A multiphase multiscale model for nutrient limited tissue growth, Part II: A simplified description

In this paper, we revisit and extend our recent work (Holden et al. (2018) A multiphase multiscale model for nutrient limited tissue growth, The ANZIAM Journal, 59(4), 499–532), that considers the derivation of an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. 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, via so-called unit cell problems that provide permeability tensors to parameterise the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell-problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.