Numerical Study on Excitation–Contraction Waves in 3D Slab-Shaped Myocardium Sample with Heterogeneous Properties

Abstract

In this study, we have performed 3D numerical simulations of the excitation and contraction of thin slab-like samples of myocardium tissue. The samples included a narrow region of almost non-excitable tissue simulating impaired myocardium. In the numerical experiments, we considered the heterogeneity of myocardium excitation and the ${\mathrm{Ca}}^{2+}$ activation of its contraction, as well as the orientation of the muscle fibers. Those characteristics varied throughout the thin wall of the sample. The simulations were performed in our numerical framework for the problems of cardiac electromechanics developed recently. The framework was previously tested for the benchmark problems in which formulations took into account only myocardium electrophysiology and passive mechanics. The study could be considered as an approbation of the framework performance with the fully coupled mathematical model of myocardium electromechanics. Here we dealt with the problems requiring a multiscale approach, taking into account cell-level electrophysiology, cell-level mechano-chemical processes, macromechanics (strain and stress) of the 3D sample, and interconnections between the levels. It was shown how the tissue heterogeneity and its strain affected the propagation of excitation–contraction waves in the sample, including, in particular, the formation of spiral waves.

1. Introduction

Mathematical modeling of heart excitation and contraction has been an actively developing tool for application in the treatment of heart disease. Among others, arrhythmia is a serious and frequently encountered problem, which could be caused by different factors, including dysfunctions of ionic currents and impaired tissue excitation or conductance, and even could be affected by myocardium deformation. The mathematical modeling of such complex cardiac dysfunctions always involves numerical simulations, as the models have to include descriptions for the complex cell-level (for a single cardiomyocyte) and tissue-level (for myocardium) processes leading to the functioning of heart chambers at the organ level. The computational implementation of the multiscale simulation of cardiac activity is not easy from the computational point of view, which encourages researchers to develop new numerical suits and optimize numerical schemes and methods.

The computation frameworks and libraries for the numerical simulations of myocardium excitation and contraction commonly have implementations of multiscale mathematical models at their core. For the correct simulation, one should take into account the mechanics of the myocardium sample undergoing large strain, cell-level models of cardiomyocyte excitation and contraction, and a macro-level model for the propagation of the excitation waves within the tissue. Heterogeneity and anisotropy of the material should also be accounted for at all levels of the modeling. The model blocks are typically interconnected with each other. In particular, the processes of excitation–contraction coupling and mechano-electric feedback (MEF) should be specified in the models. In addition, while considering a whole-organ mathematical formulation, one should couple the mechanics of the heart chambers with a model of blood circulation, specifying corresponding boundary conditions. Having a numerical framework for cardiac modeling in mind, one should note that the variety of the problem formulations and models requires the framework to allow its users to incorporate new models, specify the equations, along with their boundary and initial conditions, and operate with different geometry and meshes with ease.

A 0D model of cardiomyocyte electrophysiology commonly defines a set of ionic currents specified by a system of ordinary differential equations (ODEs). As most of the currents are very fast, the systems of ODEs are stiff and should be handled with the numerical methods applicable to that case. Excitation propagation through the tissue is commonly described by a reaction–diffusion equation for the transmembrane potential, or by a system of the equations if some other electric potentials, like the potential of external medium, are considered. The excitation models are required to reproduce the dynamics of the transmembrane potential (called action potential (AP) when it grows from its resting value) and the dependence of the action potential duration (APD) on the stimulation frequency. Complex electrophysiological simulations at the heart level are challenging from the computational point of view, even if one does not take cardiac mechanics into account [1]. Speaking of cardiac mechanics, the myocardium is generally treated as an incompressible or nearly incompressible anisotropic (orthotropic or transversely isotropic) nonlinear material with finite strains. Active stress in cardiac muscle during its contraction is much higher than its passive response to strain. Because the active forces are aligned with the fiber directions, the corresponding components of the stress tensor are especially high during the contraction. All of these factors lead to ill-conditioned stiffness matrices emerging from finite element methods. That means that the linear algebraic equations, which one has to deal with while applying solvers for the nonlinear equations, require efficient methods for the matrix preconditioning before applying a reliable iterative solver. Examples of the studies on numerical methods implemented for the simulation of cardiac electromechanics can be found in [2,3,4,5]. Moreover, the solver choice sometimes depends on the problem setup. Therefore, it would be convenient to be able to choose from several different solvers for systems of ODEs and nonlinear and linear algebraic equations. Recent studies on the MEF in myocardium, both experimental and involving numerical simulation, show that the tissue deformations could be crucial for the initiation of spiral waves or their breakup, as conductance velocity in the tissue decreases upon its stretch [6,7,8,9,10]. For example, experimental data show that the rotation dynamics of the excitation waves in the left ventricle occurring during ventricular fibrillation is associated with the propagation of deformation waves in myocardium, which was studied numerically [11]. Thus, in order to reproduce arrhythmogenic phenomena accurately, numerical studies on the problems of heart electrical activation should take cardiac mechanics into consideration, at least for the heart ventricles, as the myocardium in the ventricular walls undergoes large strains. In addition, the computational tools for cardiac simulation should provide ways to couple the macroscopic strains with cell-level models and allow their users to specify the dependencies required for the coupling without resorting to modification of the major code segments.

The computational packages, libraries, and frameworks developed by various groups for modeling in cardiology or for more general multiphysics problems include LifeX (https://gitlab.com/lifex/lifex (accessed on 30 May 2025)), OpenCARP (https://opencarp.org/ (accessed on 30 May 2025)), Cardioid (https://github.com/LLNL/cardioid (accessed on 30 May 2025)), and Chaste (https://github.com/Chaste (accessed on 30 May 2025)). Some of the libraries developed by scientific groups are based on general-purpose computing platforms, such as FEniCS (https://fenicsproject.org/ (accessed on 30 May 2025)). Some of those computational suits and libraries were used successfully to simulate both excitation and contraction of myocardium samples or heart chambers. However, there are only a few computational tools that provide the options to specify electrophysiology and mechanics model blocks and their two-way coupling in a user-friendly manner, while, in addition, providing the instruments for flexible discretization of equations in each model block.

Our study focuses on the numerical experiments with spiral waves in anisotropic myocardium samples undergoing myocardial contraction. The spiral waves can cause or accompany some types of arrhythmia, and thus their dynamics seems to be an important subject of numerical simulation. The application of a fully coupled electromechanical model in the computations could also be especially useful, as the strain and heterogeneity of the tissue structure both affect the waves’ dynamics [12,13]. In this paper, we present several 3D problems formulated for a myocardium sample of simple geometrical shape, which had a region with impaired myocardium excitation. The sample was deformed, followed by a periodic stimulation of one of its sides. For several numeric experiments, we also considered the variation of the characteristics of the regulation of myocardial contraction and the orientation of muscle fibers through the tissue. The simulations were run with our new computational framework for mathematical modeling in cardiology, CarNum [14], to test its capabilities and observe the modeled effects of the sample strain on the propagation of the excitation waves. The test problems were also supposed to demonstrate what changes in the behavior of the excitation–contraction waves we could obtain by introducing the heterogeneity of the myocardium stated above.

