Mathematical and Numerical Methods for Understanding Immune Cell Motion During Wound Healing

¹ Department of Mathematics, Slovak University of Technology, Radlinského 11, 81005 Bratislava, Slovakia
² Laboratory of Pathogens and Host Immunity (LPHI), CNRS/Université de Montpellier, Place Eugéne Bataillon, 34095 Montpellier cedex 5, France

^* Corresponding author: karol.mikula@stuba.sk

Received: 4 February 2025
Accepted: 15 November 2025

Abstract

In this paper, we propose a new workflow to analyze macrophage motion during wound healing. These immune cells are attracted to the wound after an injury and they move showing both directional and random motion. Thus, first, we smooth the trajectories and we separate the random from the directional parts of the motion. The smoothing model is based on curve evolution where the curve motion is influenced by the smoothing term and the attracting term. Once we obtain the random sub-trajectories, we analyze them using the mean squared displacement to characterize the type of diffusion. Finally, we compute the velocities on the smoothed trajectories and use them as sparse samples to reconstruct the wound attractant field. To do that, we consider a minimization problem for the vector components and lengths, which leads to solving the Laplace equation with Dirichlet conditions for the sparse samples and zero Neumann boundary conditions on the domain boundary.