Modeling glioma invasion with anisotropy- and hypoxia-triggered motility enhancement: From subcellular dynamics to macroscopic PDEs with multiple taxis

Abstract

We deduce a model for glioma invasion that accounts for the dynamics of brain tissue being actively degraded by tumor cells via excessive acidity production, but also according to the local orientation of tissue fibers. Our approach has a multiscale character: we start with a microscopic description of single cell dynamics including biochemical and/or biophysical effects of the tumor microenvironment, translated on the one hand into cell stress and corresponding forces and on the other hand into receptor binding dynamics. These lead on the mesoscopic level to kinetic equations involving transport terms with respect to all considered kinetic variables and eventually, by appropriate upscaling, to a macroscopic reaction–diffusion equation for glioma density with multiple taxis, coupled to (integro-)differential equations characterizing the evolution of acidity and macro- and mesoscopic tissue. Our approach also allows for a switch between fast and slower moving regimes, according to the local tissue anisotropy. We perform numerical simulations to assess the importance of each tactic term and investigate the influence of two models for tissue dynamics on the tumor shape. We also suggest how the model can be used to perform a numerical necrosis-based tumor grading or support radiotherapy planning by dose painting. Finally, we discuss alternative ways of including cell level environmental influences in such a multiscale modeling approach, ultimately leading in the macroscopic limit to (multiple) taxis.