The prehepatic insulin secretion rate of the pancreatic $beta$-cells is not directly measurable, since part of the secreted insulin is absorbed by the liver prior to entering the blood stream. However, C-peptide is co-secreted equimolarly and is not absorbed by the liver, allowing for the estimation of the prehepatic insulin secretion rate. We consider a stochastic differential equation model that combines both insulin and C-peptide concentrations in plasma to estimate the prehepatic insulin secretion rate. Previously this model has been analysed in an iterative deterministic set-up, where the time courses of insulin and C-peptide subsequently are used as known forcing functions. In this work we adopt a Bayesian graphical model to describe the unied model simultaneously. We develop a model that also accounts for both measurement error and process variability. The parameters are estimated by a Bayesian approach where efficient posterior sampling is made available through the use of Markov chain Monte Carlo methods. Hereby the ill-posed estimation problem inherited in the coupled differential equation model is regularized by the use of prior knowledge. The method is demonstrated on experimental data from an IntraVenous Glucose Tolerance Test (IVGTT) performed on six normal glucose-tolerant individuals.
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