MATHEMATICAL MODELING AND BAYESIAN PARAMETER ESTIMATION IN CANCER
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Recent works have highlighted the role of the differentiation of fibroblast into myofibroblast in cancer initiation and progression. Fibroblasts respond in a variety of ways to concentrations of activated transforming growth factor ($TGF\beta$). $TGF\beta$ has been reported to exhibit a double role in tumor cells progression and survival. It remains unclear how $TGF\beta$ suppresses tumor growth in normal, but in the presence of tumors cells $TGF\beta$ encourages tumor growth and differentiation of fibroblasts to myofibroblasts. In order to explore $TGF\beta$'s double role, we develop four mathematical models, based on systems of ODEs, for the dynamics of $TGF\beta$ at the single-cell level by incorporating intra- and extracellular processes as well as the autocrine signaling of $TGF\beta$. The ODE systems, given initial values and coefficients, are uniquely solvable. However, most systems are not solvable analytically. This makes it difficult to quantity the uncertainties of initial values and coefficients (called parameters), using general data-fitting methods. Most current parameter estimation methods are based on iterative local smoothing and least squares methods. In many problems, these iterative methods become stuck in local extrema particularly when the ODE systems become large and the experimental data is too sparse. This dissertation presents Simple and Hierarchical Bayesian Inference models for estimating the parameters of nonlinear ordinary differential equation systems with full or partially observed data. This present study revealed that the Simple and Hierarchical Models generally have good performance in parameters estimation of nonlinear ODE system, especially when the ODE systems are large and the experimental data is sparse. The Hierarchical model results in better accuracy and mean square error when correlation exists between variables and in the large nonlinear ODE systems. In addition, these two methods in conjunction with Adaptive MCMC (AM) algorithm schemes can improve the convergence of fit and avoid converging to local extrema. The efficiency of these two methods is illustrated by comparison with experimental time-series data of $TGF\beta$ signaling pathways in an epithelial cell line.