# Effects of the Insertion of Epicardial Anisotropy Versus Isotropy in the Inverse Problem of Electrocardiography

Emma Lagracie, Lisl Weynans, Yves Coudière

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Emma Lagracie, Lisl Weynans, Yves Coudière. Effects of the Insertion of Epicardial Anisotropy Versus Isotropy in the Inverse Problem of Electrocardiography. FIMH 2025 - 13th Functional Imaging and Modeling of the Heart International Conference, Jun 2025, Dallas (Texas), United States. pp.83-93, 10.1007/978-3-031-94559-5_8. hal-05123199

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# Effects of the Insertion of Epicardial Anisotropy versus Isotropy in the Inverse Problem of Electrocardiography

Emma Lagracie¹,²,³,\* [0009-0002-4718-2027], Lisl

Weynans¹,²,³[0000-0003-0376-8747], and Yves Coudière¹,²,³,\* [0000-0003-2687-3232]

¹ Institut de Mathématiques de Bordeaux, Talence, France

² IHU Liryc, Pessac, France

³ Inria Bordeaux, Talence, France

\* Corresponding Authors: emma.lagracie@inria.fr yves.coudiere@u-bordeaux.fr

**Abstract.** We briefly present a new epicardial source model for the inverse problem of cardiac electrocardiography. It couples a surface equation on the epicardium with the three-dimensional torso, while incorporating heart-torso fluxes and possible anisotropic conductivities in the intracellular and extracellular domains. As this capability is new for a surface model, we investigate the impact of inserting epicardial fiber directions into the surface source model on inverse reconstructions, compared with isotropy-based reconstructions. Our results show that the epicardial fibers degrade the epicardial model based ECGi reconstructions, in particular the transmembrane voltage, whereas isotropic reconstructions demonstrate significantly higher accuracy for the extracellular potential and the transmembrane voltage. We believe that, due to interactions and compensations within the myocardial volume, the solutions of the bidomain model taken on a surface are more similar to isotropic solutions than strongly anisotropic ones. In this case, taking an isotropic assumption in a surface model based ECGi is a relevant choice.

**Keywords:** ECGi · Anisotropy · Epicardial model.

## 1 Introduction

The inverse problem of electrocardiographic imaging (ECGi) aims at reconstructing the electrical source signals of the heart from potential measurements on the surface of the torso. From the reconstructions at several time instants, activation maps can be computed, representing the sequence of electrical activation of the heart, which helps understanding some cardiac rhythm disorders. The inverse problem of ECGi relies on a source model, that is used as a constraint in a minimization problem. The information injected via this source model can significantly change the reconstructions. Currently, the more widely used source models are the static part of the volume bidomain model [8], the surface equivalent dipole layer (EDL) model [4], and the surface potential model (PP), that only present a Poisson equation in the torso [2]. Among these models, only
the volume bidomain model allows to include anisotropy with non-proportional anisotropy ratio between the intracellular and extracellular mediums in the myocardium, while we do know that the heart possesses an anisotropic fiber structure. However, because the solution of the inverse problem combined with the static bidomain source model is non-unique in the myocardial volume [5], the ECGi problem is usually solved on a surface only, that encompasses the heart. In that case, the recovered transmembrane voltage and extracellular potential are unique up to a constant, but the source models used do not permit to include cardiac anisotropy patterns.

In [6] and [7], we proposed a model that couples the epicardial surface of the heart with the torso volume, and includes an electrostatic equilibrium between an effective extracellular potential and an effective transmembrane voltage on the epicardial surface. This model, and in particular the epicardial surface equation, introduces the possibility of incorporating effective anisotropic intracellular and extracellular conductivity tensors on a surface, under the assumption that cardiac fibers lie in approximately the same plane as the surface of the heart. This allows us to investigate the impact of introducing a priori anisotropy and fiber directions into the surface ECGi problem.

In this work, we compare several reconstructions of the ECGi problem, carried out under isotropic versus anisotropic conductivity assumptions. In the anisotropic case, the orientation of the fibers is chosen to be equal to the orientation of the epicardial fibers. This choice will be discussed at the end of the article. First, the epicardial model will be presented, with some examples of forward propagation. Next, we will focus on the impact of introducing anisotropy into the model on inverse problem reconstructions. We will describe the methodology adopted, based on comparison of the reconstructions with simulated “ground truth” bidomain data, then present and discuss the results.

