Multiscale Modelling and Analysis of Signalling Processes in Tissues with Non-Periodic Distribution of Cells

Mariya Ptashnyk

doi:10.1007/s10013-016-0232-9

Abstract

In this paper, a microscopic model for a signalling process in the left ventricular wall of the heart, comprising a non-periodic fibrous microstructure, is considered. To derive the macroscopic equations, the non-periodic microstructure is approximated by the corresponding locally periodic microstructure. Then, applying the methods of locally periodic homogenization (the locally periodic (l-p) unfolding operator, locally periodic two-scale (l-t-s) convergence on oscillating surfaces and l-p boundary unfolding operator), we obtain the macroscopic model for a signalling process in the heart tissue.

Highlights

We consider the multiscale analysis of microscopic problems posed in domains with non-periodic microstructures
We consider a model for a signalling process in the cardiac muscle tissue of the left ventricular wall, comprising plywood-like microstructure [25, 28]
There are two main challenges in the multiscale analysis of microscopic problems posed in domains with non-periodic perforations: (i) the approximation of the non-periodic microstructure by a locally periodic one and (ii) derivation of limit equations for the non-linear equations defined on oscillating surfaces of the microstructure

Summary

Introduction

We consider the multiscale analysis of microscopic problems posed in domains with non-periodic microstructures. There are two main challenges in the multiscale analysis of microscopic problems posed in domains with non-periodic perforations: (i) the approximation of the non-periodic microstructure by a locally periodic one and (ii) derivation of limit equations for the non-linear equations defined on oscillating surfaces of the microstructure. Assuming the C2-regularity for the rotation angle γ , we define the locally periodic microstructure which approximates the original non-periodic plywood-like structure. Applying techniques of locally periodic homogenization (locally periodic two-scale convergence (l-t-s) and l-p unfolding operator), we derive macroscopic equations for the original microscopic model. In Appendix, we summarize the definitions and main compactness results for the l-t-s convergence and l-p unfolding operator

Microscopic Model for a Signalling Process in Heart Tissue

Derivation of Macroscopic Equations