Anode/cathode make and break phenomena in a model of defibrillation

Bidomain modeling of electrical and mechanical properties of cardiac tissue

Throughout the history of cardiac research, there has been a clear need to establish mathematical models to complement experimental studies. In an effort to create a more complete picture of cardiac phenomena, the bidomain model was established in the late 1970s to better understand pacing and defibrillation in the heart. This mathematical model has seen ongoing use in cardiac research, offering mechanistic insight that could not be obtained from experimental pursuits. Introduced from a historical perspective, the origins of the bidomain model are reviewed to provide a foundation for researchers new to the field and those conducting interdisciplinary research. The interplay of theory and experiment with the bidomain model is explored, and the contributions of this model to cardiac biophysics are critically evaluated. Also discussed is the mechanical bidomain model, which is employed to describe mechanotransduction. Current challenges and outstanding questions in the use of the bidomain model are addressed to give a forward-facing perspective of the model in future studies.

Constitutively Active Acetylcholine-Dependent Potassium Current Increases Atrial Defibrillation Threshold by Favoring Post-Shock Re-Initiation

Electrical cardioversion (ECV), a mainstay in atrial fibrillation (AF) treatment, is unsuccessful in up to 10–20% of patients. An important aspect of the remodeling process caused by AF is the constitutive activition of the atrium-specific acetylcholine-dependent potassium current (I_K,ACh → I_K,ACh-c), which is associated with ECV failure. This study investigated the role of I_K,ACh-c in ECV failure and setting the atrial defibrillation threshold (aDFT) in optically mapped neonatal rat cardiomyocyte monolayers. AF was induced by burst pacing followed by application of biphasic shocks of 25–100 V to determine aDFT. Blocking I_K,ACh-c by tertiapin significantly decreased DFT, which correlated with a significant increase in wavelength during reentry. Genetic knockdown experiments, using lentiviral vectors encoding a Kcnj5-specific shRNA to modulate I_K,ACh-c, yielded similar results. Mechanistically, failed ECV was attributed to incomplete phase singularity (PS) removal or reemergence of PSs (i.e. re-initiation) through unidirectional propagation of shock-induced action potentials. Re-initiation occurred at significantly higher voltages than incomplete PS-removal and was inhibited by I_K,ACh-c blockade. Whole-heart mapping confirmed our findings showing a 60% increase in ECV success rate after I_K,ACh-c blockade. This study provides new mechanistic insight into failing ECV of AF and identifies I_K,ACh-c as possible atrium-specific target to increase ECV effectiveness, while decreasing its harmfulness.

An efficient finite element approach for modeling fibrotic clefts in the heart

Advanced medical imaging technologies provide a wealth of information on cardiac anatomy and structure at a paracellular resolution, allowing to identify microstructural discontinuities which disrupt the intracellular matrix. Current state-of-the-art computer models built upon such datasets account for increasingly finer anatomical details, however, structural discontinuities at the paracellular level are typically discarded in the model generation process, owing to the significant costs which incur when using high resolutions for explicit representation. In this study, a novel discontinuous finite element (dFE) approach for discretizing the bidomain equations is presented, which accounts for fine-scale structures in a computer model without the need to increase spatial resolution. In the dFE method, this is achieved by imposing infinitely thin lines of electrical insulation along edges of finite elements which approximate the geometry of discontinuities in the intracellular matrix. Simulation results demonstrate that the dFE approach accounts for effects induced by microscopic size scale discontinuities, such as the formation of microscopic virtual electrodes, with vast computational savings as compared to high resolution continuous finite element models. Moreover, the method can be implemented in any standard continuous finite element code with minor effort.

An intuitive safety factor for cardiac propagation

Safety factor is a useful concept for analyzing the propagation of impulses through cardiac tissue, which may have compromised ion channel function or electrical connectivity. Several formulations for its calculation have been proposed and have proved useful in one dimension; however, as we demonstrate, recent attempts to use the same formulation in multiple dimensions have led to questionable conclusions. In this study, we mathematically analyze the latest formulation of safety factor and explain its puzzling behavior. We propose a new formulation that is suitable for any dimension and can be estimated from experimental measurements. Its applicability is verified in two-dimensional simulations.

Generation of histo-anatomically representative models of the individual heart: tools and application

This paper presents methods to build histo-anatomically detailed individualized cardiac models. The models are based on high-resolution three-dimensional anatomical and/or diffusion tensor magnetic resonance images, combined with serial histological sectioning data, and are used to investigate individualized cardiac function. The current state of the art is reviewed, and its limitations are discussed. We assess the challenges associated with the generation of histo-anatomically representative individualized in silico models of the heart. The entire processing pipeline including image acquisition, image processing, mesh generation, model set-up and execution of computer simulations, and the underlying methods are described. The multifaceted challenges associated with these goals are highlighted, suitable solutions are proposed, and an important application of developed high-resolution structure-function models in elucidating the effect of individual structural heterogeneity upon wavefront dynamics is demonstrated.

