**Efficient Sampling Windows for Parameter Estimation in Population Models**

Bogacka, Barbara (1) Maciej Patan (2)

(1) Queen Mary, University of London, UK (2) University of Zielona Gora, Poland

**Introduction:** Optimum experimental design for parameter estimation in PK/PD population models has gained considerable attention in the statistical literature, see for example, [1] and [2]. The advantage of a high precision of estimation of the population parameters is clear, however, in advanced phases of clinical trials when a drug is being tested in a population of patients, it may be impossible to maintain the accurate timing of blood sampling for every patient. This can be a discouraging issue to a practitioner to apply, otherwise the best sampling schedule and it may result in an inefficient experiment and so loss of recourses. Sampling windows, that is time intervals assuring some minimum required efficiency, are a good solution to the problem. Several authors proposed various methods for deriving such windows, see [3] - [6], based on a design efficiency factor.

**Objective:** To give a method of calculating sampling windows for population models (in general, mixed non-linear models) which would not only assure a required minimum efficiency of the population parameter estimation but also would give window size reflecting the parameter sensitivity.

**Method:** We follow the approach of [7], who derived a method for calculating sampling windows for a fixed non-liner model which has such a two-fold objective. The method gives less flexibility (narrower windows) when it is important to get a reading at a time close to the optimum schedule and more flexibility (wider windows) when it is less important. For example, it gives a relatively narrow window around the optimum point during the fast absorption phase of a drug and it gives a wider window around the optimum point in the slow elimination phase. We generalize the method for population (mixed effects) non-linear models. The method is based on a condition of the Equivalence Theorem for D-optimality which we derive for such models and we show how it can be applied to assure the two-fold objective. We also explain the theory behind the method in an example of a population PK model.

**References:**

[1] Mentré, F., Mallet, A. and Baccar, D. (1997). Optimal Design in Random-effects Regression Models. Biometrika 84, 2, 429 - 442.

[2] Mentré, F., Dubruc, C. and Thénot, J.-P. (2001). Population Pharmacokinetic Analysis and Optimization of the Experimental Design for Mizolastine Solution in Children. J. Pharmocokinetics and Pharmacodynamics. 28, 3, 229-319.

[3] Green, B and Duffull, S.B. (2003). Prospective Evaluation of a D-Optimal Designed Population Pharmacokinetic Study. J. Pharmocokinetics and Pharmacodynamics. 30, 145 - 161.

[4] Pronzato, L. (2002). Information Matrices with Random Regressors. Application to Experimental Design. J. of Statistical Planning and Inference 108, 189 - 200.

[5] Johnson, P., Jones, B., Bogacka, B. and Volkov, O. (2005). Optimal PK Sampling Under the Constraint Imposed in Later Phase Clinical Trials. http://www.page-meeting.org/page/page2005/PAGE2005P61.pdf.

[6] Graham, G. and Aarons, L. (2006). Optimum Blood Sampling Time Windows for Parameter Estimation in Population Pharmacokinetic Experiments. Statistics in Medicine (in press).

[7] Bogacka, B., Johnson, P., Jones, B. and Volkov, O. D-efficient Window Experimental Designs. Submitted for publication.