A Novel Derivation of Rigorous Macroscopic Limits from a Micro-Meso Description of Signal-Triggered Cell Migration in Fibrous Environments

Abstract

In this work we upscale a prototypical kinetic transport equation which models a cell population moving in a fibrous environment with a chemo-or haptotactic signal influencing both the direction and the magnitude of the cell velocity. The presented approach to scaling does not rely on orthogonality and treats parabolic and hyperbolic scalings in a unified manner. It is shown that the steps of the formal limit procedures are mirrored by rigorous operations with finite measures provided that the measure-valued position-direction fiber distribution enjoys some spacial continuity.

A flux-limited model for glioma patterning with hypoxia-induced angiogenesis

“…Among the latter, [21] contains a deduction of macroscopic equations with flux-saturated diffusion, haptotaxis, and chemotaxis; the passage from the mesoscopic KTAP framework to the RDT system is different from the one performed here, yet still formal. Very recently, [54] started from a KTE formulation similar to that in [21], hence also related to the one in the present work, and obtained by yet another upscaling method (with rigorous convergences) an RDT with myopic diffusion for the zeroth order approximation of the solution. That PDE did not involve flux-limitation, but the equation deduced for the first order correction did.…”

“…Figure 6 showed a sharper tumor cell profile at the interface with large diffusivity drop, while the model without flux limitation generated a more uniform pattern with a tendency of smearing the interface and faster filling the areas of lower cell density. We have to stress, nevertheless, that our deduction of macroscopic PDEs is merely formal; as mentioned above, a rigorous one was done in [54], but for a simplified model and involving only limitation(s) of signal gradient(s) of the form ∇S 1+|∇S| instead of our choice encoded in φ (h, M). The obtained macroscopic model is able to reproduce, at least qualitatively, the histological patterns observed in patient biopsies.…”

Mathematical modeling of glioma invasion and therapy approaches

“…The global well-posedness of a system featuring this complexity has not been investigated and raises manifold challenges, due to the intricate couplings and nonlinearities involved in the source and motility terms. The passage from KTEs to the macroscopic dynamics has been done here in a merely formal way, the rigorous convergence is still open and might probably use some of the ideas in [55], where a much simpler system has been considered.…”

Multiscale modeling of glioma invasion: from receptor binding to flux-limited macroscopic PDEs

“…The deduction performed here is merely formal; a rigorous one, which follows a different limiting procedure and another form of flux saturation on the cell scale is addressed in a rigorous manner in [58], where there is (tactic) flux limitation only in the macroscopic PDE for the first order correction.…”