We pursued two goals in this study. On the one hand, we expected that the simulation results would be able to reveal important changes in the dynamics of the excitation–contraction waves upon the introduction of the spatial heterogeneity of the contraction regulation, since we used the model accounting for strain dependence of both the regulation and the propagation of the electrical activation in the myocardium. The features of the myocardium model used here, in particular, the presence of mechano-electric feedback, in combination with the specifics of the modeled samples determined our interest in the study. On the other hand, we intended to test and debug our framework CarNum while working on the formulation and solution of the problems of cardiac electromechanics, in which material anisotropy and heterogeneity of different myocardium cell-level properties were assumed as being important to take into account. CarNum is still under development, and this paper demonstrates its applicability to research tasks rather than stating its superiority over other packages, which differ in software architecture, approaches to solving problems, user interfaces, etc. The numerical methods used in CarNum and the related issues have already been described in [15] and will only be briefly mentioned here.

2. Materials and Methods

2.1. Numerical Implementation and CarNum Framework

We performed the simulations in the framework for cardiac electromechanics, CarNum. It was used to build the coupled numerical scheme, assemble the linear systems, and customize the exchanges and interactions between the individual submodels. CarNum has been developed on the basis of several other frameworks and libraries with flexibility in mind, providing the ability for a user to use their own discretizers and several pre-built solvers, as well as to implement custom cell-level models of cardiac electrophysiology and mechanics. The platform core is built on the parallel computing framework INMOST [16] and extended by the FEM toolkit AniFem++ (https://github.com/INMOST-DEV/INMOST-FEM (accessed on 30 May 2025)). We used heavily modified OpenCARP’s open-source library, Limpet (https://opencarp.org/doxygen/master/limpet.html (accessed on 30 May 2025)), to set the variables and equations of cell-level models in EasyML language. We used the SUNDIALS (v7.0.0) computation suit for the solvers of nonlinear algebraic and ordinary differential equations [17]. We have verified [14] the platform and the implemented solvers with electrophysiological and mechanical benchmarks. Our results of the test problems showed good agreement with the results obtained by other research groups. Parallel computations demonstrated the good performance and scalability of our software. The framework allows one to specify several parameters to set up the mathematical problem and numerical methods using the command line.

The algorithm for the problem solution and the numerical methods were as follows. The equations of continuous mechanics (the motion and incompressibility equations for the sample) and the partial differential equation for the transmembrane potential (monodomain model) were solved by the finite element method, while the cellular ODEs (for the cell-level contraction model, ionic balance for cell compartments, ionic currents, or for the phenomenological variables of simplified models) were solved independently at the points of the quadratures on the corresponding meshes. The values of the cell-level variables computed at the current time step in the elements’ points of quadrature integration were used to compute the “ionic current” terms of electrophysiological equations. Those “micro-scale” variables also included active stress developed by myocardium contractile proteins, and that stress term was included in the motion equation. To solve the ODEs of the cellular activation–contraction model, we used the CVODE solver from the SUNDIALS suit using a backward differentiation formula (BDF) of variable order. The nonlinear system of equations obtained after discretization of the continuous mechanics equations was solved by the inexact Newton method implemented in the KinSol package (numerical solvers included in the SUNDIALS suit). We solved linear systems arising at the Newton steps of the KinSol solvers, as well as the equations obtained after discretization of the monodomain equation, by the iterative BiCGStab solver with the parallelized second-order Crout-ILU preconditioner implemented in the INMOST framework [16].

The following features of CarNum that we find important and convenient for the cardiac modeling can be highlighted.

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The flexible processing of parallel computational grid meshes in common formats (inherited from the INMOST computational suit).

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The cellular block of the model (ODEs for cardiomyocyte mechanics, ionic currents or their phenomenological approximations, supplementary equations of electromechanical coupling) is specified in a human-readable language. One is able to link the block’s parameters with spatial coordinates and strains, as well as import cell-level variables into tensors of the macromechanical formulation. Macro- and micro-level blocks are interconnected by the cell-level variables.

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The framework utilizes the processing of symbolic algebraic expressions and their automatic differentiation, simplifying the mathematical formulation of the cardiology related problems, including the specification of myocardium properties and boundary conditions.

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Separate time steps can be used to solve equations of mechanics, electrophysiology, and cell-level activation–contraction models.

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Users can choose from a variety of numerical methods. Generally, methods based on backward differential formula are used to solve stiff ODEs, and inexact Newton methods are used to solve nonlinear algebraic equations that arise from PDEs for the electrical propagation and continual mechanics. Those methods utilize a custom implementation of adaptive time-step and allow one to choose one of several preconditioners used in linear iterative solvers. Parallel computation using OpenMP and MPI are supported.

2.2. Excitation–Contraction Model of Cardiomyocyte

We used the myocardium model by Syomin et al. [18] in the simulations. The model combines the phenomenological model of excitation by Aliev and Panfilov [19] with the model of cardiac mechanics developed by our group earlier. The Aliev–Panfilov model at the cell level is set by a system of two ODEs for non-dimensional transmembrane potential U and a phenomenological variable v simulating, to some extent, the ionic currents that affect the AP repolarization. The equations in non-dimensional form were as follows:

$$\begin{array}{cc}\hfill & \tau {\displaystyle \frac{\partial C_{m}U}{\partial t}}+k_{AP}U\left(U-a_{AP}\right)\left(U-1\right)+Uv=I_{stim},\hfill \\ \hfill & \tau {\displaystyle \frac{\partial v}{\partial t}}=-\left(\epsilon +{\displaystyle \frac{{\mu }_{1}v}{{\mu }_2+U}}\right)\left(v+k_{AP}U\left(U-b_{AP}-1\right)\right).\hfill \end{array}$$

(1)

Here, $C_m$ is the membrane capacitance, $a_{AP}$ sets the excitation threshold, and all other parameters do not have any physical meaning, while their values were taken from [19]. In this study, we used the model modification taking into account mechano-electric feedback (MEF) implemented through the non-instant dependence of the membrane capacitance on strain upon muscle stretching. Assuming the rate of the capacitance variation to be much slower than the rate of change in the potential, the first equation was transformed into the following equations,

$$\begin{array}{cc}\hfill & \tau C_m{\displaystyle \frac{\partial U}{\partial t}}+k_{AP}U\left(U-k_{exc}{a}_{AP}\right)\left(U-1\right)+k_{rep}Uv=I_{stim},\hfill \\ \hfill & {\displaystyle \frac{\partial C_m}{\partial t}}=k_m\left(K_m{\left({\lambda }_f-1\right)}_{+}-\left(C_m-1\right)\right),\hfill \end{array}$$

