Bayesian parameter estimation for compartmental models in biology and physics

We demonstrated that the use of Bayes' Theorem, with the Box and Draper likelihood, to estimate the parameters of compartmental models is an improvement over current "ad hoc" parameter estimation methods. This method has several novel features that are useful in parameter estimation. First, the remaining parameter uncertainty is described by the posterior density, which cannot be obtained with classical regression methods. Second, highest posterior density contours can be used to illustrate uncertainty of parameter pairs, and their shape changes can be used to describe the influence of reduced data sets and/or different models on the estimation uncertainty. Third, information from other experiments and sources can be incorporated using appropriate prior distribution. Two areas of application for this technique are biology and physics. Three biokinetic models were studied: the exchange of calcium at bone surfaces in beagles, human cerebral glucose metabolism, and the exchange of serum albumin in human. In physics, we estimated the half-lives of ²²⁶Ra, ²²²Rn and ²¹⁸Po from simulated decay data. In addition, different reduced data sets were also examined for each model to show their influence on parameter uncertainty. We applied the Bayesian method to two and three-compartment models. The presence of bimodality and "divergent" behavior in the posterior densities is new and unexpected. They are due to the likelihood and can be changed using prior information and/or the amount of data used. For example, the use of Normal priors stopped the "divergence" in the bone calcium study. However, it also introduced bimodality for all three parameters. Interestingly, the omission of the extravascular space data in the serum albumin analysis did not diminish the estimation accuracy. Instead, estimation precision was increased for three of the four parameters, as indicated by smaller contours. When open contours are present, the estimation variances are large and can be infinite. Finally, we showed that radioactive decay can be described using compartmental models and that half-lives can be estimated using Box and Draper's method. As always, the most precise estimates were obtained using data for individual compartments.