Bidomain Equations Research Articles

Uniform bounds of families of analytic semigroups and Lyapunov Linear Stability of planar fronts

AbstractWe study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology.

Transmural APD heterogeneity determines ventricular arrhythmogenesis in LQT8 syndrome: Insights from Bidomain computational modeling.

Long QT Syndrome type 8 (LQT8) is a cardiac arrhythmic disorder associated with Timothy Syndrome, stemming from mutations in the CACNA1C gene, particularly the G406R mutation. While prior studies hint at CACNA1C mutations' role in ventricular arrhythmia genesis, the mechanisms, especially in G406R presence, are not fully understood. This computational study explores how the G406R mutation, causing increased transmural dispersion of repolarization, induces and sustains reentrant ventricular arrhythmias. Using three-dimensional numerical simulations on an idealized left-ventricular model, integrating the Bidomain equations with the ten Tusscher-Panfilov ionic model, we observe that G406R mutation with 11% and 50% heterozygosis significantly increases transmural dispersion of repolarization. During S1-S4 stimulation protocols, these gradients facilitate conduction blocks, triggering reentrant ventricular tachycardia. Sustained reentry pathways occur only with G406R mutation at 50% heterozygosis, while neglecting transmural heterogeneities of action potential duration prevents stable reentry, regardless of G406R mutation presence.

A comparative study of fully implicit staggered and monolithic solution methods. Part II: Coupled excitation–contraction equations of cardiac electromechanics

A comparative study of fully implicit staggered and monolithic solution algorithms for the coupled bidomain equations of cardiac electrophysiology was discussed in Part I (JCAM 407:114021, 2022). In this follow-up study, we further employ the similar approach and present an extensive comparative analysis between the staggered and monolithic solution techniques for the coupled partial differential equations (PDEs) of cardiac electromechanics both in the monodomain and bidomain setting. For the monolithic solution technique, we adopt the work of Göktepe et al. (CM 45: 227-243, 2010) and Dal et al. (CMAME 253: 353-336, 2013), where the governing PDEs of the excitation–contraction problem are solved simultaneously and all rate equations are treated with an implicit time integration scheme. For the staggered scheme, however, we simply suggest a decoupled solution of the governing PDEs while keeping other aspects of the monolithic algorithm the same, such as utilization of the implicit time integration scheme and coupling of ordinary differential equations, which describe the state variables of cardiac electrophysiology, to the parabolic PDE of the electrophysiology. Both solution algorithms are applied to several problems, where we simulate regular planar wave propagations and scroll waves for different spatial and temporal resolution and material parameters. The comparison between the solution schemes is performed in terms of accuracy, efficiency and stability. We reveal that for regular wave propagation the results of different solution algorithms are identical. The only deviation is observed during scroll wave propagation, if the parameters controlling the mutual interaction between the mechanical and electrical field are relatively large and at the same time if the time increment is big, e.g., Δt=2.0ms. Nevertheless, the suggested staggered solution scheme yields almost identical results even for high coupling if smaller time increments are used. Besides, the staggered scheme provides an enormous gain in computational efficiency, particularly, as the number of degrees of freedom is increased and we do not encounter any stability issue arising specifically from the decoupled solution of the governing PDEs. Similar to the conclusion that was made for cardiac electrophysiology in Part I, we report that the suggested fully implicit staggered solution algorithm has a tremendous application potential for computer simulations of excitation–contraction coupling of the heart both in the monodomain and bidomain setting.

Global Strong Well-Posedness of the Stochastic Bidomain Equations with FitzHugh–Nagumo Transport

Global Strong Well-Posedness of the Stochastic Bidomain Equations with FitzHugh–Nagumo Transport

A multi-scale simulation of retinal physiology

We present a detailed physiological model of the (human) retina that includes the biochemistry and electrophysiology of phototransduction, neuronal electrical coupling, and the spherical geometry of the eye. The model is a parabolic–elliptic system of partial differential equations based on the mathematical framework of the bi-domain equations, which we have generalized to account for multiple cell-types. We discretize in space with non-uniform finite differences and step through time with a custom adaptive time-stepper that employs a backward differentiation formula and an inexact Newton method. A refinement study confirms the accuracy and efficiency of our numerical method. Numerical simulations using the model compare favorably with experimental findings, such as desensitization to light stimuli and calcium buffering in photoreceptors. Other numerical simulations suggest an interplay between photoreceptor gap junctions and inner segment, but not outer segment, calcium concentration. Applications of this model and simulation include analysis of retinal calcium imaging experiments, the design of electroretinograms, the design of visual prosthetics, and studies of ephaptic coupling within the retina.

A smoothed boundary bidomain model for cardiac simulations in anatomically detailed geometries.

