Estimating Area Under the Curve and Relative Exposure in a Pharmacokinetic Study with Data Below Quantification Limit

Abstract

Area under the drug-concentration-over-time curve (AUC) is an important endpoint for many phase I/II clinical trials and laboratory assays. Drug concentrations are measured using laboratory assays with a lower limit of quantification (LLOQ). How to calculate AUC when some drug concentration data are below the LLOQ remains as a challenge. In this article, we develop a maximum likelihood method to estimate AUC and relative exposure (i.e., ratio of two AUCs) when data below LLOQ exists. We also compare the proposed method to several commonly used methods, including imputation with model-predicted values or ad hoc values (i.e., LLOQ, LLOQ/2, or zero) through a simulation study. The proposed method gives unbiased inference. Commonly used methods can provide biased estimation, especially when a large proportion of data is below LLOQ. Application to a case study is also presented.

Key Words:

• Left-censored data
• Nonlinear mixed effects model
• Quantification limit

ACKNOWLEDGMENTS

We thank the associate editor and the two anonymous reviewers for their careful review and constructive comments, which greatly improved the content and presentation of this article.

Notes

¹Bias in percentage = bias/true value ×100%.

²MSE in percentage = MSE/true value ×100%.

Note. AUC = AUC _{0- 24h}. True values of the parameters: AUC _2mg = 1.08, AUC _3mg = 2.16, r = 0.5. MLLC1: maximum likelihood approach with AUC derived parametrically from the model; RMV1: discard BQL data and derive AUC parametrically from the model; MLL2: maximum likelihood approach with AUC derived using trapezoidal rule; RMV2: discard BQL data and derive AUC using trapezoidal rule; M0/M0.5/M1: replace BQL data with 0/0/5/1 of LLOQ and derive AUC using trapezoidal rule.

¹ r = estimated relative exposure, defined as the ratio of two area under the curve of the drug concentration, based on 1000 simulations.

²90% CP = coverage probability of 90% confidence interval of r.

Note. AUC = AUC _0-96h.

^a p Value is for hypothesis test H₀: r ≤ 0.5 vs. Ha: r > 0.5.