Stochastic electromechanical bidomain model

TL;DR

This paper introduces a stochastic bidomain model for cardiac electromechanics, combining electrophysiology and mechanics, and proves the existence of weak solutions using advanced stochastic analysis techniques.

Contribution

It develops a novel stochastic electromechanical bidomain model with an active strain framework and establishes weak solution existence, advancing mathematical understanding of cardiac tissue dynamics.

Findings

Existence of weak solutions for the stochastic model.

Development of an active strain decomposition framework.

Application of stochastic compactness and de Rham's theorem in proof.

Abstract

We analyze a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearized elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearized elastic behavior and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo--Galerkin method. We utilize a stochastic adaptation of de Rham's theorem to deduce the weak convergence of the pressure approximations.