Effects of the tissue-bath interface on the induced transmembrane potential: a modeling study in cardiac stimulation

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.

Muscle Thickness and Curvature Influence Atrial Conduction Velocities

Electroanatomical mapping is currently used to provide clinicians with information about the electrophysiological state of the heart and to guide interventions like ablation. These maps can be used to identify ectopic triggers of an arrhythmia such as atrial fibrillation (AF) or changes in the conduction velocity (CV) that have been associated with poor cell to cell coupling or fibrosis. Unfortunately, many factors are known to affect CV, including membrane excitability, pacing rate, wavefront curvature, and bath loading, making interpretation challenging. In this work, we show how endocardial conduction velocities are also affected by the geometrical factors of muscle thickness and wall curvature. Using an idealized three-dimensional strand, we show that transverse conductivities and boundary conditions can slow down or speed up signal propagation, depending on the curvature of the muscle tissue. In fact, a planar wavefront that is parallel to a straight line normal to the mid-surface does not remain normal to the mid-surface in a curved domain. We further demonstrate that the conclusions drawn from the idealized test case can be used to explain spatial changes in conduction velocities in a patient-specific reconstruction of the left atrial posterior wall. The simulations suggest that the widespread assumption of treating atrial muscle as a two-dimensional manifold for electrophysiological simulations will not accurately represent the endocardial conduction velocities in regions of the heart thicker than 0.5 mm with significant wall curvature.

Electroporation induced by internal defibrillation shock with and without recovery in intact rabbit hearts

Defibrillation shocks from implantable cardioverter defibrillators can be lifesaving but can also damage cardiac tissues via electroporation. This study characterizes the spatial distribution and extent of defibrillation shock-induced electroporation with and without a 45-min postshock period for cell membranes to recover. Langendorff-perfused rabbit hearts (n = 31) with and without a chronic left ventricular (LV) myocardial infarction (MI) were studied. Mean defibrillation threshold (DFT) was determined to be 161.4 ± 17.1 V and 1.65 ± 0.44 J in MI hearts for internally delivered 8-ms monophasic truncated exponential (MTE) shocks during sustained ventricular fibrillation (>20 s, SVF). A single 300-V MTE shock (twice determined DFT voltage) was used to terminate SVF. Shock-induced electroporation was assessed by propidium iodide (PI) uptake. Ventricular PI staining was quantified by fluorescent imaging. Histological analysis was performed using Masson's Trichrome staining. Results showed PI staining concentrated near the shock electrode in all hearts. Without recovery, PI staining was similar between normal and MI groups around the shock electrode and over the whole ventricles. However, MI hearts had greater total PI uptake in anterior (P < 0.01) and posterior (P < 0.01) LV epicardial regions. Postrecovery, PI staining was reduced substantially, but residual staining remained significant with similar spacial distributions. PI staining under SVF was similar to previously studied paced hearts. In conclusion, electroporation was spatially correlated with the active region of the shock electrode. Additional electroporation occurred in the LV epicardium of MI hearts, in the infarct border zone. Recovery of membrane integrity postelectroporation is likely a prolonged process. Short periods of SVF did not affect electroporation injury.

Evaluating intramural virtual electrodes in the myocardial wedge preparation: simulations of experimental conditions

While defibrillation is the only means for prevention of sudden cardiac death, key aspects of the process, such as the intramural virtual electrodes (VEs), remain controversial. Experimental studies had attempted to assess intramural VEs by using wedge preparations and recording activity from the cut surface; however, applicability of this approach remains unclear. These studies found, surprisingly, that for strong shocks, the entire cut surface was negatively polarized, regardless of boundary conditions. The goal of this study is to examine, by means of bidomain simulations, whether VEs on the cut surface represent a good approximation to VEs in depth of the intact wall. Furthermore, we aim to explore mechanisms that could give rise to negative polarization on the cut surface. A model of wedge preparation was used, in which fiber orientation could be changed, and where the cut surface was subjected to permeable and impermeable boundary conditions. Small-scale mechanisms for polarization were also considered. To determine whether any distortions in the recorded VEs arise from averaging during optical mapping, a model of fluorescent recording was employed. The results indicate that, when an applied field is spatially uniform and impermeable boundary conditions are enforced, regardless of the fiber orientation VEs on the cut surface faithfully represent those intramurally, provided tissue properties are not altered by dissection. Results also demonstrate that VEs are sensitive to the conductive layer thickness above the cut surface. Finally, averaging during fluorescent recordings results in large negative VEs on the cut surface, but these do not arise from small-scale heterogeneities.

