A detailed numerical treatment of the boundary conditions imposed by the skull on a diffusion–reaction model of glioma tumor growth. Clinical validation aspects

Abstract

The study of the diffusive behavior of glioma tumor growth is an active field of biomedical research with considerable therapeutic implications. An important aspect of the corresponding computational problem is the mathematical handling of boundary conditions. This paper aims at providing an explicit and thorough numerical formulation of the adiabatic Neumann boundary conditions imposed by the skull on the diffusive growth of gliomas and in particular on glioblastoma multiforme (GBM). Additionally, a detailed exposition of the numerical solution process for a homogeneous approximation of glioma invasion using the Crank–Nicolson technique in conjunction with the Conjugate Gradient system solver is provided. The entire mathematical and numerical treatment is also in principle applicable to mathematically similar physical, chemical and biological phenomena. A comparison of the numerical solution for the special case of pure diffusion in the absence of boundary conditions or equivalently in the presence of adiabatic boundaries placed in infinity with its analytical counterpart is presented. Numerical simulations for various adiabatic boundary geometries and non zero net tumor growth rate support the validity of the corresponding mathematical treatment. Through numerical experimentation on a set of real brain imaging data, a simulated tumor has shown to satisfy the expected macroscopic behavior of glioblastoma multiforme including the adiabatic behavior of the skull. The paper concludes with a number of remarks pertaining to both the biological problem addressed and the more generic diffusion–reaction context.