The role of brain hyperelasticity in growth and proliferation of Glioblastoma Multiforme

Glioblastoma Multiforme (GBM) is a highly aggressive and malignant type of brain tumour. Besides the typical hallmarks of cancer, such as uncontrolled cellular proliferation and instability, GBM also exhibits dramatic invasive potential and resistance to common therapies: even with a complete treatment including neurosurgery, chemotherapy and radiotherapy, the median survival time is about 10-16 months. Hence, there is a critical need to understand and replicate the biological complexity of the brain, in order to predict tumour evolution and arrange therapeutic strategies accordingly. For that purpose, mathematical and computational models can provide powerful instruments for investigating GBM progression: in the last decades, several models that describe brain tumour growth have been proposed, using different frameworks and accounting for different characteristics. Nevertheless, the vast majority of these models does not consider realistic mechanical and constitutive properties of brain tissue, as well as the role of stress and deformations exerted by the growing tumour. Instead, the presence of a growing mass inside the brain - known as mass effect - may be critical and dangerous for the patient: it is then important to evaluate the mechanical impact of Glioblastoma on the surrounding healthy tissue. Starting from the state-of-the-art about brain tumour modeling, in this thesis we develop a mathematical multiphase model for GBM which includes brain hyperelasticity, in order to study the effects of structural changes, deformations and stress on brain tissue due to the presence of a growing tumour. In particular, we consider the region occupied by the tumour as separated from the host tissue by a sharp moving interface: both the healthy and the diseased regions are treated as a saturated biphasic mixture, comprising a solid and a fluid phase. The solid phase is described as a Mooney-Rivlin hyperelastic material, while the fluid motion is determined using Darcy's law with anisotropic permeability; in the tumour region, we introduce proliferation and account for deformations subsequent to it: to include the mechanical effect of growth on the tumour mass in addition to the pure elastic deformation, we employ the natural configurations framework and the multiplicative decomposition of the deformation gradient tensor. We also include in our model an equation describing the evolution of the concentration of available nutrients, which are transported by the fluid and can diffuse into the anisotropic brain tissue. The mathematical model is then numerically solved using FEniCS, a Python-based PDE finite element solver, at first in a simplified geometry, then in a three-dimensional brain geometry using available data from MRI and DTI to build the computational domain and account for diffusion anisotropy. In the end, results are analyzed to investigate the effect of deformations and unnatural displacement induced on brain tissue by the growing Glioblastoma. Future developments might be focused on the inclusion of elastic or viscoelastic constitutive models of the brain in a diffuse-interface Cahn-Hilliard-type approach. Multi-scale modelling might also be used to determine how structural changes and mechanical properties at the cellular level influence the parameters at the macroscopic scale and consequently the evolution of the tumour.