Potential distribution in three-dimensional periodic myocardium--Part I: Solution with two-scale asymptotic analysis

Deriving macroscopic myocardial conductivities by homogenization of microscopic models

We derive the values for the intracellular and extracellular conductivities needed for bidomain simulations of cardiac electrophysiology using homogenization of partial differential equations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace's equation. Using directly measurable microproperties such as cellular dimensions and end-to-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cyto-architecture (Stinstra et al. in Ann. Biomed. Eng. 33:1743-1751, 2005). This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions along the myocyte membranes.

The role of mechanoelectric feedback in vulnerability to electric shock

Experimental and clinical studies have shown that ventricular dilatation is associated with increased arrhythmogenesis and elevated defibrillation threshold; however, the underlying mechanisms remain poorly understood. The goal of the present study was to test the hypothesis that (1) stretch-activated channel (SAC) recruitment and (2) geometrical deformations in organ shape and fiber architecture lead to increased arrhythmogenesis by electric shocks following acute ventricular dilatation. To elucidate the contribution of these two factors, the study employed, for the first time, a combined electro-mechanical simulation approach. Acute dilatation was simulated in a model of rabbit ventricular mechanics by raising the LV end-diastolic pressure from 0.6 (control) to 4.2 kPa (dilated). The output of the mechanics model was used in the electrophysiological model. Vulnerability to shocks was examined in the control, the dilated ventricles, and in the dilated ventricles that also incorporated currents through SAC as a function of local strain, by constructing vulnerability grids. Results showed that dilatation-induced deformation alone decreased upper limit of vulnerability (ULV) slightly and did not result in increased vulnerability. With SAC recruitment in the dilated ventricles, the number of shock-induced arrhythmia episodes increased by 37% (from 41 to 56) and the lower limit of vulnerability (LLV) decreased from 9 to 7 V/cm, while ULV did not change. The heterogeneous activation of SAC caused by the heterogeneous fiber strain in the ventricular walls was the main reason for increased vulnerability to electric shocks since it caused dispersion of electrophysiological properties in the tissue, resulting in postshock unidirectional block and establishment of reentry.

The topology of defibrillation

We describe how Art Winfree's ideas about phase singularities can be used to understand the response of cardiac tissue with a random preexisting pattern of reentrant waves (fibrillation) to a large brief current stimulus. This discussion is organized around spatial dimension, beginning with a discussion of reentry on a periodic ring, followed by reentry in a two-dimensional planar domain (spiral waves), and ending with consideration of three-dimensional reentrant patterns (scroll waves). In all cases, we show how reentrant activity is changed by the application of a shock, describing conditions under which defibrillation is successful or not. Using topological arguments we draw the general conclusion that with a generic placement of stimulating electrodes, large-scale virtual electrodes do not give an adequate explanation for the mechanism of defibrillation.

The effect of spatial scale of resistive inhomogeneity on defibrillation of cardiac tissue

Defibrillation of cardiac tissue can be viewed in the context of dynamical systems theory as the attempt to move a dynamical system from the basin of attraction of one attractor (fibrillation) to another (the uniform rest state) by applying a stimulus whose form is physically constrained. Here we give an introduction to the physical mechanism of cardiac defibrillation from this dynamical perspective and examine the role of resistive inhomogeneity on defibrillation efficacy. Using numerical simulations with rotating waves on a one-dimensional periodic ring, we study the role of the spatial scale of resistive inhomogeneity on defibrillation. For a rotating wave on a periodic ring there are three stable attractors, namely the uniform rest state, a wave traveling clockwise and a wave traveling counterclockwise. As a result, the application of a stimulus has the potential for three different outcomes, namely elimination of the wave, phase resetting of the wave, and reversal of the wave. The results presented here show that with resistive inhomogeneities of large spatial scale, all three of these transitions are possible with large amplitude shocks, so that the probability of defibrillation is bounded well below one, independent of stimulus amplitude. On the other hand, resistive inhomogeneities of small spatial scale produce a defibrillation threshold that is qualitatively consistent with that found experimentally, namely the probability of defibrillation success is an increasing function that approaches one for large enough stimulus amplitude. Extending these results to higher dimensions, we describe conditions for successful defibrillation of functional reentry with large scale spatial inhomogeneity, but find that elimination of anatomical reentry is quite difficult. With small spatial scale inhomogeneity, there are no similar restrictions.

