Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2019 DETERMINATION OF OPTIMAL PARAMETER ESTIMATES FOR MEDICAL INTERVENTIONS IN HUMAN METABOLISM AND INFLAMMATION Marcella Torres Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Other Applied Mathematics Commons, and the Systems Biology Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/5890 This Dissertation is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. ©Marcella Maria Torres, May 2019 All Rights Reserved. i DETERMINATION OF OPTIMAL PARAMETER ESTIMATES FOR MEDICAL INTERVENTIONS IN HUMAN METABOLISM AND INFLAMMATION A Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Virginia Commonwealth University. by MARCELLA MARIA TORRES M.S. Applied Mathematics, Virginia Commonwealth University, 2015 B.S. Mathematical Sciences, Minor in Physics, Virginia Commonwealth University, 2007 Director: Angela Reynolds, Associate Professor, Department of Mathematics and Applied Mathematics Virginia Commonwewalth University Richmond, Virginia May, 2019 ii Acknowledgements I am extremely grateful to my advisor, Dr. Angela Reynolds. Thank you for supporting my random walk over the applied mathematics landscape. Also, thank you - from one mother to another - for understanding my at times high level of distraction. Especially for missing the start of that one meeting that one time. I am also deeply indebted to Dr. Rebecca Segal for being a great collaborator and mentor. Your career advice has been instrumental. I would also like to thank Dr. Shobha Ghosh, committee member and collab- orator on two of my research projects. Thank you for your expertise that made it possible for me really explore parameter estimation with good data. I also wish to thank my committee members Dr. David Edwards and Dr. Ihsan Topaloglu for bringing your knowledge of statistics and mathematics and for fitting this work into spring final exams week. Special thanks to advisor Terry Green at Pierce College in Steilacoom, WA, for helping me navigate the TRIO summer bridge program in mathematics for first generation minority students. If you hadn’t opened that door for me 17 years ago I would not be here today. Finally, thank you to my husband Derek for co-parenting two small children with me over the course of this journey. Miles and Maxine, thank you for simply existing and for reminding me every day of my purpose. iii TABLE OF CONTENTS Chapter Page Acknowledgements :::::::::::::::::::::::::::::::: iii Table of Contents :::::::::::::::::::::::::::::::: iv List of Tables ::::::::::::::::::::::::::::::::::: vi List of Figures :::::::::::::::::::::::::::::::::: viii Abstract ::::::::::::::::::::::::::::::::::::: xvii 1 Introduction :::::::::::::::::::::::::::::::::: 1 2 Resistance Exercise Induced Muscle Hypertrophy ::::::::::::: 3 2.1 Introduction . .3 2.2 Methods . .5 2.2.1 Model background . .5 2.2.1.1 Glycogen Storage . .5 2.2.1.2 Extracellular Fluid . .7 2.2.1.3 Adaptive Thermogenesis . .7 2.2.1.4 Energy Partitioning . .7 2.2.2 Model development . .9 2.3 Results . 16 2.3.1 Case study . 17 2.3.2 Simulated cohort . 20 2.3.3 Parameter estimation . 24 2.3.4 Sensitivity analysis . 28 2.3.5 Discussion . 32 2.3.6 Conclusion . 34 3 Optimal Control of Body Mass Change ::::::::::::::::::: 35 3.1 Introduction . 35 3.2 Methods . 36 3.2.1 The Basic Problem in Optimal Control . 36 3.2.2 Statement of the Optimal Control Problem . 37 iv 3.2.3 Existence and Uniqueness of an Optimal Control . 38 3.2.4 Characterization of the Optimal Control Problem . 41 3.3 Results . 44 3.4 Discussion . 46 4 Qualitative Analysis of Model Behavior with Decision Trees ::::::: 49 4.1 Introduction . 49 4.2 Methods . 50 4.2.1 Simulation of data for training and testing . 50 4.2.2 Definitions and notation . 50 4.2.3 Building a tree . 51 4.2.3.1 Splitting . 52 4.2.3.2 Decision to terminate splitting . 53 4.2.3.3 Assignment of terminal node to a class . 53 4.2.4 Error rate . 54 4.2.5 Cross-validation . 54 4.3 Results . 55 4.3.1 Classifying parameter sets in a model of resistence exercise induced muscle hypertrophy . 57 4.4 Discussion . 59 5 Macrophage polarization in peritonitis ::::::::::::::::::: 62 5.1 Introduction . 62 5.2 Methods . 65 5.2.1 Model Development . 66 5.2.2 Identification and analysis of equilibrium points . 75 5.2.3 Analysis of equilibria . 77 5.2.4 Numerical simulations . 79 5.2.5 Parameter Estimation Background . 81 5.2.6 Parameter Estimation . 82 5.2.7 Goodness-of-fit measures . 84 5.3 Results . 84 5.3.1 Determination of an identifiable subset of model parameters . 84 5.3.2 Goodness-of-fit . 93 5.3.3 Sensitivity Analysis . 96 5.3.4 Simulations . 100 5.3.5 Discussion . 103 6 Macrophage polarization in atherosclerosis ::::::::::::::::: 115 v 6.1 Introduction . 115 6.2 The Inflammatory Process in Early Atherosclerosis . 115 6.2.1 The Blood . 115 6.2.2 The Gut . 116 6.2.3 The Endothelium . 117 6.2.4 Foam cells . 117 6.2.5 The Role of Neutrophils . 