**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 *E*_{θ}_{,X} (*U*) = ∫*U*(**X**,**θ**)p(**θ**)p(**X**)*d***θ***d***X**, 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

**X**

*>*

_{i}**X**

_{i-}_{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 **X**_{1} was 0.082 to 7.4 hours and for **X**_{2} 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 **X**_{1} was 0.16 to 2.7 hours for **X**_{2} was 2.0 to 8.5 hours and for **X**_{3} 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.