Mathematical models for cell migration: a non-local perspective

Abstract

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell’s motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell–cell and cell–tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue ‘Multi-scale analysis and modelling of collective migration in biological systems’.

Stable Steady-State Solutions of Some Biological Aggregation Models

“…Furthermore, these are often tied to important physical or biological questions. For good recent reviews see [13,15,11].…”

Direction-dependent turning leads to anisotropic diffusion and persistence

“…[39]). The adhesive properties of cells can be included in macroscopic models of cell movement, for example, through the inclusion of non-local flux terms [9,5]. Such a model would be significantly more complex than the model studied here, and hence we leave this for future research.…”

The role of mechanics in the growth and homeostasis of the intestinal crypt

“…Incorporating such details will be an important extension and may enable, more generally, a new approach to continuum modelling of cell migration (Chen et al. 2020 ).…”