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

被引:101
作者
Bourgault, Yves [1 ]
Coudiere, Yves [2 ]
Pierre, Charles [3 ]
机构
[1] Univ Ottawa, Dept Math & Stat, Ottawa, ON K1N 6N5, Canada
[2] CNRS, UN, ECN, Lab Math Jean Leray,UMR 6629, F-44322 Nantes 3, France
[3] Univ Pau, CNRS, UPPA, Lab Math Appl,UMR 5142, F-64013 Pau, France
基金
加拿大自然科学与工程研究理事会;
关键词
reaction-diffusion equation; bidomain model; cardiac electrophysiology; GLOBAL EXISTENCE; 3-D MODELS; EXCITATION; TISSUE; EQUATIONS; PROPAGATION; BEHAVIOR; SYSTEMS; SPREAD; WAVES;
D O I
10.1016/j.nonrwa.2007.10.007
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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.
引用
收藏
页码:458 / 482
页数:25
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