Using Parameter Estimation Techniques to Analyze a Mathematical Model in Wound Healing
Grade Level at Time of Presentation
Senior
Major
Applied Mathematics
Minor
-
Institution
Western Kentucky University
KY House District #
17
KY Senate District #
17
Faculty Advisor/ Mentor
Dr. Richard Schugart
Department
Mathematics
Abstract
Because the medical treatment of diabetic foot ulcers remains a challenge for clinicians, a quantitative approach using de-identified patient data and mathematical modeling can help researchers understand the physiology of the wounds. In this work, we utilized a previously developed mathematical model describing the interactions among matrix metalloproteinases, their inhibitors, extracellular matrix, and fibroblasts (Krishna et al., 2015). The model was modified and curve-fitted to individual patient data from Muller et al. (2008), while model parameters were estimated using ordinary least-squares. The parameter values were then analyzed using Latin Hypercube Sampling (LHS), a stratified sampling method for multidimensional parameter distribution, and Partial Rank Correlation Coefficients (PRCC), computed from a multivariable regression analysis, to describe how sensitive each parameter value is to changes in the system. Utilizing the patient data curve-fits, a mean and standard deviation may be used to formulate a normal distribution for each of the model parameters. The output of the ordinary differential equation model when the parameter values are substituted may be compared to an actual wound healing response to measure the validity of our model. A Bayesian approach, a different statistical method, is then used to analyze parameter values and establish confidence intervals. Upon comparison of the model with refined parameter estimates from the various techniques, the predictive ability of the model may be improved. The goal of this work is to quantify and understand differences between patients in order to predict future responses and individualize treatment for each patient.
Using Parameter Estimation Techniques to Analyze a Mathematical Model in Wound Healing
Because the medical treatment of diabetic foot ulcers remains a challenge for clinicians, a quantitative approach using de-identified patient data and mathematical modeling can help researchers understand the physiology of the wounds. In this work, we utilized a previously developed mathematical model describing the interactions among matrix metalloproteinases, their inhibitors, extracellular matrix, and fibroblasts (Krishna et al., 2015). The model was modified and curve-fitted to individual patient data from Muller et al. (2008), while model parameters were estimated using ordinary least-squares. The parameter values were then analyzed using Latin Hypercube Sampling (LHS), a stratified sampling method for multidimensional parameter distribution, and Partial Rank Correlation Coefficients (PRCC), computed from a multivariable regression analysis, to describe how sensitive each parameter value is to changes in the system. Utilizing the patient data curve-fits, a mean and standard deviation may be used to formulate a normal distribution for each of the model parameters. The output of the ordinary differential equation model when the parameter values are substituted may be compared to an actual wound healing response to measure the validity of our model. A Bayesian approach, a different statistical method, is then used to analyze parameter values and establish confidence intervals. Upon comparison of the model with refined parameter estimates from the various techniques, the predictive ability of the model may be improved. The goal of this work is to quantify and understand differences between patients in order to predict future responses and individualize treatment for each patient.