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

The use of two-scale asymptotic analysis allows development of a model of the steady-state potential distribution in three-dimensional cardiac muscle preserving the underlying cellular network. The myocardium is modeled as a periodic structure consisting of cylindrical cells embedded in extracellular fluid and connected by longitudinal and side junctions. The method is applicable to cardiac muscle of arbitrary extent since the periodicity of the tissue is dealt with analytically, and thus numerical computations require no more resources than a continuous volume conductor problem. The asymptotic analysis approach reveals that the potential in a periodic myocardium consists of two components. The large-scale component provides the baseline for the total solution and can be determined from the anisotropic monodomain model associated with the original periodic problem. The method provides the formula for calculating the conductivity of the equivalent monodomain model on the basis of cell geometry and conductivity distribution in the cardiac tissue. The small-scale component reflects the periodicity of the underlying structure and oscillates with periods determined by the dimensions of cardiac cells. The magnitude of these oscillations depends upon the gradient of the large-scale component. During stimulation with extracellular electrodes, the small-scale component determines both the shape and the magnitude of the transmembrane potential, while the influence of the large-scale component is negligible. Hence, the small-scale component merits closer attention in pacing and defibrillation studies, especially since the model based on two-scale asymptotic analysis provides an effective means of its computation.     

Potential distribution in three-dimensional periodic myocardium.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.     

Discrete versus syncytial tissue behavior in a model of cardiacstimulation. II. Results of simulation

For pt. I see ibid., vol. 43, no. 12, p. 1129-40 (1996). The research presented here combines mathematical modeling and computer simulation in developing a new model of the membrane polarization induced in the myocardium by the applied electric field. Employing this new model termed the “periodic” bidomain model, the steady-state distribution of the transmembrane potential is calculated on a slice of cardiac tissue composed of abutting myocytes and subjected to two point-source extracellular current stimuli. The goal of this study is to examine the relative contribution of cellular discreteness and macroscopic syncytial tissue behavior in the mechanism by which the applied electric field alters the transmembrane potential in cardiac muscle. The results showed the existence of oscillatory changes in the transmembrane potential at cell ends owing to the local resistive inhomogeneities (gap-junctions). This low-magnitude sawtooth component in the transmembrane potential is superimposed over large-scale transmembrane potential excursions associated with the syncytial (collective) fiber behavior. The character of the cardiac response to stimulation is determined primarily by the large-scale syncytial tissue behavior. The sawtooth contributes to the overall tissue response only in regions where the large-scale transmembrane potential component is small     

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.     

The magnetic field associated with a plane wave front propagating through cardiac tissue

An action potential propagating through a two-dimensional sheet of cardiac tissue produces a magnetic field. In the direction of propagation, the intracellular and extracellular current densities are equal and opposite, so the net current is zero. However, because of the unequal anisotropy ratios in the intracellular and extracellular spaces, the component of the current density perpendicular to the direction of propagation does not, in general, vanish. This line of current produces the magnetic field. The amplitude of the magnetic field is zero only if the action potential propagates parallel to or perpendicular to the fiber direction, or if the tissue has equal anisotropy ratios.     

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.     

Patterns of and mechanisms for shock-induced polarization in the heart: a bidomain analysis

This paper examines the combined action of cardiac fiber curvature and transmural fiber rotation in polarizing the myocardium under the conditions of a strong electrical shock. The study utilizes a three-dimensional finite element model and the continuous bidomain representation of cardiac tissue to model steady-state polarization resulting from a defibrillation-strength uniform applied field. Fiber architecture is incorporated in the model via the shape of the heart, an ellipsoid of variable ellipticity index, and via an analytical function, linear or nonlinear, describing the transmural fiber rotation. Analytical estimates and numerical results are provided for the location and shape of the "bulk" polarization (polarization away from the tissue boundaries) as a function of the fiber field, or more specifically, of the conductivity changes in axial and radial direction with respect to the applied electrical field lines. Polarization in the tissue "bulk" is shown to exist only under the condition of unequal anisotropy ratios in the extra- and intracellular spaces. Variations in heart geometry and, thus, fiber curvature, are found to lead to change in location of the zones of significant membrane polarization. The transmural fiber rotation function modulates the transmembrane potential profile in the radial direction. A higher gradient of the transmural transmembrane potential is observed in the presence of fiber rotation as compared to the no rotation case. The analysis presented here is a step forward in understanding the interaction between tissue structure and applied electric field in establishing the pattern of membrane polarization during the initial phase of the defibrillation shock.     