## 2 The epicardial model

We consider three variables: the effective extracellular potential $\bar{u}$ on the epicardial layer $\Gamma_H$, the effective or epicardial transmembrane voltage $v$ on $\Gamma_H$, and the extracardiac potential $u$, defined in the torso volume $\Omega_T$, as well as on the epicardial surface. The epicardial model writes

$$ \begin{aligned} \operatorname{div}(h(x)(\sigma_i + \sigma_e)\nabla\bar{u})) + \sigma_T \nabla u \cdot n &= -\operatorname{div}(h(x)(\sigma_i\nabla v)) & \text{on } \Gamma_H, \\ \operatorname{div}(\sigma_T \nabla u) &= 0 & \text{in } \Omega_T, \end{aligned} $$

coupled with the following boundary conditions:

$$ \sigma_T \nabla u \cdot n = 0 \quad \text{on } \Gamma_T, $$

$$ \sigma_e^{n,n} \frac{u - \bar{u}}{\alpha h} = \sigma_T \nabla u \cdot n \quad \text{on } \Gamma_H. $$

The boundary $\Gamma_T$ denotes the body surface, where ECGi data are taken. The number $h$ represents a fixed percentage $p$ of the wall thickness, and $\alpha$ a fixed
coefficient between 0 and 1. Note that $h$ is a function of $x \in \Gamma_H$. To build the model, we assume that, up to a small angle, the heart fibers at a distance $h$ from the epicardial surface lie in the same plane as the heart surface. This means that the transverse direction of the fiber is along the normal direction of the epicardial surface (up to a small error). In this case, $\sigma_{i/e}$ denotes the intracellular and extracellular conductivity tensors of the epicardial model, and the value $\sigma_e^{n,n}$ represents the singular value of $\sigma_e$ in the normal direction. The tensor $\sigma_T$ accounts for the torso conductivity.

Intuitively, this model results from an averaging process across a thickness $h$ (Figure 1) of the heart, from the epicardium or the pericardium. The effective extracellular potential $\bar{u}$ behaves as the average of the extracellular potential across that thickness. Via the Robin boundary conditions on $\Gamma_H$, this model allows to incorporate naturally the heart-torso fluxes as a ratio of the in-depth variation of the extracardiac potential across a the $h$-thick layer. Incorporating those fluxes is very important to reproduce the bath loading effect, while still using a surface model of the heart. Moreover, using this model allows to recover simultaneously the extracellular potential $u$ and the transmembrane voltage $v$ on the epicardium.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Sketch representation of the myocardial wall, with the endocardium, the epicardium ($\Gamma_H$) and the partial thickness $h$, between the dotted line and the epicardium black line. The torso domain $\Omega_T$ is located between the epicardium $\Gamma_H$ and the torso surface $\Gamma_T$. The fibers are represented by the red vectors, and the normal vector to the epicardium appears as a blue arrow.</figcaption></figure>

*Calibration and numerical example* The two parameters $h$ and $\alpha$ were calibrated as follows. An epicardial transmembrane voltage $v$ was computed using a full three-dimensional bidomain model. Then, solutions $(\bar{u}, u)$ of the epicardial model, simulated with different percentage $p$ of the myocardium thickness, and different $\alpha$ were compared to the extracardiac potential $u_{\text{ref}}$ of the bidomain
<figure><img src="image_3.png" /><figcaption>Fig. 2: Illustration of the variability of the extracardiac potential *u* of the epicardial model with respect to the two parameters (*p*, *α*). The extracellular epicardial potential *u*<sub>ref</sub> from a bidomain simulation is shown on the left, while the three other figures represent the solution *u* of the epicardial model obtained with three different sets of parameters.</figcaption></figure>

model. Our simulations have shown that the parameter set (*p*, *α*) has a great influence on the amplitude of the solution (ū, *u*) of the model, but do not change its shape, nor its cross correlation to the reference *u*<sub>ref</sub>. Some examples of the solutions of the model obtained with different sets of parameters are shown in figure 2. In the following, the parameters *p* and *α* are chosen equal to 0.5 and 0.1 respectively.

