Mathematical modelling of growth, reorganisation and active response in cells and tissues

The understanding of biological systems like cells and tissues is increasingly taking advantage of tools from quantitative sciences, which can provide powerful instruments to unravel complex mechanisms. In this respect, the emerging field of mechanobiology, that aims at analysing how mechanics affects the cellular and sub-cellular processes, represents a prominent example in which a combination of efforts from several disciplines is fundamental. The contribution of mathematical models to the description of biomedical phenomena can provide a notable support to the research process: indeed, they can be fruitfully employed to reproduce physiological and pathological conditions in silico, and to perform simulations to support clinical observations. Stimulated by these facts, in this Thesis we introduce some models to capture reorganisation, growth, and active processes in cells and tissues, with particular emphasis on their mechanical behaviour. First of all, we deal with the response of cells to external mechanical stimuli, motivated by experimental evidence showing that cells are able to reorganise their cytoskeleton as a reaction to external forces or deformations. Indeed, if cells are placed on a substrate that is cyclically stretched, a reorientation of the cytoskeletal fibres is observed, until a stable configuration is reached. To get insight into such a reorganisation process, which is relevant in tissue engineering and in the understanding of some diseases, we propose different types of models grounded on Continuum Mechanics. After a review of the experimental and modelling literature, we employ tools from nonlinear elasticity, active remodelling, and linear viscoelasticity to capture some relevant experimental observations. We find that strain energies belonging to a very general class all give rise to the same preferential orientations of cells on two-dimensional substrates, corresponding to the observed ones. Moreover, a remodelling framework for an anisotropic material with two fibre families is introduced and applied to the problem of cytoskeletal alignment. Viscoelastic effects are then considered to capture the effect of the deformation frequency on the cell realignment process. Then, we consider another problem related to the cellular response to mechanical stimuli, which is the active contractility of axons. In fact, experimental evidence suggests that the axonal cortex, i.e., the external coating of axons, is able to actively contract and to exert compression on the inner part. This capability seems related to the active regulation of the axon diameter which is observed in some experiments. We describe these phenomena by following an active strain approach, in which both the circumferential and axial contractility of the axonal cortex are considered. A model is derived on a thermodynamically consistent basis and used to simulate the stretching of axons and drug-induced alterations of their cytoskeletal structure, showing a good agreement with experiments. Finally, at the tissue scale, we address the problem of providing a mechanical description of brain tumour growth inside the brain. In fact, the effect of solid stresses in addition to fluid pressure has been proved to be harmful for patients. These negative repercussions are also amplified in brain tissue, which is extremely soft and confined by the skull. To study these issues, we propose a mathematical model of tumour growth based on mixture theory, to account for solid and fluid components, and morphoelasticity, to describe growth-related distortions. Both the healthy brain tissue and the tumour are treated as hyperelastic solids, so as to quantify the displacement and stress induced by cancer growth. Moreover, we perform simulations on a realistic brain geometry, reconstructed from patient-specific data, to underline the importance of a detailed mechanical description of brain tumour growth.