Membrane polarization induced in the myocardium by defibrillationfields: an idealized 3-D finite element bidomain/monodomain torso model

This study develops a three-dimensional finite element torso model with bidomain myocardium to simulate the transmembrane potential (TMP) of the heart induced by defibrillation fields. The inhomogeneities of the torso are modeled as eccentric spherical volumes with both the curvature and the rotation features of cardiac fibers incorporated in the myocardial region. The numerical computation of the finite element bidomain myocardial model is validated by a semianalytic solution. The simulations show that rotation of fiber orientation through the depth of the myocardial wall changes the pattern of polarization and decreases the amount of cardiac tissue polarized compared to the idealized analytic model with no fiber rotation incorporated. The TMP induced by transthoracic and transvenous defibrillation fields are calculated and visualized. The TMP is quantified by a continuous measure of the percentage of myocardial mass above a potential gradient threshold. Using this measure, the root mean square differences in TMP distribution produced by reversing the electrode polarity for anterior-posterior and transvenous electrode configurations are 13.6 and 28.6%, respectively. These results support the claim that a bidomain model of the heart predicts a change of defibrillation threshold with reversed electrode polarity     

The transient subthreshold response of spherical and cylindricalcell models to extracellular stimulation

The effect of extracellular stimulation on excitable tissue is evaluated using analytical models. Primary emphasis is placed on the determination of the rate of rise of the membrane potential in response to subthreshold stimulation. Three models are studied: 1) a spherical cell in a uniform electric field, 2) an infinite cylindrical fiber with a point source stimulus, and 3) a finite length cable with sealed ends and a stimulus electrode at each end. Results show that the rate of rise of the transmembrane potential was more rapid than the step response of a space-clamped membrane for all geometries considered. The response of the cylindrical fiber to extracellular stimulation is compared to previously reported studies of the cylindrical fiber response to intracellular stimulation. It is found that the location of the stimulus has little effect on the infinite fiber response. For terminated cables, however, an accurate model of stimulus response must discriminate between intracellular and extracellular stimulation.     

How electrode size affects the electric potential distribution in cardiac tissue

We investigate the effect of electrode size on the transmembrane potential distribution in the heart during electrical stimulation. The bidomain model is used to calculate the transmembrane potential in a three-dimensional slab of cardiac tissue. Depolarization is strongest under the electrode edge. Regions of depolarization are adjacent to regions of hyperpolarization. The average ratio of peak depolarization to peak hyperpolarization is a function of electrode radius, but over a broad range is close to three.     

Bipolar stimulation of a three-dimensional bidomain incorporatingrotational 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 (Φ _ϵ) and transmembrane (V_m) 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 Φ_ϵ, and V_m in the depth of the tissue, away from the epicardium. Further analysis revealed a relationship between the Laplacian of Φ _ϵ, 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     

A bidomain model with periodic intracellular junctions: aone-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 junction discontinuity between abutting myocytes. In this model the junction region of a myocyte is represented in a way that permits variation of junction size and conductivity profile. Employing spectral techniques, a method is 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 have been calculated     

The response of a spherical heart to a uniform electric field: abidomain analysis of cardiac stimulation

A mathematical model describing electrical stimulation of the heart is developed, in which a uniform electric field is applied to a spherical shell of cardiac tissue. The electrical properties of the tissue are characterized using the bidomain model. Analytical expressions for the induced transmembrane potential are derived for the cases of equal anisotropy ratios in the intracellular and interstitial (extracellular) spaces, and no transverse coupling between fibers. Numerical calculations of the transmembrane potential are also performed using realistic electrical conductivities. The model illustrates several mechanisms for polarization of the cell membrane, which can be divided into two categories, depending on if they polarize fibers at the heart surface only or if they polarize fibers both at the surface and within the bulk of the tissue. The latter mechanisms can be classified further according to whether they originate from continuous or discrete properties of cardiac tissue     

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

The goal of this simulation study is to examine, in a sheet of myocardium, the contribution of anode and cathode break phenomena in terminating a spiral wave reentry by the defibrillation shock. The tissue is represented as a homogeneous bidomain with unequal anisotropy ratios. Two case studies are presented in this article: tissue that can electroporate at high levels of transmembrane potential, and model tissue that does not support electroporation. In both cases, the spiral wave is initiated via cross-field stimulation of the bidomain sheet. The extracellular defibrillation shock is delivered via two small electrodes located at opposite tissue boundaries. Modifications in the active membrane kinetics enable the delivery of high-strength defibrillation shocks. Numerical solutions are obtained using an efficient semi-implicit predictor-corrector scheme that allows one to execute the simulations within reasonable time. The simulation results demonstrate that anode and/or cathode break excitations contribute significantly to the activity during and after the shock. For a successful defibrillation shock, the virtual electrodes and the break excitations restrict the spiral wave and render the tissue refractory so it cannot further maintain the reentry. The results also indicate that electroporation alters the anode/cathode break phenomena, the major impact being on the timing of the cathode-break excitations. Thus, electroporation results in different patterns of transmembrane potential distribution after the shock. This difference in patterns may or may not result in change of the outcome of the shock.     

