Sebastian Ueckert Comparison of Different Global Optimal Design Approximations

Sebastian Ueckert, Joakim Nyberg, Andrew C. Hooker

Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden

Objectives:To compare Monte-Carlo integration and Laplace integral approximation for global optimal design in terms of precision, runtime and best study design. Furthermore, to explore the performance of a new algorithm using the Laplace approximation, but avoiding explicit calculation of 2nd order derivatives.

Methods:
Calculation of globally optimal designs require the evaluation of an integral over the complete parameter space. In this work we compared the performance of the following four different numerical algorithms in computing ED optimal designs implemented in PopED [1]:

(I) Monte-Carlo integration with random sampling (MC-RS): integration is performed by sampling random parameter combinations, evaluating the integrand for those samples and averaging the results.
(II) Monte-Carlo integration with Latin hypercube sampling (MC-LHS): using stratified Latin hypercube instead of random samples.
(III) Laplace integral approximation (LAPLACE): integration is performed by finding the mode of the integrand and performing a second order Taylor expansion around this point [2].
(IV) Laplace integral approximation with BFGS Hessian (LAPLACE-BFGS): using the BFGS algorithm [3] to calculate an approximate Hessian during the maximization step of the LAPLACE.

A hypothetical experimental design for a drug following a simple one compartment model with IV bolus dosing, 20 individuals and 2 samples per individual was used for the comparison. Performance of the different methods was assessed in terms of optimal sampling points, runtime and objective function value. OFV obtained with MC-RS using 100,000 samples served as a reference for all methods. Variability of Monte-Carlo methods was evaluated by repeating computations for different number of random samples.

Results:
Calculated OFV differed considerably between methods with the MC-LHS method being closer to the reference. Variability in OFV was higher for MC-RS than for MC-LHS and decreased with number of samples. However, optimal sampling points found by all methods were similar. In terms of runtime LAPLACE-BFGS performed best, followed by LAPLACE and the MC methods. For MC methods, runtime was proportional to number of random samples. Computations with MC-LHS were slightly slower than with MC-RS.

Conclusions:
The LAPLACE method constitutes a fast alternative to computational intensive MC methods, however stability and application to non-normal priors has to be further investigated. Runtime of LAPLACE was further reduced by using an approximate BFGS Hessian.

References:
[1] Foracchia M, Hooker A, Vicini P & Ruggeri A: POPED, a software for optimal experiment design in population kinetics, Computer Methods and Programs in Biomedicine, vol. 74, Apr. 2004, pp. 29-46.
[2] Dodds M, Hooker A & Vicini P: Robust Population Pharmacokinetic Experiment Design, Journal of Pharmacokinetics and Pharmacodynamics, vol. 32, Feb. 2005, pp. 33-64.
[3] C.G. Broyden: The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations, IMA J Appl Math, vol. 6, Mar. 1970, pp. 76-90.