Multiscale Modelling, Analysis, and Simulation in Mechanobiology

Abstract

The main object of this thesis is the rigorous derivation of continuum models in mechanobiology via multiscale analysis. On the microscopic level, models in terms of energy functionals defined on networks / lattices are considered. Using concepts of Gamma-convergence rigorous convergence results as well as explicit homogenisation formulae can be derived. Based on a characterisation via energy functionals, appropriate macroscopic stress-strain relationships (constitutive equations) are determined. Mechanics of the membrane-bound cytoskeleton of red blood cells, and accordingly mechanics of red blood cells, are considered as one test case. The rigorous derivation of a macroscopic continuum model is based on a realistic discrete microscopic model. Simulations of optical tweezer experiments confirm the model qualitatively as well as quantitatively. For these simulations an appropriate computational framework for single cell mechanics is developed using finite element methods. It accounts explicitly for membrane mechanics and its coupling with bulk mechanics. The approach is highly flexible and can be generalised to many other cell models, also including biochemical control. As a test case considering the interactions between biological processes and mechanics, growing cell cultures are investigated. From a discrete cellular-automaton-like description macroscopic continuum models are derived. Furthermore, it is shown that the models can account for branching morphogenesis - a typical phenomenon observed in growing cell cultures, where growth is promoted by a diffusing substance.