Macroscopic limit of a kinetic model describing the switch in T cell migration modes via binary interactions

TL;DR

This paper develops a kinetic model describing how cytotoxic T lymphocytes switch from non-local search patterns to localized migration upon activation by dendritic cells, with a macroscopic limit showing diffusion and fractional diffusion behaviors.

Contribution

It introduces a coupled kinetic model incorporating binary interactions and velocity-jump processes, deriving a macroscopic limit with classical and fractional diffusion equations.

Findings

Activated CTLs exhibit diffusive movement modeled by classical diffusion.

Pre-activation CTLs follow a fractional diffusion process, indicating Lévy walk behavior.

The model links cellular interactions to macroscopic migration patterns.

Abstract

Experimental results on the immune response to cancer indicate that activation of cytotoxic T lymphocytes (CTLs) through interactions with dendritic cells (DCs) can trigger a change in CTL migration patterns. In particular, while CTLs in the pre-activation state move in a non-local search pattern, the search pattern of activated CTLs is more localised. In this paper, we develop a kinetic model for such a switch in CTL migration modes. The model is formulated as a coupled system of balance equations for the one-particle distribution functions of CTLs in the pre-activation state, activated CTLs and DCs. CTL activation is modelled via binary interactions between CTLs in the pre-activation state and DCs. Moreover, cell motion is represented as a velocity-jump process, with the running time of CTLs in the pre-activation state following a long-tailed distribution, which is consistent with a L\'evy walk, and the running time of activated CTLs following a Poisson distribution, which corresponds to Brownian motion. We formally show that the macroscopic limit of the model comprises a coupled system of balance equations for the cell densities whereby activated CTL movement is described via a classical diffusion term, whilst a fractional diffusion term describes the movement of CTLs in the pre-activation state. The modelling approach presented here and its possible generalisations are expected to find applications in the study of the immune response to cancer and in other biological contexts in which switch from non-local to localised migration patterns occurs.