Optimal monodomain approximations of the bidomain equations

Abstract

The bidomain equations are widely accepted to model the spatial distribution of the electrical potential in the heart. Although order optimal methods have been devised for discrete versions of these equations, it is still very CPU demanding to solve the equations numerically on a sufficiently fine mesh in 3D. Furthermore, the equations are hard to analyze; from a mathematical point of view, very little is known about the qualitative behavior of the solutions generated by these equations. It is well known that upon appealing to a certain relation between the extracellular and intracellular conductivities, the bidomain model can be rewritten in terms of a scalar reaction diffusion equation referred to as the monodomain model. This model is of course much easier to solve, and also the qualitative properties of the solutions are well known; such equations have been studied intensively for decades. It is the purpose of the present paper to show how the bidomain equations can be approximated in an optimal manner by the solution of a monodomain model. The key feature here is that this optimal solution can be computed without solving the bidomain model itself. The solution is obtained by putting the problem into a framework of parameter identification problems. Our results are illuminated by a series of numerical experiments.