Solving the coupled system improves computational efficiency of the bidomain equations

The bidomain equations are frequently used to model the propagation of cardiac action potentials across cardiac tissue. At the whole organ level, the size of the computational mesh required makes their solution a significant computational challenge. As the accuracy of the numerical solution cannot be compromised, efficiency of the solution technique is important to ensure that the results of the simulation can be obtained in a reasonable time while still encapsulating the complexities of the system. In an attempt to increase efficiency of the solver, the bidomain equations are often decoupled into one parabolic equation that is computationally very cheap to solve and an elliptic equation that is much more expensive to solve. In this study, the performance of this uncoupled solution method is compared with an alternative strategy in which the bidomain equations are solved as a coupled system. This seems counterintuitive as the alternative method requires the solution of a much larger linear system at each time step. However, in tests on two 3-D rabbit ventricle benchmarks, it is shown that the coupled method is up to 80% faster than the conventional uncoupled method-and that parallel performance is better for the larger coupled problem.

From mitochondrial ion channels to arrhythmias in the heart: computational techniques to bridge the spatio-temporal scales

Computer simulations of electrical behaviour in the whole ventricles have become commonplace during the last few years. The goals of this article are (i) to review the techniques that are currently employed to model cardiac electrical activity in the heart, discussing the strengths and weaknesses of the various approaches, and (ii) to implement a novel modelling approach, based on physiological reasoning, that lifts some of the restrictions imposed by current state-of-the-art ionic models. To illustrate the latter approach, the present study uses a recently developed ionic model of the ventricular myocyte that incorporates an excitation-contraction coupling and mitochondrial energetics model. A paradigm to bridge the vastly disparate spatial and temporal scales, from subcellular processes to the entire organ, and from sub-microseconds to minutes, is presented. Achieving sufficient computational efficiency is the key to success in the quest to develop multiscale realistic models that are expected to lead to better understanding of the mechanisms of arrhythmia induction following failure at the organelle level, and ultimately to the development of novel therapeutic applications.

Mechanisms of conduction slowing during myocardial stretch by ventricular volume loading in the rabbit

Acute ventricular loading by volume inflation reversibly slows epicardial electrical conduction, but the underlying mechanism remains unclear. This study investigated the potential contributions of stretch-activated currents, alterations in resting membrane potential, or changes in intercellular resistance and membrane capacitance. Conduction velocity was assessed using optical mapping of isolated rabbit hearts at end-diastolic pressures of 0 and 30 mmHg. The addition of 50 microM Gd3+ (a stretch-activated channel blocker) to the perfusate had no effect on slowing. The effect of volume loading on conduction velocity was independent of changes in resting membrane potential created by altering the perfusate potassium concentration between 1.5 and 8 mM. Bidomain model analysis of optically recorded membrane potential responses to a unipolar stimulus suggested that the cross-fiber space constant and membrane capacitance both increased with loading (21%, P = 0.006, and 56%, P = 0.004, respectively), and these changes, when implemented in a resistively coupled one-dimensional network model, were consistent with the observed slowing (14%, P = 0.005). In conclusion, conduction slowing during ventricular volume loading is not attributable to stretch-activated currents or altered resting membrane potential, but a reduction of intercellular resistance with a concurrent increase of effective membrane capacitance results in a net slowing of conduction.

Solvers for the cardiac bidomain equations

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modeling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.

An unconditionally stable numerical method for the Luo–Rudy 1 model used in simulations of defibrillation

Numerical simulations of defibrillation using the Bidomain model coupled to a model of membrane kinetics represent a serious numerical challenge. This is because very high voltages close to defibrillation electrodes demand that extreme time step restrictions be placed on standard numerical schemes, e.g. the forward Euler scheme. A common solution to this problem is to modify the cell model by simple if-tests applied to several equations and rate functions. These changes are motivated by numerical problems rather than physiology, and should therefore be avoided whenever possible. The purpose of this paper is to present a numerical scheme that handles the original model without modifications and which is unconditionally stable for the Luo-Rudy phase 1 model. This also shows that the cell model is mathematically well-behaved, even in the presence of very high voltages. Our theoretical results are illustrated by numerical computations.

Algebraic multigrid preconditioner for the cardiac bidomain model

The bidomain equations are considered to be one of the most complete descriptions of the electrical activity in cardiac tissue, but large scale simulations, as resulting from discretization of an entire heart, remain a computational challenge due to the elliptic portion of the problem, the part associated with solving the extracellular potential. In such cases, the use of iterative solvers and parallel computing environments are mandatory to make parameter studies feasible. The preconditioned conjugate gradient (PCG) method is a standard choice for this problem. Although robust, its efficiency greatly depends on the choice of preconditioner. On structured grids, it has been demonstrated that a geometric multigrid preconditioner performs significantly better than an incomplete LU (ILU) preconditioner. However, unstructured grids are often preferred to better represent organ boundaries and allow for coarser discretization in the bath far from cardiac surfaces. Under these circumstances, algebraic multigrid (AMG) methods are advantageous since they compute coarser levels directly from the system matrix itself, thus avoiding the complexity of explicitly generating coarser, geometric grids. In this paper, the performance of an AMG preconditioner (BoomerAMG) is compared with that of the standard ILU preconditioner and a direct solver. BoomerAMG is used in two different ways, as a preconditioner and as a standalone solver. Two 3-D simulation examples modeling the induction of arrhythmias in rabbit ventricles were used to measure performance in both sequential and parallel simulations. It is shown that the AMG preconditioner is very well suited for the solution of the bidomain equation, being clearly superior to ILU preconditioning in all regards, with speedups by factors in the range 5.9-7.7.

