Felipe Aguel

Computational techniques for solving the bidomain equations in three dimensions

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 (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.

The role of cardiac tissue structure in 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.

Roles of Electric Field and Fiber Structure in Cardiac Electric Stimulation

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.

Spiral Wave Attachment to Millimeter-Sized Obstacles

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 &mgr;mol/L depressed conduction velocity, increased reentry cycle length, and caused attached spiral waves to become quasi- attached 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.

IMPACT OF TRANSVENOUS LEAD POSITION ON ACTIVE-CAN ICD DEFIBRILLATION : A COMPUTER SIMULATION STUDY

Optimizing lead placement in transvenous defibrillation remains central to the clinical aspects of the defibrillation procedure. Studies involving superior vena cava (SVC) return electrodes have found that left ventricular (LV) leads or septal positioning of the right ventricular (RV) lead minimizes the voltage defibrillation threshold (VDFT) in endocardial lead→SVC defibrillation systems. However, similar studies have not been conducted for active‐can configurations. The goal of this study was to determine the optimal lead position to minimize the VDFT for systems incorporating an active can. This study used a high resolution finite element model of a human torso that includes the fiber architecture of the ventricular myocardium to find the role of lead positioning in a transvenous LEAD→can defibrillation electrode system. It was found that, among single lead systems, posterior positioning of leads in the right ventricle lowers VDFTs appreciably. Furthermore, a septal location of leads resulted in lower VDFTs than free‐wall positioning. Increasing the number of leads, and thus the effective lead surface area in the right ventricle also resulted in lower VDFTs. However, the lead configuration that resulted in the lowest VDFTs is a combination of a mid‐cavity right ventricle lead and a mid‐cavity left ventricle lead. The addition of a left ventricular lead resulted in a reduction in the size of the low gradient regions and a change of its location from the left ventricular free wall to the septal wall.

Virtual Electrode Polarization Leads to Reentry in the Far Field

Virtual Electrode Polarization. Introduction: Our previous article examined cardiac vulnerability to reentry in the near field within the framework of the virtual electrode polarization (VEP) concept. The present study extends this examination to the far field and compares its predictions to the critical point hypothesis.

Rotors and Spiral Waves in Two Dimensions

Reentrant arrhythmias are the most common and potentially life-threatening of the cardiac arrhythmias and can be anatomically or functionally based. Anatomic reentry involves pathways that encircle a large anatomic obstacle (a region that is unexcitable under any conditions, occupies a fixed area, and has a border that is structurally defined) and is generally conceptualized in terms of a tissue ring. 1 Examples involve reentry in Purkinje bundles, bundle branches, the atrioventricular node, and muscle around the tricuspid valve. 2 On the other hand, functional reentry does not require a central anatomic obstacle and underlies many ventricular and atrial tachycardias and fibrillation. Mechanisms of functional reentry have been described in terms of leading circle, spiral wave, figure-of-eight, and anisotropic reentry. 3 All of these forms of reentry may be viewed as variants of rotating waves (rotors) that exhibit a repetitive, self-sustained, circulating spread of activation in which different behaviors emerge, depending on the particular combination of cellular excitatory/refractory properties and tissue structure.

Influence of Anisotropy on Local and Global Measures of Potential Gradient in Computer Models of Defibrillation

AbstractA heart–torso model including fiber orientation is used to calculate electric field strength in an active-can transvenous defibrillation system and estimate errors due to inadequate description of the anisotropy of the myocardium. Using a minimum potential gradient (5 V/cm) in a critical mass (95%) of the tissue, the estimated defibrillation voltage threshold for a right ventricular transvenous lead placement differs by only 4.5% when using isotropic myocardial conductivity compared to a model with realistic fiber architecture. In addition, pointwise comparisons of the two solutions reveal differences of 10.8% rms in potential gradient strength and 31.6% rms in current density magnitude in the myocardium, resulting in a change in the location of the low gradient regions. These results suggest that if a minimum potential gradient throughout the heart is necessary to avoid reinitiation of fibrillatory wave fronts, then isotropic models are adequate for modeling the electric field in the heart. Alternatively, the model demonstrates the use of physiologically based descriptions of anisotropy and fiber orientation, which will soon allow simulations of shock induced membrane polarization during defibrillation.

ADVANCES IN MODELING CARDIAC DEFIBRILLATION

This paper presents the development of a bidomain model of electrical defibrillation and post-shock arrhythmogenesis of the rabbit heart. The model incorporates a realistic representation of cardiac geometry, fiber structure, and membrane ionic processes. Fibrillation is induced in the rabbit heart with burst pacing. Defibrillation shocks of various strengths are delivered to the model in an attempt to terminate the arrhythmia. Two sets of defibrillation simulations are conducted based on the waveform of the defibrillation shock: monophasic and biphasic. The simulations presented here demonstrate the feasibility of modeling realistically the effect of the defibrillation shock on the heart.

Chapter 31 – Modeling Cardiac Defibrillation

Despite the large body of research devoted to ventricular defibrillation, the mechanisms by which a strong electric shock delivered to the heart terminates lethal disturbances in ventricular rhythm are still a subject of considerable debate. Experimental evidence strongly suggests 1 2 3 4 5 that the shock induces regions of positive and negative change in transmembrane potential termed virtual electrode polarization ; this evidence is also supported by numeric simulations of the defibrillation process. 6 7 8 9 10 11 The virtual electrode polarization is highly nonuniform and is strongly dependent on the underlying tissue structure. 3 12 13 Cells experiencing this shock-induced change in transmembrane potential rapidly alter the course of their action potential in a nonlinear fashion. The changes in individual cell electrical behavior result in altered overall electrical behavior of the myocardium: certain regions could remain refractory for considerable intervals of time, whereas excitability might be partially or completely restored in others. The outcome of the shock depends on this postshock transmembrane potential distribution, but more importantly, on the spatial interactions that ensue thereafter and the new wave fronts that are elicited as a result of these spatial interactions.