title: A complete mathematical study of a 3D model of heterogeneous and anisotropic glioma evolution.
abstract: Glioma is the most aggressive type of brain cancer. Several mathematical models have been developed towards identifying the mechanism of tumor growth. The most successful models have used variations of the diffusion-reaction equation, with the recent ones taking into account brain tissue 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 related heterogeneity and anisotropy issues. To this end, this paper introduces a complete mathematical framework on how to derive the solution of the equation using different numerical approximation of finite differences. It indicates how different proliferation rate schemes can be incorporated in this solution and presents a comparative study of different numerical approaches.
authors: Institute of Computer Science, Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. roniotis@ics.forth.gr
authors: Alexandros
authors: A
authors: Roniotis
authors: Kostas
authors: K
authors: Marias
authors: Vangelis
authors: V
authors: Sakkalis
authors: George D
authors: GD
authors: Tsibidis
authors: Michalis
authors: M
authors: Zervakis
mesh_terms: D000465:Algorithms; D016880:Anisotropy; D001932:Brain Neoplasms / Q000175:diagnosis* / Q000473:pathology; D002465:Cell Movement; D019295:Computational Biology / Q000379:methods*; D003198:Computer Simulation; D003936:Diagnosis, Computer-Assisted / Q000379:methods; D004058:Diffusion; D005910:Glioma / Q000175:diagnosis* / Q000473:pathology; D006801:Humans; D021621:Imaging, Three-Dimensional; D016014:Linear Models; D008962:Models, Theoretical; D015203:Reproducibility of Results; D013997:Time Factors