Multiphase modelling of glioma pseudopalisading under acidosis

We propose a multiphase modeling approach to describe glioma pseudopalisade patterning under the influence of acidosis. The phases considered at the model onset are glioma, normal tissue, necrotic matter, and interstitial fluid in a void-free volume with acidity represented by proton concentration. We start from mass and momentum balance to characterize the respective volume fractions and deduce reaction-cross diffusion equations for the space-time evolution of glioma, normal tissue, and necrosis. These are supplemented with a reaction-diffusion equation for the acidity dynamics and lead to formation of patterns which are typical for high grade gliomas. Unlike previous works, our deduction also works in higher dimensions and involves less restrictions. We also investigate the existence of weak solutions to the obtained system of equations and perform numerical simulations to illustrate the solution behavior and the pattern occurrence.