even worse, atrial and ventricular fibrillation are characterised by an incessant interaction between short-lived wavebreaks, for which a theory is hitherto largely lacking.

In recent years, it was suggested independently by Tomii et al. (2021) and Arno et al. (2021) that the classical phase singularity concept may not be an accurate description for the structure of a linear rotor core. Here, the zig-zag pattern emerges since the classical spiral wave tip, i.e., the end point of the wave front, is moving along the border of a region that has not yet recovered from the previous activation. As soon as this region recovers, the spiral wave tip turns over about in a so-called pivot point and then travels in the opposite direction, after which this process is repeated. The mentioned authors (Arno et al., 2021; Tomii et al., 2021) remarked that the parts of the medium at both sides of the linear rotor core have a different phase, as these regions were excited at different times. Hence, Tomii et al. (2021) coined that across the conduction block line at the centre of a linear-core rotor, there is a phase discontinuity. We also noted this (Arno et al., 2021), but remarked that due to electrotonic effects, i.e., diffusion of transmembrane potential, a boundary layer of finite width will be formed, across which the phase appears to change rapidly in space. The situation is reminiscent of domain walls or other interfaces in physics, and therefore we refer to them as ‘phase defects’ (Arno et al., 2021). The diffusive effect also explains why the phase defects were not identified before: If you have a line along which phase is discontinuous and smoothen the variables slightly, the resulting phase function tends to become continuous except in isolated points that happen to have the values $V=V_*$, $R=R_*$, where all phases meet. These points are exactly the phase singularities that have been used in cardiac analysis for nearly three decades now (Gray et al., 1995). While many of these points will be true phase singularities, a diffusive smoothing applied to a conduction block may introduce pairs of false phase singularities, which in our opinion explains the non-robust detection of phase singularity pairs in conduction block regions (Arno et al., 2021; Rodrigo et al., 2017).

The concept of a ‘phase defect’ in excitable media is able to link different objects already known in the field and will reveal new ones, as can be seen below. From the explanation above, the phase defect can represent a conduction block without a rotor attached to it, and the conduction block that is present in a linear-core rotor. In hindsight, Krinsky’s reverbators (Krinsky, 1966, 1968) were also phase defects, as across the interface, the phase was also discontinuous. Note that conduction blocks are key features in activation maps (Janse et al., 1980). In clinical acquisitions, it is common to record the local activation times (LATs) and draw the isochrones. Where different isochrones coincide, a conduction block and hence a phase defect is present. This situation is similar to cartography, where coincident isolines mark a cliff, i.e., a discontinuity in altitude.

The aim of this paper is to provide a more detailed theoretical framework that reconciles the concepts of wave fronts, wave backs, conduction blocks, rotors with different core types, fragmented wave fronts (wavelets) etc. The key concept will be the phase defect mentioned above, and we will show that tracking how such phase defect is connected to wave fronts and backs opens up new ways for pattern analysis. Our exposition has been designed to unravel the critical processes that cause, sustain or terminate complex arrhythmias. We purposely use terminology from particle physics, as in this work we bring several analogies from that discipline to a cardiac context.

Conceptual framework

We build up the framework by defining and motivating subsequently: phase, phase defects, quasiparticles, charges and diagrams. A sketch illustrating the main quantities in the following definitions can be found in Fig. 2.

Figure 2: Sketch illustrating the main quantities in the definitions for the topological framework on a linear-core rotor. Isochrones at different LATs $t$ are used to define the phase $\varphi$, which in turn is used to define the wave front ($\mathrm{F}$) and back ($\mathrm{B}$). At the core of the vertex, the phase $\varphi$ is discontinuous. The threshold $\rho > \rho_*$ of the phase defect density is used to define the phase defect region $Z$. The angle $\psi$ with the $x$-axis is the direction of wave propagation.

Phase

Winfree wrote the following motivation to describe biological media with a phase concept (Winfree & Strogatz, 1983): “After excitation, [the phase] initially grows linearly with time, but after a while, it lingers near a value of $2\pi$, which is equivalent to phase 0, the completely recovered state.” So, when looking at an action potential as a function of time, we require that $\varphi(t) = 0$ for $t<0$ and $\varphi(t) = 2\pi$ for large $t$. Phase definition Eq. 1 is one possibility. It is certainly not unique, since different variables $V$, $R$ can be chosen, as well as their threshold values $V_*$, $R_*$. Two other choices were given in by Kabus et al. (2022), designed to exhibit no sharp phase variation around the wave front.

To make the link with isochrone maps, we here define the local activation time (LAT) as the last time when the transmembrane potential $V$ rose above a threshold value $V_*$: $$\begin{aligned} t_\mathrm{LAT}({{\bm{{r}}}},t) = \operatorname{argmax}_{t_0 < t} \{ V({{\bm{{r}}}},t_0) = V_* \wedge \partial_t V({{\bm{{r}}}},t_0) > 0 \} \end{aligned}\qquad{(2)}$$

The elapsed time since last activation is then $\tau = t - t_\mathrm{LAT}$. We now propose a linear activation time-based phase: $$\begin{aligned} \varphi({{\bm{{r}}}}, t) = \begin{cases} 2\pi \frac{\tau({{\bm{{r}}}}, t)}{\tau_0} & \tau({{\bm{{r}}}}, t) < \tau_0 \\ 2 \pi & \tau({{\bm{{r}}}}, t) \geq \tau_0 \end{cases} \end{aligned}\qquad{(3)}$$ Hence, the resting state has phase 0, and after activation phase grows linearly with time over a time interval $\tau_0$, after which the tissue is assumed to be fully recovered. Then the tissue has phase $2\pi$, which is equivalent to phase $0$. This simple phase has its limitations: It is identical to zero at the foot of the upstroke, when $V$ is still lower than $V_*$. After a time interval $\tau_0$, that we choose equal to the local action potential duration in that point, the medium is assumed to be fully recovered. Still, it suffices for the formal introduction of the concepts below.

Phase defects

By construction, $\varphi$ is continuous over a wave front, since there the phase amounts to $0$. It is also continuous over a wave back, where $\varphi$ is slightly lower than $2\pi$. At conduction block sites, however, the impeding wave does not make $V$ cross the threshold value $V_*$ from below anymore, and hence $\varphi$ will be discontinuous across a conduction block. As a consequence of the phase definition Eq. 3, a true phase discontinuity forms, of co-dimension 1. In an inhomogeneous medium where action potential duration depends on position, the edge of the inhomogeneity can take any shape, such that the discontinuity of $\varphi$ needs not be a line but can take any shape: Closed or branching phase defect structures may also form.

To numerically detect the phase defect, we compute the phase defect density $\rho({{\bm{{r}}}},t)$ over all pixels of the grid (Kabus et al., 2022; Tomii et al., 2021): $$\begin{aligned} \rho({{\bm{{r}}}}_a,t) = \frac{1}{2 N_a} \sum_{b \in \mathcal{N}(a)} [1 - \cos( \varphi({{\bm{{r}}}}_a,t) - \varphi({{\bm{{r}}}}_b,t) )] \end{aligned}\qquad{(4)}$$ Here, $\mathcal{N}(a)$ is the set of nearest neighbours in the Cartesian grid next to the point ${{\bm{{r}}}}_a$. $N_a$ is the number of such neighbours; it is everywhere four except near the domain boundaries or inexcitable obstacles. When the phases locally differ much, $\rho$ will have a value close to $1$. If all excitable elements excite simultaneously, $\rho=0$. The set $\mathrm{Z}$ of points inside a phase defect can be identified numerically as the set where $\rho$ exceeds a chosen threshold value $\rho_*$ (Kabus et al., 2022): $$\begin{aligned} \mathrm{Z}(t) = \{ {{\bm{{r}}}} \;|\; \rho({{\bm{{r}}}}, t) > \rho_* \} \end{aligned}\qquad{(5)}$$ In this manner, the phase defect region $\mathrm{Z}$ forms an open subset of the domain; it is not a one-dimensional curve anymore. The contours $\mathcal{C}$ obeying $\rho = \rho_*$ encircling each phase defect are well-defined and will be used below.

It is also possible to start from the classical activation phase, rescale it via a sigmoid function and then compute the phase defect density (Kabus et al., 2022). In that case, the phase defect also becomes an open set $\mathrm{Z}$, surrounded by a continuous contour $\mathcal{C}$.

Special points

The essence of our analysis is to divide the medium in three regions and to identify the points where all these regions meet, see Fig. 1.

First, we compute the phase defect density $\rho$ via Eq. 4 to find all points within the phase defect $\mathrm{Z}$, see Eq. 5. These points will be colored in gray.

Let $0<\varphi_\mathrm{F}<\varphi_\mathrm{B}<2\pi$, where $\varphi_\mathrm{F}$ is a small phase threshold close to $0$, and $\varphi_\mathrm{B}$ a large phase threshold close to $2\pi$, which must be small enough that a jump from $\varphi_\mathrm{B}$ to $\varphi_\mathrm{F}$ is considered a phase defect following Eq. 4 and Eq. 5. The set of points where the phase is between these values and outside a phase defect is the excited region ($\mathrm{E}$): $$\begin{aligned} \mathrm{E}(t) = \{ {{\bm{{r}}}}\;|\; \rho({{\bm{{r}}}}, t) < \rho_* \wedge \varphi({{\bm{{r}}}}, t) \in (\varphi_\mathrm{F}, \varphi_\mathrm{B}) \} \end{aligned}\qquad{(6)}$$ and is depicted in yellow in Fig. 1. The set of points outside of that range and phase defects is called unexcited ($\mathrm{U}$) and coloured white: $$\begin{aligned} \mathrm{U}(t) = \{ {{\bm{{r}}}}\;|\; \rho({{\bm{{r}}}}, t) < \rho_* \wedge \varphi({{\bm{{r}}}}, t) \not\in [\varphi_\mathrm{F}, \varphi_\mathrm{B}] \} \end{aligned}\qquad{(7)}$$

Where the excited region meets the unexcited one, the wave front ($\mathrm{F}$) and wave back ($\mathrm{B}$) are found, denoted in cyan and magenta, respectively: $$\begin{aligned} \mathrm{F}(t) = \{ {{\bm{{r}}}}\;|\; \rho({{\bm{{r}}}}, t) < \rho_* \wedge \varphi({{\bm{{r}}}}, t) = \varphi_\mathrm{F} \} \\ \mathrm{B}(t) = \{ {{\bm{{r}}}}\;|\; \rho({{\bm{{r}}}}, t) < \rho_* \wedge \varphi({{\bm{{r}}}}, t) = \varphi_\mathrm{B} \} \end{aligned}\qquad{(8)}$$

For completeness, we remark that two other boundaries of co-dimension 1 exist: the border between $\mathrm{Z}$ and $\mathrm{U}$, and the border between $\mathrm{Z}$ and $\mathrm{E}$. Together they form the aforementioned contours $\mathcal{C}$.

In the classical theory of excitation, the wave front meets the wave back in a single point, and this point coincides with the classical phase singularity (Clayton et al., 2006; Gray et al., 1995; Zykov, 1987).

In our case, wave fronts and wave backs will always end on a phase defects and medium boundaries. We call the point where a wave front and a phase defect meet, a ‘head’ (denoted in blue) and a point where a wave back and a phase defect meet, a ’tail’ (denoted in red). Essentially, heads and tails are the loci where the three distinct zones ($\mathrm{E}$, $\mathrm{U}$ and $\mathrm{Z}$) meet. As such, any continuous deformation of the boundaries between these zones will not affect the existence of such special points. Therefore, we call heads and tails topologically protected. A similar reasoning holds under time evolution: If the excited, unexcited and phase defect regions change their boundaries in a continuous manner, the head point can move in time, but is only lost if it meets another head or a medium boundary. An analogous reasoning justifies to call the tail points to be topologically conserved.

From Fig. 1E, we see that heads can exist with two different chiralities: If one circumscribes the head point counterclockwise and meets the regions $\mathrm{Z}$, $\mathrm{E}$, $\mathrm{U}$ in that order, we call it a ‘positive’ head, denoted $\mathrm{h}^+$. If we find the order $\mathrm{Z}$, $\mathrm{U}$, $\mathrm{E}$, we call it a ’negative’ head, written $\mathrm{h}^-$. These intuitive notions will be formalized below, to grant them the meaning of a ‘charge’.

For now, let us return to the depiction of a linear-core rotor in Fig. 1D. We observe yet another type of special point, namely the end point of the phase defect line. Such point are already known in literature as the loci around which the wave turns suddenly, and hence we also call them ‘pivots’. Using the LAT-based phase definition Eq. 3, it follows that the phase defects are truly curves of discontinuous phase. Hence these lines can have distinct end points, the pivots. The phase defect lines may also branch spatially; in such case a phase defect ‘joint’ will emerge, which however falls outside the scope of our present exposition. Pivots also exist in two flavours, see Fig. 1E: If the wave made a sudden half-turn to the left at that pivot, we call the pivot point a $\mathrm{p}_+$, with the plus sign marking counterclockwise rotation. If the half-turn was to the right, we call it a $\mathrm{p}_-$. We will discuss the topological preservation of pivots below, after the formal introduction of pivot charge.

Two topological charges: rotation in state space vs. rotation in physical space

We will here demonstrate that heads and tails carry half-integer charges that are identical to the classical ‘charge’ of a phase singularity. Pivots carry a charge that is also half-integer, but of a different kind.

First, consider a portion of the contour $\mathcal{C}$ that surrounds a phase defect in a counterclockwise sense, i.e., with the phase defect to the left. See Fig. 2. With arc length parameter $\ell$, we evaluate the classical charge density: $$\begin{aligned} Q = - \frac{1}{2\pi} \int d \varphi = - \frac{1}{2\pi} \int {{\bm{{\nabla}}}} \varphi \cdot {{\bm{{d \ell}}}}. \end{aligned}\qquad{(9)}$$ Here, $\varphi$ can be any phase related to the local state of cells; both Eq. 1 and Eq. 3 are possible, just like other choices (Clayton et al., 2006; Gray et al., 1995; Tomii et al., 2021). The minus sign is included such that a counterclockwise rotating spiral has $Q=+1$. Authors usually evaluate Eq. 9 around a closed contour, to find that $Q$ always takes an integer value if $\varphi$ is well-defined over the entire contour. However, if one takes only a portion of the contour, one can find the charge density $dQ = - \frac{d\varphi}{2\pi}$. Part of the phase change occurs during excitation, and part during the biological recovery process. If one assumes these changes to take place at the wave front and wave back, a phase change of $\pm \pi$ will be found each time the contour encircling the phase defect meets a wave front or wave back. These values correspond to localized charges of $\pm \frac{1}{2}$. Therefore, we here choose to ascribe a topological charge of $\pm \frac{1}{2}$ to heads and tails, with the sign chosen as in Fig. 1E. In principle, this choice can be made rigorous by defining the phase in the region $\mathrm{U}$ to be 0 and the phase over the region $\mathrm{E}$ to be $\pi$. Elegantly, the sum of all head and tail charges when circumscribing a phase defect equals the classical topological charge $Q$. To distinguish it from the pivot charge below, we call this the $Q$-charge. From the above, it follows that the $Q$-charge in a way ‘measures’ the amount of rotation in state space.

Second, we introduce the pivot charge as follows. Define the angle of wave propagation as: $$\begin{aligned} \psi = \operatorname{atan2}( \partial_y t_\mathrm{LAT}, \partial_x t_\mathrm{LAT}). \end{aligned}\qquad{(10)}$$ where the LAT is continuous and differentiable. $\psi$ is the angle with the positive $x$-direction under which the last arrived activation wave travelled. Now, in full analogy with Eq. 9, step along the contour $\mathcal{C}$ while keeping the phase defect to the left: $$\begin{aligned} P = \frac{1}{2\pi} \int d \psi = \int {{\bm{{\nabla}}}}\psi \cdot {{\bm{{d \ell}}}} \end{aligned}\qquad{(11)}$$

Let us verify how the $P$-charge density $d P = \frac{d\psi}{2\pi}$ behaves in standard cases. First, take the neighbourhood of the positive pivot at the top-right panel of Fig. 1E. There, the wave turns over $+\pi$ near the pivot, such that $P=+\frac{1}{2}$. This corresponds to the intuitive notion of ‘half a turn to the left’ that we advocated above. Similarly, a negative pivot $\mathrm{p}_-$ can now be proven to have $P=-\frac{1}{2}$. Third, consider a classical spiral wave core denoted as $\mathrm{c}^+$ in the bottom row of Fig. 1E. Near the classical phase singularity, phase gradients will become arbitrarily large, such that the classical phase singularity is also a special case of a phase defect: It is a phase defect of small size (Arno et al., 2021), whose exact diameter depends on the chosen $\rho_*$. Evaluating $P$ around a closed contour that circumscribes the phase singularity counterclockwise delivers: $P=+1$. Noteworthily, even circular rotor cores form over time from conduction blocks: Initially, a finite conduction block line is created, e.g. by an S1-S2 stimulus protocol, with two pivots each having $P=+\frac{1}{2}$. During successive wave rotations, this phase defect shrinks to a point. At that time, the two pivots with $P=+\frac{1}{2}$ as well as the head and tail with each $Q=+\frac{1}{2}$ merge to form a stable state, with $P=+1$ and $Q=+1$, that we call a ‘core particle’ here.

Let us now investigate when the $P$-charge is conserved. Hereto, we apply Stokes’ theorem to Eq. 11 on a closed contour $\mathcal{C}$ around a region $S$. $$\begin{aligned} P = \frac{1}{2\pi} \oint d \psi =\oint {{\bm{{\nabla}}}}\psi \cdot {{\bm{{d \ell}}}} = \iint {{\bm{{\nabla}}}} \times {{\bm{{\nabla}}}}\psi \cdot {{\bm{{d s}}}} =0 \end{aligned}\qquad{(12)}$$ The last equality follows from the properties of curl and gradient. The second-last equality requires that the conditions for Stokes’ theorem are fulfilled: That ${{\bm{{\nabla}}}}\psi$ is everywhere in $S$ defined, and that it has continuous first derivatives. Result Eq. 12 implies that $P$ will not change its value if the contour is continuously deformed outside regions with phase defects or phase singularities (where $\psi$ is undefined), or where $\psi$ suddenly changes its orientation. The latter case happens at excitation sources, or on the boundary at the position where it was hit by a wave front. Given these limitations, the $P$-charge is conserved. E.g. a closed contour may contain two opposite rotating linear-core rotors, i.e., with $2\mathrm{p}_+$ and $2\mathrm{p}_-$ in it. The total $P=\frac{1}{2}+\frac{1}{2}-\frac{1}{2}-\frac{1}{2}=0$. If these rotors merge to form a single conduction block line, a $\mathrm{p}_+$, $\mathrm{p}_-$ pair annihilates, leaving the second $\mathrm{p}_+, \mathrm{p}_-$ inside the contour, and still $P=\frac{1}{2}-\frac{1}{2}=0$.

In conclusion, the $P$-charge measures the amount of rotation in physical space. It is thus different from the $Q$-charge, carried by heads and tails. Therefore, when using the detailed notation for charges (see Table 1), we denote the $P$-charge in subscript: $\mathrm{p}_\pm$.

Quasiparticles

Until here, we have identified three different special points in excitable media, that can carry half-integer ‘charges’ and that seem to be preserved under time evolution (for now disregarding boundaries and obstacles). We propose to refer to these special points as quasiparticles and call them ‘cardions’. Examples of the seven different cardions that we know now of are given in Fig. 1E. While three cardions are elementary, heads, tails and pivots, four of them arise as bound states between the elementary ones.

Bound states

Strikingly, our analysis of activation patterns in simulation and experiments revealed that the heads, tails and pivots tend to form bound states, see Fig. 1E and Table 1. The situation reminds us of the hydrogen atom consisting of a proton and an electron, or the proton itself, which is made up of quarks in particle physics.

The first bound state consists of a head and a tail of equal $Q$-charge. Already in Fig. 1D, it can be seen that while the tail traces a phase defect, a head can follow it directly, exciting the newly-recovered tissue created by the tail particle. The two special points propagate as a pair along the phase defect, with an excitable gap between them. In the classical viewpoint, this corresponds to a wave front and wave back ending in one point, the classical ‘spiral tip’. In three spatial dimensions, the tips form a filament curve (Clayton et al., 2006; Winfree et al., 1996) around which the wave rotates. Therefore, we annotate the bound state of head and tail with f and call it a filament point and annotate it in the colour purple. The convention of assigning half a $Q$-charge to heads and tails has the useful property that the sum of these charges will be equal to the classical topological charge $Q=\frac{1}{2}+ \frac{1}{2} = 1$ of the filament. As the absolute charge of a filament particle is always the same, we abbreviate the ‘full’ notation f$^{\pm \pm}$ to $\mathrm{f}^{\pm}$, see second column in Table 1. In this simpler notation, we write the charges of all particles with superscript, for example, $\mathrm{p}^+$, rather than $\mathrm{p}_+$.

The second bound state is the core particle, which we introduced above as an example of $P$-charge. This core particle occurs with rigidly rotating spiral waves, where the phase defect is small and head and tails are close to each other. Since these cores typically form from shrinking of a phase defect line containing two pivots of the same charge, the core particle consists of a head, a tail and two pivots (where the latter are no more distinguishable). This ‘core particle’, denoted $\mathrm{c}^{\pm\pm}_{\pm\pm}$ or shorter $\mathrm{c}^{\pm}$ and indicated by an orange circle, see the bottom panel of Fig. 1E.

The next two bound states have no classical counterpart to our knowledge. Consider a propagating wave front that hits a wave back. Such event will generally occur in a single point first, creating a small phase defect. The end point of one side of the wave front will then travel along the existing wave back, and in the process, the conduction block region grows to become a phase defect line. Hence, there exists a state of a head, tail, and pivot of the same chirality that all move together. We call this a ‘growth’ particle and denote it with a teal circle. It has $Q=\pm 1$ and $P=\pm \frac{1}{2}$, with the same sign chosen. Hence, we identify quasiparticles g$^{++}_+$ and $\mathrm{g}^{--}_{-}$, or $\mathrm{g}^+$, $\mathrm{g}^-$ in shortened notation (see Table 1).

Finally, we were ourselves struck by the observation that the shrinking and growing of phase defect lines occurs in an asymmetric manner. When a wave back travels along a phase defect such that the tissue has recovered at both sides of the defect line, the phase defect line will retract, see Fig. 1E, bottom right. Such retraction implies that a tail and a pivot of same chirality travel together. We denote this as a ‘shrink’ particle s$^\pm_\pm$, shortened s$^\pm$, and mark it in brown. Table 1 summarises the hitherto discovered quasiparticles and their bound states in excitable media.

Medium boundaries and conservation laws

Until here, we have disregarded inexcitable obstacles in the medium, which also includes the medium boundary itself.

Let us first consider how the $Q$-charge behaves in the presence of obstacles. In the example of Fig. 1D, the total $Q$-charge in the medium disregarding the medium boundaries would be $-\frac{1}{2}-\frac{1}{2} = -1$. However, we now argue that it might be useful to also regard medium boundaries as a kind of phase defect. If we do this, e.g. by colouring the medium boundaries also in grey, it follows that every wave front is either closed or has two endpoints (heads) with different chirality. A similar reasoning holds for the wave backs, which are either closed, or end in two tails of opposite charge. Then, by construction the total $Q$-charge for both all heads in $H$, the set of heads, and tails in $T$, the set of tails, must be zero: $$\begin{aligned} \sum_{h \in H} Q_h &= 0 \\ \sum_{t \in T} Q_t &= 0 \end{aligned}\qquad{(13)}$$ These relations only hold if medium boundaries are considered as phase defects.

Taking the sum of the previous two statements proves that the total $Q$-charge in a two-dimensional medium is also always zero: $$\begin{aligned} \sum_{j \in H \cup T} Q_j &= 0 \end{aligned}\qquad{(14)}$$ considering medium boundaries as phase defects. This result is stronger than the conservation of phase singularities, which only holds in the bulk of the medium.

In the viewpoint that includes medium boundaries and inexcitable obstacles, the classical annihilation of a rotor core at the medium boundary can be regarded as the head-tail pair becoming a pair of a head and tail which are travelling apart from each other on the boundary. Also, the generation of a single rotor via S1-S2 stimulation now includes the boundary: The S2 pulse introduces a pair of heads, one of which joins a tail particle of the S1 pulse to form a filament quasiparticle.

An additional reason for including the medium boundary in the description lies in the interpretation of electrical signals. In leading order, the unipolar electrical potential measured by an electrode is determined by the angle subtended by the wave fronts and wave backs, seen from the electrode (Holland & Arnsdorf, 1977). The measured signal is thus only determined by the boundaries of the wave fronts and backs, namely the heads and tail curves in three dimensions (Arno et al., 2024b). Therefore, the concepts of heads and tails may also prove useful when improving efficient inverse methods to recover spatiotemporal activation sequences from patient measurements (Ramanathan et al., 2004).

Second, we consider how the pivot charge behaves in the presence of obstacles. If a wave front travels clockwise along the edge of a square domain, it will turn 90 ° to the right when crossing the corner. Thus, according to Eq. 11, the corners of a square domain carry a pivot charge of $\pm\frac{1}{4}$. Since the edge of obstacles can turn over any angle, pivot charges are not always half-integer, but can take on the boundary of the medium arbitrary values in $(-1,1)$. Another example is a wave hitting an obstacle within the medium, as shown in Fig. 3. Suppose the wave encircling the obstacle counterclockwise travels over an angle $\psi_\mathrm{l} >0$, while the wave travelling in the other direction around the obstacle over an angle $\psi_\mathrm{r} < 0$, with $|\psi_\mathrm{l}| + |\psi_\mathrm{r}| = 2\pi$ . Then, the $P$-charge of this obstacle will become via Eq. 11: $P = \frac{\psi_\mathrm{l} + \psi_\mathrm{r}}{2\pi} \in [-1, 1]$. In this situation, the $P$-charge is not conserved. Nonetheless, the value of the integral Eq. 11 contains useful information: If it has $P = \pm 1$, a rotor is attached to the obstacle, while $P = 0$ indicates a symmetric conduction block. Intermediate values of $P$ indicate whether the wave-obstacle interaction is close to the formation of a rotor attached to the obstacle or not.

Figure 3: Non-integer $P$-charges are possible for paths around inexcitable obstacles.

In summary, $P$-charge is not conserved in the presence of medium boundaries. This was also the case for the $Q$-charge with classical phase singularities, as they can simply vanish by collision with the medium boundary.

Graphical depiction of processes via Feynman-like diagrams

In the last step of our conceptual framework, we describe the evolution of cardions via diagrams similar to Feynman diagrams (Feynman, 1949). A first example is given in Fig. 1F. A $\mathrm{g}^+$, $\mathrm{g}^-$ pair is born, after which the growth particles each split into triples of head, tail, and pivot. In this graphical depiction, time passes from left to right and positions in space are represented along a single, vertical axis. The arrows represent the continuous existence of the quasiparticle in space and time, until an event happens in which the particle merges, splits or annihilates. The premise of this work is that these diagrams can offer insight in the pattern’s dynamics, see the examples below.

Results

Figure of eight formation in-silico

Fig. 4 shows a numerical simulation in the smoothed Karma model (Karma, 1993; Marcotte & Grigoriev, 2017) for atrial fibrillation, in which a single rotor spontaneously breaks up into multiple vortices. We analyze the creation of a vortex pair, known in cardiology literature as a figure-of-eight re-entry (Wu et al., 1994). In the top two rows, phase singularities detected by the Kuklik method (Kuklik et al., 2016) are annotated on top of the normalised transmembrane voltage and phase. The vortex pair with labels 2 and 3 is situated away from a central persisting rotor, labelled 1. In the third row of panels, heads, tails and pivots, as well as their bound states are annotated. When the wave front hits the wave back, a pair of $\mathrm{g}^+$, $\mathrm{g}^-$ emerges. Each particle lasts until the vortex makes a pivoting turn around the formed conduction block line, which marks the decay of the $\mathrm{g}$-particles into triples of head, tail, and pivot. Just after $t={465~\mathrm{{m}{s}}}$, the tail pair annihilates, as the wave back detaches from the conduction block line to form a single wave back again. The recovery of the tissue at the location of the initial collision marks the splitting of the phase defect line, which involves the birth of a shrink pair, just before $t={510~\mathrm{{m}{s}}}$. Thereafter, both phase defect lines shrink until the head, tail, and two pivots are so close to each other that they effectively form a core particle. We conclude that the birth of a rotor pair, which is a single event in the classical theory, in fact consists of seven subsequent events, revealed by the vertices in the diagram at the bottom of Fig. 4.

Figure 4: Diagrammatic analysis of the creation of two spiral waves in a simulation of break-up (Karma, 1993, 1994). First row: Snapshots of the normalized transmembrane potential with the phase singularities highlighted which were detected by the Kuklik method (Kuklik et al., 2016), and in the second row, corresponding phase maps (Gray et al., 1998; Kabus et al., 2022). Third row: Identification of the quasiparticles in subsequent snapshots. The resulting diagram at the bottom shows that the birth of a rotor pair involves seven topological interactions, including the creation of a shrink pair, and the splitting of a conduction block line. No heads, tails, or pivots are drawn at the boundary since it is an inset, not the medium boundary. The vortex with label 1 is excluded from the diagram of topological interactions. The same color scheme as in Fig. 1 is used consistently in this article.

Figure of eight creation in-vitro under burst pacing

The creation of a figure-of-eight spiral pair in a monolayer of human atrial myocytes (Harlaar et al., 2022) is thoroughly analyzed via the quasiparticles and diagrams in Figs. 5-9, see also the Methods section below. By adding a voltage-sensitive dye to the cell culture, the local transmembrane voltage could be optically mapped (Salama et al., 1987), which is shown here after normalization to $[0,1]$. The culture was stimulated at a high frequency from the top left corner. After several transient wave breaks, i.e., conduction blocks, a pair of oppositely rotating spiral waves was formed. The experiment presents an in-vitro realization of the onset of an arrhythmia, an event to be prevented in patients.

Figure 5: Feynman-like diagram of the creation of two spiral waves via two merging conduction block lines (CBLs) in an optical voltage mapping experiment. The detailed quasiparticle viewpoint shows that both events are part of a single complex multi-stage process of up to twelve quasiparticles at the same time interacting in 93 reactions. The stages A, B, D, & E are shown in more detail in Figs. 5-9, respectively. The formation of the second CBL in panel C is analogous to the first one in panel A and Fig. 6.

Fig. 5 presents a diagrammatic overview of the figure-of-eight spiral pair formation, while Figs. 6-9 present parts of this process in more detail in the same style as Fig. 4.

As can be seen in Fig. 6, the initial conduction block produces a growth pair just before $t={4056~\mathrm{{m}{s}}}$, in the same way as in the in-silico case above, which then break apart into a head, tail and pivot each. The tails and heads annihilate and leave the persisting pair of pivots behind, marking the end points of the conduction block line. Later on, a secondary conduction block forms in the same way, also producing a pivot-pair, Fig. 5C.

Figure 6: Creation of a persisting pair of pivot particles due to conduction block as the first step of figure-of-eight spiral formation in the optical mapping experiment, Fig. 5A. At first, when the wave front runs into the wave back, a pair of growth particles is formed, which then decay into a pivot, head, and tail each. The heads and tails annihilate, while the pivot particles persist.

The process causing the the mere conduction block to develop into a dangerous figure-of-eight reentry, which can be life-threatening in peoples’ hearts, can be seen in Fig. 7 After the wave back hits the phase defect region of the conduction block, a tail pair is produced. Each tail then meets the persisting pivot particles to produce shrink particles that begin to shorten the conduction block line at $t={4164~\mathrm{{m}{s}}}$. However, shortly afterwards at $t={4170~\mathrm{{m}{s}}}$, the wave front also hits the phase defect, leading to a pair of heads, that combine with the shrinks to growth particles. Driven by this pair, the conduction block line grows again, until the growth particles decay into a head, tail and pivot each. Just after $t={4194~\mathrm{{m}{s}}}$, the tails annihilate, as well as the heads just after $t={4212~\mathrm{{m}{s}}}$, while again two pivot particles persist. As the growing outweighs the shrinking in this case, in total, the phase defect length accumulated. The distance between the pivots increased from ${(2.85\ ±\ 0.50)~\mathrm{{m}{m}}}$ at $t={4146~\mathrm{{m}{s}}}$ to ${(4.65\ ±\ 0.50)~\mathrm{{m}{m}}}$ at $t={4212~\mathrm{{m}{s}}}$. This process repeats six times, each time making the phase defect longer.

Figure 7: Accumulation of conduction block is the process of shrinking due to a wave back hitting a phase defect line and growing due to a wave front, which, in total, makes the conduction block grow. This process is observed another five times in the data during the following burst pacing pulses leading up to figure-of-eight spiral formation in the optical mapping experiment, Fig. 5B.

At $t={4854~\mathrm{{m}{s}}}$, also the second phase defect has formed, and both will merge, as visualized in Fig. 8. The merging process starts when both defects grow with a pair of $\mathrm{g}^+$ and $\mathrm{g}^-$ each. Then, the growth particles decay in triplets of head, tail and pivot and the tail-pairs of both phase defect sites annihilate just as in the accumulation stage Fig. 7). Meanwhile, the $\mathrm{h}^-$ of the large phase defect and the $\mathrm{h}^+$ of the smaller phase defect annihilate with each other just after $t={4866~\mathrm{{m}{s}}}$, as well as their respective $\mathrm{p}^-$ and $\mathrm{p}^+$, merging the two phase defects into one larger U-shaped phase defect line. At $t={4890~\mathrm{{m}{s}}}$, a pair of pivots and heads each with opposing charges remain. The pivots now have a distance of ${(6.72\ ±\ 0.50)~\mathrm{{m}{m}}}$.

Figure 8: The merger of two conduction blocks takes place when the two phase defects grow close enough such that their particles annihilate with each other, creating a larger phase defect line. This is another intermediate step of figure-of-eight spiral formation in the optical mapping experiment, Fig. 5D.

Finally, in Fig. 9, it can be seen how the recovery of the initial zone of contact creates a shrink pair, which causes the splitting of the conduction block line just before $t={4902~\mathrm{{m}{s}}}$. Shortly after that, each shrink particle decays into a pivot and tail. At $t={4932~\mathrm{{m}{s}}}$, the system has evolved to a state with two nearly parallel conduction block lines, i.e., phase defects that comprise an effective functional isthmus around which a figure-of-eight rotor pair can revolve. Each of the phase defects have a head, a tail, and two pivot particles, all with the same charge. The phase defect with a clockwise rotating rotor has $Q=-1$ and $P=-1$, while the counter-clockwise rotor has $Q=+1$ and $P=+1$. From $t={4956~\mathrm{{m}{s}}}$, the pivots remain in place and the head and tail propagate along the phase defect line. Zooming out, these particles behave like a pair of cores.

Figure 9: Splitting of a large conduction block as the final step of figure-of-eight spiral formation in the optical mapping experiment, Fig. 5E. When a wave back meets the phase defect, a pair of shrink particles is formed, halving the U-shaped phase defect line with $Q=0$ and $P=0$. When the shrink particles decay, two rotor cores with $Q=\pm 1$ and $P=\pm 1$ are formed.

