A population-based Bayesian approach to the minimal model of glucose and insulin homeostasis

The minimal model was proposed in the late 1970s by Bergman et al. (Am. J. Physiol. 1979; 236(6):E667) as a powerful model consisting of three differential equations describing the glucose and insulin kinetics of a single individual. Considering the glucose and insulin simultaneously, the minimal model is a highly ill-posed estimation problem, where the reconstruction most often has been done by non-linear least squares techniques separately for each entity. The minimal model was originally specified for a single individual and does not combine several individuals with the advantage of estimating the metabolic portrait for a whole population. Traditionally it has been analysed in a deterministic set-up with only error terms on the measurements. In this work we adopt a Bayesian graphical model to describe the coupled minimal model that accounts for both measurement and process variability, and the model is extended to a population-based model. The estimation of the parameters are efficiently implemented in a Bayesian approach where posterior inference is made through the use of Markov chain Monte Carlo techniques. Hereby we obtain a powerful and flexible modelling framework for regularizing the ill-posed estimation problem often inherited in coupled stochastic differential equations. We demonstrate the method on experimental data from intravenous glucose tolerance tests performed on 19 normal glucose-tolerant subjects.