Bidomain Model Of Cardiac Tissue Research Articles

Six Conductivity Values to Use in the Bidomain Model of Cardiac Tissue.

The aim of this work is to produce a consistent set of six conductivity values for use in the bidomain model of cardiac tissue. Studies in 2007 by Hooks etal. and in 2009 by Caldwell etal. have found that, in the directions longitudinal:transverse:normal (l:t:n) to the cardiac fibers, ratios of bulk conductivities and conduction velocities are each approximately in the ratio 4:2:1. These results are used here as the basis for a method that can find sets of six normalized bidomain conductivity values. It is found that the ratios involving transverse and normal conductivities are quite consistent, allowing new light to be shed on conductivity in the normal direction. For example, it is found that the ratio of transverse to normal conductivity is much greater in the intracellular (i) than the extracellular (e) domain. Using parameter values from experimental studies leads to the proposal of a new nominal six conductivity dataset: gil=2.4, gel=2.4, git=0.35, get=2.0, gin=0.08, and gen=1.1 (all in mS/cm). When it is used to model partial thickness ischaemia, this dataset produces epicardial potential distributions in accord with experimental studies in an animal model. It is, therefore, suggested that the dataset is suitable for use in numerical simulations. Since the bidomain approach is the most commonly used method for modeling cardiac electrophysiological phenomena, new information about conductivity in the normal direction, as well as a consistent set of six conductivity values, is valuable for researchers who perform simulation studies.

Sensitivity and specificity of substrate mapping: an in silico framework for the evaluation of electroanatomical substrate mapping strategies.

Voltage mapping is an important tool for characterizing proarrhythmic electrophysiological substrate, yet it is subject to geometric factors that influence bipolar amplitudes and thus compromise performance. The aim of this study was to characterize the impact of catheter orientation on the ability of bipolar amplitudes to accurately discriminate between healthy and diseased tissues. We constructed a 3-dimensional, in silico, bidomain model of cardiac tissue containing transmural lesions of varying diameter. A planar excitation wave was stimulated and electrograms were sampled with a realistic catheter model at multiple positions and orientations. We carried out validation studies in animal experiments of acute ablation lesions mapped with a clinical mapping system. Bipolar electrograms sampled at higher inclination angles of the catheter with respect to the tissue demonstrated improvements in both sensitivity and specificity of lesion detection. Removing low-voltage electrograms with concurrent activation of both electrodes, suggesting false attenuation of the bipolar electrogram due to alignment with the excitation wavefront, had little effect on the accuracy of voltage mapping. Our results demonstrate possible mechanisms for the impact of catheter orientation on voltage mapping accuracy. Moreover, results from our simulations suggest that mapping accuracy may be improved by selectively controlling the inclination of the catheter to record at higher angles with respect to the tissue.

Monodomain shear wave propagation and bidomain shear wave dispersion in an elastic model of cardiac tissue

Cardiac tissue elastically deforms under an applied stress, permitting shear waves to propagate through the heart. Traditionally, this behavior has been modeled with a monodomain approach, in which the mechanical properties of the intracellular and extracellular spaces are averaged together. We consider a mechanical bidomain model of cardiac tissue in which the mechanics of the intracellular and extracellular spaces are considered individually with the two spaces coupled by a spring constant. We find two normal modes of oscillation: one in which the intracellular and extracellular spaces oscillate together (a monodomain mode) and the other in which they oscillate in opposition (a bidomain mode). These two modes have unique dispersion relationships. In the extreme approximation of equal shear moduli and mass densities of the intracellular and extracellular spaces, the dispersion in the bidomain mode depends on the spring constant, while it does not in the monodomain mode.

How ischaemic region shape affects ST potentials in models of cardiac tissue

This numerical study uses a simple bidomain model of cardiac tissue to compare the effect of three different ischaemic region geometries (rectangular, cylindrical and semi-ellipsoidal) on the extracellular epicardial potentials during the ST segment. Results are obtained using anisotropic conductivities based on experimentally derived data. The model is then altered, to include heterogeneous conductivities in the ischaemic region and larger border zone widths, in order to better reproduce realistic scenarios. Initial results for the rectangular and cylindrical ischaemic shapes show a central depression over the ischaemic region, for low ischaemic thicknesses, which separates into three depressions as the ischaemic thickness increases. For ischaemic thicknesses above 70% an elevation appears over the ischaemic region and this increases in magnitude as the ischaemia becomes transmural. Results for the semi-ellipsoidal shape, however, differ, with the central depression separating into only two depressions as the thickness increases. Changing the conductivity inside the ischaemic region significantly affects results for each geometry, with depression staying over the ischaemic region for much higher levels of ischaemia (up to 90% thickness). Increasing the intramural border zone thickness did not significantly affect the epicardial potential distributions.

