Marie Wijk (1), Roeland Wasmann (1), Karen Jacobson (2), Elin Svensson (3,4), Paolo Denti (1)
(1) Division of Clinical Pharmacology, Department of Medicine, University of Cape Town, Cape Town, South Africa, (2) Section of Infectious Diseases, Boston University School of Medicine and Boston Medical Centre, Boston, MA, USA, (3) Department of Pharmacy, Uppsala University, Uppsala, Sweden, (4) Department of Pharmacy, Radboud University Medical Center, Nijmegen, Netherlands
Objectives: Adequate handling of data below the lower limit of quantification (BLQ) is key for parameter estimation in population pharmacokinetic (PK) analyses. The M3 method [1], which considers BLQ data as censored and computes the likelihood of the data being BLQ has been shown to introduce least bias and is the de facto gold standard [2,3]. However, M3 suffers from limitations in “real world” scenarios, i.e. increasing run times and failing to minimize consistently, and evaluation of M3 on more complex models (e.g., $DES, high non-linearity) is lacking [4,5]. Alternative frequently used methods are excluding the BLQ data (M1) or imputing a value (e.g., M6 and M7). M6 imputes half of the lower limit of quantification (LLOQ) and handles BLQ values in a series by only fitting the BLQ sample closest to the maximal concentration and discarding the rest, while M7 imputes 0 for all values [1]. All these methods may introduce bias [4,6]. Of note, M6 and M7 do not account for the fact that imputed values carry additional uncertainty compared to observed ones, which may increase bias. This work aims to assess the stability and accuracy of BLQ handling methods in a complex model.
Methods: We first evaluated stability (consistent convergence to the same solution) of the methods on real data, then reassessed stability and accuracy on simulated data. We compared M1, M3, M6, M7 and modifications of M6 and M7 (M6+ and M7+) including extra additive error (i.e., LLOQ/2) for imputed data, to inform the model that those observations are less informative. Isoniazid PK data for 100 individuals from the study TRUST (NIAID R01AI119037) [7] was used as real data, with plasma samples collected at steady-state pre-dose and 1.5, 3, 5, and 8 h post-dose. The LLOQ was 0.105 mg/L and 23% of 491 samples were BLQ. A previously developed NONMEM model (ADVAN13, FOCE-I) fit this data well: two-compartment disposition with transit compartment absorption, first-order elimination with hepatic extraction and a mixture model to assign N-acetyltransferase 2 metabolizer status (affecting clearance) as genetic data was missing. Residual error was described by a combined additive and proportional model, with the additive component constrained to >= 20% of LLOQ. Parallel retries were run (n=10) with 30% tweaking of initial estimates and the resulting SD of OFV was used to compare stability [8]. This was repeated on simulated data to determine bias and accuracy against the “true” estimates. We used a slightly modified version of the model with slower absorption and richer sampling, thus increasing the proportion of BLQs to 34%. Model stability was tested with parallel retries. To compare accuracy in parameter estimates we used bias [4] and relative root mean squared error (rRMSE) obtained with stochastic simulation and estimation (SSE). 50 datasets with the richer sampling schedule were simulated and estimation was executed with 30% variability in initial estimates.
Results: On the real data, M3 converged to very different solutions in each retry (OFV SD=71). M7 was also unstable and estimated implausible parameter estimates, whereas the other methods converged to similar solutions (OFV SD<1). Similar results were seen on simulated data, except for M1 being more unstable (OFV SD=8.1) and M6+ and M7+ getting stuck in local minima in 1-2 retries, increasing their SD from 0.1 and 0.9 to 2.9 and 4.3, respectively. When comparing bias and accuracy using SSE, M3 performed best. M7 had very high rRMSEs (40% to >1000%) and M1 and M6 performed poorly in most parameters. M6+ and M7+ performed similarly to M3, and better in clearance of both slow and fast/intermediate metabolizers (rRMSE 5.8-7.5% vs ~16%).
Conclusions: Using M3 is suboptimal for model development when dealing with complex models, due to its lack of consistency in converging towards a stable solution. This is evidenced by the high variability in OFV produced by parallel retries with perturbed initial estimates and makes it challenging to use drop in OFV as a measure of significance for covariates. The imputation methods (except M7) showed superior stability in both the real and simulated data but may introduce bias and the imputation method should be chosen carefully. Including extra additive error as in M6+ and M7+ mitigates the impact of the imputation and performs similarly to M3 in terms of accuracy but with significantly improved model stability. In summary, we propose M6+ and M7+ as attractive alternatives to M3 when developing complex models.
References:
[1] Beal, S. L. (2001). Ways to fit a PK model with some data below the quantification limit. Journal of pharmacokinetics and pharmacodynamics, 28, 481-504.
[2] Bergstrand, M., & Karlsson, M. O. (2009). Handling data below the limit of quantification in mixed effect models. The AAPS journal, 11, 371-380.
[3] Ahn, J. E., Karlsson, M. O., Dunne, A., & Ludden, T. M. (2008). Likelihood based approaches to handling data below the quantification limit using NONMEM VI. Journal of pharmacokinetics and pharmacodynamics, 35, 401-421.
[4] Keizer, R. J., Jansen, R. S., Rosing, H., Thijssen, B., Beijnen, J. H., Schellens, J. H., & Huitema, A. D. (2015). Incorporation of concentration data below the limit of quantification in population pharmacokinetic analyses. Pharmacology research & perspectives, 3(2), e00131.
[5] Bauer, R. J. (2019). NONMEM tutorial part II: estimation methods and advanced examples. CPT: pharmacometrics & systems pharmacology, 8(8), 538-556.
[6] Duval, V., & Karlsson, M. O. (2002). Impact of omission or replacement of data below the limit of quantification on parameter estimates in a two-compartment model. Pharmaceutical research, 19, 1835-1840.
[7] PAGE 30 (2022) Abstr 10155 [www.page-meeting.org/?abstract=10155]
[8] https://uupharmacometrics.github.io/PsN/docs.html
Reference: PAGE 32 (2024) Abstr 10958 [www.page-meeting.org/?abstract=10958]
Poster: Methodology - New Modelling Approaches