Spatial localization of cardiac optical mapping with multiphoton excitation

Depth and radius of regions interrogated by cardiac optical mapping with a laser beam depend on photon travel inside the heart. It would be useful to limit the range of depth and radius interrogated. We modeled the effects of a condensing lens to concentrate laser light at a target depth inside the heart, and near infrared excitation to increase penetration and produce two-photon absorption. A Monte Carlo simulation that incorporated a 0.55-NA lens, and absorption and scattering of 1064- or 488-nm laser light in 3-D cardiac tissue indicated the distribution of excitation fluence inside the tissue. A subsequent simulation incorporating absorption and scattering of transmembrane voltage-sensitive fluorescence (wavelength 669 nm) indicated locations from which fluorescence photons exiting the tissue surface originated. The results indicate that mapping at depths up to 300 microm in hearts can provide significant improvement in localization over existing cardiac optical mapping. The estimated interrogation region is sufficiently small to examine cardiac events at a cellular or subcellular scale and may allow mapping at various depths in the heart.

Evidence for roles of the activating function in electric stimulation

The hypothesis that the activating function drives transmembrane voltage changes (delta Vm) has been tested in hearts. Optical delta Vm were measured during activating functions produced with nonuniform and uniform transparent electrodes. When a nonuniform electrode was used to produce [equation: see text], the signs of delta Vm and [equation: see text] matched. The extracellular voltage gradients, often assumed important, did not predict delta Vm. When a uniform electrode was used to eliminate [equation: see text], the signs of delta Vm matched the signs of [equation: see text] estimated from variations in heart width. Demonstration of the activating function as a determinant of stimulation may improve research and therapy that use electric stimulation.

Progressive depolarization: a unified hypothesis for defibrillation and fibrillation induction by shocks

Experimental studies of defibrillation have burgeoned since the introduction of the upper limit of vulnerability (ULV) hypothesis for defibrillation. Much of this progress is due to the valuable work carried out in pursuit of this hypothesis. The ULV hypothesis presented a unified electrophysiologic scheme for linking the processes of defibrillation and shock-induced fibrillation. In addition to its scientific ramifications, this work also raised the possibility of simpler and safer means for clinical defibrillation threshold testing. Recent results from an optical mapping study of defibrillation suggest, however, that the experimental data supporting the ULV hypothesis could instead be interpreted in a manner consistent with traditional views of defibrillation such as the critical mass hypothesis. This review will describe the evidence calling for such a reinterpretation. In one regard the ULV hypothesis superseded the critical mass hypothesis by linking the defibrillation and shock-induced fibrillation processes. Therefore, this review also will discuss the rationale for developing a new defibrillation hypothesis. This new hypothesis, progressive depolarization, uses traditional defibrillation concepts to cover the same ground as the ULV hypothesis in mechanistically unifying defibrillation and shock-induced fibrillation. It does so in a manner consistent with experimental data supporting the ULV hypothesis but which also takes advantage of what has been learned from optical studies of defibrillation. This review will briefly describe how this new hypothesis relates to other contemporary viewpoints and related experimental results.

Syncytial heterogeneity as a mechanism underlying cardiac far-field stimulation during defibrillation-level shocks

INTRODUCTION

The mechanisms by which a defibrillation shock directly stimulates regions of cardiac tissue distal to the stimulus electrodes ("far-field" stimulation) are still not well understood. Existing hypotheses have proposed that intercellular discontinuities and/or fiber curvatures induce the requisite membrane polarizations. This article hypothesizes a third potential mechanism: one based on the existence and influences of syncytial (anatomic) heterogeneities inherent throughout the bulk myocardium itself.

METHODS AND RESULTS

We simulated the effects of such heterogeneities in a model of a two-dimensional region of passive cardiac tissue subjected to uniform 1 V/cm longitudinal or transverse field stimuli. Heterogeneities were manifested via random spatial variations of intracellular volume fractions (fi) over multiple length scales, with mean fi of 80% and standard deviation of fi (sigma[fi]) ranging from 0% to 10%. During field stimulation, many interspersed and variously shaped and sized islands of hyperpolarization and depolarization developed across the tissue, with their locations and extents correlated to the spatial gradients of the underlying heterogeneities. Increases in sigma(fi) correspondingly increased the shock-induced magnitudes of resulting membrane polarizations. The ratio of maximal polarizations for equivalent longitudinal and transverse shocks approximated 2:1 across all sigma(fi) tested. At sigma(fi) = 5%, these maximal induced polarizations were 17.4 +/- 2.4 mV and 8.18 +/- 1.5 mV, respectively. Assuming an excitation threshold of 25 mV, these data suggest corresponding diastolic thresholds of 1.47 +/- 0.20 V/cm and 3.14 +/- 0.50 V/cm, respectively.

CONCLUSION

This study predicts that syncytial heterogeneities inherent within cardiac tissue could represent a significant-and heretofore unappreciated-mechanism underlying field-induced polarizations throughout the bulk myocardium.