(2)

where ${\left(x\right)}_{+}=\left(\right|x|+x)/2$, ${\lambda }_f=\sqrt{\left(\mathbb{F}\mathbf{f},\mathbb{F}\mathbf{f}\right)}$ is the strain of muscle units, sarcomeres, with deformation gradient $\mathbb{F}=\mathbb{I}+\nabla \mathbf{u}$ and fiber direction given by the vector $\mathbf{f}$, $\parallel \mathbf{f}\parallel =1$. $k_{rep}$ is equal to 1 in the Aliev–Panfilov model, while in this study we varied this “repolarization rate” parameter to change the APD through the sample tissue (see Section 2.3). We also varied this parameter and an additional multiplier $k_{exc}$ to make the AP very short and the excitation threshold higher in the tissue regions with simulated impaired excitability.

The mechanical block was described by a system of ODEs for a set of state variables for contractile and regulatory proteins in sarcomeres, an ensemble-average distortion (muscle microscopic strain) of the formed contractile structures, cross-bridges, per sarcomere, the concentration of Ca²⁺ ions in different cell compartments, and a few other variables, which affected the calcium currents and provided correct functioning of excitation–contraction coupling. The general forms of the equations for cell-level mechanics and electromechanical coupling were

$$\begin{array}{cc}\hfill & \dot{n}=f_{cb1}\left(n,A_1,\delta \right),\hfill \\ \hfill & \dot{\theta }=f_{cb2}\left(n,\theta ,A_1,\delta \right),\hfill \\ \hfill & \dot{\delta }=f_{\delta }\left({\dot{\lambda }}_f,n,A_i\right),\hfill \\ \hfill & {\dot{A}}_i=f_A\left(n\theta ,A_i,\delta ,c_{cyt},{\lambda }_f\right),\hfill \\ \hfill & {\dot{c}}_{cyt}=f_{c1}\left(U,c_{cyt},c_{oth},{\lambda }_f\right).\hfill \\ \hfill & {\dot{\mathbf{c}}}_{\mathbf{oth}}={\mathbf{f}}_{\mathbf{c}\mathbf{2}}\left(U,c_{cyt},{\mathbf{c}}_{\mathbf{oth}}\right).\hfill \end{array}$$

(3)

Here, n is the non-dimensional concentration of the contractile proteins interacting through the formation of cross-bridges, $\theta$ is a fraction of the cross-bridges being in the additionally deformed strongly bound state, and $\delta$ is the cross-bridges’ average distortion. $A_1$ and $A_2$ are the non-dimensional concentrations of complexes formed by muscle regulatory proteins binding ${\mathrm{Ca}}^{2+}$ ions, and $c_{cyt}$ is the concentration of free ${\mathrm{Ca}}^{2+}$ in the cytoplasm. The equations for $A_1$ and $A_2$ take into account the length-dependence and so called cooperativity of the muscle activation. ${\mathbf{c}}_{\mathbf{oth}}$ notes a vector of variables for ${\mathrm{Ca}}^{2+}$ concentration in other cell compartments and additional variables regulating the calcium currents. The equations for ${\mathbf{c}}_{\mathbf{oth}}$ describe calcium-induced calcium release and were sufficient to reproduce the dependencies of the peak force and $c_{cyt}$ on the stimulation frequency, as well as the speed-up in muscle relaxation at increased stimulation frequency. Active stress $T_{act}$ generated by the cross-bridges depended on their number (n), strain (the average strain $\delta$ and additional constant strain h of the strongly bound cross-bridges, whose fraction among all attached cross-bridges equals $\theta$), and the sarcomere length (can be substituted by the fiber strain ${\lambda }_f$).

$$T_{act}=k_{act}n\left(\delta +\theta h\right)W_{ov}\left({\lambda }_f\right).$$

(4)

The right-hand side of (4) includes the coefficient $k_{act}$ defined from the stiffness of a cross bridge, the maximal number of cross bridges per sarcomere, and the number of contractile filaments per unit area. $W_{ov}$ is a function of sarcomere length corresponding to the normalized length of the overlap zone of thick and thin contractile filaments, which affects the maximal number of binding sites for the cross-bridges. Details of the model of cell-level mechanics and electromechanical coupling can be found in Appendix A.1.

2.3. Mathematical Formulation for the Simulation of a Slab-like Myocardial Sample

The mathematical problem was specified by the following partial differential equations. Electrical propagation in the tissue domain $\mathrm{\Omega }$ was described by the monodomain electrophysiology model, which was the first equation of (2) expanded to the tissue scale.

$$C_m{\displaystyle \frac{\partial U}{\partial t}}-\nabla ·(\mathbf{D}\nabla U)+I_{ion}(U,v)=I_{stim}\left(t\right), \mathbf{X}\in \mathrm{\Omega }$$

(5)

with an insulating boundary condition on the surface with normal vector $\mathbf{N}$,

$$(\mathbf{D}\nabla U)·\mathbf{N}=0,\mathbf{X}\in \partial \mathrm{\Omega },$$

(6)

where the diffusion term emerges with the conductivity tensor $\mathbf{D}$ given by

$$\mathbf{D}=D_{iso} \mathbb{I}+D_{aniso} \mathbf{f}\otimes \mathbf{f},$$

which contains the terms for isotropic $D_{iso}$ and anisotropic (aligned with the muscle fibers) $D_{aniso}$ conductivity. The continuum mechanics were described by the equilibrium equation for the first Piola–Kirchhoff stress tensor $\mathbf{P}$ with the assumption of quasi-stationary contraction.

$$\sum _{j=1}^3{\nabla }_j{P}^{ij}=0, i\in \overline{1, 3}, \mathbf{X}\in \mathrm{\Omega },$$

(7)

with the general form of boundary conditions specified as

$$\mathbf{P}\mathbf{N}+(k_{\Vert }\mathbf{N}\otimes \mathbf{N}+k_{\perp }(\mathbb{I}-\mathbf{N}\otimes \mathbf{N}))(\mathbf{u}-{\mathbf{u}}_{\mathbf{spr}})=-p_{ext}J{\mathbb{F}}^{-T}\mathbf{N}, \mathbf{X}\in \partial \mathrm{\Omega }.$$

(8)

