Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics

Rıdvan Yaprak; Erhan Coşkun

doi:10.47000/tjmcs.1740108

Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics

Abstract

This study investigates an ordinary differential equation model with lumped parameters that describes the interaction between wound tissue and slough tissue. We extend this model to a partial differential equation framework to better represent cellular level characteristics and examine how varying tissue diffusivities affect the healing process. We analyze the well-posedness of the system, including the existence, uniqueness, and positivity of solutions, under homogeneous Neumann boundary conditions and given initial distributions. Furthermore, we provide numerical simulations for various wound healing scenarios. The results indicate that acute wounds tend to heal quickly, while chronic wounds suffer from disruptions that inhibit successful healing. We also explore the impact of diffusion on the stability of equilibrium points in the earlier model and demonstrate that the system does not exhibit Turing instability. Additionally, we analyze how various cellular properties, particularly differences in diffusion rates, affect the healing process.

Keywords

Ethical Statement

The author/authors declare that there are no conflicts of interest regarding the publication of this article.

References

Adam, J.A., A simplified model of wound healing (with particular reference to the critical size defect), Mathematical and Computer Modelling,30(1999), 23–32.
Arnold, J.S., Adam, J.A., A simplified model of wound healing II: The critical size defect in two dimensions, Mathematical and Computer Modelling, 30(1999), 47–60.
Allen, L.J.S., An Introduction to Mathematical Biology, Pearson/Prentice Hall, 2007.
Ames, W.F., Numerical Methods for Partial Differential Equations, 3rd ed., Academic Press, 1992.
Angel, D., Slough: what does it mean and how can it be managed, Wound Practice and Research, 27(2019), 164–167.
Baktir, G., Wound repair and experimental wound models, Experimed, 9(2019), 130–137.
Bardsley, W.G., Sattar, A., Armstrong, J.R., Shah, M., Brosnan, P. et al., Quantitative analysis of wound healing, Wound Repair and Regeneration, 3(1995), 426–441.
Bowden, L.G., Maini, P.K., Moulton, D.E., Tang, J.B., Wang, X.T. et al., An ordinary differential equation model for full thickness wounds and the effects of diabetes, Journal of Theoretical Biology, 361(2014), 87–100.

Bradley, M., Cullum, N., Sheldon, T., The debridement of chronic wounds: a systematic review, Health Technology Assessment, 3(1999), 1–78.
Cardinal, M., Eisenbud, D.E., Armstrong, D.G., Zelen, C., Driver, V. et al., Serial surgical debridement: A retrospective study on clinical outcomes in chronic lower extremity wounds, Wound Repair and Regeneration, 17(2009), 306–311.
Cruywagen, G.C., Murray, J.D., On a tissue interaction model for skin pattern formation, Journal of Nonlinear Science, 2(1992), 217–240.
Dale, P.D., Maini, P.K., Sherratt, J.A., Mathematical modeling of corneal epithelial wound healing, Mathematical Biosciences, 124(1994), 127–147.
Doerler, M., Reich-Schupke, S., Altmeyer, P., St¨ucker, M., Impact on wound healing and efficacy of various leg ulcer debridement techniques, JDDG: Journal der Deutschen Dermatologischen Gesellschaft, 10(2012), 624–631.
Fick, A., Ueber Diffusion, Annalen der Physik 170(1855), 59–86.
Haberman, R., Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems, 5th ed., Pearson, Boston, 2012.
Hundsdorfer, W., Verwer, J.G., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer, Springer Series in Computational Mathematics, vol. 33, 2003.
Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, 2009.
Jones, M.A., Song, B., Thomas, D.M., Controlling wound healing through debridement, Mathematical and Computer Modelling, 40(2004), 1057–1064.
Liu, W.L., Jiang, Y.L., Wang, Y.Q., Li, Y.X., Liu, Y.X., Combined debridement in chronic wounds: A literature review, Chinese Nursing Research 4(2017), 5–8.
Martin, R.H., Pierre, M., Nonlinear reaction-diffusion systems, Mathematics in Science and Engineering, 185(1992), 363–398.
The MathWorks, Inc., MATLAB R2023a, Natick, Massachusetts, 2023.
Murray, J.D., Mathematical Biology: II. Spatial Models and Biomedical Applications, 3rd ed., Interdisciplinary Applied Mathematics, vol. 18, Springer, New York, 2003.
Orgill, D.P., Blanco, C., Biomaterials for Treating Skin Loss, Woodhead Publishing, 2009.
Sherratt, J.A., Murray, J.D., Models of epidermal wound healing, Proceedings of the Royal Society B: Biological Sciences, 241(1990), 29–36.
Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer, 1994.
Thomas, D.C., Tsu, C.L., Nain, R.A., Arsat, N., Lah, N.A.S.N. et al., The role of debridement in wound bed preparation in chronic wound: A narrative review, Annals of Medicine and Surgery, 71(2021), 2049–0801.
Turing, A.M., The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(1952), 37–72.
Wouwer, A.V., Saucez, P., Vilas, C., Simulation of ODE/PDE Models with MATLAB®, OCTAVE and SCILAB: Scientific and Engineering Applications, Springer International Publishing, 2014.

Details

Primary Language

English

Subjects

Numerical Analysis, Partial Differential Equations, Biological Mathematics

Journal Section

Research Article

Authors

Rıdvan Yaprak ^*
0000-0002-6417-4929
Türkiye

Erhan Coşkun
0000-0002-5382-8843
Türkiye

Publication Date

October 13, 2026

Submission Date

July 11, 2025

Acceptance Date

March 6, 2026

Published in Issue

Year 2026 Volume: 18 Number: 3

DOI

https://doi.org/10.47000/tjmcs.1740108

IZ

https://izlik.org/JA88WB96WM

Cite

RIS / Bibtex

APA

Yaprak, R., & Coşkun, E. (2026). Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics. Turkish Journal of Mathematics and Computer Science, 18(3), 621-638. https://doi.org/10.47000/tjmcs.1740108

AMA

1.Yaprak R, Coşkun E. Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics. TJMCS. 2026;18(3):621-638. doi:10.47000/tjmcs.1740108

Chicago

Yaprak, Rıdvan, and Erhan Coşkun. 2026. “Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics”. Turkish Journal of Mathematics and Computer Science 18 (3): 621-38. https://doi.org/10.47000/tjmcs.1740108.

EndNote

Yaprak R, Coşkun E (October 1, 2026) Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics. Turkish Journal of Mathematics and Computer Science 18 3 621–638.

IEEE

[1]R. Yaprak and E. Coşkun, “Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics”, TJMCS, vol. 18, no. 3, pp. 621–638, Oct. 2026, doi: 10.47000/tjmcs.1740108.

ISNAD

Yaprak, Rıdvan - Coşkun, Erhan. “Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics”. Turkish Journal of Mathematics and Computer Science 18/3 (October 1, 2026): 621-638. https://doi.org/10.47000/tjmcs.1740108.

JAMA

1.Yaprak R, Coşkun E. Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics. TJMCS. 2026;18:621–638.

MLA

Yaprak, Rıdvan, and Erhan Coşkun. “Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 3, Oct. 2026, pp. 621-38, doi:10.47000/tjmcs.1740108.

Vancouver

1.Rıdvan Yaprak, Erhan Coşkun. Modelling the Impact of Cellular Diffusion on the Wound–Slough Healing Dynamics. TJMCS. 2026 Oct. 1;18(3):621-38. doi:10.47000/tjmcs.1740108