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

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## 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.”

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