Here, $k_{\perp }\ge 0$, $k_{\Vert }\ge 0$ are stiffness coefficients for normal and tangential mounting springs, and displacements ${\mathbf{u}}_{\mathbf{spr}}$ were either zero or equal to the displacements of the initially free boundaries of the sample after its preliminary stretch or inflation, preceding their fixation. $p_{ext}$ is an external pressure at the boundary, and $\mathbb{F}=\mathbb{I}+\nabla \mathbf{u}$ and $J=\det \mathbb{F}$ are the deformation gradient and its third invariant. The stress tensor included the isotropic stress defined by a strain-energy function for the hyperelastic isotropic material, the stress caused by incompressibility (optional), and the anisotropic term for the stress caused by the forces applied along the muscle fibers. That last term was the sum of the nonlinear response of the elastic protein titin and the active tension developed by the contractile proteins in muscle ($T_{act}$ in the Equation (4)). The expression for the titin stress was modified compared to the one from [18] and was described in [15]. The stress–strain curve was fitted by the exponential polynomial function of ${\lambda }_f$ to avoid values close to zero emerging in the denominator of the original expression during the numerical simulations and to make the myocardium stiffer at sarcomeres shortening. The isotropic part of the stress–strain relationship was specified by a modified Fung model for hyperelastic materials [20]. The details can be found in Appendix A.2.

The initial conditions for the cell-level ODEs were chosen from the solution of the 0D problem of the isometric contraction at the sarcomere length of 2.25 µm with the transitions to the isotonic contraction at fixed load. Under the load, the sarcomeres shortened to approximately 1.8 µm and then stretched to their initial length; after that, the contraction mode switched back to the isometric one. Dimensional minimal and maximal values for AP were $-20$ and 80 mV.

The interconnections between the model blocks are shown in Figure 1, and a summary of the model equations and associated processes is presented in

Table 1. Mathematical formulation of the model blocks.

Equations (Reference Numbers)	Physical Processes	Mesh Associated Variables	Numerical Solvers
PDEs: (5), (6)	AP propagation in the tissue	Nodal (P1, linear basis function) values of the action potential	Solution of the corresponding system of linear equations by iterative BiCGStab solver with the parallelized second-order Crout-ILU preconditioner
PDEs: (7), (8)	Continuum mechanics: equilibrium equations	Nodal (P2, quadratic basis functions) displacements	Solution of the corresponding system of nonlinear equations by the inexact Newton method, solution of linear equations at the Newton method’s iterations by iterative BiCGStab solver with the parallelized second-order Crout-ILU preconditioner
ODEs: the second eq. in (1), the second eq. in (2)	Phenomenological description of ionic currents through the cell membrane and strain-dependence of the membrane capacitance	Values of v and $C_m$ in integration points of the mesh cells (elements)	Solution of ODEs system by BDF of variable order.
ODEs: (3)	Interaction of muscle contractile and regulatory proteins, ${\mathrm{Ca}}^{2+}$ currents	Values of the state variables describing muscle contraction and its regulation, calcium concentrations. The variables are solved for integration points of the mesh cells (elements)	Solution of ODEs system by BDF of variable order.

.

The sample of myocardial tissue studied in the numerical experiments had a simple shape of a thin parallelepiped slab with a size of 9 × 9 × 1 cm. The material was considered to be transversally isotropic, and the anisotropy was defined by the orientation of the cardiac fibers, whose distribution differed between the numerical experiments. The left edge plane of the slab ($X=0$) and one of the wide planes ($Z=0$) were fixed by stiff linear elastic springs with a stiffness of 10 kPa for the left side and 50 kPa for the right one ($X,Y,Z$ are Lagrangian reference coordinates). For the first test series of numerical experiments, the pressure was applied to the right plane ($X=L=9 \mathrm{cm}$) of the sample. The pressure was grown linearly in time up to 1 kPa during 100 ms in order to stretch the sample to approximately the length of $1.1L$. After that, the pressure was fixed and the right side was fastened to its current position by springs, more flexible than the ones used at the left side (2 kPa). All other planes were free of load. After the stretching, an external stimulus of 3 ms duration was applied with a period of 0.5 s, initiating the sample contraction. A non-excitable region, surrounded by the gray zone, was simulated by the parameters $k_{exc}$ and $k_{rep}$, which provided high excitation threshold and fast repolarization. The details on the parameters and equations for their spatial variation are presented in Appendix A.3. We performed the computations of the myocardial sample contraction with variation of the following factors. The first one was the fiber orientation, which was horizontal and linearly changing through the Z-axis from ${80}^{\circ }$ at $Z=0$ to $-{60}^{\circ }$ at $Z=1 \mathrm{cm}$; the second one was the MEF, which was taken as the strain-dependent membrane capacitance (the second equation in Equation (2)) or turned off ($C_m$ being constant).

Another series of numerical experiments was set with the sample more similar to a part of the ventricular wall. For that purpose, the initial slab was inflated by incrementally increasing pressure applied to one of its wide surfaces. The pressure was increased linearly up to 0.05 kPa. Two edges ($X=0$ and $X=L$) of the pressured surface were fixed by springs with a stiffness of 10 kPa. The orientation of the muscle fibers was specified in the initial non-inflated slab: the fibers were located in the XY plane and had their angle with the X-axis changing from ${80}^{\circ }$ in the pressured inner, “endo”, surface to $-{60}^{\circ }$ in the unloaded “epi” surface. We introduced a heterogeneity of the cell-level properties of the electromechanical model to reproduce two effects observed in the myocardium of the left ventricle: the variations of the APD and length-dependence of the contraction activation from the endocardial layer to the epicardial layer. The activation length-dependence was specified by the model parameters, which defined the dependence of the kinetics of calcium-troponin complexes on $lambd{a}_f$ (Equation (A3) in Appendix A.1) to reproduce experimental data [21]. The authors of that paper showed how the ${\mathrm{Ca}}^{2+}$ sensitivity differed through the ventricular wall. The sensitivity was measured by a difference in levels of calcium concentrations corresponding to half-maximal force (${\mathrm{EC}}_{50}$) or by a difference of ${\mathrm{pCa}}_{50}=-{\mathrm{lgEC}}_{50}$ in steady-state isometric contraction at two different sarcomere lengths. Another effect taken into account was the APD variation: shorter AP in epicardium and longer AP in endocardium. In the case of the detailed electrophysiological model, that effect would be partially provided by the length-dependent change in the ${\mathrm{Ca}}^{2+}$ regulation. However, the Aliev–Panfilov model used here does not account for this type of calcium-excitation feedback. Therefore, we varied the APD through the repolazitation rate $k_{rep}$. The modified parameters and the introduced spatial dependencies are presented in Appendix A.3. To observe the heterogeneity effect more clearly without making the sample thicker (larger in the Z-direction), which would require more cells of the computational mesh, we decreased the isotropic conductance in some of the simulations with the sample.

