Modelling three-dimensional growth of brain tumours from time series of scans

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 formulati on of mathematical models that an analysis of the spatio-temporal tumour gr owth revealed on each patient serial scans can lead to a quantification of parameters characterising the proliferative and expensive dynamic of the br ain tumour. This paper reviews some of the results and limitations encounte red in modelling the different stages of a brain tumour growth, namely befo re 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 tum our growth patterns obtained with various combinations of nonlinear growth rates and cellular diffusion laws are considered and compared to real MRI s cans taken in a patient with a glioblastoma and having undergone radiothera py. From these simulations, we characterise the effects of different therap ies on survival durations, with special attention to the effect of cell dif fusion inside the resected brain region when surgical resection of the tumo ur is carried out.