Virtual electrode theory explains pacing threshold increase caused by cardiac tissue damage

The virtual electrode polarization (VEP) effect is believed to play a key role in electrical stimulation of heart muscle. However, under certain conditions, including clinically, its existence and importance remain unknown. We investigated the influence of acute tissue damage produced by continuous pacing with strong current (40-mA, 4-ms biphasic pulses with 4-Hz frequency for 5 min) on stimulus-generated VEPs and pacing thresholds. A fluorescent optical mapping technique was used to obtain stimulus-induced transmembrane potential distribution around a pacing electrode applied to the ventricular surface of a Langendorff-perfused rabbit heart ( n = 5). Maps and pacing thresholds were recorded before and after tissue damage. Spatial extents of electroporation and cell uncoupling were assessed by propidium iodide ( n = 2) and connexin43 ( n = 3) antibody staining, respectively. On the basis of these data, passive and active three-dimensional bidomain models were built to determine VEP patterns and thresholds for different-sized areas of the damaged region. Electrophysiological results showed that acute tissue damage led to disappearance of the VEP with an associated significant increase in pacing thresholds. Damage was expressed in electroporation and cell uncoupling within a ∼1.0-mm-diameter area around the tip of the electrode. According to computer simulations, cell uncoupling, rather than electroporation, might be the direct cause of VEP elimination and threshold increase, which was nonlinearly dependent on the size of the damaged region. Fiber rotation with depth did not substantially affect the numerical results. The study explains failure to stimulate damaged tissue within the concepts of the VEP theory.

Shock-Induced Epicardial and Endocardial Virtual Electrodes Leading to Ventricular Fibrillation via Reentry, Graded Responses, and Transmural Activation

INTRODUCTION

The mechanism of ventricular fibrillation (VF) induction by T wave shocks has been attributed to reentry, propagated graded responses (PGR), and triggered activity. The limitation of recording transmembrane potential (V(m)) from only a single surface has hampered efforts to elucidate the relative role of these phenomena and their relationship to shock-induced virtual electrodes.

METHODS AND RESULTS

V(m) patterns from epicardial and endocardial surfaces of isolated sheep right ventricles were recorded with two CCD cameras for monophasic (M) and biphasic (B) shocks delivered at various coupling intervals (CI) from a unipolar mesh electrode on the epicardium. VF was induced via (1) the formation of reentry following make or break excitation; (2) propagated graded responses during apparent isoelectric window; and (3) breakthrough activation patterns coincident with endocardial-to-epicardial gradients in V(m). M shocks depolarized both surfaces at long CIs and polarized epicardial and endocardial surfaces oppositely at short CIs. At intermediate CIs, postshock V(m) patterns could lead to reentry on one surface or endocardial-to-epicardial gradients resulting in breakthrough. B induced VF less than M for short and intermediate CIs due to more homogeneous end-shock V(m) patterns. However, at long CIs these homogeneous patterns resulted in more VF induction because B left the tissue closer to the V(m) threshold for propagation.

CONCLUSION

Postshock activity occurred either immediately via epicardial or endocardial reentry, or after a delay caused by transmural propagation or propagated graded responses. These findings could explain the isoelectric window and focal activation patterns observed on the epicardium following VF induction shocks.

Vortex cordis as a mechanism of postshock activation: arrhythmia induction study using a bidomain model

INTRODUCTION

The ventricular apex has a helical arrangement of myocardial fibers called the "vortex cordis." Experimental studies have demonstrated that the first postshock activation originates from the ventricular apex, regardless of the electrical shock outcome; however, the related underlying mechanism is unclear. We hypothesized that the vortex cordis contributes to the initiation of postshock activation. To clarify this issue, we numerically studied the transmembrane potential distribution produced by various electrical shocks.

METHODS AND RESULTS

Using an active membrane model, we simulated a two-dimensional bidomain myocardial tissue incorporating a typical fiber orientation of the vortex cordis. Monophasic or biphasic shock was delivered via two line electrodes located at opposite tissue borders. Transmembrane potential distribution during the monophasic shock at the center of the vortex cordis showed a gradient high enough to initiate postshock activation. The postshock activation from the center of the vortex cordis was not suppressed, regardless of the initiation of spiral wave reentry. Spiral wave reentry was induced by the monophasic shock when the center area of the vortex cordis was partially excited by the nonuniform virtual electrode polarization. Postshock activation following the biphasic shock also originated from the center of the vortex cordis, but it tended to be suppressed due to the narrower excitable gap around the center of the vortex cordis. The electroporation effect, which was maximal at the center of the vortex cordis, is another possible mechanism of postshock activation.

CONCLUSION

Our simulations suggest that the vortex cordis may cause postshock activation.

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.

