Functional Data Analytic Inference for Systems Governed By Differential Equations with Applications

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  • March 22, 2019
Creator
  • Mandal, Siddhartha
    • Affiliation: Gillings School of Global Public Health, Department of Biostatistics
Abstract
  • The objective of this dissertation research is to develop formal statistical methodology for analyzing systems governed by ordinary differential equations (ODE). Ordinary differential equations are commonly used to describe a wide variety of biological and physiological phenomena. They arise in the description of gene regulatory networks, study of HIV dynamics and other infectious diseases and toxicology . This work is motivated by physiologically based pharmacokinetic (PBPK) models in toxicology which are deterministic models used to describe chemical kinetics in human or animal physiology. These models relate the concentration of chemicals in tissues and blood to their rates of change and physiological parameters, such as tissue volume and blood flow, and metabolic parameters among others, through a system of ODEs. Usual strategies of analyzing such models involve non-linear least squares methodology which can potentially be computationally intensive. Often, some of the existing procedures for modeling ODEs do not necessarily account for inter and intra-individual variability that are common in multi-subject experiments. Using functional data analytic methods, in this dissertation research, we provide a formal statistical framework for drawing statistical inferences regarding subject specific and population specific parameters in models governed by a system of ODE. One of the main features of the proposed methodology is to cast the problem in a constrained inferential framework and thus avoid solving the differential equations, which is often challenging and time consuming. Such a formulation allows for the possibility that all components of the ODE may not adequately describe the underlying biological phenomena. The proposed framework also allows the researcher to estimate both within and between subject variability, while drawing statistical inferences at the individual as well as the population level. We make as few assumptions as possible while taking into account the underlying structure in the data. The proposed framework allows researchers to compare parameters among several populations, such as different dose groups, while adjusting covariates, whether discrete or continuous. Such inferences were not possible until now. We illustrate the proposed methodology using some simulated data sets as well as a real data set on benzene concentration in exhaled breath.
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  • ... in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Biostatistics.
Advisor
  • Sen, Pranab Kumar
Degree granting institution
  • University of North Carolina at Chapel Hill
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