A tumor invasion model for heterogeneous cancer cell populations: mathematical analysis and numerical methods


Tumor invasion of tissue is the first step in cancer metastasis and thus a process that is co-responsible for most deaths in cancer patients. In this thesis we consider two macroscopic tumor invasion models from the literature and derive a third one that takes the presence of recently discovered stem-like cells within tumors into account. The modeling considers microscopic events on the cancer cell receptors and tissue remodeling by fibroblast cells. In addition to the model derivation, we prove global in time existence of classical solutions in two space dimensions for a slightly simplified version of the new model. We show in numerical experiments in one and two space dimensions that the new model can qualitatively reproduce the biomedical understanding of the invasion by the two considered types of cancer cells.Since Keller-Segel type models of tumor invasion such as our new model are associated with very rich dynamics, their numerical simulation requires elaborate methods. These constitute another core theme of this thesis. Here we design and study, in particular, combined finite volumes/finite differences with implicit-explicit time discretization in the first place, adaptive mesh refinement methods in the second place and a new mass-transport scheme in the third place.

PhD Thesis