In brief, the quasiparticle framework here reveals that the establishment of the figure-of-eight re-entry is a multi-step process that takes over ${800~\mathrm{{m}{s}}}$ to take place. The process features tight interplay between dynamically changing conduction blocks and wave front ends (heads). The formalism unifies the concepts of rotors and conduction block lines, and reveals how an initially flat conduction block line first takes a U-shape ($t={4890~\mathrm{{m}{s}}}$) and then becomes a functional isthmus. The details of this interaction are not visible in the classical phase singularity analysis, which would only reveal the creation, vanishing, and recreation of a phase singularity pair.

Discussion

An extended quasiparticle viewpoint on excitation

The above analysis and examples show that complex excitation patterns feature certain preserved special points. Noteworthily, these are critical points in space, rather than critical points of evolution equations, which also satisfy topological rules (Glass, 1975). While tips, pivots and cores have been observed before, the growth and shrink quasiparticles emerge from the framework itself. This situation is to a certain degree reminiscent of the ‘Standard Model’ in physics, which is based on a small set of fundamental particles (Mann, 2010) and explains almost all experimental results in particle physics. The theoretical framework presented in this work gains its power from simple geometrical arguments, as the heads and tails are points where three zones meet: excited, unexcited, and phase defect. The concept of ‘head’ itself has been implicitly used before, e.g. in Krinsky’s quantitative theory of reverbators (Krinsky, 1968; Krinsky et al., 1992) for inhomogeneous domains. Our framework is much wider valid, as it can describe long and short-lived rotors, wavelets and conduction blocks in homogeneous and inhomogeneous excitable media.

Cardions

In physics, it is customary to give localised and preserved structures a name with the suffix ‘-on’, with the first part of the name referring to the nature of the system of the (quasi-)particle (Merriam-Webster, n.d.). Examples include not only most particles in the Standard Model (proton, electron, baryon, fermion, photon) but also other moving structures such as phonons, solitons and excitons. In analogy, we here propose to name the here-identified quasiparticles as ‘cardions’, as they were first observed in cardiac excitation patterns.

Dynamical transitions

The motivation for this study was to investigate the interactions between short-lived rotors and approaching wave fronts during arrhythmogenesis (Clayton et al., 2006). We go beyond the observation that apparent phase singularities co-locate with conduction blocks (Arno et al., 2021; Rodrigo et al., 2017), by looking at a finer spatial scale. We discovered multiple interactions of the cardions in the data and represented them schematically. Our initial analyses show new mechanistic insight, like the dynamical isthmus formation in Fig. 9. It can be expected that the automated construction of those diagrams for larger fibrillation domains (see Fig. 5) will allow an enhanced classification and characterisation of those patterns.

Links between particle physics and cardiology

The quasiparticle viewpoint allows us to introduce more concepts from particle physics in the world of excitable media. First, note that the terms ‘creation’ and ‘annihilation’ also applied to the classical viewpoint (Clayton et al., 2006; Gray et al., 1998). There, a single quasiparticle was present, i.e., the spiral wave centre (phase singularity), bearing a positive or negative charge ($Q=\pm 1$) depending on its rotation sense. In our viewpoint, creation and annihilation also occurs with heads, tails and pivots. Since cardiac tissue is, in reality, non-homogeneous, a wave front will often create a head pair near obstacles, e.g. areas of fibrosis, which annihilates shortly afterwards. Second, the difference between linear-core and circular-core rotors follows from the mutual interaction between pivots, which can be expressed with spiral wave response functions (Biktasheva & Biktashev, 2003; Dierckx et al., 2017; Marcotte & Grigoriev, 2016). If nearby pivot charges of the same charge attract each other, they form a core particle. Conversely, when they repel each other, a linear-core rotor emerges. However, such a phase defect cannot grow larger than the width of a travelling pulse in the medium, since otherwise, the defect line will break by the creation of a new $\mathrm{p}_+$, $\mathrm{p}_-$ pair. Therefore, pivot particles are in our opinion subjected to confinement, analogous to quarks in quantum chromodynamics (K. G. Wilson, 1974). This confinement may partially explain why the here-identified quasiparticles remained unnoticed during simple dynamical regimes, i.e., at time intervals where no new vortices or conduction blocks form or disappear.

Finally, we note that in Fig. 1B, there are two apparent symmetries. First, if one exchanges excited with unexcited regions, wave fronts become wave backs, heads become tails and vice versa. This corresponds to flipping the voltage axis around the value $V_*$ from Eq. 1. Second, one can reverse the arrow of time, which preserves $\mathrm{U}$ and $\mathrm{E}$ regions, but interchanges wave backs with fronts and heads with tails. However, when looking at the dynamics, there is a clear distinction between heads and tails. As heads lie on a wave front, they move at the conduction velocity in the medium. Since the repolarisation of the tissue occurs at a fixed delay after the depolarisation, the wave back is a phase wave, and wave backs and tails can move at any speed. Also, the medium can remain in the unexcited state for a unlimited time but only a limited time in the excited state. This effective breaking of symmetry also manifests in the growth and decay states. In the cases we have yet observed in simulations and experiment, the growth of a phase defect line each time involved the three different quasiparticles. Namely, a conduction block arises when a wave front (h) hits a wave back (t) and in the process, a phase defect is formed (p). Conversely, the vanishing of a phase defect only requires that the phases at both sides return to full recovery. We took the threshold of the wave back to mark the state where a standard impeding wave front can re-excite the tissue. Then, only a tail and pivot particle are needed. In case of a heterogeneous medium, we anticipate that also other situations can occur, but these fall outside the scope of our present investigation. Noteworthily, a statistical analysis of occurring processes in the topological approach of Marcotte & Grigoriev (2017) also demonstrated asymmetry between favoured processes in the creation and annihilation processes of spiral cores.

Limitations

The parallels drawn here between biological excitation and theoretical physics are only at the conceptual level. Feynman diagrams in physics are representations of a path integral, allowing to structure tedious calculations. To use the diagrams in excitable context for quantitative calculations and predictions could be a next research step. Even then, the mathematical implications will be different from the physics case. For this reason, we refer to the diagrams here consistently as ‘Feynman-like’. Likewise, the terms of confinement, quasiparticle interaction, standard model and symmetry breaking are powerful analogues when used in complex systems, but will at a more detailed level have different meaning than their original physics use.

The local state of a cell was here assumed to lie along a cycle, such that it can be labelled with a single phase variable $\varphi({{\bm{{r}}}},t)$. Capturing specific memory effects, such as alternans (Karma, 1994) or after-depolarisations in cardiac tissue (Weiss et al., 2010) will require an extension of the formalism, e.g. using phase-amplitude equations (D. Wilson, 2020).

We are also not certain if all possible cardions states have been described here. We already noted that phase defects can have side branches, and the ‘joints’ where this happens will also be cardions, to be described elsewhere. In Fig. 8, co-localisation of a $\mathrm{h}$ and a $\mathrm{p}$ particle is seen in the frames. However, in our experience this state is transient and therefore not considered a bound state here. Nonetheless, it is possible that other regimes exist where $\mathrm{h}$-$\mathrm{p}$ pairs may travel together as a bound state.

The concept of $P$-charge was introduced in this work, and our understanding of it is currently incomplete. The $P$-charge is currently undefined before the first wave crosses the medium. The precise conditions under which it is conserved or not need to be further elucidated. For isolated, thin, conduction block lines, $P$-charge seems to be conserved: Basic features in the patterns observed are linear rotor cores ($P=\pm 1$) or mere conduction blocks ($P=0$).

Finally, we have applied the framework here already on in-silico and in-vitro datasets. An obvious next step would be to also analyse surface patterns in cardiac tissues, and to post-process phase maps or LAT maps obtained in patients. We encourage the scientific community to work together on these steps and are providing basic numerical methods for this on GitLab, see below. We are currently working on automated algorithms to detect cardions in datasets.

Applications to cardiac arrhythmia

The above examples revealed dynamics at a finer level than in the phase singularity picture. Conducting similar analyses on existing datasets during arrhythmias is likely to give new clues on underlying mechanisms. More specifically, counting the elementary interactions (Marcotte & Grigoriev, 2017), computing statistics of potentially large diagrams, and deriving interaction laws between the quasiparticles could prove useful. The value of the $P$-charge for obstacles in the medium could also serve as a ‘distance’ to the rotor regime, and thus quantify how far a regime is from arrhythmia, complementing other risk scores in literature (Arevalo et al., 2016).

The pivot sites, here identified as quasiparticles, are actively being investigated by clinicians as possible ablation targets (Seitz et al., 2017). We believe that the analysis of the clinical data and the pivot ablation within our framework could help to answer if, why, and when these points are appropriate ablation targets.

Outlook

In this manuscript, we have exposed several new concepts, such as the pivot charge, different quasiparticles and a diagrammatic approach to better understand their interactions. Continuing research by ourselves and colleagues will be needed to elaborate these concepts.

Within the context of continuous excitable media, several extensions are possible. Since we do not suppose an underlying evolution model, the analysis could also be performed to the surface of three-dimensional media. A next step is to analyse optical voltage mapping surface recordings of the heart muscle. Such analysis can be applied to any excitable surface pattern that is sufficiently sampled in space and time. Our original motivation is to perform predictive calculations in those patterns: Can one find the critical size of a wave break, or the timescale at which a rotor produces multiple wavelets? In three spatial dimensions, our preliminary findings (Arno et al., 2024b) indicate that the cardions become string-like objects (Verschelde et al., 2007). As such, cardions in three dimensions will refine the concept of a rotor filament (Clayton et al., 2006; Winfree, 1973), and may be used in the future to further investigate the three-dimensional organisation of turbulence in complex systems.

Conclusion

We presented a conceptual framework revising the classical theory of excitable media, intentionally designed for complex regimes: short-lived pivoting motion and prominent conduction blocks. By classifying the endpoints of wave fronts, wave backs, and conduction blocks as charged quasiparticles, Feynman-like diagrams can be created from phase maps or local activation times. We believe this framework has the potential to become a useful analysis tool in excitable media, with applications within and beyond the cardiac electrophysiology context.

Methods

In silico data generation

The synthetic data from Fig. 1A were obtained by Euler-forward stepping of the Bueno-Orovio-Cherry-Fenton model (Bueno-Orovio et al., 2008), with finite-differences grid size ${0.5~\mathrm{{m}{m}}}$ and time step ${0.1~\mathrm{{m}{s}}}$ in a biventricular heart geometry using an S1S2 stimulation protocol in the Ithildin framework (Kabus et al., 2024). The data from Fig. 1B are obtained in the same way but for a square piece of tissue with grid size ${0.3~\mathrm{{m}{m}}}$. For the fibrillation-like data from Fig. 4, the Karma model (Karma, 1993, 1994) modified according to Marcotte & Grigoriev (2017) was used, with step size ${1~\mathrm{{m}{m}}}$ and time step ${0.1~\mathrm{{m}{s}}}$.

In vitro data generation

Monolayers of fully functional human atrial myocytes were generated as described in detail by Harlaar et al. (2022). A voltage-sensitive dye was added to the culture, after which a real-time recording can be made of the intensity of emitted light, which is a measure of the local transmembrane potential (Salama et al., 1987). The recording has a resolution of $100 \times 100$ pixels with pixel size ${0.25~\mathrm{{m}{m}}}$. The sampling time between frames was ${6~\mathrm{{m}{s}}}$.

Data analysis

Local activation times were converted to phase following Eq. 3 with time constant $\tau_0 = {110~\mathrm{{m}{s}}}$ for the in vitro data and ${70~\mathrm{{m}{s}}}$ for the in-silico data. We use the phase values $\varphi_\mathrm{F} = {0~\mathrm{rad}}$ and $\varphi_\mathrm{B} = {5~\mathrm{rad}}$. Phase defects are identified following Eq. 5 with the threshold $\rho_* = 0.2$. The positions of tips, heads, tails, pivots and compound particles were drawn manually on the resulting figures.

Data availability

The simulation output and pre-processed optical voltage mapping data we have used for this article can be found on Zenodo (DOI: 10.5281/zenodo.13379783). This repository also contains the scripts and Python modules to generate the figures found in this article. The source code for the Ithildin finite-differences solver (Kabus et al., 2024) (https://gitlab.com/heartkor/ithildin), its Python module (Kabus et al., 2022) (https://gitlab.com/heartkor/py_ithildin), and a Python module to create the Feynman-like diagrams (https://gitlab.com/heartkor/pdl-feynman) can also be found on GitLab.

Addenda

Acknowledgments:

The authors are grateful to D.A. Pijnappels, A.A.F. de Vries and N. Harlaar for collecting and sharing the hiAM dataset. The authors thank J. Ector and T. De Coster for helpful discussions. The authors thank A.A.F. de Vries for useful comments on the manuscript. The authors thank A. Gobeyn for coining the term ‘cardion’ for the collection of quasiparticles.

Funding:

L. Arno is funded by a FWO-Flanders fellowship, grant 117702N. D. Kabus is supported by KU Leuven grant GPUL/20/012. H. Dierckx was supported by KU Leuven grant STG/019/007.

Competing interests:

The authors declare no competing interests.

Copyright:

Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Author contributions:

LA: Conceptualisation, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Writing – review & editing, Visualisation. DK: Conceptualisation, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualisation. HD: Conceptualisation, Methodology, Resources, Writing – original draft, Writing – review & editing, Supervision, Project administration, Funding acquisition.

References

Allessie, M. A., Bonke, F. I. M., & Schopman, F. J. G. (1973). Circus movement in rabbit atrial muscle as a mechanism of tachycardia. Circulation Research, 33(1), 54–62. https://doi.org/10.1161/01.RES.33.1.54

Aras, K. K., Kay, M. W., & Efimov, I. R. (2017). Ventricular fibrillation: Rotors or foci? both! Circulation: Arrhythmia and Electrophysiology, 10(12), e006011. https://doi.org/10.1161/circep.117.006011

Arevalo, H. J., Vadakkumpadan, F., Guallar, E., Jebb, A., Malamas, P., Wu, K. C., & Trayanova, N. A. (2016). Arrhythmia risk stratification of patients after myocardial infarction using personalized heart models. Nature Communications, 7(1), 11437. https://doi.org/10.1038/ncomms11437

Arno, L., Kabus, D., & Dierckx, H. (2024a). Analysis of complex excitation patterns using Feynman-like diagrams. Scientific Reports, 14(1), 28962. https://doi.org/10.1038/s41598-024-73544-z

Biktashev, V. N., Holden, A. V., & Zhang, H. (1994). Tension of organizing filaments of scroll waves. Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, 347(1685), 611–630. https://doi.org/10.1098/rsta.1994.0070

Biktasheva, I. V., & Biktashev, V. N. (2003). Wave-particle dualism of spiral waves dynamics. Physical Review E, 67(2), 026221. https://doi.org/10.1103/PhysRevE.67.026221

Bray, M.-A., & Wikswo, J. P. (2003). Examination of optical depth effects on fluorescence imaging of cardiac propagation. Biophys. J., 85, 4134–4145. https://doi.org/10.1016/S0006-3495(03)74825-5

Christoph, J., Chebbok, M., Richter, C., Schroder-Schetelig, J., Bittihn, P., Stein, S., Uzelac, I., Fenton, F. H., Hasenfuss, G., Gilmour, R. F., & Luther, S. (2018). Electromechanical vortex filaments during cardiac fibrillation. Nature, 555, 667–672. https://doi.org/10.1038/nature26001

Cronin, E. M., Bogun, F. M., Maury, P., Peichl, P., Chen, M., Namboodiri, N., Aguinaga, L., Leite, L. R., Al-Khatib, S. M., Anter, E., Berruezo, A., Callans, D. J., Chung, M. K., Cuculich, P., d’Avila, A., Deal, B. J., Della Bella, P., Deneke, T., Dickfeld, T.-M., Hadid, C., Haqqani, H. M., Kay, G. N., Latchamsetty, R., Marchlinski, F., Miller, J. M., Nogami, A., Patel, A. R., Pathak, R. K., Sáenz Morales, L. C., Santangeli, P., Sapp, J. L., Sarkozy, A., Soejima, K., Stevenson, W. G., Tedrow, U. B., Tzou, W. S., Varma, N., Zeppenfeld, K., ESC Scientific Document Group, Asirvatham, S. J., Sternick, E. B., Chyou, J., Ernst, S., Fenelon, G., Gerstenfeld, E. P., Hindricks, G., Inoue, K., Kim, J. J., Krishnan, K., Kuck, K.-H., Avalos, M. O., Paul, T., Scanavacca, M. I., Tung, R., Voss, J., Yamada, T., & Yamane, T. (2019). 2019 HRS/EHRA/APHRS/LAHRS expert consensus statement on catheter ablation of ventricular arrhythmias. EP Europace, 21(8), 1143–1144. https://doi.org/10.1093/europace/euz132

Dierckx, H., Verschelde, H., & Panfilov, A. V. (2017). Measurement and structure of spiral wave response functions. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(9). https://doi.org/10.1063/1.4999606

Fenton, F. H., & Cherry, E. M. (2008). Models of cardiac cell. Scholarpedia, 3(8), 1868. https://doi.org/10.4249/scholarpedia.1868

Feynman, R. P. (1949). The theory of positrons. Physical Review, 76, 749–759. https://doi.org/10.1103/PhysRev.76.749

Glass, L. (1975). A topological theorem for nonlinear dynamics in chemical and ecological networks. Proc. Natl. Acad. Sci. U.S.A., 72(8), 2856–2857. https://doi.org/10.1073/pnas.72.8.2856

Gray, R. A., Jalife, J., Panfilov, A. V., Baxter, W. T., Cabo, C., Davidenko, J. M., & Pertsov, A. M. (1995). Mechanisms of cardiac fibrillation. Science, 270(5239), 1222–1223. https://doi.org/10.1126/science.270.5239.1222

Gray, R. A., Pertsov, A. M., & Jalife, J. (1998). Spatial and temporal organization during cardiac fibrillation. Nature, 392(6671), 75–78. https://doi.org/10.1038/32164

Henry, H., & Hakim, V. (2000). Linear stability of scroll waves. Physical Review Letters, 85(25), 5328. https://doi.org/10.1103/PhysRevLett.85.5328

Holland, R. P., & Arnsdorf, M. F. (1977). Solid angle theory and the electrocardiogram: Physiologic and quantitative interpretations. Progress in Cardiovascular Diseases, 19(6), 431–457. https://doi.org/10.1016/0033-0620(77)90009-3

Jalife, J., Gray, R. A., Moeley, G. E., & Davidenko, J. M. (1998). Self-organization and the dynamical nature of ventricular fibrillation. Chaos, 8, 79–93. https://doi.org/10.1063/1.166289

Janse, M. J., van Capelle, F. J. L., Morsink, H., Kleber, A. G., Wilms-Schopman, F. J. G., Cardinal, R., Naumann D’Alnoncourt, C., & Durrer, D. (1980). Flow of injury current and patterns of excitation during early ventricular arrhythmias in acute regional myocardial ischemia in isolated porcine and canine hearts. Evidence for 2 different arrhythmogenic mechanisms. Circulation Research, 47, 151–165. https://doi.org/10.1161/01.RES.47.2.151

Kapral, R., & Showalter, R. (1995). Chemical waves and patterns. Kluwer.

Karma, A. (1993). Spiral breakup in model equations of action potential propagation in cardiac tissue. Physical Review Letters, 71(7), 1103. https://doi.org/10.1103/PhysRevLett.71.1103

Karma, A. (1994). Electrical alternans and spiral wave breakup in cardiac tissue. Chaos: An Interdisciplinary Journal of Nonlinear Science, 4(3), 461–472. https://doi.org/10.1063/1.166024

Keener, J. P. (1988). The dynamics of three-dimensional scroll waves in excitable media. Physica D: Nonlinear Phenomena, 31(2), 269–276. https://doi.org/10.1016/0167-2789(88)90080-2

Krinsky, V. I. (1966). Spread of excitation in an inhomogeneous medium (state similar to cardiac fibrillation). Biophysics, 11, 776–784.

Krinsky, V. I. (1968). Fibrillation in excitable media. Problemy Kibernetiki, 20, 59–73.

Krinsky, V. I., Efimov, I. R., & Jalife, J. (1992). Vortices with linear cores in excitable media. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 437(1901), 645–655. https://doi.org/10.1098/rspa.1992.0084

Kuklik, P., Zeemering, S., van Hunnik, A., Maesen, B., Pison, L., Lau, D. H., Maessen, J., Podziemski, P., Meyer, C., Schäffer, B., et al. (2016). Identification of rotors during human atrial fibrillation using contact mapping and phase singularity detection: Technical considerations. IEEE Transactions on Biomedical Engineering, 64(2), 310–318. https://doi.org/10.1109/tbme.2016.2554660

Lee, S., Khrestian, C. M., Sahadevan, J., & Waldo, A. L. (2020). Reconsidering the multiple wavelet hypothesis of atrial fibrillation. Heart Rhythm, 17(11), 1976–1983. https://doi.org/10.1016/j.hrthm.2020.06.017

Mann, R. (2010). An introduction to particle physics and the standard model. CRC Press. https://doi.org/10.1201/9781420083002

Marcotte, C. D., & Grigoriev, R. O. (2016). Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(9), 093107. https://doi.org/10.1063/1.4962644

Merriam-Webster. (n.d.). -on – noun suffix (2). In Merriam-Webster.com dictionary. Retrieved April 26, 2024, from https://www.merriam-webster.com/dictionary/-on#dictionary-entry-6

Moe, G. K., & Abildskov, J. A. (1959). Atrial fibrillation as a self-sustaining arrhythmia independent of focal discharge. American Heart Journal, 58(1), 59–70. https://doi.org/10.1016/0002-8703(59)90274-1

Mujović, N., Marinković, M., Lenarczyk, R., Tilz, R., & Potpara, T. S. (2017). Catheter ablation of atrial fibrillation: An overview for clinicians. Advances in Therapy, 34, 1897–1917. https://doi.org/10.1007/s12325-017-0590-z

Narayan, S. M., Krummen, D. E., Shivkumar, K., Clopton, P., Rappel, W.-J., & Miller, J. M. (2012). Treatment of atrial fibrillation by the ablation of localized sources. Journal of the American College of Cardiology, 60(7), 628–636. https://doi.org/10.1016/j.jacc.2012.05.022

Panfilov, A. V., & Dierckx, H. (2017). Theory of cardiac arrhythmias. In Cardiac electrophysiology. From cell to bedside, 7rd edition (pp. 325–335). Elsevier.

Ramanathan, C., Ghanem, R. N., Jia, P., Ryu, K., & Rudy, Y. (2004). Noninvasive electrocardiographic imaging for cardiac electrophysiology and arrhythmia. Nature Medicine, 10(4), 422–428. https://doi.org/10.1038/nm1011

Rodrigo, M., Climent, A. M., Liberos, A., Fernandez-Avilas, F., Berenfeld, O., Atienza, F., & Guillem, M. S. (2017). Technical considerations on phase mapping for identification of atrial reentrant activity in direct- and inverse-computed electrograms. Circulation: Arrhythmia and Electrophysiology, 10(9), e005008. https://doi.org/10.1161/circep.117.005008

Salama, G., Lombardi, R., & Elson, J. (1987). Maps of optical action potentials and NADH fluorescence in intact working hearts. American Journal of Physiology-Heart and Circulatory Physiology, 252(2), H384–H394. https://doi.org/10.1152/ajpheart.1987.252.2.H384

Samol, A., Hahne, K., & Monnig, G. (2016). Atrial fibrillation and silent stroke: Links, risks, and challenges. Vascular Health and Risk Management, 2016, 65. https://doi.org/10.2147/vhrm.s81807

Seitz, J., Bars, C., Théodore, G., Beurtheret, S., Lellouche, N., Bremondy, M., Ferracci, A., Faure, J., Penaranda, G., Yamazaki, M., et al. (2017). Wholly patient-tailored ablation of atrial fibrillation guided by spatio-temporal dispersion of electrograms in the absence of pulmonary veins isolation. Journal of the American College of Cardiology, 69(3), 303. https://doi.org/10.1016/j.jacc.2016.10.065

Shibata, N., Inada, S., Nakazawa, K., Ashihara, T., Tomii, N., Yamazaki, M., Honjo, H., Seno, H., & Sakuma, I. (2022). Mechanism of ventricular fibrillation: Current status and problems. Advanced Biomedical Engineering, 11, 117–135. https://doi.org/10.14326/abe.11.117

Smith, D. L., Lucey, B., Waller, L. A., Childs, J. E., & Real, L. A. (2002). Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes. Proceedings of the National Academy of Sciences, 99(6), 3668–3672. https://doi.org/10.1073/pnas.042400799

Trayanova, N. A., Popescu, D. M., & Shade, J. K. (2021). Machine learning in arrhythmia and electrophysiology. Circulation Research, 128(4), 544–566. https://doi.org/10.1161/circresaha.120.317872

Verschelde, H., Dierckx, H., & Bernus, O. (2007). Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue. Physical Review Letters, 99(16), 168104. https://doi.org/10.1103/PhysRevLett.99.168104

Weiss, J. N., Garfinkel, A., Karagueuzian, H. S., Chen, P.-S., & Qu, Z. (2010). Early afterdepolarizations and cardiac arrhythmias. Heart Rhythm, 7(12), 1891–1899. https://doi.org/10.1016/j.hrthm.2010.09.017

WHO. (2020). The top 10 causes of death. https://www.who.int/news-room/fact-sheets/detail/the-top-10-causes-of-death

Wilson, D. (2020). Phase-amplitude reduction far beyond the weakly perturbed paradigm. Physical Review E, 101(2), 022220. https://doi.org/10.1103/PhysRevE.101.022220

Wilson, K. G. (1974). Confinement of quarks. Physical Review D, 10(8), 2445–2459. https://doi.org/10.1103/PhysRevD.10.2445

Winfree, A. T. (1973). Scroll-shaped waves of chemical activity in three dimensions. Science, 181(4103), 937–939. https://doi.org/10.1126/science.181.4103.937

Winfree, A. T., Caudle, S., Chen, G. L., McGuire, P., & Szilagyi, Z. (1996). Quantitative optical tomography of chemical waves and their organizing centers. Chaos: An Interdisciplinary Journal of Nonlinear Science, 6(4), 617–626. https://doi.org/10.1063/1.166208

Winfree, A. T., & Strogatz, S. H. (1983). Singular filaments organize chemical waves in three dimensions: I. Geometrically simple waves. Physica D: Nonlinear Phenomena, 8(1-2), 35–49. https://doi.org/10.1016/0167-2789(83)90309-3

Wu, D., Yeh, S.-J., Wang, C.-C., Wen, M.-S., & Lin, F.-C. (1994). Double loop figure-of-8 reentry as the mechanism of multiple atrioventricular node reentry tachycardias. American Heart Journal, 127(1), 83–95. https://doi.org/10.1016/0002-8703(94)90513-4

Zhabotinsky, A. M., & Zaikin, A. N. (1973). Autowave processes in a distributed chemical system. Journal of Theoretical Biology, 40(1), 45–61. https://doi.org/10.1016/0022-5193(73)90164-1

Zykov, V. S. (1987). Simulation of wave processes in excitable media. Manchester University Press.

The Ithildin library for efficient numerical solution of anisotropic reaction-diffusion problems in excitable media

Thu, 19 Sep 2024 00:00:00 +0000

This article was previously published in PLOS ONE 19(9): e0303674 (Kabus, Cloet, et al., 2024) and is a chapter of my dissertation. The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Authors:

Desmond Kabus^1,2 (ORCID: 0000-0002-6965-5211)
Marie Cloet¹ (ORCID: 0000-0002-8974-6401)
Christian Zemlin³ (ORCID: 0000-0001-5834-5544)
Olivier Bernus⁴ (ORCID: 0000-0003-3917-5791)
Hans Dierckx¹ (ORCID: 0000-0003-0899-8082)

Institutions:

Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Etienne Sabbelaan 53, 8500 Kortrijk, Belgium
Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Albinusdreef 2, 2333 ZA Leiden, The Netherlands
Division of Cardiothoracic Surgery, Department of Surgery, University of Washington School of Medicine, 660 South Euclid Avenue, 63110 St Louis, MO, United States of America
Univ. Bordeaux, Inserm, Centre de Recherche Cardio-Thoracique de Bordeaux U1045, IHU Liryc, Hôpital Xavier Arnozan, Avenue du Haut Lévêque, 33600 Pessac, France

Correspondence: h.dierckx@kuleuven.be

DOI: 10.1371/journal.pone.0303674

Editor: Rafael Sachetto Oliveira, Universidade Federal de Sao Joao del-Rei, Brazil

Keywords:

excitable media
cardiac electrophysiology
open-source
finite differences
anisotropic diffusion
reaction-diffusion equation
C++

Abstract:

Ithildin is an open-source library and framework for efficient parallelized simulations of excitable media, written in the C++ programming language. It uses parallelization on multiple CPU processors via the message passing interface (MPI). We demonstrate the library’s versatility through a series of simulations in the context of the monodomain description of cardiac electrophysiology, including the S1S2 protocol, spiral break-up, and spiral waves in ventricular geometry. Our work demonstrates the power of Ithildin as a tool for studying complex wave patterns in cardiac tissue and its potential to inform future experimental and theoretical studies. We publish our full code with this paper in the name of open science.

Popular summary:

We present Ithildin, an open-source library for reaction-diffusion systems such as the electrical waves in cardiac tissue controlling the heart beat. We demonstrate the versatility of Ithildin by example simulations in various cell models and geometries, from simple 2D simulations to detailed ones in ventricular geometry. Our simulations highlight key features of Ithildin, such as recording pseudo-electrograms or filament trajectories. We hope that our work will contribute to the growing understanding of cardiac electrophysiology and inform future experimental and theoretical studies.

Introduction

With the Ithildin framework, we want to open up new gateways in the numerical simulation of reaction-diffusion systems, such as the electrical activation patterns in the heart.¹

Our motivation to write a reaction-diffusion solver comes from the numerical study of electrical patterns inside the heart (Clayton et al., 2011). These patterns, which are incompletely understood, are a main cause of death and even as a chronic disease, they complicate people’s lives. In the past decades, computer models of arrhythmia have allowed mechanistic insight in the origin and control of arrhythmias (Fenton & Karma, 1998). On the longer term, it is thought that digitized versions of patients' hearts could help offer better diagnostics and planning of procedures; such personalized heart models are called cardiac digital twins (Gillette et al., 2021; Koopsen et al., 2024; Niederer et al., 2019; Trayanova et al., 2020). In view of open science, we have decided to share the code that has been steadily developed in our group since 2007 with the scientific community.

Figure 1: Ithildin can be used to solve reaction-diffusion problems in excitable media. Required inputs for the software are: the diffusion tensor and geometry of a medium—such as the heart muscle, a reaction term—the so-called model, and source terms—typically a stimulation protocol. Ithildin can then calculate the evolution of the model variables in the medium over time. During calculation, Ithildin records relevant spatio-temporal data and metadata, as well as detecting rotor cores as filaments. The data visualized here are taken from several simulations which will be discussed below.

A flowchart outlining the functionality of Ithildin can be found in Fig. 1. Ithildin is designed to comply with the 2011 version of the ISO-C++ standard (ISO, 2011), but it compiles with all newer versions, including the current 2023 ISO-C++ standard (ISO, 2014, 2017, 2020, 2023). The software facilitates forward Euler and Runge-Kutta finite-difference solutions for reaction-diffusion systems in $N$-dimensional space, such as the monodomain equation for cardiac electrophysiology, with specified boundary conditions (Clayton et al., 2011).

The framework offers quick computation through CPU parallelization using OpenMPI (Graham et al., 2006). It also boasts decent documentation of available features, made accessible through Doxygen (van Heesch, 2023). Ithildin writes easy-to-parse YAML log files to document simulation setups (döt Net et al., 2023). Additionally, it allows convenient output of frames of recorded variables at regular intervals in the form of NumPy NPY files (Harris et al., 2020). The software supports easy and powerful post-processing with the Python module for Ithildin (Kabus et al., 2022), including integration with Scientific Python (Virtanen et al., 2020), 2D visualization with Matplotlib (Hunter, 2007), and 3D visualization with ParaView (Ahrens et al., 2005).

Ithildin also allows the recording of pseudo-electrograms (EGMs) and state variables at full numerical time resolution, as well as the tracking of filaments, which represent the instantaneous rotation axes of rotors. The software features a flexible setup for in-silico experiments, also called simulations, through a simple class-based C++ interface.

Various types of geometries are implemented, ranging from a simple 1D cable and spirals in 2D tissue to whole-heart geometry and even 4D hyperspace. The space can be partitioned to use multiple cell models in the same experiment via Model_multi. Realistic stimulation protocols can be added as Stimulus objects and may be started by a Trigger. Ithildin also includes a logging system with minimal impact on computation speed and various levels of verbosity.

In this paper, we provide an overview of this framework, guiding the reader through its components. Results from several in-silico experiments are presented as the main components of Ithildin are introduced. The details of these so-called simulations are outlined towards the end of this paper in section 5, along with a tabular overview in Table 4.

Essential numerical methods

Reaction-diffusion system

The diffusion of electrical signals in cardiac tissue can be modelled as a reaction-diffusion system, where the diffusion tensor ${{\bm{{D}}}}$ represents the anisotropic properties of the medium. This tensor encapsulates the spatial orientation of fibers in the medium and the effects of inhomogeneities on signal propagation. With the local unit vectors along the fibers ${{\bm{{e}}}}_{\text{f}}$, normal to fibers in the sheet plane ${{\bm{{e}}}}_{\text{s}}$, and normal to both of these ${{\bm{{e}}}}_{\!\times\!} = {{\bm{{e}}}}_{\text{f}} \times {{\bm{{e}}}}_{\text{s}}$, forming a orthonormal basis, the fiber orientation is encoded using diffusivities $D_{{\text{f}}, {\text{s}}, {\!\times\!}}{{\left( {{\bm{{x}}}} \right)}}$ in each of these directions (Clayton et al., 2011): $$ {{\bm{{D}}}} = D_{\text{f}} {{\bm{{e}}}}_{\text{f}} {{{{\bm{{e}}}}_{\text{f}}}^\mathrm{T}} + D_{\text{s}} {{\bm{{e}}}}_{\text{s}} {{{{\bm{{e}}}}_{\text{s}}}^\mathrm{T}} + D_{\!\times\!} {{\bm{{e}}}}_{\!\times\!} {{{{\bm{{e}}}}_{\!\times\!}}^\mathrm{T}} \qquad{(1)}$$

The core equation governing the evolution of the state variable vector ${{\underline{{u}}}}$ is the reaction-diffusion equation: $$ \partial_t {{\underline{{u}}}} = {{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}} + {{\underline{{r}}}}{{\left( {{\underline{{u}}}} \right)}} \qquad{(2)}$$ or in index notation: $$ \partial_t u_m{{\left( t,{{\bm{{x}}}} \right)}} = \textstyle\sum_{m'} P_{mm'} \; \textstyle\sum_{n} \partial_{n} \textstyle\sum_{n'} D_{nn'}{{\left( {{\bm{{x}}}} \right)}} \; \partial_{n'} u_{m'}{{\left( t, {{\bm{{x}}}} \right)}} + r_{m}{{\left( {{\underline{{u}}}};{{\bm{{x}}}} \right)}} \qquad{(3)}$$ for $m, m' \in {{\left\{1,...,M \right\}}}$ with the number of state variables $M$, and $n, n' \in {{\left\{1,...,N \right\}}}$ with the number of spatial dimensions $N$, using the notation $\partial_n = \partial_{x_n}$ for the spatial partial derivatives.

