The role of spatial interactions in creating the dispersion of transmembrane potential by premature electric shocks

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.

Diastolic field stimulation: the role of shock duration in epicardial activation and propagation

Detailed knowledge of tissue response to both systolic and diastolic shock is critical for understanding defibrillation. Diastolic field stimulation has been much less studied than systolic stimulation, particularly regarding transient virtual anodes. Here we investigated high-voltage-induced polarization and activation patterns in response to strong diastolic shocks of various durations and of both polarities, and tested the hypothesis that the activation versus shock duration curve contains a local minimum for moderate shock durations, and it grows for short and long durations. We found that 0.1-0.2-ms shocks produced slow and heterogeneous activation. During 0.8-1 ms shocks, the activation was very fast and homogeneous. Further shock extension to 8 ms delayed activation from 1.55 ± 0.27 ms and 1.63 ± 0.21 ms at 0.8 ms shock to 2.32 ± 0.41 ms and 2.37 ± 0.3 ms (N = 7) for normal and opposite polarities, respectively. The traces from hyperpolarized regions during 3-8 ms shocks exhibited four different phases: beginning negative polarization, fast depolarization, slow depolarization, and after-shock increase in upstroke velocity. Thus, the shocks of >3 ms in duration created strong hyperpolarization associated with significant delay (P < 0.05) in activation compared with moderate shocks of 0.8 and 1 ms. This effect appears as a dip in the activation-versus-shock-duration curve.

A mechanism for the upper limit of vulnerability

BACKGROUND

The strongest shock that induces reentry in the heart is the upper limit of vulnerability (ULV). In order to understand defibrillation, one must know what causes the ULV.

OBJECTIVE

The goal of this study was to examine the mechanism of the upper limit of vulnerability.

METHODS

Numerical simulations of cardiac tissue were performed using the bidomain model. An S2 shock was applied during the refractory period of the S1 action potential, and results using a smooth curving fiber geometry were compared with results using a smooth plus random fiber geometry.

RESULTS

When using a smooth fiber geometry only, no ULV was observed. However, when a random fiber geometry was included, the ULV was present. The difference arises from the fate of the shock-induced break wave front when it reaches the edge of the tissue hyperpolarized by the shock (the virtual anode).

CONCLUSION

Our numerical simulations suggest that local heterogeneities throughout the tissue may be crucial for determining the fate of the shock-induced wave front at the edge of the virtual anode, and therefore play an important role in the mechanism underlying the ULV.

Construction of a computer model to investigate sawtooth effects in the Purkinje system

The sawtooth effect refers to how one end of a cardiac cell is depolarized, while the opposite end is hyperpolarized, upon exposure to an exogenous electric field. Although hypothesized, it has not been observed in tissue. The Purkinje system is a one-dimensional (1-D) cable-like system residing on the endocardial surface of the heart and is the most obvious candidate for the manifestation of this phenomenon. This paper describes a computer modeling study of the effect of electric fields on the Purkinje system. Starting with a three-dimensional geometrically realistic, finite element, ventricular description, a Purkinje system is constructed which adheres to general physiological principles. Electrical activity in the Purkinje is described by use of 1-D cubic Hermite finite elements. Such a formulation allows for accurate modeling of loading effects at the Purkinje-myocyte junctions, and for preserving the discrete nature of the system. The response of a strand of Purkinje cells to defibrillation strength shocks is computed under several conditions. Also, the response of the isolated Purkinje network is illustrated. Results indicate that the geometry of the Purkinje system is the greatest determinant for far field excitation of the system. Given parameters within the plausible physiological range, the 1-D nature of the Purkinje system may lead to sawtooth potentials which are large enough to affect excitation. Thus, the Purkinje system is capable of affecting the defibrillation process, and warrants further experimentation to elucidate its role.

