[Paper Review] Multiscale modeling of glioma pseudopalisades: contributions from the tumor microenvironment

Pawan Kumar, Jing Li+1 more|arXiv (Cornell University)|Jan 1, 2020

Mathematical Biology Tumor Growth57 references19 citations

TL;DR

This paper proposes a multiscale kinetic model of glioblastoma pseudopalisades, deriving a reaction-diffusion-taxis system with pH-taxis from active particle dynamics. Simulations show that pseudopalisade-like patterns emerge under parabolic scaling (undirected tissue), but not under hyperbolic scaling (directed tissue), suggesting brain tissue may be undirected for glioma migration, and that these patterns are not Turing-type but arise from degenerate diffusion and repellent pH-taxis.

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 patterns1 (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.

Motivation & Objective

• To understand the formation of glioma pseudopalisades, a hallmark of glioblastoma multiforme (GBM), through a mechanistic multiscale model.
• To investigate how hypoxia-induced acidity and tumor microenvironment interactions drive pattern formation in GBM.
• To determine whether pseudopalisades are Turing-type patterns or emerge via alternative mechanisms such as pH-taxis and degenerate diffusion.
• To compare the impact of different scaling limits (parabolic vs. hyperbolic) on macroscopic pattern formation in glioma dynamics.
• To establish mathematical rigor by proving global existence and boundedness of classical solutions for the derived macroscopic system.

Proposed method

• The model is derived from the kinetic theory of active particles (KTAP), incorporating cell-level responses to hypoxia and acidity.
• An upscaling process transforms mesoscopic particle dynamics into a macroscopic system of reaction-diffusion-taxis partial differential equations (PDEs).
• The resulting PDE system includes a repellent pH-taxis term, modeling glioma cells migrating away from acidic, hypoxic regions.
• Two scaling limits are analyzed: parabolic (for undirected tissue) and hyperbolic (for directed tissue), leading to different macroscopic PDEs.
• Mathematical analysis proves existence and uniqueness of global bounded classical solutions for the macroscopic system.
• Numerical simulations compare pattern formation under different diffusion coefficient profiles, including degenerate and non-degenerate cases.

Experimental results

Research questions

• RQ1 Can pseudopalisade-like patterns in glioblastoma be explained by a multiscale model incorporating pH-taxis and hypoxia-induced cell migration?
• RQ2 Is the formation of pseudopalisades a Turing-type instability, or does it result from non-Turing mechanisms such as degenerate diffusion and repellent taxis?
• RQ3 How does the choice of scaling (parabolic vs. hyperbolic) affect the emergence of spatial patterns in glioma cell density?
• RQ4 What role does tissue anisotropy or directionality play in glioma migration, as inferred from the model’s scaling limits?
• RQ5 How do different cell-level responses to hypoxia (e.g., metabolic shift, protease secretion) influence the macroscopic pattern formation?

Key findings

• Pseudopalisade-like patterns form in simulations under parabolic scaling (undirected tissue), indicating that brain tissue may be effectively undirected for glioma migration.
• No such patterns emerge under hyperbolic scaling (directed tissue), suggesting that directed tissue structures do not support pseudopalisade formation in this model.
• The patterns observed are not of Turing type, as the condition for Turing instability (negative determinant in linear stability analysis) cannot be satisfied for biologically relevant parameters, especially when α < 1 (typical for GBM).
• Repellent pH-taxis enhances the likelihood of pattern formation but does not enable Turing instability under standard parameter ranges.
• Strongly degenerate diffusion coefficients (e.g., vanishing on subintervals) promote the formation of localized hypercellular structures resembling pseudopalisades.
• The model proves global existence and boundedness of classical solutions for the derived macroscopic PDE system, ensuring mathematical well-posedness.