118 6.2.6 Treatments . 119 6.3 A Selection of ODE and PDE Models . 119 6.4 Model Development . 122 6.5 Discussion . 137 7 Conclusion :::::::::::::::::::::::::::::::::: 138 7.1 Summary . 138 7.2 Future directions . 142 Appendix A Code for the resistance exercise model :::::::::::::: 144 A.1 SBML model file . 144 A.2 Matlab code for Latin Hypercube Sampling . 163 A.2.1 ODE_LHS_NHANESmale.m (ODEs) . 163 A.2.2 Parameter_settings_LHS_NHANESmale.m (parameter baseline values and initial conditions) . 165 A.2.3 Model_LHS_NHANESmale.m (main file) . 166 A.2.4 LHS_Call.m (LHS algorithm file) . 167 Appendix B Matlab code for the optimal control problem :::::::::: 170 Appendix C Peritonitis model ::::::::::::::::::::::::: 183 C.1 Experimental data . 183 C.2 Code for the PottersWheel Matlab toolbox (model definition file) . 183 C.3 XPP files . 186 C.3.1 Simplified model used for numerical analysis of equilibria . 186 C.3.2 Full model . 187 Appendix D SBML code for the atherosclerosis model :::::::::::: 189 References :::::::::::::::::::::::::::::::::::: 222 Vita :::::::::::::::::::::::::::::::::::::::: 248 vi LIST OF TABLES Table Page 1 Variables, parameters, and constants in the model of Hall et al. [1]. ::::::::::::::::::::::::::::::::::::: 6 2 Comparison of initial conditions and parameter values for an average US male versus female aged 20-39 years [41]. ::::::: 13 3 Model inputs for experiment comparing the effects of a hypocaloric diet only to diet with CE or RE. Initial conditions are specific to a hypothetical individual weighing 100 kg with 27.2% body fat. Pa- rameters appear in Eqn. 2.1 and were chosen such that a modest LM gain in response to RE was simulated. ::::::::::::::::::: 18 4 Parameter set data. LHS was restricted to these parameter ranges to ensure that physiologically reasonable simulations were generated given initial conditions. Sample means are close to the midpoints we expect for a uniform distribution, so a sample size of n = 100 was deemed sufficient. ::::::::::::::::::::::::::::::: 22 5 Cohort statistics. Mean and standard deviation calculated for ab- solute change in LM and FM after 12 weeks and 24 weeks of RE. ::::: 22 6 Initial conditions and parameter estimates for group data. :::: 25 7 PRCC. Time point t = 730 days. *Significant (p < 0:10). ::::::::: 31 8 Description of parameters and estimates for the full model. :::::::: 71 9 Pairwise collinearity indices. Pairs of parameters were considered collinear (highly correlated) if CI > 20. :::::::::::::::::: 91 10 All identifiable parameter subsets of size 6. ::::::::::::::::: 91 11 Parameter values and 95% pointwise confidence intervals for identifiable model. Remaining parameters were fixed at values given in Table 8. :::::::::::::::::::::::::::::::::: 94 vii 12 Goodness-of-fit statistics. In the full model, 24 parameters were estimated. After identifiability analysis, estimated parameters were reduced to 6 and the remaining parameters were fixed prior to fitting. The reduction in estimated parameters improved the weighted least squares merit function value (χ2), increased p-value on a χ2 test indi- cating that the identifiable model sufficiently explains the data, and lowered the estimated amount of information lost between the model and the data by the Aikake Information Criterion (AIC) measure. :::: 97 13 Description of model variables. :::::::::::::::::::::::: 124 14 Description of parameters and values. ::::::::::::::::: 129 15 Experimental data from mouse model of peritonitis. At each time point, cells were harvested from a sample of n mice. Aver- age cell counts for neutrophils (N), M1 macrophages (M1), and M2 macrophages (M2) are given in units of (107) cells. Standard error of p the mean for each cell type x is calculated as σx = σx= n. :::::::: 183 viii LIST OF FIGURES Figure Page 1 Model behavior and predictions for initial body measure- ments corresponding to the average US male [41] perform- ing RE with varying parameter values and energy intakes (A) Daily change in LM in response to RE is plotted against total LM. As LM accumulates the rate of increase slows and approaches zero. (B) Predicted time course of accumulation of LM in response to RE is plotted for different sets of parameter values shown in the legend. Physical activity level, PAL, was set to 1.6 for each simulation in order to compare the effects of varying parameters in the muscle hypertrophy term in Eqn. 2.1. ::::::::::::::::::::::::::::::: 10 2 Effect of an 800 kcal/day decrease in energy intake on total lean body mass. :::::::::::::::::::::::::::::: 12 3 Effects of initial conditions on model predictions. Initial con- ditions and parameters used to generate simulations in (A)-(D) are shown in Table 2. (A) A comparison of predicted LM in response to RE for the same parameter set and varying energy intakes: maintenance, a hypocaloric diet, and a hypercaloric diet.