Cytotoxic T lymphocytes (CTLs) belong to our bodies’ most effective weapons in the fight against distorted cells, such as cells infected by viruses or cancer cells. By expressing specific T-cell receptors, these T cells can bind to and destroy, e.g., cancerous cells that show the respective antigen on their surface. The way CTLs become sensitive to an antigen is via an activation process, which is often induced by an encounter with a dendritic cell (DC) presenting said antigen.

Hence, there exist inactive CTLs, active CTLs and dendritic cells, the three types of cells behave in different ways and the groups interact with each other. Earlier studies have worked on a mathematical way to describe the motion of single cells. Such individual-based models have concluded that, while activated cells move according to classic Brownian motion, their inactive counterparts can be described by a Lévy walk, i.e. a random walk with a heavy-tailed distribution of movement distances [1]. In consequence, inactive cells move further distances and once activated, their motion turns into a local search pattern.

Even if the behaviour of an individual cell can be modelled, from a computational point of view, it is not preferable to simulate large amounts of individual cells. Hence it is natural to investigate the macroscopic limit of the kinetic model using analytic mathematical methods. The recently published article ‘Macroscopic limit of a kinetic model describing the switch in T cell migration modes via binary interactions’, European Journal of Applied Mathematics, 2021 by Gissell Estrada-Rodriguez (Sorbonne Université, Paris) and Tommaso Lorenzi (Politecnico di Torino) [2] addresses this topic.

In the manuscript, the authors start from the above-mentioned model of random walk motion in activated and inactivated CTLs and DCs. Encounters between cells of different groups in their model lead to either ‘conservative interactions’ (which only changes the velocities of the two cells) or ‘population-switching interactions’ once a DC activates a previously inactive CTL. The authors then prove that the particle distribution functions satisfy certain transport equations. This behaviour on the mesoscopic scale is steered by the type of the particles’ motion (local or non-local), an interaction term and (for the CTLs only) in- respectively outflow terms proportional to the rate at which particles switch from an inactive into an activated state upon interaction.

By scaling the system in time and space, eventually a macroscopic limit of the cell motion behaviour is derived. This model consists in three coupled balance equations for the densities of DCs and inactive and active CTLs. The final balance terms are steered by constants encapsulating the microscopic behaviour of the cells (i.e., cell velocity and constants determining the running time probability distributions) and reflect physical intuitions about the system.

Despite the need to model several aspects in the behaviour of the different groups of cells and between them, the authors have managed to derive a framework of the cells’ movement at a macroscopic level. Naturally, since the immune system is a highly complex body of interacting agents, this model provides a basis for several generalizations. Among other possible further directions of research, the authors name the extension of their model to populations like cancerous cells or DCs without tumour antigen information, or the inclusion of other types of cell motion like locally conditioned behaviour or sub-diffusive or ballistic random walks.

[1]: Harris, T., Banigan, E., Christian, D. et al. (2012). Generalized Lévy walks and the role of chemokines in migration of effector CD8+ T cells. Nature 486, 545–548. doi.org/10.1038/nature11098

[2]: Estrada-Rodriguez, G., & Lorenzi, T. (2021). Macroscopic limit of a kinetic model describing the switch in T cell migration modes via binary interactions. European Journal of Applied Mathematics, 1-27. doi:10.1017/S0956792521000358