A continuum mechanical model to predict the growth of Glioblastoma Multiforme and the deformation of white matter tracts

Glioblastoma Multiforme (GBM) is one of the most malignant types of brain cancer and it exhibits a strong resistance to common therapies. Therefore, it is crucial to investigate its progression, in order to acquire more details about its anisotropic nature, which follows the orientation of surrounding white matter tracts. Mathematical models of cerebral tumour growth can help in understanding the physiology and the progression of this disease, for the purpose of predicting the evolution of tumour shape and volume and quantifying its aggressiveness. The aim of the present work is to reproduce the evolution of this highly malignant brain tumour evaluating its mechanical impact on the surrounding healthy tissue. A mathematical multiphase model for GBM, based on Continuum Mechanics, is developed, where both the healthy and the diseased regions are treated as a saturated biphasic mixture, comprising a solid and a fluid phase. Moreover, it is considered the region occupied by the tumour as separated from the host tissue by a sharp moving interface. The cell phase is supposed to behave as a Mooney-Rivlin hyperelastic solid, with different material parameters between the healthy and the diseased zone. Instead, the liquid phase is considered constitutively as an ideal fluid. With the aim to describe the mechanical effect of tumour growth onto tissue deformation, theory for materials with evolving natural configurations and the multiplicative decomposition of the deformation gradient tensor are employed. For what concerns the growth tensor, which appears in this decomposition, we focus on its anisotropic evolution, in order to enforce the different cases of monodirectional, planar and spherical growth. Furthermore, it is necessary to introduce in the model an equation describing the evolution of nutrients in the domain, since their amount affects the cells capability to duplicate. The preferential directions for nutrient diffusion and cancer cell motion and growth are obtained, at the initial time step, through DTI imaging. Then, in order to take into account the modification of the preferred directions according to the brain tissue deformation, a push-forward of the corresponding Lagrangian tensor is performed. After having set the mechanical model for Glioblastoma growth, we solve it through numerical simulations. For this purpose, the Lagrangian formulation is derived from the Eulerian model. Later, a weak formulation of the Lagrangian model is obtained in order to numerically solve the model using FEniCS, a Python-based PDE finite element solver. At the beginning, the code is tested on a simplified geometry in order to verify its stability and effectiveness. Afterwards, the numerical simulations on the real three-dimensional brain geometry are performed, using available data from MRI and DTI to build the computational domain and account for patient-specific anisotropy. From a numerical point of view, the obtained algorithm is stable and it allows to represent discontinuous deformation gradients, through the use of a mesh conforming to the material host-tumour interface. On the other hand, from a modelling point of view, with respect to available models in the literature, the anisotropy has also been included in tumour growth and the model is able to describe how the brain tracts are modified due to the tumour mass expansion.