A stochastic hierarchical model for low grade glioma evolution

TL;DR

This paper introduces a stochastic hierarchical model combining cellular-level diffusion processes and macroscopic equations to understand the progression of low-grade gliomas, focusing on how microscopic and macroscopic parameters influence tumor evolution.

Contribution

It develops a novel multiscale model linking cellular diffusion dynamics with tumor progression, providing insights into the transition from low to high-grade gliomas.

Findings

The jump rate function significantly affects tumor progression.

Diffusion coefficients influence the diffusive behavior of glioma cells.

Numerical tests reveal the relationship between microscopic parameters and malignancy onset.

Abstract

A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using piecewise diffusion Markov processes (PDifMPs) at the cellular level, we derive an equation for the density of the transition probability of this Markov process using the generalised Fokker-Planck equation. Then a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.