Oblique propagation of activation allows the detection of uncoupling microstructures from cardiac near field behavior

Wave fronts of cardiac activation, when propagating oblique to the fiber axis, reveal small fractionations and distortions in extracellular potential waveforms Phi(e) as well as in parameters derived from Phi(e) like dPhi(e)/dt and the gradient of Phi(e), the cardiac near field E. dPhi(e)/dt shows multiple deflections and E changes its morphology forming abnormal or even two or multiple subsequent loops. We analyze segments of such irregular loops of E obtained from in-vitro experiments and from computer simulation of a 2D-tissue sheet with a longitudinal oriented obstacle. In computer simulations we found that the individual sections of E reflect fairly well individual pathways of activation separated and delayed by the presence of a structural obstacle similar to connective tissue embedded longitudinally in heart tissue. Electrophysiological experiments with papillary muscles confirm this near field behavior.

Mechanistic enquiry into the effect of increased pacing rate on the upper limit of vulnerability

The goal of this study is to investigate the mechanisms responsible for the increase in the upper limit of vulnerability (ULV; highest shock strength that induces arrhythmia) following the increase in pacing rate. To accomplish this goal, the study employs a three-dimensional bidomain finite element model of a slice through the canine ventricles. The preparation was paced eight times at a basic cycle length (BCL) of either 80 or 150ms followed by delivery of shocks of various strengths and timings. Our results demonstrate that the shock strength, which induced an arrhythmia 50% of the time, increased 20% for the faster pacing compared to the slower pacing. Analysis of the mechanisms underlying the increased vulnerability revealed that delayed post-shock activations originating in the tissue depths appear as breakthrough activations on the surfaces of the preparation following an isoelectric window (IW). However, the IW duration was consistently shorter in the faster-paced preparation. Consequently, breakthrough activations appeared on the surfaces of this preparation earlier, when the tissue was less recovered, resulting in higher probability of unidirectional block and reentry. This explains why shocks of the same strength were more likely to result in arrhythmia induction when delivered to a preparation that was rapidly paced.

Do Intramural Virtual Electrodes Facilitate Successful Defibrillation? Model-Based Analysis of Experimental Evidence

INTRODUCTION

Recent computer model and experimental studies have suggested that microscopic intramural collagenous planes may facilitate successful defibrillation through the generation of shock-induced virtual electrodes deep within the ventricular wall. Evidence supporting the existence of intramural virtual electrodes has been drawn from several recent studies, which map shock-induced membrane potential (Vm) over the cut transmural surface of dissected segments of porcine left ventricle (LV). The artificially created transmural boundary in these experiments is impermeable to intracellular current. It is not known how this constraint limits the interpretation of these experiments in terms of the shock response of the intact ventricle.

METHODS AND RESULTS

This study uses a realistic 3D computer model of LV myocardium to aid experimental interpretation. The model incorporates a microstructural description of intramural cleavage plane discontinuities measured by confocal microscopy of rat LV. Electrical shocks are applied across the model tissue, with and without introduced transmural boundaries. Shocks of varying strength (4-40 V/cm) are also applied to the model and the response analyzed. Results show that shock-induced Vm changes (deltaVm) on a transmural tissue boundary are significantly different to deltaVm of the intact ventricle, and the extent of difference depends on boundary orientation. However, the presence and qualitative behavior of intramural virtual electrodes is preserved irrespective of boundary placement. The model also confirms experimental observations that most rapid transmural activation occurs for shocks of strength 5-10 V/cm. Two distinct mechanisms suppress virtual electrode propagation, and hence slow tissue activation, outside of this optimal shock strength range.

CONCLUSIONS

This study supports the hypothesis that distributed microscopic intramural virtual electrodes contribute to rapid activation of the ventricular wall during defibrillation.

Defibrillation of the heart: insights into mechanisms from modelling studies

Despite its critical role in restoring cardiac rhythm and thus in saving human life, cardiac defibrillation remains poorly understood. Further mechanistic inquiry is hampered by the inability of presently available experimental techniques to resolve, with sufficient accuracy, electrical behaviour confined to the depth of the ventricles. The objective of this review article is to demonstrate that realistic 3-D simulations of the ventricular defibrillation process in close conjunction with experimental observations are capable of bringing a new level of understanding of the electrical events that ensue from the interaction between fibrillating myocardium and applied shock. The article does this by reviewing the results of two studies, one on vulnerability to electric shocks and another on defibrillation. An overview of the modelling tools used in these studies is also provided.

Cell and tissue responses to electric shocks

AIM

Existing models of myocardial membrane kinetics have not been able to reproduce the experimentally-observed negative bias in the asymmetry of transmembrane potential changes (DeltaV(m)) induced by strong electric shocks. The goals of this study are (1) to demonstrate that this negative bias could be reproduced by the addition, to the membrane model, of electroporation and an outward current, I(a), part of the K(+) flow through the L-type Ca(2+)-channel, and (2) to determine how such modifications in the membrane model affect shock-induced break excitation in a 2D preparation.

METHODS AND RESULTS

We conducted simulations of shocks in bidomain fibres and sheets with membrane dynamics represented by the Luo-Rudy dynamic model (LRd'2000), to which electroporation (LRd + EP model) and the outward current, I(a), activated upon strong shock-induced depolarization (aLRd model) was added. Assuming I(a) is a part of K(+) flow through the L-type Ca(2+)-channel enabled us to reproduce both the experimentally observed rectangularly-shaped positive DeltaV(m) and the value of near 2 of the negative-to-positive DeltaV(m) ratio. In the sheet, I(a) not only contributed to the negative bias in DeltaV(m) asymmetry at sites polarized by physical and virtual electrodes, but also restricted positive DeltaV(m). Electroporation, in its turn, was responsible for the decrease in cathode-break excitation threshold in the aLRd sheet, compared with the other two cases, as well as for the occurrence of the excitation after the shock-end rather than during the shock.