Because it includes heart-torso fluxes and conductivity tensors on the epicardial surface, the epicardial model offers the possibility to incorporate heterogeneity or anisotropy with an unequal anisotropy ratio between the intracellular and extracellular conductivities on the heart surface. This ability is novel for a surface model of the heart. It is then very appealing to investigate the effects of a priori conductivity, in ECGi surface problem. Based on simulated data with realistic fiber orientations, we compare reconstructions obtained with isotropic and strongly anisotropic epicardial conductivities with each other and with the reference. We also study the activation maps obtained from these reconstructions.

# 3 Effects of the epicardial conductivity in ECGi

## 3.1 Method

**Data and reference** We solved the three dimensional anisotropic bidomain - Mitchell-Shaeffer [9] model, with different earliest activation sites (corresponding to the different cases presented in the results). For Ω<sub>H</sub> and Ω<sub>T</sub> the (volume) heart and extracardial domains respectively, the bidomain - Mitchell-Shaeffer model
writes: for $t \in [0, T]$,

$$
\begin{array}{ll}
- \operatorname{div}(\sigma_i \nabla(u + v)) + \partial_t v + f(v, h) = I_{\text{stim}} & \text{in } \Omega_H, \\
- \operatorname{div}((\sigma_e + \sigma_i)\nabla u) - \operatorname{div}(\sigma_i \nabla v) = 0 & \text{in } \Omega_H, \\
- \operatorname{div}(\sigma_T \nabla u) = 0 & \text{in } \Omega_T, \\
[u] = 0 & \text{on } \Gamma_H, \\
\sigma_T \nabla u \cdot n = \sigma_e \nabla u \cdot n & \text{on } \Gamma_H, \\
\sigma_i \nabla(u + v) \cdot n = 0 & \text{on } \Gamma_H, \\
\sigma_T \nabla u \cdot n = 0 & \text{on } \partial\Omega,
\end{array}
$$

where

$$
f(v, h, t) = \frac{1}{\tau_{in}}hv^2(1-v) - \frac{v}{\tau_{out}},
$$
$$
\partial_t h = \begin{cases} \frac{1-h}{\tau_{open}} & \text{if } v < v_{gate}, \\ \frac{-v}{\tau_{close}} & \text{else,} \end{cases}
$$

and the Mitchell-Shaeffer model parameters are taken identical to those given in the original paper [9]. Electrical potentials simulated with the Mitchell-Shaeffer model are non-dimensional. The current $I_{\text{stim}}$ is non-zero at the initiation of an activation propagation, and identically null after. We used a tetrahedral mesh of a myocardium (17000 nodes, mesh size: 3 mm) surrounded by a torso mesh (20000 nodes). The body surface was constituted of around 2000 nodes (mesh size: 25 mm). This geometry was obtained from segmented patient's data. For each propagation scenario, associated to an earliest activation site, the extracardiac potential on the boundary of the torso was extracted to be used as BSPM data for the inverse problem. Note that the inverse model (the epicardial model) and the forward model (the bidomain model) are neither the same, nor have the same mesh (see below for the inverse model mesh). Thus the inverse crime is avoided, and a modeling error is introduced in the inversion procedure. We chose not to add noise on the data, in order to focus on the effects of the conductivity only on the reconstructions.

*Numerical method* We used the $P^1$ Lagrange finite element method for the spatial discretization of the equations, and a semi-implicit backward differentiation of order two scheme for the time discretization.

*Insertion of ruled-based fibers* We used the ruled-based method from [1] to incorporate fiber orientation in the bidomain simulations. We underline that, because the myocardial mesh includes the ventricles but also the right ventricular outflow tract, the fiber orientation close to the right ventricular outflow tract is not realistic. In the bidomain model, the eigenvalues of $\sigma_i$ were taken to be 3 in the longitudinal direction and 0.3 in the two transverse directions. For the extracellular conductivity the eigenvalues were 3 in the longitudinal direction, and 1.2 in the two transverse directions. The torso was taken homogeneous of conductivity 2.
**Inverse reconstructions** For each time iteration corresponding to a BSPM data, the values of $v$ and $u$ were reconstructed by solving the inverse problem (see below) based on the epicardial model with and without anisotropy.

**Meshes** We used a surface epicardial mesh (18000 nodes) coupled with a volume torso mesh of 15000 nodes in the volumes and 1200 body surface nodes. Note that the mesh, even on the common torso surface, is distinct from the mesh used for generating the data and reference electrical signals $u$ and $v$.

