Mathematical and numerical methods for understanding immune cell motion during wound healing

TL;DR

The paper develops a robust workflow to study macrophage migration during wound healing by first smoothing trajectories with an evolving-curve model that retains directional motion while suppressing random components. It introduces a novel attracting term based on evolving original-segment lengths and uses self-intersection criteria to determine stopping, enabling adaptive parameter selection. Random parts of motion are characterized via mean squared displacement, revealing subdiffusive dynamics across three datasets. The smoothed velocities serve as sparse samples to reconstruct the wound attractant field by solving a Laplace equation with Dirichlet data on samples and zero-flux Neumann conditions, yielding a continuous vector field that highlights directional flow toward the wound and provides a quantitative link between cell motion and the chemoattractant landscape.

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.

Figures (29)

• Figure 1: Visualization of curve discretization. The green dots represent the grid points and the red dots represent the endpoints of the segments.
• Figure 2: Original trajectory (light blue line) and results of smoothing for $\lambda=10$ (red line) and $\lambda=50$ (blue line). The black boxes indicate the regions chosen for the zooms. Middle and bottom: zooms of the smoothing process.
• Figure 3: Top: original trajectory (light blue curve) and regions of self-intersections (black rectangles). Bottom: zooms of the rectangles. Grid points of the trajectory (blue asterisks), grid points where the self-intersection is detected (red asterisks), and grid points belonging to a self-intersecting part (green asterisks).
• Figure 4: Original trajectory (light blue line) and results of smoothing (green lines) for $\delta=0.003$ (top) and $\delta=0.01$ (bottom). When no further self-intersections are detected, the algorithm performs $50$ additional curve evolution steps before stopping.
• Figure 5: Original trajectory (light blue line) and results of smoothing (green lines) for $\delta=0.003$ (top) and $\delta=0.01$ (bottom). When no further self-intersections are detected, the algorithm performs $50$ additional curve evolution steps before stopping.