The bidomain equations are the most complete description of cardiac electrical activity. Their numerical solution is, however, computationally demanding, especially in three dimensions, because of the fine temporal and spatial sampling required. This paper methodically examines computational performance when solving the bidomain equations. Several techniques to speed up this computation are examined in this paper. The first step was to recast the equations into a parabolic part and an elliptic part. The parabolic part was solved by either the finite-element method (FEM) or the interconnected cable model model (ICCM). The elliptic equation was solved by FEM on a coarser grid than the parabolic problem and at a reduced frequency. The performance of iterative and direct linear equation system solvers was analyzed as well as the scalability and parallelizability of each method. Results indicate that the ICCM was twice as fast as the FEM for solving the parabolic problem, but when the total problem was considered, this resulted in only a 20% decrease in computation time. The elliptic problem could be solved on a coarser grid at one-quarter of the frequency at which the parabolic problem was solved and still maintain reasonable accuracy. Direct methods were faster than iterative methods by at least 50% when a good estimate of the extracellular potential was required. Parallelization over four processors was efficient only when the model comprised at least 500,000 nodes. Thus, it was possible to speed up solution of the bidomain equations by an order of magnitude with a slight decrease in accuracy.

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.

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.

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Background-Functional reentry in the heart takes the form of spiral waves. Drifting spiral waves can become pinned to anatomic obstacles and thus attain stability and persistence. Lidocaine is an antiarrhythmic agent commonly used to treat ventricular tachycardia clinically. We examined the ability of small obstacles to anchor spiral waves and the effect of lidocaine on their attachment. Methods and Results-Spiral waves were electrically induced in confluent monolayers of cultured, neonatal rat cardiomyocytes. Small, circular anatomic obstacles (0.6 to 2.6 mm in diameter) were situated in the center of the monolayers to provide an anchoring site. Eighty reentry episodes consisting of at least 4 revolutions were studied. In 36 episodes, the spiral wave attached to the obstacle and became stationary and sustained, with a shorter reentry cycle length and higher rate. Spiral waves could attach to obstacles as small as 0.6 mm, with a likelihood for attachment that increased with obstacle size. After attachment, both conduction velocity of the wave-front tip and wavelength near the obstacle adapted from their pre-reentry values and increased linearly with obstacle size. In contrast, reentry cycle length did not correlate significantly with obstacle size. Addition of lidocaine 90 mol/L depressed conduction velocity, increased reentry cycle length, and caused attached spiral waves to become quasiattached to the obstacle or terminate. Conclusions-Anchored spiral waves exhibit properties of both unattached spiral waves and anatomic reentry. Their behavior may be representative of functional reentry dynamics in cardiac tissue, particularly in the setting of monomorphic tachyarrhythmias.

A variety of chemical and biological nonlinear excitable media, including heart tissue, can support stable, self-organized waves of activity in a form of rotating single-arm spirals. In particular, heart tissue can support stationary and meandering spirals of electrical excitation, which have been shown to underlie different forms of cardiac arrhythmias. In contrast to single-arm spirals, stable multiarm spirals (multiple spiral waves that rotate in the same direction around a common organizing center) have not been demonstrated and studied yet in living excitable tissues. Here, we show that persistent multiarm spirals of electrical activity can be induced in monolayer cultures of neonatal rat heart cells by a short, rapid train of electrical point stimuli applied during single-arm-spiral activity. Stable formation is accomplished only in monolayers that show a relatively broad and steep dependence of impulse wavelength and propagation velocity on rate of excitation. The resulting multiarm spirals emit waves of electrical activity at rates faster than for single-arm spirals and exhibit two distinct behaviors, namely ''arm-switching'' and ''tip-switching.'' The phenomenon of rate acceleration due to an increase in the number of spiral arms possibly may underlie the acceleration of functional reentrant tachycardias paced by a clinician or an antitachycardia device.R otating single-arm spirals can exist in different chemical(1-4) and biological systems (5-8). If a spiral is stationary or weakly meandering, it represents a persistent source of highly periodic activity with a period of rotation that is determined by the properties of excitation, recovery, and diffusion in the medium. In the heart, stable single-arm spirals can underlie periodic activity such as monomorphic tachycardia, whereas unstable spirals that continuously form and break up are shown to underlie aperiodic and lethal heart activity, namely fibrillation (9). So far, stable rotating structures with a higher degree of organization, such as multiarmed spirals, have been demonstrated experimentally in chemically (2) or light-controlled (10) Belousov-Zhabotinsky reactions, in starving social amoebae Dictyostelium discoideum (11), and in FitzHugh-Nagumo-type computer models of homogenous weakly excitable media (12)(13)(14). The modes of induction of these multiarmed spirals [i.e., use of an unexcitable central obstacle that is subsequently removed (2, 10) and spontaneous formation from dense multiple wavebreaks (12) or from preset spatial distributions of excited and recovered medium (13, 14)] are not applicable to cardiac tissue. Moreover, healthy cardiac tissue exhibits normal excitability, in contrast to two-variable FitzHugh-Nagumo models, which require weak excitability to support stable multiarm spirals (12-15). In a normally excitable medium, the existence of a stable multiarm spiral as an entity distinct from negligibly interacting adjacent single spirals was questioned by Winfree (16,17), who considered that in this case distinct rates of rotat...

This study investigated roles of the variation of extracellular voltage gradient (VG) over space and cardiac fibers in production of transmembrane voltage changes (DeltaV(m)) during shocks. Eleven isolated rabbit hearts were arterially perfused with solution containing V(m)-sensitive fluorescent dye (di-4-ANEPPS). The epicardium received shocks from symmetrical or asymmetrical electrodes to produce nominally uniform or nonuniform VGs. Extracellular electric field and DeltaV(m) produced by shocks in the absolute refractory period were measured with electrodes and a laser scanner and were simulated with a bidomain computer model that incorporated the anterior left ventricular epicardial fiber field. Measurements and simulations showed that fibers distorted extracellular voltages and influenced the DeltaV(m). For both uniform and nonuniform shocks, DeltaV(m) depended primarily on second spatial derivatives of extracellular voltages, whereas the VGs played a smaller role. Thus, 1) fiber structure influences the extracellular electric field and the distribution of DeltaV(m); 2) the DeltaV(m) depend on second spatial derivatives of extracellular voltage.

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