**Computation of $h(x)$** The thickness of the myocardium wall $h_{full}(x)$ was computed as a distance function between the endocardium and the epicardium with the heat method [3]. Then $h$ was taken to be $0.5h_{full}$.

**Fiber orientation and conductivity** For the isotropic model, we took $\sigma_i = \sigma_e = 3$ Id on the heart surface, ie the largest eigenvalue of the two bidomain conductivity tensors in the heart. This choice allows to preserve the scale of the reconstructions. For the anisotropic model, the (volumic) conductivity tensors used in the bidomain simulations were interpolated on the epicardial mesh, and thus were identical. In both cases, the torso conductivity was chosen identical to the bidomain simulations, and the value $\sigma_e^{n,n}$ was taken equal to 1.2.

**Formulation of the inverse problem** The inverse problem is formulated as a constrained minimization problem. We minimize the functional

$$J(u, \bar{u}, v) = \frac{1}{2} \int_{\Gamma_T} |u - z_T|^2 + \frac{\varepsilon}{2} \int_{\Gamma_H} \sigma_i \nabla v \cdot \nabla v,$$

where $z_T$ is the data, under the constraint of satisfying the epicardial model. The term $\frac{\varepsilon}{2} \int_{\Gamma_H} \sigma_i \nabla v \cdot \nabla v$ is a Tikhonov order 1 - type regularization term, with $\varepsilon$ the regularization parameter. We use a Lagrangian formulation for the minimization problem, and write its optimality conditions system. This system is solved using the finite element method. The regularization parameter is chosen by an L-curve technique.

**Activation time computation and comparison metrics** After recovering the extracellular potential $u$ and the transmembrane voltage $v$ simultaneously on the epicardium over a series of time steps, several post-processing methods are used to compute activation maps. Those maps are then compared with the ground truth maps obtained from the bidomain simulations.

**Spatial derivative** Before computing time-derivative, the temporal signals are smoothed with a Gaussian filter of order 0. Then, the spatio-temporal derivatives ($\partial_{st}$, product of time derivative and gradient norm) of $u$ and $v$ are computed as in [11] and [7]. Activation times are then defined as the time of maximal (resp. minimal) spatio-temporal derivative on $v$ (resp. $u$).

**Threshold** As we are able to recover also the transmembrane voltage, we can define the activation time as the time iteration at which a predefined threshold is crossed. Using the methods described in [7] and [10], the transmembrane voltage $v$ is scaled to obtain a coherent time course with values contained between 0 and 1.3 at all times. Then several activation maps were computed with thresholds
distributed between 0.7 and 1. The final activation map was defined as the
average of all these maps.

*Comparison Metrics* As in [7], three errors were defined to estimate the accu-
racy of the recovered activation maps: the $L^2$ relative error ($L^2$ relative, mesh
ponderated root mean square error), the Pearson's Correlation Coefficient (CC),
and the Slowness Coefficient (SC, cross correlation of the surface gradients).

3.2 Results

<figure><img src="image_4.png" /><figcaption>Fig. 3: Examples of reconstructions of the extracellular potential *u* (top row) and the transmembrane voltage *v* (bottom row) at two fixed time ((a), (b)) using an isotropic model on the left and an anisotropic model with epicardial fibers on the center. The reference solutions are located on the right column.</figcaption></figure>

Figures 3a and 3b show reconstructions of *u* and *v* on the epicardium at two increasing time steps. The most striking observation is that the amplitude of the transmembrane voltage reconstructed with epicardial fibers (*v*<sub>fib</sub>) is too large. What's more, this amplitude changes radically over time. In figure 3a, *v*<sub>fib</sub> has an amplitude of almost 3, whereas in figure 3b, later in the same propagation sequence, *v*<sub>fib</sub> can take values above 5. We recall here that in the Mitchell-Shaeffer model, transmembrane voltage values range from 0 to 1. In addition, the shape of the activated part of *v*<sub>fib</sub> is elongated in the fiber direction, an elongation not observed in the bidomain transmembrane voltage. In comparison, the reconstructions performed with the isotropic model are satisfactory: their amplitude is correct, and their shape is in agreement with the reference, within the usual framework of the inverse problem. We can add that the extracellular potentials resulting from the isotropic *u*<sub>iso</sub> and anisotropic *u*<sub>fib</sub> reconstructions are quite similar, up to a few oscillations, whereas the associated transmembrane tensions differ radically. The impact of fiber incorporation in the epicardial model is mainly observed on *v*.
<figure><img src="image_5.png" /><figcaption>Fig. 4: Activation (AT) maps (ms) computed as the average of the threshold method and the $\partial_{stu}$ method, for three different activation sequences (cases 1-3). The maps are computed on the isotropic reconstructions (center row) and on the epicardial fibers reconstructions (bottom row). The reference maps are on the top.</figcaption></figure>