Previously, we conducted a comprehensive study on the convergence of the computations of cardiac electromechanics problems, similar to those presented here, published in [15]. Here, for the sake of faster computations, we considered larger spatial steps than the optimal ones found in the convergence study. The chosen mesh sizes sufficed to show the qualitative and relative quantitative effects of various factors on the propagation of excitation–contraction waves. Our computational mesh for the numerical simulations shown below in Section 3 had the following specifications. The time steps for the cell-level ODEs, equations of the monodomain model, and equilibrium equations were 0.01 ms, 0.01 ms, and 1 ms, respectively. The mesh element size was equal to 2.5 mm. Linear discretization was used for the AP variables, and quadratic discretization was used for the displacements, together with fifth-order quadrature integration formulas. Relative tolerances of ${10}^{-8}$ and ${10}^{-12}$ were used for the nonlinear and linear solvers. Overall, 1 s of simulated contraction took approximately 7 h to compute on the computing cluster with 16 MPI tasks and 4 OpenMP processes per task.

3. Results

3.1. Excitation–Contraction of Uniaxially Stretched Slab-like Myocardial Sample

First, we ran the computations of the sample contraction with fiber orientation aligned with the X-axis. The sample was stretched along the fiber direction prior to the first stimulus. We analyzed the waves’ behavior in the simulated excitation–contraction cycles, initiated by 2 Hz stimulation of the left side ($X=0$) of the sample. This problem setup resulted in relatively large strain of the initially elongated fibers, as well as high strain gradients arising during the sample contraction. The results also showed the concentrations of stress and strain in the boundary areas of the “scar”. So, the computations demonstrated both characteristic propagation of the excitation waves passing around the non-excitable region and mechanical stress and strain accompanying the excitation. Thus, we believe that this numerical experiment, which involved the model of myocardial electromechanics, was illustrative as a test problem. The computations were also useful for the debugging process of the framework implementation of the numerical methods. Analyzing the results, we focused on the waves’ detachment from the non-excitable region and checked how the MEF affected the contraction slowing the conduction speed. Figure 2 and Figure 3 show the AP maps and the maps of relative sarcomere strain at 150 ms and 300 ms after a periodic external stimulus.

At the time instant of 150 ms after stimulation, the fiber elongations (minimal per element and relative to initial relaxed fiber length) above the “scar” area varied from 3% to 9% over 2 cm (along the X-axis), while the fiber lengthened by a maximum of 22% in the right side of the sample and shortened by 3% in the left side. It should be recalled that the sample was stretched by 10% before the stimulation, and thus the area behind the wave front shortened, while the relaxed fibers located in the right inactivated area elongated. After the excitation–contraction wave rotated around the “scar”, a region of high strains of 20% was formed near the non-excitable zone with a boundary of shortened (by 10%) fibers, which can be seen at the time instant of 300 ms after the stimulus.

While the first wave of electrical activation ran through the “scar” gray zone without any visible disturbance and almost did not change its horizontal direction, the following waves detached from the “scar” and tended to rotate around its boundary as soon as a stable periodic mode for the variable v, which controls the repolarization in Aliev–Panfilov model, was set up. However, when the MEF was turned off in the cell model through the usage of a constant cell capacitance instead of a non-instant length-dependent one, no wave rotation was observed (Figure 4).

The results demonstrated that the MEF influence taken into account in the simulations shown in Figure 2 and Figure 3 led to an increase in the membrane capacitance ahead of the wavefront by 22–25% due to the initial stretching. That resulted in a slowdown in the excitation propagation. In particular, in the computations with the MEF, the wavefront reached the right edge of the sample in 150 ms after the stimulation compared to 110 ms in the simulations without the MEF taken into account.

We also performed similar numerical experiments with the horizontally pre-stretched myocardium sample, but varying the orientation of the muscle fibers linearly through the coordinate Z as described in Section 2.3. In those simulations, the fiber orientation affected the wave propagation, changing its direction rather predictably. Figure 5 shows the snapshot taken 220 ms after the stimulation, which was more suitable to compare to the simulation results for the slab with horizontally oriented fibers (Figure 2b), 300 ms after the stimulus, because of the difference in wave velocity. The variation in fiber orientation resulted in a slightly slower wave propagation in the horizontal direction (155 ms to reach the right edge) compared to the simulation shown in Figure 2, despite the fact that the MEF effect was weaker. Although the cell capacitance ahead of the wavefront was high (increased by 35%) in the middle (over Z-axis) of the slab, it increased by only 5% in the “epi” side and by less than 1% in the “endo” side. However, due to the non-horizontal fiber orientation, the vertical propagation of the front was much faster than in the case of horizontally oriented fibers, as the wave reached the bottom side of the sample in 155 ms instead of 240 ms after the moment of passing through the top side of the “scar”. The detachment of the excitation wave was less clear, noticeably tending to a rotation mode only in the “epi” side, at the $Z=L,{\alpha }_f=-{60}^{\circ }$ surface. The wave propagated along the boundary of the “scar” nearly without a detachment in the “endo” surface, where the fibers were oriented almost vertically. The wavefront dynamics are shown by the small magenta colored squares approximately representing the trajectory of the point with local maximum curvature on the wavefront.

3.2. Excitation–Contraction of Convex Slab-like Myocardial Sample

We then proceeded to the series of simulations of the excitation and contraction of the convex slab, embowed by pressure applied to the $Z=0$ surface, whose value was growing linearly up to 0.05 kPa for the duration of 50 ms before the first external stimulus. Myocardial regulatory and excitation properties and the orientation of muscle fibers were changing through the thin wall of the sample. Below, we would like to present the results demonstrating noticeable effects, caused by the variations of the following factors: the fibers’ orientation, excitation threshold, and mechano-electric feedback. We also ran the simulations without the heterogeneity of APD and activation length-dependency to see its effect on the propagation of the excitation–contraction waves specifically. Hereafter, we refer to the simulation of 2 Hz stimulation of the convex slab with “normal” excitation threshold, the heterogeneity of fibers’ orientation, heterogeneity of cell-level properties, and with the MEF taken into account as a “default” one. The excitation waves obtained in the default simulation are shown in Figure 6.

The results were similar to those obtained in the simulations of the horizontally pre-stretched sample with linearly varying fiber orientation. However, the wave detachment and its consequent rotation, observed in the “epi” surface of the sample, were even less prominent, presumably because of faster propagation speed caused by lower elongation of the majority of the fibers during the slab inflation, in comparison with the case of its uniaxial stretch. Although the fibers with zero spiral angle, located in the mid-wall of the slab, were stretched by higher values than in the “endo” and “epi” layers, the strains in that area seemed to have a negligible effect on the waves’ dynamics. The increase in the capacitance of cardiomyocyte membrane reached 15% in the middle layers of the sample, decreasing towards its “endo” and “epi” sides to values of about 3% and less than 1%, respectively. Excitation waves reached the right edge of the slab in 135 ms, and the trajectory of the local curvature maximum of the wavefront was observed to be located even closer to the “scar” boundary than in the previous simulation, which is presumably a result of the very weak MEF influence.

