# Methods to Determine Optimal Sampling Windows for Nonlinear Mixed Effects Models

Background: D-optimality is often used to design an optimum experiment for population pharmacokinetic (PK) studies. However, obtaining samples at specific time points is not always feasible due to logistic constraints. Therefore many D-optimal designs may be executed sub-optimally and risk attaining uninformative data. Sampling windows (a time range around each of the D-optimal time points) can provide flexibility in the sampling times and provides regions of planned and controlled “sub-optimality”. However, there is no analytical solution for sampling windows for nonlinear mixed effects models, such as population PK models, and all current methods are either slow or require significant assumptions. This research proposes two methods to determine the sampling windows for a population PK experiment.

Aim: To develop and assess two methods for determining sampling windows that can be applied to population pharmacokinetic studies.

Methods: Two methods were proposed to determine the sampling windows with predefined efficiency for time points provided by the D-optimality criterion. 1) A naïve adaptive approach which determines the sampling windows adaptively. 2) A recursive random sampling approach based on Markov chain Monte Carlo techniques. Both methods were applied to locate sampling windows for a population PK experiment that follows a one compartment first order input and first order output PK model (termed Bateman PK model). Both methods were coded in MATLAB and used POPT to evaluate the population Fisher information matrix.

Results: The naïve adaptive approach can be used to locate the next sampling window given the current and previous sampling time for the Bateman PK model. The window for the first sample in this method is conditioned on the subsequent samples taken at the D-optimal time. In theory if the first sample is taken at a time that is far from optimal then the subsequent windows may not suffice. The MCMC approach successfully located the sampling windows for all sampling times and all windows satisfied the predefined efficiency. The MCMC method converged almost immediately and appeared to be insensitive to the initial starting conditions.

Conclusion: Two methods for determining sampling windows were proposed. The MCMC method provided quick and accurate results and its properties warrant further evaluation.

#### Lee Kien Foo

• Multimedia University