The figure 4 presents the activation maps (AT maps) recovered for the three propagation sequences (cases 1-3) with isotropic and anisotropic with epicardial fibers models. In this figure, the activation times are computed as the average between the threshold activation map and the $\partial_{stu}$ activation map, thus it combines information on the transmembrane voltage $v$ with information on the extracellular potential $u$. There is a clear superiority of the maps reconstructed with the isotropic a priori over the maps reconstructed with the epicardial fibers.

If we look at the table 1 that gives the similarity between the reconstructed maps and the reference maps for the three metrics and the three AT maps computation methods, we can observe that the AT maps based on $u$ have comparable accuracy for both isotropic and anisotropic case. However, the AT maps based on measures on $v$ are of very poor accuracy for the anisotropic case, in comparison with the isotropic case. This can be explained by the fact that $v_{fib}$ has a large time-dependent amplitude, thus degrading the activation maps either based on its derivatives or on its scaled amplitude across time. We believe that the strong ratio of anisotropy between the transverse and longitudinal directions of the fibers, inserted in the source model, indirectly forces the recovered transmembrane voltage to change amplitude depending on the orientation of the activation front relative to the orientation of the fiber field. Indeed, supposing we have reconstructed a visually isotropic extracellular potential, then, in
<table><caption>Table 1: Errors for the isotropic and with epicardial fibers AT maps reconstructions, in the three test cases, and for three different methods of AT computation. The L² relative error, cross correlation (CC) and slowness coefficient (SC) are given.</caption><thead><tr><th rowspan="2">AT method</th><th rowspan="2">Type of error</th><th colspan="2">Case 1</th><th colspan="2">Case 2</th><th colspan="2">Case 3</th></tr><tr><th>Isotropic</th><th>Epi fibers</th><th>Isotropic</th><th>Epi fibers</th><th>Isotropic</th><th>Epi fibers</th></tr></thead><tbody><tr><td rowspan="3">∂<sub>st</sub>u</td><td>L<sup>2</sup> relative</td><td>0.2</td><td>0.21</td><td>0.1</td><td>0.11</td><td>0.11</td><td>0.1</td></tr><tr><td>CC</td><td>0.91</td><td>0.9</td><td>0.97</td><td>0.97</td><td>0.97</td><td>0.98</td></tr><tr><td>SC</td><td>0.88</td><td>0.87</td><td>0.89</td><td>0.87</td><td>0.8</td><td>0.7</td></tr><tr><td rowspan="3">Threshold</td><td>L<sup>2</sup> relative</td><td>0.1</td><td>0.83</td><td>0.1</td><td>0.61</td><td>0.14</td><td>0.53</td></tr><tr><td>CC</td><td>0.98</td><td>0.16</td><td>0.98</td><td>0.78</td><td>0.94</td><td>0.65</td></tr><tr><td>SC</td><td>0.99</td><td>0.59</td><td>0.99</td><td>0.86</td><td>0.91</td><td>0.84</td></tr><tr><td rowspan="3">∂<sub>st</sub>v</td><td>L<sup>2</sup> relative</td><td>0.2</td><td>0.44</td><td>0.21</td><td>0.19</td><td>0.23</td><td>0.33</td></tr><tr><td>CC</td><td>0.95</td><td>0.65</td><td>0.91</td><td>0.94</td><td>0.89</td><td>0.76</td></tr><tr><td>SC</td><td>0.96</td><td>0.71</td><td>0.84</td><td>0.83</td><td>0.64</td><td>0.73</td></tr></tbody></table>

the strongly anisotropic model, the corresponding transmembrane voltage must compensate for the anisotropy of conductivity by its amplitude. While the action potential propagates, the orientation of the activation front with respect to the fibers changes, resulting in change in amplitude of $v_{fib}$.

In either case, even considering the intrinsic variability between the tests cases, at fixed time or for activation maps, the isotropic a priori shows superior reconstructions compared to the epicardial fiber a priori reconstructions.

