Chapter 12 - Microstructural Influences on Growth and Transport in Biological Tissue—A Multiscale Description

Abstract

The detailed understanding of growth and transport dynamics within biological tissue is made particularly challenging by the complex and multiscale nature of this medium. For this reason so-called effective descriptions are frequently sought. These offer coarse-scale models that still accommodate aspects of microscale dynamics. When considering tissue growth, such formulations must accommodate the continuous growth and remodeling processes that occur in response to environmental cues. As a model system for investigating relevant phenomena, in this chapter we consider nutrient-limited growth of a porous medium (with broad application to vascularized tumor growth). Using asymptotic homogenization we derive the macroscale equations that describe a ‘double porous medium’ whose flow is influenced by both the tissue microstructure and growth that occurs in response to nutrient transport governed by an advection–reaction equation. The coupled flow and transport dynamics are demonstrated by numerical experiments indicating the influence of microscale structure and transport phenomena on the macroscale dynamics. The importance of slip, tortuosity, and of nutrient-limited growth are considered.