The role of electroporation in defibrillation

Electric shock is the only effective therapy against ventricular fibrillation. However, shocks are also known to cause electroporation of cell membranes. We sought to determine the impact of electroporation on ventricular conduction and defibrillation. We optically mapped electrical activity in coronary-perfused rabbit hearts during electric shocks (50 to 500 V). Electroporation was evident from transient depolarization, reduction of action potential amplitude, and upstroke dV/dt. Electroporation was voltage dependent and significantly more pronounced at the endocardium versus the epicardium, with thresholds of 229+/-81 versus 318+/-84 V, respectively (P=0.01, n=10), both being above the defibrillation threshold of 181.3+/-45.8 V. Epicardial electroporation was localized to a small area near the electrode, whereas endocardial electroporation was observed at the bundles and trabeculas throughout the entire endocardium. Higher-resolution imaging revealed that papillary muscles (n=10) were most affected. Electroporation and conduction block thresholds in papillary muscles were 281+/-64 V and 380+/-79 V, respectively. We observed no arrhythmia in association with electroporation. Further, preconditioning with high-energy shocks prevented reinduction of fibrillation by 50-V shocks, which were otherwise proarrhythmic. Endocardial bundles are the most susceptible to electroporation and the resulting conduction impairment. Electroporation is not associated with proarrhythmic effects and is associated with a reduction of vulnerability.

Virtual electrode polarization in the far field: implications for external defibrillation

We recently suggested that failure of implantable defibrillation therapy may be explained by the virtual electrode-induced phase singularity mechanism. The goal of this study was to identify possible mechanisms of vulnerability and defibrillation by externally applied shocks in vitro. We used bidomain simulations of realistic rabbit heart fibrous geometry to predict the passive polarization throughout the heart induced by external shocks. We also used optical mapping to assess anterior epicardium electrical activity during shocks in Langendorff-perfused rabbit hearts (n = 7). Monophasic shocks of either polarity (10-260 V, 8 ms, 150 microF) were applied during the T wave from a pair of mesh electrodes. Postshock epicardial virtual electrode polarization was observed after all 162 applied shocks, with positive polarization facing the cathode and negative polarization facing the anode, as predicted by the bidomain simulations. During arrhythmogenesis, a new wave front was induced at the boundary between the two regions near the apex but not at the base. It spread across the negatively polarized area toward the base of the heart and reentered on the other side while simultaneously spreading into the depth of the wall. Thus a scroll wave with a ribbon-shaped filament was formed during external shock-induced arrhythmia. Fluorescent imaging and passive bidomain simulations demonstrated that virtual electrode polarization-induced scroll waves underlie mechanisms of shock-induced vulnerability and failure of external defibrillation.

Virtual electrode-induced reexcitation: A mechanism of defibrillation

Mechanisms of defibrillation remain poorly understood. Defibrillation success depends on the elimination of fibrillation without shock-induced arrhythmogenesis. We optically mapped selected epicardial regions of rabbit hearts (n=20) during shocks applied with the use of implantable defibrillator electrodes during the refractory period. Monophasic shocks resulted in virtual electrode polarization (VEP). Positive values of VEP resulted in a prolongation of the action potential duration, whereas negative polarization shortened the action potential duration, resulting in partial or complete recovery of the excitability. After a shock, new propagated wavefronts emerged at the boundary between the 2 regions and reexcited negatively polarized regions. Conduction velocity and maximum action potential upstroke rate of rise dV/dt (max) of shock-induced activation depended on the transmembrane potential at the end of the shock. Linear regression analysis showed that dV/dt(max) of postshock activation reached 50% of that of normal action potential at a V(m) value of -56.7+/-0.6 mV postshock voltage (n=9257). Less negative potentials resulted in slow conduction and blocks, whereas more negative potentials resulted in faster conduction. Although wavebreaks were produced in either condition, they degenerated into arrhythmias only when conduction was slow. Shock-induced VEP is essential in extinguishing fibrillation but can reinduce arrhythmias by producing excitable gaps. Reexcitation of these gaps through progressive increase in shock strength may provide the basis for the lower and upper limits of vulnerability. The former may correspond to the origination of slow wavefronts of reexcitation and phase singularities. The latter corresponds to fast conduction during which wavebreaks no longer produce sustained arrhythmias.

