American Institute of Mathematical Sciences

Advanced
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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

[1]

Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179
[2]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[3]

Haiyan Wang, Shiliang Wu. Spatial dynamics for a model of epidermal wound healing. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1215-1227. doi: 10.3934/mbe.2014.11.1215
[4]

Sophia A. Maggelakis. Modeling the role of angiogenesis in epidermal wound healing. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 267-273. doi: 10.3934/dcdsb.2004.4.267
[5]

Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386
[6]

John A. Adam. Inside mathematical modeling: building models in the context of wound healing in bone. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 1-24. doi: 10.3934/dcdsb.2004.4.1
[7]

Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057
[8]

M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián. Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. Communications on Pure & Applied Analysis, 2004, 3 (4) : 653-662. doi: 10.3934/cpaa.2004.3.653
[9]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013
[10]

Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273
[11]

M.J. Lopez-Herrero. The existence of weak solutions for a general class of mixed boundary value problems. Conference Publications, 2011, 2011 (Special) : 1015-1024. doi: 10.3934/proc.2011.2011.1015
[12]

R. Kannan, S. Seikkala. Existence of solutions to some Phi-Laplacian boundary value problems. Conference Publications, 2001, 2001 (Special) : 211-217. doi: 10.3934/proc.2001.2001.211
[13]

Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243
[14]

John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273
[15]

Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457
[16]

John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515
[17]

G. Kamberov. Prescribing mean curvature: existence and uniqueness problems. Electronic Research Announcements, 1998, 4: 4-11.

[18]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[19]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
[20]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences