Variability and singularity arising from poor compliance in a pharmacokinetic model II: the multi-oral case.

We propose a stochastic model for the drug concentration in the case of multiple oral doses and in a situation of poor patient adherence. Our model is able to take into account an irregular drug intake schedule. This article is the second in a series of three. It presents a multi-oral version of the results given in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15-39, 2013), that dealt with the multi-IV bolus case. Under the assumption that the irregular dosing schedule follows a Poisson law, we study features of the drug concentration that have practical implications, such as its variability and the regularity of its cumulative probability distribution, which describes its predictive power with respect to the mean behaviour. We consider four variants: continuous-time, with either deterministic or random doses, and discrete-time, also with either deterministic or random doses. Our computations allow one to assess in a precise way the effect of various significant parameters such as the mean rate of intake, the elimination rate, the absorption rate and the mean dose. They quantify how much poor adherence will affect the efficacy of therapy. To appreciate this impact, we provide detailed comparisons with the variability of concentration in two reference situations: a fully adherent patient and a population of fully adherent patients with log-normally distributed pharmacokinetic parameters. Besides, the discrete-time versions of our models reveal unexpected links with objects which have been studied in the mathematical literature under the name of infinite Bernoulli convolutions (Erdós, Am J Math 61:974-975, 1939). This allows us to quantify the fact that, when the random dosing schedule is too sparse, the concentration behaves in a very erratic way. Our results complement the ones in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15-39, 2013) and help understanding the consequences of poor adherence. They may have practical outcomes in terms of drug dosing and scheduling.

Variability and singularity arising from poor compliance in a pharmacokinetic model I: the multi-IV case.

We consider a simple multi-IV model for drug concentration in the case of poor patient compliance. The model is a stochastic one, and is thus able to take into account an irregular drug intake schedule. Under some assumptions, we study features of the drug concentration relevant for practical purposes such as its variability or the regularity of its cumulative probability distribution. We consider five variants: random instants for drug intake with either deterministic or random doses, both in continuous and discrete-time settings, plus a model with stochastically varying elimination rate. Our computations make it possible to assess in a precise way the effect of various significant parameters such as the mean rate of intake, the elimination rate, and the mean dose. They also quantify how much poor compliance will affect the regimen: in that view, we provide precise comparisons with the variability of concentration in the cases of (a) a fully compliant patient and (b) a population of fully compliant patients with lognormally distributed elimination rates. The time discretized version of our models reveal unexpected links with measures known as infinite Bernoulli convolutions. Our findings help in understanding the consequences of poor compliance, and may have practical outcomes in terms of drug dosing and scheduling.

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.