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  - , 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. , . 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 , 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.
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