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Glioma is one of the most aggressive types of brain tumor. Several mathematical models have been developed during the past two decades, toward simulating the mechanisms that govern the development of glioma. The most common models use the diffusion-reaction equation (DRE) for simulating the spatiotemporal variation of tumor cell concentration. Nevertheless, despite the applications presented, there has been little work on studying the details of the mathematical solution and implementation of the 3-D diffusion model and presenting a qualitative analysis of the algorithmic results. This paper presents a complete mathematical framework on the solution of the DRE using different numerical schemes. This framework takes into account all characteristics of the latest models, such as brain tissue heterogeneity, anisotropic tumor cell migration, chemotherapy, and resection modeling. The different numerical schemes presented have been evaluated based upon the degree to which the DRE exact solution is approximated. Experiments have been conducted both on real datasets and a test case for which there is a known algebraic expression of the solution. Thus, it is possible to calculate the accuracy of the different models.

Glioma, especially glioblastoma, is a leading cause of brain cancer fatality involving highly invasive and neoplastic growth. Diffusive models of glioma growth use variations of the diffusion-reaction equation in order to simulate the invasive patterns of glioma cells by approximating the spatiotemporal change of glioma cell concentration. The most advanced diffusive models take into consideration the heterogeneous velocity of glioma in gray and white matter, by using two different discrete diffusion coefficients in these areas. Moreover, by using diffusion tensor imaging (DTI), they simulate the anisotropic migration of glioma cells, which is facilitated along white fibers, assuming diffusion tensors with different diffusion coefficients along each candidate direction of growth. Our study extends this concept by fully exploiting the proportions of white and gray matter extracted by normal brain atlases, rather than discretizing diffusion coefficients. Moreover, the proportions of white and gray matter, as well as the diffusion tensors, are extracted by the respective atlases; thus, no DTI processing is needed. Finally, we applied this novel glioma growth model on real data and the results indicate that prognostication rates can be improved., (© 2012 IEEE)