New finite difference formulations for general inhomogeneous anisotropic bioelectric problems

Due to its low computational complexity, finite difference modeling offers a viable tool for studying bioelectric problems, allowing the field behaviour to be observed easily as different system parameters are varied. Previous finite difference formulations, however, have been limited mainly to systems in which the conductivity is orthotropic, i.e., a strictly diagonal conductivity tensor. This in turn has limited the effectiveness of the finite difference technique in modeling complex anatomies with arbitrarily anisotropic conductivities, e.g., detailed fiber structures of muscles where the fiber can lie in any arbitrary direction. Here, the authors present both two-dimensional and three dimensional finite difference formulations that are valid for structures with an inhomogeneous and nondiagonal conductivity tensor. A data parallel computer, the connection machine CM-5, is used in the finite difference implementation to provide the computational power and memory for solving large problems. The finite difference grid is mapped effectively to the CM-5 by associating a group of nodes with one processor. Details on the new approach and its data parallel implementation are presented together with validation and computational performance results. In addition, an application of the new formulation in providing the potential distribution inside a canine torso during electrical defibrillation is demonstrated.