Abstract

The derivation of continuum models which represent underlying discrete or microscale phenomena is emerging as an important part of mathematical biology: integration between subcellular, cellular and tissue-level behaviour is crucial to understanding tissue growth and mechanics. I will consider the application of a multiscale method to two problems on this theme.

Firstly a new macroscale description of nutrient-limited tissue growth, which is formulated as a microscale moving-boundary problem within a porous medium, is introduced. A multiscale homogenisation method is employed to enable explicit accommodation of the influence of the underlying microscale tissue structure, and its evolution, on the macroscale dynamics.

A challenging consideration in continuum models of tissue is the accommodation of (spatially-discrete) cell-signalling events, a feature of which being the progression of moving fronts of cell-signalling activity across a lattice. New (continuum) analyses of monotone waves in a discrete diffusion equation are presented, and extended to modulated fronts exhibited in cell signalling models.