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