Model B was a two-compartment model with intravenous dosing. In the simulated datasets BQL observations were located with absolute predominance in the absorption phase. Model A was a one compartment model with oral dosing and absorption transit compartments ( 14). Three distinctly different models were used for simulation. This publication includes a suggested extension to the traditional VPC to better diagnose models in the presence of BQL samples. The presence of non-randomly censored data such as BQL samples should be taken into account when producing VPCs in order to not introduce bias in comparison of observations and model predictions.
Visual predictive check (VPC) is a valuable tool in diagnosing the performance of mixed effect models ( 9). However BQL samples also frequently occur in the absorption phase of PK models and in various ways in PD models for continuous biomarkers. The influence of BQL samples and the performance of the likelihood based methods have previously only been assessed for the case where BQL samples occur in the terminal elimination phase of a one- or two-compartment PK model ( 5, 7, 8). The use of this feature has been reported to improve the performance of the M3 method compared to the original implementation ( 3, 7). The possibility for simultaneous modeling of continuous and categorical data is improved in NONMEM VI and easily implemented by the use of the indication variable F_FLAG. An alternative to using M4 is to log-transform the dependent variable and use the somewhat simpler M3 approach. M4 differs from M3 by the fact that it conditions on the observations being greater than zero. The likelihood for BQL observations are maximized with respect to the model parameters and the likelihood for an observation is taken to be the likelihood that it is indeed below LOQ. The methods M3 and M4 also suggested by Beal are based on simultaneous modeling of continuous and categorical data where the BQL observations are treated as categorical data. This approach can be implemented in NONMEM VI by utilization of the YLO functionality ( 6). The method referred to as M2 in the publication applies conditional likelihood estimation to the observations above LOQ and the likelihood for the data being above LOQ are maximized with respect to the model parameters. In 2001, Stuart Beal published an overview of ways to fit a PK model in the presence of BQL data ( 5). Common approaches for handling of concentration measurements reported as BQL, such as discharging the information or substitution with the LOQ divided by two, have been shown to introduce bias in parameter estimates ( 2– 4). Samples associated with a signal less than LOQ are by this standard reported as below the quantification limit (BQL). The limit of 20% CV originates primarily from the FDA Guidance for Industry Bioanalytical Method Validation ( 1). The new standard VPCs was found to identify model misfit more clearly than VPCs of data above LOQ only.īioanalytical laboratories traditionally define the lower limit of quantification (LOQ) as the lowest concentration on the standard curve that is associated with a coefficient of variation (CV) of less than 20%. Results following substitution of BQL observations with LOQ/2 were in some cases shown to introduce bias and were always suboptimal to the best method. In the tested examples this method generated overall unbiased parameter estimates. Best performance was seen when the likelihood of being below LOQ was incorporated into the model. Omission of BQL data was associated with substantial bias in parameter estimates for all tested models even for seemingly small amounts of censored data. An improved standard for VPCs was suggested to better evaluate simulation properties both for data above and below LOQ.
Different approaches for handling of BQL data were compared with estimation of the full dataset for 100 simulated datasets following models A, B, and C.
The third model, C, an indirect response model illustrated a case where the variable of interest in some cases decreases below the LOQ before returning towards baseline. Model A was used to represent a case with BQL observations in an absorption phase of a PK model whereas model B represented a case with BQL observations in the elimination phase. Three typical ways in which BQL can occur in a model was investigated with simulations from three different models and different levels of the limit of quantification (LOQ). The purpose of this study is to investigate the impact of observations below the limit of quantification (BQL) occurring in three distinctly different ways and assess the best method for prevention of bias in parameter estimates and for illustrating model fit using visual predictive checks (VPCs).