CONCLUSIONS

The incorporation of electroporation and I(a) in a membrane model ensures match between simulation results and experimental data. The use of the aLRd model results in a lower threshold for shock-induced break excitation.

Cardiac structure and electrical activation: models and measurement

1. Our group has developed finite element models of ventricular anatomy that incorporate detailed structural information. These have been used to study normal electrical activation and re-entrant arrhythmia. 2. A model based on the actual three-dimensional microstructure of a transmural left ventricular (LV) segment predicts that cleavage planes between muscle layers may give rise to non-uniform, anisotropic electrical propagation and also provide a substrate for bulk resetting of the myocardium during defibrillation. 3. The model predictions are consistent with the results of preliminary experiments in which a novel fibre-optic probe is used to record transmembrane potentials at multiple intramural sites in the intact heart. Extracellular potentials are recorded at adjacent LV sites in these studies. 4. We conclude that structural discontinuities in ventricular myocardium may play a role in the initiation of re-entrant arrhythmia and discuss future studies that address this hypothesis.

Asymmetry in membrane responses to electric shocks: insights from bidomain simulations

Models of myocardial membrane dynamics have not been able to reproduce the experimentally observed negative bias in the asymmetry of transmembrane potential changes (DeltaVm) induced by strong electric shocks delivered during the action potential plateau. The goal of this study is to determine what membrane model modifications can bridge this gap between simulation and experiment. We conducted simulations of shocks in bidomain fibers and sheets with membrane dynamics represented by the LRd'2000 model. We found that in the fiber, the negative bias in DeltaVm asymmetry could not be reproduced by addition of electroporation only, but by further addition of hypothetical outward current, Ia, activated upon strong shock-induced depolarization. Furthermore, the experimentally observed rectangularly shaped positive DeltaVm, negative-to-positive DeltaVm ratio (asymmetry ratio) = approximately 2, electroporation occurring at the anode only, and the increase in positive DeltaVm caused by L-type Ca2+-channel blockade were reproduced in the strand only if Ia was assumed to be a part of K+ flow through the L-type Ca2+-channel. In the sheet, Ia not only contributed to the negative bias in DeltaVm asymmetry at sites polarized by physical and virtual electrodes, but also restricted positive DeltaVm. Inclusion of Ia and electroporation is thus the bridge between experiment and simulation.

Refractory Gradient is Responsible for the Increase in Ventricular Vulnerability Under Sodium Channel Blockade

BACKGROUND

Previous studies have shown that sodium channel (I(Na)) blockade increases ventricular vulnerability; however, there were differences in the degree of the increase. Because the vulnerable window (VW) is altered by the type of preshock refractory gradient (RG), the hypothesis was that the differences in the arrhythmogenesis of I(Na) blockade result from the different types of preshock RG employed.

METHODS AND RESULTS

Simulations of regio(Na)l electric shock following constant pacing stimuli in 2-dimensional bidomain myocardial sheets under I(Na) blockade were conducted using 3 types of preshock RG: longitudinally tilted (LRG), transversely tilted (TRG), and non-tilted RG (NRG). The increase in the degree of I(Na) blockade almost linearly decreased the conduction velocity. The action potential duration in the LRG and TRG cases was non-linearly shortened with the increase in INa blockade because of electrotonic influences, whereas in the case of NRG it was slightly prolonged. In both LRG and TRG cases, the VW for reentry induction by electric shock was considerably widened by the INa blockade; however, this was not the case for NRG in which the VW was rather narrowed by the INa blockade.

CONCLUSION

The type of preshock RG alters the degree of the increase in ventricular vulnerability under INa blockade.

Parallel Multigrid Preconditioner for the Cardiac Bidomain Model

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.

Computational tools for modeling electrical activity in cardiac tissue

Computer models offer many attractive benefits. However, the modeling of cardiac tissue is computationally expensive due to several physical constraints which result in fine spatiotemporal discretization over large spatiotemporal regions. Our laboratory has been actively trying to develop new techniques to make large scale cardiac simulations tractable over the past 15 years. This paper describes the latest modeling software that our group has developed, called Carp (Cardiac arrhythmias research package). It is designed to run in both shared memory and clustered computing environments. Carp aims to be modular and flexible by following a plug-in framework. This allows the latest models and most efficient solvers to be incorporated as well as enabling run-time selection of techniques. Performance results are given for a large-scale simulation which utilized a comprehensive membrane ionic current description.

From myocardial cell models to action potential propagation

Membrane equations that describe sarcolemmal currents and ion transfer processes are important building blocks for theoretical studies of action potential propagation in cardiac tissue. Introduction of such ionic models into cellular and tissue networks allows analyses of passive contributions associated with tissue structure to be considered alongside active contributions from myocytes themselves in studies involving arrhythmia initiation, maintenance and termination. Maturation of contemporary membrane equations that attempt to replicate voltage clamp experiments from different species and tissue types with specific examples of modifications to extend those equations for simulations under conditions of rapid pacing, myocardial ischemia and remodeling following myocardial infarction are considered. Additionally, the integrating of membrane equations into models where coupling to represent current flow paths associated with the anisotropic tissue structure is described.