This manuscript presents a novel finite difference method to solve cardiac bidomain equations in anatomical models of the heart. The proposed method employs a smoothed boundary approach that represents the boundaries between the heart and the surrounding medium as a spatially diffuse interface of finite thickness. The bidomain boundary conditions are implicitly implemented in the smoothed boundary bidomain equations presented in the manuscript without the need of a structured mesh that explicitly tracks the heart-torso boundaries. We reported some significant examples assessing the method's accuracy using nontrivial test geometries and demonstrating the applicability of the method to complex anatomically detailed human cardiac geometries. In particular, we showed that our approach could be employed to simulate cardiac defibrillation in a human left ventricle comprising fiber architecture. The main advantage of the proposed method is the possibility of implementing bidomain boundary conditions directly on voxel structures, which makes it attractive for three dimensional, patient specific simulations based on medical images. Moreover, given the ease of implementation, we believe that the proposed method could provide an interesting and feasible alternative to finite element methods, and could find application in future cardiac research guiding electrotherapy with computational models.

Epicardial Dispersion of Repolarization Promotes the Onset of Reentry in Brugada Syndrome: A Numerical Simulation Study

The Brugada syndrome (BrS) is a cardiac arrhythmic disorder responsible for sudden cardiac death associated with the onset of ventricular arrhythmias, such as reentrant ventricular tachycardia and fibrillation. The mechanisms which lead to the onset of such electrical disorders in patients affected by BrS are not completely understood, yet. The aim of the present study is to investigate by means of numerical simulations the electrophysiological mechanisms at the basis of the morphology of electrocardiogram (ECG) and the onset of reentry associated with BrS. To this end, we consider the Bidomain equations coupled with the ten Tusscher–Panfilov membrane model, on an idealized wedge of human right ventricular tissue. The results have shown that: (1) epicardial dispersion of repolarization, generated by the coexistence of regions of early and late repolarization, due to different modulation of the I_mathrm{{CaL}} current, produces ECG waveforms exhibiting qualitatively the typical BrS morphology, characterized by ST elevation and partially negative T-waves; (2) epicardial dispersion of repolarization promotes the onset of reentry during the implementation of the programmed stimulation protocol, because of the conduction block occurring when a premature beat reaches the border of late repolarizing regions; and (3) the modulation of the I_mathrm{{to}} current affects the duration of reentry, which becomes sustained with a remarkable increase of I_mathrm{{to}} in the subepicardial layers.

Strong solutions and asymptotic behavior of bidomain equations with random noise

In this paper, we study the conditions for the existence of strong solutions (both local and global) for stochastic bidomain equations. To this end, we use a priori energy estimates and Serrin-type theorems. We further address the asymptotic behavior of the solutions, which includes the analysis of small stochastic perturbations and large deviations. In a separate section we specify the support of the invariant measure, whose existence was established in [M. Hieber, O. Misiats and O. Stanzhytskyi, On the bidomain equations driven by stochastic forces, Discrete Contin. Dyn. Syst. 40 (11) (2020) 6159–6177].

An In-Silico model for evaluating the directional shock vectors in terminating and modulating rotors

Out-of-hospital cardiac arrest (OHCA) accounts for a majority of mortality worldwide. Survivability from an OHCA highly depends on timely and effective defibrillation. Most of the OHCA cases are due to ventricular fibrillation (VF), a lethal form of cardiac arrhythmia. During VF, previous studies have shown the presence of spatiotemporally organized electrical activities called rotors and that terminating these rotor-like activities could modulate or terminate VF in an in-hospital or research setting. However, such an approach is not feasible for OHCA scenarios. In the case of an OHCA, external defibrillation remains the main therapeutic option despite the low survival rates. In this study, we evaluated whether defibrillation effectiveness in an OHCA scenario could be improved if a shock vector directly targets rotor-like, spatiotemporal electrical activities on the myocardium. Specifically, we hypothesized that the position of defibrillator pads with respect to a rotor’s core axis and shock current density could influence the likelihood of rotor termination and thereby result in successful defibrillation. We created a bidomain cardiac model based on porcine heart data using Aliev–Panfilov bidomain equations. We simulated localized rotors, which we attempted to terminate using different defibrillation pad orientations relative to the rotor axis (i.e., perpendicular, parallel, and oblique). In addition, we gradually increased current densities for each defibrillation pad orientation from 4 to 12 A/m2. We repeated the above defibrillation procedure for rotors originating from four different locations on the ventricles. The shock parameters and the outcomes were analyzed using a Generalized Linear Mixed Model (GLMM) with Logistic Regression to link rotor termination with the defibrillation pad orientation and current density. Our results suggest the highest average likelihood of terminating rotors during VF is when defibrillator pads are placed perpendicular to the rotor axis (0.99 ± 0.03), with an average current density of 7.2 A/m2, compared to any other orientation (parallel: 0.76 ± 0.26 and oblique: 0.08 ± 0.12). Our simulations suggest that optimal defibrillator pad orientation, combined with sufficient current density magnitude, could improve the likelihood of rotor termination during VF and thereby improving defibrillation success in OHCA patients.

