# Connecting Local and Global Sensitivities in a Mathematical Model for Wound Healing

## Abstract

The process of wound healing is governed by complex interactions between proteins and the extracellular matrix, involving a range of signaling pathways. This study aimed to formulate, quantify, and analyze a mathematical model describing interactions among matrix metalloproteinases (MMP-1), their inhibitors (TIMP-1), and extracellular matrix in the healing of a diabetic foot ulcer. De-identified patient data for modeling were taken from Muller et al. (Diabet Med 25(4):419–426, 2008), a research outcome that collected average physiological data for two patient subgroups: “good healers” and “poor healers,” where classification was based on rate of ulcer healing. Model parameters for the two patient subgroups were estimated using least squares. The model and parameter values were analyzed by conducting a steady-state analysis and both global and local sensitivity analyses. The global sensitivity analysis was performed using Latin hypercube sampling and partial rank correlation analysis, while local analysis was conducted through a classical sensitivity analysis followed by an SVD-QR subset selection. We developed a “local-to-global” analysis to compare the results of the sensitivity analyses. Our results show that the sensitivities of certain parameters are highly dependent on the size of the parameter space, suggesting that identifying physiological bounds may be critical in defining the sensitivities.

## Keywords

Parameter estimation Wound healing Sensitivity analysis Latin hypercube sampling Patient-specific modeling Proteolytic enzyme model## Notes

### Acknowledgments

NK was supported under a Gatton Academy Research Internship Grant. HP was supported by a Western Kentucky University (WKU) Faculty-Undergraduate Student Engagement (FUSE) Award #14-SP141, a WKU Honors Development grant, and a Gatton Alumni Scholarship Award. CC was supported by a National Science Foundation Research Experiences for Undergraduates (REU) grant DBI-1004665. RS was partially supported as a Sabbatical Fellow at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation through NSF Award #DBI-1300426, with additional support from The University of Tennessee, Knoxville. Additional financial support was also provided by the WKU Office of Sponsored Programs, the WKU Office of Academic Affairs, the WKU Office of Research, the WKU Ogden College Dean’s Office, the WKU Biotechnology Center, the WKU Bioinformatics Science Center, the WKU Applied Research and Technology Program, the WKU Biology and Mathematics Departments, the University of Chicago Department of Mathematics, and the Gatton Academy of Mathematics and Science in Kentucky. Permission was granted by Wiley to use the data given in Muller et al. as “reuse of this article is permitted in accordance with the Creative Commons Deed, Attribution 2.5, which does not permit commercial exploitation.”

