The development of brain tumours, after diagnosis, is routinely recorded by different medical imaging techniques like computerised tomography (CT) or magnetic resonance imaging (MRI). However, it is only through the formulation of mathematical models that an analysis of the spatio-temporal tumour growth revealed on each patient serial scans can lead to a quantification of parameters characterising the proliferative and expensive dynamic of the brain tumour. This paper reviews some of the results and limitations encountered in modelling the different stages of a brain tumour growth, namely before and after diagnosis and therapy. It extends an original two-dimensional approach by considering three-dimensional growth of brain tumours submitted to the spatial constraints exerted by the skull and ventricles boundaries. Considering the dynamic of both the pre- and post-diagnosis stages, the tumour growth patterns obtained with various combinations of nonlinear growth rates and cellular diffusion laws are considered and compared to real MRI scans taken in a patient with a glioblastoma and having undergone radiotherapy. From these simulations, we characterise the effects of different therapies on survival durations, with special attention to the effect of cell diffusion inside the resected brain region when surgical resection of the tumour is carried out.
Inverse Mathematical Models for Brain Tumour Growth