Parallel Newton--Krylov BDDC and FETI-DP Deluxe Solvers for Implicit Time discretizations of the Cardiac Bidomain Equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

Asymptotic Behavior of Fronts and Pulses of the Bidomain Model

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we investigate the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen-Cahn equation and the bidomain FitzHugh-Nagumo equation in two spatial dimension. In previous work, it was shown that planar fronts of the bidomain Allen-Cahn equation can become unstable in contrast to the classical Allen-Cahn equation. We find that, after the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical, by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. In our computational studies of the bidomain FitzHugh-Nagumo pulse solution, we show that the pulses can also become unstable much like the bidomain Allen-Cahn fronts. However, unlike the bidomain Allen-Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choice of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by the development of a numerical scheme that allows for the accurate computation of the bidomain equation in a two dimensional strip domain of infinite extent.

Discrete mass conservation and stability analysis of the ocean-atmosphere model with coupling conditions

In this work we considered bi-domain partial differential equations (PDEs) with two coupling interface conditions. The one domain is corresponding to the ocean and the second is to the atmosphere. The two coupling conditions are used to linked the interaction between these two regions. As we know that almost every engineering problem modeled via PDEs. The analytical solutions of these kind of problems are not easy, so we use numerical approximations. In this study we discuss the two essential properties, namely mass conservation and stability analysis of two types of coupling interface conditions for the ocean-atmosphere model. The coupling conditions arise in general circulation models used in climate simulations. The two coupling conditions are the Dirichlet-Neumann and bulk interface conditions. For the stability analysis, we use the Godunov–Ryabenkii theory of normal-mode analysis. The main emphasis of this work is to study the numerical properties of coupling conditions and an important point is to maintain conservativity of the overall scheme. Furthermore, for the numerical approximation we use two methods, an explicit and implicit couplings. The implicit coupling have further two algorithms, monolithic algorithm and partitioned iterative algorithm. The partitioned iterative approach is complex as compared to the monolithic approach. In addition, the comparison of the numerical results are exhibited through graphical illustration and simulation results are drafted in tabular form to validate our theoretical investigation. The novel characteristics of the findings from this paper can be of great importance in science and ocean engineering.

A comparative study of fully implicit staggered and monolithic solution methods. Part I: Coupled bidomain equations of cardiac electrophysiology

This manuscript presents an extensive comparative analysis between a staggered and a monolithic solution technique for the bidomain equations of cardiac electrophysiology in the context of finite element method. For the monolithic solution technique, we adopt the work of Dal et al. (CMBBE 15: 645-656, 2012), where the parabolic and elliptic partial differential equations (PDEs) of the bidomain model are solved simultaneously and all rate equations are treated with an implicit time integration scheme. For the staggered scheme, however, we simply suggest a decoupled solution of the parabolic and elliptic PDE while keeping other aspects of the monolithic algorithm the same, such as utilization of the implicit time integration scheme, coupling of ordinary differential equations, which describe the state variables of cardiac electrophysiology, to the parabolic PDE and consideration of the transmembrane potential and the extracellular potential as degrees of freedom. Both solution algorithms are applied to several problems where we simulate regular planar wave propagations, scroll waves and externally applied electrical fields for different spatial and temporal resolution and material parameters. The comparison between the solution schemes is performed in terms of accuracy, efficiency and stability. We reveal that the suggested staggered solution scheme yields almost identical results with the monolithic scheme for physiological conductivity parameters and small time increments. Besides, the staggered scheme provides an enormous gain in computational efficiency, particularly, as the number of degrees of freedom is increased and we do not encounter any stability issue arising from the decoupled solution of the PDEs. Therefore, we conclude that the suggested fully implicit staggered solution algorithm has a tremendous application potential for the bidomain equations of cardiac electrophysiology.

Parallel space-time adaptive numerical simulation of 3D cardiac electrophysiology

The bidomain equations form the state-of-the-art model of cardiac electrophysiology and describe normal or pathological propagation of the excitation wave through cardiac tissue. If based on mechanistic cell models and applied to anatomically realistic geometries, their discretization requires very fine resolutions both in space and time and renders the numerical solution an extremely challenging computational problem. In this paper, we address this challenge by combining space-time adaptive discretization with dynamic load balancing for parallel computing. We use multilevel finite element methods for the spatial adaptation of the mesh, embedded Rosenbrock time integration for an adaptive choice of step sizes, and non-overlapping domain decomposition for the parallelization. In order to further reduce the computational costs, we exploit the sparsity of local matrices during the assembly of global FEM matrices which ensure an optimal usage of memory. In numerical tests, we demonstrate the feasibility of our approach that a substantial reduction factor of computing time is achieved w.r.t the static grid computation and a good parallel efficiency is achieved.

