2011, 8(2): 253-261. doi: 10.3934/mbe.2011.8.253

A mathematical model for chronic wounds

1. 

Mathematical Biosciences Institute and Department of Mathematics, Ohio State University, Columbus, OH 43210, United States

2. 

Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, United States

Received  March 2010 Revised  August 2010 Published  April 2011

Chronic wounds are often associated with ischemic conditions whereby the blood vascular system is damaged. A mathematical model which accounts for these conditions is developed and computational results are described in the two-dimensional radially symmetric case. Preliminary results for the three-dimensional axially symmetric case are also included.
Citation: Avner Friedman, Chuan Xue. A mathematical model for chronic wounds. Mathematical Biosciences & Engineering, 2011, 8 (2) : 253-261. doi: 10.3934/mbe.2011.8.253
References:
[1]

C. K. Sen, G. M. Gordillo, S. Roy, 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. doi: 10.1111/j.1524-475X.2009.00543.x.

[2]

R. F. Diegelmann and M. C. Evans, Wound healing: An overview of acute, fibrotic and delayed healing, Front Biosci., 9 (2004), 283-289. doi: 10.2741/1184.

[3]

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. doi: 10.1016/j.clindermatol.2006.12.005.

[4]

A. J. Singer and R. A. Clark, Cutaneous wound healing, N. Engl. J. Med., 341 (1999), 738-746. doi: 10.1056/NEJM199909023411006.

[5]

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

[6]

P. D. Dale, J. A. Sherratt and P. K. Maini, A mathematical model for collagen fibre formation during foetal and adult dermal wound healing, Proc. Biol. Sci., 263 (1996), 653-660. doi: 10.1098/rspb.1996.0098.

[7]

G. J. Pettet, H. M. Byrne, D. L. S. Mcelwain and J. Norbury, A model of wound-healing angiogenesis in soft tissue, Math. Biosci., 136 (1996), 35-63. doi: 10.1016/0025-5564(96)00044-2.

[8]

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. doi: 10.1098/rspb.1996.0217.

[9]

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. doi: 10.1080/10273660008833045.

[10]

R. C. Schugart, A. Friedman, R. Zhao and C. K. Sen, Wound angiogenesis as a function of tissue oxygen tension: A mathematical model, Proc. Nat. Acad. Sci. U.S.A., 105 (2008), 2628-2633. doi: 10.1073/pnas.0711642105.

[11]

Y. Dor, V. Djonov and E. Keshet, Induction of vascular networks in adult organs: Implications to proangiogenic therapy, Ann. N.Y. Acad. Sci., 995 (2003), 208-216. doi: 10.1111/j.1749-6632.2003.tb03224.x.

[12]

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. doi: 10.1016/S0962-8924(03)00022-9.

[13]

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. doi: 10.1006/bulm.2002.0293.

[14]

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. doi: 10.1016/j.mcm.2005.05.008.

[15]

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. doi: 10.1006/bulm.1998.0042.

[16]

C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, Proc. Nat. Acad. Sci. USA, 106 (2009), 16782-16787. doi: 10.1073/pnas.0909115106.

[17]

A. Friedman, C. Huang and J. Yong, Effective permeability of the boundary of a domain, Comm. Partial Differential Equations, 20 (1995), 59-102.

[18]

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 pre-clinical model of chronic ischemic wound, Physiol. Genomics, 37 (2009), 211-224. doi: 10.1152/physiolgenomics.90362.2008.

[19]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040. doi: 10.1137/090772630.

show all references

References:
[1]

C. K. Sen, G. M. Gordillo, S. Roy, 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. doi: 10.1111/j.1524-475X.2009.00543.x.

[2]

R. F. Diegelmann and M. C. Evans, Wound healing: An overview of acute, fibrotic and delayed healing, Front Biosci., 9 (2004), 283-289. doi: 10.2741/1184.

[3]

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. doi: 10.1016/j.clindermatol.2006.12.005.

[4]

A. J. Singer and R. A. Clark, Cutaneous wound healing, N. Engl. J. Med., 341 (1999), 738-746. doi: 10.1056/NEJM199909023411006.

[5]

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

[6]

P. D. Dale, J. A. Sherratt and P. K. Maini, A mathematical model for collagen fibre formation during foetal and adult dermal wound healing, Proc. Biol. Sci., 263 (1996), 653-660. doi: 10.1098/rspb.1996.0098.

[7]

G. J. Pettet, H. M. Byrne, D. L. S. Mcelwain and J. Norbury, A model of wound-healing angiogenesis in soft tissue, Math. Biosci., 136 (1996), 35-63. doi: 10.1016/0025-5564(96)00044-2.

[8]

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. doi: 10.1098/rspb.1996.0217.

[9]

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. doi: 10.1080/10273660008833045.

[10]

R. C. Schugart, A. Friedman, R. Zhao and C. K. Sen, Wound angiogenesis as a function of tissue oxygen tension: A mathematical model, Proc. Nat. Acad. Sci. U.S.A., 105 (2008), 2628-2633. doi: 10.1073/pnas.0711642105.

[11]

Y. Dor, V. Djonov and E. Keshet, Induction of vascular networks in adult organs: Implications to proangiogenic therapy, Ann. N.Y. Acad. Sci., 995 (2003), 208-216. doi: 10.1111/j.1749-6632.2003.tb03224.x.

[12]

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. doi: 10.1016/S0962-8924(03)00022-9.

[13]

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. doi: 10.1006/bulm.2002.0293.

[14]

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. doi: 10.1016/j.mcm.2005.05.008.

[15]

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. doi: 10.1006/bulm.1998.0042.

[16]

C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, Proc. Nat. Acad. Sci. USA, 106 (2009), 16782-16787. doi: 10.1073/pnas.0909115106.

[17]

A. Friedman, C. Huang and J. Yong, Effective permeability of the boundary of a domain, Comm. Partial Differential Equations, 20 (1995), 59-102.

[18]

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 pre-clinical model of chronic ischemic wound, Physiol. Genomics, 37 (2009), 211-224. doi: 10.1152/physiolgenomics.90362.2008.

[19]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040. doi: 10.1137/090772630.

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