Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth

Abstract

We work with a well-known model of reaction–diffusion type for brain tumour growth and accomplish full 3-dimensional (3d) simulations of the tumour in time on two types of imaging data, the 3d Shepp–Logan head phantom image and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository. The source term is such that we have logistic growth. These simulations are obtained using standard finite difference approximations with novel calculations to increase speed and accuracy. Moreover, biological background to the model, its well-posedness together with a variational formulation are given. The variational formulation enable the feasibility of different derivations and modifications of the model.