Brain tumours frequently grow along the fibres of the white matter or along vessels. Depending on the amount and orientation of such preferential pathways, brain cancers may appear very different from an individual to another and may invade different neurological areas. Furthermore, the tumour growth-induced compression may also corrupt brain functions in the healthy region and constrict the flow of cerebrospinal fluid in the brain ventricles. It is therefore important to reproduce through mathematical and computational models the patient-specific heterogeneity of brain microstructure and the mechanical behaviour of the brain tissue in order to correctly predict the progression of the pathology, the extent of the injured areas, and the effect of therapies. Motivated by these needs, we develop a mathematical multiphase model that explicitly includes brain hyperelasticity and couples solid and fluid stresses with brain tumour anisotropic growth and healthy tissue reorganization. The simulations of our model are implemented in a personalised three-dimensional framework that allows the incorporation of the realistic brain geometry, and the patient-specific diffusion and permeability tensors reconstructed from neuroimaging data. Our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding tissue, quantifying the displacements inside the healthy region and the deformation of brain ventricles, and to modify the preferential directions of diffusion and cell motion as a consequence of the mechanical deformation. Finally, we also account for tumour response to radiation therapy and chemotherapy.