Edward J. Vigmond

Cholinergic Atrial Fibrillation in a Computer Model of a Two-Dimensional Sheet of Canine Atrial Cells With Realistic Ionic Properties

Classical concepts of atrial fibrillation (AF) have been rooted in Moe’s multiple-wavelet hypothesis and simple cellular-automaton computer model. Recent experimental work has raised questions about the multiple-wavelet mechanism, suggesting a discrete “driver region” underlying AF. We reexplored the theoretical basis for AF with a 2-dimensional computer model of a 5×10-cm sheet of atrial cells with realistic ionic and coupling properties. Vagal actions were formulated based on patch-clamp studies of acetylcholine (ACh) effects. In control, a single extrastimulus resulted in a highly meandering unstable spiral wave. Simulated electrograms showed fibrillatory activity, with a dominant frequency (DF, 6.5 Hz) that correlated with the mean rate. Uniform ACh reduced core meander of the spiral wave by ≈70% (as measured by the standard deviation of spiral-wave tip position) and accelerated the DF to 17.0 Hz. Simulated vagally induced refractoriness heterogeneity caused wavefront breakup as accelerated reentrant activity in regions of short refractoriness impinged on regions unable to respond in a 1:1 fashion because of longer refractoriness. In 7 simulations spanning the range of conditions giving sustained AF, 5 were maintained by single dominant spiral waves. On average, 3.0±1.3 wavelets were present (range, 1 to 7). Most wavelets were short-lived and did not contribute to AF maintenance. In contrast to predictions of the multiple-wavelet hypothesis, but in agreement with recent experimental evidence, our model indicates that AF can result from relatively stable primary spiral-wave generators and is significantly organized. Our results suggest that vagal AF may arise from ACh-induced stabilization of the primary spiral-wave generator and disorganization of the heterogeneous tissue response. The full text of this article is available at http://www.circresaha.org.

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.

Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model

The bidomain equations are considered to be one of the most complete descriptions of the electrical activity in cardiac tissue, but large scale simulations, as resulting from discretization of an entire heart, remain a computational challenge due to the elliptic portion of the problem, the part associated with solving the extracellular potential. In such cases, the use of iterative solvers and parallel computing environments are mandatory to make parameter studies feasible. The preconditioned conjugate gradient (PCG) method is a standard choice for this problem. Although robust, its efficiency greatly depends on the choice of preconditioner. On structured grids, it has been demonstrated that a geometric multigrid preconditioner performs significantly better than an incomplete LU (ILU) preconditioner. However, unstructured grids are often preferred to better represent organ boundaries and allow for coarser discretization in the bath far from cardiac surfaces. Under these circumstances, algebraic multigrid (AMG) methods are advantageous since they compute coarser levels directly from the system matrix itself, thus avoiding the complexity of explicitly generating coarser, geometric grids. In this paper, the performance of an AMG preconditioner (BoomerAMG) is compared with that of the standard ILU preconditioner and a direct solver. BoomerAMG is used in two different ways, as a preconditioner and as a standalone solver. Two 3-D simulation examples modeling the induction of arrhythmias in rabbit ventricles were used to measure performance in both sequential and parallel simulations. It is shown that the AMG preconditioner is very well suited for the solution of the bidomain equation, being clearly superior to ILU preconditioning in all regards, with speedups by factors in the range 5.9-7.7

Parallel multigrid preconditioner for the cardiac bidomain model

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.

Reentry in a Morphologically Realistic Atrial Model

Reentry in Morphologically Realistic Atria. Introduction: Atrial fibrillation is the most common cardiac arrhythmia. In ablation procedures, identification of the reentrant pathways is vital. This has proven difficult because of the complex morphology of the atria. The purpose of this study was to ascertain the role of specific anatomic structures on reentry induction and maintenance.

Towards predictive modelling of the electrophysiology of the heart

The simulation of cardiac electrical function is an example of a successful integrative multiscale modelling approach that is directly relevant to human disease. Today we stand at the threshold of a new era, in which anatomically detailed, tomographically reconstructed models are being developed that integrate from the ion channel to the electromechanical interactions in the intact heart. Such models hold high promise for interpretation of clinical and physiological measurements, for improving the basic understanding of the mechanisms of dysfunction in disease, such as arrhythmias, myocardial ischaemia and heart failure, and for the development and performance optimization of medical devices. The goal of this article is to present an overview of current state‐of‐art advances towards predictive computational modelling of the heart as developed recently by the authors of this article. We first outline the methodology for constructing electrophysiological models of the heart. We then provide three examples that demonstrate the use of these models, focusing specifically on the mechanisms for arrhythmogenesis and defibrillation in the heart. These include: (1) uncovering the role of ventricular structure in defibrillation; (2) examining the contribution of Purkinje fibres to the failure of the shock; and (3) using magnetic resonance imaging reconstructed heart models to investigate the re‐entrant circuits formed in the presence of an infarct scar.

