2008 - Marseille - France

PAGE 2008: Methodology
Sergei Leonov

Stochastic pharmacokinetic models: selection of sampling times

V. V. Fedorov (1), S. L. Leonov (1), V. A. Vasiliev (2)

(1) GlaxoSmithKline, Collegeville, U.S.A.; (2) Tomsk State University, Russia

Introduction and Objectives: We discuss pharmacokinetic (PK) studies which are described by compartmental models. Traditionally, ordinary differential equations (ODE) are used for PK modeling, and two sources of randomness are introduced, measurement errors and population variability. In this presentation we focus on the intrinsic variability induced by the random terms in stochastic differential equations (SDE). Unlike the ODE-based models, the intrinsic variability leads to a "within-subject" correlation, or autocorrelation, between values of a stochastic PK process at different time points. This means that in serial sampling schemes, starting from certain sample sizes, the gain of information from adding extra observations for a given patient will diminish.

Methods and Results: Using the techniques of stochastic calculus, see [1] - [3], we find closed-form expressions for the mean and covariance function for a number of PK processes generated by SDE. Special attention is given to those cases where trajectories of the stochastic system are positive which is important from a biological perspective, cf. [4], [5]. The formulae for the covariance function allow us to address the problem of optimal design, i.e. finding sequences of sampling times that guarantee the most precise estimation of unknown model parameters. We use the first-order optimization algorithm, see [6], to construct D-optimal designs for a number of examples, including cases where experimental costs are taken into account.

Conclusions: We emphasize that all three sources of variability should be considered in stochastic PK models: within-subject, between-subject, and measurement errors. We recommend cost-based designs which allow for a meaningful comparison of sampling schemes with different numbers of samples.

[1] Øksendal  B. (1992). Stochastic Differential Equations. Springer, Berlin.
[2] Vasiliev VA, Dobrovidov AV, Koshkin GM (2004). Nonparametric Estimation of Functionals of Stationary Sequences Distributions. Nauka, Moscow.
[3] Bishwal J.P.N. (2007).  Parameter Estimation in Stochastic Differential Equations. Springer, New York.
[4] Anisimov VV, Fedorov VV, Leonov SL (2007).  Optimal design of pharmacokinetic studies described by stochastic differential equations. In: Lopez-Fidalgo J, Rodriguez-Diaz JM, Torsney B (Eds), mODa 8 - Advances in Model-Oriented Design and Analysis, Physica-Verlag, Heidelberg, pp. 9-16.
[5] Overgaard RV, Jonsson N, Tornoe CW, Madsen H. (2005). Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm. J. Pharmacokinet. Pharmacodyn.,  32 (1), 85-107.
[6] Fedorov VV, Hackl P. (1997), Model-Oriented Design of Experiments. Springer, New York.

Reference: PAGE 17 (2008) Abstr 1248 [www.page-meeting.org/?abstract=1248]
Oral Presentation: Methodology