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 making use of DTI data and accounting 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 w.r.t. all 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 investigate the behavior of solutions w.r.t. various scenarios of tissue dynamics and the dominance of each of the tactic terms, also suggesting how the model can be used to perform a numerical necrosis-based tumor grading or support radiotherapy planning by dose painting. We also provide a discussion about alternative ways of including cell level environmental influences in such multiscale modeling approach, ultimately leading in the macroscopic limit to (multiple) taxis.

Multiscale modeling of glioma pseudopalisades: contributions from the tumor microenvironment

“…Ω ⊂ R N is considered to be a bounded domain with sufficiently smooth boundary ∂ Ω, all involved constants are positive, M 0 ∈ L ∞ (Ω), M 0 ≡ 0, M 0 , S 0 ≥ 0, and S 0 ∈ W 1,∞ (Ω). The no-flux boundary conditions are obtained through the upscaling procedure (as done e.g., in [14] for a related problem). For the tumor diffusion tensor D T we require Assumption 5.1…”

Mathematical modeling of glioma invasion: acid- and vasculature mediated go-or-grow dichotomy and the influence of tissue anisotropy

“…histological patterns [92] or tumor size [75]), however, we focus here on grading by the amount of necrosis relative to the whole tumor volume, in view of [29,36], where the tumor volume by itself was found to have no influence on overall survival. Following [15], we define the time-dependent grade G(t) ∈ [0, 1] of the simulated tumor via:…”

“…these parameters. The tumor and EC densities are plotted after 560 days of evolution for three different values of λ 0 (expressed in s −1 ), i.e., 10 −4 , 10 −3 and 10 −2 , and for four pairs (s, σ) of speed values (expressed in µm• h −1 ), i.e., (15,20), (20,15), (30,20), and (30,25). The simulations suggest that vascularization at the tumor site requires a sufficiently large glioma turning rate λ 0 accompanied by relatively large EC speed σ.…”

“…(", $) (15,20), (20,15), (30,20), and (30,25). All values belong to the parameter ranges reported in Appendix A.…”

“…Figure 4: Comparison between the evolution of tumor (first row of each box) and endothelial cells (second row of each box) for three different values of λ0 (s −1 ): 10 −4 , 10 −3 and 10 −2 , and four pairs (s, σ) of speed values (µm• h −1 ):(15,20),(20,15),(30,20), and(30,25). All values belong to the parameter ranges reported in Appendix A.…”