A note on stress-driven anisotropic diffusion and its role in active deformable media

Christian Cherubini,Simonetta Filippi,Alessio Gizzi,Ricardo Ruiz-Baier

doi:10.1016/j.jtbi.2017.07.013

Abstract

We introduce a new model to describe diffusion processes within active deformable media. Our general theoretical framework is based on physical and mathematical considerations, and it suggests to employ diffusion tensors directly influenced by the coupling with mechanical stress. The proposed generalised reaction-diffusion-mechanics model reveals that initially isotropic and homogeneous diffusion tensors turn into inhomogeneous and anisotropic quantities due to the intrinsic structure of the nonlinear coupling. We study the physical properties leading to these effects, and investigate mathematical conditions for its occurrence. Together, the mathematical model and the numerical results obtained using a mixed-primal finite element method, clearly support relevant consequences of stress-driven diffusion into anisotropy patterns, drifting, and conduction velocity of the resulting excitation waves. Our findings also indicate the applicability of this novel approach in the description of mechano-electric feedback in actively deforming bio-materials such as the cardiac tissue.

Highlights

Excitable media, whether of biological type or not, represent complex nonlinear systems which are often of electrochemical nature, and can typically be coupled to several multi-physical factors as heat transfer or solid and/or fluid mechanics
Cardiac contraction results from the combination of a complex emerging behaviour where subcellular ion dynamics induce the overlapping of protein filaments, rapidly scaling up to both the cellular and tissue scales through a process known as excitation-contraction coupling and, as main topic of the present work, its reverse effect known as the mechano-electric feedback (MEF) (Kaufmann and Theophile, 1967; Kohl and Sachs, 2001)
In this note we present a novel formulation for the description of soft active deformable media within the context of coupled reaction-diffusion-mechanics systems, and employ nonlinear cardiac dynamics as a main motivating example

Summary

Introduction

Whether of biological type or not, represent complex nonlinear systems which are often of electrochemical nature, and can typically be coupled to several multi-physical factors as heat transfer or solid and/or fluid mechanics. In this note we present a novel formulation for the description of soft active deformable media within the context of coupled reaction-diffusion-mechanics systems, and employ nonlinear cardiac dynamics as a main motivating example. Here we have found that an anisotropic and inhomogeneous diffusivity is naturally induced by mechanical deformations, affecting the nonlinear dynamics of the spatiotemporal excitation wave This important fact implies that the present formulation can recover and generalise a large class of electromechanical models based on basic FitzHugh-Nagumo-type descriptions (Aliev and Panfilov, 1996; Panfilov and Keldermann, 2005). Our assessment is conducted for stretched tissues, focusing on appropriate physical indicators as conduction velocity, propagation patterns and spiral dynamics, and carefully identifying conditions leading to the stability of the coupled system

A stress-assisted electromechanical model

Discussion