Mathematical Models and Methods related to Cell–Extracellular Matrix Interactions

This Doctor of Philosophy Dissertation in Matematica per le Scienze dell’Ingegneria is composed by two distinct parts having in common the study of the interactions between cells and extracellular-matrix. This is the result of a three year working project supervised by professors Davide Ambrosi (Politecnico di Milano) and Luigi Preziosi (Po- litecnico di Torino). In the spirit of the aims and scopes of the Doctoral degree, both modelling and applications of analytical/numerical methods constitute the backbone of such a work; in fact, the first part emphasizes the latter aspect whilst the second part is more focused on the former. Specifically: 1. Part I deals with the mathematical aspects of Force Traction Microscopy. This is an inversion method that allows to reconstruct the stress field acting on a substratum knowing the displacement of the substrate itself, measured at some points. The formulation of this problem in the Inverse/Optimal Control framework together with its analytical and numerical study constitutes the real aim of this part of dissertation. Remarkably, some contents of this part had been submitted [70]. 2. Part II deals with the mathematical modeling of cell-tissue adhesion. Such an adhesive property of living tissue seems to be of importance in the developing of biological process. Moreover, quite a huge amount of experimental data can be found in the specialistic literature. A Continuum Mechanics point of view is adopted to describe adhesion and the outcome of our model is qualitatively compared with experiments. Remarkably, some contents of this part had been published [59].