# American Institute of Mathematical Sciences

November  2012, 17(8): 2691-2712. doi: 10.3934/dcdsb.2012.17.2691

## A three dimensional model of wound healing: Analysis and computation

 1 Department of Mathematics and Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, United States, United States 2 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556

Received  March 2011 Revised  May 2011 Published  July 2012

This paper is concerned with a three-dimensional model of wound healing. The boundary of the wound is a free boundary, and the region surrounding it is viewed as a partially healed tissue, satisfying a viscoelastic constitutive law for the velocity v. In the partially healed region the densities of several types of cells and the concentrations of several chemical species satisfy a coupled system of parabolic equations, whereas the tissue density satisfies a hyperbolic equation. The parabolic equations include advection by the velocity v and chemotaxis/haptotaxis terms. We prove existence and uniqueness of a smooth solution of the free boundary problem, for some time interval $0\leq t\leq T$, $T>0$. We also simulate the model equations to demonstrate the difference in the healing rate between normal wounds and chronic (or ischemic) wounds.
Citation: Avner Friedman, Bei Hu, Chuan Xue. A three dimensional model of wound healing: Analysis and computation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2691-2712. doi: 10.3934/dcdsb.2012.17.2691
##### References:
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##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II,, Communications on Pure and Applied Mathematics, 17 (1964), 35. [2] A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis,, Bull. Math. Biol., 60 (1998), 857. [3] H. M. Byrne, M. A. J. Chaplain, D. L. Evans and I. Hopkinson, Mathematical modelling of angiogenesis in wound healing: Comparison of theory and experiment,, J. Theor. Med., 2 (2000), 175. [4] X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth,, SIAM Journal on Mathematical Analysis, 35 (2003), 974. [5] R. F. Diegelmann and M. C. Evans, Wound healing: An overview of acute, fibrotic and delayed healing,, Front. Biosci., 9 (2004), 283. [6] Y. Dor, V. Djonov and E. Keshet, Induction of vascular networks in adult organs: Implications to proangiogenic therapy,, Annals of the New York Academy of Sciences, 995 (2003), 208. [7] Y. Dor, V. Djonov and E. Keshet, Making vascular networks in the adult: Branching morphogenesis without a roadmap,, Trends in Cell Biology, 13 (2003), 131. [8] A. Friedman, A multiscale tumor model,, Interfaces and Free Boundaries, 10 (2008), 245. [9] A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds,, SIAM J. Math. Anal., 42 (2010), 2013. [10] A. Friedman and G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis,, Math. Models Methods Appl. Sci., 15 (2005), 95. [11] A. Friedman and C. Xue, A mathematical model for chronic wounds,, Mathematical Biosciences and Engineering, 8 (2011), 253. [12] S. R. McDougall, A. R. A. Anderson, M. A. J. Chaplain and J. A. Sherratt, Mathematical modelling of flow through vascular networks: Implications for tumour-induced angiogenesis and chemotherapy strategies,, Bull. Math. Biol., 64 (2002), 673. [13] N. B. Menke, K. R. Ward, T. M. Witten, D. G. Bonchev and R. F. Diegelmann, Impaired wound healing,, Clinics in Dermatology, 25 (2007), 19. [14] G. Pettet, M. A. J. Chaplain, D. L. S. Mcelwain and H. M. Byrne, On the role of angiogenesis in wound healing,, Proc. R. Soc. Lond. B, 263 (1996), 1487. [15] G. J. Pettet, H. M. Byrne, D. L. S. Mcelwain and J. Norbury, A model of wound-healing angiogenesis in soft tissue,, Mathematical Biosciences, 136 (1996), 35. [16] S. Roy, S. Biswas, S. Khanna, G. Gordillo, V. Bergdall, J. Green, C. B. Marsh, L. J. Gould and C. K. Sen, Characterization of a preclinical model of chronic ischemic wound,, Physiological Genomics, 37 (2009). [17] R. C. Schugart, A. Friedman, R. Zhao and C. K. Sen, Wound angiogenesis as a function of tissue oxygen tension: A mathematical model,, PNAS, 105 (2008), 2628. [18] C. K. Sen, G. M. Gordillo, S. R., R. Kirsner, L. Lambert, T. K Hunt, F. Gottrup, G. C. Gurtner and M. T. Longaker, Human skin wounds: A major and snowballing threat to public health and the economy,, Wound Repair Regen., 17 (2009), 763. [19] A. J. Singer and R. A. Clark, Cutaneous wound healing,, N. Engl. J. Med., 341 (1999), 738. [20] A. Stephanou, S. R. McDougall, A. R. A. Anderson and M. A. J. Chaplain, Mathematical modelling of flow in 2D and 3D vascular networks: Applications to anti-angiogenic and chemotherapeutic drug strategies,, Mathematical and Computer Modelling, 41 (2005), 1137. [21] F. Werdin, M. Tennenhaus, H. Schaller and H. Rennekampff, Evidence-based management strategies for treatment of chronic wounds,, Eplasty, 9 (2009). [22] C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds,, PNAS, 106 (2009), 16782.
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