Here, ${{\underline{{u}}}}$ is the state variable vector and ${{\underline{{r}}}}({{\underline{{u}}}})$ accounts for the reaction term and is called the model. We refer to the first component of ${{\underline{{u}}}}$ as $u$, which for electrophysiological models is the transmembrane voltage $V_\text{m}$ or a rescaled version of it, see also section 4.2. For two-variable models, the second component of ${{\underline{{u}}}}$ is often referred to as the restitution or recovery variable $v$. In the term representing diffusion, ${{\bm{{D}}}}$ is determined by the geometry of the medium and the presence of inhomogeneities, see section 4.1. The projection matrix ${{\underline{{P}}}}$ is typically a diagonal matrix describing whether or not a variable is diffused. For instance, only the first variable of the AP96 model (Aliev & Panfilov, 1996) is diffused, such that ${{\underline{{P}}}} = {\operatorname{diag}{{\left( 1, 0 \right)}}}$.

Different notation is used to distinguish between vectors ${{\bm{{x}}}}$ and matrices ${{\bm{{D}}}}$ in physical space in bold font, and underlined vectors ${{\underline{{u}}}}$ and matrices ${{\underline{{P}}}}$ with respect to state variables. We use lowercase letters for vectors and uppercase for matrices. An overview of the most relevant quantities in Ithildin is given in Table 1.

symbol	dimension	unit	name
$N$	$\in\mathbb N$	1	number of spatial dimensions
$M$	$\in\mathbb N$	1	number of state variables
$t$	$\in \mathbb R_+$	ms	time
${{\bm{{x}}}}$	$\in \Omega \subset \mathbb R^N$	mm	space
${{\underline{{u}}}}{{\left( t, {{\bm{{x}}}} \right)}}$	$\in \mathbb R^M$	$\star$ (various units)	state variables
${{\underline{{r}}}}{{\left( {{\bm{{x}}}}, {{\underline{{u}}}} \right)}}$	$\in \mathbb R^M$	$\star$/ms	reaction term, model function
${{\bm{{D}}}}{{\left( {{\bm{{x}}}} \right)}}$	$\in \mathbb R^{N\times N}$	mm²/ms	diffusion tensor
${{\underline{{P}}}}$	$\in \mathbb R^{M\times M}$	1	projection matrix

Table 1: Quantities in the reaction-diffusion problem.

Ithildin obtains approximate solutions of the reaction-diffusion equation (Eq. 2) via a finite-differences approach: Time $t$ and space ${{\bm{{x}}}}$ are discretized on a grid and values ${{\underline{{u}}}}{{\left( t, {{\bm{{x}}}} \right)}}$ are associated with the vertices of this grid. We choose a fixed temporal resolution, the time step $\Delta t$, and constant spatial grid spacing $\Delta {{\bm{{x}}}}$. The values ${{\underline{{u}}}}{{\left( t + \Delta t, {{\bm{{x}}}} \right)}}$ at a subsequent time-step are computed based on the previous ones, according to discretized versions of the governing equations, i.e., the reaction-diffusion equation (Eq. 2), together with boundary and initial conditions.

Time integration

Starting from an initial state, the state variable vector ${{\underline{{u}}}}$ is integrated over time using a so-called time stepping scheme leading to an approximate solution of the reaction-diffusion system using finite differences. Ithildin implements two main stepping schemes to choose from: forward Euler and the classic Runge-Kutta method (RK4) (Euler, 1794; Press, 2007).

Defining ${{\underline{{f}}}}$ as the right hand side of the reaction-diffusion equation (Eq. 2), the forward Euler method takes the form (Press, 2007): $$ {{\underline{{u}}}} {{\left( t + \Delta t, {{\bm{{x}}}} \right)}} = {{\underline{{u}}}} {{\left( t, {{\bm{{x}}}} \right)}} + \Delta t \, {{\underline{{f}}}} {{\left( t, {{\bm{{x}}}}; {{\underline{{u}}}} \right)}} + O{{\left( \Delta t^2 \right)}} \qquad{(4)}$$ This method is the default time integration scheme in Ithildin. Despite its numerical error being of order $O{{\left( \Delta t^2 \right)}}$, with a sufficiently small time step, the accuracy of the Euler method is adequate for our use cases.

Due to the Courant-Friedrichs-Lewy condition (CFL), a stability criterion for the integration of the reaction-diffusion equation, $\Delta t$ needs to be chosen sufficiently small (Courant et al., 1928). Ithildin automatically chooses an appropriate time step based on the CFL condition for the different supported geometries, cf. section 4.1. For example, for the most simple implemented geometry contained in Ithildin, i.e., isotropic diffusion (see Geometry_Iso in section 4.1 and Table 2), the CFL condition is enforced by setting (Li et al., 1994): $$ \Delta t < {{\left[ 2\max{{{\underline{{P}}}}} \sum_{n=1}^{N} \frac{D_{nn}}{\Delta x_n^2} \right]}}^{-1} \qquad{(5)}$$ where $D_{nn}$ are the diagonal components of ${{\bm{{D}}}}$ and $\max{{{\underline{{P}}}}}$ is the maximum value of ${{\underline{{P}}}}$.

Higher accuracy at the cost of more computations per time step, can be achieved with the RK4 method (Press, 2007): $$\begin{aligned} {{\underline{{u}}}} {{\left( t + \Delta t, {{\bm{{x}}}} \right)}} &= {{\underline{{u}}}} {{\left( t, {{\bm{{x}}}} \right)}} + \frac{1}{6}\Delta{{\underline{{u}}}}_1 + \frac{1}{3}\Delta{{\underline{{u}}}}_2 + \frac{1}{3}\Delta{{\underline{{u}}}}_3 + \frac{1}{6}\Delta{{\underline{{u}}}}_4 + O{{\left( \Delta t^5 \right)}} \\ \Delta{{\underline{{u}}}}_1 &:= \Delta t\, {{\underline{{f}}}} {{\left( t, {{\bm{{x}}}}; {{\underline{{u}}}} \right)}} \\ \Delta{{\underline{{u}}}}_2 &:= \Delta t\, {{\underline{{f}}}} {{\left( t + \textstyle\frac{1}{2}\Delta t, {{\bm{{x}}}}; {{\underline{{u}}}} + \textstyle\frac{1}{2}\Delta{{\underline{{u}}}}_1 \right)}} \\ \Delta{{\underline{{u}}}}_3 &:= \Delta t\, {{\underline{{f}}}} {{\left( t + \textstyle\frac{1}{2}\Delta t, {{\bm{{x}}}}; {{\underline{{u}}}} + \textstyle\frac{1}{2}\Delta{{\underline{{u}}}}_2 \right)}} \\ \Delta{{\underline{{u}}}}_4 &:= \Delta t\, {{\underline{{f}}}} {{\left( t + \Delta t, {{\bm{{x}}}}; {{\underline{{u}}}} + \Delta{{\underline{{u}}}}_3 \right)}} \end{aligned}\qquad{(6)}$$ Note that ${{\underline{{f}}}}$ needs to be evaluated four times for the RK4 method and only once for the Euler method. While RK4 is still an explicit method subject to instability at too large $\Delta t$, a larger $\Delta t$ value than for the Euler method can typically be used.

The stepping scheme to be used can be chosen in Ithildin on a per-variable level via Model::steppings.

Numerical spatial derivatives

For the numerical solution of the reaction-diffusion equation (Eq. 2), the spatial derivative in its right hand side must be computed, i.e., the diffusion operator ${{\underline{{P}}}}\nabla\cdot{{\bm{{D}}}}\nabla{{\underline{{u}}}}$. This is implemented as weighted sums of the value of ${{\underline{{u}}}}$ at neighboring vertices on the grid of the discretized domain. The weights for the calculation of the stencil depend on the chosen type of diffusion (section 4.1). The two main types in this software are a first order stencil, including only the nearest neighbors, and a second order stencil, including also the next to nearest neighbors.

In the simplest case (see Geometry_Iso in section 4.1 and Table 2), we consider isotropic and homogeneous diffusivity. The diffusion operator can then be computed via the Laplacian operator $\nabla^2$. This is done with a 5-point stencil for the 2D case, a 7-point stencil for the 3D case, etc., see also Fig. 2, panel (a). Consequently, the weights are calculated as: $$ \begin{aligned} w_n &= \frac{1}{\Delta x_n} \quad \text{for }1\leq n\leq 2 N \\ w_0 &= -\sum_{n=1}^{2N}w_n \end{aligned} \qquad{(7)}$$ where $N$ is the number of dimensions and $\Delta x_n$ is the grid resolution in the direction of the neighbor corresponding to weight $w_i$, e.g., $\Delta x_5 = \Delta z$. Note that the indices correspond to those displayed in Fig. 2.

Figure 2: Stencils for isotropic diffusion (a) and orthotropic diffusion (b) with numbering of the involved grid points.

For the more general orthotropic diffusion, the stencil for numerical differentiation includes the nearest and diagonal neighbors of a grid point, see Fig. 2, panel (b). The weights for orthotropic diffusion are obtained by a combination of central differences approximations to derivatives and linear interpolation of values between two grid points. For more details, the reader is referred to the documentation of Geometry_OrtAniso.

Implementation

The source code of Ithildin is written in the C++ programming language, following the 2011 version of the ISO-C++ standard (ISO, 2011). This was chosen to facilitate programming at a level relatively close to the hardware, but also using some of the useful data-structures that are contained in the standard template library (STL).

Ithildin is designed to run in parallel on multiple CPU cores using MPI, specifically OpenMPI (Graham et al., 2006). Using the mpirun command, multiple instances, also known as processes, of the same compiled executable are started that run across different processor cores. The processes are then coordinated such that there is one so-called manager process, that manages a bunch of so-called worker processes. The manager does an equal share of the computational work, just like a worker. The only difference is that the manager distributes and directs information to be exchanged from one process to another. In Ithildin, the memory is not shared across processes, instead each process works on its share of the computational domain. We split the domain in the $x$-direction, such that each process is responsible for computations on a roughly equal share of vertices inside the to-be-simulated medium. Even when some parts of the domain are classified as exterior points, i.e., points on which no calculations need to be performed, they are taken into account when splitting the domain between processes. This splitting is possible because the reaction-diffusion systems to be studied with Ithildin are local, meaning that the temporal evolution at each time $t$ and each point ${{\bm{{x}}}}$ in space depends only on the current state vector ${{\underline{{u}}}}{{\left( t, {{\bm{{x}}}} \right)}}$ at that position and its spatial derivatives (Eq. 2). Internally, a layer of so-called ghost points is added around each process’ part of the domain such that the spatial derivatives can still be calculated in the same way as for any other point in the domain (section 2.3). The values ${{\underline{{u}}}}$ on these ghost points are exchanged with the neighboring processes, as coordinated by the manager process. Additional ghost points are also used to enforce Neumann boundary conditions, which is done by setting the weights for the calculation of the numerical spatial derivatives accordingly.

To run a simulation in Ithildin, the user needs to define a main() function that is to be called by all subprocesses. This is usually done using a C++ file calling the required components of the Ithildin library defining the main() function, a so-called main file. Ithildin can be installed as a shared library on Unix-based systems that is required by executables obtained from compiling main files. Alternatively, it is also possible to statically compile the Ithildin library with a main file into a stand-alone executable.

Besides the necessary preparations for using MPI, running a simulation using Ithildin’s Sim class requires three main components, which are instances of three classes:

Model: the reaction term ${{\underline{{r}}}}{{\left( {{\underline{{u}}}} \right)}}$, typically a cell model,
Geometry: the discretized diffusion term ${{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}}$ for a chosen geometry of the medium, and
Source: the stimulus protocol to use as well as inhomogeneities in the medium.

In the following section 4, an overview is given for each of these classes and their derived classes. Combining all of the components, an illustrative, minimal main file can be obtained, in which a planar wave crosses the medium in positive $x$-direction:

#include "ithildin.h"

int main(int argc, char** argv){
 // 0. initialize MPI
 Mpiclass mpi(argc, argv);

 // 1. select reaction term
 Model_SmooKa model;

 // 2. select diffusion term
 vector<int> size{30, 30, 1};
 vector<float> dx{1., 1., 1.};
 Geometry_Iso geom{&mpi, size, dx, &model};

 // (set up simulation)
 Sim sim{1., 10, &mpi, &model, &geom, "example"};

 // 3. define stimulus protocol
 Source sour{&model, &geom, &mpi, &sim};
 sour.stimulate({{0}, {1.}, Shape::Rect({0, 0}, {5, 0})});

 // (run simulation)
 return sim.run(&sour);
}

More detailed examples for main files can be found in the appendix, where we set up the numerical examples used throughout this paper.

We consider the geometry, the model and the source to be the inputs of Ithildin, see also Fig. 1. Upon running the simulation, Ithildin produces a variety of outputs, as files in easy-to-parse standardized data formats. The names of these files all begin with the so-called stem, consisting of a descriptive series name and a serial number, which defaults to a time-stamp. In the following, an overview of the usual output files of Ithildin by suffix appended to the stem is provided:

_log.yaml: The log file contains metadata describing the setup of the simulation, as well as metadata about the conditions under which the simulation was run. This file is always output by Ithildin and is considered the central file of the results, as it points to the relevant other files that are only conditionally written during the simulation. While earlier versions of Ithildin used a non-standard format for the log files, in current versions, the YAML format is used, making it easy to parse by both: humans and machines (döt Net et al., 2023).
_main.cpp: A copy of the C++ code in the main file may be included in the results for reproducibility.
_git.diff: If run in a Git-repository with changes since the last commit, a patch file of these changes will be included in the results.
_*.txyz.npy: For each of the state variables ${{\underline{{u}}}}$, a so-called var file in the NumPy NPY format will be written (Harris et al., 2020). These files contain the $N+1$-dimensional floating-point number array of the evolution of a state variable $u$ in the whole computational grid over time. Note that the order of indices is $(t, x, y, z)$, meaning that time is the slowest varying index and the $x$-axis the second slowest varying index. This order was chosen because the domain is split across processes along the $x$-axis, such that the processors can open and write to these files sequentially. For 2-dimensional simulations, the third spatial dimension, the $z$-axis, is one vertex thick, such that each var file still has four dimensions. For higher-dimensional simulations, more axes are added, leading to more dimensions in the var files.
_inhom.txyz.npy: The inhom field, see details in section 4.3, is also stored in the NPY format, but with integer values, and only for the initial time-step. This file hence has the shape $(1, N_x, N_y, N_z)$ with $N_n$ denoting the number of vertices in each of the spatial dimensions.
_hist*.csv: Comma-separated value (CSV) files describing the temporal evolution of ${{\underline{{u}}}}{{\left( t,{{\bm{{x}}}}_\text{s} \right)}}$ at a chosen sensor position ${{\bm{{x}}}}_\text{s}\in\Omega\subset\mathbb{R}^{N}$, see details in section 4.5.
_egm*.csv: CSV files containing the recorded pseudo-EGM $\Phi{{\left( t,{{\bm{{x}}}}_\text{e} \right)}}$ at a chosen electrode location ${{\bm{{x}}}}_\text{e}\in\mathbb{R}^{N}$, see details in section 4.6.
_tipdata.yaml: This YAML file is written if filament-tracking is turned on, see also section 4.7, and contains the detected phase singularities in regular time intervals.

While we have selected these file formats to be easy to read using a wide variety of software, we have also developed the Python module for Ithildin to facilitate interacting with the results of an Ithildin simulation and converting them to a variety of file formats, for instance writing files in the extensible data model and format (XDMF) that can be used to view simulation results in ParaView (Ahrens et al., 2005; Kabus et al., 2022; Kabus, De Coster, et al., 2024; Mark et al., 2007). The Python module also offers post-processing and analysis methods, for instance the computation of action potential duration (APD), conduction velocity (CV), various phases, phase defect detection, functions acting on filaments and filament trajectories, and several plotting functions (Kabus et al., 2022; Kabus, De Coster, et al., 2024).

Application-focused numerical methods

Diffusion term

The diffusion term in the reaction-diffusion equation (Eq. 2) is stated as ${{\underline{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\underline{{u}}}}$. The conduction in cardiac tissue and hence the diffusion is stronger along the fiber direction than normal to the fibers (Clayton et al., 2011). This is encoded in the diffusion matrix ${{\bm{{D}}}}$.

As an example, in Fig. 3, the fiber direction is drawn on the surface of the ventricular geometry used in Sim. 4. The fibers are additionally colored by their fiber helix angle (Streeter et al., 1969). The voxel-based representation of this geometry was obtained by cutting a human heart into slices, digitizing and stacking them (Hren, 1996; ten Tusscher et al., 2007).

Figure 3: Ventricle geometry with fiber direction, colored by the fiber helix angle.

While the projection matrix ${{\underline{{P}}}}$ is managed by the Model class, see section 4.2, the diffusion matrix ${{\bm{{D}}}}$ and the handling of the spatial derivatives is implemented in the Geometry class. The most important things the Geometry class takes care of, are:

Initialization of the computational grid, along with the strides and pointers, which are important for efficient computing;
Computation of the entries of the diffusion tensor, based on the main directions of diffusion and the respective diffusion values;
Computation and storage of the weights for the stencils of the numerical spatial derivatives (section 2.3) respecting the Neumann boundary conditions; and
Handling the upper bound for the time step due to the CFL condition (section 2.2).

The Geometry class is a base class and should not be used directly to construct a computational domain. Instead, there are several subclasses, each representing a different type of diffusion or extrinsic shape. An overview of the Geometry subclasses is given in Table 2.

class	description	dim $N$	points in stencil
`Geometry_Iso`	isotropic diffusion	1..3	$2N+1$
`Geometry_ND`	isotropic diffusion in higher dimensions (Cloet et al., 2023)	$N\in\mathbb{N}$	$2N+1$
`Geometry_OrtAniso`	orthotropic diffusion	2..3	$\begin{cases}9 & N=2 \\ 19 & N=3\end{cases}$
`Surface`	2D domain with extrinsic curvature	2	9
`QuadricSurface`	quadric surface with rotated parallel fibers	2	9
`Ellipsoid`	ellipsoidal surface with rotated parallel fibers	2	9
`Hyperboloid`	hyperboloidal surface with rotated parallel fibers	2	9
`Paraboloid`	paraboloidal surface with rotated parallel fibers (Dierckx et al., 2013)	2	9

Table 2: Overview of tissue geometries supported by Ithildin.

The Geometry_ND class is a subclass of Geometry_Iso and the Ellipsoid, Hyperboloid, and Paraboloid classes are subclasses of QuadricSurface.

The details on how the extrinsic and intrinsic curvature affect the diffusion tensor and hence the weights for the discretized differential operator, are discussed in the documentation of the code. There, the different ways to initialize the available domain types are given as well.

Note that some additional features are implemented in the Geometry class, that do not have a direct link with the diffusion term, such as inhomogeneities (section 4.3) and filament detection (section 4.7).

Reaction term

The Model class serves as the base class for all cardiac electrophysiology models in the Ithildin framework. It encapsulates common functions and variables used across different models. Key features of the Model class include:

Implementation of the reaction term ${{\underline{{r}}}}{{\left( {{\underline{{u}}}} \right)}}$ in the reaction-diffusion equation.
Storage of model-specific metadata such as relevant citations.
Handling of variable-related information, including their names, indices, and resting values.
Management of the values of the projection matrix ${{\underline{{P}}}}$.

Derived classes extend the Model class to implement specific cardiac electrophysiology models in the reactionterm function.

An overview of the cell models that are currently included in the source code of this project is provided in Table 3. More models may be added as additional classes.

class	description	#vars	references
`Model_1VarPoly`	one-variable polynomial model	1
`Model_AP`	Aliev-Panfilov, continuous epsilon	2	Aliev & Panfilov (1996)
`Model_AP2`	Aliev-Panfilov, discontinuous epsilon	2	Aliev & Panfilov (1996)
`Model_AP3`	Aliev-Panfilov, discontinuous epsilon, simple recovery	2	Aliev & Panfilov (1996); Pravdin et al. (2015); Dierckx et al. (2015)
`Model_AP4`	Aliev-Panfilov, smoothened epsilon, simple recovery	2	Aliev & Panfilov (1996); Pravdin et al. (2015); Dierckx et al. (2015)
`Model_BO`	Bueno-Orovio 2008 4 var	4	Bueno-Orovio et al. (2008)
`Model_Ba`	Barkley	2	Barkley (1991)
`Model_FHNa`	FitzHugh-Nagumo (a), 2 var	2	FitzHugh (1961); Nagumo et al. (1962)
`Model_FHNb`	FitzHugh-Nagumo (b), 2 var	2	FitzHugh (1961); Nagumo et al. (1962)
`Model_FHNc`	FitzHugh-Nagumo (c), 2 var	2	FitzHugh (1961); Nagumo et al. (1962)
`Model_FK`	Fenton-Karma 3 var	3	Fenton & Karma (1998)
`Model_Kaz`	Kazantsev PRE 2003 spiking neuron	2	Kazantsev et al. (2003)
`Model_LRI`	Luo-Rudy Phase I	8	Luo & Rudy (1994)
`Model_MS`	Mitchell Schaeffer	2	Mitchell (2003)
`Model_SmooKa`	Smooth Karma by Marcotte 2017	2	Marcotte & Grigoriev (2017); Byrne et al. (2015); Karma (1993); Karma (1994)
`Model_TP06`	Ten Tusscher & Panfilov 2006	19	ten Tusscher & Panfilov (2006); Niederer et al. (2019)

Table 3: Overview of cell models included in Ithildin.

The Ithildin framework also introduces several ModelWrapper classes that provide additional functionality and allow for the combination of models. These wrappers enable the recording of diffusion terms, reaction terms, local activation times (LAT), and local deactivation times (LDT) as additional state variables. This is done by adding code to the reaction term calculations of the underlying model. The wrappers inherit from the base ModelWrapper class, which is a wrapper that leaves the model unchanged. The primary ModelWrapper classes are:

ModelWrapper_RecordDiffusion records diffusion terms of selected variables as additional variables.
ModelWrapper_RecordReaction records reaction terms of selected variables as additional variables.
ModelWrapper_RecordActivationTime records LAT for selected variables.
ModelWrapper_RecordDeactivationTime records LDT for selected variables.
ModelWrapper_RescaleTimeSpace linearly rescales the wrapped model in time and space.
ModelWrapper_RescaleVars linearly rescales selected state variables of the wrapped model.

The Model_multi class enables the combination of multiple submodels into a single model. The behavior of the combined model may vary depending on the location ${{\bm{{x}}}}$ and is determined by one of its submodels. Which model is to be used depends on the integer value of the inhom field, which is used to describe spatial inhomogeneities, such as obstacles. Inhomogeneities will be further explained in the following, cf. section 4.3. This class is particularly useful for simulating scenarios where different regions of cardiac tissue exhibit distinct behaviors.

The Ithildin C++ framework provides a structured and modular approach to modeling cardiac electrophysiology. The Model class serves as the base for different models, while various ModelWrapper classes and the Model_multi class offer extended functionalities for recording and combining different model aspects.

Inhomogeneities

In simulations of cardiac electrophysiology, accurately modeling the spatial properties of the cardiac tissue is essential. Inhomogeneities represent variations in the tissue’s characteristics, such as its electrical conductivity or cellular properties, that influence the propagation of electrical signals.

An inhomogeneity in the Ithildin framework is a distinct region within the simulation domain with different properties compared to its surroundings. In the context of cardiac electrophysiology, these properties could correspond to variations in the electrical conductivities of cells, cellular properties, or even the absence of excitable cells altogether. Inhomogeneities are defined by an integer field called inhom associated with each point in the domain. Grid points with a non-zero inhom value are considered interior points, indicating that the reaction-diffusion equation needs to be solved on these points. If the selected reaction term is a Model_multi (see also section 4.2), for an inhom value of $n$, the $n$th submodel will be used for this point. For example, in Fig. 4, the inhom field for Sim. 1 is visualized. Two different cell models are used for the values 1 and 2. The points where inhom has the value 0, are considered exterior points. The set of all interior points is the physical domain $\Omega$. At the boundary of the physical domain, Neumann boundary conditions are applied.

Figure 4: Inhomogeneities in Sim. 1. The field inhom describes where unexcitable obstacles or exterior points are located with the value inhom = 0 and which cell model is to be used inside if inhom > 0. This figure also visualizes the filament trajectories colored by time, cf. section 4.7. Birth of a tip is denoted by stars, their deaths by crosses, and tips that are still around at the end of the simulation by points. The three trajectories with the longest lifetime are numbered by indices 0 through 2.

To incorporate inhomogeneities into the simulation, the framework provides methods to add them to the domain. These methods allow specifying the shape, location, and properties of each inhomogeneity. For example, a rectangular inhomogeneity could be added by specifying its width, height and location.

Stimulation protocols

Ithildin provides a flexible way to define stimulation protocols using the Stimulus and Trigger classes, as well as the scheduling functionality of the Source class.

The Stimulus class enables the specification of temporal and spatial characteristics of voltage-based or current-based stimuli. It allows defining which state variables should be affected directly, the associated values, and whether the stimulus sets the variable directly or whether it is additive and hence behaving like a current source. Temporal modulation of stimulus strength is achieved through the amplitude function, while spatial constraints are managed by the shape function, an instance of the Shape class.

The Shape class describes a geometric shape via its characteristic function which can be used to define regions of interest in a simulation domain. The core principle is that the function evaluates a given position vector and returns a value: 1 if the position is inside the defined shape, and 0 if it is outside. However, values in the range ${{\left[ 0,1 \right]}}$ may also be used to create a smooth transition. A smoothly varying characteristic function may be useful to create more-realistic stimuli that deposit current in a smooth profile. This simple yet powerful concept forms the basis for constructing intricate spatial configurations.

The Shape class provides several pre-defined shape functions, though additional shapes can easily be added by defining a characteristic function:

Ellipsoid: Defined by radii, a center and optionally the Euler angles, this shape represents a general three-dimensional ellipsoid with the specified orientation.
Sphere: A special case of an ellipsoid where all radii are equal, forming a three-dimensional sphere.
Ellipse in $xy$-plane: This two-dimensional shape resembles an ellipse lying on the $xy$-plane, defined by radii and a center.
Cylinder along $z$-axis: Representing a three-dimensional cylinder centered along the $z$-axis, this shape is defined by a radius and a center. In 2D, it defines a disk.
Rectangular cuboid: Defining a three-dimensional region, this shape is specified by two opposing corner points, creating a cuboid. In 2D, it defines a rectangle.
Half plane: A plane defined by an origin and an outward normal vector splits the three-dimensional space into a half-space. In 2D, a straight line splits the plane in a similar way.

Characteristic functions can also be loaded from files in the NPY format (Harris et al., 2020). This feature facilitates the incorporation of custom shapes derived from external data sources.

The scheduling functionality via Source::schedule offers the ability to execute functions at specified points in time during the simulations. This feature greatly enhances experimental flexibility by allowing the execution of arbitrary code snippets at chosen moments during simulations. The scheduler is especially useful for introducing dynamic changes to the simulation environment, such as modifying stimuli or conditions mid-simulation.

The Trigger class provides a means to orchestrate actions based on specific conditions. Triggers encapsulate the decision-making process of when to execute a particular action, influenced by condition checks and coordination modes. Different coordination modes allow for the synchronization of trigger actions across multiple processes, facilitating complex simulations of activation waves. These modes are to trigger on each process individually once the condition is met, once the condition is met in a specific process, once the condition is met in any process, or once it is true in all processes.

Within the framework of cardiac electrophysiology, triggers are essential for defining stimulation protocols, e.g. for the S1S2 protocol, which is illustrated in Fig. 5: After a first excitation wave passes a sensor position, a second wave is triggered behind a part of the first waveback to stimulate spiral waves.

Figure 5: The S1S2 protocol illustrated for Sim. 2. The first and third row display the transmembrane voltage $u$ at selected frames in time, and the second and fourth row the state space phase (Arno et al., 2021; Kabus et al., 2022). The first stimulus is applied in the first frame in the hatched region at the left edge. The sensor location is marked with a cross. Right after the third frame, the sensor triggers the second stimulus in the hatched region at the bottom edge. A spiral wave forms. The phase singularity at the center of the spiral is tracked as the curve, which is dotted for the entire trajectory of the phase singularity, and solid for its trajectory since the previous frame.

Recording temporal evolution of variables at sensor positions

Besides the sensors for triggering stimuli, sensors in the context of this simulation framework are components that monitor and record the state variables of the simulated system at chosen positions during the simulation. We call this the history at a given sensor position.

These sensors are used to gather data about the behavior of the system at particular time intervals, regulated by the sensorlag parameter. This parameter controls the frequency at which sensor data is collected, allowing for flexibility in recording intervals. Notably, the recording frequency set by sensorlag need not align with the simulation’s time step or the duration between frames. The data will be recorded at the first time step after the specified sensorlag duration. This high-resolution temporal data can be used to study individual points in the medium in detail.

The recorded data are then written to designated comma separated value (CSV) output files associated with each sensor. These files are used to store the collected data over the course of the simulation.

The first four panels of Fig. 6 contain time traces of the transmembrane voltage $u$, the restitution variable $v$, the recorded value of the diffusion term, and the LAT at a given sensor position for Sim. 1. The times of the three stimuli are indicated by the black vertical lines. The other panels will be explained in the subsequent sections.

Figure 6: Extracted time traces of Sim. 1. The first two panels (a, b) contain the model variables $u$ and $v$, followed by data computed by model wrappers, namely the diffusion term $\nabla \cdot {{\bm{{D}}}} \nabla u$ (c) and the LAT (d), at an interior point, cf. section 4.5. Panel (e) contains the pseudo-EGM at an exterior point, next to the 2D domain, cf. section 4.6. The last panel (f) contains the number of rotors detected via phase singularities, cf. section 4.7. The black vertical lines indicate times at which a stimulus was applied.

Pseudo-EGMs

The EGM is a measurement of the potential generated by the charge distribution in cardiac tissue over time at a point in space, outside the tissue. In theory, this is a measurement of the extracellular potential ${\Phi_\text{e}}$. However, since Ithildin is a monodomain solver, the extracellular potential is not part of the model equations (Clayton et al., 2011). The code therefore calculates an approximation of the extracellular potential at points outside the mesh using the Egm class. To obtain this approximation, the pseudo-bidomain theory is used (Bishop & Plank, 2011). The approximation, referred to as a pseudo-EGM, uses the simplifications that the intracellular and extracellular conductivities are proportional, such that there is an explicit formula to calculate the extracellular potential, and that the bath conductivity is homogeneous.

The extracellular potential ${\Phi_\text{e}}$ at position ${{\bm{{x}}}}_\text{e}$ over time is then calculated as: $$ {\Phi_\text{e}}{{\left( {{\bm{{x}}}}_\text{e},t \right)}} = \frac{ \tau_\text{e} }{ 4\pi } \int_{\Omega}\mathrm{d}{{\bm{{x}}}}\; \frac{ \nabla \cdot {{\bm{{D}}}} \nabla u({{\bm{{x}}}},t) }{ {{\left\lVert {{\bm{{x}}}}_\text{e} - {{\bm{{x}}}} \right\rVert}} } \qquad{(8)}$$ where $\Omega$ denotes the computational domain, i.e., the simulated heart muscle tissue, $u$ is a state variable of the model representing the transmembrane potential, , which is usually encoded as the first variable in the state vector ${{\underline{{u}}}}$. Furthermore, ${{\bm{{D}}}}$ is the diffusion matrix from the reaction-diffusion equation (Eq. 2), and $\tau_\text{e}$ is a proportionality factor with units ${\mathrm{{m}{s}}}$.

Essentially, the used approximation for the EGM is a convolution of the diffusion term $\nabla \cdot {{\bm{{D}}}} \nabla u$ for the first variable with a kernel ${{\left\lVert {{\bm{{x}}}}_\text{e} - {{\bm{{x}}}} \right\rVert}}^{-1}$. Depending on the choice of the diffusion tensor, made by the user in the Geometry class (section 4.1), a different prefactor of the integral is required. Hence, the prefactor $\frac{\tau_\text{e}}{4\pi}$ is user-defined and can be changed using the function set_prefactor.

The integral in Eq. 8 is discretized and its calculation is implemented such that the additional amount of storage and number of calculations is limited as much as possible. For instance, as the required diffusion term $\nabla \cdot {{\bm{{D}}}} \nabla V_\text{m}$ is already computed during the forward stepping of the reaction-diffusion equation, it can be stored as an additional state variable using a ModelWrapper_RecordDiffusion and subsequently used in the pseudo-EGM calculation. This model wrapper is essential for the functioning of the Egm class and hence is a requirement when setting up a simulation with pseudo-EGM calculation.

The result is a CSV file with the pseudo-EGM data for each electrode that is defined by the user. The computed pseudo-EGM at a given position for Sim. 1 is displayed in the fifth panel of Fig. 6.

Filaments

Formally, filaments can be understood as a line of wave break, i.e., a line where an activation and recovery surface come together (Clayton et al., 2006; Winfree, 1994). The activation surface can be seen as the wavefront, while the recovery surface can be seen as the waveback. When considering an excitable system in 2D, filaments become tips, being the point of intersection between the activation and recovery curve. Since a point cannot be excited and recovering at the same time, points on a filament are also called phase singularities.

Using this definition of filament points, detection algorithms have been designed. Our code relies on the one described by Fenton & Karma (1998). Additionally, this algorithm has been extended to grids in any dimension, where the generalization of filaments are called superfilaments (Cloet et al., 2023).

The filament point detection algorithm is included in the Geometry class. Its goal is to compute the points where the wavefront and waveback meet. While looping over all coordinate planes and grid points, it is checked whether there is an intersection of isolines in the adjacent voxel faces of a grid point. The location of the filament point is then estimated by bi-linear interpolation. An illustrative sketch of this method is given in Fig. 7.

Figure 7: Illustration of the filament point tracking algorithm. Each coordinate plane adjacent to a grid point $(i,j,k)$ is checked for an intersection of two surfaces (purple and orange lines) which are usually isosurfaces of state variables in ${{\underline{{u}}}}$. In this example, a filament point (dot) will be found in the $yz$-face.

The last panel in Fig. 6 displays the number of rotors over time for Sim. 1, which are found via filament detection. It can be seen that at the S2 stimulus, a single rotor is formed, and subsequently pairs of rotors as figure-of-eight spiral pairs.

More details of this process can be seen in Fig. 4 which shows the trajectories of these filaments in Sim. 1 tracked using the Python module for Ithildin. The trajectories are colored by time and the formation of a new tip is denoted by stars and their decay by crosses. Tips that still persist at the final frame of the simulation are indicated by points. The tip trajectory denoted with index 0 is formed by the S1S2 protocol and meanders around the medium. It persists until the end of the simulation. The figure-of-eight spiral wave pair denoted by indices 1 and 2 is formed by a conduction block breaking up. The spiral tip with index 2 runs into the boundary to disintegrate there.

