Advancements Leading to the Development of the Bidomain Model of Defibrillation

Historically, overwhelming electrical artifacts had prevented researchers from recording during as well as shortly after the shock. A breakthrough in mapping cardiac activity associated with defibrillation occurred during the last decade of the twentieth century with the introduction of potentiometric dyes, which allowed continuous recording of activity before, during, and after the shock. At the same time, the theoretical electrophysiology community adopted a novel modeling methodology, the bidomain model; its theoretical underpinnings and applications are comprehensively explored in this book. The bidomain model (see the chapter by Henriquez and Ying for more details) is a continuum representation of the myocardium that takes into consideration current distribution resulting from a particular characteristic of cardiac tissue—the fact that the two spaces comprising the myocardium, the intra- and extracellular, are both anisotropic but to a different degree. The myocardium is thus characterized with "unequal anisotropy ratios." Since the bidomain model accounts for the current flow in the interstitium, it became instantly a powerful modeling tool in the study of stimulation of cardiac tissue (see the chapter by Janks and Roth), where current delivered in the extracellular space finds its way across the membranes of cardiac cells.

The first significant achievement of this new approach was the study of the passive (i.e., the ionic currents are not accounted for) shock-induced change in transmembrane potential following a strong unipolar stimulus (near-field effects). Bidomain simulations by Sepulveda et al.2 demonstrated that the tissue response in the vicinity of a strong unipolar stimulus involved simultaneous occurrence of positive (depolarizing) and negative (hyperpolarizing) effects in close proximity. This finding of virtual electrodes was in stark contrast with the established view that tissue responses should only be depolarizing (hyperpolarizing) if the stimulus was cathodal (anodal).3 Optical mapping studies that followed convincingly confirmed these theoretical predictions.4 Since then, virtual electrode polarization (VEP) has been documented in experiments involving various stimulus configurations.5-9

The next big contribution of passive bidomain modeling was the detailed analysis of VEP etiology and its dependence on cardiac tissue structure and the configuration of the applied field; both were shown to be major determinants of the shape, location, polarity, and intensity of the shock-induced polarization.7'10-14 In particular, theoretical considerations led to the recognition of two types of VEP: (1) surface VEP, which penetrates the ventricular wall over a few cell layers, due to current redistribution near the boundaries separating myocardium from blood cavity or surrounding bath, and (2) bulk VEP throughout the ventricular wall.11'15 Analysis of the bidomain equations revealed that a necessary condition for the existence of the bulk VEP is the unequal anisotropy in the myocardium. Sufficient conditions include either spatial nonuniformity in applied electric field or nonuniformity in tissue architecture, such as fiber curvature, fiber rotation, fiber branching and anastomosis, and local changes in tissue conductivity due to resistive heterogeneities. Additional detail regarding the formation of VEP and the structural mechanisms that drive it can be found in the chapter by Tung.

How do cells respond to externally imposed changes in their transmembrane potential, such as those predicted by the passive bidomain model? The cellular response to shock-induced VEP depends on its magnitude and polarity, as well as on the electrophysiological state of the cell at the time of shock delivery. Action potential duration can be either extended (by positive VEP) or shortened (by negative VEP) to a degree that depends on VEP magnitude and shock timing, with strong negative VEP completely abolishing (deexciting) the action potential, thus creating postshock excitable areas in the virtual anode regions. Two-dimensional (myocardial sheet) simulations with the bidomain model that incorporated for the first time ionic fluxes through cell membranes (active bidomain modeling) resulted in the recognition of the importance of the distribution of transmembrane potential established by the shock to the origin of the postshock activations. Analyzing results of two-dimensional (2D) bidomain simulations of near-field behavior (behavior in the vicinity of a stimulus), Roth16 demonstrated that the close proximity of a deexcited region and a virtual cathode could result in an excitation at shock end, called break excitation (i.e., at the break of the shock); the virtual cathode serves as an electrical stimulus, eliciting a regenerative depolarization and a propagating wave in the newly created excitable area. Break excitations arise at the borders between oppositely polarized regions provided that the transmembrane potential gradient across the border spans the threshold for regenerative depolarization.17 The finding of break excitations, combined with the fact that positive VEP can result in regenerative depolarization in regions where tissue is at or near diastole (make excitation; takes place at the onset of the shock), resulted in a novel understanding of how a strong stimulus can result in the development of new activations.

These findings provided the background for the development of a comprehensive set of mechanisms aiming to explain the success or failure of a defibrillation shock based on the application of the bidomain model. This set of mechanisms includes the generating of VEP in the 3D ventricles by the defibrillation electrodes (far-field effects) and the initiation of postshock activations, the origin of which, if any, depends on shock strength and waveform. Developing this understanding required large-scale simulations of defib-rillation in the whole heart, where geometry and fiber orientation play a major role in activity during and after the shock. This necessitated technological advancement in the application of the bidomain model and the numerical approaches employed in simulating defibrillation. The sections below present the algorithms used to solve the bidomain equations in 3D as well as examples of mechanistic insight obtained from the analysis of the simulations.

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