Multiscale Modeling of Glioma Invasion: From Receptor Binding to Flux-Limited Macroscopic PDEs

Abstract

We propose a novel approach to modeling cell migration in an anisotropic environment with biochemical heterogeneity and interspecies interactions, using as a paradigm glioma invasion in brain tissue under the influence of hypoxia-triggered angiogenesis. The multiscale procedure links single-cell and mesoscopic dynamics with population level behavior, leading on the macroscopic scale to flux-limited glioma diffusion and multiple taxis. We verify the nonnegativity of regular solutions (provided they exist) to the obtained macroscopic PDE-ODE system and perform numerical simulations to illustrate the solution behavior under several scenarios.

Modelling non-local cell-cell adhesion: a multiscale approach

“…Similar to Zhigun and Surulescu ( 2022 ), Dietrich et al ( 2022 ), Corbin et al ( 2021 ), we begin with a detailed description of the deterministic part of the cell movement on the microscale which includes acceleration due to external forces. Let a population of a large number of cells be modelled as points with position and velocity coordinates Following Newton’s second law, we set up an initial value problem (IVP) for an ODE system that describes their motion: where , , , and are some continuous functions.…”

“…As in Dietrich et al ( 2022 ) and Zhigun and Surulescu ( 2022 ), the term on the right-hand side of ( 3.1b ) is included to describe the acceleration (rather, deceleration) due to the viscous force. Following Stokes’ law, we take it to be proportional to the velocity of the cell.…”

“…However, a few works have introduced such a transport term into the model (Zhigun and Surulescu 2022 ; Dietrich et al. 2022 ; Corbin et al. 2021 ; Chauvière et al.…”

“…So far, no non-local interactions of activity variables on the microscale have been considered. Inspired by M4–M4 as well as approaches to microscale modelling of the deterministic portion of cell motion preceding a KTE in Dietrich et al ( 2022 ), Zhigun and Surulescu 2022 and to KTE upscaling in Zhigun and Surulescu ( 2022 ), we use a multiscale approach to formally derive two classes of non-local CCA models: firstly, a basic model without CAM binding dynamics in Sect. 3 and, secondly, a considerably more involved model which includes such dynamics in Sect.…”

A genuinely hybrid, multiscale 3D cancer invasion and metastasis modelling framework

“…In addition to the aforementioned simplifications, the geometrical representation of the various organs considered in this paper has been abstractly described as simple 3D polyhedrons of variable density. However, our multiple organ modelling framework could also account for more realistic organ geometries and structures, we refer for instance to [40, 41, 42].…”