Mechanical bidomain model of cardiac tissue

Intracellular and extracellular spaces are separately considered in an electrical bidomain model of tissue. We propose a mechanical bidomain model separately considering the intracellular and extracellular spaces, coupled through a linear restoring force proportional to the displacement difference of the two spaces. We consider a mechanically passive model of heart fibers (no tension) with an action potential, and an electrically passive model (no action potential) in tissue with an ischemic boundary. We find the pressure and displacement fields arising from our consideration of a bidomain instead of a monodomain and note interesting characteristics evident only with a bidomain approach.

Low Energy Defibrillation in Human Cardiac Tissue: A Simulation Study

We aim to assess the effectiveness of feedback-controlled resonant drift pacing as a method for low energy defibrillation. Antitachycardia pacing is the only low energy defibrillation approach to have gained clinical significance, but it is still suboptimal. Low energy defibrillation would avoid adverse side effects associated with high voltage shocks and allow the application of implantable cardioverter defibrillator (ICD) therapy, in cases where such therapy is not tolerated today. We present results of computer simulations of a bidomain model of cardiac tissue with human atrial ionic kinetics. Reentry was initiated and low energy shocks were applied with the same period as the reentry, using feedback to maintain resonance. We demonstrate that such stimulation can move the core of reentrant patterns, in the direction that depends on the location of the electrodes and the time delay in the feedback. Termination of reentry is achieved with shock strength one-order-of-magnitude weaker than in conventional single-shock defibrillation. We conclude that resonant drift pacing can terminate reentry at a fraction of the shock strength currently used for defibrillation and can potentially work where antitachycardia pacing fails, due to the feedback mechanisms. Success depends on a number of details that these numerical simulations have uncovered.

Convergence of a finite volume scheme for the bidomain model of cardiac tissue

We analyze a finite volume scheme for the bidomain model, which is a degenerate reaction–diffusion system modeling electrophysiological waves in cardiac tissue. We establish existence and uniqueness of solutions to the finite volume scheme, and show that it converges to a weak solution of the bidomain model. The convergence proof is based on deriving series of a priori estimates and using a general L p compactness criterion.

Forward Euler Stability of the Bidomain Model of Cardiac Tissue

The bidomain model describes the anisotropic electrical properties of cardiac tissue. One common numerical technique for solving the bidomain equations is the explicit forward Euler method. In this communication we derive a relationship between the time and space steps that ensures the stability of this numerical method.

Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue

We prove well-posedness (existence and uniqueness) results for aclass of degenerate reaction-diffusion systems. A prototype system belongingto this class is provided by the bidomain model, which is frequently used tostudy and simulate electrophysiological waves in cardiac tissue. The existenceresult, which constitutes the main thrust of this paper, is proved by means ofa nondegenerate approximation system, the Faedo-Galerkin method, and thecompactness method.

Approximate solution to the bidomain equations for electrocardiogram problems

Simulating the electrocardiogram requires specifying the transmembrane potential distribution within the heart and calculating the potential on the surface of the body. Often, such calculations are based on the bidomain model of cardiac tissue. A subtle but fundamental problem arises when considering the boundary between the cardiac tissue and the surrounding volume conductor. In general, one finds that two potentials--the extracellular potential in the tissue and the potential in the surrounding bath--obey three boundary conditions, implying that the potentials are overdetermined. In this paper, we derive a general method for handling bidomain boundary conditions that eliminates this problem. The gist of the method is that we add an additional term to the transmembrane potential that falls exponentially with depth into the tissue. The purpose of this term is to satisfy the third boundary condition. Then, we take the limit as the length constant associated with this extra term goes to zero. Our result is two boundary conditions that approximately account for the full set of three boundary conditions at the tissue surface.