Composite Backward Differentiation Formula for the Bidomain Equations.

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.

Reduced order method for finite difference modeling of cardiac propagation

Abstract Efficient numerical simulation of cardiac electrophysiology is crucial for studying the electrical properties of the heart tissue. The cardiac bidomain model is the most widely accepted representation of the electrical behaviour of the heart muscle. The bidomain model offers fast cardiac potential variation, which can lead to high computational cost due to the required large grid sizes and small time steps. In this paper, the complexity of the finite difference approximation of the bidomain equations is reduced with the model order reduction technique. Proper orthogonal decomposition, a projection-based algorithm, is used to efficiently approximate the original high fidelity cardiac bidomain model with a low-dimensional system of equations. The low-dimensional basis functions are computed first from the ‘snapshots,’ which contain the solutions of the full-order system for different temporal and spatial parameters. Galerkin projection of the original cardiac bidomain system onto the subspace of the reduced order basis functions reduces the size of the linear system. Numerical results confirm the efficiency of the proposed reduced order modeling technique, reducing the simulation time by a factor of 9.54, while maintaining an RMS error of 0.769 mV between the original full order solution and the reduced order POD solution.

The periodic version of the Da Prato\u2013Grisvard theorem and applications to the bidomain equations with FitzHugh\u2013Nagumo transport

In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal {{L}}^p-regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo,Aliev–Panfilov, or Rogers–McCulloch, it is proved that this set of equations admits a unique, strongT-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T-periodic forces.

Numerical simulation of mathematical heart model in COMSOL

Electrical activity is essential for the cardiac cell to perform its function. Mathematical modeling of cardiac electrical activity is performed from the cell, tissue and organ levels through to the body surface level. The electrical activity of the cardiac as a whole is thus characterized by a complex multiscale structure. The most complete model of such a complex setting is the anisotropic bidomain model that consists of a system of two degenerate parabolic reaction diffusion equations describing the intra and extracellular potentials in the cardiac muscle, coupled with a system of ordinary deferential equations describing the ionic currents flowing through the cellular membrane. This study describes an anatomically realistic 3D Bidomain model of whole-heart electrical activity. The heart was embedded in a human torso, incorporating spontaneous activation with heterogeneous action potential (AP) morphologies throughout the heart. The aim of this study is the development of a geometrically simple and computationally efficient 3D model of heart. In this paper a finite element formulation, model and simulation of Bidomain equation has been conducted. The FitzHugh-Nagumo (FHN) equations were incorporated into Bidomain model of cardiac electrical activity, which was comprised of a simplified geometry of the whole heart with the torso as an extracellular volume conductor. Laplace equation for the torso also considered. Simplified 3D cardiac model was implemented using COMSOL Multiphysics 5.0 finite element software. Electrical potential at different point on torso is measured. Propagation of electrical excitation on heart surface is also observed. This study represents the first stage toward the development of an accurate computer model of heart activation.  Â

On the Bidomain equations driven by stochastic forces

The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

Experimental Validation of Image-Based Modeling of Torso Surface Potentials During Acute Myocardial Ischemia.

Myocardial ischemia is an early clinical indicator of several underlying cardiac pathologies, including coronary artery disease, Takatsobu cardiomyopathy, and coronary artery dissection. Significant progress has been made in computing body-surface potentials from cardiac sources by solving the forward problem of electrocardiography. However, the lack of in vivo studies to validate such computations from ischemic sources has limited the translational potential of such models. To resolve this need, we have developed a large-animal experimental model that includes simultaneous recordings within the myocardium, on the epicardial surface, and on the torso surface during episodes of acute, controlled ischemia. Following each experiment, magnetic resonance images were obtained of the anatomy and electrode locations to create a subject-specific model for each animal. From the electrical recordings of the heart, we identified ischemic sources and used the finite element method to solve a static bidomain equation on a geometric model to compute torso surface potentials. Across 33 individual heartbeats, the forward computed torso potentials showed only moderate agreement in both pattern and amplitude with the measured values on the torso surface. Qualitative analysis showed a more encouraging pattern of elevations and depressions shared by computed and measured torso potentials. Pearson's correlation coefficient, root mean squared error, and absolute error varied significantly by heartbeat (0.1642 ± 0.223, 0.10 ± 0.03mV, and 0.08 ± 0.03mV, respectively). We speculate several sources of error in our computation including noise within torso surface recordings, registration of electrode and anatomical locations, assuming a homogeneous torso conductivities, and imposing a uniform "transition zone" between ischemic and non-ischemic tissues. Further studies will focus on characterizing these sources of error and understanding how they effect the study results.