A new concept for multiscale modelling of growing cell cultures

Abstract

Dynamics of growing or actively deforming tissues in biological systems cannot be understood without taking mechanical interactions into account. In this work, we propose a new multiscale approach for modelling growing tissues using analytical up-scaling techniques originally developed for crystals. Adopting this approach corresponding macroscopic continuum models can be derived on the basis of the microscopic models (individual based models). Assuming isotropy these macroscopic models based on energy functionals can be formulated in the framework of multiple natural configurations often used in modelling growing tissues. In the case of anisotropic growth our ansatz shows that constitutive relations depending only on mechanical deformations, as in the case of isotropic growth, are not sufficient. They depend also on the growth itself. The explicit form of the dependence can be recovered via homogenisation formulae inheriting most details of the microscopic models. This new concept of a multiscale modelling approach unifying individual based and sub-cellular element models with continuum models offers an new perspective for mathematical modelling and simulation in many biological systems.