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Bifurcation analysis of critical values for wound closure outcomes in wound healing experiments

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Abstract

A nonlinear partial differential equation containing a nonlocal advection term and a diffusion term is analyzed to study wound closure outcomes in wound healing experiments. There is an extensive literature of similar models for wound healing experiments. In this paper we study the character of wound closure in these experiments in terms of the sensing radius of cells and the force of cell-cell adhesion. We prove a bifurcation result which differentiates uniform closure of the wound from nonuniform closure of the wound, based on a critical value \(\lambda _\star \) of the force of cell-cell adhesion parameter \(\lambda \). For \(\lambda < \lambda _\star \) the steady state solution \(u\equiv 1\) of the model is stable and the wound closes uniformly. For \(\lambda > \lambda _\star \) the steady state solution \(u\equiv 1\) of the model is unstable and the wound closes nonuniformly. We provide numerical simulations of the model to illustrate our results.

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Acknowledgements

The authors would like to thank the reviewers for their careful review and valuable suggestions. The authors are very grateful to Professor Yongtao Zhang for helpful discussions about the numerical simulations in this paper. The code for the numerical simulations are available upon request.

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  1. Mathematics Department, Vanderbilt University, Nashville, TN, 37240, USA

    Glenn Webb & Xinyue Evelyn Zhao

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  1. Glenn Webb
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  2. Xinyue Evelyn Zhao
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Correspondence to Glenn Webb.

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Webb, G., Zhao, X.E. Bifurcation analysis of critical values for wound closure outcomes in wound healing experiments. J. Math. Biol. 86, 66 (2023). https://doi.org/10.1007/s00285-023-01896-7

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