Simulations of optical mapping during electroporation

Experiments using optical mapping suggest that electroporation occurs in cardiac tissue when the transmembrane potential, Vm, is observed to be significantly less than +/- 400 mV. Our hypothesis, which we test by numerical simulation, is that Vm is greater than +/- 400 mV at the tissue surface, but optical mapping underestimates Vm because it averages over depth. Results indicate a significant underestimation of Vm. Experimental studies indicate a depolarization of the resting transmembrane potential, Vrest, after a strong shock. In a homogeneous model, electroporation only occurs near the tissue surface. Just as Vm during the stimulus is underestimated due to averaging, we hypothesize that the depolarization of Vrest is also underestimated.

Field stimulation of cardiac fibers with random spatial structure

Polarization of individual cells ("sawtooth") has been proposed as a mechanism for field stimulation and defibrillation. To date, the modeling work has concentrated on the myocardium with periodic spatial structure; this paper investigates potentials arising in cardiac fibers with random spatial structure. Ten different random fibers consisting of cells with varying length (l(c) = 100 +/- 50 microm), diameter (d(c) = 20 +/- 10 microm), thickness of extracellular space (t(e) = 1.18 +/- 0.59 microm), and junctional resistance (R(j) = 2 +/- 1 M(omega)) are studied. Simulations demonstrate that randomizing spatial structure introduces to the field-induced potential (V(m)) a randomly varying baseline, which arises due to polarization of groups of cells. This polarization appears primarily in response to randomizing t(e); R(j), l(c), and d(c) have less influence. The maximum V(m) increases from 3.5 mV in a periodic fiber to 20.5+/-4.7 mV in random fibers (1 V/cm field applied for 5 ms). Field stimulation threshold E(th) decreases from 6.9 to 1.59 +/- 0.43 V/cm, which is within the range of experimental measurements. Thresholds for normal and reversed field polarities are statistically equivalent: 1.59 +/- 0.43 versus 1.44 +/- 0.41 V/cm (p value = 0.453). Thus, V(m) arising due to random structure of the myocardium may play an important role in field stimulation and defibrillation.

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.

The effect of plunge electrodes during electrical stimulation of cardiac tissue

The mechanism for far-field stimulation of cardiac tissue is not known, although many hypotheses have been suggested. This paper explores a new hypothesis: the insulated plunge electrodes used in experiments to map the extracellular potential may affect the transmembrane potential when an electric field is applied to cardiac tissue. Our calculation simulates a 10-mm-diameter sheet of passive tissue with a circular insulated plunge electrode in the middle of it, ranging in diameter from 0.05 to 2 mm. We calculate the transmembrane potential induced by a 500-V/m electric field. Our results show that a transmembrane potential is induced around the electrode in alternating areas of depolarization and hyperpolarization. If the electric field is oriented parallel to the myocardial fibers, the maximum transmembrane potential is 89 mV. A layer of fluid around the electrode increases the transmembrane potential. We conclude that plunge electrodes may introduce artifacts during experiments designed to study the response of the heart to strong electric shocks.

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.

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).

Nonuniform responses of transmembrane potential during electric field stimulation of single cardiac cells

The response of cellular transmembrane potentials (V(m)) to applied electric fields is a critical factor during electrical pacing, cardioversion, and defibrillation, yet the coupling relationship of the cellular response to field intensity and polarity is not well documented. Isolated guinea pig ventricular myocytes were stained with a voltage-sensitive fluorescent dye, di-8-ANEPPS (10 microM). A green helium-neon laser was used to excite the fluorescent dye with a 15-micrometers-diameter focused spot, and subcellular V(m) were recorded optically during field stimulation directed along the long axis of the cell. The membrane response was measured at the cell end with the use of a 30-ms S1-S2 coupling interval and a 10-ms S2 pulse with strength of up to approximately 500-mV half-cell length potential (field strength x one-half the cell length). The general trends show that 1) the response of V(m) at the cell end occurs in two stages, the first being very rapid (<1 ms) and the second much slower in time scale, 2) the rapid response consists of hyperpolarization when the cell end faces the anode and depolarization when the cell end faces the cathode, 3) the rapid response varies nonlinearly with field strengths and polarity, being relatively larger for the hyperpolarizing responses, and 4) the slower, time-dependent response has a time course that varies in slope with field strength. Furthermore, the linearity of the dye response was confirmed over a voltage range of -280 to +140 mV by simultaneous measurements of optically and electrically recorded V(m). These experimental findings could not be reproduced by the updated, Luo-Rudy dynamic model but could be explained with the addition of two currents that activate outside the physiological range of voltages: a hypothetical outward current that activates strongly at positive potentials and a second current that represents electroporation of the cell membrane.