To look further into the heterogeneity, we focused on examining the effects of the distribution of the muscle regulatory properties and APD in the sample myocardium. To observe the effects, we compared the results of the “default” simulation with the results of computations for the sample, in which only the fiber orientation was changing through the sample’s shortest dimension. The comparison demonstrated almost no difference in the propagation of the action potential and only a slight difference in contractility, which could be estimated by the concentration of activated regulatory complexes in the overlap region of thin and thick muscle filaments, $A_1$ (Equation (3)). The difference in speed of the action potential propagation was less than 0.5%, and the difference in the peak activation level locally reached 9%, while being less than 3% on average. In an attempt to reveal more significant effects of heterogeneity, we also ran the computations with five times slower isotropic conductance ($D_{iso}=0.1 {\mathrm{mm}}^2/\mathrm{ms}$ instead of $D_{iso}=0.5 {\mathrm{mm}}^2/\mathrm{ms}$). The results of those simulations demonstrated more noticeable difference between the samples with constant and transmurally varying properties, although the difference was still very small, except for some boundary areas. While the cell-level values in the “epi” surface differ by less than 3%, the shortened AP propagating in the “endo” surface, as well as higher $A_1$ values near the right ($X=L$) boundary of the heterogeneous sample, compared to the homogeneous sample, were observed. Higher levels of contractility are presumably caused by the increased length-dependence of ${\mathrm{Ca}}^{2+}$-activation. The values of $A_1$ 250 ms after the stimulation are shown in Figure 7, where the maximal difference in peak activation levels was equal to 15%.

The time courses of ${\mathrm{Ca}}^{2+}$ concentration and $A_1$ in an element in the upper right corner of the sample are presented in Figure 8. Although the solution had not reached a periodic mode during 3 s of 2 Hz stimulation, the result plots could still be used to compare the sample contractility showing the small difference of the variables between the “default” simulation and the simulation of the homogeneous sample. The values for the homogeneous sample were slightly lower than for the heterogeneous one, which was presumably caused by a longer APD in the “endo” layers of the heterogeneous sample. The maximal difference between those cell-level variables seemed to increase along with their peak values, while the simulation slowly approached the periodic mode. The differences in peak calcium concentrations and activation levels measured in the point were 7% and 6%, respectively.

We then proceeded to the simulations in which we could obtain the results featuring general effects of the sample-wide increased excitation threshold and our implementation of MEF. Both modifications result in expected changes in the propagation speed of excitation–contraction waves, leading to different behaviors of the waves upon passing through the “scar” area. Figure 9 shows the results of the simulation with uniformly increased excitation threshold $k_{exc}$ in Equation (2) (the parameter $k_{exc0}=0.1$ was changed to $k_{exc0}=0.15$). Action potential values 250 ms after another stimulus are presented, as the horizontal region of the wavefront at that time point was almost at the same location as the wavefront for the “default” simulation 190 ms after the stimulus. The results are shown in Figure 9.

The propagation speed of the waves was slower compared to the “default” simulation, as it took about 165 ms after the stimulus for the wave to reach the right edge of the sample. The waves’ detachment in the “epi” surface of the sample was more pronounced than in all previous simulations, which is shown by the trajectory points located much further from the “scar” boundary in its upper half. In the “endo” side, the excitation waves only tended slightly to the transition to the rotation mode, though that transition was more distinctive than in the “default” numerical experiments. Proceeding to the discussion of the computations with the MEF “turned off”, the absence of the strain-dependence of the membrane capacitance resulted only in mild speed-up of the excitation waves (135 ms from the stimulus to reach the right side). However, the consequences of the “turned off” MEF were different in the numerical experiments with either decreased isotropic conductance or horizontal orientation of the muscle fibers. In the absence of the MEF, those factors resulted in an almost uniform propagation of the waves after they passed the impaired region instead of the wave detachment from the “scar”, as the wave was propagating through the gray zone of the impaired region near its bottom border. The effect is shown in Figure 10. We suppose that the reason for that wave passing through the “scar” area might be the non-impaired conductivity of the modeled scar. Apparently, both horizontal orientation of the fibers and reduced isotropic conductivity resulted in the diffusion of the excitation across the “scar” under our model assumptions, limited to high excitation threshold and repolarization rates of the scar tissue.

A summary on the performed simulations and their results is presented in

Table 2. Summary on the numerical simulations performed during the study.

Preliminary Deformation	Heterogeneity over Z-Axis	Additional Conditions	Dynamics of Excitation–Contraction Waves
Horizontally stretched slab	Uniform myocardium properties over Z-axis; horizontal fiber orientation	MEF is taken into account	Apparent detachment of the wave from the “scar” region and its further rotation. Moderate slowdown in the wave propagation because of the MEF.
	Uniform myocardium properties over Z-axis; horizontal fiber orientation	No MEF taken into account	The wave propagation along the scar border without detachment, faster propagation speed.
	Uniform myocardium properties over Z-axis; linear variation of the fiber orientation	MEF is taken into account	Apparent detachment of the wave from the “scar” region and its further rotation, which was much less pronounced in the “endo” side. The wave propagation in horizontal direction was slower because of the varying fiber orientation, in spite of the weaker MEF effect (fibers’ preliminary strains and cell capacitance were lower in average than at horizontal fibers).
Convex embowed slab	Linear variation of APD and the length-dependence of the contraction activation over Z-axis; linear variation of the fiber orientation	MEF is taken into account; simulations at normal and reduced isotropic conductance	The wave detachment was less pronounced and was almost absent at the “endo” side, where the waves propagated along the “scar” boundary. The wave propagation was faster than in the simulations presented above in rows 1 and 3 due to very weak MEF effect (the fiber strains and cell capacitance were much lower on average).
	Uniform myocardium properties over Z-axis; linear variation of the fiber orientation	MEF is taken into account; simulations at normal and reduced isotropic conductance	There is no difference in propagation of the excitation waves comparing to the previous case, except for the APD. While comparing the simulations under the condition of reduced isotropic conductance, a small difference in myocardial contractile properties was observed, which did not lead to the difference in excitation because of the weak MEF effect.
	Linear variation of APD and the length-dependence of the contraction activation over Z-axis; linear variation of the fiber orientation	MEF is taken into account; increased excitation threshold	The wave detachment and rotation were more pronounced than in the other simulations, while the horizontal wave propagation was much slower.
	Linear variation of APD and the length-dependence of the contraction activation over Z-axis; linear variation of the fiber orientation	No MEF taken into account; simulations at normal and reduced isotropic conductance	Only a slight increase in propagation speed at normal isotropic conductance was observed. However, at reduced isotropic conductance the wave detachment was barely distinguished, and the waves passed through the bottom part of the “scar” region.
	Linear variation of APD and the length-dependence of the contraction activation over Z-axis; horizontal fiber orientation	No MEF taken into account	A significant increase in propagation speed was observed. The waves passed through the “scar” region without rotation.

