Bayesian Estimation of Optimal Sampling Times for Pharmacokinetic Models
Duffull, S B (1), G Graham (2), K Mengersen (3)
(1) University of Queensland, Brisbane, Australia; (2) Pfizer, Sandwich, UK; (3) QUT, Brisbane, Australia.
Introduction: Optimal design techniques are gaining acceptance as a tool for designing pharmacokinetic and pharmacokinetic-pharmacodynamic studies. These designs are based on finding the maximum of a scalar function of the information matrix (usually the determinant). The optimum design is then reported in terms of sampling windows. We explore a Bayesian method for estimating the sampling windows.
Objectives: To explore the use of an MCMC approach to estimation of sampling windows for the design of a pharmacokinetic study.
Methods: Optimal study design was explored for two 1-compartment fixed effects models within an MCMC framework. These were M1: a simple intravenous bolus model with two parameters (V, k) and M2: a Bateman function with three parameters (V, k, ka). A Markov chain was constructed that has the optimal design as its stationary distribution from which the pre-posterior distribution of the sampling times can be generated. We used the Metropolis Hastings algorithm to explore the posterior distribution of the sampling times. The pre-posterior mean utility of the sampling times X is defined by the integral Eθ,X (U) = ∫U(X,θ)p(θ)p(X)dθdX, where p(θ) is the prior distribution of the parameters, p(X) is the prior distribution of the sampling times and U(X,θ) is a utility function defined by: U(X,θ) = prod(((diag(M –1(X, θ))) 0.5 θ –1) 2) and M is the information matrix. Maximizing this expected utility corresponds to minimizing the product of the squared relative standard errors. The credible interval on X is calculated by determining the quantiles of the MCMC samples on X. We chose a uniform distribution for the prior of the sampling times, while constraining Xi > Xi-1 for i > 2, and assumed the pharmacokinetic parameters were log-normally distributed, with mean (V, k) = (ln(20, ln(0.1)) and mean (V, k, ka) = (ln(20), ln(0.1), ln(1)) and a 30% CV for all parameters for models M1 and M2, respectively. The Markov chain was run for a total of 20000 samples where the first 1000 were discarded. Two sampling times were optimized for M1 and three sampling times for M2.
Results: For M1, the 95% credible interval for X1 was 0.082 to 7.4 hours and for X2 was 8.8 to 22 hours. The posterior mode of X was 0.053 and 14.6 hours. The upper 95% credible interval of the posterior distribution of the asymptotic standard errors was < 10% for both parameters. For M2, the 95% credible interval for X1 was 0.16 to 2.7 hours for X2 was 2.0 to 8.5 hours and for X3 was 8.7 to 22 hours. The posterior mode of X was (0.44, 2.7, 12). The upper boundary of the 95% credible interval of the asymptotic estimates of the standard errors for all parameters was < 10%.
Conclusion: A MCMC method for determining optimal sampling windows is described. This method incorporates prior uncertainty on the parameter values as well as a prior on the sampling times. The method provides both the credible interval of the sampling windows for each design point as well as the marginal pre-posterior distribution of the asymptotic standard errors.