Phase defects

While a phase singularity can be seen as points where all phases meet—in mathematics called a pole, a phase defect is a point at which there is a discrete jump from one phase value to another in an otherwise continuously varying phase. While phase defects are well known in physics, they were only recently identified in excitable media, within linear-core rotors and conduction block regions (Arno et al., 2021; Tomii et al., 2021). Phase defects are lines in 2D and surfaces in 3D.

In Ithildin, the phase defect detection is done with its Python module. For instance, during simulation, Ithildin can record the local activation times (LAT) which can then be used to compute the activation time phase in post-processing (Arno et al., 2021; Kabus et al., 2022). A variety of methods exist to then localize the phase defect (Kabus et al., 2022; Tomii et al., 2021).

Two examples for phase defect detection in Sim. 1 are given in Fig. 8 and Fig. 9. Both display four frames over time of the transmembrane voltage $V_\text{m}$, the activation time phase ${\varphi}$, and the phase defect $\rho$ computed via the cosine method (Kabus et al., 2022; Tomii et al., 2021). In Fig. 8, the phase defect of a single spiral wave is tracked. It can be seen that the phase defect extends due to conduction block such that the spiral wave moves across the domain. In Fig. 9, the break-up of a conduction block line into a figure-of-eight spiral wave pair can be seen. The conduction block line is a phase defect of zero topological charge (Arno et al., 2024a; Goryachev & Kapral, 1996; Mermin, 1979) which, in this case, reaches a critical length breaking apart into two oppositely charged spiral waves with much shorter phase defect lines.

In Fig. 10, we present the final frame of Sim. 4 in ventricular geometry, visualized with ParaView (Ahrens et al., 2005). It is colored by the normalized transmembrane voltage. Both, the classical tip and the phase defect surface are visualized. The spiral waves revolve around the phase defect surfaces.

Figure 8: Phase defect detection for a single meandering spiral in Sim. 1. By recording the activation times of the transmembrane voltage $V_\text{m}$ (top row), the activation time phase ${\varphi}$ (Kabus et al., 2022; Tomii et al., 2021) can be computed (middle row). Where this phase is discontinuous, a phase defect is localized. This is visualized as the phase defect density $\rho$ (bottom row). In these four frames, a single rotor is tracked shortly after its formation. Due to conduction block, which can be seen as an extended phase defect line, the spiral moves through the medium. Afterwards, the spiral remains mostly stationary, leading to a shorter phase defect line.

Figure 9: Phase defect detection during figure-of-eight spiral wave pair creation in Sim. 1. Visualization in the same style as Fig. 8. A long conduction block line breaks apart into two rotors.

Figure 10: Final frame of Sim. 4 of the BOCF model in ventricle geometry. The classical filament is plotted as the purple lines. The phase defect surface is contained in the gray contour.

Further documentation

More complete documentation of Ithildin can be generated with Doxygen (van Heesch, 2023) and can be also found online, see the data availability statement for details. The documentation contains detailed information on how to get started installing and working with ithildin.

Results

Ithildin is used for numerical experiments in various use cases. For instance, it was used to study the structure of the core of rotors emerging in in-silico cardiac-electrophysiology cell models, leading to the description of phase defect lines (Arno et al., 2021, 2024a, 2024b; Kabus et al., 2022). Also, higher-dimensional rotors waves were simulated and the emerging super-filaments were detected using Ithildin (Cloet et al., 2023). In the creation of novel data-driven cell models using state space expansion, Ithildin was used to generate synthetic training data sets (Kabus, De Coster, et al., 2024).

Five simulations were conducted for this paper to illustrate the features of Ithildin. An overview of the simulations is given in Table 4, along with references to the figures that were generated with these data sets while details on their simulation setup can be found in their C++ code in the appendix.

simulation	Sim. 1	Sim. 2	Sim. 3	Sim. 4	Sim. 5
cell model	`SmooKa`	`AP`	`BO`	`BO`	`TP06`
parameter set	default	default	EPI	EPI	EPI
references	²	³	⁴	⁵	⁶
dimensions	2	2	2	3	3
anisotropy type	isotropic	isotropic	isotropic	orthotropic	orthotropic
diffusivity $P_{uu}$	0.031	1.6	0.12	0.12	0.1
ratio $D_\parallel / D_\perp$	1.0	1.0	1.0	4.0	7.6
inhomogeneities	sphere, rectangle, waves	none	none	biventricular geometry	none
grid size	100×70	120×120	450×450	168×208×231	various
spacing [mm]	$0.2, 0.2$	$1, 1$	$0.3, 0.3$	$0.43, 0.43, 0.5$	various
time step [ms]	0.1	0.1	0.1	0.1	various
stim. times [ms]	0, 600, 921	0, 426	0, 394.1	0	0
duration [ms]	2000.1	838.6	680.1	600.1	150.0
used in	Figs. 1, 4, 6, 8, 9	Fig. 5	Sim. 4	Figs. 1, 3, 10	Fig. 11

Table 4: Overview of the numerical simulations used in this paper.

Cardiac electrophysiology benchmark

Niederer et al. (2011) proposed a benchmark problem that is now used by the in-silico modelling community to validate and compare cardiac electrophysiology solvers (Campos et al., 2016; Niederer et al., 2011). Ithildin passes the benchmark as implemented in Sim. 5. In the benchmark problem an excitation wave travels through a cuboid-shaped medium following the cell model by ten Tusscher & Panfilov (2006). In Fig. 11, we present the results from the benchmark in a similar way as in the original publication introducing the benchmark (Niederer et al., 2011): Point $P_1={{{{\left[ 0,0,0 \right]}}}^\mathrm{T}}$ is the corner of the cuboid where the stimulus is applied and $P_8={{{{\left[ 20,7,3 \right]}}}^\mathrm{T}}{\mathrm{{m}{m}}}$ the furthest-away opposite corner. Consider the plane going through $P_1$ and $P_8$ as well as $P_{10}={{{{\left[ 0,7,1.5 \right]}}}^\mathrm{T}}{\mathrm{{m}{m}}}$. We call the distance along the long axis of the plane slicing through the cubioid $\xi_1$ and the short axis $\xi_2$. In the panels (a), the LAT on this plane is shown for the benchmark simulation at high and low resolution in space, $\Delta x = {0.1~\mathrm{{m}{m}}}$ and $\Delta x = {0.5~\mathrm{{m}{m}}}$ respectively, at the same temporal resolution $\Delta t = {0.005~\mathrm{{m}{s}}}$. Panel (b) also shows the LAT on the line from $P_1$ to $P_8$ for the different spatial resolutions $\Delta x \in {{\left\{{0.1~\mathrm{{m}{m}}}, {0.2~\mathrm{{m}{m}}}, {0.5~\mathrm{{m}{m}}} \right\}}}$ at the same temporal resolution. In panel (c), the LAT value at $P_8$ is compared across all the combinations of spatial and temporal resolution. Just like for most other solvers which are compared in the benchmark paper, it can be seen that at the coarsest spatial resolution, the excitation wave is slowed down significantly in the transversal direction (Niederer et al., 2011). Similarly to the other finite-difference solvers, the simulation fails at $\Delta t = {0.05~\mathrm{{m}{m}}}$ and $\Delta x = {0.1~\mathrm{{m}{m}}}$ (Niederer et al., 2011), due to numerical instability. When a check of the CFL condition is enabled (Eq. 5), Ithildin suggests to lower $\Delta t$ to given this spatial resolution, leading to a simulation at which no instability is observed.

Panel (d) of Fig. 11 shows the speed-up in computation time from parallelization by comparing the computation time for Sim. 5 at $\Delta x = {0.2~\mathrm{{m}{m}}}$ and $\Delta t = {0.01~\mathrm{{m}{s}}}$ on an 8-core Intel i7-10875H processor using 1, 2, 4, and 8 processes. In the double-logarithmic plot, it can be seen with linear regression that the computational speed increases almost linearly going from one to four processes, $t_\text{sim} \propto N_\text{proc}^{-p}$ with $p \approx 1$. Going to eight processes leads to diminishing returns, resulting in a smaller speed-up in computational time, as the overhead due to the boundary-exchange between processes grows. On this system using eight processes, at $\Delta x = {0.2~\mathrm{{m}{m}}}$ and $\Delta t = {0.01~\mathrm{{m}{s}}}$, solving the benchmark problem takes ${65.733~\mathrm{s}}$ of computation time in Ithildin, while it takes ${611.231~\mathrm{s}}$ in cbcbeat (Rognes et al., 2017). For both codes, we have turned off output to the disk and included the initial setup of the problem, such as compilation and memory allocation.

Figure 11: Results of the cardiac electrophysiology benchmark. The benchmark was proposed by Niederer et al. (2011) and is implemented in Sim. 5. To ease comparisons with the paper introducing the benchmark, we present our results in a similar style. In panels (a)-(c), we present the LAT in the medium at different spatial and temporal resolutions. Coloring according to LAT is consistent across these panels. In panel (d), we measure the computational speed of Ithildin at different number of used processes. A linear fit is shown in the double-logarithmic plot which excludes the data point at $N_\text{proc} = 8$.

Discussion

The Ithildin framework is a tool for the simulation of cardiac electrophysiology. The code offers a number of assets that allow for simulations targeting a variety of phenomena. For example, there are many instances available to set the local anisotropy of the myocardium via the Geometry class. Please refer to the example presented in section 4.1 for further details. The Geometry class also allows for the simulation and filament tracking in a domain with an arbitrary number of spatial dimensions (Arno et al., 2021, 2024a, 2024b; Cloet et al., 2023; Kabus et al., 2022).

These include the ability to define inhomogeneities, which can be used to model domains of any shape and with any kind of obstacles. This is used in Sim. 1 and Sim. 4. Furthermore, the user can define a variety of stimulation protocols, of which some are showcased in the example simulations. We also note the following options: EGM calculation, the possibility of recording the temporal evolution of variables at sensor positions, and the Python module for post-processing and analysis.

It is evident that Ithildin represents but one instance of software designed for the simulation of cardiac electrophysiology. Another finite-differences solver is BeatBox (Antonioletti et al., 2017), which also relies on domain splitting. A non-exhaustive list of examples of established simulation software based on the finite element method (FEM) are Chaste (Cooper et al., 2020), openCARP (Plank et al., 2021), lifex-ep (Africa et al., 2023), cbcbeat (Rognes et al., 2017), GEMS (Arens et al., 2018), CEPS (CARMEN, 2024), and simcardems (Finsberg et al., 2023). Besides the methods of finite differences, elements, and volumes, approaches from computational fluid dynamics such as the lattice Boltzmann method may be used (Campos et al., 2016). Several of these packages have more advanced features than our software. In addition, some have broader applications than cardiac electrophysiology. More software can be found in (Niederer et al., 2011).

Limitations of our software are the fact that, in the context of cardiac electrophysiology, Ithildin only supports the monodomain model as it is a reaction-diffusion problem conforming with Eq. 2, and the fact that the domain decomposition happens solely in one coordinate direction. Further limitations are that Ithildin is fundamentally voxel-based and hence does not support tetrahedral meshes, and that only Neumann boundary conditions are implemented. Additionally, it is only maintained by a small research team. This paper serves to enhance the visibility of the software and to invite fellow cardiac modelers to use and contribute to the project. We believe that the GitLab environment is an effective medium for facilitating interaction between users, identifying issues and suggesting improvements.

Our work with Ithildin has demonstrated its ability to study complex wave patterns in cardiac tissue (Arno et al., 2021, 2024a, 2024b; Cloet et al., 2023; Kabus et al., 2022; Kabus, De Coster, et al., 2024), but it is important to acknowledge the limitations of the fully explicit numerical scheme used: High resolution in time and space may be required for numerical stability. Nevertheless, there are use cases where Ithildin is practical in the context of clinical applications. For example, it can be used to study rotor waves and their implications for cardiac function, which have important implications for diagnosing and treating cardiac diseases. Additionally, Ithildin can be used to develop new models or simulations that are tailored to specific experimental conditions or clinical scenarios.

Conclusion

In this work, we introduced Ithildin, an open-source library that allows for numerical simulation and analysis of rotor waves. We demonstrated the versatility of Ithildin through a series of simulations, including spiral break-up in the Smooth-Karma model, the S1S2 protocol in the AP96 model, and 2D and 3D spiral waves in the BOCF model in ventricular geometry.

Our simulations highlighted several key features of Ithildin, such as the different implemented geometries and reaction terms, inhomogeneities, and stimuli, as well as recording data such as the pseudo-EGM or filament trajectories. These findings contribute to the growing understanding of rotor waves in cardiac electrophysiology and have the potential to inform future experimental and theoretical studies.

Overall, our work demonstrates the power of Ithildin as a tool for studying complex wave patterns in cardiac tissue. We hope that this library will be useful to researchers seeking to better understand the dynamics of rotor waves and their implications for cardiac function.

Data availability

The source code of version 3.5.1 of the Ithildin software implemented for this paper is publicly available at https://gitlab.com/heartkor/ithildin and has been archived on Zenodo (DOI: 10.5281/zenodo.12799245). This archive also contains the data generated by the simulations used throughout this paper. Documentation of the code is publicly available at https://heartkor.gitlab.io/ithildin/.

Appendix

This section contains C++ code to run the four simulations that are used throughout this paper. An overview of their parameters is given in Table 4.

Sim. 1: Spiral break-up in the Smooth-Karma model

This is the source code to the Ithildin simulation used as the main example throughout this paper.

#include "ithildin.h"

int main(int argc, char** argv){
 Mpiclass mpi(argc, argv);

 // define model
 Model_SmooKa model1;

 // define model for obstacles
 Model_SmooKa model2;
 model2.hidden_eps *= 1.5;

 // rescale time and space
 const float scale_time = 2.4; // ms/t.u.
 const float scale_space = 0.273; // mm/s.u.
 ModelWrapper_RescaleTimeSpace model1a{
 &model1, scale_time, scale_space, 2,
 };
 ModelWrapper_RescaleTimeSpace model2a{
 &model2, scale_time, scale_space, 2,
 };

 // rescale u to voltage, such that: u is in range [0, 1]
 const float ucrit = 0.5;
 const float ustim = 0.75;
 ModelWrapper_RescaleVars model1b{&model1a, "u", 0.25, 0.};
 ModelWrapper_RescaleVars model2b{&model2a, "u", 0.25, 0.};

 // record diffusion term for EGMs
 ModelWrapper_RecordDiffusion model1c{&model1b};
 model1c.record("u");
 ModelWrapper_RecordDiffusion model2c{&model2b};
 model2c.record("u");

 // record local activation time
 ModelWrapper_RecordActivationTime model1d{&model1c};
 model1d.record("u", ucrit);
 ModelWrapper_RecordActivationTime model2d{&model2c};
 model2d.record("u", ucrit);

 // combine models
 Model_multi model{{&model1d, &model2d}};

 // geometry parameters
 const size_t Nx = 100;
 const size_t Ny = 70;
 const float dx = 0.2;
 const float dy = dx;
 const float Lx = Nx*dx;
 const float Ly = Ny*dy;

 // set up geometry
 vector<int> size{Nx, Ny, 1};
 vector<float> deltas{dx, dy, 1.};
 Geometry_Iso geom{&mpi, size, deltas, &model};
 geom.add_inhom(
 Shape::Sphere(0.25*Ly, {0.7*Lx, 0.3*Ly}), 2
 );
 geom.add_inhom(
 Shape::Rect({0.2*Lx, 0.7*Ly}, {0.3*Lx, 0.8*Ly}), 2
 );
 geom.add_inhom(Shape([&](vector<float> p) {
 return p[1] < 0.1*Ly*(1. + 0.7*sin(14.*p[0]/Ly));
 }, "waves"), 0);

 // set up simulation and timing
 const int Nframe = 100; // number of frames
 const float framedur = 20.; // time between frames
 const float sampledur = 2.; // sampling time for time traces
 Sim sim{framedur, Nframe, &mpi, &model, &geom, "everything", 0};

 // filament tracking
 model.define_tip("u", ucrit, "v", 0.95);
 sim.enable_filament_recording(0, sampledur);

 // recording temporal data at a specific location
 sim.add_history_point({0.5*Lx, 0.5*Ly});
 sim.sensorlag = sampledur;

 // record EGMs
 sim.add_egm_electrode({0.5*Lx, 0.5*Ly, 1}, sampledur, "diffusu");
 const float lambda = 0.3; // proportionality factor between \
 // intra- and extracellular conduction matrix
 const float beta = 1e5; // 1/m; area-to-volume ratio
 const float capacitance = 1e-2; // F/m²; specific cell \
 // membrane capacitance
 const float conductivity = 0.5; // S/m; lumped conductivity
 for(Egm* egm : sim.egm_collection) {
 egm->construct_kernel(lambda, beta, capacitance, conductivity);
 }

 // set up stimulus protocol
 Source sour(&model, &geom, &mpi, &sim);
 sour.set_val("u", ustim);

 // S0: initial stimulus
 sour.stimulate(Shape::Rect({0, 0}, {0.1*Lx, Ly}));

 // wait until S1S2 protocol starts
 const float timeS1 = 600.; // ms
 sour.schedule.emplace(timeS1, [&]() {
 // S1: first stimulus of S1S2
 sour.stimulate(Shape::Rect({0, 0}, {0.1*Lx, Ly}));

 // S2: second stimulus of S1S2
 sour.setupS2(Point::Phys(&geom, &mpi, 0.5*Lx, 0.9*Ly), 0, ucrit,
 [&]() -> void {
 sour.stimulate(Shape::Rect({0, 0}, {Lx, 0.5*Ly}));
 }
 );
 });

 return sim.run(&sour);
}

Sim. 2: 2D spiral wave in the AP96 model

This example illustrates the S1S2 protocol in an easy-to-interpret setting.

#include "ithildin.h"

int main(int argc, char** argv){
 Mpiclass mpi(argc, argv);

 Model_AP model0(0.15, 0.002, 8, 0.2, 0.3, 20);

 // rescale time and space [@aliev1996simple]
 const float scale_time = 12.9; // ms/t.u.
 const float scale_space = 1.; // mm/s.u.
 ModelWrapper_RescaleTimeSpace model{
 &model0, scale_time, scale_space, 2,
 };

 vector<int> size = {120, 120, 1};
 vector<float> dx = {1.0, 1.0, 1.0};
 float Lx = size[0]*dx[0];
 float Ly = size[1]*dx[1];
 Geometry_Iso geom{&mpi, size, dx, &model};

 float framedur = 1.*scale_time;
 int Nframe = 65;
 Sim sim{framedur, Nframe, &mpi, &model, &geom, "s1s2", 0};
 sim.enable_filament_recording(0, framedur);

 Source sour{&model, &geom, &mpi, &sim};
 sour.set_val("u", 0.75);

 // S1
 sour.stimulate(Shape::Rect({0, 0}, {0.1f*Lx, Ly}));

 // S2
 const Point sensor = Point::Phys(&geom, &mpi, 0.6f*Lx, 0.5f*Ly, 0);
 LOG("sensor", sensor.get_phys());
 sour.setupS2(sensor, 0, 0.5, [&]() -> void {
 sour.stimulate(Shape::Rect({0, 0}, {Lx, 0.2f*Ly}));
 });

 return sim.run(&sour);
}

Sim. 3: 2D spiral wave in the BOCF model

This simulation is used to get an initial state for a 3D simulation in ventricular geometry in the following example.

#include "ithildin.h"

int main(int argc, char** argv){
 Mpiclass mpi(argc, argv);

 // define model
 Model_BO modelbo{1};
 const float ucrit = 0.5;
 const float ustim = 1.0;

 // record local activation time
 ModelWrapper_RecordActivationTime model{&modelbo};
 model.record("u", ucrit);

 // set up geometry
 vector<int> size{450, 450, 1};
 vector<float> dx{0.3, 0.3, 1.};
 const float Lx = size[0]*dx[0];
 const float Ly = size[1]*dx[1];
 Geometry_Iso geom = Geometry_Iso(&mpi, size, dx, &model);

 // set up simulation and timing
 const float framedur = 20.;
 const int Nframe = 34;
 Sim sim{framedur, Nframe, &mpi, &model, &geom, "bocf2d", 0};

 // set up stimulus protocol
 Source sour(&model, &geom, &mpi, &sim);
 sour.set_val("u", ustim);

 // S1: first stimulus of S1S2
 sour.stimulate(Shape::Rect({0, 0}, {0.1*Lx, Ly}));

 // S2: second stimulus of S1S2
 const Point sensor = Point::Phys(&geom, &mpi, 0.75*Lx, 0.5*Ly);
 LOG("sensor", sensor.get_phys());
 sour.setupS2(sensor, 0, ucrit, [&]() -> void {
 sour.stimulate(Shape::Rect({0, 0}, {0.75*Lx, 0.75*Ly}));
 });

 return sim.run(&sour);
}

Sim. 4: 3D spiral wave in the BOCF model

This 3D spiral wave is stimulated by extending a 2D spiral wave and placing it on ventricular geometry.

#include "ithildin.h"

int main(int argc, char** argv){
 Mpiclass mpi(argc, argv);

 // choose model
 Model_BO modelbo{1};
 const float ucrit = 0.5;
 const float ustim = 1.0;

 // record local activation time
 ModelWrapper_RecordActivationTime model{&modelbo};
 model.record("u", ucrit);

 // set up geometry
 vector<float> Ds {1, 0.25, 0.25};
 vector<int> size {168, 208, 231};
 vector<float> deltas {0.43, 0.43, 0.5};
 Geometry_OrtAniso geom{
 size, deltas, "ventricles", &model, Ds, &mpi,
 };

 // set up simulation
 float framedur = 20.;
 int Nframe = 30;
 Sim sim{framedur, Nframe, &mpi, &model, &geom, "bocf3d", 0};

 // filament tracking
 sim.enable_filament_recording(0, framedur);

 // set up stimuli
 Source sour{&model, &geom, &mpi, &sim};

 // transform 2D simulation onto 3D ventricles, PD on RV
 NDArray<float> transform; transform.read("bocf2d3d.transform.npy");
 NDArray<float> vars = sour.read_from_stem("bocf2d_0", 34);
 const vector<size_t> shape{
 vars.get_shape(0), size[0], size[1], size[2],
 };
 vars = vars.interpolate(&mpi, shape,
 [&transform](std::vector<float> x){
 if(x.size() != 4) { ERROR("invalid shape"); }
 std::vector<float> z{1., x[1], x[2], x[3]};
 std::vector<float> y(4, 0);
 for(size_t i=0; i<4; i++) {
 for(size_t j=0; j<4; j++) {
 y[i] += transform(i,j)*z[j];
 }
 }
 return std::vector<float>{x[0], y[1], y[2], y[3]};
 }
 );
 sour.set_frame(vars);

 return sim.run(&sour);
}

Sim. 5: Cardiac electrophysiology benchmark

This is the benchmark problem as proposed by Niederer et al. (2011) using the cell model by ten Tusscher & Panfilov (2006).

#include "ithildin.h"

int main(int argc, char** argv){
 std::vector<float> arg = strings_to_floats(argc, argv);
 Mpiclass mpi(argc, argv);

 // arg 1: serial number
 const int serialnr = (arg.size() > 1) ? round(arg.at(1)) : 0;
 MASTER_INFO("serialnr", serialnr);

 // arg 2: discretization {0.1, 0.2, 0.5}
 const float dx = (arg.size() > 2) ? arg.at(2) : 0.2; // mm

 // arg 3: PDE time steps {0.005, 0.01, 0.05}
 const float dt = (arg.size() > 3) ? arg.at(3) : 0.05; // ms

 // define model
 Model_TP06 model_tp06;
 ModelWrapper_RecordActivationTime model{&model_tp06};
 model.record("V", 0);

 // constants
 const float C_m = 0.01; // µF/mm² (membrane capacitance)
 const float chi = 140; // 1/mm (surface-to-volume ratio)

 // diffusivities intra/extra longitudinal/transversal
 const float scale = 1 / (chi * C_m); // mm³/µF
 const float Dil = scale*0.17; // mm²/ms
 const float Dit = scale*0.019; // mm²/ms
 const float Del = scale*0.62; // mm²/ms
 const float Det = scale*0.24; // mm²/ms

 // diffusivities longitudinal/transversal
 const float Dl = Dil*Del / (Dil+Del); // mm²/ms
 const float Dt = Dit*Det / (Dit+Det); // mm²/ms

 // diffusivity tensor only scales, Pmat stores magnitude
 vector<float> D{1., Dt/Dl, Dt/Dl}; // 1
 model.set_Pmat(Dl); // mm²/ms

 // domain size
 const float Lx = 20; // mm
 const float Ly = 7; // mm
 const float Lz = 3; // mm

 // set up geometry
 vector<float> deltas{dx, dx, dx};
 vector<int> size{
 int(Lx/deltas[0])+1, int(Ly/deltas[1])+1, int(Lz/deltas[2])+1,
 };
 vector<float> tissue_angles{0., 0.};
 vector<float> diffpars{1, 1, 0};
 Geometry_OrtAniso geom{
 size, deltas, &model, tissue_angles, diffpars, D, &mpi,
 };

 // limit output
 for(size_t ivar=0; ivar<model.get_Nvar(); ivar++)
 { geom.set_bwritevar(ivar, 0); }
 geom.set_bwritevar(model.get_ivar("V"), 1);
 geom.set_bwritevar(model.get_ivar("latV"), 1);

 // set up simulation and timing
 const int Nframe = 15; // number of frames
 const float framedur = 10.; // time between frames in ms
 Sim sim{
 dt, framedur, Nframe, 0, &mpi, &model, &geom,
 "niederer2011benchmark", serialnr, false,
 };

 // set up stimulus
 Source sour(&model, &geom, &mpi, &sim);
 const float I_stim = 50; // mA/cm³ (stimulus current)
 const float i_stim = I_stim * scale; // A/F = V/s = mV/ms
 sour.set_val("V", i_stim); // mV/ms
 sour.stimulate({
 Shape::Rect({0, 0, 0}, {1.5, 1.5, 1.5}),
 1.0, // amplitude factor
 2.0, // ms; duration
 true, // flag: additive stimulus
 });

 return sim.run(&sour);
}

Addenda

Acknowledgments:

We are grateful to Daniël A. Pijnappels, Antoine A.F. de Vries and our other collaborators at the LUMC, as well as Louise Arno, Lore Leenknegt, and Nathan Dermul at KU Leuven in Kortrijk for useful feedback and discussions.

While no generative AI has been used to write the Ithildin source code, generative AI has been used by the authors to aid in the writing of the text, specifically the Large Language Model (LLM) implementations GitHub Copilot, ChatGPT using GPT-3.5 and GPT-4o, as well as Mistral 0.2 via the Ollama software. The authors confirm that they have followed the current ethical publishing practices of PLOS One.

Funding:

DK is supported by KU Leuven grant GPUL/20/012. MC is supported by KU Leuven grant STG/19/007 and FWO-Flanders fellowship, grant 11PMS24N. HD is supported by KU Leuven grant STG/19/007. OB is supported by Agence Nationale de la Recherche grant ANR-IHUA-04. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests:

The authors have declared that no competing interests exist.

Copyright:

© 2024 Kabus et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Author contributions:

DK: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing – original draft, Writing – review & editing, Visualization, Project administration. MC: Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing – original draft, Writing – review & editing, Visualization. CZ: Software. OB: Software. HD: Conceptualization, Software, Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition.

References

Ahrens, J., Geveci, B., Law, C., Hansen, C., & Johnson, C. (2005). ParaView: An end-user tool for large-data visualization. The Visualization Handbook, 717, 50038–50031. https://doi.org/10.1016/b978-012387582-2/50038-1

Aliev, R. R., & Panfilov, A. V. (1996). A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals, 7(3), 293–301. https://doi.org/10.1016/0960-0779(95)00089-5

Arno, L., Kabus, D., & Dierckx, H. (2024a). Analysis of complex excitation patterns using Feynman-like diagrams. Scientific Reports, 14(1), 28962. https://doi.org/10.1038/s41598-024-73544-z

Barkley, D. (1991). A model for fast computer simulation of waves in excitable media. Physica D: Nonlinear Phenomena, 49(1-2), 61–70. https://doi.org/10.1016/0167-2789(91)90194-e

Bishop, M. J., & Plank, G. (2011). Bidomain ECG simulations using an augmented monodomain model for the cardiac source. IEEE Transactions on Bio-Medical Engineering, 58(8). https://doi.org/10.1109/TBME.2011.2148718

Campos, J. O., Oliveira, R. S., dos Santos, R. W., & Rocha, B. M. (2016). Lattice boltzmann method for parallel simulations of cardiac electrophysiology using GPUs. Journal of Computational and Applied Mathematics, 295, 70–82. https://doi.org/10.1016/j.cam.2015.02.008

CARMEN. (2024). Cardiac ElectroPhysiology Simulator (CEPS). IHU Liryc, Inria. https://carmen.gitlabpages.inria.fr/ceps/

Cloet, M., Arno, L., Kabus, D., Van der Veken, J., Panfilov, A. V., & Dierckx, H. (2023). Scroll waves and filaments in excitable media of higher spatial dimension. Physical Review Letters, 131(20), 208401. https://doi.org/10.1103/PhysRevLett.131.208401

Courant, R., Friedrichs, K., & Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100(1), 32–74. https://doi.org/10.1007/BF01448839

Dierckx, H., Arens, S., Li, B.-W., Weise, L. D., & Panfilov, A. V. (2015). A theory for spiral wave drift in reaction-diffusion-mechanics systems. New Journal of Physics, 17(4), 043055. https://doi.org/10.1088/1367-2630/17/4/043055

Dierckx, H., Brisard, E., Verschelde, H., & Panfilov, A. V. (2013). Drift laws for spiral waves on curved anisotropic surfaces. Physical Review E, 88(1), 012908. https://doi.org/10.1103/PhysRevE.88.012908

döt Net, I., Müller, T., Antoniou, P., Aro, E., Smith, T., Evans, C. C., & Ben-Kiki, O. (2023). YAML ain’t markup language, revision 1.2.2. YAML Language Development Team. https://yaml.org/spec/1.2.2/

Euler, L. (1794). Institutiones calculi integralis (Vol. 4). Academia Imperialis Scientiarum.

FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466. https://doi.org/10.1016/s0006-3495(61)86902-6

Goryachev, A., & Kapral, R. (1996). Spiral waves in chaotic systems. Physical Review Letters, 76(10), 1619–1622. https://doi.org/10.1103/PhysRevLett.76.1619

Hren, R. (1996). A realistic model of the human ventricular myocardium: Application to the study of ectopic activation [PhD thesis, Dalhousie University]. https://dalspace.library.dal.ca/handle/10222/55139

Hunter, J. D. (2007). Matplotlib: A 2D graphics environment. Computing in Science & Engineering, 9(3), 90–95. https://doi.org/10.1109/MCSE.2007.55

ISO. (2011). ISO/IEC 14882:2011: Information technology — programming languages — C++ (Third, p. 1338). International Organization for Standardization.

ISO. (2014). ISO/IEC 14882:2014: Information technology — programming languages — C++ (Fourth, p. 1358). International Organization for Standardization.

ISO. (2017). ISO/IEC 14882:2017: Programming languages — C++ (Fifth, p. 1605). International Organization for Standardization.

ISO. (2020). ISO/IEC 14882:2020: Programming languages — C++ (Sixth, p. 1853). International Organization for Standardization.

ISO. (2023). ISO/IEC 14882:2023: Programming languages — C++ (Seventh, p. 2101). International Organization for Standardization.

Karma, A. (1993). Spiral breakup in model equations of action potential propagation in cardiac tissue. Physical Review Letters, 71(7), 1103. https://doi.org/10.1103/PhysRevLett.71.1103

Karma, A. (1994). Electrical alternans and spiral wave breakup in cardiac tissue. Chaos: An Interdisciplinary Journal of Nonlinear Science, 4(3), 461–472. https://doi.org/10.1063/1.166024

Kazantsev, V. B., Nekorkin, V. I., Binczak, S., & Bilbault, J. M. (2003). Spiking patterns emerging from wave instabilities in a one-dimensional neural lattice. Physical Review E, 68(1), 017201. https://doi.org/10.1103/PhysRevE.68.017201

Luo, C., & Rudy, Y. (1994). A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circulation Research, 74(6), 1071–1096. https://doi.org/10.1161/01.RES.74.6.1071

Mark, E. et al. (2007). Enhancements to the extensible data model and format (XDMF). 2007 DoD High Performance Computing Modernization Program Users Group Conference, 322–327. https://doi.org/10.1109/HPCMP-UGC.2007.30

Mermin, N. D. (1979). The topological theory of defects in ordered media. Reviews of Modern Physics, 51(3), 591–648. https://doi.org/10.1103/RevModPhys.51.591

Mitchell, C. (2003). A two-current model for the dynamics of cardiac membrane. Bulletin of Mathematical Biology, 65(5), 767–793. https://doi.org/10.1016/S0092-8240(03)00041-7

Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50(10), 2061–2070. https://doi.org/10.1109/jrproc.1962.288235

Niederer, S. A., Lumens, J., & Trayanova, N. A. (2019). Computational models in cardiology. Nature Reviews Cardiology, 16(2), 100–111. https://doi.org/10.1038/s41569-018-0104-y

Pravdin, S., Dierckx, H., Markhasin, V. S., & Panfilov, A. V. (2015). Drift of scroll wave filaments in an anisotropic model of the left ventricle of the human heart. BioMed Research International, 2015. https://doi.org/10.1155/2015/389830

Press, W. H. (2007). Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press.

Streeter, D. D., Spotnitz, H. M., Patel, D. P., Ross, J., & Sonnenblick, E. H. (1969). Fiber orientation in the canine left ventricle during diastole and systole. Circulation Research, 24(3), 339–347. https://doi.org/10.1161/01.RES.24.3.339

ten Tusscher, K. H. W. J., Hren, R., & Panfilov, A. V. (2007). Organization of ventricular fibrillation in the human heart. Circulation Research, 100(12). https://doi.org/10.1161/CIRCRESAHA.107.150730

Tolkien, J. R. R. (1954). The Lord of the Rings: The Fellowship of the Ring. Allen & Unwin.

van Heesch, D. (2023). Doxygen 1.9.7. Doxygen. https://www.doxygen.nl

Winfree, A. T. (1994). Electrical turbulence in three-dimensional heart muscle. Science, 266, 1003–1006. https://doi.org/10.1126/science.7973648

In this way, it is similar to its namesake in the Lord of the Rings, where Ithildin is an Elven substance that reveals a hidden gateway to another realm after a spell is cast (Tolkien, 1954). ↩︎
Marcotte & Grigoriev (2017); Byrne et al. (2015); Karma (1993); Karma (1994) ↩︎
Aliev & Panfilov (1996) ↩︎
Bueno-Orovio et al. (2008) ↩︎
Bueno-Orovio et al. (2008) ↩︎
ten Tusscher & Panfilov (2006); Niederer et al. (2019) ↩︎

Fast creation of data-driven low-order predictive cardiac tissue excitation models from recorded activation patterns

Wed, 03 Jan 2024 00:00:00 +0000

This article was previously published in Computers in Biology and Medicine 169 (2024) 107949 (Kabus, De Coster, et al., 2024) and is a chapter of my dissertation. The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Authors:

Desmond Kabus^1,2 (ORCID: 0000-0002-6965-5211)
Tim De Coster² (ORCID: 0000-0002-4942-9866)
Antoine A.F. de Vries² (ORCID: 0000-0002-4787-7155)
Daniël A. Pijnappels² (ORCID: 0000-0001-6731-4125)
Hans Dierckx¹ (ORCID: 0000-0003-0899-8082)

Institutions:

Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Etienne Sabbelaan 53, 8500 Kortrijk, Belgium
Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Albinusdreef 2, 2333 ZA Leiden, the Netherlands

Correspondence: h.dierckx@kuleuven.be

DOI: 10.1016/j.compbiomed.2024.107949

Keywords:

data-driven modelling
surrogate modelling
machine learning
digital twin
excitable media
cardiac electrophysiology
optical voltage mapping

Highlights:

Models of excitation waves can be created in minutes from experiment to fitting.
One variable in space and time is sufficient to create a working excitation model.
A polynomial can predict excitation waves based on useful extracted features.
Spiral waves in heart muscle tissue can be predicted from focal wave data.

