Parameter estimation of renal models
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Two parameters of the delay time and the feedback gain play important roles in understanding an important mechanism—Tubuloglomerular feedback in renal hemodynamics. A few mathematical models have been developed, where each emphasizes different parts of the renal system but all of them include these two parameters and they define these two parameters differently. An inverse problem, estimating the two parameters by combining the information of the experimental data and the simulated data from mathematical models has brought to our attention. Ditlevsen et al. estimated these two parameters by the least square distance between the spectral densities of experimental data and simulated data from their renal model for normatensive and hypertensive rats in 2005 and 2007. The main work of this thesis is to use different statistical estimation methods, Bayes linear method (BLM), maximal likelihood estimation method based on generalized-polynomial chaos expansion (MLE-GPC) and markov chain monte carlo method (MCMC), to estimate the delay time and the feedback gain in a partial differential equation model developed by Layton and Pitman. The idea of BLM is to update beliefs about adjusted expectations and adjusted variances of quantities. MLE-GPC and MCMC methods focus on probability density functions of quantities. Likely regions, consisting of plausible values of the parameters, are estimated instead of a single plausible value for each parameter and further confident intervals in Ditlevsen's papers.