Different ways of incorporating data below the quantification limit in datasets for parameter estimation

Background: The handling of data below the lower limit of quantification (LLOQ) and the limit of detection (LOD) continues to be a challenge in pharmacokinetic analyses. The LLOQ is the lowest concentration of an analyte in a sample which can be measured with a predefined acceptable level of accuracy and precision (± 20 %). Whereas the LOD is the lowest concentration of an analyte that the assay can reliably differentiate from background noise[1]. Often concentrations below these quantification limits (BQL) will not be reported by analytic chemists.

Beal[2] showed in 2001 that fixing all BQL concentrations to zero or discarding these data points results in bias. He suggested some more complex, but also some straightforward ways to handle this BQL data.

Objectives: Evaluating parameter estimates depending on different methods of handling data below the limit of detection.

Methods: Our group recently collected a total of 241 blood samples in a clinical pharmacokinetic study which were analysed using NONMEM version 5 (level 1.1; GloboMax LLC, Hanover, MD, USA). The data set was used to develop a simultaneous model for parent and metabolite.[3] The concentrations of parent and metabolite were analysed simultaneously by HPLC and both had the same LOD of 0.04 mg/L.[4] Forty six per cent of the parent and twenty eight percent of the metabolite concentrations were below the LOD. The entire profile after administration of the capsule for five patients was below LOD and therefore the data from this occasion was omitted from estimation.

Four methods were assessed for modelling of the PK dataset. These methods (corresponding to methods 1, 5, 6 & 4 from Beal[2]), respectively were M1) where all values less than LOD were discarded, M5) where all values less than LOD were assigned to half of LOD, M6) where the closest missing value that is less than LOD was assigned to half the LOD and all previous (if during absorption) or subsequent (if during elimination) missing samples were deleted, and M4) where the contribution of the expectation of each missing concentration to the likelihood was estimated.

Results and Discussion: For the PK analysis of the observed data one of the simpler methods (M5) was used for the modelling process. The final model was evaluated using a visual predictive check and a bootstrap procedure providing confidence in the estimates for the population pharmacokinetic parameters of the final model.

The parameter estimates of the final model (using M5) were compared to the parameter estimates of the three other methods. All models resulted in successful convergence under FOCE and INTERACTION, however for the M4 model the fixed effect parameters for volume of the peripheral compartment (V3) for the parent and the inter-compartment clearance (Q) were fixed and the residual error model was changed to a combined proportional and additive error model to account for the more complicated error structure.

The parameter estimates from the final model (M5) were similar compared to the different methods 4 and 6. An expected bias of under-prediction for clearance was shown using M1 and therefore BQL samples cannot be ignored. Negligible differences in the parameter estimates were shown comparing the M5 method and the more complex M4 method. Furthermore, there was little difference between the parameters estimated with M5 and M6. Method 4 required a more complex set up and run times were significantly longer. It is apparent from the residual error estimates (comparing all methods) that using a nonlinear mixed-effects modelling approach can account for the reduced accuracy and precision of BQL data.

Conclusion: To incorporate missing data, using the simple Beal method 5 (using half LOD for all samples below LOD) provided comparable results to the more complex but theoretically better Beal method 4 (integration method).


  1. Shah VP, Midha KK, Findlay JWA, Hulse JD, J MI. Bioanalytic method validation – a revisit with a decade of progress. Pharmaceut Res 2000;17:1551-7.
  2. Beal SL. Ways to fit a PK model with some data below the quantification limit. J Pharmacokinet Pharmacodyn 2001;28:481-504.
  3. Hennig S, Waterhouse TH, Bell SC, France M, Wainwright CE, Miller H, et al. A d-optimal designed population pharmacokinetic study of oral itraconazole in adults cystic fibrosis patients. Br J Clin Pharmacol 2006;Oct 30 (online ahead of print).
  4. Redmann S, Charles BG. A rapid HPLC method with fluorometric detection for determination of plasma itraconazole and hydroxy-itraconazole concentrations in cystic fibrosis children with allergic bronchopulmonary aspergillosis. Biomed Chromatogr 2006;20:343-8.

Stefanie Hennig