Computational modeling of physiological systems

in recent years, the development of information-gathering technologies such as high-throughput technology in genomics and proteomics, and the introduction of related techniques for managing and mining large-scale data, have facilitated a dramatic increase in the degree of quantification applied to

The effect of simplifying assumptions in the bidomain model of cardiac tissue: Application to ST segment shifts during partial ischaemia

In this study various electrical conductivity approximations used in bidomain models of cardiac tissue are considered. Comparisons are based on epicardial surface potential distributions arising from regions of subendocardial ischaemia situated within the cardiac tissue. Approximations studied are a single conductivity bidomain model, an isotropic bidomain model and equal and reciprocal anisotropy ratios both with and without fibre rotation. It is demonstrated both analytically and numerically that the approximations involving a single conductivity bidomain, an isotropic bidomain or equal anisotropy ratios (ignoring fibre rotation) results in identical epicardial potential distributions for all degrees of subendocardial ischaemia. This result is contrary to experimental observations. It is further shown that by assuming reciprocal anisotropy ratios, epicardial potential distributions vary with the degree of subendocardial ischaemia. However, it is concluded that unequal anisotropy ratios must be used to obtain the true character of experimental observations.

Art Winfree and the bidomain model of cardiac tissue

This paper reviews Art Winfree's contributions to the bidomain model of cardiac tissue. Specifically, he first predicted quatrefoil reentry, he showed that an S1 refractory gradient is not required for an S2 stimulus to induce reentry, and his work on spiral wave meandering led to studies on how the path of the tip of a spiral wave is influenced by tissue anisotropy.

Numerical Simulations of Fractionated Electrograms and Pathological Cardiac Action Potential

The aim of this work is twofold. First we focus on the complex phenomenon of electrogram fractionation, due to the presence of discontinuities in the conduction properties of the cardiac tissue in a bidomain model. Numerical simulations of paced activation may help to understand the role of the membrane ionic currents and of the changes in cellular coupling in the formation of conduction blocks and fractionation of the electrogram waveform. In particular, we show that fractionation is independent ofINAalterations and that it can be described by the bidomain model of cardiac tissue. Moreover, some deflections in fractionated electrograms may give nonlocal information about the shape of damaged areas, also revealing the presence of inhomogeneities in the intracellular conductivity of the medium at a distance.The second point of interest is the analysis of the effects of space–time discretization on numerical results, especially during slow conduction in damaged cardiac tissue. Indeed, large discretization steps can induce numerical artifacts such as slowing down of conduction velocity, alteration in extracellular and transmembrane potential waveforms or conduction blocks, which are not predicted by the continuous bidomain model. Several possible numerical and physiological explanations of these effects are given. Essentially, the discrete system obtained at the end of the approximation process may be interpreted as a discrete model of the cardiac tissue made up of isopotential cells where the effective intracellular conductivity tensor depends on the space discretization steps; the increase of these steps results in an increase of the effective intracellular resistance and can induce conduction blocks if a certain critical value is exceeded.

The Pinwheel Experiment Re-revisited

Virtual electrode induced phase singularity hypothesis explains the origin of cardiac arrhythmias caused by artificial electrical induction of rotors, i.e. vortex-like self-sustained sources of activity. This mechanism is thought to underlie both stimulus-induced arrhythmias and shock defibrillation therapy. In this paper, we extend this hypothesis to three dimensions using the bidomain model of cardiac tissue. We predict that virtual electrode polarization can produce three topologically distinct types of rotors anchored to: (1) transmural I-shaped scroll wave filaments; (2) near-surface U-shaped scroll wave filaments; and (3) intramural O-shaped scroll wave filaments.

Time dependence of anodal and cathodal refractory periods in cardiac tissue.

Mehra et al. (PACE 1980; 3:526) observed that immediately after implantation of a pacing electrode in a dog heart, the anodal refractory period (RP) is shorter than the cathodal RP, but after several weeks the anodal RP becomes longer than the cathodal RP. We examine this experiment using numerical simulations based on the bidomain model of cardiac tissue and a Beeler-Reuter membrane. Our hypothesis is that accumulation of inexcitable tissue around the electrode following implantation causes the effective size of the electrode to increase and that this increase is the mechanism underlying the change in RP. We calculate that the anodal RP is shorter than the cathodal RP for both large and small electrodes. However, for large electrodes the threshold for anode "break" stimulation is greater than 8 mA. Mehra et al. defined RP experimentally as the interval at which the threshold stimulus strength becomes greater than 8 mA. If we restrict the stimulus current in our calculations to less than 8 mA, we exclude anode break stimulation from our calculation of the RP. In that case, our results are consistent with Mehra et al. and suggest that their observation resulted from their definition of RP.

