Abstract-Our understanding of the electrophysiological properties of the heart is incomplete. We have investigated two issues that are fundamental to advancing that understanding. First, there has been widespread debate over the mechanisms by which an externally applied shock can influence a sufficient volume of heart tissue to terminate cardiac fibrillation. Second, it has been uncertain whether cardiac tissue should be viewed as an electrically orthotropic structure, or whether its electrical properties are, in fact, isotropic in the plane orthogonal to myofiber direction. In the present study, a computer model that incorporates a detailed three-dimensional representation of cardiac muscular architecture is used to investigate these issues. We describe a bidomain model of electrical propagation solved in a discontinuous domain that accurately represents the microstructure of a transmural block of rat left ventricle. From analysis of the model results, we conclude that (1) the laminar organization of myocytes determines unique electrical properties in three microstructurally defined directions at any point in the ventricular wall of the heart, and (2)

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SUMMARYThis paper reports on some problems that can arise with the use of regularized derivative boundary integral equations. It concentrates on developing a formulation for the simple Laplace equation using a cubic Hermite interpolation and shows how certain combinations of derivative and conventional boundary integral equations can result in a solution scheme severely lacking in stability. With some simple two-and three-dimensional geometries, the derivative equations on their own do not provide enough information to solve a Dirichlet problem. Even combinations of the conventional and derivative equations fail for some simple geometries. We conclude that the only consistently successful combination is that of the conventional equation with the tangential derivative equation, which showed cubic convergence of results with mesh refinement. Numerical results are presented for this scheme in both two and three dimensions.KEY WORDS derivative boundary integral equations; Hermite interpolation

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Abstmct-The spreading of excitation in ventricular myocardium is modelled by treating the thin region of rapidly depolarizing tissue 84 a propagating wavefront. The model determines tissue excitation time using an eikonal equation that includes the effects of wavefront orientation in the myocardial structure and wavefront curvature.Use of a Petrov-Galerkin finite element method with a no-inflow boundary condition enables the eikonal equation to be solved on reasonably coarse meshes of cubic Hermite elements. The method is applied successfully on a model of the complete canine ventricular myocardium.