The Continuous Mathematics Based Glioblastoma Oncosimulator: Application of an explicit three dimensional numerical treatment of the skull-glioblastoma Neumann boundary condition on real anatomical data

The Continuous Mathematics Based Glioblastoma Oncosimulator is a platform for simulating, investigating, better understanding, and exploring the natural phenomenon of glioma tumor growth. Modelling of the diffusive-invasive behaviour of glioma tumour growth may have considerable therapeutic implications. A crucial component of the corresponding computational problem is the numerical treatment of the adiabatic Neumann boundary conditions imposed by the skull on the diffusive growth of gliomas and in particular glioblastoma multiforme (GBM). In order to become clinically acceptable such a numerical handling should ensure that no potentially life-threatening glioma cells disappear artificially due to oversimplifying assumptions applied to the simulated region boundaries. However, no explicit numerical treatment of the 3D boundary conditions under consideration has appeared in the literature to the best of the authors´ knowledge. Therefore, this paper aims at providing an outline of a novel, explicit and thorough numerical solution to this problem. Additionally, a brief exposition of the numerical solution process for a homogeneous approximation of glioma diffusion-invasion using the Crank - Nicolson technique in conjunction with the Conjugate Gradient system solver is outlined. The entire mathematical and numerical treatment is also in principle applicable to mathematically similar physical, chemical and biological diffusion based spatiotemporal phenomena which take place in other domains for example embryonic growth and general tissue growth and tissue differentiation. A comparison of the numerical solution for the special case of pure diffusion in the absence of boundary conditions with its analytical counterpart has been made. In silico experimentation with various adiabatic boundary geometries and non zero net tumour growth rate support the validity of the corresponding mathematical treatment. Through numerical experimentation on a set of real brain imaging data,- a simulated tumour has shown to satisfy the expected macroscopic behaviour of glioblastoma multiforme, on concrete published clinical imaging data, including the adiabatic behaviour of the skull. The paper concludes with a number of remarks pertaining to the potential and the limitations of the diffusion-reaction approach to modelling multiscale malignant tumour dynamics.