Effects of electroporation on the transmembrane potential distribution in a two-dimensional bidomain model of cardiac tissue.

Defibrillation shocks, when delivered through internal electrodes, establish transmembrane potentials (Vm) large enough to electroporate the membrane of cardiac cells. The effects of such shocks on the transmembrane potential distribution are investigated in a two-dimensional rectangular sheet of cardiac muscle modeled as a bidomain with unequal anisotropy ratios. The membrane is represented by a capacitance Cm, a leakage conductance g(l) and a variable electroporation conductance G, whose rate of growth depends exponentially on the square of Vm. The stimulating current Io, 0.05-20 A/m, is delivered through a pair of electrodes placed 2 cm apart for stimulation along fibers and 1 cm apart for stimulation across fibers. Computer simulations reveal three categories of response to Io: (1) Weak Io, below 0.2 A/m, cause essentially no electroporation, and Vm increases proportionally to Io. (2) Strong Io, between 0.2 and 2.5 A/m, electroporate tissue under the physical electrode. Vm is no longer proportional to Io; in the electroporated region, the growth of Vm is halted and in the region of reversed polarity (virtual electrode), the growth of Vm is accelerated. (3) Very strong Io, above 2.5 A/m, electroporate tissue under the physical and the virtual electrodes. The growth of Vm in all electroporated regions is halted, and a further increase of Io increases both the extent of the electroporated regions and the electroporation conductance G. These results indicate that electroporation of the cardiac membrane plays an important role in the distribution of Vm induced by defibrillation strength shocks.

Bipolar stimulation of a three-dimensional bidomain incorporating rotational anisotropy.

A bidomain model of cardiac tissue was used to examine the effect of transmural fiber rotation during bipolar stimulation in three-dimensional (3-D) myocardium. A 3-D tissue block with unequal anisotropy and two types of fiber rotation (none and moderate) was stimulated along and across fibers via bipolar electrodes on the epicardial surface, and the resulting steady-state interstitial (phi e) and transmembrane (Vm) potentials were computed. Results demonstrate that the presence of rotated fibers does not change the amount of tissue polarized by the point surface stimuli, but does cause changes in the orientation of phi e and Vm in the depth of the tissue, away from the epicardium. Further analysis revealed a relationship between the Laplacian of phi e, regions of virtual electrodes, and fiber orientation that was dependent upon adequacy of spatial sampling and the interstitial anisotropy. These findings help to understand the role of fiber architecture during extracellular stimulation of cardiac muscle.

Nonsustained reentry following successive stimulation of cardiac tissue through a unipolar electrode.

Using numerical simulations, we predict that nonsustained reentry occurs following a strong, premature stimulus through a unipolar electrode. Our simulations were based on the bidomain model of cardiac tissue, and the active membrane properties were represented by the Beeler-Reuter model. An outwardly propagating wavefront was excited by an initial stimulus (S1). A second stimulus (S2) was then applied through the same electrode. Nonsustained reentry or reentrant-like behavior followed the S2 stimulus for both cathodal and anodal stimulation, and were associated with "break" stimulation but not with "make" stimulation. The direction of spiral-wave rotation was reversed when the polarity of the stimulus was reversed. These complex dynamics occur only for a narrow window of S1-S2 intervals. During anodal S2 stimulation, two different modes of reentry exist. Our simulations also explain the "no response" phenomenon. Our mathematical model predicts that both anodal and cathodal unipolar S2 stimulation results in reentry. This behavior arises from an interaction of virtual anodes and cathodes surrounding the stimulating electrode.

Electrical conductivity values used with the bidomain model of cardiac tissue.

Electrical conductivities in the bidomain model of cardiac tissue are expressed as functions of four parameters. These expressions allow simulations to be performed using nominal, equal, and reciprocal anisotropy without introducing undesired effects, such as length constant variations. Relative values of the bidomain conductivities are estimated to be: sigma iL = 1, sigma iT = 0.1, sigma eL = 1, and sigma eT = 0.4.