Solvability analysis and numerical approximation of linearized cardiac electromechanics

Abstract : This paper is concerned with the mathematical analysis of a coupled elliptic-parabolic system modeling the interaction between the propagation of electric potential and subsequent deformation of the cardiac tissue. The problem consists in a reaction-diffusion system governing the dynamics of ionic quantities, intra and extra-cellular potentials, and the linearized elasticity equations are adopted to describe the motion of an incompressible material. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we prove existence of weak solutions to the underlying coupled reaction-diffusion system and uniqueness of regular solutions. The proof of existence is based on a combination of parabolic regularization, the Faedo-Galerkin method, and the monotonicity-compactness method of J.L. Lions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with a numerical example illustrating the convergence of the method and some features of the model.
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Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2015, 25, pp.959-993. 〈10.1142/S0218202515500244〉
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Boris Andreianov, Mostafa Bendahmane, Alfio Quarteroni, Ricardo Ruiz Baier. Solvability analysis and numerical approximation of linearized cardiac electromechanics. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2015, 25, pp.959-993. 〈10.1142/S0218202515500244〉. 〈hal-00865585v3〉

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