Defining electrical communication in skeletal muscle resistance arteries: a computational approach

Vascular cells communicate electrically to coordinate their activity and control tissue blood flow. To foster a quantitative understanding of this fundamental process, we developed a computational model that was structured to mimic a skeletal muscle resistance artery. Each endothelial cell and smooth muscle cell in our virtual artery was treated as the electrical equivalent of a capacitor coupled in parallel with a non‐linear resistor representing ionic conductance; intercellular gap junctions were represented by ohmic resistors. Simulations revealed that the vessel wall is not a syncytium in which electrical stimuli spread equally to all constitutive cells. Indeed, electrical signals spread in a differential manner among and between endothelial cells and smooth muscle cells according to the initial stimulus. The predictions of our model agree with physiological data from the feed artery of the hamster retractor muscle. Cell orientation and coupling resistance were the principal factors that enable electrical signals to spread differentially along and between the two cell types. Our computational observations also illustrated how gap junctional coupling enables the vessel wall to filter and transform transient electrical events into sustained voltage responses. Functionally, differential electrical communication would permit discrete regions of smooth muscle activity to locally regulate blood flow and the endothelium to coordinate regional changes in tissue perfusion.

Automatically Generated, Anatomically Accurate Meshes for Cardiac Electrophysiology Problems

Significant advancements in imaging technology and the dramatic increase in computer power over the last few years broke the ground for the construction of anatomically realistic models of the heart at an unprecedented level of detail. To effectively make use of high-resolution imaging datasets for modeling purposes, the imaged objects have to be discretized. This procedure is trivial for structured grids. However, to develop generally applicable heart models, unstructured grids are much preferable. In this study, a novel image-based unstructured mesh generation technique is proposed. It uses the dual mesh of an octree applied directly to segmented 3-D image stacks. The method produces conformal, boundary-fitted, and hexahedra-dominant meshes. The algorithm operates fully automatically with no requirements for interactivity and generates accurate volume-preserving representations of arbitrarily complex geometries with smooth surfaces. The method is very well suited for cardiac electrophysiological simulations. In the myocardium, the algorithm minimizes variations in element size, whereas in the surrounding medium, the element size is grown larger with the distance to the myocardial surfaces to reduce the computational burden. The numerical feasibility of the approach is demonstrated by discretizing and solving the monodomain and bidomain equations on the generated grids for two preparations of high experimental relevance, a left ventricular wedge preparation, and a papillary muscle.

Image-based models of cardiac structure with applications in arrhythmia and defibrillation studies

The objective of this article is to present a set of methods for constructing realistic computational models of cardiac structure from high-resolution structural and diffusion tensor magnetic resonance images and to demonstrate the applicability of the models in simulation studies. The structural image is segmented to identify various regions such as normal myocardium, ventricles, and infarct. A finite element mesh is generated from the processed structural data, and fiber orientations are assigned to the elements. The Purkinje system, when visible, is modeled using linear elements that interconnect a set of manually identified points. The methods were applied to construct 2 different models; and 2 simulation studies, which demonstrate the applicability of the models in the analysis of arrhythmia and defibrillation, were performed. The models represent cardiac structure with unprecedented detail for simulation studies.

Endothelial Ca2+ wavelets and the induction of myoendothelial feedback

When arteries constrict to agonists, the endothelium inversely responds, attenuating the initial vasomotor response. The basis of this feedback mechanism remains uncertain, although past studies suggest a key role for myoendothelial communication in the signaling process. The present study examined whether second messenger flux through myoendothelial gap junctions initiates a negative-feedback response in hamster retractor muscle feed arteries. We specifically hypothesized that when agonists elicit depolarization and a rise in second messenger concentration, inositol trisphosphate (IP(3)) flux activates a discrete pool of IP(3) receptors (IP(3)Rs), elicits localized endothelial Ca(2+) transients, and activates downstream effectors to moderate constriction. With use of integrated experimental techniques, this study provided three sets of supporting observations. Beginning at the functional level, we showed that blocking intermediate-conductance Ca(2+)-activated K(+) channels (IK) and Ca(2+) mobilization from the endoplasmic reticulum (ER) enhanced the contractile/electrical responsiveness of feed arteries to phenylephrine. Next, structural analysis confirmed that endothelial projections make contact with the overlying smooth muscle. These projections retained membranous ER networks, and IP(3)Rs and IK channels localized in or near this structure. Finally, Ca(2+) imaging revealed that phenylephrine induced discrete endothelial Ca(2+) events through IP(3)R activation. These events were termed recruitable Ca(2+) wavelets on the basis of their spatiotemporal characteristics. From these findings, we conclude that IP(3) flux across myoendothelial gap junctions is sufficient to induce focal Ca(2+) release from IP(3)Rs and activate a discrete pool of IK channels within or near endothelial projections. The resulting hyperpolarization feeds back on smooth muscle to moderate agonist-induced depolarization and constriction.