Models of defibrillation of cardiac tissue

Heterogeneities, such as gap junctions, defects in periodical cellular lattices, intercellular clefts and fiber curvature allow one to understand the effect of an electric field in cardiac tissue. They induce membrane potential variations even in the bulk of the myocardium, with a characteristic sawtooth shape. The sawtooth potential, induced by heterogeneities at large scales (tissue strands) can be more easily observed, and lead to stronger effects than the one induced at the cellular level. In the generic model of propagation in cardiac tissue (FitzHugh), 4 mechanisms of defibrillation were found, two mechanisms based on excitation (E(A),E(M)), and two-on de-excitation (D(A),D(M)). The lowest electric field is required by an E(M) mechanism. In the Beeler-Reuter ionic model, mechanism D(M) is impossible. We critically review the experimental basis of the theory and propose new experiments. (c) 1998 American Institute of Physics.

On the mechanism of ventricular defibrillation

The mechanism of ventricular defibrillation can be considered at many different levels. The highest level is considered at strength of the shock given through the defibrillation electrodes. At the next level, the mechanism of defibrillation can be examined in terms of the electrical field that the shock produces throughout the ventricles. Other levels include the effects this electric field has on the activation sequences and on the cellular action potentials that either initiate or inhibit the early sites of activation following the shock. Yet another level considers the mechanism by which the shock field initiates new action potentials or prolongs the action potential by changing the transmembrane potential during the shock. Finally, the subcellular level is considered, which involves the response of the individual ion channels to the shock. This review gives a brief overview of some salient features of defibrillation at each of these mechanistic levels.

Impedance to defibrillation countershock: does an optimal impedance exist?

Defibrillation is thought to occur because of changes in the transmembrane potential that are caused by current flow through the heart tissue. Impedance to electric countershock is an important parameter because it is determined by the magnitude and distribution of the current that flows for a specific shock voltage. The impedance is comprised of resistive contributions from: (1) extra-tissue sources, which include the defibrillator, leads, and electrodes; (2) tissue sources, which include intracardiac and extra-cardiac tissue; and (3) the interface between electrode and tissue. Tissue sources dominate the impedance and probably contribute to the wide range of impedance values presented to the defibrillation pulse. Because impedance is not constant within or between subjects, defibrillators must be designed to accommodate these differences without compromising patient safety or therapeutic efficacy. Experimental investigations in animals and humans suggest that impedance changes at several different time scales ranging from milliseconds to years. These alterations are believed to be a result of both electrochemical and physiological mechanisms. It is commonly thought that impedance is optimized when it has been decreased to a minimum, since this allows the most current flow for a given voltage shock. However, if the impedance is lowered by changing the location or size of the electrodes in such a way that current flow is decreased in part of the heart even though current flow is increased elsewhere, then the total voltage, current, and energy needed for defibrillation may increase, not decrease, even though impedance is decreased. A simple boundary element computer model suggests that the most even distribution of current flow through the heart is achieved for those electrode locations in which the impedance across the heart is at or near the maximum cardiac impedance for any location of these particular electrodes. Thus, the optimum shock impedance is achieved when impedance is minimized for extra-tissue and extra-cardiac tissue sources and is at or near a maximum for intracardiac tissue sources.

An approximate solution to the periodic bidomain equations in one dimension

An approximate, computationally tractable solution is proposed for the potentials in the bidomain model with periodic intracellular junctions (the periodic bidomain model). This new approach is based on the one-dimensional rigorous spectral method described previously by Trayanova and Pilkington (IEEE Trans. Biomed. Eng., May 1993). The total solution to the one-dimensional periodic bidomain problem is decomposed in the spectral domain into solutions to (1) the single-fiber classical bidomain problem in which the intracellular conductivity value incorporates the average contribution from cytoplasm and junction and (2) the "junctional" potential problem due to the presence of junctions at discrete locations alone. Solving for the junctional term rigorously requires most of the numerical effort in the solution for the periodic bidomain potentials. Here the junctional potential is found approximately with little numerical effort. A comparison between the rigorous and the approximate solutions serves as a justification for the proposed approximate solution procedure. The procedure outlined in this paper is applicable to higher spatial dimensions where both tissue anisotropy and junctional inhomogeneities play a role in establishing the transmembrane potential distribution.

A bidomain model with periodic intracellular junctions: a one-dimensional analysis

The classical bidomain model of cardiac tissue views the intracellular and extracellular (interstitial) spaces as two coupled but separate continua. In the present study, the classical bidomain model has been extended by introducing a periodic conductivity in the intracellular space to represent the junctional discontinuity between abutting myocytes. In this model the junctional region of a myocyte is represented in a way that permits variation of junction size and conductivity profile. Employing spectral techniques, a new method was developed for solving the coupled differential equations governing the intracellular and extracellular potentials in a tissue preparation of finite dimensions. Different spectral representations are used for the aperiodic intra- and extracellular potentials (finite Fourier integral transform) and for the periodic intracellular conductivity (Fourier series). As a first application of the method, the response of a 50-cell, single interior fiber to a defibrillating current is examined under steady-state conditions. Transmembrane as well as intra- and extracellular potential distributions along the fiber were calculated.

