Incorporation of concentration data below the limit of quantification in population pharmacokinetic analyses
Ron J Keizer, Robert S Jansen, Hilde Rosing, Jos H Beijnen, Jan HM Schellens, Alwin DR Huitema
Department of Pharmacy and Pharmacology, Slotervaart Hospital / The Netherlands Cancer Institute
Introduction: In population pharmacokinetic (PopPK) analyses, the modeler is often confronted with concentrations below the lower limit of quantification (LLOQ), generally censored as "BLQ". The LLOQ is determined by a specified level of bias and precision (usually 20%) while concentration below the LLOQ may still by quantifiable albeit with increased bias and precision. Concentrations below the limit of detection (LOD) can not be quantified. In general the LLOQ is in the order of 3-5 times the LOD. Several methods have been proposed to handle these data, such as discarding them, replacing them with LLOQ/2, or the use of likelihood-based methods.(1-3) However, we hypothesize that using the actual concentration data extrapolated below the LLOQ has superior performance over the established methods, and decreases bias and imprecision of parameter estimates.
Objectives: Investigate the validity of using extrapolated BLQ concentration data in PopPK analyses, and compare performance to established BLQ methods.
Methods: First, to quantify the contribution of analytical error on overall residual error an analytical error model was constructed and fitted to results from analytical method validations from our own laboratory and from analyses published in literature. This model allowed description of the precision of the analytical methods over the entire concentration range (scaled using the LLOQ). Another model was defined which described a ‘worst-case' analytical method that just complied with FDA standards. The analytical error model was combined with a proportional error model (20% error) to account for model misspecification.
Using these residual error models, simulation and re-estimation analyses were performed using R, NONMEM and Perl, for various levels of BLQ censoring (10%, 20% and 40%), and several i.v. and oral PK models. The performance (in terms of RMSE, and run success) was evaluated for the following BLQ approaches: ‘Discard', in which BLQ data was discarded, ‘LLOQ/2': all BLQ data in the absorption phase were substituted with LLOQ/2, and in the elimination phase the first BLQ observation was substituted with LLOQ/2 while subsequent points were discarded, ‘M3': simultaneously modelling of the continuous data above the LLOQ and binary data below the LLOQ (1,2), ‘All data': using all concentration data, including BLQ concentrations, as continuous data (but discarding data below the limit of detection). Subsequently, the influence of several additional factors was investigated: the use of NONMEM7 instead of NONMEM6, the use of the new SAEM estimation method in NONMEM7, and the use of another approach ‘M3LOD' in which the M3 method was only used for points below the LOD.
Results: For all evaluated PK models and levels of censoring, RMSE values were lowest using the ‘All data' method. Performance of the M3 method was generally better than the ‘LLOQ/2' or ‘Discard' method, while differences between all methods were small at the lowest level of censoring. Using the ‘M3' method, low percentages of runs were reported as successful (<50%) and even lower percentages of covariance steps were performed (<30%), although a considerable percentage of runs did produce parameter estimates (~90%).
The ‘worst-case' analytical error model showed bias and precision comparable to the situation of the ‘average' analytical error: RMSE values were lowest for the ‘All data' method, except at the highest level of BLQ censoring where the ‘M3' provided better results. NONMEM7 using either the Laplacian or the SAEM method provided similar performance to NONMEM6, although the percentages of successful runs was about 20% higher. The ‘M3LOD‘ approach resulted in slightly larger RMSE values and more unsuccessful runs compared to the ‘All data' method.
Conclusion: The incorporation of BLQ concentration data showed superior performance in terms of bias and precision over established BLQ methods. This indicates that the use of BLQ data as a continuous data source is a valid approach in PopPK modelling. Investigations in the use of this approach for real PK datasets is in progress.
References
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2. Ahn J, Karlsson M, Dunne A, Ludden T. Likelihood based approaches to handling data below the quantification limit using NONMEM VI. Journal of Pharmacokinetics and Pharmacodynamics. 2008 Aug 7;
3. Bergstrand M, Karlsson MO. Handling data below the limit of quantification in mixed effect models. AAPS J. 2009 Jun;11(2):371-380.
