## Honors College Capstone Experience/Thesis Projects

Mathematics

Thesis

#### Abstract

A mathematical model for the healing response of diabetic foot ulcers was developed using averaged data (Krishna et al., 2015). The model contains four major factors in the healing of wounds using four separate differential equations with 12 parameters. The four differential equations describe the interactions between matrix metalloproteinases (MMP-1), tissue inhibitors of matrix metalloproteinases (TIMP-1), the extracellular matrix (ECM) of the skin, and the fibroblasts, which produce these proteins. Recently, our research group obtained the individual patient data that comprised the averaged data. The research group has since taken several approaches to analyze the model with the individual patient data. One approach was to introduce mixed modeling techniques on certain parameters in that model. Mixed effects modeling is an analytical tool useful for the repeated measurement of data with subjects, patients, etc., that have random affects that deviate from a specified norm. This is accomplished by taking a parameter that is shared across all data sets and splitting it into a fixed variable and a random variable. Then all data sets are modeled so that the fixed variable is the same for all patients and the random variable is a unique modifier that accounts for the differences across patients. Another approach has been to use an optimal design technique to identify which times are ideal for data gathering for the model. For this project a Standard Error (SE) optimal design method was chosen with the goal of minimizing the sum of squared normalized standard errors. Our project worked to combine these techniques by first introducing mixed modeling parameter values into an SE-optimal design algorithm and then comparing the results collected to the standard algorithm. Other optimal design techniques were used with and without these mixed modeling parameters to see if a certain technique was better than the others. Finally, we worked toward improving our estimates for mixed modeling parameters by attempting to implement in MONOLIX. Our focus for this part was to develop test cases that could be implemented in MONOLIX.

Richard Schugart, Ph.D.

#### Disciplines

Applied Mathematics | Other Applied Mathematics

COinS