Spreading of excitation in 3-D models of the anisotropic cardiac tissue. I. Validation of the eikonal model

In this work we investigate, by means of numerical simulations, the performance of two mathematical models describing the spread of excitation in a three dimensional block representing anisotropic cardiac tissue. The first model is characterized by a reaction-diffusion system in the transmembrane and extracellular potentials v and u. The second model is derived from the first by means of a perturbation technique. It is characterized by an eikonal equation, nonlinear and elliptic in the activation time psi(x). The level surfaces psi(x) = t represent the wave-front positions. The numerical procedures based on the two models were applied to test functions and to excitation processes elicited by local stimulations in a relatively small block. The results are in excellent agreement, and for the same problem the computation time required by the eikonal equation is a small fraction of that needed for the reaction-diffusion system. Thus we have strong evidence that the eikonal equation provides a reliable and numerically efficient model of the excitation process. Moreover, numerical simulations have been performed to validate an approximate model for the extracellular potential based on knowledge of the excitation sequence. The features of the extracellular potential distribution affected by the anisotropic conductivity of the medium were investigated.

Cardiac electrophysiological experiments in numero, Part III: Simulation of arrhythmias and pacing

This paper is the third and final part of a series of articles reviewing mathematical and computer models of the electrophysiological processes. This section reviews the arrhythmia simulation and discusses models of arrhythmogenic processes, fibrillation and defibrillation, and of heart-pacemaker interaction. The models of arrhythmogenesis are classified into three main sections: models of reentry and vortex reentry, models of myocardial electrotonic interactions, and models of macroreentrant supraventricular tachycardias. This final part of the review discusses the future potential of mathematical and computer models of different cardiac processes.

Action potential propagation in a thick strand of cardiac muscle

A theoretical model of action potential propagation in a thick strand of cardiac muscle is presented. The calculation takes into account the anisotropic and syncytial properties of the tissue, the presence of the interstitial space, the effect of the surrounding tissue bath, and the variation of the potential both along the strand length and across the strand cross section. The bidomain model is used to represent the electrical properties of the tissue, and the Ebihara-Johnson model is used to represent the properties of the active sodium channels. The calculated wave front is curved, with the action potential at the surface of the strand leading that at the center. The rate of rise of the action potential and the time constant of the action potential foot vary with depth into the tissue. The velocity of the wave front is nearly independent of strand radius for radii greater than 0.5 mm. The conduction velocity decreases as the volume fraction of the interstitial space decreases. In the limit of tightly packed cells, an action potential propagates quickly over the surface of the strand; the bulk of the tissue is then excited by a slow inward wave front initiated on the surface. This model does not predict an increase in conduction velocity when cells are tightly packed, a hypothesis that has been proposed previously to explain the fast conduction velocity in Purkinje fibers of some species.

Potential distribution in three-dimensional periodic myocardium--Part II: Application to extracellular stimulation

Modeling potential distribution in the myocardium treated as a periodic structure implies that activation from high-current stimulation with extracellular electrodes is caused by the spatially oscillating components of the transmembrane potential. This hypothesis is tested by comparing the results of the model with experimental data. The conductivity, fiber orientation, the extent of the region, the location of the pacing site, and the stimulus strength determined from experiments are components of the model used to predict the distributions of potential, potential gradient, and the transmembrane potential throughout the region. Next, assuming that a specific value of the transmembrane potential is necessary and sufficient to activate fully repolarized myocardium, the model provides an analytical relation between large-scale field parameters, such as gradient and current density, and small-scale parameters, such as transmembrane potential. This relation is used to express the stimulation threshold in terms of gradient or current density components and to explain its dependence upon fiber orientation. The concept of stimulation threshold is generalized to three dimensions, and an excitability surface is constructed, which for cardiac muscle is approximately conical in shape. The numerical values of transmembrane potential and stimulation thresholds calculated using asymptotic analysis are in agreement with the results of animal experiments, confirming the validity of this approach to study the electrophysiology of periodic cardiac muscle.

Modelling the periodicity of cardiac muscle

The intractable problem of modelling cardiac muscle of arbitrary extent while preserving cellular structure has been solved using an analytical rather than numerical approach with a method called two-scale asymptotic analysis. In this method, the myocardium was modelled as a collection of bundles arranged periodically in space and connected by junctions, and the distribution of the steady-state potential and current density was determined. The potential both along and across fibers was found to contain a distinct periodic component that determines the transmembrane potential. The magnitude of the transmembrane potential depends on the gradient of applied potential, the dimensions of the bundles, and their internal conductivity. Current flows primarily in the extracellular space, and the extracellular pathway also determines the apparent conductivity of cardiac muscle.