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

A three dimensional model of wound healing: Analysis and computation

Avner Friedman 1, , Bei Hu 2, and  Chuan Xue 1,

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.
Keywords: existence and uniqueness of solutions., free boundary problems, ischemia, Wound healing, viscoelasticity.
Mathematics Subject Classification: Primary: 92C50, 35R35, 35B40,.
Citation: Avner Friedman, Bei Hu, Chuan Xue. A three dimensional model of wound healing: Analysis and computation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2691-2712. doi: 10.3934/dcdsb.2012.17.2691
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), 35-92.

[2]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899.

[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), 175-197.

[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), 974-986.

[5]

R. F. Diegelmann and M. C. Evans, Wound healing: An overview of acute, fibrotic and delayed healing, Front. Biosci., 9 (2004), 283-289.

[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), 208-216.

[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), 131-136.

[8]

A. Friedman, A multiscale tumor model, Interfaces and Free Boundaries, 10 (2008), 245-262.

[9]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040.

[10]

A. Friedman and G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci., 15 (2005), 95-107.

[11]

A. Friedman and C. Xue, A mathematical model for chronic wounds, Mathematical Biosciences and Engineering, 8 (2011), 253-261.

[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), 673-702.

[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), 19-25.

[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), 1487-1493.

[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), 35-63.

[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), 211.

[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), 2628-2633.

[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), 763-771.

[19]

A. J. Singer and R. A. Clark, Cutaneous wound healing, N. Engl. J. Med., 341 (1999), 738-746.

[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), 1137-1156.

[21]

F. Werdin, M. Tennenhaus, H. Schaller and H. Rennekampff, Evidence-based management strategies for treatment of chronic wounds, Eplasty, 9 (2009), e19.

[22]

C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.

show all references

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), 35-92.

[2]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899.

[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), 175-197.

[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), 974-986.

[5]

R. F. Diegelmann and M. C. Evans, Wound healing: An overview of acute, fibrotic and delayed healing, Front. Biosci., 9 (2004), 283-289.

[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), 208-216.

[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), 131-136.

[8]

A. Friedman, A multiscale tumor model, Interfaces and Free Boundaries, 10 (2008), 245-262.

[9]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040.

[10]

A. Friedman and G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci., 15 (2005), 95-107.

[11]

A. Friedman and C. Xue, A mathematical model for chronic wounds, Mathematical Biosciences and Engineering, 8 (2011), 253-261.

[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), 673-702.

[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), 19-25.

[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), 1487-1493.

[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), 35-63.

[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), 211.

[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), 2628-2633.

[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), 763-771.

[19]

A. J. Singer and R. A. Clark, Cutaneous wound healing, N. Engl. J. Med., 341 (1999), 738-746.

[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), 1137-1156.

[21]

F. Werdin, M. Tennenhaus, H. Schaller and H. Rennekampff, Evidence-based management strategies for treatment of chronic wounds, Eplasty, 9 (2009), e19.

[22]

C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.

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