Inverse Mathematical Models for Brain Tumour Growth

This equation describes the change over time of the normalised tumour cell density u as a consequence of two biological phenomena: proliferation and diffusion.

We discuss a mathematical method for the inverse problem of locating the brain tumour source (origin) based on the reaction-diffusion model. Our approach consists in recovering the initial spatial distribution of the tumour cells  starting from a later state , which can be given by a medical image. We use the nonlinear Landweber regularization method to solve the inverse problem as a sequence of well-posed forward problems.

We give full 3-dimensional simulations of the tumour in time on two types of data, the 3d Shepp-Logan phantom and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository (IBSR). These simulations are obtained using standard finite difference discretisation of the space and time-derivatives, generating a simplistic approach that performs well. We also give a variational formulation for the model to open the possibility of alternative derivations and modifications of the model. Simulations with synthetic images show the accuracy of our approach for locating brain tumour sources.