Approximate analytical solutions to the bidomain equations with unequal anisotropy ratios

Abstract

The anisotropic electrical properties of cardiac tissue are described by the bidomain model. In this model, the ratio of the electrical conductivities parallel to and perpendicular to the myocardial fibers is greater in the intracellular space than in the extracellular space, resulting in a condition called unequal anisotropy ratios. No analytical solutions exist in this case. In this paper, we present approximate analytical solutions to the bidomain equations. The gist of our method is a perturbation expansion in a parameter that is defined as one minus the ratio of the anisotropy ratios in the extracellular and intracellular spaces. Three applications are considered: stimulation of the tissue by an electrode, an expanding action potential wave front, and injury currents. In the first application, the first-order perturbation term of the transmembrane potential depends on orientation by a second-order Legendre polynomial and induces adjacent regions of depolarization and hyperpolarization. In the second and third applications, the extracellular potential outside a wave front or an injured region depends on orientation by a second-order Legendre polynomial and creates regions of positive extracellular potential in the direction parallel to the fibers.