The mathematical path to develop a heterogeneous, anisotropic and 3-dimensional glioma model using finite differences

Several mathematical models have been developed to express glioma growth behavior. The most successful models have used the diffusion-reaction equation, with the most recent ones taking into account spatial heterogeneity and anisotropy. However, to the best of our knowledge, there hasn´t been any work studying in detail the mathematical solution and implementation of the 3D diffusion model, addressing all related heterogeneity and anisotropy issues. This paper presents a complete mathematical framework on how to derive the solution of the equation using different numerical schemes of finite differences. Moreover, the derived mathematics can be customized to incorporate various cell proliferation schemes. Lastly, a comparative study of the numerical scheme helps us select the best of them and then apply it to real clinical data.