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4. Discussion

4.1. Major Results and Limitations

The problems’ formulations and their solutions presented here should be considered not as the computations that reproduce some important phenomena of myocardial excitation and contraction but rather as the numerical test experiments demonstrating how our computational framework deals with the modeling of coupled electromechanics in an anisotropic and heterogeneous myocardium sample. The main effects shown in the simulations are the tendency of the excitation–contraction waves to transform into spiral waves as they pass around a narrow obstacle. We have also shown how the wave propagation depended on some conditions varied by the model parameters. The demonstrated effects are well known and have been studied thoroughly in experiments and numerical simulations [22,23,24]. It is also known that the effects grow stronger when the excitation takes place in the tissue with an obstacle with impaired excitability or under localized or global slowed conductance [13,25]. We observed that the waves’ behavior at the boundary of non-excitable regions in the “epi” and “endo” sides of the deformed slab-formed samples differed from each other, and the difference was larger at slower “transmural” (Z-axis oriented) propagation, when the isotropic term of conduction was decreased. Still, it could be said that the influence of fiber orientation and APD on the excitation was also rather predictable, as those factors changed the conduction velocity, including its direction near the “scar” boundary. That resulted in alternations of the wavefront’s curvature, which is known to affect the initiation of spiral waves at fast-paced stimulation [26]. Even considering those phenomena as being well-studied, the application of the fully coupled electromechanical model allowed us to specify the heterogeneity in myocardium contraction and to make an attempt to examine how the variation of the properties of the contraction and electromechanical coupling in the sample would affect the sample’s excitation and strains. Although we were not able to show any clear cases of re-entry formation caused by the contraction heterogeneity, the simulations seem to be useful as example problems. The successful simulations demonstrated that we should be able to set up and solve similar problems for myocardium in the ventricular wall using our computational framework. We would also like to note that the heterogeneity of the cell-level properties did affect the distribution of the activated regulatory units. That resulted in different fields of strain and stress, especially at the higher relation of anisotropic and isotropic conductance. With mechano-electric feedback of any kind being taken into account, this difference in strain might lead to disturbances in the excitation under some conditions, as it did, to some extent, in our numerical experiments with strain-dependent cardiomyocyte capacitance.

We would also like to comment on the “scar” tissue in our simulations. Although a scar region is usually considered as an insulated area, we used a less strict formulation here. Only the coefficients of excitation threshold and repolarization rate were increased in the scar zone, varying continuously from the values in the normal myocardium through the gray zone tissue. The equations of the cell-level model were solved for the scar region as if it was a part of the myocardial tissue and not a separate region of connective tissue. However, in the simulations, due to high values of the excitation threshold and fast repolarization, the action potential was never high enough to provide the calcium influx into the cell cytoplasm, and thus no active tension was generated. The transmembrane potential was also allowed to change due to the diffusion across the scar boundary, which could affect the behavior of the waves near the scar border differently, compared to the scar with no-flow boundary conditions. Normal conductivity of the scar region might be a reason for the absence of the waves’ rotation in the simulation with the heterogeneous convex slab and the MEF being “turned off”.

Another remark should be made about the contraction mode obtained in the performed simulations. While the simulations were run at the stimulation frequency of 2 Hz, the initial conditions for the cell-level equations were set to the diastolic values obtained for the 1 Hz periodic uniaxial contraction. Those conditions, including, in particular, low (for the 2 Hz contraction) diastolic ${\mathrm{Ca}}^{2+}$ concentration in sarcoplasmic reticulum, together with simplifications of the model (absence of the balance equations for sodium and potassium concentrations) resulted in low values of free cytosolic ${\mathrm{Ca}}^{2+}$ during the first several seconds of the simulation, while the peak ${\mathrm{Ca}}^{2+}$ values were slowly increasing to much higher values of the periodic solution. Thus, active tension and myocardium strain were rather low in the case of the convex slab, in which sarcomere shortening was less than 10%. Presumably, the effect of non-uniform strain-dependence of the contraction activation would be more significant at higher strain values. However, we believe that the qualitative effects shown in the results would be similar even at higher strains.

Discussing the choice of spatial grid size in our study, we would like to make the following remark. Several studies on the modeling of the propagation of excitation waves in myocardium have shown that a computation grid size of less than 500 µm should be used to obtain an accurate solution for the reaction–diffusion equations, while the motion equations have more relaxed requirements. In this study, we performed the simulations using a coarse grid with a spatial size of 2.5 mm, which can presumably lead to a notable difference in the results compared to the computations on a finer mesh. This difference can also involve the different effects of the mechano-electric feedback at the same model parameters because of the error in the conduction velocity. However, our study did not aim to estimate the precise values of characteristics, such as fibers’ strains or excitation threshold, at which the wave rotation becomes apparent. Moreover, we suppose that the simplified phenomenological excitation model used here would not allow us to obtain accurate results even on a fine grid. To investigate possible numerical errors, we performed a test simulation on a homogeneous, except for the “scar” region, uniaxially pre-stretched sample on a mesh that was twice as fine as the one used in our study, with an element size of 1.25 mm. Some results are presented in the Supplementary Materials File S1.

4.2. CarNum Performance and Perspectives

In the study, we used the new cardiac modeling framework CarNum for the numerical simulations. The framework and its dependencies were entirely implemented in C/C++. The core methods of our computational platform were based on their implementations in the massively parallel framework INMOST and inherited its MPI/OpenMP parallelization properties with the computations running on high-performance computing systems. The CarNum platform was designed to solve the problems of coupled cardiac mechanics by dividing them into different physical processes (blocks), where continuous mechanics, electrical activation propagation, and cellular electrophysiology and mechanics were the major ones. This structure allows one to organize the computations of the individual model blocks with different space and time steps, as well as to use different discretization strategies for the different model blocks [15]. The linking of the blocks to each other and the organization of their interactions is managed by the software’s user, which provides great flexibility. When setting up the model by the platform’s methods, the user should provide descriptions of functional dependencies and parameters included in the model blocks. For interaction with user-provided functions and equations, the platform offers tools for the processing of symbolic expressions and methods of automatic differentiation. To make rapid implementation of individual submodels possible, some special capabilities have been implemented, which include support for a human-readable language for the descriptions of cellular models, expressions of the strain-energy functions used in passive continual mechanics, and configuration files. The interaction with the fully specified model is implemented with the use of configuration files and command line arguments.

