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

Abstract : 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.
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Nonlinear Analysis: Real World Applications, Elsevier, 2009, 10 (1), pp.458-482
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Yves Bourgault, Yves Coudière, Charles Pierre. Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Analysis: Real World Applications, Elsevier, 2009, 10 (1), pp.458-482. 〈hal-00101458v2〉

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