## References

- Alon U (2006) An introduction to systems biology: design principles of biological Circuits. Chapman & Hall/CRC mathematical and computational biology. Taylor & Francis, LondonGoogle Scholar
- Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an hiv model, as an example. Int Stat Rev 62(2):229–243CrossRefMATHGoogle Scholar
- Bode W, Fernandez-Catalan C, Grams F, Gomis-Rüth FX, Nagase H, Tschesche H, Maskos K (1999) Insights into MMP–TIMP interactions. Ann N Y Acad Sci 878:73–91CrossRefGoogle Scholar
- Conover WJ, Iman RL (1981) Rank transformations as a bridge between parametric and nonparametric statistics. Am Stat 35(3):124–129. doi: 10.1080/00031305.1981.10479327 MATHGoogle Scholar
- Eisenberg MC, Hayashi MA (2014) Determining identifiable parameter combinations using subset profiling. Math Biosci 256:116–126. doi: 10.1016/j.mbs.2014.08.008 CrossRefMathSciNetMATHGoogle Scholar
- Ellwein LM, Tran HT, Zapata C, Novak V, Olufsen MS (2008) Sensitivity analysis and model assessment: mathematical models for arterial blood flow and blood pressure. J Cardiovasc Eng 8:94–108CrossRefGoogle Scholar
- Eslami M (1994) Theory of sensitivity in dynamic systems: an introduction. Springer, BerlinCrossRefGoogle Scholar
- Frank PM (1978) Introduction to system sensitivity theory. Academic Press, New YorkMATHGoogle Scholar
- Geris L, Schugart RC, van Oosterwyck H (2010) In silico design of treatment strategies in wound healing and bone fracture healing. Philos Trans A Math Phys Eng Sci 368(1920):2683–2706CrossRefGoogle Scholar
- Golub GH, van Loan CF (1989) Matrix computations, 2nd edn. The Johns Hopkins University Press, BaltimoreMATHGoogle Scholar
- Hamby DM (1994) A review of techniques for parameter sensitivity analysis of environmental models. Environ Monit Assess 32(2):135–154CrossRefGoogle Scholar
- Helton JC, Davis FJ (2002) Illustration of sampling-based methods for uncertainty and sensitivity analysis. Risk Anal 22(3):591–622. doi: 10.1111/0272-4332.00041 CrossRefGoogle Scholar
- Iman RL, Conover WJ (1979) The use of the rank transform in regression. Technometrics 21(4):499–509. doi: 10.1080/00401706.1979.10489820 CrossRefGoogle Scholar
- International Atomic Energy Agency (1989) Evaluating the reliability of predictions made using environmental transfer models. International Atomic Energy AgencyGoogle Scholar
- Jørgensen LN (2003) Collagen deposition in the subcutaneous tissue during wound healing in humans: a model evaluation. APMIS Suppl 115:1–56Google Scholar
- Lawrence WT (1998) Physiology of the acute wound. Clin Plast Surg 25(3):321–340MathSciNetGoogle Scholar
- Li M, Moeen Rezakhanlou A, Chavez-Munoz C, Lai A, Ghahary A (2009) Keratinocyte-releasable factors increased the expression of MMP1 and MMP3 in co-cultured fibroblasts under both 2d and 3d culture conditions. Mol Cell Biochem 332(1–2):1–8Google Scholar
- Lobmann R, Schultz G, Lehnert H (2005) Proteases and the diabetic foot syndrome: mechanisms and therapeutic implications. Diabetes Care 28(2):461–471CrossRefGoogle Scholar
- Marino S, Hogue IB, Ray CJ, Kirschner DE (2008) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 254(1):178–196. doi: 10.1016/j.jtbi.2008.04.011 CrossRefMathSciNetGoogle Scholar
- Maskos K (2005) Crystal structures of MMPs in complex with physiological and pharmacological inhibitors. Biochimie 87(3–4):249–263CrossRefGoogle Scholar
- Mast B (1992) The skin. In: Cohen IK, Diegelmann RF, Lindblad WJ (eds) Wound healing: biochemical and clinical aspects, Chap. 22. W. B. Saunders Co., LondonGoogle Scholar
- MathWorks: How globalsearch and multistart work—matlab & simulink.http://www.mathworks.com/help/gads/how-globalsearch-and-multistart-work.html
- McKay MD, Beckman RJ, Conover WJ (1979) Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245MathSciNetMATHGoogle Scholar
- Muller M, Trocme C, Lardy B, Morel F, Halimi S, Benhamou PY (2008) Matrix metalloproteinases and diabetic foot ulcers: the ratio of MMP-1 to TIMP-1 is a predictor of wound healing. Diabet Med 25(4):419–426CrossRefGoogle Scholar
- Olufsen MS, Ottesen JT (2013) A practical approach to parameter estimation applied to model predicting heart rate regulation. J Math Bio 67(1):39–68CrossRefMathSciNetMATHGoogle Scholar
- Parks WC, Wilson CL, López-Boado YS (2004) Matrix metalloproteinases as modulators of inflammation and innate immunity. Nat Rev Immunol 4(8):617–629CrossRefGoogle Scholar
- Pilcher BK, Dumin JA, Sudbeck BD, Krane SM, Welgus HG, Parks WC (1997) The activity of collagenase-1 is required for keratinocyte migration on a type 1 collagen matrix. J Cell Biol 137:1445–1457CrossRefGoogle Scholar
- Pope SR, Ellwein LM, Zapata CL, Novak V, Kelley CT, Olufsen MS (2009) Estimation and identification of parameters in a lumped cerebrovascular model. Math Biosci Eng 6(1):93–115CrossRefMathSciNetMATHGoogle Scholar
- Sawicki G, Marcoux Y, Sarkhosh K, Tredget EE, Ghahary A (2005) Interaction of keratinocytes and fibroblasts modulates the expression of matrix metalloproteinases-2 and -9 and their inhibitors. Mol Cell Biochem 269(1–2):209–216CrossRefGoogle Scholar
- Sheehan P, Jones P, Caselli A, Giurini JM, Veves A (2003) Percent change in wound area of diabetic foot ulcers over a 4-week period is a robust predictor of complete healing in a 12-week prospective trial. Diabetes Care 26(6):1879–1882CrossRefGoogle Scholar
- Sherratt JA, Dallon JC (2002) Theoretical models of wound healing: past successes and future challenges. C R Biol 325(5):557–564CrossRefGoogle Scholar
- Stamenkovic I (2003) Extracellular matrix remodelling: the role of matrix metalloproteinases. J Pathol 200:448–464CrossRefGoogle Scholar
- Stein M (1987) Large sample properties of simulations using latin hypercube sampling. Technometrics 29(2):143–151. doi: 10.1080/00401706.1987.10488205 CrossRefMathSciNetMATHGoogle Scholar
- Valdez-Jasso D, Haider MA, Banks HT, Bia D, Zocalo Y, Armentano R, Olufsen MS (2008) Viscoelastic mapping of the arterial ovine system using a kelvin model. IEEE Trans Biomed Eng 56:210–219CrossRefGoogle Scholar
- Wahl SM, Hunt DA, Wakefield LM, McCartney-Francis N, Wahl LM, Roberts AB, Sporn MB (1987) Transforming growth factor type beta induces monocyte chemotaxis and growth factor production. Proc Natl Acad Sci USA 84(16):5788–5792CrossRefGoogle Scholar
- Werner S, Krieg T, Smola H (2007) Keratinocyte-fibroblast interactions in wound healing. J Invest Dermatol 127(5):998–1008CrossRefGoogle Scholar
- Yager DR, Kulina RA, Gilman LA (2007) Wound fluids: a window into the wound environment? Int J Low Extrem Wounds 6(4):262–272CrossRefGoogle Scholar
- Zhang Q, Gould LJ (2014) Hyperbaric oxygen reduces matrix metalloproteinases in ischemic wounds through a redox-dependent mechanism. J Invest Dermatol 134:237–246CrossRefGoogle Scholar