Multiscale modeling of glioma pseudopalisades: contributions from the tumor microenvironment

Abstract

Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around the occlusion site of a capillary are typical for GBM and hint on poor prognosis of patient survival. We propose a multiscale modeling approach in the kinetic theory of active particles framework and deduce by an upscaling process a reaction-diffusion model with repellent pH-taxis. We prove existence of a unique global bounded classical solution for a version of the obtained macroscopic system and investigate the asymptotic behavior of the solution. Moreover, we study two different types of scaling and compare the behavior of the obtained macroscopic PDEs by way of simulations. These show that patterns (not necessarily of Turing type), including pseudopalisades, can be formed for some parameter ranges, in accordance with the tumor grade. This is true when the PDEs are obtained via parabolic scaling (undirected tissue), while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related .

Keywords: Directed/undirected tissue; Glioblastoma; Global existence; Hypoxia-induced tumor behavior; Kinetic transport equations; Long time behavior; Multiscale modeling; Pseudopalisade patterns; Reaction-diffusion-taxis equations; Uniqueness; Upscaling.

Fig. 1

Initial conditions. Upper row: set (4.8) for tumor cell density a and acidity distribution b, lower row: set (4.9) for tumor cell density c and acidity distribution d

Fig. 2

a: Macroscopic tissue density (Experiments 1, 2) and b: mesoscopic tissue distribution for Experiment 2, for a given fiber direction

Fig. 3

Tumor (upper row) and acidity (lower row) at several times for Experiment 1 and initial conditions (4.8)

Fig. 4

Tumor (upper row) and acidity (lower row) at several times for Experiment 1 and initial conditions (4.9)

Fig. 5

Tumor (upper row) and acidity (lower row) at several times for Experiment 2 and initial conditions (4.8)

Fig. 6

Tumor (upper row) and acidity (lower row) at several times for Experiment 2 and initial conditions (4.9)

Fig. 7

Tumor (upper row) and acidity (lower row) at several times for Experiment 2, initial conditions (4.9), and stronger proton buffering

Fig. 8

Tumor (upper row) and acidity (lower row) at several times for Experiment 2, initial conditions (4.9), and source term ${\mu }_0(1-M)\frac{M}{1+S}$ instead of that in (4.2). All parameters as in Table 1

Fig. 9

Difference between tumor (upper row) and acidity (lower row) at several times computed for System (4.2), (4.3) with and without pH-taxis in the framework of Experiment 2, initial conditions (4.9)

Fig. 10

Mean fiber orientation ${\mathbb{E}}_q$ a and zoom near crossing of fiber strands b for $q_h$ as in (4.6) with $\delta =0.2$. c: mesoscopic tissue density $q_h$ for direction $\xi =\pi /2$

Fig. 11

Tumor (upper row) and acidity (lower row) at several times for $q_h$ as in (4.6) with $\delta =0.2$ and initial conditions (4.9). Solutions of system (3.30), (2.9) obtained by hyperbolic scaling

Fig. 12

Zoomed mean fiber orientation ${\mathbb{E}}_q$ (a), fractional anisotropy FA b for $q_h$ as in (4.6) with $\delta =1$. Subfigure c: mesoscopic tissue density $q_h$ for direction $\xi =\pi /2$

Fig. 13

Tumor (upper row) and acidity (lower row) at several times for $q_h$ as in (4.6) with $\delta =1$ and initial conditions (4.9). Solutions of system (3.30), (2.9) obtained by hyperbolic scaling

Fig. 14

Patterns (in 1D) for tumor (left column) and acidity (middle column) for several choices of the tumor diffusion coefficient $D_T(x)$ (plots of the latter shown in the right column). Simulations are done for the dimension-free system (4.2), (4.3), the maximum simulation time corresponds to 10 months in dimensional framework