Multiphase modelling of glioma pseudopalisading under acidosis

Pawan Kumar; Christina Surulescu; Anna Zhigun; Pawan Kumar; Christina Surulescu; Anna Zhigun

doi:10.3934/mine.2022049

Mathematics in Engineering

2022, Volume 4, Issue 6: 1-28. doi: 10.3934/mine.2022049

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Multiphase modelling of glioma pseudopalisading under acidosis

1.
Felix-Klein-Zentrum für Mathematik, Technische Universität Kaiserslautern, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany
2.
School of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT7 1NN, Northern Ireland, UK

Received: 25 May 2021 Accepted: 26 August 2021 Published: 16 November 2021
MSC : 35K20, 35K55, 35Q92, 92C15, 92C50

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.
- glioma pseudopalisade patterns,
- multiphase model,
- reaction-cross diffusion equations
Citation: Pawan Kumar, Christina Surulescu, Anna Zhigun. Multiphase modelling of glioma pseudopalisading under acidosis[J]. Mathematics in Engineering, 2022, 4(6): 1-28. doi: 10.3934/mine.2022049

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Abstract

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.

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