Abstract:

Excitable systems give rise to important phenomena such as heat waves, epidemics and cardiac arrhythmias. Understanding, forecasting and controlling such systems requires reliable mathematical representations. For cardiac tissue, computational models are commonly generated in a reaction-diffusion framework based on detailed measurements of ionic currents in dedicated single-cell experiments.

Here, we show that recorded movies at the tissue-level of stochastic pacing in a single variable are sufficient to generate a mathematical model. Via exponentially weighed moving averages, we create additional state variables, and use simple polynomial regression in the augmented state space to quantify excitation wave dynamics. A spatial gradient-sensing term replaces the classical diffusion as it is more robust to noise. Our pipeline for model creation is demonstrated for an in-silico model and optical voltage mapping recordings of cultured human atrial myocytes and only takes a few minutes.

Our findings have the potential for widespread generation, use and on-the-fly refinement of personalised computer models for non-linear phenomena in biology and medicine, such as predictive cardiac digital twins.

Introduction

A wide variety of phenomena qualify as excitable systems, such as epidemics (Noble, 1974) or electrical excitation in the brain (Bressloff, 2014) or heart (Clayton et al., 2011). Reliable and accurate mathematical modelling of these systems is necessary to forecast their evolution in time. These predictions are useful for decision making to steer these systems to a desired state.

To diagnose and treat heart rhythm disorders, a common health burden, the scientific community is creating sophisticated models for different processes in the heart with the goal of building a digital twin of each patient’s heart, i.e., a full recreation of their heart in the computer (Niederer et al., 2019; Trayanova et al., 2020). One of them is the contraction of the heart, which is controlled by excitation waves travelling through the myocardium (Clayton et al., 2011). An in-silico model of cardiac tissue electrophysiology is a mathematical model that is able to describe the dynamics of the excitation patterns (Clayton et al., 2011). Typically, in-silico models for cardiac electrophysiology fall into one of two categories. On the one hand, there are the mathematical models that focus mostly on the overall dynamics in the tissue (Aliev & Panfilov, 1996; Barkley, 1991; Bueno-Orovio et al., 2008; Fenton & Karma, 1998; FitzHugh, 1961; Marcotte & Grigoriev, 2017; Nagumo et al., 1962). While most of these models have continuous state variables, there are also cellular automata where each cell can only be in one of finitely many states (Moe et al., 1964). On the other hand there are detailed models that aim to model each current relevant for the specific use case of the model across the cell membrane (Courtemanche et al., 1998; Majumder et al., 2016; Paci et al., 2013). Typically, these models are fit using in-vitro data obtained from patch-clamping of single cells, or from optical voltage or Ca²⁺ mapping of monolayers or of the epicardium. Capturing cell memory is also crucial in the modelling of cardiac tissue. This may for instance be done via ionic current gates opening and closing or exponentially decaying memory variables (Fox et al., 2002; Wei et al., 2015). A common property in the models focusing on the overall dynamics rather than the individual ion currents is that the state variables used in them often do not have an equivalent in the physical tissue. These internal, hidden states are not fit to actual data, instead they are often chosen as a means to an end.

A so-called surrogate model is a mathematically simpler model that is fit to an existing, more detailed ionic model. For instance, Bueno-Orovio et al. (2008) provide parameter sets for their model to act as a surrogate for the models by Priebe & Beuckelmann (1998) and ten Tusscher et al. (2004). Surrogate modelling can be used to create patient-specific models much more quickly, such that they can be used in a clinical setting after a relatively short amount of time, e.g. during a hospital visit (Gillette et al., 2021). The current models of cardiac tissue electrophysiology are typically not fit to actual recordings of the wave dynamics. We are in this work fitting models to such data in a controlled case, i.e., dense data from isotropic monolayers, which also enables us to directly compare forward simulation to experiment.

Machine learning has become a useful tool in cardiac electrophysiology. The eikonal equation has been solved using physics-informed neural networks, which means that the local arrival time of the excitation wave can be recovered well for the entire domain based on relatively few data points (Costabal et al., 2020). This method is limited, however, to only model excitation, not recovery and re-excitation. Physics-informed neural networks have also been used to solve the reaction-diffusion systems for Aliev & Panfilov (1996) dynamics, as well as recovering one or two parameters of this model from data (Martin et al., 2022). Reservoir computing approaches, such as echo-state networks, can also be used to predict the temporal evolution of cardiac action potentials (Shahi et al., 2022). None of these methods are yet used to create full standalone tissue models for cardiac electrophysiology in a data-driven way.

A recent step forward in in-vitro models for cardiac electrophysiology was the conditional immortalisation of human atrial myocytes (Harlaar et al., 2022). This resulted in cell lines allowing the generation of near infinite numbers of fully functional human atrial myocytes called hiAMs. These hiAMs overcome a major problem associated with the use of heart muscle cells of animals for in-vitro modelling, namely the existence of species specific differences in electrophysiological behaviour including differences in the action potential duration (APD) and shape, conduction velocity (CV), and wave propagation (Clayton et al., 2011; Harlaar et al., 2022; Joukar, 2021; Majumder et al., 2016; O’Hara & Rudy, 2012). No in-silico version of the electrophysiological behaviour of this cell line has yet been published.

In this work, we present a general method to create data-driven in-silico tissue models from just one recorded variable over two-dimensional space and time. The method is more generally applicable and faster than the conventional creation of new models. Our method requires only minimal assumptions and works by expanding the data to a multi-dimensional state space. Models created with our method are designed directly at the tissue scale, instead of needing extra steps transitioning from the cell kinetics to tissue dynamics. We both re-create an existing in-silico model with our method as a surrogate model, and use this method to build an in-silico tissue model for hiAMs only from optical voltage mapping (OVM) data. Note that our model creation pipeline is not specific to this cell line and may be applied to OVM data in general. The generated models are specialised to the tissues for which data is contained in the data set. However, since model generation is cheap and robust, models for different tissues can easily be created.

The steps of our method are outlined graphically in Fig. 1. In section 2, we describe how the data are acquired (section 2.1), how they are processed, expanded with additional state variables (section 2.2), statistically binned in multi-dimensional state space histograms (section 2.3), and how the data-driven models are finally fit to them (section 2.4). In section 3, we present results from the models that are generated with this method. Finally, in sections 4, 5, we wrap up by summarising the method and discussing its advantages and shortcomings, as well as giving an outlook on how the models fit by our method can be improved.

Figure 1: Outline of the model fitting procedure. In the first step, relevant features are extracted from the spatio-temporal data $u$. Then the state space is discretised and histogrammed. The value of the time-derivative $\dot u$ of the main variable is analysed statistically in this state space. A model can then be fit to the state space and used to predict the evolution of the medium.

Methods

Data acquisition

In this work, we study spatio-temporal data from two excitable media: an in-vitro data set from OVM of monolayers of cardiomyogenically differentiated hiAMs and an in-silico data set from finite-differences simulations solving the mono-domain equations (Clayton et al., 2011). In this section, we outline how these data are generated and processed.

Optical voltage mapping

To obtain a two-dimensional recording of the electrical activity of tissue as an in-vitro data set, we performed OVM of hiAM monolayers in 10- wells of six-well culture plates (Harlaar et al., 2022). The recorded data consist of one variable, i.e., the measured difference in intensity of the light emitted by the voltage-sensitive dye, over space and time. We have used a MiCAM05-Ultima camera by SciMedia with a resolution of 100x100 pixels. With the particular configuration of lenses, each square pixel has a size of $\Delta x = {250~\mathrm{{u}{m}}}$. For the recordings, we use the lowest supported duration between frames of $\Delta t = {1~\mathrm{{m}{s}}}$. We record ${61440}$ consecutive frames, which corresponds to slightly more than a minute and lines up nicely in the camera’s file format. For modelling purposes, we assume that the monolayer is approximately homogeneous and isotropic.

To excite the tissue, we place a bi-polar stimulation electrode with two poles with a diameter of and inter-electrode spacing of at the well’s boundary, and stimulate with $N_\text{pulse}=200$ current pulses of random amplitude, random duration, and random timing. We call this procedure the stochastic burst pacing (SBP) protocol, inspired by stochastic pacing experiments done by Krogh-Madsen et al. (2005). The amplitudes $A_n$ are sampled from a uniform distribution $\operatorname{U}{{\left( {-0.8~\mathrm{{m}{A}}}, {0.8~\mathrm{{m}{A}}} \right)}}$. The times $T_n = T_{\text{u}} + T_{\text{e}}$ between subsequent pulses are sampled from a combination of a uniform and exponential distribution: $$ \begin{aligned} T_{\text{u}} &\sim \operatorname{U}{{\left( {0~\mathrm{{m}{s}}}, {50~\mathrm{{m}{s}}} \right)}} \\ T_{\text{e}} &\sim \operatorname{Exp}{{\left( 1/{200~\mathrm{{m}{s}}} \right)}} \end{aligned} \qquad{(1)}$$ where $\operatorname{U}{{\left( a, b \right)}}$ denotes the uniform distribution on the interval ${{\left[ a, b \right]}}$ and $\operatorname{Exp}{{\left( \lambda \right)}}$ the exponential distribution with decay rate $\lambda$: $$ \operatorname{Exp}{{\left( \lambda \right)}}{{\left( t \right)}} \propto \begin{cases} {{\mathrm{e}^{-\lambda t}}} & t \ge 0 \\ 0 & t < 0 \end{cases} \qquad{(2)}$$

We choose this probability distribution of durations such that, on the one hand, it covers a typical range of values, but on the other, occasionally, durations are sampled that are much longer than usual.

The durations $\tau_n = \tau_{\text{u}} + \tau_{\text{e}}$ of stimulation are sampled from a similar probability distribution: $$ \begin{aligned} \tau_{\text{u}} &\sim \operatorname{U}{{\left( {0~\mathrm{{m}{s}}}, {10~\mathrm{{m}{s}}} \right)}} \\ \tau_{\text{e}} &\sim \operatorname{Exp}{{\left( 1/{40~\mathrm{{m}{s}}} \right)}} \end{aligned} \qquad{(3)}$$

The stimulus current then takes the form: $$ I_\text{stim}{{\left( t \right)}} = \sum_{n=1}^{N_\text{pulse}} A_n {{\left[ \operatorname{H}{{\left( t - t_n \right)}} - \operatorname{H}{{\left( t - t_n - \tau_n \right)}} \right]}} \qquad{(4)}$$ with the start time of the $n$-th pulse: $$ t_n = \sum_{m=1}^{n} {{\left[ T_{m} + \tau_{m} \right]}} \qquad{(5)}$$ and the Heaviside function: $$ \operatorname{H}{{\left( t \right)}} = \begin{cases} 0 & t < 0 \\ \frac{1}{2} & t = 0 \\ 1 & t > 0 \end{cases} \qquad{(6)}$$

We preprocess these data in a similar way as in a previous publication (Kabus et al., 2022) by Gaussian blurring with a kernel size of three grid points and rescale the data to zero-mean and unit variance.

To bring the resting state close to zero, we subtract the baseline of the data. As baseline, we take the most common recorded intensity in each 4x4 pixel neighbourhood over a time span of This is based on the assumption that this most common value corresponds to the resting state.

The final preprocessing step for this data set is to smooth the data using an exponential moving average (EMA) with $\alpha = 0.1$, which, for time $t \ge 0$ with frame duration $\Delta t$ and the signal over time $s{{\left( t \right)}}$ at each position in space, is iteratively defined as: $$ \tilde s{{\left( t \right)}} = \begin{cases} \alpha s{{\left( t \right)}} + {{\left[ 1 - \alpha \right]}} \tilde s{{\left( t - \Delta t \right)}} & t > 0 \\ s{{\left( t=0 \right)}} & t = 0 \end{cases} \qquad{(7)}$$ which, within first-order time discretisation, is equivalent to: $$ \tilde s{{\left( t \right)}} = {{\mathrm{e}^{-a t}}} s{{\left( 0 \right)}} + a \int_0^t {{\mathrm{e}^{a {{\left[ t' - t \right]}}}}} s{{\left( t' \right)}} \;\mathrm{d} t' \qquad{(8)}$$ with $a = \frac{\alpha}{\Delta t} > 0$.

We refer to this recording as the OVM data set and $u{{\left( t, {{\bm{{x}}}} \right)}}$ denotes the unit-less signal after preprocessing scaled such that the resting state is close to $u=0$ and the excited state around $u=1$. This signal $u$ corresponds to the electrical activity of the cells, i.e., it is linked to the transmembrane voltage over space and time.

Only for qualitative comparisons after a data-driven model is fit to the OVM data set, a second OVM recording is used. In this second recording, several minutes after rotor formation through burst pacing, the resulting single, stable spiral wave is observed. This experiment is also performed with hiAMs in the same six-well format as before and at the same spatial resolution but at a lower temporal resolution of between frames. We preprocess these data in the same way as the main OVM data set.

Numerical simulation

To obtain a fully controllable testing data set, we ran an in-silico experiment similar to the in-vitro experiment in section 2.1.1. We use the finite-differences method to solve the reaction-diffusion system using a rescaled version of the minimalist two-variable tissue model by Aliev & Panfilov (1996), or in short the AP96 model, whose equations we will briefly summarise (Eq. 11).

As the diffusivity is proportional to the square of the CV, we set it such that the medium has a measured CV close to that of hiAMs (Clayton et al., 2011; Harlaar et al., 2022): $$ \begin{aligned} D &= {0.0625~\mathrm{{m}{m}{^2}{/}{m}{s}}} \\ \Rightarrow \text{CV} &= {0.17~\mathrm{{m}{/}{s}}} \end{aligned} \qquad{(9)}$$ We rescale the entire model over time such that its APD has a value of roughly the APD of hiAMs (Harlaar et al., 2022): $$ r = \frac{\text{APD}_\text{AP96}}{\text{APD}_\text{hiAM}} = \frac{{25.58}}{{100~\mathrm{{m}{s}}}} = {0.2558~\mathrm{\frac{1}{{m}{s}}}} \qquad{(10)}$$

The time derivatives are rescaled by this time scaling factor $r$ (Aliev & Panfilov, 1996): $$ \begin{aligned} \partial_t u &= \nabla \cdot D \nabla u - r k u {{\left[ u - a \right]}} {{\left[ u - 1 \right]}} - r u v + \frac{i_\text{stim}}{C} \\ \partial_t v &= - r \epsilon v - r \epsilon k u {{\left[ u - a - 1 \right]}} \\ \epsilon &= \epsilon_0 + \frac{\mu_1 v}{u + \mu_2} \end{aligned} \qquad{(11)}$$

By doing so, we introduced the millimetre as the spatial unit and the millisecond for time into the AP96 model, which uses arbitrary space and time units. The variables $u{{\left( t, {{\bm{{x}}}} \right)}}$ and $v{{\left( t, {{\bm{{x}}}} \right)}}$ remain unit-less. The value $u=0$ corresponds to the resting state and when excited, $u$ approaches the value $u=1$.

For all other parameters, $a$, $\epsilon_0$, $k$, $\mu_1$, and $\mu_2$, which are all unitless, we use the default values, as in the original publication by Aliev & Panfilov (1996). An overview of all parameters can be found in Table 1.

Parameter	Value
$a$	0.15
$\epsilon_0$	0.002
$k$	8
$\mu_1$	0.2
$\mu_2$	0.3
$D$
$r$	${0.2558~\mathrm{{m}{s}}}$

Table 1: Parameters of the rescaled AP96 model.

We have run a simulation over ${43104}$ frames at $\Delta t = {1~\mathrm{{m}{s}}}$ per frame at the same resolution as for the in-vitro OVM data and with the same SBP protocol. The length of the simulation was chosen such that the tissue can fully recover after the last of the 200 pulses of the SBP protocol. The electrode was placed off-centre at the side of one edge of the medium. For the time integration, we used a simple forward Euler step with numerical time step ${0.1~\mathrm{{m}{s}}}$ and a second-order central finite-differences five-point stencil for the Laplacian $\nabla \cdot D \nabla$. The square simulated tissue was homogeneous and isotropic with no-flux boundary conditions.

We refer to this synthetic recording as the AP96 data set. We only include the variable $u{{\left( t, {{\bm{{x}}}} \right)}}$ in the data set, as in the in-vitro data set, i.e., the OVM data set. For surrogate model fitting, we hence also only use one variable related to the transmembrane voltage of the tissue and not the internal state of the cell, which in the AP96 model is encapsulated in the second variable $v{{\left( t, {{\bm{{x}}}} \right)}}$.

In the subsequent steps, only these data will be used, partially as training data and partially as testing data. However, we also generate auxiliary data sets for the AP96 model that are only used for qualitative comparisons with the data-driven models after fitting.

To generate a recording of a rotor using the AP96 model, we ran another two-dimensional simulation at the same resolution in space and time with initial conditions that led to a spiral wave: The transmembrane voltage was set to a value of $u = {0.95}$ within the rectangle ${{\left[ 0, {0.5}L_x \right]}}\times{{\left[ {0.3}L_y, {0.5}L_y \right]}}$, and the second variable to $v = 1$ within a second rectangle ${{\left[ 0, {0.5}L_x \right]}}\times{{\left[ {0.4}L_y, {0.6}L_y \right]}}$, where $L_x = L_y = {100~\mathrm{{m}{m}}}$. Gaussian blurring with a radius of four grid lengths was applied to avoid steep gradients at the edge of the rectangles.

We also measured restitution curves, i.e., APD and CV as functions of the cycle length (CL) of stimulation. For this, we ran a one-dimensional cable simulation of the model with the same spatial and temporal resolution. At the left end of the cable, we stimulated by instantaneously setting $u$ to a value above the excitation threshold to trigger a wave travelling to the right. We repeatedly stimulated starting at a CL of decreasing in steps of until we reached a CL where not every stimulus resulted in a pulse that traveled through the entire cable, i.e., where 2:1 conduction block begins.

State space augmentation

The multi-dimensional space spanned by all variables describing the state of the modelled medium, ${{\bm{{u}}}} = {{{{\left[ u, v, ... \right]}}}^\mathrm{T}}$, is called state space in contrast to the physical space spanned by ${{\bm{{x}}}}$ and time $t$.

An in-silico tissue model usually prescribes a unique reaction ${{\bm{{R}}}}$ at each position ${{\bm{{u}}}}$ in the state space. We write a more generic version of Eq. 11, the mono-domain description (Clayton et al., 2011): $$ \begin{aligned} \partial_t {{\bm{{u}}}} &= {{\bm{{P}}}} \nabla \cdot {{\bm{{D}}}} \nabla {{\bm{{u}}}} + {{\bm{{R}}}} {{\left( {{\bm{{u}}}} \right)}} \end{aligned} \qquad{(12)}$$ with the diffusivity matrix ${{\bm{{D}}}}$, where ${{\bm{{P}}}}$ is a diagonal matrix describing which variables in ${{\bm{{u}}}}$ are to be diffused.

For the two-variable AP96 model, with ${{\bm{{u}}}} = {{{{\left[ u, v \right]}}}^\mathrm{T}}$ and ${{\bm{{P}}}} = \operatorname{diag}{{\left( 1, 0 \right)}}$, the state space of the AP96 model is shown in Fig. 2, coloured only by the component in $u$ of the reaction term ${{\bm{{R}}}}$. It can be seen that the second variable $v$ can be used to distinguish cases with positive and negative reaction at the same value of $u$. A model that can describe both excitation from a resting state and recovery to it, is only possible with at least two variables, such that the inertial manifold can be a loop in the state space.

Figure 2: Reaction in state space of the AP96 model. The reaction term ${{\bm{{R}}}}$ is a vector field defining a reaction at each point ${{\bm{{u}}}}$ in the state space. Colouring is according to the $u$-component of the reaction term $R_u$. The trajectory through the state space from excitation of one point is shown as an example.

In both types of data sets considered in this work, we only have data for one variable available, namely $u$. To still be able to differentiate between excitation and recovery, we extract additional information from $u$ over space and time to use as additional states for the state space, so-called augmentations. A well-designed augmented state space unfolds the data such that cells in the tissue with different histories are identified by different augmented state vectors.

As we want to find models that can be used as forward models in a similar way as in Eq. 12, we impose two more requirements on the states with which we augment the state space:

Augmentations should be local in space, i.e., they should only use states within a neighbourhood around each point.
Causality must be respected, i.e., only data from previous points in time and the current point in time may be used to compute each augmentation.

The two augmentations that we found to be most useful for model creation are presented in the remainder of this section.

Exponential moving averages

The goal of the state space augmentation is to capture that the model can evolve differently from the same value of $u$. This value may increase at the front of waves, during excitation, or decrease at the wave backs, during recovery.

One way to sense excitation and recovery is by expanding the state space with augmented states that capture the passage of time. A naïve way to do so may be to just include a time-delayed signal, which is a useful augmentation in phase mapping (Gray et al., 1995). This is, however, not very robust to noise, as noise in the past may directly impact the present. Another option may be to use Hilbert transforms, but as their computation involves an integral over all of time, they violate causality and are therefore not meeting our requirements (Kuklik et al., 2014).

The EMA (Eq. 7) is robust to noise and typically lags behind the signal by an amount of time inversely proportional to the constant $\alpha$. It can intuitively be used to differentiate between excitation and recovery for signals $u$ in-between the resting state and excited state: If an EMA with a suitable scale $\alpha$ is lower than $u$, it is likely that the medium is currently getting excited; if it is higher, the cells are likely recovering.

Multiple EMAs can be used to be able to capture more advanced behaviour: With an appropriately chosen $\alpha$, we can differentiate between states where the tissue is activated a lot and where it is mostly resting. With such an EMA, restitution characteristics can be captured.

Here, we augment the AP96 data with a single EMA $\tilde u$ with $\alpha = {0.03}$. For $\tilde u$ in the OVM data, we use $\alpha = {0.008}$.

Standard deviation as a proxy for the absolute gradient

We also conjecture that information about the slopes and higher derivatives in space of $u$ is useful for a well fitting data-driven model. In the reaction-diffusion description, the Laplacian is used to capture curvature effects in space. Because it is a second-order derivative, it is very sensitive to noise. The absolute value of the gradient in combination with other state space variables can also encode much information about the slopes of $u$. We use a proxy for the absolute gradient in a neighbourhood with high robustness with respect to noise. Such a proxy is the standard deviation (SD) of the signal $u$ in the local neighbourhood up to a radius $R$ around ${{\bm{{x}}}}$.

It can be shown analytically that the SD is proportional to the absolute gradient of a plane $h{{\left( {{\bm{{x}}}} \right)}} = {{{{\bm{{w}}}}}^\mathrm{T}} {{\bm{{x}}}} + b$, which evaluates to ${{\left\lVert \nabla h \right\rVert}} = {{\left\lVert {{\bm{{w}}}} \right\rVert}}$. The SD in a spherical neighbourhood $\mathcal N$ up to a radius $R$ around the origin can be found via integration in polar coordinates $r$, ${\varphi}$: $$ \begin{aligned} \sigma_h^2 &= {{\left\langle {{\left[ h-b \right]}}^2 \right\rangle}} = \frac {1}{\int_{\mathcal N} \mathrm{d} {{\bm{{x}}}}\;} \int_{\mathcal N} \mathrm{d} {{\bm{{x}}}}\; {{\left[ {{{{\bm{{w}}}}}^\mathrm{T}} {{\bm{{x}}}} \right]}}^2 \\ \sigma_h^2 &= \frac {1}{\pi R^2} \int_{0}^{R} \mathrm{d} r\; \int_{0}^{2\pi} \mathrm{d} {\varphi}\; {{\left[ w_x r\cos{\varphi}+ w_y r\sin{\varphi}\right]}}^2 \\ \sigma_h &= \frac {R}{2} {{\left\lVert {{\bm{{w}}}} \right\rVert}} \end{aligned} \qquad{(13)}$$ Analogously, a similar relation can be shown in three dimensions: $$ \sigma_h = \frac{R}{\sqrt{5}} {{\left\lVert {{\bm{{w}}}} \right\rVert}} \qquad{(14)}$$

Assuming that the signal $u$ varies slow enough that a linear function can approximate it well within the chosen neighbourhood, the SD can be used to compute its absolute gradient. Despite the relatively high time-derivative $\dot u$ of the cardiac action potential during its upstroke, and hence $\nabla u$, we can make this assumption if the resolution is sufficiently high.

Note that the SD has the useful property, that it can still be computed at or near the boundary of the medium, just based on fewer samples than for fully interior points.

To capture gradient information, we augment the AP96 data set with an SD variable for the neighbourhood up to a radius $R = 5 \Delta x$ around each point ${{\bm{{x}}}}$ and denote it by $g$. For $g$ in the OVM data set, we use a larger neighbourhood with $R = 8 \Delta x$ for the SD calculation.

State space histograms

In the usual dynamics of the excitable systems we study in this work, not all points ${{\bm{{u}}}}$ in the state space are equally likely. Some state vectors ${{\bm{{u}}}}$ may be quite common as being part of the inertial manifold, while others might be completely absent in the data.

For example, for the AP96 model with ${{\bm{{u}}}} = {{{{\left[ u, v \right]}}}^\mathrm{T}}$, it can be seen in Fig. 3 that a part of the state space is not visited and most of the samples are on the inertial manifold, a closed approximately one-dimensional loop. We conjecture that in higher dimensions the state space is even more sparse, e.g. in case of detailed ionic models, some of which contain more than ten state variables.

Figure 3: Likelihood of each point in the state space of the AP96 model for the numerical simulation generating the data set used for surrogate model fitting.

With the chosen state space expansion (section 2.2), the state vector is ${{\bm{{u}}}} = {{{{\left[ u, \tilde u, g \right]}}}^\mathrm{T}}$. We discretise the state space to 100 bins in each dimension and collect the data in a multi-dimensional histogram. This results in three-dimensional state space histograms for the two data sets with ${10}^6$ bins each. For efficient memory usage, these histograms are computationally implemented using multi-dimensional sparse arrays. Regions in the state space containing no data do not take up memory in this way. We can then fit the models to the average value inside each bin instead of to all data points. This efficient memory management enables building even higher-dimensional state spaces, though in this work we require only three variables.

The time-derivative $\dot u$ fully defines the reaction of the data-driven models developed in this work. The reason for this is that all other variables in the state vector ${{\bm{{u}}}}$ can be derived from the one variable $u$ over space and time.

In a well chosen state space, for each state vector ${{\bm{{u}}}}$, the reaction $\dot u$ is the same, i.e., the states are well separated. Therefore, we calculate the mean value of $\dot u$ computed using finite-differences at each bin in the discretised state space, as well as its SD. This mean value of $\dot u$ takes a similar role as $R_u$ for the AP96 model and “colours” the state space in a comparable way to $R_u$ in Fig. 2.

Model fitting

A data-driven model will be fit to the data in state space as a scalar-valued function $f$: $$ f : {{\bm{{u}}}} \mapsto \dot u \qquad{(15)}$$ such that the model consists of the state space expansion ${{\bm{{u}}}}$ by calculating the EMA (Eq. 7), the SD within the neighbourhoods around each point (section 2.2.2), as well as the forward Euler step: $$ u{{\left( t+\Delta t, {{\bm{{x}}}} \right)}} = u{{\left( t, {{\bm{{x}}}} \right)}} + \Delta t f{{\left( {{\bm{{u}}}}{{\left( t, {{\bm{{x}}}} \right)}} \right)}} \qquad{(16)}$$

As the model function $f$, we choose a low-order polynomial which is linear with respect to each of the components: $$ f{\left({{\bm{{u}}}}\right)} = w_{\tilde{u} g u} \tilde{u} g u + w_{\tilde{u} g} \tilde{u} g + w_{\tilde{u} u} \tilde{u} u + w_{\tilde{u}} \tilde{u} + w_{g u} g u + w_{g} g + w_{u} u + w \qquad{(17)}$$

Assuming that the SD $g$ approximates the absolute gradient ${{\left\lVert \nabla u \right\rVert}}$ of the main state variable $u$ well, we hence essentially describe the propagation of the activation waves using advection, instead of diffusion (Eq. 12): $$\begin{aligned} \partial_t u &= f{{\left( {{\bm{{u}}}} \right)}} = f_0{{\left( u, \tilde u \right)}} + c_1 f_1{{\left( u, \tilde u \right)}} {{\left\lVert \nabla u \right\rVert}} \\ f_0{{\left( u, \tilde u \right)}} &= w + w_u u + w_{\tilde u} \tilde u + w_{\tilde u u} \tilde u u \\ f_1{{\left( u, \tilde u \right)}} &= w_{g} + w_{g u} u + w_{\tilde u g} \tilde u + w_{\tilde u g u} \tilde u u \\ \tilde u{{\left( t \right)}} &= {{\mathrm{e}^{-a t}}} u{{\left( 0 \right)}} + a \int_0^t {{\mathrm{e}^{a {{\left[ t' - t \right]}}}}} u{{\left( t' \right)}} \;\mathrm{d} t' \end{aligned}\qquad{(18)}$$ with real-valued scalar constants $a, c_1, w_{...}$ and $g \approx c_1 {{\left\lVert \nabla u \right\rVert}}$. For an appropriate choice of parameters, obtained for instance through fitting, Eq. 18 describe a model of excitation in an advection-regression framework.

To prevent the fit model from leaving the valid state space, after each forward Euler step, we clip the predicted $u$ to the range ${{\left[ u_{\min},u_{\max} \right]}}$ for which we have training data, the so-called valid range. Note that this also limits the EMA $\tilde u$ to that range, while the SD $g$ can still leave the range observed in the data, i.e., for larger than typical slopes.

In each step forward in time, $t \leftarrow t + \Delta t$, we update the state vector ${{\bm{{u}}}} = {{{{\left[ u, \tilde u, g \right]}}}^\mathrm{T}}$ of the data-driven model according to the following pseudocode:

Compute the new value of $u$ at each point ${{\bm{{x}}}}$ using the polynomial $f$ (Eq. 16; Eq. 17): $$ \begin{aligned} &u \leftarrow u + \Delta t f{{\left( {{\bm{{u}}}} \right)}} \\ &u \leftarrow \max{{\left( u_{\min}, \min{{\left( u, u_{\max} \right)}} \right)}} \end{aligned} \qquad{(19)}$$
Update the EMA $\tilde u$ based on its old value and the updated $u$ (Eq. 7): $$ \tilde u \leftarrow \alpha u + {{\left[ 1 - \alpha \right]}} \, \tilde u \qquad{(20)}$$
Calculate the SD $g$ of the updated $u$ (section 2.2.2): $$ g \leftarrow {{\left\langle u^2 - {{\left\langle u \right\rangle}}^2 \right\rangle}}^{\frac{1}{2}} \qquad{(21)}$$ where ${{\left\langle \bullet \right\rangle}}$ denotes the average of all points in the spherical neighbourhood $\mathcal N$ of each position ${{\bm{{x}}}}$ up to a radius $R$.

To find the optimal model function $f$, we use the least-squares algorithm to find the polynomial of the given order which minimises the loss: $$ \mathcal L{{\left( f \right)}} = \sum_{{{\bm{{u}}}} \in \mathcal U} N{{\left( {{\bm{{u}}}} \right)}} {{\left[ \dot u{{\left( {{\bm{{u}}}} \right)}} - f{{\left( {{\bm{{u}}}} \right)}} \right]}}^2 + \sum_{{{\bm{{u}}}} \in \mathcal R} f^2{{\left( {{\bm{{u}}}} \right)}} \qquad{(22)}$$

where $\mathcal U$ denotes the centres of all bins in the discretised state space containing at least one sample, i.e., one data point, and $N{{\left( {{\bm{{u}}}} \right)}}$ is the number of samples in the bin ${{\bm{{u}}}}$. To regularise the model space in favour of more physically realistic models, we add additional co-location points $\mathcal R$ outside of $\mathcal U$ at which we assume $f$ should be close to zero. We choose these co-location points $\mathcal R$ by Latin hypercube sampling and only keep points for which no data is contained in the data set $\mathcal U$ (McKay et al., 2000). We choose the number $N_{\mathcal R}$ of co-location points $\mathcal R$ based on the number $N_{\mathcal U}$ of non-empty bins in the data set $\mathcal U$: $$ N_{\mathcal R} = {0.5} \, N_{\mathcal U} \qquad{(23)}$$

As training data spanning the state space $\mathcal U$ that the model is fit to, we use all frames up to just before the last pulse of the SBP in each recording. The frames of the last pulse of each recording are then used as testing data, i.e., as a reference to compare with the predicted frames of the data-driven model. We only include points at least 20 grid points away from the boundary to minimise boundary effects on the used data.

Besides this first simulation that can be directly and quantitatively compared to the reference for both data sets, we also run simulations using the data-driven models to compare them qualitatively with the reference model, i.e., the AP96 model for the synthetic data set, and the in-vitro model for the OVM data. As the second simulation, for both data-driven models, we run a similar rectangle-based protocol to the one used to stimulate spirals for the reference AP96 model (section 2.1.2). Here, we use the same stimulation values for $u$, but $\tilde u = 0.6$ inside the second rectangle and $\tilde u = 0.1$ outside of it. Thirdly, we also run a one-dimensional cable simulation to obtain APD and CV restitution curves for the fit surrogate AP96 model with the same restitution stimulation protocol as outlined for the reference AP96 model in section 2.1.2

Results

In this section, we present how the state space expansion and data-driven models as described in section 2, deal with the two data sets. Since the results for the in-silico synthetic data are easier to interpret, we begin with those data.

Surrogate model for synthetic data

State space

For the data set from numerical simulation of the AP96 model (cf. section 2.1.2), the three-dimensional state space spanned by ${{\bm{{u}}}} = {{{{\left[ u, \tilde u, g \right]}}}^\mathrm{T}}$ is quite sparse. ${95695}$ bins in this space contain at least one data point, which corresponds to ${9~\mathrm{\%}}$ of all bins. Of these bins, ${40~\mathrm{\%}}$ contain at least 10 data points. Non-empty bins contain ${230}$ data points on average.