Separation between virtual sources modifies the response of cardiac tissue to field stimulation

INTRODUCTION

While it is now understood that the tissue geometry and the electric field distribution are important in generating virtual electrodes, the effects of interaction between a collection of electrodes have not been examined. To develop a basis for understanding such interactions, we have studied a single pair of oppositely polarized virtual sources. Although such oppositely polarized pairs of virtual electrodes can be generated by a variety of field distributions and tissue geometries, we examine one simple system that incorporates the salient features of source interaction.

METHODS AND RESULTS

Our model system is a homogeneous tissue strip stimulated by a uniform extracellular field. To clarify virtual source interaction, we show that field stimulated tissue can be equivalently polarized by a set of intracellular current sources with magnitude and distribution defined by the generalized activating function. In our model system, an intracellular current source is produced at one edge of the tissue and an intracellular current sink at the other. Therefore, the tissue length acts to modulate the overlap, or interaction, between the polarizations arising from each source. To quantify the effects of source interaction, the chronaxie and rheobase values of the strength-duration relation were determined for source separations varying between 1.0 cm and 100 microm (active membrane dynamics were modeled with the Luo-Rudy phase I formulation). At all separations >3.0 mm, the chronaxie was constant at 3.09 msec and the rheobase was 0.38 V/cm. Under 0.2 mm, the chronaxie decreased to 0.55 msec while the rheobase increased linearly with the inverse of source separation. The dependence of these parameters on separation primarily reflects passive electrotonic interactions between the two virtual electrodes. However, the exact values are strongly dependent upon active tissue properties-largely the inward rectifier potassium channel and activation of the sodium current.

CONCLUSION

Tissue excitation in response to field stimulation is strongly modulated by the proximity of, and therefore the interaction between, oppositely polarized virtual electrode sources.

Spatiotemporal effects of syncytial heterogeneities on cardiac far-field excitations during monophasic and biphasic shocks

INTRODUCTION

It has recently been postulated that syncytial (anatomic) heterogeneities inherent within cardiac tissue might represent a significant mechanism underlying field-induced polarization of the bulk myocardium. This simulation study examines and characterizes the spatiotemporal excitatory dynamics associated with this newly hypothesized mechanism.

METHODS AND RESULTS

Two-dimensional regions of syncytially heterogeneous cardiac tissue were simulated with active membrane kinetics. Heterogeneities were manifested via random spatial variations of intracellular volume fractions over multiple length scales. Excitation thresholds were determined for uniform rectangular monophasic (M) and symmetric biphasic (B) far-field stimuli, from which strength-duration and strength-interval relationships were constructed. For regions measuring 5.4 x 5.4 mm, baseline diastolic thresholds for longitudinal (L) and transverse (T) shocks of 5-msec total duration averaged (in V/cm, n = 10) M-L = 2.87+/-0.26, M-T = 6.71+/-0.83, B-L = 3.22+/-0.25, and B-T = 7.93+/-0.51. These thresholds decreased by 15% to 25% when the region sizes were increased to 10.8 x 10.8 mm. Strength-duration relationships correlated strongly with the Weiss-Lapicque hyperbolic relationship, with rheobases and chronaxies of 2.33 V/cm and 1.15 msec for M-L stimuli, and 2.28 V/cm and 2.04 msec for B-L stimuli. Strength-interval relationships for M-L and B-L stimuli decreased monotonically with increasing coupling intervals, with similar minimum coupling intervals at absolute refractoriness. However, the B-L thresholds were substantially less sensitive to changes in coupling intervals than their M-L counterparts.

