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

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 (lt-s) convergence on oscillating surfaces and l-p boundary unfolding operator), we obtain the macroscopic model for a signalling process in the heart tissue.

Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model

“…We are convinced that this work will further allow the analysis of more complex models by laying the ground of the bidomain equations 2-scale analysis. More precisely, among the modeling ingredients that could fit in our context, one could consider: heterogeneous concentrations of ionic species inside the cells, influences of heart mechanical deformations [53,24,18], gap junctions [27] and the cardiac microscopic fiber structure in the context of local 2-scale convergence [12,51].…”

Apprehending the effects of mechanical deformations in cardiac electrophysiology: A homogenization approach

“…The periodicity assumption is not exactly adapted to the global fiber structure of the heart, because the fiber direction varies spatially. However, we can argue that periodicity holds "locally" in the sense of [58], and we will derive the corresponding straightforward extension in the next section, while starting here with exact periodicity. Our homogenization Ansatz is then that each main unknown of the microscopic electrophysiology problem defined on the reference domain is the sum of…”

“…The homogenized equations obtained above can be shown to also hold in the framework of so-called locally periodic media. We refer to [58] for a mathematical definition and treatment of locally periodic homogenization by 2-scale convergence. This framework is particularly well-suited to the modeling of the heart because of its fiber structure of smoothly varying orientation, see Figure 4, and we refer e.g.…”

“…to [8] for a detailed mathematical description of this structure . In terms of homogenization this corresponds to assuming that the cell geometry Y is a regular function of ξ with "slow variations", see [58] for a precise definition. Applying these concepts, the above homogenized equations (51)-(54) are exactly preserved, up to the fact that A m , |Y (α) |, |Y | and T (α) are now functions of ξ.…”