In Fig. 4, we present a three-dimensional view of the state space and two different two-dimensional projections of the state space; to $u$ and $g$, and to $u$ and $\tilde u$, respectively. Panel A shows that most data points lay on a loop-like inertial manifold, while a large volume of the state space is still covered by data points (panel B). Colouring by the mean of $\dot u$, we can see that the two halves of the loop nicely distinguish between excitation and recovery, see the blue and red shades in panel C. Note that the colouring of the state space in panel D corresponds to the SD $\sigma$ and not to the SD $g$. To be more specific, this panel is coloured by the SD $\sigma$ of the samples of the time-derivative $\dot u$ in the bins, instead of the SD $g$ of the signal $u$ in the neighbourhoods around each point ${{\bm{{x}}}}$, which is one of the axes spanning the state space in Fig. 4. In large parts of these projections, the SD $\sigma$ is much lower than the mean absolute value of $\dot u$, showing that the model’s possible states are well-separated in the chosen state space ${{\bm{{u}}}}$.

Figure 4: Visualisation of the three-dimensional state space for the AP96 data set and projections onto two dimensions. (A): By colouring the space by the number $N$ of points in each bin, it can be seen that most points lay on a loop. (B): Colouring by $\log_{10}{N}$, we can see that still a large volume of the space is covered by data. (C): Looking at the mean of $\dot u$ at each bin in this projection, we can see that in one side of the loop, $u$ increases while it decreases at the other side. (D): The standard deviation of $\dot u$ is indicative of whether the states have been chosen well.

Forward model

A model in the context of this work consists of a “colouring” of the entire three-dimensional state space representing the $\dot u$ at each possible state ${{\bm{{u}}}}$.

The data-driven surrogate model $f$ is fit to the three-dimensional state space as outlined in section 2.4 resulting in: $$ \begin{aligned} f{\left({{\bm{{u}}}}\right)} =& - {0.22407}\, \tilde{u} g u - {0.82589}\, \tilde{u} g + {0.0054601}\, \tilde{u} u + {0.021655}\, \tilde{u} \\& + {0.31082}\, g u + {0.23077}\, g - {0.0081478}\, u - {0.0097110} \end{aligned} \qquad{(24)}$$

Subsequently, a forward simulation is performed with the model, leading to a prediction of $u$ based on the history until that point which can be compared to the reference evolution of $u$ in Fig. 5. This simple model displays excitation and recovery, as well as propagation of the wave. The predicted evolution is similar to its reference: The values for the plateau-phase and resting state are close to the reference values. The depolarisation at the wave front is a bit too late, leading to predicted values that are too low. Likewise, the repolarisation at the wave back is too late, such that there the predicted values are too high.

Figure 5: Prediction of the surrogate model over time compared to the reference solution for the AP96 data set. The difference between the prediction and reference is displayed in the bottom row.

Using the method by Bayly et al. (1998), CV has been measured to be ${(0.1403\ ±\ 0.0036)~\mathrm{{m}{/}{s}}}$ for the prediction, and ${(0.1621\ ±\ 0.0065)~\mathrm{{m}{/}{s}}}$ for the reference. This lower propagation velocity leads to the observed delay of wave front and back.

We also computed the average APD of the activity in the central 60x60 pixels of the frame with the polarisation threshold of $u = 0.5$: For the prediction, we measure an APD of ${(56.68\ ±\ 0.48)~\mathrm{{m}{s}}}$, which is slightly longer than the reference of ${(47.25\ ±\ 0.44)~\mathrm{{m}{s}}}$.

At the boundary of the domain, the method is still able to compute the evolution well despite less neighbouring data points being used for the calculation of the SD $g$.

Prediction of a spiral wave

Using similar initial conditions for the data-driven surrogate AP96 model and its reference, we have run simulations leading to spiral waves. In Fig. 6, the resulting evolutions of $u$ over space and time are juxtaposed. It can be seen that the overall wave dynamics are similar, despite the data-driven model only being fit to data from SBP focal waves, i.e., spherical waves growing from a small source region with different timing and amplitudes. The surrogate model predicts wave fronts and backs at similar scales as the reference. The timing of the predicted spiral is also similar to the reference. However, the prediction differs from the reference in the APD of the initial rotation, at the rotor core, and in memory effects. Such a memory effect can clearly be seen in the reference at $t={150~\mathrm{{m}{s}}}$: The recovery to the resting state is much faster in the rectangle where the second variable $v$ of the reference model has been set to a higher value in the initial state of the simulation. This causes a slightly different evolution afterwards.

Figure 6: Spiral wave resulting from similar initial conditions for the data-driven surrogate model for the AP96 data set compared to its reference model. The data-driven model can predict the shape of the spiral wave despite only focal waves being included in the training data it is fit to.

Restitution curves

We have also performed one-dimensional cable simulations with the same stimulation protocol for the data-driven surrogate model compared to the rescaled AP96 model for reference. In Fig. 7, the resulting restitution curves of CV (panel A) and APD (panel B) as functions of the CL are shown. The overall shape of the curves for the low-order surrogate model is similar to the reference. However, the predicted CVs are lower than the reference values.

Figure 7: CV and APD restitution curves for the reference AP96 model and its data-driven surrogate model.

Data-driven model for optical mapping data

State space

For the OVM data set of hiAMs (cf. section 2.1.1), the three-dimensional state space spanned by ${{\bm{{u}}}} = {{{{\left[ u, \tilde u, g \right]}}}^\mathrm{T}}$ is also sparse: ${152417}$ bins in this space contain at least one data point, which corresponds to just ${15~\mathrm{\%}}$ of all bins. Of these bins, ${71~\mathrm{\%}}$ contain at least 10 data points. Non-empty bins on average contain ${949}$ data points.

As can be seen in Fig. 8, for these data, we are also observing a loop-like inertial manifold with two halves as in the other data set.

Figure 8: Visualisation of the three-dimensional state space for the OVM data set and projections onto two dimensions in the same style as in Fig. 4.

Forward model

For the OVM data set, the fit polynomial for the data-driven model is: $$ \begin{aligned} f{\left({{\bm{{u}}}}\right)} =& - {0.41392}\, \tilde{u} g u - {0.31842}\, \tilde{u} g + {0.026308}\, \tilde{u} u + {0.0010949}\, \tilde{u} \\& + {0.19341}\, g u + {0.15352}\, g - {0.014174}\, u - {0.0044818} \end{aligned} \qquad{(25)}$$

The prediction by the forward model for the OVM data set is depicted in Fig. 9. It models the overall behaviour of the tissue, i.e., excitation, recovery, and wave propagation, at similar time scales: The APD for an excitation threshold of $u=0.5$ of ${(140.1\ ±\ 5.1)~\mathrm{{m}{s}}}$ for the prediction is longer than the APD of ${(127\ ±\ 36)~\mathrm{{m}{s}}}$ measured for the reference. The CV for the prediction is measured at ${(0.148\ ±\ 0.012)~\mathrm{{m}{/}{s}}}$, and at ${(0.21\ ±\ 0.13)~\mathrm{{m}{/}{s}}}$ for the reference. The large uncertainty in the APD and CV measurements for the reference data is due to small regions of inhomogeneity in the recording. Due to the uncertainty and disparity in APD and CV values between prediction and reference, no reliable restitution curves can been measured for this model.

Figure 9: Prediction of the data-driven model over time compared to the reference solution for the OVM data set. The difference between the prediction and reference is displayed in the bottom row.

It can be seen that such artefacts due to inhomogeneity in the monolayer and other more complex effects are difficult to predict for the model, which only gets the state ${{\bm{{u}}}}{{\left( t_0, {{\bm{{x}}}} \right)}}$ at the initial time as input. Yet, similar structures are found in the prediction.

For the OVM data, the repolarisation at the wave back is much slower than the depolarisation at the wave front. For the synthetic data set AP96 re- and depolarisation happen at roughly the same time scale. Still, the one EMA variable $\tilde u$ is enough to capture these processes in both cases.

Prediction of a spiral wave

In Fig. 10, we compare a rotor in another OVM recording of hiAMs and a rotor predicted by the in-silico data-driven model. The model has been fit to data from only focal waves. To stimulate a spiral wave, we use the rectangle-based initial conditions outlined in section 2.1.2. Like for the synthetic data set (AP96, section 3.1), the tissue model is able to predict the overall spiral dynamics. Note that the spiral wave in the OVM recording has stabilised for several minutes before $t={0~\mathrm{{m}{s}}}$, while the rotor is just forming in the simulation with the data-driven model. The APD predicted by the model is much longer than the one observed APD by OVM. This leads to the wave front running into the wave back in the simulation.

Figure 10: Comparison of a spiral wave for the data-driven model for the OVM data set for hiAMs to a OVM recording of hiAMs showing a stable rotor. The data-driven model can predict the overall dynamics of the spiral wave despite only focal waves being included in the training data it is fit to.

Discussion

In this paper, we have shown that enough information can be extracted from just one spatio-temporal variable to build a state space, which enables the creation of in-silico models that can predict the further dynamics of the system. In layman’s terms, from a single movie showing the behaviour of excitation and recovery, a predictive model can be constructed. Just the information from OVM is already enough to fit a data-driven model to the dynamics. With our approach, it is even possible to fit a stable model to a single excitation wave, which, in turn, does not generalise well to other waves. For this reason, the SBP protocol is designed to ensure that less likely stimulation patterns are observed in the data, too. The resulting models are able to replicate the excitation patterns of focal waves included in the training data and can even predict the dynamics of spiral waves, a pattern that the models are not trained on.

While the variables with which we augment the state space should be seen as preliminary and not optimal choices, the models created with these variables already are able to capture crucial behaviour of the tissue, namely excitation, recovery and wave propagation. Both short and long-term memory can be encoded with the use of EMAs as state variables, which can easily be updated on the fly (Eq. 7). It remains to be seen if more or other state space augmentations can be used to be able to replicate higher-order effects, such as wave front curvature and to obtain more accurate restitution behaviour.

Through state space expansion with EMAs and SDs, we have found a way to encode useful information to describe the mechanics of excitation waves. It is remarkable, that, after “unfolding” with the proper state space expansion, we are able to capture with simple polynomial regression the kinetics of excitation waves sufficiently in the sense that we observe excitation and recovery with an APD similar to the data, and wave propagation at a similar CV.

The polynomial model $f$ is fit using a basic supervised machine learning method. However, this is just a small part of the method presented in this work. We have shown that using unsupervised learning, i.e., data transformations, much insight can be gained about the structure of the data. The insights obtained from presenting the data sets in the chosen three-dimensional state spaces as done in this work make it almost trivial to fit the data-driven model to the data. Fitting a function in the chosen state space is easier to interpret than black box approaches common in machine learning.

We have chosen to use an advection-based description of the wave propagation. This is due to the computation of the diffusion term $\nabla \cdot {{\bm{{D}}}} \nabla u$ in Eq. 12 being quite sensitive to small variations in $u$ and hence less robust to noisy data. The absolute gradient ${{\left\lVert \nabla u \right\rVert}}$ as approximated using the SD $g$ can deal with noise much better by design. Solving the forward equations in the advection-regression framework requires less computational effort in forward simulations than most traditional in-silico models and other novel approaches utilising convolutional and physics-informed neural networks. In the numerical solution of reaction-diffusion systems (Eq. 12), a sufficiently small numerical time step must be chosen according to a stability criterion. Typically, this leads to numerical time steps of the order of magnitude in two-dimensional simulations of in-silico tissue models of cardiac electrophysiology. Forward Euler stepping our model in the advection-regression framework can be done at much larger time steps of up to with the trade-off of fitting less to the original data. Also the computation of the polynomial $f$, EMAs and SDs result in an overall computationally cheaper model than typical tissue models.

While this was not needed for this use case, in further research, more sophisticated general function approximators may be tried for $f$, such as neural networks. Although the Latin hypercube sampling enables some basic regularisation of the solution, it remains to be seen how to keep the fit functions simple enough such that they generalise well. Another limitation that could be addressed when using more flexible function approximators is that, in some cases, the polynomial model $f$ updates the state $u$ such that it tries to leave the valid range.

While other augmentations were tried in the development of this method, such as pseudo-electrograms, the last APD and diastolic interval at each point over time, or the absolute value of the gradient fit to a neighbourhood of each point using the least-squares method, we are here presenting two augmentations we found most useful, the EMA $\tilde u$ as well as the SD $g$.

We have not yet taken into account boundary effects in the model creation procedure. Some information about the gradient of the main variable $u$ is encoded in the SD $g$ in the neighbourhood of each point in the medium. The SD $g$ can still be computed near the boundary by treating those points just like interior points but with fewer samples. How to encode boundary conditions in this model could be investigated in follow-up studies.

The procedure to generate in-silico tissue models is designed to work well in a setting where not much is known about the wave propagation in an in-vitro tissue model. Our method only requires very little assumptions to uncover the dynamics. Fitting is not limited to just in-vitro OVM data of focal waves in cardiac tissue excitation models, but can also be done in general where excitation wave dynamics can be observed in two dimensional space over time. An advantage of our method is that models for any observed behaviour can be created and fit to data which is in contrast to classical phenomenological models (Aliev & Panfilov, 1996; Barkley, 1991; Bueno-Orovio et al., 2008; Fenton & Karma, 1998; FitzHugh, 1961; Marcotte & Grigoriev, 2017; Nagumo et al., 1962). Fitting parameters of such existing models can get the model closer to observed data. However, if a behaviour falls outside the regime of these models, such methods will not work (Martin et al., 2022). With our method, a data-driven model can be fit flexibly with relative ease within minutes and subsequently used to predict the further evolution of the tissue. This puts our model creation pipeline more in line with recent deep learning approaches (Costabal et al., 2020; Martin et al., 2022; Shahi et al., 2022). However, our approach is typically faster in fitting due to the polynomial model function and faster in forward stepping because of the SD-based spatial coupling. Depending on the complexity of the dynamics, a simple approach to modelling such as ours may be enough. Promising use cases to apply the proposed method with minimal changes are the peristaltic motion of for instance embryonic hearts or the digestive tract (Moorman & Christoffels, 2003), or the spread of lichens which has been observed to form spiral waves (Müller & Tsuji, 2019). For more complex dynamics, more than three state space variables may be needed, or an extension using a neural network as the model function $f$. For very complex dynamics, the aforementioned, more extensive deep learning methods may be necessary (Aliev & Panfilov, 1996; Barkley, 1991; Bueno-Orovio et al., 2008; Fenton & Karma, 1998; FitzHugh, 1961; Marcotte & Grigoriev, 2017; Nagumo et al., 1962). Possible extensions of our data-driven model include an online-learning approach, where the model’s predictive power is continuously improved while new data comes in. Such a model could, for instance, be fit to patient recordings, such as intra-cardiac electrograms or electrocardiograms on the body surface, and then be used to predict cardiac arrhythmias before they happen.

Conclusion

In this work, we presented a method to create simple data-driven in-silico tissue models from in-vitro OVM data and synthetic data that capture the basic behaviour of the corresponding excitable media. These models can predict the activation patterns included in the training data, i.e., only focal waves, and generalise to unseen activation patterns, such as spiral waves. A useful feature of this model creation method is its ability to extract all relevant information that the models are fit to from just one state variable over space and time, such as the transmembrane voltage of cardiomyocytes which can be obtained from a single experiment, e.g. an OVM recording.

Data availability

The source code of the software implemented for this paper is publicly available at https://gitlab.com/heartkor. The functionality is broken up into several separate packages, some of which depend on each other.

The model creation pipeline is implemented as the Distephym Python module, distephym 1.0.0. It stores the augmented state spaces in sparse histograms, specifically implemented for this publication as another module, sparsealgos 1.0.0.

Reading, writing, and analysis of OVM recordings is implemented in the Sappho Python module, sappho 1.0.0.

The Ithildin C++ framework, ithildin 3.2.2, is a finite-differences, parallelised, reaction-diffusion solver written in C++. It will be released in a separate, dedicated publication (Cloet et al., 2023; Kabus et al., 2024). We have used Ithildin to generate the synthetic (i.e., AP96) data set.

For the other forward simulations in this work, we have written another simple finite-differences reaction-diffusion solver in Python, ithilmin 1.0.0, that can easily be combined with other Python modules. It was used for forward runs of the fit data-driven models.

In conjunction with this paper, we are also releasing a new version of the Ithildin Python module, py_ithildin 0.6.1, (Kabus et al., 2022). This module was used for the processing of in-vitro/in-silico data sets. Results from both reaction-diffusion solvers, ithildin and ithilmin, can be analysed with this Python module.

The source code of these projects and the AP96 and OVM data sets used in this work have been archived on Zenodo (DOI: 10.5281/zenodo.8183722).

Addenda

Acknowledgments:

We are grateful to our collaborators at the LUMC, especially to Sven O. Dekker and Juan Zhang for providing access to hiAM monolayers and the OVM setup, and to Niels Harlaar for the spiral wave OVM recordings, respectively. Also, we would like to thank all members of Team HeartKOR at KULAK for valuable help during writing, in particular, Marie Cloet.

Funding:

DK is supported by KU Leuven grant GPUL/20/012. HD is supported by KU Leuven grant STG/19/007. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The hiAMs were developed as part of the research programme “More Knowledge with Fewer Animals” (MKMD, project 114022503, to AAFdV), which was financed by the Netherlands Organisation for Health Research and Development (ZonMw) and by the Dutch Society for the Replacement of Animal Testing (dsRAT).

Competing interests:

The authors have declared that no competing interests exist.

Copyright:

Author contributions:

DK: Conceptualisation, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Writing – review & editing, Visualisation. TDC: Methodology, Writing – review & editing. AAFdV: Conceptualisation, Resources, Writing – review & editing. DAP: Conceptualisation, Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition. HD: Conceptualisation, Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition.

References

Aliev, R. R., & Panfilov, A. V. (1996). A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals, 7(3), 293–301. https://doi.org/10.1016/0960-0779(95)00089-5

Barkley, D. (1991). A model for fast computer simulation of waves in excitable media. Physica D: Nonlinear Phenomena, 49(1-2), 61–70. https://doi.org/10.1016/0167-2789(91)90194-e

Bressloff, P. C. (2014). Waves in neural media. Lecture Notes on Mathematical Modelling in the Life Sciences, 18–19. https://doi.org/10.1007/978-1-4614-8866-8

FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466. https://doi.org/10.1016/s0006-3495(61)86902-6

Fox, J. J., Bodenschatz, E., & Gilmour, R. F. (2002). Period-doubling instability and memory in cardiac tissue. Physical Review Letters, 89(13). https://doi.org/10.1103/physrevlett.89.138101

Joukar, S. (2021). A comparative review on heart ion channels, action potentials and electrocardiogram in rodents and human: Extrapolation of experimental insights to clinic. Laboratory Animal Research, 37(1), 1–15. https://doi.org/10.1186/s42826-021-00102-3

Krogh-Madsen, T., Schaffer, P., Skriver, A. D., Taylor, L. K., Pelzmann, B., Koidl, B., & Guevara, M. R. (2005). An ionic model for rhythmic activity in small clusters of embryonic chick ventricular cells. American Journal of Physiology-Heart and Circulatory Physiology, 289(1), H398–H413. https://doi.org/10.1152/ajpheart.00683.2004

Kuklik, P., Zeemering, S., Maesen, B., Maessen, J., Crijns, H. J., Verheule, S., Ganesan, A. N., & Schotten, U. (2014). Reconstruction of instantaneous phase of unipolar atrial contact electrogram using a concept of sinusoidal recomposition and hilbert transform. IEEE Transactions on Biomedical Engineering, 62(1), 296–302. https://doi.org/10.1109/TBME.2014.2350029

McKay, M. D., Beckman, R. J., & Conover, W. J. (2000). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 42(1), 55–61. https://doi.org/10.1080/00401706.2000.10485979

Moe, G. K., Rheinbolt, W. C., & Abildskov, J. A. (1964). A computer model of atrial fibrillation. Am. Heart J., 67, 200–220. https://doi.org/10.1016/0002-8703(64)90371-0

Moorman, A. F. M., & Christoffels, V. M. (2003). Cardiac chamber formation: Development, genes, and evolution. Physiological Reviews. https://doi.org/10.1152/physrev.00006.2003

Müller, S. C., & Tsuji, K. (2019). Appearance in nature. Spirals and Vortices: In Culture, Nature, and Science, 31–66. https://doi.org/10.1007/978-3-030-05798-5_2

Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50(10), 2061–2070. https://doi.org/10.1109/jrproc.1962.288235

Niederer, S. A., Lumens, J., & Trayanova, N. A. (2019). Computational models in cardiology. Nature Reviews Cardiology, 16(2), 100–111. https://doi.org/10.1038/s41569-018-0104-y

Noble, J. V. (1974). Geographic and temporal development of plagues. Nature, 250(5469), 726–729. https://doi.org/10.1038/250726a0

O’Hara, T., & Rudy, Y. (2012). Quantitative comparison of cardiac ventricular myocyte electrophysiology and response to drugs in human and nonhuman species. American Journal of Physiology-Heart and Circulatory Physiology, 302(5), H1023–H1030. https://doi.org/10.1152/ajpheart.00785.2011

Priebe, L., & Beuckelmann, D. J. (1998). Simulation study of cellular electric properties in heart failure. Circulation Research, 82(11), 1206–1223. https://doi.org/10.1161/01.res.82.11.1206

Wei, N., Mori, Y., & Tolkacheva, E. G. (2015). The role of short term memory and conduction velocity restitution in alternans formation. Journal of Theoretical Biology, 367, 21–28. https://doi.org/10.1016/j.jtbi.2014.11.014

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Cover photo: Am Graben, Kaufbeuren, Bayern, Deutschland

Escaping rainy autumn to Guadeloupe

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The Guadeloupe archipelago has some of the most beautiful beaches that I have ever seen.

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Lipari Islands

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Numerical methods for the detection of phase defect structures in excitable media

Tue, 12 Jul 2022 00:00:00 +0000

This article was previously published in PLOS ONE 17(7): e0271351 (Kabus et al., 2022) and is a chapter of my dissertation. The full thesis is available online as a PDF or can be read chapter by chapter on this website.

Authors:

Desmond Kabus^1,2,3 (ORCID: 0000-0002-6965-5211)
Louise Arno^1,3 (ORCID: 0000-0001-9384-519X)
Lore Leenknegt^1,3 (ORCID: 0000-0003-4667-3473)
Alexander V. Panfilov^4,5,6 (ORCID: 0000-0003-2643-642X)
Hans Dierckx¹ (ORCID: 0000-0003-0899-8082)

Institutions:

Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Leiden, The Netherlands
iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven, Belgium
Department of Physics and Astronomy, Ghent University, Ghent, Belgium
Laboratory of Computational Biology and Medicine, Ural Federal University, Ekaterinburg, Russia
World-Class Research Center “Digital biodesign and personalized healthcare”, Sechenov University, Moscow, Russia

Correspondence: h.dierckx@kuleuven.be

DOI: 10.1371/journal.pone.0271351

Editor: Ivan Kryven, Utrecht University, The Netherlands

Abstract:

Electrical waves that rotate in the heart organize dangerous cardiac arrhythmias. Finding the region around which such rotation occurs is one of the most important practical questions for arrhythmia management. For many years, the main method for finding such regions was so-called phase mapping, in which a continuous phase was assigned to points in the heart based on their excitation status and defining the rotation region as a point of phase singularity. Recent analysis, however, showed that in many rotation regimes there exist phase discontinuities and the region of rotation must be defined not as a point of phase singularity, but as a phase defect line. In this paper, we use this novel methodology and perform a comparative study of three different phase definitions applied to in-silico data and to experimental data obtained from optical voltage mapping experiments on monolayers of human atrial myocytes. We introduce new phase defect detection algorithms and compare them with those that appeared in literature already. We find that the phase definition is more important than the algorithm to identify sudden spatial phase variations. Sharp phase defect lines can be obtained from a phase derived from local activation times observed during one cycle of arrhythmia. Alternatively, similar quality can be obtained from a reparameterization of the classical phase obtained from observation of a single timeframe of transmembrane potential. We found that the phase defect line length was (35.9 ± 6.2) mm in the Fenton-Karma model and (4.01 ± 0.55) mm in cardiac human atrial myocyte monolayers. As local activation times are obtained during standard clinical cardiac mapping, the methods are also suitable to be applied to clinical datasets. All studied methods are publicly available and can be downloaded from an institutional web-server.

Introduction

The heart is a self-organizing dynamical system for which the mechanical contraction is regulated by waves of electrical activation travelling through the cardiac wall. During cardiac arrhythmia complex electrical patterns emerge that often result into a rotating pattern, either circling around an obstacle or around its own wave back (Allessie et al., 1973; Gray et al., 1998). These vortices are also known as rotors, or spiral waves in two dimensions (2D), or scroll waves in three dimensions (3D).

After the experimental observation of such structures in animal hearts during ventricular tachycardia (Gray et al., 1995), it was conjectured that rotors can sustain several heart rhythm disorders. However, the precise dynamics of rotors, the structure of the rotor core and the most efficient manner to remove them from the heart remain incompletely understood.

A quantitative description of spiral wave motion requires localizing it in space. In first approximation, the region around which the rotor revolves is called the spiral wave core. The location of a spiral wave at a given time can be further narrowed down to a single point, usually called the spiral wave tip. Different methods exist to define the tip, e.g. as the point where wave front and wave back merge (Zykov, 1987), as the point on a line of constant voltage that does not change instantaneously, as an intersection between two isolines of different variables (Fenton & Karma, 1998) or as a point singularity of the activation phase (Bray & Wikswo, 2002; Gray et al., 1998).

By following the tip position of a single spiral wave over time, a disc-like or star-like shape emerges, known as the spiral wave core. Different types of cores have been observed (Winfree, 1991), and the non-circular cores are referred to as meandering cores. Among the meandering cores, simulations of detailed ionic models for cardiac tissue typically show so-called linear cores, as depicted in Fig. 1.

Figure 1: Qualitatively different tip trajectories of spiral waves in modelled cardiac tissue. The first row displays the Aliev-Panfilov (AP) model (Aliev & Panfilov, 1996), the second row the Fenton-Karma (FK) model (Fenton & Karma, 1998) with modified Luo-Rudy I (MLR-I) parameters, the third row Bueno-Orovio-Cherry-Fenton (BOCF) (Bueno-Orovio et al., 2008), and the last row displays optical voltage mapping data derived from monolayers of conditionally immortalized human atrial myocytes (hiAMs) (Harlaar et al., 2022). In section 3, we outline in more detail how these data sets have been obtained. The first 3 columns show different snapshots in time and are colored according to local transmembrane voltage. Corresponding spiral wave tip trajectories are shown in the last column. Darker coloring of the trajectory corresponds to later points in time.

The linear-core regime arises in systems with long action potential duration (APD): If the tip is next to a region of refractory tissue, it will follow this interface until meeting a point where the tissue has recovered. As a result, the tip moves along an almost straight line, interleaved with turning points. These dynamics have not only been observed in simulations, but were also reported in experiments (Efimov et al., 1999; Janse et al., 1980) in the form of a line of conduction block. For this reason, it is important to elucidate the spatial distribution of the core of the rotor.

In 3D, a spiral wave or rotor becomes a structure called a scroll wave. Within the scroll wave, the collection of spiral wave tips forms a filament curve (Clayton et al., 2006). In numerical simulations with linear cores and a few experimental observations (Efimov et al., 1999), the straight segment in the linear core extends to a ribbon-like filament (Efimov et al., 1999). However, these spatially extended filaments have not been substantially included in theory development, as was done for circular-core filaments (Keener, 1991).

Recent works (Arno et al., 2021; Tomii et al., 2021) have proposed to treat linear cores fundamentally different from circular cores. Specifically, classical phase analysis (Bray & Wikswo, 2002; Gray et al., 1998) of cardiac activation patterns assumes that there is a phase singularity (PS) located near the spiral wave tip. However, when the wave front reaches a part of tissue that is for instance not fully recovered yet, a so-called conduction block will form. On both sides of this conduction block line, there is a different phase, such that the authors of (Arno et al., 2021; Tomii et al., 2021) argue that the phase representation resembles more a phase defect line (PDL), i.e. a line where the phase changes abruptly from one side to the other. In other words, we go from a phase singularity (PS) to a branch cut known in complex analysis. Making the distinction between phase defects (PDs) and point singularities could have potential clinical use, as current analysis methods in simulation, experiment and of clinical data are only aiming to localize point singularities of phase.

The aim of this paper is to provide and compare several methods to numerically calculate the phase and the corresponding PDs. We include a quantitative analysis of the results and the performance of the algorithms.

A summary of our workflow is presented in Fig. 2: An image (e.g. of the transmembrane potential $u$) is in the first step converted into a phase ${\varphi}$. Where jumps in phase are detected, the PD density $\rho$ (see below) will be much larger than zero. If desired, the field $\rho({{\bm{{x}}}},t)$ can be further processed to yield localized PDLs.

Figure 2: An example of the steps done in the process of constructing the PD density $\rho$. (A) We start from the first phase state variable $u$, often the normalized and hence unitless transmembrane potential $V$. (B) Next, from this $u$ and possibly other state variables, the phase, here ${\varphi}^{\text{LAT}}$, is calculated. How to compute it will be introduced below in Eq. 3. (C) Finally from the phase, the PD density is produced based on one of the PDL detection algorithms; here, the phase coherence (PC) method was used, which will be introduced below in section 3.3.3. These quantities are plotted over a 2D square domain of myocardial tissue in physical space. Color is used to represent the values of the quantities mentioned in the top left corner. The same coloring will be used throughout this manuscript.

This manuscript is organized as follows: First, we briefly review the concepts of phase, PSs and PDs (section 2). In the methods section 3, we outline our simulation methods and present several methods to trace PDs in excitation patterns. Results of these methods and their performance are presented in section 4. We conclude this paper with a discussion and outlook (section 5 and section 6).

Theoretical background

The concept of phase

As remarked long ago by Winfree (1974), many biological processes take values within a cycle rather than on the line of real numbers. Cardiac excitation is such an example, since during an action potential, the cell membranes depolarize and repolarize in normal circumstances along a predefined sequence, tracing out a closed loop in state space. To keep track of the relative state of cells along this cycle, the concept of phase can be used. In addition to the classical phase definition, called activation phase ${\varphi}^{\text{act}}$ below, an alternative phase based on local activation times (LAT) was defined by Arno et al. (2021). In this paper, a third phase ${\varphi}^{\text{skew}}$ will be defined below as an approximation of ${\varphi}^{\text{act}}$ when LAT are not available (Eq. 7).

Next, we will briefly review the previously defined phases ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{LAT}}$.

The activation phase ${\varphi}^{\text{act}}$ is the phase as seen in a space spanned by two observables $V({{\bm{{x}}}},t)$ and $R({{\bm{{x}}}},t)$ in the system (Bray & Wikswo, 2002; Clayton et al., 2006; Gray et al., 1998). We henceforth assume that $V$ is representing the activation or depolarization of the medium, i.e. in cardiac context, we take $V$ to be the normalized transmembrane potential.

Even if there is only one variable $V$ observed, its time-delayed version (Gray et al., 1995), time derivative (Fenton & Karma, 1998), or Hilbert transform (Bray & Wikswo, 2002) can be used as a linearly independent variable $R$. In numerical simulations, all state variables of the system can be observed, and any pair can be chosen as $(V,R)$. Then, one usually defines the activation phase as the polar angle of a state in the $(V,R)$-plane, relative to a reference point $(V_*, R_*)$ that lies within the cycle: $$ {\varphi}^{\text{act}}= \operatorname{arctan2}(R-R_*, V-V_*) + c \qquad{(1)}$$ Here, the polar angle is returned by the two-argument inverse tangent: $\operatorname{arctan2}(y,x) = \arctan(y/x)$ if $x\geq 0$ and $\arctan(y/x) + \pi \operatorname{mod}2\pi$ if $x<0$. A constant $c$ can furthermore be added to make ${\varphi}^{\text{act}}= 0$ correspond to the resting state. The left column of Fig. 3 visualizes ${\varphi}^{\text{act}}$ for the four data sets introduced in Fig. 1.

Figure 3: Illustration of the three different phases for one frame of four data sets. The color code represents the phase in modelled myocardial tissue in a 2D square domain in the same scale as in Fig. 2 B.

We recently proposed a second definition of phase that is based on the local activation time (LAT) of tissue (Arno et al., 2021). The LAT, which is commonly used clinically, is defined as the time $t^{\text{arrival}}$ when the tissue locally depolarizes, i.e. when the transmembrane voltage $V$ at that point exceeds a value $V_*$.

In noisy circumstances (see Fig. 12 below), care is taken to estimate LAT in a robust manner by using an alternative definition of LAT with two thresholds: It has another condition that needs to be met for LAT to be updated. This condition is that $V$ must first decrease below a second threshold $V_*^\downarrow$ before an increase above the first threshold $V_*$ is considered as depolarization at the wave front and therefore as the trigger of an update of LAT. Unless noted otherwise, we use the first definition of LAT with just one threshold.

The LAT relative to the current time is the elapsed time: $$ t^{\text{elapsed}}= t - t^{\text{arrival}} \qquad{(2)}$$ In other words, elapsed time is the time since the start of the last local activation.

The LAT phase ${\varphi}^{\text{LAT}}$ is just a mapping of $t^{\text{elapsed}}$ onto the interval $[0,2\pi)$ by applying a sigmoidal function: $$ {\varphi}^{\text{LAT}}= 2\pi\tanh(t^{\text{elapsed}}/\tau) \qquad{(3)}$$ where $\tau$ is the characteristic time of the cyclic process. Here, we take $\tau$ equal to half of the typical APD in the medium. Note that another sigmoid function instead of $\tanh$ could also have been chosen in Eq. 3. The right column of Fig. 3 visualizes ${\varphi}^{\text{LAT}}$ for the same four data sets.

An asset of ${\varphi}^{\text{LAT}}$ is that the curves of equal ${\varphi}^{\text{LAT}}$ are precisely the isochrones that cardiologists work with during endocardial catheter mapping. Also, if $\tau$ is chosen appropriately, repolarized (recovered) tissue will have ${\varphi}^{\text{LAT}}\approx 2\pi$, such that this phase is not changing abruptly at the wave front and wave back, in contrast to ${\varphi}^{\text{act}}$. A disadvantage of ${\varphi}^{\text{LAT}}$ is that it requires a dense temporal sampling (either in simulation or experiment), since otherwise staircase artifacts emerge. As a last remark, care should be taken when initializing ${\varphi}^{\text{LAT}}$ at the start of the observation window: If the previous LAT are unknown, so is ${\varphi}^{\text{LAT}}$ until a propagating wave has swept through the medium.

Phase singularities (PSs) and phase defects (PDs)

The analysis of spatial distributions of phase makes use of several concepts of complex numbers and complex analysis (Arfken & Weber, 1995), such as contours, PSs and branch cuts. We now briefly review these concepts in the context of cardiac excitation.