CONCLUSION

This study provides strong additional support for and understanding of the syncytial heterogeneity hypothesis and its manifested properties. Furthermore, these results predict that syncytial heterogeneities of even modest proportions could represent a significant mechanism contributing to the far-field excitation process.

Postshock potential gradients and dispersion of repolarization in cells stimulated with monophasic and biphasic waveforms

INTRODUCTION

Even though the clinical advantage of biphasic defibrillation waveforms is well documented, the mechanisms that underlie this greater efficacy remain incompletely understood. It is established, though, that the response of relatively refractory cells to the shock is important in determining defibrillation success or failure. We used two computer models of an isolated ventricular cell to test the hypothesis that biphasic stimuli cause a more uniform response than the equivalent monophasic shocks, decreasing the likelihood that fibrillation will be reinduced.

METHODS AND RESULTS

Models of reciprocally polarized and uniformly polarized cells were used. Rapid pacing and elevated [K]o were simulated, and either 10-msec rectangular monophasic or 5-msec/5-msec symmetric biphasic stimuli were delivered in the relative refractory period. The effects of stimulus intensity and coupling interval on response duration and postshock transmembrane potential (Vm) were quantified for each waveform. With reciprocal polarization, biphasic stimuli caused a more uniform response than monophasic stimuli, resulting in fewer large gradients of Vm (only for shock strengths < or = 1.25x threshold vs < or = 2.125x threshold) and a smaller dispersion of repolarization (1611 msec2 vs 1835 msec2). The reverse was observed with uniform polarization: monophasic pulses caused a more uniform response than did biphasic stimuli.

CONCLUSION

These results show that the response of relatively refractory cardiac cells to biphasic stimuli is less dependent on the coupling interval and stimulus strength than the response to monophasic stimuli under conditions of reciprocal polarization. Because this may lead to fewer and smaller spatial gradients in Vm, these data support the hypothesis that biphasic defibrillation waveforms will be less likely to reinduce fibrillation. Further, published experimental results correlate to a greater degree with conditions of reciprocal polarization than of uniform polarization, providing indirect evidence that interactions between depolarized and hyperpolarized regions play a role in determining the effects of defibrillation shocks on cardiac tissue.

The induction of reentry in cardiac tissue. The missing link: How electric fields alter transmembrane potential

This review examines the initiation of reentry in cardiac muscle by strong electric shocks. Specifically, it concentrates on the mechanisms by which electric shocks change the transmembrane potential of the cardiac membrane and create the physiological substrate required by the critical point theory for the initiation of rotors. The mechanisms examined include (1) direct polarization of the tissue by the stimulating current, as described by the one-dimensional cable model and its two- and three-dimensional extensions, (2) the presence of virtual anodes and cathodes, as described by the bidomain model with unequal anisotropy ratios of the intra- and extracellular spaces, (3) polarization of the tissue due to changing orientation of cardiac fibers, and (4) polarization of individual cells or groups of cells by the electric field ("sawtooth potential"). The importance of these mechanisms in the initiation of reentry is examined in two case studies: the induction of rotors using successive stimulation with a unipolar electrode, and the induction of rotors using cross-field stimulation. These cases reveal that the mechanism by which a unipolar stimulation induces arrhythmias can be explained in the framework of the bidomain model with unequal anisotropy ratios. In contrast, none of the examined mechanisms provide an adequate explanation for the induction of rotors by cross-field stimulation. Hence, this study emphasizes the need for further experimental and theoretical work directed toward explaining the mechanism of field stimulation. (c) 1998 American Institute of Physics.