Asymptotics and bioavailability in multicompartment pharmacokinetic models with enterohepatic circulation

Analysing discrete as well as continuous linear autonomous pharmacokinetic models, it is shown that their asymptotic behaviour is independent of the rates of kinetic processes and timing of drug application. Consequently, for the description of pharmacokinetic endpoints, i.e. the total amounts of drug eliminated through different organs under various ways of administration, in such a model the knowledge of total amounts delivered to individual compartments and its transition probability matrix P=[p(ij)] is sufficient.A design and analysis of a 9-compartment pharmacokinetic model with enterohepatic circulation (EHC), avoiding several common simplifications, test the applicability of our method. The central compartment of the model is the liver acting as filter and linking the systemic and enterohepatic circulation. Explicit formulas are given for pharmacokinetic endpoints of the model using the elements of the transition probability matrix P. Conversely, the transition probabilities are determined in terms of certain measurable pharmacokinetic endpoints and the flow rates through the kidneys, liver and the cardiac output, contributing that way to the structural identifiability problem. As a further consequence, the bioavailability of the drug with and without EHC can be determined and the efficiency of EHC expressed as the 'probability' of the enterohepatic cycle.Finally, we apply our method to analyse and compare various pharmacokinetic models, describing the EHC of drugs, based on some previously published articles.