The detection of rotor cores from a spatial map of phase can be performed by calculating the total phase difference along a closed loop (contour) ${\mathcal{C}}$ in the medium, usually taken on the cardiac surface: $$ \Delta {\varphi}= \oint_{\mathcal{C}} d{\varphi}\qquad{(4)}$$ Since the first and last point have the same phase, the resulting phase difference $\Delta {\varphi}$ will return an integer multiple of $2\pi$. Hence, one defines the topological charge circumscribed by the contour as: $$ Q = \frac{\Delta {\varphi}}{2\pi} = \oint_{\mathcal{C}} \frac{d{\varphi}}{2\pi} \qquad{(5)}$$

Now, in the classical theory (Gray et al., 1995), one assumes that the phase function ${\varphi}({{\bm{{x}}}})$ at a given time $t$ is continuous nearly everywhere, except in a few points where the phase is undefined. When using ${\varphi}^{\text{act}}$, it can be seen that these points will correspond to $V=V_*, R=R_*$. In the immediate vicinity of such points, all phases are present (both in the $(V,R)$ plane and in the spatial phase map), hence this point is called a phase singularity (PS). Note that the existence of a PS, where all different phases touch, within the contour region $\mathcal{C}$ is implied by the assumption of the original theory that the phase is a continuous function except at the PS.

In the generalized theory (Arno et al., 2021; Tomii et al., 2021), however, the phase is allowed to be discontinuous near a conduction block line, which also happens in the core of a linear-core spiral. Then, the contour $\mathcal C$ cannot be shrunk to surround a single point, without having the contour cross an interface where ${\varphi}$ changes abruptly. We call such transition zones phase defects (PD): phase defect lines (PDLs) in 2D and phase defect surfaces (PDSs) in 3D. The discontinuous behavior of phase is more easily noted with ${\varphi}^{\text{LAT}}$ than with ${\varphi}^{\text{act}}$, since ${\varphi}^{\text{act}}$ also shows strong gradients near the wave front and wave back (Arno et al., 2021). As a result, we are convinced that PS detection algorithms that assume a continuous phase distribution are behaving non-robustly near such spatial phase transition, which motivates this work. Here, we will provide PD detection methods that are explicitly discriminating the PD structures, either as a sharp line, or in a probabilistic manner using PD density.

Methods

Data generation and collection

Numerical methods for pattern generation

The methods for PD detection developed here are designed to operate on excitation patterns, regardless of their generation. However, we here test the methods on numerical simulations in a cardiac monodomain setting. That is, in a rectangular Cartesian grid, we modeled forward evolution of a column matrix of state variables $\bm{u}({{\bm{{x}}}},t)$ according to a reaction-diffusion equation (Clayton et al., 2011): $$ \partial_t \bm{u}= \bm{P} \Delta \bm{u}+ \bm{F}(\bm{u}) \qquad{(6)}$$ Here, the first component $u_1$ of $\bm{u}$ equals the normalized transmembrane potential $u_1 = u = V$, and $\bm{P} = \operatorname{diag}(1,0,...,0)$ in order to enable wave propagation by diffusion of $u$. The number of state variables in $\bm{u}$ varies between the different mathematical models of cardiac myocytes, which are encoded in non-linear reaction functions $\bm{F}(\bm{u})$. To assess the reliability of our methods, we tried several reaction kinetics: Linear cores are known to occur with the Bueno-Orovio-Cherry-Fenton (BOCF) model for human ventricles (Bueno-Orovio et al., 2008) with parameter set ‘PB’ mimicking Priebe and Beuckelmann kinetics (Priebe & Beuckelmann, 1998), and with the Fenton-Karma (FK) model (Fenton & Karma, 1998) with modified Luo-Rudy I (MLR-I) parameters. In addition, we investigated how the methods perform when operating on a circular spiral wave core by applying them to the Aliev-Panfilov (AP) model (Aliev & Panfilov, 1996).

All simulations were executed on a 2D isotropic square domain of myocardial tissue using Neumann boundary conditions and a 5-point stencil for the Laplacian. Integrating in time is done using forward Euler stepping, with values per model given in Table 1. There, we also present the typical APD values and other parameters that we use for the calculation of the phases.

Table 1: Overview of the performed experiments with relevant parameters.

The spiral wave was generated by applying a classical S1S2 protocol: The western side of the square domain is initially excited to generate a plane wave. When the central point of the medium has finished repolarization, the south-western quarter of the domain behind the traveling wave is stimulated, such that a spiral wave (cardiac rotor) is formed. This protocol may also be applied rotated or mirrored.

Three frames of the first model variable $u$ for each of the three chosen models resulting from numerical simulation were displayed in Fig. 1.

Optical voltage mapping experiments

To test our methods on in-vitro measurements, in contrast to the in silico simulations, we used optical voltage mapping data derived from monolayers of conditionally immortalized human atrial myocytes (hiAMs) following cardiomyogenic differentiation of these cells (Harlaar et al., 2022). A voltage-sensitive dye is added to the culture, after which a real-time recording can be made of the intensity of emitted light, which is a measure of the local transmembrane potential. For the used recordings, pixel size was and the sampling time between frames was The full protocols for cellular differentiation of hiAMs and the optical voltage mapping experiments can be found in the previous publication (Harlaar et al., 2022).

Gaussian smoothing with a kernel size of three grid points has been applied to each frame. The data have been rescaled such that each grid point has unit variance in time. Then, arbitrary units have been defined such that the resting state corresponds to optical activity $u$ equal to zero, and the excited state to $u=1$. To calculate the LAT phase ${\varphi}^{\text{LAT}}$, we use a value of $\text{APD} = {100~\mathrm{{m}{s}}}$. This corresponds to a value in between the measured values of $\text{APD}_{50} = {(36.4\ ±\ 7.7)~\mathrm{{m}{s}}}$ and $\text{APD}_{80} = {(136\ ±\ 12)~\mathrm{{m}{s}}}$ (Harlaar et al., 2022). Three frames of one of these recordings can be seen in Fig. 1 M-O.

As a second variable for calculating ${\varphi}^{\text{act}}$ for this data set, we use the time-delayed version of $u$ by which corresponds to roughly of the duration of a rotation. The other parameters needed to calculate the phases can be found in Table 1.

Ethics statement. This study was conducted with approval of the institutional review board of the Leiden University Medical Center (P08.087) and in compliance with the International Code of Medical Ethics of the World Medical Association.

A third phase definition

The aforementioned phase definitions ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{LAT}}$ have each their downsides: Gradients in ${\varphi}^{\text{act}}$ not only show PD but also wave fronts, and ${\varphi}^{\text{LAT}}$ requires intense sampling over time of the medium. We propose to combine the advantages of ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{LAT}}$ in a new phase, ${\varphi}^{\text{skew}}$. This phase is designed as a computationally cheap approximation of the elapsed time phase ${\varphi}^{\text{LAT}}$ that can be calculated using ${\varphi}^{\text{act}}$. By construction, it does not require the whole history of $V$, but can instead be calculated using $(V,R)$ at one point in time. In essence, ${\varphi}^{\text{skew}}$ is a re-parameterization of ${\varphi}^{\text{act}}$: $$ {\varphi}^{\text{skew}} = h({\varphi}^{\text{act}}), \;\;\; h(0) = 0, \;\;\; h(2\pi) = 2\pi \qquad{(7)}$$ where $h$ is furthermore monotonically rising. Essentially, the cycle visited by cells during the action potential is now labeled in a more free manner than the classical polar coordinates in the $(V,R)$-plane. As can be seen in Fig. 4 by plotting ${\varphi}^{\text{act}}$ vs. ${\varphi}^{\text{LAT}}$, ${\varphi}^{\text{LAT}}$ is also a re-parameterization of ${\varphi}^{\text{skew}}$.

Figure 4: Correlation of the different phases. Phase data have been taken from the one snapshot in time of the Fenton-Karma simulation in Fig. 3. Points for two phase values with darker shading correspond to higher logarithmic probability density. (A) The LAT phase ${\varphi}^{\text{LAT}}$ stays close to zero while the state space phase ${\varphi}^{\text{act}}$ is already increasing. It then transitions to $2\pi$ much quicker and stays relatively close to that value afterwards. (B) The skewed phase ${\varphi}^{\text{skew}}$ is designed as a re-parameterization of the state space phase ${\varphi}^{\text{act}}$ such that it approximates this behavior using a piecewise linear function. (C) The skewed phase ${\varphi}^{\text{skew}}$ correlates more with the LAT phase ${\varphi}^{\text{LAT}}$, getting closer to the dotted line where ${\varphi}^{\text{skew}}={\varphi}^{\text{LAT}}$.

In principle, one could fit a function $h$ to a plot of ${\varphi}^{\text{LAT}}$ vs. ${\varphi}^{\text{act}}$, but we opt for a different approach here and define a piece-wise linear function $h$: $$ h({\varphi}) = \begin{cases} 0 & {\varphi}\in [0, {\varphi}_0] \\ 2\pi \frac{{\varphi}-{\varphi}_0}{{\varphi}_1-{\varphi}_0} & {\varphi}\in ({\varphi}_0, {\varphi}_1) \\ 2\pi & {\varphi}\in [{\varphi}_1, 2\pi] \end{cases} \qquad{(8)}$$ Values for ${\varphi}_0$ and ${\varphi}_1$ were manually chosen for the different reaction kinetics models used, see section 2.1 and Table 1.

Below, we will apply different PD detection methods on the three phases ${\varphi}^{\text{act}}$, ${\varphi}^{\text{LAT}}$ and ${\varphi}^{\text{skew}}$, to see which performs best in visualizing PD structures.

Phase defect detection algorithms

Requirements for phase defect detection algorithms

The aim of this paper is to provide and evaluate numerical methods that can be used as a successor of classical PS detection algorithms, but directed towards the detection of PDs instead. The following factors are taken into account when proposing the methods.

Zero or finite thickness. Due to the formation of a physical boundary layer (either by electrotonic effects or numerical smoothing, see (Arno et al., 2021)), a PD has a finite width in practice. Therefore, we see two options. A first option is to see the PD as an idealized structure with zero thickness, situated near the steepest spatial variation of phase or a spatial discontinuity in the LAT. A second option is to accommodate for the finite transition width, and describe the PD in a probabilistic manner, e.g. by regarding the phase gradient as a kind of PD density, below denoted as $\rho$. If desired, the PDL extent can then be determined by putting a threshold on $\rho$, a process which becomes easier if this density is normalized between 0 and 1.

Vertex-based or edge-based detection. Our algorithms take phase data on a set of nodes as input. We discriminate methods based on whether their output is on the nodes, edges or faces of the grid. Since the PDs have co-dimension one, it is natural to consider them as being situated on edges of the computational grid, either in 2D or 3D. However, the result of an edge-based method is not located on the original grid, such that methods that return values on the vertices of the grid (i.e. collocated with local phase data) can also be useful. Both edge-based and vertex-based methods are in contrast with PS detection: since PS have co-dimension 2, they are naturally calculated on the faces of the grid (Fenton & Karma, 1998; Kuklik et al., 2014). Below, we provide for most PD detection methods an edge-based and vertex-based variant. We currently test our methods on a 2D Cartesian grid only and leave the extension to 3D and irregular meshes to future work.

Taking phase differences. Since phase is a cyclic variable, phase differences should be taken with care. Spatial derivatives are implemented in such a manner that an integer multiple of $2\pi$ is added in order to bring the result as close as possible to zero. Also trigonometric functions are adjusted such that they are indifferent to $2\pi$ differences. In some methods, the complex number $z = e^{\mathrm i{\varphi}}$ is calculated and the absolute value $|z|$ is taken afterwards to obtain the phase instead of just using ${\varphi}$. This will make sure that a large jump in phase is not just attributed to the phase being cyclic.

Performance. In the results section (section 4), the different algorithms are compared in computational speed and relative performance. The location of PDs depends on the choice of algorithms for phase and PD calculation. To still make a comparison between methods possible, we designed an in silico experiment where the ground truth location of a PDL is known.

In the remainder of this section, nine different PD localization methods will be briefly presented. For the vertex-based algorithms, the output is a discretized scalar field: a non-negative PD density $\rho({{\bm{{x}}}},t)$ at time $t$ that is defined in the points where phase is available to calculate the defect from (here either ${\varphi}^{\text{act}}, {\varphi}^{\text{LAT}}$ or ${\varphi}^{\text{skew}}$). For the edge-based algorithms, the output is a number $\sigma_{ab}$ computed from the pair of phases ${\varphi}_a = {\varphi}({{\bm{{x}}}}_a)$, ${\varphi}_b = {\varphi}({{\bm{{x}}}}_b)$ found at the vertices connected by that edge, which could be regarded as making up a vector field.

Interpolation between vertex-based and edge-based methods

In what follows, ${{\bm{{x}}}}_a, {{\bm{{x}}}}_b, ...$ are positions of vertices $a$ and $b$, $\rho_a =\rho({{\bm{{x}}}}_a)$, ${\varphi}_a = {\varphi}({{\bm{{x}}}}_a)$ and $\sigma_{ab}$ is a quantity defined on the edge of the mesh between ${{\bm{{x}}}}_a$ and ${{\bm{{x}}}}_b$. The set of neighbor vertices connected to vertex $a$ is $\mathcal{N}(a)$, containing $N_a$ elements. In a 2D Cartesian grid, $N_a = 4$ inside the medium, but on the boundary of the domain or near obstacles, this value will be lower. In this case, the following formulas can still be applied with lower $N_a$. In a 3D Cartesian grid, $N_a = 6$. See Fig. 5 for graphical depiction of $\rho_a$ and $\sigma_{ab}$ on a portion of a Cartesian grid.

Figure 5: Overview of vertex- and edge-centered quantities. For vertices $a$ and $b$ at positions ${{\bm{{x}}}}_a$ and ${{\bm{{x}}}}_b$, for which the phase ${\varphi}$ has been calculated, we denote by $\rho_a$ the vertex-centered PD density, and $\rho_b$, respectively. For the edge $a\leftrightarrow b$ between the two vertices $a$ and $b$, we denote the edge-centered PD density by $\sigma_{ab}$. The vertices that $a$ is connected to are $a$’s neighbourhood $\mathcal N(a)$.

If a quantity arises naturally along an edge (e.g. a gradient), it can be interpolated onto the vertex grid using $$ \rho_a = \frac{1}{N_a} \sum_{b \in \mathcal{N}(a)} \sigma_{ab} \qquad{(9)}$$

Conversely, if a quantity is found at vertices, it can be allocated to the edges using linear interpolation: $$ \sigma_{ab} = \frac{1}{2} (\rho_a + \rho_b) \qquad{(10)}$$

To distinguish between methods, the name of the method will be added in superscript, e.g. $\sigma^\text{CM}$, $\rho^\text{PC}$, etc.

Overview of phase defect detection algorithms

In the following, we will present several different methods to detect PDs. A tabular overview of all the methods is given in Table 2.

abbr.	name	rationale	centered	calc. time	source
CM	Cosine Method	Jumps in phase mod $2\pi$ are okay, in between: PDL.	on edges	${3.74~\mathrm{s}}$	¹
GLAT	Gradient of Local Activation Time	LAT jumps at PDL and wave front.	on edges	${2.13~\mathrm{s}}$
RPG	Real Phase Gradient	Phase jumps at PDL.	on edges	${4.90~\mathrm{s}}$	²
CPG	Complex Phase Gradient	Phase jumps at PDL.	on edges	${13.31~\mathrm{s}}$
PC	Phase Coherence	Phase is not coherent at PDL.	on edges	${10.03~\mathrm{s}}$
DM	Dipole Moment	Points on opposite side of PDL have opposite “charge”.	on vertices	${13.74~\mathrm{s}}$
SVF	Spatial Vector Field	Rotation of gradient is close to zero, except at PDL.	on vertices	${8.37~\mathrm{s}}$
AM	Angular Momentum	Use the classical topological charge also known as angular momentum.	on vertices	${1.47~\mathrm{s}}$	³
IPM	Inflection Point Method	Change of sign of the phase Hessian.	on edges	${10.62~\mathrm{s}}$

Table 2: Overview of existing and proposed algorithms for PD detection. Calculation times are for 500 frames in a medium of $450\times 450$ pixels on an Intel Core i7-10875H processor in a Numpy implementation without specific optimization for speed.

Cosine method (CM)

Tomii et al. (2021) introduced the following quantity to visualize PDs along an edge: $$ \tilde\sigma_{ab}^\text{CM} = \cos({\varphi}_a - {\varphi}_b) \qquad{(11)}$$ This method returns a value in $[-1,1]$, where low values indicate the presence of a PD.

To derive a normalized PD density with values in $[0,1]$, we modify this to: $$ \sigma_{ab}^\text{CM} = \frac{1}{2}[1 - \cos({\varphi}_a - {\varphi}_b)] \qquad{(12)}$$ We define a vertex-based version of this quantity via Eq. 9, which we denote with $\rho_a^\text{CM}$.

Gradient of local activation time (GLAT)

It is expected that around a PD, the LAT does not vary smoothly but instead jumps across this line. This implies that the gradient in the neighbourhood of the PDL should be much larger than further away where neighbouring vertices are activated subsequently.

This relation is easily expressed using edges: $$ \sigma_{ab}^\text{GLAT} = t^{\text{arrival}}({{\bm{{x}}}}_a) - t^{\text{arrival}}({{\bm{{x}}}}_b) \qquad{(13)}$$

Note that we are using a second condition here: If the elapsed time since excitation of either vertices is exactly zero, we still set $\sigma_{ab}^\text{GLAT}$ to zero. The rationale here is that the jump is due to the wave front passing instead of pointing to a PD.

A vertex-based variant is found by averaging over all edges leaving the same vertex, see Eq. 9, applied to the absolute value of the LAT difference: $$ \rho^\text{GLAT}_a = \frac{1}{N_a} \sum_{b \in \mathcal{N}(a)} |\sigma^\text{GLAT}_{ab}| \qquad{(14)}$$

Real phase gradient (RPG)

In previous work (Arno et al., 2021), we considered phase gradients, disregarding $2\pi$ phase differences: $$ \sigma_{ab}^\text{RPG} = | \operatorname{mod}({\varphi}_a-{\varphi}_b+\pi, 2\pi) - \pi | \qquad{(15)}$$ returning values in $[-\pi,\pi]$. The normalized cosine method (Eq. 12) can be seen as a mapping of this interval to a PD density taking values in $[0,1]$.

Complex phase gradient (CPG)

The next method works in a similar fashion, but to avoid the modulo operation, we look for gradients in the complex number $z = \mathrm e^{\mathrm i {\varphi}}$: $$ \sigma_{ab}^\text{CPG} = | e^{i{\varphi}_a} - e^{i{\varphi}_b} | \qquad{(16)}$$ From this a vertex-based density can be computed using Eq. 9.

Phase coherence (PC)

Inspired by the literature of phase oscillators (Kuramoto, 1984), we define the phase coherence of a vertex $a$ with neighbours $b$ as: $$ p_a = \frac{1}{N_a} \left| \sum_{b \in \mathcal{N}(a)} e^{i {\varphi}_b} \right| \qquad{(17)}$$

The PD density is then defined as $$ \rho_a^\text{PC} = 1 - p_a \qquad{(18)}$$ such that $\rho$ is large when the coherence is low, since a low phase coherence is expected near a PDL. This index $\rho^\text{PC}$ is normalized in $[0,1]$ with large values indicating high PD probability.

Note that applying the PC method to only two vertices connected via an edge delivers $$ \sigma_{ab}^\text{PC} = \frac{1}{2} |e^{i{\varphi}_a} + e^{i{\varphi}_b}| \qquad{(19)}$$ returning $\sqrt{1+\tilde{\sigma}^{CM}_{ab}}$.

Dipole moment (DM)

Along a PD, the phase values that surround a given point are expected to be divided into two groups, one on either side of the PDL. We could perhaps detect this splitting by using the concept of the dipole moment of a charge distribution, where the complex number $z = \mathrm e^{\mathrm i {\varphi}}$ takes the role of charge: $$ {{\bm{{p}}}}_a = {{\bm{{p}}}}({{\bm{{x}}}}_a) = \sum_{{{\bm{{x}}}}_b\in\mathcal N(a)} [ {{\bm{{x}}}}_b - {{\bm{{x}}}}_a ] e^{i {\varphi}_b} \qquad{(20)}$$

The point-based PD density is then found by taking the norm of this complex vector: $$ \rho^\text{DM}_a = ||{{\bm{{p}}}}_a || = \sqrt{{{\bm{{p}}}}_a^* \cdot {{\bm{{p}}}}_a} \qquad{(21)}$$ where $^*$ denotes complex conjugation.

When two vertices are connected by one edge, the edge-based implementation will recreate the CPG method. For this reason, no direct implementation of the latter was done. Still, edge-based values can be computed via interpolation, see Eq. 10.

Spatial vector field (SVF)

When Stokes’ law is applied to the expression of topological charge, one finds $$ Q = \int_C \nabla {\varphi}\cdot {{\bm{{d\ell}}}} = \iint_S \nabla \times \nabla {\varphi}\cdot {{\bm{{ds}}}} \qquad{(22)}$$ where $C$ is the boundary curve to the region $S$. Since for a continuous field ${\varphi}$, the rotation of the gradient $\nabla \times \nabla {\varphi}$ vanishes everywhere, a continuous phase field cannot bear non-zero topological charge. Nevertheless, computing $Q$ for all faces of the grid has been used to find PSs (Gray et al., 1995).

Inspired by the right-hand side of Eq. 22, and replacing ${\varphi}$ by $z=e^{i{\varphi}}$ to get easier differentiation, we propose: $$ \rho^\text{SVF} = || \nabla \times \nabla z || \qquad{(23)}$$ The motivation for this method is that a PD is essentially a discontinuity in the field ${\varphi}$. At such a discontinuity the rotation of the gradient may be different from zero. In our current implementation, we calculate the gradient in a vertex-based manner, e.g. $\partial_x u(x,y) = [u(x+dx,y) - u(x-dx,y)]/(2dx)$, such that the result is also vertex-based.

Angular momentum (AM)

A classical method to detect the central region in a spiral wave is using the pseudo-vector: (Winfree et al., 1996) $$ {{\bm{{l}}}} = \nabla V \times \nabla R \qquad{(24)}$$ Far away from the spiral core, the activation resembles a plane wave, making $\nabla V$ nearly parallel to $\nabla R$, such that ${{\bm{{l}}}} \approx {{\bm{{0}}}}$ except near the core of the spiral. Since PS can be seen as a limit of a PDL with vanishing length, we will visualize $$ \rho^\text{AM} = || {{\bm{{l}}}} || \qquad{(25)}$$ An edge-based method can be derived by taking: $$ \sigma_{ab}^\text{AM} = (\rho_a^\text{AM} + \rho_b^\text{AM})/2 \qquad{(26)}$$

Inflection point method (IPM)

Given that a PDL in practice connects two regions of different phase in an abrupt but continuous manner, it is interesting to look where the phase transition is the steepest, and localize the PD there.

For 1D functions, an inflection point is found where $f'(x)$ changes sign. This can be translated to the condition $f''(x) = 0$. To find the same region for a 2D function, we express that we want an inflection point when stepping in the direction of the local phase gradient. With ${{\bm{{e}}}}_g$ as the normalized gradient vector: $$ \nabla {\varphi}= {{\bm{{g}}}} = g {{\bm{{e}}}}_g \qquad{(27)}$$ the spatial derivative in the gradient direction is $\partial_g = {{\bm{{e}}}}_g \cdot \nabla$. With this, $\partial_g{\varphi}= g$, and the concavity in the direction of the gradient becomes: $$ F({{\bm{{x}}}}) = \sum_{i,j} g^i g^j \partial^2_{ij} {\varphi}= \sum_{i,j} g^i g^j H_{ij} \qquad{(28)}$$ where the Hessian of the phase is $H_{ij} = \partial^2_{ij} {\varphi}$. Hence, the PD can be found as the set of points where $F({{\bm{{x}}}}) =0$. This method is unlike the mentioned algorithms above, since it immediately returns a line, i.e. PDL of zero thickness. Note that in practice, one needs to impose a minimal value of $||\rho^{GLAT}||$ such that the background region with low PDL density is filtered out.

To compare this method to the other algorithms, we color the edges where $F({{\bm{{x}}}})$ changes sign with the phase gradient along that edge, i.e. $$ \sigma_{ab} = \sigma^{RPG}_{ab}\ H[ - F({{\bm{{x}}}}_a) F({{\bm{{x}}}}_b) ] \qquad{(29)}$$ with the Heaviside function $H$, which takes the value 1 for inputs larger than zero and 0 otherwise.

Visual representation of the methods

For the methods that return a vertex-based PD density $\rho$, we simply color the pixels in the rectangular grid according to $\rho$. For the methods that return an edge-based PD indicator $\sigma$, we color the dual grid, i.e. we color every point in the plane according to its nearest edge. This results in a coloring of the plane using pixels that are 45 ° tilted and centered around the midpoint of edges in the original grid. In this way, interpolation between edges and vertices does not affect the presented results.

Post-processing of phase defects

Having obtained a PD density $\rho({{\bm{{x}}}})$ using one of the methods, we keep only points above a threshold value $\rho_c$ to obtain a set of points on the PDL, and connect it using the minimal spanning tree graph algorithm. Thereafter, the smallest branches of each tree are cut to gain a discrete representation of a PDL, centered at the vertices of the image grid.

To measure PDL length $L$, the PDL points are connected by line segments; the sum of their lengths is taken as an estimate to the PDL length.

To measure PDL precession speed, we first selected a spatial region where only one PDL was seen during the timespan of interest. Then, we performed principal component analysis (PCA) to the point cloud of the PDL at all time instances to obtain the main vector of alignment ${{\bm{{e}}}}_1$. The angle between this vector and the positive $x$-axis is taken to be $\beta$, after which linear regression of $\beta(t) = \beta(0) + \omega t$ yields an estimate for the precession frequency $\omega$ and the precession period $T = \frac{2\pi}{\omega}$.

Performance tests

Data sets with Gaussian additive noise

The data sets we are working with in this paper are quite smooth in the sense that there is very little noise in them. This is due to the data from in-silico simulations being smooth by design and the optical voltage mapping data being pre-processed with Gaussian smoothing before calculating the phase and phase defects (section 3.1.2).

To gauge how well the algorithms are able to deal with noise, we have added noise following a normal distribution with different signal to noise ratios (SNRs) to the data sets.

In the context of our work, we define SNR as the ratio of the standard deviation of the signal $u$ to the standard deviation of the noise $n$ $$ \text{SNR} = \frac{\operatorname{stddev}{u}}{\operatorname{stddev}{n}} = \frac{\sqrt{{{\left\langle (u - {{\left\langle u \right\rangle}})^2 \right\rangle}}}}{\sqrt{{{\left\langle (n - {{\left\langle n \right\rangle}})^2 \right\rangle}}}} \qquad{(30)}$$ with ${{\left\langle f \right\rangle}}= \sum_{k=1}^N f({{\bm{{x}}}}_k, t_k)$ on all $N$ points of the space-time grid.

Data sets at lower spatial resolution

In many contexts, the only available data are at much lower spatial resolution than the in-silico simulations and optical mapping experiments considered in this work. For instance, there are electrode arrays that are inserted via balloon catheters with 64 electrodes (Schilling et al., 1998).

To still be able to assess the effectiveness of the phase and PD algorithms at such resolutions, we down-sample our data by a factor of $N\in\mathbb N$ by pooling $N\times N$ grid points together using the arithmetic mean. Vertices that are not inside the medium are not included in this mean.

Data sets: recovery of an obstacle

There is no ground truth in the exact location of the PD as it depends on the choice of algorithms for phase and PD calculation. Still, experiments are possible where the expected location of a PDL is known. We have conducted such an experiment in-silico where we add an elongated, thin obstacle for a rotor to attach to.

For this, we have taken the last frame from the experiment using the FK model as the initial state of this experiment and placed such an obstacle at the core of the observed rotor.

For the length of the obstacle, we have chosen roughly the previously observed PDL length. We choose its width to be less than eight grid lengths $8\Delta x$, such that when we downsample the data by a factor of eight as in section 3.6.2, all pixels in the medium are again active. To put it briefly, the obstacle is designed such that it is too thin to be detected directly at that lower spatial resolution. Instead, we use phase defect detection to recover the obstacle.

For this experiment, we used the parameters for the FK model as in Table 1, except for the time step $\Delta t = {0.1~\mathrm{{m}{s}}}$ and duration

To recover the obstacle from the simulation, we have used the different phase and PD algorithms to calculate PD densities $\rho({{\bm{{x}}}}, t)$. Recall that all methods are designed such that high $\rho({{\bm{{x}}}}, t)$ corresponds to high likelihood of a PD being located there. We calculate a prediction of the location of the obstacle $\chi({{\bm{{x}}}})$ based on $\rho({{\bm{{x}}}}, t)$ as follows:

For a duration of at least one rotation of the spiral, calculate the mean value of the PD density $\rho({{\bm{{x}}}}, t)$ at each point in space. Call this quantity $\bar\rho({{\bm{{x}}}})$.
As $\rho({{\bm{{x}}}}, t)$ is close to zero except for at the phase defect, the distribution of values in $\bar\rho({{\bm{{x}}}})$ will also be heavily skewed towards lower values. Therefore, we clip $\bar\rho({{\bm{{x}}}})$ to the interval from the 40th percentile to the 99th percentile to get rid of both small fluctuations around zero and outliers.
To obtain $\chi({{\bm{{x}}}})$, we finally rescale to the interval $[0,1]$ and then round to $\{0,1\}$. This quantity can be thought of as an approximation of the characteristic function $\hat\chi({{\bm{{x}}}})$ of the set of points in the obstacle.

Finally, to be able to judge how well the obstacle has been recovered, we calculate the classification error as the fraction of misclassified points by the prediction $\chi({{\bm{{x}}}})$ with respect to the ground truth $\hat\chi({{\bm{{x}}}})$.

Results

We here apply the different proposed detection methods for the three phases ${\varphi}^{\text{act}}$, ${\varphi}^{\text{LAT}}$, and ${\varphi}^{\text{skew}}$. We do this in three cardiac monodomain models and compare performance of the methods (section 3.1.1). Finally, we apply a selection of methods to an experimental dataset obtained by optical voltage mapping of a monolayer culture of cardiac cells (section 3.1.2).

Comparison of different phase definitions

Fig. 3 shows the three phase definitions applied to a snapshot of the three monodomain models, the Aliev-Panfilov (AP) model (Aliev & Panfilov, 1996), the Fenton-Karma (FK) model (Fenton & Karma, 1998), the Bueno-Orovio-Cherry-Fenton (BOCF) model (Bueno-Orovio et al., 2008), and an optical voltage mapping experiment.

The AP model shown in the first row of Fig. 3 produces a rigidly rotating spiral. With ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{skew}}$, a PS is seen. However, due to the thresholding on $V$ used to determine LAT, the inner part of the core region is never excited, such that ${\varphi}^{\text{LAT}}$ shows an abrupt change at the trajectory of the classical PS, which will be picked up as a PD below.

In the simulations with linear core (FK and BOCF models), ${\varphi}^{\text{act}}$ shows sudden transitions at the rotor core and the wave front, while ${\varphi}^{\text{skew}}$ and ${\varphi}^{\text{LAT}}$ only show a distinct phase gradient near the conduction block line.

The optical voltage mapping experiment in Fig. 3 J-L shows apparent PSs for ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{skew}}$, but an extended PD for ${\varphi}^{\text{LAT}}$. Hence, at first sight, it resembles the AP spiral, but this relation will be further investigated below using the PD detection techniques outlined above.

Fig. 4 shows a scatter plot between the different phases for the FK frame shown in Fig. 3. We took the convention that the phase at the resting state is 0. The skewed phase ${\varphi}^{\text{skew}}$ with parameters tuned as outlined above resembles the elapsed time phase ${\varphi}^{\text{LAT}}$ much closer than the state space phase ${\varphi}^{\text{act}}$. In short, ${\varphi}^{\text{skew}}$ is an approximation to ${\varphi}^{\text{LAT}}$ that does not require observation of the system during the previous excitation sequence.

Comparison of phase defect detection methods in simulations

We have applied all PD detection methods (section 3.3) to all experiments, for all phases. For definiteness, we only show the result for the FK model in Fig. 6 and Fig. 7, but the others can be found in the Supplementary Materials (Fig. 17, Fig. 18, Fig. 19, and Fig. 20).

Figure 6: Overview of PD detection methods for one snapshot of the FK data set. The PD on vertices $\rho$ or edges $\sigma$ is measured in arbitrary units. The same coloring as in Fig. 2 C is used here. As the PDL has a width of only a few grid points, we zoom in around the turning point to get a better view of the structure on the grid. For reference, we also show the corresponding frame of the transmembrane voltage $V = u$ in panel A.

Figure 7: Overview of more PD detection methods for one snapshot of the FK data set as in Fig. 6. Each row shows a detection algorithm, applied to the three different phase definitions.

In the following, we will present a selection of the results exhibiting the common features and our main observations regarding the different methods.

In general, all methods return low densities away from the wave front and PDL and higher values near the region of interest, although the precise PD density distribution is different between the methods.

In Fig. 6 B, the GLAT method clearly shows the conduction block line at the rotor core. Since LAT is discontinuous there, the set of points is thin such that small gaps can be seen. At the rightmost part, the PDL doubles, since the process of reaching and leaving the rightmost turning point, both leave a discontinuity in LAT. Moreover, the PDL’s precise location depends on the chosen threshold $V_*$.

The AM method (Fig. 6 C) locates only the site where the wave front meets on the PDL. Also it can be seen that $\rho^\text{AM}$ is located at the wave front, though with much lower magnitude. This is consistent with this method traditionally being used for PS detection.

The other methods in Figs. 6, 7 are phase-based. In each case, the wave front is most clearly seen as an artifact using ${\varphi}^{\text{act}}$, less visible using ${\varphi}^{\text{skew}}$ and absent in ${\varphi}^{\text{LAT}}$.

The CM, RPG and CPG methods (Fig. 6 D-L) and PC and DM methods (Fig. 7 A-F) give all qualitatively similar results: With ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{LAT}}$, not only the PDL but also the end point of the wave front (tip) is stressed. The ${\varphi}^{\text{LAT}}$-variant distinctly shows the PDL, as the wave front is filtered out by the definition of ${\varphi}^{\text{LAT}}$.

The IPM method shows a line that is only one pixel wide, as it was designed to localize the PD at the site of steepest phase variation.

Finally, the SVF method yields many points in the region of interest, but the result is noisy even in this idealized simulation.

Comparison of phase defect detection methods in an optical voltage mapping experiment

We also applied the different phase definitions and detection methods to the excitation sequence observed in a hiAM monolayer, as detailed in section 3.1.2. In Figs. 8, 9, we show the results of this process for all of those methods. Fig. 8 A shows the optical intensity at a given time in a multiple-spiral state. The non-phase methods GLAT and AM in Fig. 8 B-C show non-zero densities at several positions that are similar in both methods. The four most intense points, at which either a PS or PDL could be present, are confirmed by the other methods (CM, RPG, CPG, PC, DM, IPM) using ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{skew}}$. When using ${\varphi}^{\text{LAT}}$, the same 4 points are prominent, but they extend to a line (PDL) since the LAT and ${\varphi}^{\text{LAT}}$ keep track of the recent history of excitation. Compared to the simulation data, more background structures are seen in the optical voltage mapping data, such as borders of excited regions and a staircase effect in LAT due to the time sampling.

Figure 8: Overview of PD detection methods for one snapshot of the optical voltage mapping data. The data are presented in the same way as in Fig. 6.

Figure 9: Overview of more PD detection methods for one snapshot of the optical voltage mapping data as in Fig. 8.

