A simplified multiphase multiscale model for tissue growth

TL;DR

This paper develops a simplified multiscale model for tissue growth within porous scaffolds, enabling easier computation by decoupling microscale cell problems from macroscale tissue dynamics.

Contribution

It introduces a linearized, decoupled multiscale model that reduces computational complexity while capturing key tissue growth features within scaffolds.

Findings

The model includes Darcy flow and nutrient concentration equations.

Decoupling of cell problems simplifies multiscale analysis.

The approach retains microscale structure influence on macroscale behavior.

Abstract

In this paper, we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. As in our recent work (Holden \textit{et al.} "A multiphase multiscale model for nutrient limited tissue growth", The ANZIAM Journal, 2018, doi:10.1017/S1446181118000044) the underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearisation of the underlying multiphase model (whose nonlinearity poses significant challenge for such analyses), we obtain, by means of multiple-scales homogenisation, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics. The model we obtain comprises Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. These are coupled to underlying Stokes-type cell problems that provide permeability tensors to parameterise the macroscale description. In Holden \textit{et al.}, the cell problems retain macroscale dependence, posing significant computational challenges; here, we obtain a decoupled system whereby the quasi-steady cell-problems may be solved separately from the macroscale description, thereby greatly reducing the complexity associated with fully-coupled multiscale descriptions. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.S