Effects of Elevated Extracellular Potassium Ion Concentration on Anodal Excitation of Cardiac Tissue

INTRODUCTION

Anodal excitation of cardiac tissue occurs by two mechanisms: "make" and "break." Anodal strength-interval curves are divided into two sections, with break excitation occurring at short intervals and make at long intervals. Our goal is to determine how an elevated extracellular potassium ion concentration, [K]o, affects the mechanism of anodal excitation and influences the anodal strength-interval curve.

METHODS AND RESULTS

Computer simulations of unipolar stimulation were performed using the bidomain model, with membrane kinetics governed by the Luo-Rudy model. The diastolic threshold for anodal stimulation first decreased and then increased with increasing [K]o, reaching a minimum value at [K]o = 12 mM. The mechanism for diastolic anodal excitation was make for all [K]o values except 13.3 mM, in which case it was break. For low [K]o (4 and 8 mM) the break section of the anodal strength-interval contained a "dip," but for high [K]o (12 and 13 mM), the dip disappeared.

CONCLUSION

High [K]o predisposes cardiac tissue to break excitation, which is thought to play an important role in reentry induction and defibrillation. Because fibrillation raises extracellular [K]o levels, break excitation may play a more important role in defibrillation than is suggested by simulations and experiments using normal [K]o values.

Effects of elevated extracellular potassium on the stimulation mechanism of diastolic cardiac tissue

During cardiac disturbances such as ischemia and hyperkalemia, the extracellular potassium ion concentration is elevated. This in turn changes the resting transmembrane potential and affects the excitability of cardiac tissue. To test the hypothesis that extracellular potassium elevation also alters the stimulation mechanism, we used optical fluorescence imaging to examine the mechanism of diastolic anodal unipolar stimulation of cardiac tissue under 4 mM (normal) and 8 mM (elevated) extracellular potassium. We present several visualization methods that are useful for distinguishing between anodal-make and anodal-break excitation. In the 4-mM situation, stimulation occurred by the make, or stimulus-onset, mechanism that involved propagation out of the virtual cathodes. For 8-mM extracellular potassium, the break or stimulus termination mechanism occurred with propagation out of the virtual anode. We conclude that elevated potassium, as might occur in myocardial ischemia, alters not only stimulation threshold but also the excitation mechanism for anodal stimulation.

The effect of conductivity values on ST segment shift in subendocardial ischaemia

The aim of this study was to investigate the effect of different conductivity values on epicardial surface potential distributions on a slab of cardiac tissue. The study was motivated by the large variation in published bidomain conductivity parameters available in the literature. Simulations presented are based on a previously published bidomain model and solution technique which includes fiber rotation. Three sets of conductivity parameters are considered and an alternative set of nondimensional parameters relating the tissue conductivities to blood conductivity is introduced. These nondimensional parameters are then used to study the relative effect of blood conductivity on the epicardial potential distributions. Each set of conductivity parameters gives rise to a distinct set of epicardial potential distributions, both in terms of morphology and magnitude. Unfortunately, the differences between the potential distributions cannot be explained by simple combinations of the conductivity values or the resulting dimensionless parameters.

New concepts in transthoracic defibrillation

The transition of biphasic waveforms from ICDs to external defibrillators constitutes a significant technological advances for transthoracic defibrillation. Impedance compensation has enabled the delivery of defibrillating current adapted to each patient and each shock in the same patient. Optimally designed biphasic waveforms have been shown clinically to have greater efficacy in the termination of VF when compared with monophasic waveforms, and because peak current delivery is less, these waveforms are likely to be less injurious to myocardial function. Advances in the understanding of the mechanisms of fibrillation and defibrillation have identified the electrophysiologic events that initiate and sustain VF and the effects of defibrillation shocks on those events. Definition of the role of VEP and postshock excitation has clarified the mechanisms by which shocks can either fail or succeed. The ability of the second phase of optimal biphasic waveform shocks to exploit recruited sodium channels in negatively polarized areas and thus induce rapid propagation of postshock excitation assures uniform depolarization and prevention of re-entry. This appears to be the major mechanism of greater efficacy of biphasic waveforms. It seems certain that continuing investigation of virtual electrodes will enhance our understanding of defibrillation and optimal waveforms. At the same time, much more needs to be known regarding translation of these experimental observations to mechanisms of defibrillation in human hearts with long-standing underlying structural heart disease, which often arises of multiple factors. This represents a major challenge in defibrillation research.

Computational techniques for solving the bidomain equations in three dimensions

The bidomain equations are the most complete description of cardiac electrical activity. Their numerical solution is, however, computationally demanding, especially in three dimensions, because of the fine temporal and spatial sampling required. This paper methodically examines computational performance when solving the bidomain equations. Several techniques to speed up this computation are examined in this paper. The first step was to recast the equations into a parabolic part and an elliptic part. The parabolic part was solved by either the finite-element method (FEM) or the interconnected cable model model (ICCM). The elliptic equation was solved by FEM on a coarser grid than the parabolic problem and at a reduced frequency. The performance of iterative and direct linear equation system solvers was analyzed as well as the scalability and parallelizability of each method. Results indicate that the ICCM was twice as fast as the FEM for solving the parabolic problem, but when the total problem was considered, this resulted in only a 20% decrease in computation time. The elliptic problem could be solved on a coarser grid at one-quarter of the frequency at which the parabolic problem was solved and still maintain reasonable accuracy. Direct methods were faster than iterative methods by at least 50% when a good estimate of the extracellular potential was required. Parallelization over four processors was efficient only when the model comprised at least 500,000 nodes. Thus, it was possible to speed up solution of the bidomain equations by an order of magnitude with a slight decrease in accuracy.