Shock-induced figure-of-eight reentry in the isolated rabbit heart

The patterns of transmembrane potential on the whole heart during and immediately after fibrillation-inducing shocks are unknown. To study arrhythmia induction, we recorded transmembrane activity from the anterior and posterior epicardial surface of the isolated rabbit heart simultaneously using 2 charge-coupled device cameras (32,512 pixels, 480 frames/second). Isolated hearts were paced from the apex at a cycle length of 250 ms. Two shock coils positioned inside the right ventricle (-) and atop the left atrium (+) delivered shocks at 3 strengths (0.75, 1.5, and 2.25 A) and 6 coupling intervals (130 to 230 ms). The patterns of depolarization and repolarization were similar, as is evident in the uniformity of action potential duration at 75% repolarization (131.4¿8.3 ms). At short coupling intervals (<180 ms), shocks hyperpolarized a large portion of the ventricles and produced a pair of counterrotating waves, one on each side of the heart. The first beat after the shock was reentrant in 90% of short coupling interval episodes. At long coupling intervals (>180 ms), increasingly stronger shocks depolarized an increasingly larger portion of the heart. The first beat after the shock was reentrant in 18% of long coupling interval episodes. Arrhythmias were most often induced at short coupling intervals (98%) than at long coupling intervals (35%). The effect and outcome of the shock were related to the refractory state of the heart at the time of the shock. Hyperpolarization occurred at short coupling intervals, whereas depolarization occurred at long coupling intervals. Consistent with the "critical point" hypothesis, increasing shock strength and coupling interval moved the location where reentry formed (away from the shock electrode and pacing electrode, respectively).

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

INTRODUCTION

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.

METHODS AND RESULTS

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.

CONCLUSION

These results indicate that electroporation of the cardiac membrane plays an important role in the distribution of Vm induced by defibrillation strength shocks.

Effect of a bath on the epicardial transmembrane potential during internal defibrillation shocks

Using a three-dimensional model of cardiac tissue, we consider a rectangular slab of tissue. We examine the effect of a defibrillating shock from an intracavitary electrode upon the epicardial transmembrane potential (Vm) for two cases: one in which the epicardium is bounded by air and another in which it is bounded by a conductive bath. We find that the inclusion of the bath changes the polarity of the steady-state Vm in the epicardial region that is closest to the shock electrode. In addition, the magnitude of Vm is increased dramatically if the bath is present; the degree of hyperpolarization increases twenty-fivefold, while the degree of depolarization increases elevenfold. The remaining bulk of the cardiac tissue is relatively unaffected by the inclusion of the bath.

Virtual electrode effects in transvenous defibrillation-modulation by structure and interface: evidence from bidomain simulations and optical mapping

INTRODUCTION

Our goal in this combined modeling and experimental study was to gain insight into the transmembrane potential changes in defibrillation conditions, namely, when shocks are delivered by an implantable cardioverter defibrillator (ICD). Two hypotheses concerning the presence and characteristics of virtual electrode effects (VEE) during an ICD shock were tested numerically and experimentally: (H1) anisotropy-dependent VEE are induced over a considerable portion of the "bulk" myocardium; and (H2) surface (epicardial and endocardial) VEE are generated under special tissue bath conditions and are not fully anisotropy determined.

METHODS AND RESULTS

Optical mapping was performed on Langendorff-perfused rabbit hearts (n = 4) stained with di-4-ANEPPS. Monophasic shocks were applied during the plateau phase of an action potential through a 9-mm long distal electrode in the right or left ventricle and a 6-cm proximal electrode positioned 3 cm posteriorly to the heart. We modeled the experiment using an ellipsoidal bidomain heart with transmural fiber rotation, placed in a perfusing bath, and subjected to defibrillation shocks delivered by an electrode configuration as described. Our numerical simulations demonstrated VEE occupying a significant portion of the myocardium in the conditions of unequal anisotropy ratios for the intra- and extracellular domains. Statistically significant differences in epicardial polarization patterns were predicted numerically and confirmed experimentally when the interface conditions varied.

CONCLUSION

The present study concludes that VEE are present in transvenous defibrillation. They are shaped by the combined effect of cardiac tissue characteristics and interface conditions. Because of their size, VEE might contribute significantly to defibrillation outcome.

The role of cardiac tissue structure in defibrillation

The purpose of this paper is to investigate the relationship between cardiac tissue structure, applied electric field, and the transmembrane potential induced in the process of defibrillation. It outlines a general understanding of the structural mechanisms that contribute to the outcome of a defibrillation shock. Electric shocks defibrillate by changing the transmembrane potential throughout the myocardium. In this process first and foremost the shock current must access the bulk of myocardial mass. The exogenous current traverses the myocardium along convoluted intracellular and extracellular pathways channeled by the tissue structure. Since individual fibers follow curved pathways in the heart, and the fiber direction rotates across the ventricular wall, the applied current perpetually engages in redistribution between the intra- and extracellular domains. This redistribution results in changes in transmembrane potential (membrane polarization): regions of membrane hyper- and depolarization of extent larger than a single cell are induced in the myocardium by the defibrillation shock. Tissue inhomogeneities also contribute to local membrane polarization in the myocardium which is superimposed over the large-scale polarization associated with the fibrous organization of the myocardium. The paper presents simulation results that illustrate various mechanisms by which cardiac tissue structure assists the changes in transmembrane potential throughout the myocardium. (c) 1998 American Institute of Physics.