The effect of externally applied electrical fields on myocardialtissue

The explanation of the response to electrical stimulus by macroscopic regions of cardiac tissue in terms of the behavior of membrane ion channels requires mathematical models that span the range of spatial scales from the single cardiac cell to the entire heart. This is accomplished by the bidomain model, which successfully characterizes the electrical properties of the heart and the effect of externally applied electric fields on myocardial tissue, Recently the bidomain model has been used to make several specific, testable predictions: (1) a 4-fold symmetric magnetic field pattern is associated with an expanding wave front, (2) a region of positive interstitial potential precedes an expanding wave front in the direction parallel to the myocardial fibers, (3) the rate of rise of an action potential depends on the direction of propagation in superfused tissue, (4) the wave front in superfused strands of tissue is curved (5) a “dog bone” shaped region of depolarization exists under a unipolar cathode, (6) depolarized regions along the fiber direction adjacent to a unipolar anode are responsible for anodal stimulation, (7) interactions between adjacent depolarized and hyperpolarized tissue cause anode-break and cathode-break stimulation, (8) reentry can be induced by successive stimulation using a single unipolar cathode, and (9) a mechanism for far field stimulation depends on fiber curvature. These predictions have been verified qualitatively in every case where they have been tested experimentally. In some cases, such as the mechanisms for reentry induction and far field stimulation, the necessary experiments have not yet been performed     

Analytic solution of the anisotropic bidomain equations for myocardial tissue: the effect of adjoining conductive regions

The anisotropic bidomain model for the propagation of electrical activation in the human myocardium H consists of coupled elliptic-parabolic partial differential equations for the transmembrane potential Vm, intracellular potential phi(i), and extracellular potential phi(e) in H, together with quasi-static equations for the potential distribution phiB in the surrounding (passive) isotropic extracardiac regions B. Four local parameters sigma((i,e) (l,t)) specify the conductivities in the longitudinal (l) and transverse (t) directions with respect to cardiac muscle fibers. Continuous current flow is required at the interface S(H) between H and B. We derive analytic formulas for Vm, phi(e), phi(i), and phiB for plane wave propagation in a uniformly anisotropic slab surmounted by a homogeneous region of conductivity sigmaB. No assumptions are required regarding the anisotropy ratios of the conductivity coefficients. The properties of these solutions are examined with a view to providing insight into the effect of the passive region B on the propagation of Vm and phi(e) in H. We show that for a suitably chosen boundary condition, the problem can be reduced to solving the bidomain equations in H alone.     

Effects of the propagation velocity of a surface depolarizationwave on the extracellular potential of an excitable cell

Extracellular electric fields have been proposed as a mechanism for electrical coupling between excitable cells. This study deals with the extracellular potential produced by an isolated excitable spherical cell due to a traveling depolarization wave on the cell's surface. Both uniform and nonuniform propagation velocity profiles are considered. Using boundary element methods, the extracellular potential was computed. The polarity of the extracellular potential was found to be space-dependent. The peak extracellular potential increased when a) the propagation velocity decreased, b) the rise time of the depolarization decreased, and c) the extracellular resistivity increased     

Discrete versus syncytial tissue behavior in a model of cardiacstimulation. I. Mathematical formulation

This paper presents a model describing the steady-state response of a two-dimensional (2-D) slice of myocardium to extracellular current injection. The model incorporates continuous representation of the multicellular, syncytial cardiac tissue based on the bidomain model. The classical bidomain model is modified by introducing periodic conductivities to better represent the electrical properties of the intracellular space. Thus, junctional discontinuity between abutting myocytes is reflected in the macroscopic representation of cardiac tissue behavior. Since a solution to the resulting coupled differential equations governing the intracellular and extracellular potentials in the tissue preparation is not computationally tractable when traditional numerical approaches, such as finite element or finite difference methods are used, spectral techniques are employed to reduce the problem to the solution of a set of algebraic equations for the transform of the bidomain potentials. Further, the solution to the “periodic” bidomain problem in the Fourier space is decomposed into two separate solutions: one for the classical-bidomain potentials where it is assumed that the intracellular conductivity values along and across cells incorporate the average contribution from cytoplasm and junction, and another for the junctional potential component. The decomposition of the total solution allows to approximately solve for the junctional component thus achieving high overall computational efficiency. The results of simulation are presented in an accompanying paper (see ibid., vol. 43, no. 12, p. 1141-50, 1996)