Properties of PDLs in-silico and in-vitro

The presented methods allow to characterize the observed PDLs in terms of length $L$ and orientation angle $\beta$, which is a further step in the quantitative analysis of excitation patterns.

Fig. 10 shows the length over time of one PDL, in simulations and experiment. The PDL is detected using the LAT phase ${\varphi}^{\text{LAT}}$ as input for the phase coherence method (PC). We observe that the PDL length varies over time. Its time-averaged value is summarized in Table 3.

Figure 10: Length of detected PDLs over time. For one of the PDLs detected by the PC method for ${\varphi}^{\text{LAT}}$, we show how its length changes over time. This length fluctuates around an average value.

model / experiment	average length $L$	precession period $T$
AP simulation
FK simulation
BOCF simulation
hiAM optical voltage mapping

Table 3: Statistics of one PDL’s length and precession over time. We observe similar values for other PDLs in the data.

In both simulations and experiment, we also estimated the precession period $T$ of the PDL, see Table 3 and Fig. 11. The orientation of the PDL changes almost linearly in all cases, hence, we observe quite low variance along the fit linear functions. On the one hand, in the FK and BOCF simulations, we see that $\beta$ almost stays constant, but slightly precesses in one direction. On the other hand, in the AP model and optical voltage mapping data, the precession takes place in a much shorter period of time: $T$ is just 1.5 to 3 times longer than the APD in both of those cases.

Figure 11: PDL orientation over time. For these figures, we use the same PDLs as in Fig. 10, Table 3.

Robustness to noisy data

We have also investigated how the methods perform under noisy conditions. For this we considered noisy data which were obtained as outlined in section 3.6.1. We then ran the different PD detection algorithms on those sets as well.

A general observation is that if the input data $V$ is noisy, so are the state space phases ${\varphi}^{\text{act}}$ and ${\varphi}^{\text{skew}}$, which then can be seen in $\rho$ as well.

Another effect can be observed for ${\varphi}^{\text{LAT}}$, as it is based on LAT: When $V$ increases such that it crosses the threshold $V_*$, the value of LAT is updated to the current time. This can be triggered by noise which is especially critical right after $V$ decreases falling below the threshold. A random fluctuation due to noise can then push it above the threshold again.

To counter this effect, we use a second threshold value $V_*^\downarrow$ in the calculation of LAT and ${\varphi}^{\text{LAT}}$ (section 2.1). A good value for $V_*^\downarrow$ can be obtained by decreasing $V_*$ by an offset that is proportional to the SNR. This value also depends on the wave back in the tissue model.

$V_*^\downarrow$ must be chosen low enough such that it suppresses LAT triggering due to noise at the wave back, but high enough that the tissue always repolarizes below it before the tissue can be excited again.

Figure 12: Effect of additive white Gaussian noise on the detection of PDs in the FK data set. The PD has been determined using the PC method applied to the LAT phase ${\varphi}^{\text{LAT}}$ with two thresholds $V_*^\downarrow=0.5$ and $V_*=0.65$. In the three columns, we vary the SNR. The data are presented in the same way and same point in time as in Fig. 6.

As an example, we show the PD $\rho$ as determined by the PC method based on the LAT phase ${\varphi}^{\text{LAT}}$ for the FK data set at three different levels of noise in Fig. 12. We choose $V_*^\downarrow=0.5$ and $V_*=0.65$. It can clearly be seen that this method succeeds to locate the PDL for SNRs above a critical value. In this example this critical value is around 10 db. Here it can be seen that at various pixels in the medium a random fluctuation due to noise has pushed the input data from below $V_*^\downarrow$ to above $V_*$. At this stage it is still quite clear where the PDL is located. With more noise than this, however, this effect takes over. This leads to being unable to distinguish the PDL from the noise artifacts in the very noisy case with $\text{SNR} = {5~\mathrm{dB}}$.

Performance at lower spatial resolution

Figure 13: Performance of PD detection at different spatial resolutions for the BOCF data set. The PD has been determined using the PC method applied to the LAT phase ${\varphi}^{\text{LAT}}$. The data are presented in the same way as in Fig. 6.

We also applied the PD detection algorithms to the data sets at different, lower spatial resolutions (section 3.6.2). In Fig. 13, we present a frame of the input data $V$ and the phase defect $\rho$ for PC method and the LAT phase ${\varphi}^{\text{LAT}}$. It can be seen that even at those resolutions, the PDL can still successfully be identified. For grid lengths larger than the length of a PDL, only few pixels have high enough $\rho$ to be considered a PD. This illustrates that PSs and PDLs can not be distinguished from one another when resolution is too low.

Also note that the jump in phase due to the wave front and back passing through an area is larger at lower spatial resolution. As a PD is a large jump in phase, our methods detect this jump as well. The wave front is therefore harder to distinguish from a PD at low resolutions. This effect is much stronger when using ${\varphi}^{\text{act}}$ but can be reduced by using ${\varphi}^{\text{LAT}}$ or ${\varphi}^{\text{skew}}$.

Recovery of an obstacle

In section 3.6.3, we designed an experiment such that a ground truth location for a PDL is known. An elongated obstacle was placed such that a rotor could attach to it.

Looking at the resulting data, we see that in fact the rotor has attached to this obstacle. A PDL formed around the obstacle.

With these recordings in $u$ and $v$ for the FK model, we moved on to calculate the phases ${\varphi}$, PD densities $\rho$, and the approximation of the characteristic function $\chi$ of the obstacle for all different methods. In Figs. 14, 15, we present the recovered obstacles by each of the methods.

Figure 14: Recovery of the location of an obstacle based on each of the PD detection methods. As outlined in section 3.6.3, we calculate an approximation of the characteristic function $\chi({{\bm{{x}}}})$ of the obstacle for each of the methods. We then calculate the classification error comparing this prediction to the ground truth $\hat\chi({{\bm{{x}}}})$.

Figure 15: Recovery of the location of an obstacle based on each of the PD detection methods as in Fig. 14.

It can be seen that all methods perform well at recovering the obstacle leading to a classification error of around or less than We have therefore validated the methods and shown that they are able to locate PDLs.

The AM method is also able to predict the obstacle well, even though it is based on PSs. This is because PSs follow the PDL. While the PD based methods show the full extent of the line, the PS based AM method only returns a point-like region.

While there is only functional re-entry in the optical voltage mapping data set, we can still use the same algorithm to recover the long-term location of the PDL in the OM data using the PC method and LAT phase. As can be seen in Fig. 16, the cores of the spirals can successfully be detected using this method. Our prediction $\chi({{\bm{{x}}}})$ of the sites of functional obstacles via PDLs can be compared to so-called driver domains, specifically to driver-density maps which are based on PSs (Haissaguerre et al., 2014).

Figure 16: Prediction of the characteristic function $\chi$ of the effective obstacles based on $\rho^\text{PC}({\varphi}^{\text{LAT}})$ for the OM data set. Due to the functional re-entry, the PDLs in the centres of the spirals observed in this recording effectively form functional obstacles. These differ from anatomical obstacles in the way that functional ones could excite and do in fact excite in this case before the rotors form. We recover those obstacles via $\chi$ as outlined in section 3.6.3.

Discussion

In this paper, we provide and compare several numerical methods to detect a PDL, a recently proposed structure present at the core of a rotor as an alternative for the classical PSs (Arno et al., 2021; Tomii et al., 2021). Here, we attempt to improve the simple PDL detection methods from these works (CM & RPG) and tested them on simulations and experimental data.

Several phase-based algorithms were applied, not only the classical phase ${\varphi}^{\text{act}}$ but also the recently introduced LAT-based LAT phase ${\varphi}^{\text{LAT}}$ (Arno et al., 2021), since the LAT better keeps the spatio-temporal activation and therefore more clearly shows extended PDs. In addition to a systematic comparison between detection methods, we also introduce a third phase, the skewed phase ${\varphi}^{\text{skew}}$ in this work. ${\varphi}^{\text{skew}}$ was designed as a way to estimate ${\varphi}^{\text{LAT}}$ from a single snapshot. This is useful in the post-processing of data from experiments or simulations, where a sparse time-sampling was used.

As ${\varphi}^{\text{act}}$ transitions on small scales at the wave front and back, they may wrongly be identified as PDLs for this phase definition. Although ${\varphi}^{\text{skew}}$ filters the wave front and back better than ${\varphi}^{\text{act}}$, we still find that direct measurements of LAT and ${\varphi}^{\text{LAT}}$ produce better-resolved PDLs (cf. Fig. 6). Still, in the regime of fast depolarization, wave front and wave back can be considered PDs. These can be distinguished from PDLs due to conduction blocks via two criteria:

Wave fronts propagate in space.
Wave front and wave back connect always the same phase values, while PDLs connect other values.

When comparing the different methods to convert phase into a PD, we find a good performance and strong correlation between the previously coined cosine method (Tomii et al., 2021) and the real phase gradient method (Arno et al., 2021). Both methods are based on the same idea (measuring angular differences along a circle) and therefore the correlation comes as no surprise. Qualitatively similar performance is found by related methods (CPG, CM, PC). The IPM method also works well. Some advanced methods such as DM, and SVF actually performed worse in terms of contrast and noise suppression. Of special interest is the classical AM method (Winfree et al., 1996), which consistently finds the end point of the wave front, even if lying on a PDL, and the GLAT method. The gradient of LAT correctly identifies the PDL and locates it very sharply (by construction); however, its precise location depends on the chosen threshold ($V_*$) to classify tissue as excited or not. Regarding the calculation time, we find that the CM, AM and GLAT are the fastest and therefore recommended to use for processing larger datasets, e.g. extended in time or in three spatial dimensions.

To calculate the state space phase for the OM data, we have used a time-delayed version of the observed variable $u$ to be able to calculate the state space phase ${\varphi}^{\text{act}}$. Note that this traditional approach is unable to detect PDLs, as it gives distinct points where ${\varphi}^{\text{act}}$ is high, corresponding to classical PSs. This explains why line shaped PDs were not investigated closer before. In the panels of Figs. 8, 9 that use the LAT phase, it is however apparent that regions with different LAT do indeed touch each other and therefore form a PD. Depending on the time resolution of the LAT map, staircase artifacts can be seen. However these can be filtered away by thresholding at an appropriately high value of $\rho$, as at those artifacts $\rho$ is still much lower than at the PD.

To show the power of these methods, we applied them to a simulation and an OM experiment to find the length and orientation of PDLs over time. Here we conclude that using a method sensitive to PDLs allows to also identify linear rotor cores in experiment. However, having identified linear cores (PDLs) in optical voltage mapping of intact rabbit hearts (Arno et al., 2021) and human immortalized atrial myocyte cultures (in this work) does not allow to draw general conclusions. Therefore, we propose to use the suggested methods also on other datasets, starting with existing optical voltage mapping results. Then, the presented methods can be used to characterize PDL size and rotation. Such measurement would give another handle to judge the degree to which mathematical models of the heart resemble reality, in addition to e.g. reproducing restitution curves and observing basic spiral dynamics in terms of meander and stability.

We have verified that the proposed methods also work well when only noisy data or data at low resolution is available. In another experiment where the location of a thin, elongated obstacle was known as ground truth, we have seen that all proposed methods successfully are able to recover its location. We conjecture that therefore the methods also work well to recover PDLs in the setting of anatomical re-entry. More work is needed to better characterize the defect line analogues of functional and anatomical re-entry. In certain cases, the anchoring site may be a hybrid version of this classical distinction: Part of the linear rotor core can lie at an obstacle, or a functional region (PS or PDL) can be attracted to a inhomogeneity in the medium, to stay in place there, see Fig. 16.

The methods used here are available as Python scripts from our online repositories, see the data availability statement. Please cite this paper when using the implementation. Note that some methods were originally introduced elsewhere: The AM method (Winfree et al., 1996), the CM method (Tomii et al., 2021) and RPG method (Arno et al., 2021). The algorithms have currently been tested on dense Cartesian grids in 2D, but can naturally be extended to 3D and time, and unstructured grids (meshes), where the distinction between vertex-based densities $\rho$ and edge-based densities $\sigma$ will play a more prominent role.

Conclusion

In this work, we demonstrate that in order to visualize PDs in dense 2D data, it is recommended to use LAT-based methods or to use the skewed-phase to derive it from snapshots. Several algorithms were proposed to highlight the PDs visually, for which the simple methods (CM, RPG, CPG, PC) were most effective.

We applied the methods to simulations and an optical voltage mapping experiment; in the latter case we found that in a hiAM cell-culture, the average PDL length in a multi-spiral state was and precession period $T$ was . We made our detection methods publicly available on our institutional repository and hope it can serve to further help understanding the building blocks of cardiac excitation patterns.

Supporting information

In the supplementary material, we provide figures of PD densities $\rho$, $\sigma$ for the different methods for AP and BOCF reaction kinetics, to enable a full comparison between methods (Figs. 17-20).

The numerical methods implemented for this paper are available as a Python module at https://gitlab.com/heartkor/py_ithildin. The Python scripts used to generate the figures in this paper are available at https://gitlab.com/heartkor/scripts-pdl-detection. Finally, we have archived the simulation output and pre-processed optical voltage mapping data on which the scripts were applied on Zenodo (DOI: 10.5281/zenodo.6477532). This archive also contains the Python module and scripts.

Please cite this paper when using the implementation and/or the data.

Figure 17: Overview of PD detection methods for one snapshot of the AP data set. The data are presented in the same way as in Fig. 6.

Figure 18: Overview of more PD detection methods for one snapshot of the AP data set as in Fig. 17.

Figure 19: Overview of PD detection methods for one snapshot of the BOCF data set. The data are presented in the same way as in Fig. 6.

Figure 20: Overview of more PD detection methods for one snapshot of the BOCF data set as in Fig. 19.

Addenda

Acknowledgments:

We are grateful to Sven O. Dekker, Niels Harlaar, Daniël A. Pijnappels and Antoine A.F. de Vries for providing optical voltage mapping data of cardiomyogenically differentiated hiAM monolayers. Moreover, we thank Tim De Coster for helpful comments on the analogy between a PDL and the spiral wave tip trajectory.

HD and AVP conceived the study. DK, LA and LL ran the numerical simulations and implemented the detection algorithms. All authors wrote parts of the manuscript and assisted in internal reviewing.

Funding:

DK is supported by KU Leuven grant GPUL/20/012. LA was funded by a KU Leuven FLOF grant and a FWO-Flanders fellowship, grant 1177022N; LL was funded by KU Leuven and FWO-Flanders, grant G025820N. Research at Sechenov University was financed by the Ministry of Science and Higher Education of the Russian Federation within the framework of state support for the creation and development of World-Class Research Centers “Digital biodesign and personalized healthcare” 075-15-2020-926. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests:

The authors have declared that no competing interests exist.

Copyright:

© 2022 Kabus et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Peer review history:

PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: https://doi.org/10.1371/journal.pone.0271351.

Author contributions:

DK: Data curation, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. LA: Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft. LL: Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft. AVP: Conceptualization, Methodology, Project administration, Resources, Supervision, Writing – original draft, Writing – review & editing. HD: Conceptualization, Methodology, Project administration, Resources, Supervision, Writing – original draft, Writing – review & editing.

References

Aliev, R. R., & Panfilov, A. V. (1996). A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals, 7(3), 293–301. https://doi.org/10.1016/0960-0779(95)00089-5

Arfken, G. B., & Weber, H. J. (1995). Mathematical methods for physicists; 4th ed. Academic Press.

Efimov, I. R., Sidorov, V., Cheng, Y., & Wollenzier, B. (1999). Evidence of three-dimensional scroll waves with ribbon-shaped filament as a mechanism of ventricular tachycardia in the isolated rabbit heart. Journal of Cardiovascular Electrophysiology, 10(11), 1452–1462. https://doi.org/10.1111/j.1540-8167.1999.tb00204.x

Gray, R. A., Pertsov, A. M., & Jalife, J. (1998). Spatial and temporal organization during cardiac fibrillation. Nature, 392(6671), 75–78. https://doi.org/10.1038/32164

Haissaguerre, M., Hocini, M., Denis, A., Shah, A. J., Komatsu, Y., Yamashita, S., Daly, M., Amraoui, S., Zellerhoff, S., Picat, M.-Q., Quotb, A., Jesel, L., Lim, H., Ploux, S., Bordachar, P., Attuel, G., Meillet, V., Ritter, P., Derval, N., Sacher, F., Bernus, O., Cochet, H., Jais, P., & Dubois, R. (2014). Driver domains in persistent atrial fibrillation. Circulation, 130(7), 530–538. https://doi.org/10.1161/circulationaha.113.005421

Keener, J. P. (1991). The effects of discrete gap junction coupling on propagation in myocardium. Journal of Theoretical Biology, 148(1), 49–82. https://doi.org/10.1016/S0022-5193(05)80465-5

Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence. Springer-Verlag. https://link.springer.com/book/10.1007/978-3-642-69689-3

Priebe, L., & Beuckelmann, D. J. (1998). Simulation study of cellular electric properties in heart failure. Circulation Research, 82(11), 1206–1223. https://doi.org/10.1161/01.res.82.11.1206

Schilling, R. J., Peters, N. S., & Davies, D. W. (1998). Simultaneous endocardial mapping in the human left ventricle using a noncontact catheter: Comparison of contact and reconstructed electrograms during sinus rhythm. Circulation, 98(9), 887–898. https://doi.org/10.1161/01.CIR.98.9.887

Winfree, A. T. (1974). Patterns of phase compromise in biological cycles. Journal of Mathematical Biology, 1(1), 73–93. https://doi.org/10.1007/BF02339491

Winfree, A. T. (1991). Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media. Chaos: An Interdisciplinary Journal of Nonlinear Science, 1(3), 303–334. https://doi.org/10.1063/1.165844

Zykov, V. S. (1987). Simulation of wave processes in excitable media. Manchester University Press.

Tomii et al. (2021) ↩︎
Arno et al. (2021) ↩︎
Winfree et al. (1996) ↩︎

Summer School in Bordeaux

Sun, 03 Jul 2022 21:20:44 +0200

French culture

LIRYC Summer School

New things and old things in Copenhagen

Tue, 05 Apr 2022 19:03:44 +0200

Impressions of Danish culture

Modern architecture

Unfortunately, many buildings that are built nowadays look like UFOs, that landed right in the middle of European old towns.

Older architecture

I personally prefer buildings in the old architectural styles.

EHRA conference

A few days in Luxembourg

Sat, 13 Nov 2021 15:42:18 +0100

Island hopping near Istria

Mon, 26 Apr 2021 11:13:43 +0200

Your browser doesn't support HTML5 video. Here is a link to the video instead.

Dolphins sighted!

This sailing trip was organised by SIRUB, the sailing club of Ruhr-University Bochum. If you live in the Ruhr area and want to learn sailing, feel free to visit SIRUB.com.

What I still want to see in New Zealand

Sat, 25 Apr 2020 13:31:00 +0200

This blog post is only available in German.

The Fall of New Zealand

Thu, 16 Apr 2020 14:37:00 +0200

This blog post is only available in German.

Water ist the mountains' blood

Sat, 28 Mar 2020 13:07:00 +1300

This blog post is only available in German.

The raw, vertical West Coast

Fri, 13 Mar 2020 20:45:00 +1300

This blog post is only available in German.

The North of the South

Tue, 03 Mar 2020 12:44:00 +1300

This blog post is only available in German.

Lots of Leeway

Mon, 24 Feb 2020 14:30:00 +1300

This blog post is only available in German.

From Mount Taranaki to the Interislander

Tue, 21 Jan 2020 17:17:00 +1300

This blog post is only available in German.

Summer, sun, christmas

Thu, 02 Jan 2020 23:00:00 +1300

This blog post is only available in German.

Roadtrip of the Rings

Sun, 22 Dec 2019 17:00:00 +1300

This blog post is only available in German.

Boat trip at the end of the blueberry season

Tue, 03 Dec 2019 22:38:00 +1300

This blog post is only available in German.

Blueberry picking in the Far North

Mon, 11 Nov 2019 14:55:00 +1300

This blog post is only available in German.

Finding paradise in the Bay of Islands

Fri, 25 Oct 2019 20:24:00 +1300

This blog post is only available in German.

In the beginning, there was Auckland

Sun, 20 Oct 2019 20:44:00 +1300

This blog post is only available in German.

The breathtaking Blue Mountains

Thu, 10 Oct 2019 14:23:00 +1100

This blog post is only available in German.

Sydney -- metropolis on the other side of the globe

Fri, 04 Oct 2019 12:02:00 +1000

This blog post is only available in German.

How to use Syncthing with Wi-Fi Direct on Android

Sat, 28 Sep 2019 10:45:00 +1000

We can use the Android mobile hotspot feature to create a network that we will use to sync files.

1. Turn off mobile data usage

Since we don’t want and don’t need to use the internet to sync files, we can turn off mobile data.

2. Turn on Wi-Fi hotspot

Open up a mobile hotspot on your phone. You will need to choose the name of your network and a password, if you’ve never used this feature before.

Connect your PC to the Wi-Fi network you just created and make sure, Syncthing is running on the PC.

3. Change run conditions

If you open up the Syncthing app on your Android phone, you will probably get a message that it is not currently running. We need to change this.

Mainly, we need to make sure that it also runs without an internet connection. To do this turn on the setting to let it run in flight mode.

You might also need to change other settings. For example, it can be useful to turn on sync when there is no charging cable (AC cable) is connected.

That’s it

If you set up the run conditions and the Wi-Fi connection successfully, Syncthing should now start syncing:

This has been tested with Syncthing 1.2.2 on Android 8.1 and Arch Linux.

Designs for RUB shirts und pullovers

Sat, 15 Jun 2019 22:49:00 +0200

This blog post is only available in German.

Simple JavaScript gallery

Sun, 26 May 2019 19:42:00 +0200

In some recent blog posts, I added quite a lot of images. So, I thought it would be nice, to be able to click on an image to view it full screen and then to scroll through all images on the page. I did not find a minimal script that did just that, all options were very bloated, so I just wrote my own slideshow script.

The script is so simple, that this blog post contains all the code, so you can take it, adapt it, and make it your own.

Example

An example can be found here: SIRUB.com

Features

click on any image on the page to view it full screen
use $\leftarrow$/$\rightarrow$ or k/j or h/l or a/d or w/s to flip through all images
hit Escape or q or tap anywhere to leave the gallery and return to the page

Installation

The installation is simple:

Just include the gallery.js file at the end of your html page

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<script src="gallery.js"></script>

add this to your css stylesheet:

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#gallery {
 max-width: 100%;
 max-height: 100%;
 position: fixed;
 left: 50%;
 top: 50%;
 transform: translate(-50%, -50%);
 border: 100rem solid black;
 background: white;
}

you do not have to change how you embed your images, all <img> tags are used for the gallery:

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<img src="photo1.jpg" />
<img src="photo2.jpg" />

Source code

Here are the contents of gallery.js:

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/*
 Simple JavaScript Gallery v1.0
 by Desmond Kabus
 https://www.kabus.eu/blog/2019/05-26-simple-js-gallery.html
*/

const imgs = document.images;
var img_index_open = -1;

function close() {
 var gallery = document.getElementById("gallery");
 if ( gallery )
 { document.body.removeChild(gallery); }
 img_index_open = -1;
}

function open(i) {
 close();
 if( 0 <= i && i < imgs.length )
 {
 img_index_open = i;
 var gallery = imgs[img_index_open].cloneNode(true);
 gallery.style = "width:auto; height:auto;";
 gallery.id = "gallery";
 gallery.onclick = close;
 document.body.appendChild(gallery);
 }
}

for ( let i = 0; i < imgs.length; i++ ) {
 imgs[i].onclick = function(){ open(i) };
}

window.addEventListener("keydown", function (event) {
 if ( img_index_open < 0 || img_index_open >= imgs.length )
 { return; }
 else if ( ["Escape", "q"].includes(event.key) )
 { close(); }
 else if ( ["ArrowLeft","ArrowUp","h","k","w","a"].includes(event.key) )
 { open(img_index_open - 1); }
 else if ( [" ","ArrowRight","ArrowDown","l","j","s","d"].includes(event.key) )
 { open(img_index_open + 1); }
});

Sailing trip in the Baltic Sea

Tue, 21 May 2019 23:21:00 +0200

This blog post is only available in German.

How to Choose a New Phone

Sun, 28 Apr 2019 21:02:00 +0200

My new phone: the Samsung Galaxy S9, codename starlte with LineageOS and KISS Launcher

Introduction

It comes down to this: Is there anything that’s an absolute deal breaker and what are your must-haves?

For me it is absolutely crucial that LineageOS is available for my phone, which is a free and open-source phone operating system based on the Android Open Source Project. For most people, this is probably not what they want since you void your warranty installing it, but I enjoy it. It has no bulky apps you cannot remove and avoids the weirdness the phone manufacturers force upon you. You could use your phone entirely without closed-source Google apps, of which nobody really knows what they are doing. They could be used to spy on you. Or not. But because you can not look into their source code, you will never really know.

There are some other things, I would like. I really do not get, why so many phones do not have a headphone jack anymore. Another trend, I think is really annoying is phones with glass backs. It’s bad enough that one side easily breaks when you drop your phone, so its ridiculous to put glass on the other. But people really like wireless charging. I like the concept of charging with coils, but so far the efficiency and speed of charging lacks compared to a simple cable. I’d prefer a more sturdy phone with a metal back.

So to summarize, this was my wishlist for my ideal phone:

LineageOS
dual SIM
not gigantic
JerryRigEverything tested and found durable
expandable or large storage
headphone jack

This stuff would be nice, but is not required:

small size
waterproof
fingerprint scanner
good camera
relatively cheap
USB-C
fast charging
large battery

I actually do not need/want this:

wireless charging
multiple cameras on the back
glass back

But my perfect phone does not exist.

Analysing some data

Fortunately, the LineageOS project provides some data that can easily be analysed to make a better choice. I just read the yaml files with python, applied some filters and drew some plots. The most eye-opening result has been this graph of approximate size of the screen compared to the area of the front side of the phone:

The Samsung Galaxy S9, codename starlte, immediately caught my attention. It has a very large screen for its size.

I then looked into the details and realized, I liked what I saw: Headphone jack, waterproof, small with a very good, notch-less screen (another stupid trend). It basically ticked a lot of the right boxes for me, except for having a glass back and having a relatively small battery compared with the competition.

It is also praised by The Verge. But they also list some flaws – which are all addressed by LineageOS.

Actually buying the phone

While there are other nice phones designed to be cheap by less known manufacturers, starlte was actually the flagship phone by Samsung in the year 2018 and has since steadily gotten cheaper. It now costs about half the price of when it was originally released.

So I compared online and waited for a good deal and then ordered it, just a few days later, it arrived.

Checking out the original software

I was actually quite surprised, about how well One UI, Samsungs current modified version of Android, suits the phone. It can be set up to showcase the very high contrast of the screen and has a bunch of nice gimmicks. I also liked the heart-rate sensor which I found to work quite well and the feature of One UI, that you can pull down on many screens such that you can reach everything using just one hand, blowing up the headers which are filled with pretty typography.

Unfortunately, there is a lot of stuff installed that can not be removed: lots of Google Apps, a full Microsoft Office, and several social networks like Facebook and LinkedIn, which I do not even use. And, of course, Bixby, the most annoying smart assistant with a dedicated button on the side of the phone, which can only be remapped if you create a Samsung account which I do not want.

Assimilating the new phone to my needs

After I checked out, that everything works on my phone out of the box, I wanted to really make starlte my own. I think you only really own something if you have voided its warranty, so that’s what I did by installing another operating system.

After successfully backing up the original Android image, the install process was quite easy, but there was a step missing in these instructions: I had to download a firmware update which I was made aware of by an actually helpful error message. It is always a bit thrilling to not know if the phone still works after the installation, but, if you faithfully follow the instructions by LineageOS, it should work out.

This time, I decided to install the smallest Open GApps to have the most bare-bones version of the proprietary Google Play Services (aka Google Services Framework, GSF), which many apps rely on; they do not function without them. I think it is crazy that so many developers use them, not knowing, what they actually do in the background. Google of course can use them to track you and your position, as well as other behavioural patterns, without your knowledge. I would really like it, if more people would write their apps without this dependency. On my old phone, the Motorola Moto G 2014 / titan, I had no Google apps at all and therefore had to miss out on some apps. Now on my new phone, starlte, I installed a few Google apps, but I am not signed in to Google, making it a bit harder for them to track me.

To get good, open-source Android apps with no dependencies on GSF, you might want to check out F-Droid, an alternative FOSS app store for Android.

Samsung Galaxy S9 with LineageOS

Now, with LineageOS, I’m very happy. I got rid of all the stuff I didn’t need, to get a clean, simple phone.

The Verge listed some flaws in their review, but these have all been improved upon by LineageOS:

battery life has improved
there are way fewer Gimmicks
no more Bixby
“Samsung has a poor history of updating its phones”, but LineageOS does not. I am certain that the next few versions of Android will be brought to starlte by the awesome people creating LineageOS

Unfortunately, the heart-rate sensor is not supported, as well as the different apertures of the camera, since that would require drivers from Samsung, which are not publicly available; these are the only gimmicks I will miss.

Getting a phone case

I would not have thought that this would be this difficult. First I looked at some phone cases in a shop in the city and then realized that again ordering online is way cheaper. So I ordered the official Galaxy S9 Hyperknit Cover by Samsung and it fit the phone a little too well: The power button was constantly pressed, such that it tried to turn itself off all the time. This is unacceptable, so I returned it.

Now, I use the Galaxy S9 Alcantara Cover by Samsung and like it very much. It is soft, warm, and grippy. The buttons are also satisfyingly clicky.

Conclusion

I was probably overthinking the whole process, but I am really glad with the outcome. I like my new phone very much, it has a great camera, great software due to LineageOS, looks nice, is just the right size for me, and has an amazing screen. I hope that it will serve me well in the years to come, I used my last phone titan for almost four years, so the bar is set!

Sun Protection Factor to the Extreme

Mon, 08 Apr 2019 23:05:00 +0200

It is well known that the sun protection factor SPF does not describe how much longer you can stay in the sun but rather how much of the damaging UV radiation is blocked. To get proper protection from UV radiation, you have to reapply sunscreen multiple times a day and still avoid staying in direct sunlight for extended periods of time. In this short post, I want to assume an idealized world, in which sunscreen simply extends the time you can stay in the sun without getting burnt by the SPF.

(Disclaimer: This is not medical advice!)

The time you can stay in the sun without getting burnt strongly depends on the UV index $U$, which is a dimensionless quantity proportional to the intensity of the damaging UV radiation. Many weather services provide this information in their weather forecasts, it is a great way of knowing when you can skip sunscreen and when you definitely need it.

The time $T$ skin can – on average – naturally resist getting burnt for radiation with UV index $U=8$ depends on the Fitzpatrick skin type:

skin type	protection time without sunscreen $T$	emoji
1	10 min	👦🏻
2	15 min	🧑🏻
3	20 min	👩🏼
4	30 min	👨🏽
5	90 min	🧒🏾
6	120 min	👧🏿

Assume, that the sun protection factor SPF $S$ simply extends the time as a factor at $U=8$: $$ T_8 = S T $$

Since the time until you get burnt is anti-proportional to the amount of radiation you are exposed to, we get for other UV indices $U$: $$ T_U = \frac{8}{U} T_8 = \frac{8}{U} S T $$ $$ \text{time with sunscreen} = \frac{8}{\text{UV index}} \times \text{SPF} \times \text{skin type time} $$

For example, we can now calculate the protected time using sunscreen with the typical SPFs used in the EU for Fitzpatrick skin type 2 with a sun self-resisting time of 15 min at UV index 8:

$S$	$U=1$	$U=2$	$U=3$	$U=4$	$U=5$	$U=6$	$U=7$	$U=8$	$U=9$	$U=10$
1	2.0 h	60 min	40 min	30 min	24 min	20 min	17 min	15 min	13 min	12 min
6	12.0 h	6.0 h	4.0 h	3.0 h	2.4 h	2.0 h	1.7 h	1.5 h	1.3 h	1.2 h
10	20.0 h	10.0 h	6.7 h	5.0 h	4.0 h	3.3 h	2.9 h	2.5 h	2.2 h	2.0 h
15	>24 h	15.0 h	10.0 h	7.5 h	6.0 h	5.0 h	4.3 h	3.8 h	3.3 h	3.0 h
20	>24 h	20.0 h	13.3 h	10.0 h	8.0 h	6.7 h	5.7 h	5.0 h	4.4 h	4.0 h
25	>24 h	>24 h	16.7 h	12.5 h	10.0 h	8.3 h	7.1 h	6.2 h	5.6 h	5.0 h
30	>24 h	>24 h	20.0 h	15.0 h	12.0 h	10.0 h	8.6 h	7.5 h	6.7 h	6.0 h
50	>24 h	>24 h	>24 h	>24 h	20.0 h	16.7 h	14.3 h	12.5 h	11.1 h	10.0 h

Table: Protected time $T_U$ for skin type 2

If you want to find the minimum SPF $S$ you need for a given UV index $U$ and a given exposure time of $T_U$: $$ S = \frac 1 8 \frac{T_U}{T_8} U $$ $$ \text{SPF} = 0.125 \times \frac{\text{exposure time}}{\text{skin type time}} \times \text{UV index} $$ and for skin type 2 and a time of 8 h, we get a rule of thumb: $$ S = 4 U $$ $$ \text{SPF} = 4 \times \text{UV index} $$

So for UV index 8, this rule suggests SPF 32, which is sensible. In conclusion, it is probably better to use higher SPF and do not forget to reapply sunscreen, especially after swimming. Now, the hot time of the year can begin!

Siebenschläfer ärgere dich nicht

Sun, 31 Mar 2019 19:32:00 +0200

Mensch ärgere dich nicht is a German game, designed for four players.

By the way: In German, “sieben” means “seven” and a Siebenschläfer (edible dormouse) is someone who sleeps a little too much.

Personally, I played it only once; it is quite frustrating to play with four pawns each, you just keep kicking each other off all the time. Maybe it is better if you only use two or three pawns each…

Well, I just want to share my design; feel free to print it and play it yourself! I’d love to hear your feedback.

Download it here as a:

SVG file (vector graphics)
PNG file (works everywhere)

You’ll need the Sansation font to display the SVG file.

If you are wondering which emoji I used, check out Twemoji by Twitter.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

First Blog Entry

Wed, 13 Mar 2019 11:39:00 +0100

In contrast to my day job, where I work with scientific literature and citations everyday, I cannot guarantee the correctness of everything I post here. I’ll try to research for my posts as thoroughly as I think is appropriate; just take everything I post with a grain of salt. A curious reader might want to verify it with other sources. After all, I do this just for fun; mainly for myself and hopefully for others as well. Also, I might express my personal opinion in the posts¹; all blog posts reflect my view on the world.

My blog does not have any rules. I will just write about whatever I want in my free time.

I do not want to add a comments section to this blog, since it creates unnecessary bloat which slows down the site and creates lots of other annoyances. So, if you found anything you like or if you have feedback, check out how to follow me.

To stay up to date with the latest posts, check out the blog index, the rss feed or follow me on the few social media I like.

which is a no-no in scientific publications ↩︎