## 4 Discussion and conclusion

We have presented a novel epicardial surface model for ECGi that may include fiber orientation, and heterogeneous conductivities. Heterogeneity in the conductivity tensors may be used to model scars, for example. We exploited this model to evaluate the effect of incorporating a priori fiber orientation in comparison with isotropy in surface model-based ECGi. In the tests cases presented, the isotropy assumption in the source model yields very satisfactory reconstructions for the extracellular potential and transmembrane voltage, as well as high-quality activation maps, irrespective of the computational method. On the contrary, incorporating epicardial fibers into the model degrades the quality of the epicardial reconstructions. In particular, it significantly degrades the recovered transmembrane voltage, and consequently the recovered activation map. Indeed, the amplitude of the reconstructed transmembrane voltage is distorted by the high anisotropy ratio between the longitudinal and transverse directions of $\sigma_i$, and its shape is distorted by the fiber orientation. In future work, more scenarios should be tested to confirm this pattern.

Since the epicardium is in contact with the torso and its electrical signals are the most observable signals of the heart from the torso's point of view, we may have expected our results to be the other way around. There are two elements to explain these results. Firstly, there could be a more relevant choice of fiber
orientation on the epicardium, reflecting the effective mean conductivity in a
small depth of the heart, from its surface. Secondly, there are three-dimensional
effects through the cardiac wall that the surface model cannot represent by its
intrinsic nature. The high anisotropy of one layer of cardiac tissue (here, the
epicardial layer) does not account well for the global behavior of a thick layer of
cardiac tissue, which includes fiber rotation, compensation effects and volumic
interactions. Thus, the anisotropy of the surface model has to be carefully chosen
to be representative of the effective volume behaviour. This may explain why the
absence of anisotropy in the surface model based ECGi yields better results than
strong epicardial anisotropy.

**Acknowledgments.** This study received financial support from the French Govern-
ment as part of the "Investments of the Future" program managed by the National
Research Agency (ANR), Grant reference ANR-10-IAHU-04.

**Disclosure of Interests.** The authors have no competing interests to declare that are relevant to the content of this article.

References

1. Bayer, J.D., Blake, R.C., Plank, G., Trayanova, N.A.: A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models. Annals of biomedical engineering **40**, 2243-2254 (2012)

2. Bear, L., Cuculich, P.S., Bernus, O., Efimov, I., Dubois, R.: Introduction to non-invasive cardiac mapping. *Cardiac electrophysiology clinics* **7**(1), 1–16 (2015)

3. Crane, K., Weischedel, C., Wardetzky, M.: The heat method for distance computation. *Communications of the ACM* **60**(11), 90–99 (2017)

4. Geselowitz, D.B.: Description of cardiac sources in anisotropic cardiac muscle: application of bidomain model. *Journal of electrocardiology* **25**, 65–67 (1992)

5. Kalinin, V., Shlapunov, A., Ushenin, K.: On uniqueness theorems for the inverse problem of electrocardiography in the sobolev spaces. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik **103**(1), e202100217 (2023)

6. Lagracie, E., Bourgault, Y., Coudière, Y., Weynans, L.: A depth-averaged heart model for the inverse problem of cardiac electrophysiology. *Inverse Problems* (2024)

7. Lagracie, E., Weynans, L., Coudière, Y.: Assessment of a threshold method for computing activation maps from reconstructed transmembrane voltages. In: 2024 Computing in Cardiology (CinC)

8. Lines, G., Buist, M., Grottum, P., Pullan, A., Sundnes, J., Tveito, A.: Mathematical models and numerical methods for the forward problem in cardiac electrophysiology. *Computing and Visualization in Science* **5**, 215–239 (2003)

9. Mitchell, C.C., Schaeffer, D.G.: A two-current model for the dynamics of cardiac membrane. *Bulletin of Mathematical Biology* **65**(5), 767–793 (2003)

10. Schuler, S., Potyagaylo, D., Dössel, O.: Delay-based regularization for ecg imaging of transmembrane voltages. In: 2019 Computing in Cardiology (CinC). pp. Page-1. IEEE (2019)
11. Schuler, S., Schaufelberger, M., Bear, L.R., et al.: Reducing line-of-block artifacts in cardiac activation maps estimated using ecg imaging: A comparison of source models and estimation methods. IEEE Transactions on Biomedical Engineering 69(6), 2041–2052 (2021)