Modeling glioblastoma growth and inhomogeneous tumor invasion with explicitly numerically treated neumann boundary conditions

A couple of multiscale spatiotemporal simulation models of glioblastoma multiforme (GBM) growth and invasion into the surrounding normal brain tissue is presented. Both models are based on a continuous and subsequently finite mathematical approach centered around the non-linear partial differential equation of diffusion-reaction referring to glioma tumour cells. A novel explicit, strict and thorough numerical treatment of the three dimensional adiabatic Neumann boundary conditions imposed by the skull is also included in both models. The first model assumes a homogeneous representation of normal brain tissue whereas the second one, assuming an inhomogeneous representation of normal brain tissue, distinguishes between white matter, grey matter and cerebrospinal fluid. The predictions of the tumour doubling time by both models are compared for specific data sets. Clinical observational data regarding the range of the GBM doubling time values are utilized in order to ensure the realism of both models and their predictions. We assume that the inhomogeneous normal brain tissue representation is a virtual rendering of reality more credible than its homogeneous counterpart. The simulation results for the cases considered show that using the homogeneous normal brain based model may lead to an error of up to 10% for the first 25 simulated days in relation to the predictions of the inhomogeneous model. However, the error drops to less than 7% afterwards. This observation suggests that even by using a homogeneous brain based model and a realistic weighted average value of its diffusion coefficient, a rough but still informative estimate of the expected tumour doubling time can be achieved. Additional in silico experimentation aiming at statistically testing and eventually further supporting the validity of this hypothesis is in progress. It is noted that the values of the diffusion coefficients and the cell birth and death rates of the model are amenable to refinement and personalization by exploiting the histological and molecular profile of the patient. Work on this aspect is in progress.