Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch models. (C) 2007 Elsevier Ltd. All rights reserved.