Solvability analysis and numerical approximation of linearized cardiac electromechanics

Boris Andreianov (1, 2) , Mostafa Bendahmane (3) , Alfio Quarteroni (4, 5) , Ricardo Ruiz Baier (4, 6)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
3 IMB - Institut de Mathématiques de Bordeaux
4 CMCS - Chair of Modelling and Scientific Computing
5 MOX - Modeling and Scientific Computing [Milano]
6 ISTE - Institut des sciences de la terre [Lausanne]

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.

Mots clés: Electro--mechanical coupling; Bidomain equations; Active deformation; Weak solutions; Weak compactness method; Weak-strong uniqueness; Finite element approximation; Convergence of approximations

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Analyse numérique [cs.NA]

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, 2015, 25, pp.959-993. ⟨10.1142/S0218202515500244⟩. ⟨hal-00865585v2⟩