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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 
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 tumorinduced 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 elliptichyperbolic 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 tumourinduced 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 woundhealing 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 antiangiogenic and chemotherapeutic drug strategies,, Mathematical and Computer Modelling, 41 (2005), 1137. Google Scholar 
[21] 
F. Werdin, M. Tennenhaus, H. Schaller and H. Rennekampff, Evidencebased 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 tumorinduced 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 elliptichyperbolic 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 tumourinduced 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 woundhealing 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 antiangiogenic and chemotherapeutic drug strategies,, Mathematical and Computer Modelling, 41 (2005), 1137. Google Scholar 
[21] 
F. Werdin, M. Tennenhaus, H. Schaller and H. Rennekampff, Evidencebased 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 
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