**Optimal design in population kinetic experiments by set-valued methods**

P. Gennemark (1,2), A. Danis (1), J. Nyberg (3), A.C. Hooker (3), W. Tucker (1)

(1) Dept. of Mathematics, Uppsala University, SE-751 06 Uppsala, Sweden; (2) Mathematical sciences, University of Gothenburg, SE-412 96 Göteborg, Sweden; (3) Dept. of Pharmaceutical Biosciences, Uppsala University, SE-751 06 Uppsala, Sweden

**Objectives:** Optimal experimental design is of importance to obtain accurate and precise parameter estimates from sparse data in population kinetic analysis studies. In traditional methods the structure and statistical properties of the model are mostly assumed known, while the parameters are either assumed known (local optimal design), or known to the level of statistical distributions (robust approach). The objective of this work is to describe and evaluate how set-valued methods [1,2] can be applied to solve optimal design problems.

**Methods:** We use set-valued methods based on interval analysis; all variables and parameters are represented as intervals rather than real numbers. The evaluation of a specific design is based on multiple simulations and set-valued parameter estimations of designs from the search domain. In the parameter estimation, the output for each parameter consists of a range that is consistent with data [3,4].

**Results:** We propose a heuristic method for optimal experimental design of population pharmacometric experiments based on a set-valued approach. The method is evaluated on several optimal design problems collected from the literature [5]. The method requires no prior information in form of point estimates for the parameters, since the parameters are represented by intervals and can incorporate any level of uncertainty. Notably, no numerical integration is required as in traditional robust optimal design methods. Sampling times and covariates like doses can be represented by intervals, which gives a direct way of optimizing with rigorous sampling/dose intervals that can be useful in clinical practice. General problems with parameter estimation in non-linear models are avoided in set-valued parameter estimation (no distributional assumptions regarding parameters; no model linearization; problems with local minima are avoided). For non-identifiable problems, e.g., with infinitely many solutions, set-valued parameter estimation brackets all solutions, while the traditional maximum likelihood method only outputs one solution.

**Conclusions:** Main advantages of the proposed method are that no prior point estimates for the parameters are required, the method works on underdetermined problems, and that sampling times and covariates like doses can be represented by intervals.

**References:**

[1] Tucker W. Validated numerics for pedestrians. In European Congress of Mathematics, Eur. Math. Soc., Zurich, Austria. 2005;851-860.

[2] Moore RE. Interval Analysis, Prentice-Hall, Englewood Cliffs, New Jersey; 1966.

[3] Hooker AC, Tucker W. Parameter estimation in non-linear mixed effects models using interval analysis PAGE 17 (2008) Abstr 1369 [www.page-meeting.org/?abstract=1369]. ISSN 1871-6032.

[4] Danis A, Ueckert S, Hooker AC, Tucker W. Set-valued methods for estimation of parameters in population models. Submitted abstract for PAGE 20 (2011).

[5] Foracchia M, Hooker A, Vicini P, Ruggeri A. POPED, a software for optimal experiment design in population kinetics. Comput Methods Programs Biomed. 2004;74:29-46.