The structure and features of the CarNum framework allowed us to switch quickly between different mathematical formulations, changing the boundary conditions and stimulation strategies, and to introduce tissue regions with special properties of the myocardium. We also used the framework’s flexibility to modify the equations of the cell model, continuous mechanics, and the variables shared between the blocks of micromechanics and macromechanics with relatively small effort, while debugging and correcting the mathematical setup and its implementation. We believe that our platform proved to be both effective and efficient while dealing with the electromechanical problems studied here, and it can be modified further to simulate more complex objects, such as heart chambers.

4.3. Peculiarities of Numerical Implementation

Our initial attempts to run the computations for the electromechanical problems stated here were unsuccessful because of a divergence of the nonlinear solver. We found out that the divergence was caused by a very fast variation of the microscopic strain $\delta$. The change rate of $\delta$ depended on the sarcomere strain rate and the fast cross-bridge kinetics. The first problem was that the active stress in myocardium depended on $\delta$ explicitly, leading to the dependence of the time derivative of the stress term on the strain rate. It was shown earlier [4] that the usage of the microscopic strain from a previous time step in a similar problem resulted in cycling through repeated consequential increments and decrements in strain, as the solution was unable to satisfy the equilibrium equation. This problem could be solved by the usage of the unknown (at the current time step) strain rate, expressed through unknown nodal displacements, instead of the one from the previous step. However, here we encountered another problem. The active stress depended also on the number of the cross-bridges and the fraction of strongly bound cross-bridges, while the variation of both of those variables depended on $\delta$. Moreover, the $\delta$ variation depended on the cross-bridge kinetics, and these kinetics, together with the muscle active stress, reacted significantly to such low changes in $\delta$ as 2 nm, while the rate of sarcomeric strain could reach 1 µm/s and higher. This caused the numerical methods to perform poorly at the reasonable time-steps of 1 ms. We tried several approximation approaches, and our final workaround was to replace $\delta$ in the equilibrium equations by the value of $\delta$ obtained in the steady-state solution of the system (3). The constant strain rate was expressed in terms of the unknown nodal displacements. Then $\delta$ in the cell-level model block was calculated after the step of solution of the balance equations for macromechanics in accordance with its ODE. The time-derivative of ${\lambda }_f$ in the right-hand part of the ODE for $\delta$ was expressed as $\left({\lambda }_f^t-{\lambda }_f^{t-1}\right)/{\Delta }_t$, where the time indexes and time increments corresponded to the macromechanics time-mesh. We had also found that solving the ODE for $\theta$ in (3) (see also (A1)) instead of its quasi-steady-state approximation, which was used earlier in [18] and other studies on that model, resulted in better convergence due to the not instantaneous dependence of $\theta$ on $\delta$ in that case.

4.4. Novelty

The mathematical formulations of the problems of cardiac excitation and contraction presented here are rather simplistic, and similar numerical studies with 2D patches or thin 3D slabs, including ones with more sophisticated structure and conditions than in our paper, have been performed by different scientific groups [7,27,28]. However, we would like to highlight the following aspects of the study.

The problems examined in the study were formulated and solved to initially test our new computational framework. The results demonstrated that the framework performed well for 3D computations of coupled electromechanical problems. When performing a numerical analysis of myocardium excitation, researchers generally do not take into account muscle active contraction, and only a few research groups consider more or less detailed models of cardiac mechanics in the simulations. In addition, only a few groups presented the simulations of fully coupled cardiac electromechanics (see, for example, [29,30]). The effects of the MEF type considered here were studied in 3D simulations, but without active muscle contraction in consideration [10,31]. One can notice that the qualitative effects observed here in the sample of the simplistic geometrical form could be examined in 2D simulations, as our group did earlier. However, it was important to examine if the framework could deal with mechanics-related 3D problems, in particular, when the material anisotropy that was heterogeneous over the sample depth (Z-dimension) was taken into account.

As we have plans to apply our software to the case of the heart ventricles contracting within the circulatory system, we needed first to demonstrate that the simulations operated properly for a sample with more simple geometry under different conditions. We have performed the important numerical tests showing that we were able to consider the heterogeneity of the sample’s material and introduce customizable parameters of the model’s blocks without significant changes of the program code. Even though the heterogeneity taken into account for the cubic sample did not lead to results that would be as meaningful as the ones that could be obtained in the simulations of heart chamber contraction, this study’s results are still useful and significant. The investigation of the test problems, which were numerically studied here, has provided insight into the possible difficulties that could arise while proceeding to the planned 3D simulation of the ventricles’ function. The study also helped us to optimize the introduction of space-dependent and strain-dependent parameters of the cell-level models. The results demonstrated that the strain-dependence of the cardiomyocyte capacitance affects the conditions under which arrythmogenic phenomena occur within the myocardium. The variation of some parameters of the muscle activation combined with the variation in the fibers’ orientation was also shown to have an effect on the dynamics of excitation–contraction waves.

5. Conclusions

In this study, we focused on a series of spatially simple problems related to myocardium excitation and contraction, which could potentially reveal the influence of strain and distribution of some myocardial properties in tissue on the propagation of excitation–contraction waves. At the same time, the results could help us to improve our computational framework and identify its strengths and weaknesses when applied to the problems requiring a two-way coupled model of cardiomyocyte electromechanics. We have presented the results of numerical simulations of the excitation–contraction waves in a thin-walled slab-shaped myocardial sample with heterogeneity of some of tissue properties and a narrow region imitating a non-excitable scar. Two different initial deformations were applied to the samples. The uniaxial stretch was useful to examine the effects of elongation of the muscle fibers, and the inflation of the sample imitated the region of the ventricular wall to some extent. The results of the simulations helped us to debug and modify the modules of our computational framework to deal with the problems of cardiac electromechanics more effectively, coupling macro- and microscopic deformations of a simulated sample and to set any changes of the tissue properties in specified locations. In addition, we could examine if the tissue heterogeneity, the fibers’ orientation, muscle strain, and the increased excitation threshold affect propagation of the excitation–contraction waves through the sample. While the fibers’ orientation strongly affected the waves’ behavior, the effects of non-constant APD and length-dependence of contraction activation were almost negligible. However, the effects were slightly stronger in the case of slower conductivity across the muscle fibers. At the thicker myocardial tissue and larger strains, those factors could be rather important for correct simulations of myocardium electromechanics. The clear effects of the increased excitation threshold and the MEF were also shown in the numerical studies. We plan to obtain more strongly-pronounced results by applying our computational framework to simulations of ventricular contraction within the cardiovascular system.