Browsing by Author "Robert H. Martin, Committee Member"

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    A Multiple Inhibin Model of the Human Menstrual Cycle
    (2008-06-18) Pasteur, Roger Drew II; James F. Selgrade, Committee Chair; John E. Franke, Committee Member; Robert H. Martin, Committee Member; Ernest L. Stitzinger, Committee Member
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    Non-parametric Parameter Estimation and Clinical Data Fitting with a Model of HIV Infection
    (2005-07-29) Adams, Brian Michael; Robert H. Martin, Committee Member; Hien T. Tran, Committee Member; H.T. Banks, Committee Chair; Marie Davidian, Committee Member
    The focus of this dissertation is to develop a combined mathematical and statistical modeling approach for analyzing clinical data from an HIV acute infection study. We amalgamate two existing models from the literature to create a nonlinear differential equation model of in-host infection dynamics that is capable of predicting sustained low-level viral loads and multiple stable equilibria. Using this example system of differential equations we demonstrate two contrasting parameter identification problem formulations for estimating the distribution of model parameters across a population: the first at the individual patient level and the second directly at the population level itself. In the latter case one leverages data from all patients to estimate a probability density function representing the distribution. We discuss well-posedness and computational implementation for such inverse problems. Directly estimating the distribution in this way may offer computational advantages over estimating parameters for individual patients. In the context of the model, we implement the Expectation Maximization (EM) Algorithm for maximum likelihood estimation to handle patient measurements censored by assay resolution limits. This censored data method is beneficial since with it we do not arbitrarily assign values for measurements below the limit of detection, but rather compute their expected value based on the dynamics model and conditioned on the knowledge that they are censored. In addition, in both inverse problem contexts (estimating a vector of parameters for a single patient and the distribution of a parameter across all patients) we develop and apply methods for estimating variability of the resulting parameter estimates by using sensitivity analysis to calculate confidence intervals. We validate each of the methods with simulated data and demonstrate typical results. Finally we present results for the application of the methods to actual clinical data and give examples of conclusions that one might draw from them. This model fitting approach may help clinicians better understand patient behaviors and notably, could alert them to the expected long-term trend for a particular patient.
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    Sensitivity Methods for Dynamical Systems
    (2008-07-25) Ernstberger, Stacey Leigh; H. T. Banks, Committee Chair; Robert H. Martin, Committee Member; Mansoor A. Haider, Committee Member; Hien T. Tran, Committee Member