Journal of Mathematical Biology

, Volume 74, Issue 4, pp 809–841 | Cite as

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

  • Lisandro J. Fermín
  • Jacques Lévy-Véhel


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.


Pharmacokinetic Compliance Drug dosing interval Multiple doses Variability Irregularity 

Mathematics Subject Classification

60G55 60J75 90B36 



The research of L.J. Fermín was supported by project DIGITEO DIM, ANIFRAC: Uncertainties in processes with fractal characteristics, by a research grant from the project DIUV 2/2011 of the Universidad of Valparaíso, and partially supported by projects Anillo ACT1112 CONICYT-CHILE, and MathAmSud 16MATH03.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.CIMFAV, Facultad de IngenieríaUniversidad de ValparaísoValparaisoChile
  2. 2.Regularity Team, INRIA Saclay-Ile-de-FranceOrsayFrance

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