Artifacts, assumptions, and ambiguity: Pitfalls in comparing experimental results to numerical simulations when studying electrical stimulation of the heart

Insidious experimental artifacts and invalid theoretical assumptions complicate the comparison of numerical predictions and observed data. Such difficulties are particularly troublesome when studying electrical stimulation of the heart. During unipolar stimulation of cardiac tissue, the artifacts include nonlinearity of membrane dyes, optical signals blocked by the stimulating electrode, averaging of optical signals with depth, lateral averaging of optical signals, limitations of the current source, and the use of excitation-contraction uncouplers. The assumptions involve electroporation, membrane models, electrode size, the perfusing bath, incorrect model parameters, the applicability of a continuum model, and tissue damage. Comparisons of theory and experiment during far-field stimulation are limited by many of these same factors, plus artifacts from plunge and epicardial recording electrodes and assumptions about the fiber angle at an insulating boundary. These pitfalls must be overcome in order to understand quantitatively how the heart responds to an electrical stimulus. (c) 2002 American Institute of Physics.

Shock-induced arrhythmogenesis in the myocardium

The focus of this article is the investigation of the electrical behavior of the normal myocardium following the delivery of high-strength defibrillation shocks. To achieve its goal, the study employs a complex three-dimensional defibrillation model of a slice of the canine heart characterized with realistic geometry and fiber architecture. Defibrillation shocks of various strengths and electrode configurations are delivered to the model preparation in which a sustained ventricular tachycardia is induced. Instead of analyzing the post-shock electrical events as progressions of transmembrane potential maps, the study examines the evolution of the postshock phase singularities (PSs) which represent the organizing centers of reentry. The simulation results demonstrate that the shock induces numerous PSs the majority of which vanish before the reentrant wavefronts associated with them complete half of a single rotation. Failed shocks are characterized with one or more PSs that survive the initial period of PS annihilation to establish a new postshock arrhythmia. The increase in shock strength results in an overall decrease of the number of PSs that survive over 200 ms after the end of the shock; however, the exact behavior of the PSs is strongly dependent on the shock electrode configuration. (c) 2002 American Institute of Physics.

Cardiac microstructure: implications for electrical propagation and defibrillation in the heart

Our understanding of the electrophysiological properties of the heart is incomplete. We have investigated two issues that are fundamental to advancing that understanding. First, there has been widespread debate over the mechanisms by which an externally applied shock can influence a sufficient volume of heart tissue to terminate cardiac fibrillation. Second, it has been uncertain whether cardiac tissue should be viewed as an electrically orthotropic structure, or whether its electrical properties are, in fact, isotropic in the plane orthogonal to myofiber direction. In the present study, a computer model that incorporates a detailed three-dimensional representation of cardiac muscular architecture is used to investigate these issues. We describe a bidomain model of electrical propagation solved in a discontinuous domain that accurately represents the microstructure of a transmural block of rat left ventricle. From analysis of the model results, we conclude that (1) the laminar organization of myocytes determines unique electrical properties in three microstructurally defined directions at any point in the ventricular wall of the heart, and (2) interlaminar clefts between layers of cardiomyocytes provide a substrate for bulk activation of the ventricles during defibrillation.

New currents in electrical stimulation of excitable tissues

Electric fields can stimulate excitable tissue by a number of mechanisms. A uniform long, straight peripheral axon is activated by the gradient of the electric field that is oriented parallel to the fiber axis. Cortical neurons in the brain are excited when the electric field, which is applied along the axon-dendrite axis, reaches a particular threshold value. Cardiac tissue is thought to be depolarized in a uniform electric field by the curved trajectories of its fiber tracts. The bidomain model provides a coherent conceptual framework for analyzing and understanding these apparently disparate phenomena. Concepts such as the activating function and virtual anode and cathode, as well as anode and cathode break and make stimulation, are presented to help explain these excitation events in a unified manner. This modeling approach can also be used to describe the response of excitable tissues to electric fields that arise from charge redistribution (electrical stimulation) and from time-varying magnetic fields (magnetic stimulation) in a self-consistent manner. It has also proved useful to predict the behavior of excitable tissues, to test hypotheses about possible excitation mechanisms, to design novel electrophysiological experiments, and to interpret their findings.

Modelling induction of a rotor in cardiac muscle by perpendicular electric shocks

A strong, properly timed shock applied perpendicularly to a propagating wavefront causes a rotor in the canine myocardium. Experimental data indicate that the induction of this rotor relies on the shock exciting tissue away from the electrodes. The computational study reproduced such direct excitation in a two-dimensional model of a 2.7 x 3 cm sheet of cardiac muscle. The model used experimentally measured extracellular potentials to represent 100 and 150 V shocks delivered through extracellular electrodes. The shock-induced transmembrane potential was computed according to two mechanisms, the activating function and the unit-bundle sawtooth potential. The overall process leading to initiation of a rotor was the same in model and experiment. For the 100 V shock, the directly excited region extended 2.26 cm away from the electrode; the centre of the rotor ('critical point') was 1.28 cm away, where the electric field Ecr was 4.54 Vcm(-1). Increasing the shock strength to 150 V moved the critical point 1.02 cm further and decreased Ecr by 0.39 Vcm(-1). The results are comparable with experimental data. The model suggests that the unit-bundle sawtooth is responsible for the creation of the directly excited region, and the activating function is behind the dependence of Ecr on shock strength.

