Variability and singularity arising from a Piecewise-Deterministic Markov Process applied to model poor patient compliance in the multi-IV case

We propose a Piecewise-Deterministic Markov Process (PDMP) to model the drug concentration in the case of multiple intravenous-bolus (multi-IV) doses and poor patient adherence situation: the scheduled time and doses of drug administration are not respected by the patient, the drug administration considers switching regime with random drug intake times. We study the randomness of drug concentration and derive probability results on the stochastic dynamics using the PDMP theory, focusing on two aspects of practical relevance: the variability of the concentration and the regularity of its stationary probability distribution. The main result show as the regularity of the concentration is governed by a parameter, which quantifies in a precise way the situations where drug intake times are too scarce concerning the elimination rate. Our approach is novel for the study of the regularity of the stationary distribution in PDMP models. This article extends the results given in [J. Lévy-Véhel and P.E. Lévy-Véhel, Variability and singularity arising from poor compliance in a pharmacodynamical model I: The multi-IV case, J. Pharmacokinet. Pharmacodyn. 40 (2013), pp. 15–39], by considering more realistic irregular dosing schedules. The computations permit precise assessment of the effect of various significant parameters such as the mean rate of intake, the elimination rate, and the mean dose. They quantify how much poor adherence will affect the regimen. Our results help to understand the consequences of poor adherence.