Non-compartmental estimation of pharmacokinetic parameters in serial sampling designs

Abstract

Pharmacokinetic studies are commonly analyzed using a two-stage approach where the first stage involves estimation of pharmacokinetic parameters for each subject separately and the second stage uses the individual parameter estimates for statistical inference. This two-stage approach is not applicable in sparse sampling situations where only one sample is available per subject. Nonlinear models are often applied to analyze pharmacokinetic data assessed in such serial sampling designs. Modelling approaches are suitable provided that the form of the true model is known, which is rarely the case in early stages of drug development. This paper presents an alternative approach to estimate pharmacokinetic parameters based on non-compartmental and asymptotic theories in the case of serial sampling when a drug is given as an intravenous bolus. The statistical properties of estimators of the pharmacokinetic parameters are investigated and evaluated using Monte Carlo simulations.

Appendix

Proof of Lemma 1

By the multivariate central limit theorem [e.g., 26, p. 16] the sequence $\sqrt{n}\left(\user2{T}_{n}-\varvec{\mu}\right)$ converges in distribution to N(0, Σ). Let $\varphi_{\theta}\left( \user2{T}_{n}\right) :{\mathbb{R}}^{2J}\rightarrow {\mathbb{R}}^{1}$ be defined as $\varphi_{\theta}\left(\overline{X}_{1},\ldots, \overline{X}_{J},\overline{Y}_{1},\ldots,\overline{Y}_{J}\right) =\hat{\theta}$ with the partial derivatives at $\varvec{\mu}$ denoted by $\varphi_{\theta}^{\prime}$ and given in Table 1 and let $\varphi_{\theta }\left( \varvec{\mu}\right) :{\mathbb{R}}^{2J}\rightarrow {\mathbb{R}}^{1}$ be the corresponding parameter θ.

Because {t _j } is strictly monotone increasing and μ _j  > 0 results in λ > 0, all elements of $\varphi_{\theta}^{\prime}$ are continuous at $\varvec{\mu}$ and the sequence $Z_{n}=\sqrt{n}\left( \varphi _{\theta }\left( \user2{T}_{n}\right)-\varphi_{\theta}\left( \varvec{\mu }\right) \right)$ converges in distribution to $N\left( 0,\varphi _{\theta }^{\prime }\Sigma \left( \varphi _{\theta }^{\prime }\right) ^{t}\right)$ by the multivariate delta method [e.g., 26, pp. 25–26]. □