Success and failure of biphasic shocks: results of bidomain simulations

The mechanisms behind the superiority of optimal biphasic defibrillation shocks over monophasic are not fully understood. This simulation study examines how the shock polarity and second-phase magnitude of biphasic shocks influence the virtual electrode polarization (VEP) pattern, and thus the outcome of the shock in a bidomain model representation of ventricular myocardium. A single spiral wave is initiated in a two-dimensional sheet of myocardium that measures 2 x 2 cm(2). The model incorporates non-uniform fiber curvature, membrane kinetics suitable for high strength shocks, and electroporation. Line electrodes deliver a spatially uniform extracellular field. The shocks are biphasic, each phase lasting 10 ms. Two different polarities of biphasic shocks are examined as the first-phase configuration is held constant and the second-phase magnitude is varied between 1 and 10 V/cm. The results show that for each polarity, varying the second-phase magnitude reverses the VEP induced by the first phase in an asymmetric fashion. Further, the size of the post-shock excitable gap is dependent upon the second-phase magnitude and is a factor in determining the success or failure of the shock. The maximum size of a post-shock excitable gap that results in defibrillation success depends on the polarity of the shock, indicating that the refractoriness of the tissue surrounding the gap also contributes to the outcome of the shock.

Entrainment by an extracellular AC stimulus in a computational model of cardiac tissue

INTRODUCTION

Cardiac tissue can be entrained when subjected to sinusoidal stimuli, often responding with action potentials sustained for the duration of the stimulus. To investigate mechanisms responsible for both entrainment and extended action potential duration, computer simulations of a two-dimensional grid of cardiac cells subjected to sinusoidal extracellular stimulation were performed.

METHODS AND RESULTS

The tissue is represented as a bidomain with unequal anisotropy ratios. Cardiac membrane dynamics are governed by a modified Beeler-Reuter model. The stimulus, delivered by a bipolar electrode, has a duration of 750 to 1,000 msec, an amplitude range of 800 to 3,200 microA/cm, and a frequency range of 10 to 60 Hz. The applied stimuli create virtual electrode polarization (VEP) throughout the sheet. The simulations demonstrate that periodic extracellular stimulation results in entrainment of the tissue. This phase-locking of the membrane potential to the stimulus is dependent on the location in the sheet and the magnitude of the stimulus. Near the electrodes, the oscillations are 1:1 or 1:2 phase-locked; at the middle of the sheet, the oscillations are 1:2 or 1:4 phase-locked and occur on the extended plateau of an action potential. The 1:2 behavior near the electrodes is due to periodic change in the voltage gradient between VEP of opposite polarity; at the middle of the sheet, it is due to spread of electrotonic current following the collision of a propagating wave with refractory tissue.

CONCLUSION

The simulations suggest that formation of VEP in cardiac tissue subjected to periodic extracellular stimulation is of paramount importance to tissue entrainment and formation of an extended oscillatory action potential plateau.

Success and failure of the defibrillation shock: insights from a simulation study

INTRODUCTION

This simulation study presents a further inquiry into the mechanisms by which a strong electric shock fails to halt life-threatening cardiac arrhythmias.

METHODS AND RESULTS

The research uses a model of the defibrillation process that represents a sheet of myocardium as a bidomain. The tissue consists of nonuniformly curved fibers in which spiral wave reentry is initiated. Monophasic defibrillation shocks are delivered via two line electrodes that occupy opposite tissue boundaries. In some simulation experiments, the polarity of the shock is reversed. Electrical activity in the sheet is compared for failed and successful shocks under controlled conditions. The maps of transmembrane potential and activation times calculated during and after the shock demonstrate that weak shocks fail to terminate the reentrant activity via two major mechanisms. As compared with strong shocks, weak shocks result in (1) smaller extension of refractoriness in the areas depolarized by the shock, and (2) slower or incomplete activation of the excitable gap created by deexcitation of the negatively polarized areas. In its turn, mechanism 2 is associated with one or more of the following events: (a) lack of some break excitations, (b) latency in the occurrence of the break excitations, and (c) slower propagation through deexcited areas. Reversal of shock polarity results in a change of the extent of the regions of deexcitation, and thus, in a change in defibrillation threshold.

CONCLUSION

The results of this study indicate the paramount importance of shock-induced deexcitation in both defibrillation and postshock arrhythmogenesis.

A numerically efficient model for simulation of defibrillation in an active bidomain sheet of myocardium

Presented here is an efficient algorithm for solving the bidomain equations describing myocardial tissue with active membrane kinetics. An analysis of the accuracy shows advantages of this numerical technique over other simple and therefore popular approaches. The modular structure of the algorithm provides the critical flexibility needed in simulation studies: fiber orientation and membrane kinetics can be easily modified. The computational tool described here is designed specifically to simulate cardiac defibrillation, i. e., to allow modeling of strong electric shocks applied to the myocardium extracellularly. Accordingly, the algorithm presented also incorporates modifications of the membrane model to handle the high transmembrane voltages created in the immediate vicinity of the defibrillation electrodes.

