What good is maths in studies of wound healing?

Abstract

Wound healing is an aspect of normal physiology that we all take for granted until it goes wrong, such as, for example, the scarring that results from a severe burn, or those patients who suffer from debilitating chronic wounds that fail to heal. Ever since wound repair research began as a discipline, clinicians and basic scientists have collaborated to try and understand the cell and molecular mechanisms that underpin healthy repair in the hope that this will reveal clues for the therapeutic treatment of pathological healing. In recent decades mathematicians and physicists have begun to join in with this important challenge. Here we describe examples of how mathematical modeling married to biological experimentation has provided insights that biology alone could not fathom. To date, these studies have largely focused on wound re-epithelialization and inflammation, but we also discuss other components of wound healing that might be ripe for similar interdisciplinary approaches.

Keywords: Computer modeling; Mathematical biosciences.

Graphical abstract

Figure 1

Schematic of a healing wound A schematic of a healing wound. Epithelial cells migrate down between scab (brown) and healthy granulation tissue (yellow) to seal the wound gap. Cut epithelial appendages (eg hair stumps) can contribute to this wound re-epithelialization. Inflammatory cells (dark blue) are drawn to the wound by chemoattractants released from wound edge cells and other “damage” signals. Inflammatory cells roll along the activated luminal surface and then extravasate from vessels adjacent to the wound (left inset), and then orchestrate the activities of many cell lineages at the wound, including the sprouting wound angiogenesis of endothelial cells and the laying down of collagen fibrils by fibroblasts (pale blue) to make wound scar (right inset).

Figure 2

Tracking epithelial cell migration in mouse wound healing Imaging data from the Greco lab show tracks of epithelial cells as they move toward the wound. Cells closer to the wound edge migrate further toward the wound and so have longer tracks. High-resolution nuclear imaging enables the analysis of cell divisions and their orientation bias (Park et al., 2017).

Figure 3

Modeling the wound-induced calcium wave in the Drosophila notum (A) The Hutson and Page-McCaw lab’s study utilized a diffusion-reaction model to better understand how the wound calcium signal is transduced. The model begins with the release of protease from lysed cells in the wound. This protease diffuses and can cleave pro-Gbp leading to the release and diffusion of Gbps. Gbps binds to Mthl10 leading to intracellular calcium release. (B) Resulting reaction-diffusion model. (C) The output of the model in purple compared with the experimental data (O’Connor et al., 2021).

Figure 4

Modeling of the wound inflammatory response in the Drosophila pupal wing (A) Still image from a movie of a wound (white dashed line) made to the Drosophila pupal wing. Macrophages (green cytoplasm with red stingerRFP nuclei) are attracted to the wound by, as yet, unknown damage signals. a’ – tracks in a. (B) A diffusion equation with a fixed time period models the source and movement of the signal. Parameters of the equation can be estimated using Bayesian inference. a-concentration of attractant, D – diffusion coefficient, $\text{τ}$ – signal production time. (C-C’ and D-D’) A simulation of wound chemoattractant using the parameters after 10mins and 30mins whereby warmer colors indicate a higher concentration (adapted with permission from Weavers et al., 2016).