An S1 gradient of refractoriness is not essential for reentry induction by an S2 stimulus

This communication reports a numerical simulation of reentry induction by successive stimulation (S1-S2) that does not require an S1 gradient of refractoriness. The S1 action potential is uniform in space, so before the S2 stimulus there is no refractory gradient. Nevertheless, a unipolar S2 stimulus initiates quatrefoil reentry. The result supports the growing realization that virtual electrodes, hyperpolarization, de-excitation, and break excitation may be important during reentry induction.

Virtual electrodes and deexcitation: new insights into fibrillation induction and defibrillation

Previous models of fibrillation induction and defibrillation stressed the contribution of depolarization during the response of the heart to a shock. This article reviews recent evidence suggesting that comprehending the role of negative polarization (hyperpolarization) also is crucial for understanding the response to a shock. Negative polarization can "deexcite" cardiac cells, creating regions of excitable tissue through which wavefronts can propagate. These wavefronts can result in new reentrant circuits, inducing fibrillation or causing defibrillation to fail. In addition, deexcitation can lead to rapid propagation through newly excitable regions, resulting in the elimination of excitable gaps soon after the shock and causing defibrillation to succeed.

Role of virtual electrodes in arrhythmogenesis: pinwheel experiment revisited

INTRODUCTION

Recent experimental evidence demonstrates that a point stimulus generates a nonuniform distribution of transmembrane potential (virtual electrode pattern) consisting of large adjacent areas of depolarization and hyperpolarization. This simulation study focuses on the role of virtual electrodes in reentry induction.

METHODS AND RESULTS

We simulated the electrical behavior of a sheet of myocardium using a two-dimensional bidomain model with straight fibers. Membrane kinetics were represented by the Beeler-Reuter Drouhard-Roberge model. Simulations were conducted for equal and unequal anisotropy ratios. S1 wavefront was planar and propagated parallel or perpendicular to the fibers. S2 unipolar stimulus was cathodal or anodal. With regard to unequal anisotropy, for both cathodal and anodal stimuli, the S2 stimulus negatively polarizes some portion of membrane, deexciting it and opening an excitable pathway in a region of otherwise unexcitable tissue. Reentry is generated by break excitation of this tissue and subsequent propagation through deexcited and recovered areas of myocardium. Figure-of-eight and quatrefoil reentry are observed, with figure-of-eight most common. Figure-of-eight rotation is seen in the direction predicted by the critical point hypothesis. With regard to equal anisotropy, reentry was observed for cathodal stimuli only at strengths > -95 A/m.

CONCLUSION

The key to reentry induction is the close proximity of S2-induced excited and deexcited areas, with adjacent nonexcited areas available for propagation.

Sinusoidal stimulation of myocardial tissue: effects on single cells

INTRODUCTION

Cardiac tissue subjected to sinusoidal stimulus is characterized by action potentials (APs) that have extended plateau phases, sustained for the duration of the stimulus. Extended action potential durations (APDs) are beneficial because they disrupt wandering wavelets in the fibrillating heart. To investigate the mechanisms by which periodic stimulus affects cardiac tissue, particularly the development of sustained depolarization, computer simulations of single cardiac cells exposed to alternating current (AC) are performed.

METHODS AND RESULTS

Two modes of stimulation of the cell are examined: external field stimulation and transmembrane current injection. Several membrane models, including Luo-Rudy I and II, are used in the simulations. External AC field stimuli increase the APD of the single cell. The extended plateau of the cellular AP is characterized by periodic oscillations that are 1:2 phase locked with the applied stimulus. This specific behavior is due to the variations in stimulus magnitude and polarity along the cell border, which elicit opposite electrical responses from the cell sides. These pointwise responses are averaged in the macroscopic cellular response and result in sustained oscillatory depolarization that lasts for the duration of the stimulus. In contrast, the cell undergoing current injection does not develop an extended APD.

CONCLUSION

The simulations demonstrate that variation of membrane potential within a cell is of paramount importance to the formation of an extended AP plateau in response to AC stimulation.

Roles of electric field and fiber structure in cardiac electric stimulation

This study investigated roles of the variation of extracellular voltage gradient (VG) over space and cardiac fibers in production of transmembrane voltage changes (DeltaV(m)) during shocks. Eleven isolated rabbit hearts were arterially perfused with solution containing V(m)-sensitive fluorescent dye (di-4-ANEPPS). The epicardium received shocks from symmetrical or asymmetrical electrodes to produce nominally uniform or nonuniform VGs. Extracellular electric field and DeltaV(m) produced by shocks in the absolute refractory period were measured with electrodes and a laser scanner and were simulated with a bidomain computer model that incorporated the anterior left ventricular epicardial fiber field. Measurements and simulations showed that fibers distorted extracellular voltages and influenced the DeltaV(m). For both uniform and nonuniform shocks, DeltaV(m) depended primarily on second spatial derivatives of extracellular voltages, whereas the VGs played a smaller role. Thus, 1) fiber structure influences the extracellular electric field and the distribution of DeltaV(m); 2) the DeltaV(m) depend on second spatial derivatives of extracellular voltage.