Daniel Kaschek (1), Henning Schmidt (1)
(1) IntiQuan AG, Basel, Switzerland
Objectives: The M3 method is known to NONMEM users as a standard method for handling of below the limit of quantification (BLOQ) concentration data [1, 2]. The same concept is implemented in Monolix, denoted as handling of left-censored data [3]. In the M3 method, the likelihood of observing a BLOQ concentration at a certain time point is computed based on the model prediction at that time point. When using NONMEM, the model prediction typically is based on a compartmental pharmacokinetic (PK) or pharmacokinetic-pharmacodynamic (PK/PD) model. However, the method itself is much more general and applicable to simpler situations such as the estimation of a mean and standard deviation from multiple replicates, e.g., for data-exploratory analysis or non-compartmental analysis (NCA).
Methods: For the estimation of the mean µ and standard deviation σ from multiple replicates we assumed an underlying normal distribution. Furthermore, we assumed that the observed sample could have both observations above and below the LLOQ. Given the probability density of the normal distribution, we formulated the likelihood of obtaining the specifically measured values above the LLOQ and the number of values below the LLOQ. Using twice the negative logarithm of the likelihood as an objective function, we employed numerical optimization to estimate the parameters µ and σ from the normal distribution. The entire approach was implemented in a short and simple to use function in R.
We applied the implemented function to simulated profiles of a two-compartmental distribution model with linear elimination. Trial simulation (N = 1000) was performed for cohorts of 12 subjects with inter-individual variability and residual variability for different elimination scenarios. For each scenario, the clearance parameter was chosen such that the typical prediction at the last observed time point was above, at, or below the LLOQ. Subsequently, the geometric mean of the simulated cohort sample at the last time point was determined by our implementation of the M3 method. Alternatively, the geometric mean was determined from the same sample with BLOQ observations removed (M1 method) or with BLOQ observations imputed by LLOQ/2. The three methods were compared by their geometric mean ratio (GMR) with the geometric mean from the uncensored sample in the simulation.
Results: The results showed that for scenarios with a low proportion of BLOQ observations at the last time point (< 5%), all three methods adequately estimated the geometric mean concentration as indicated by GMRs between 0.99 and 1.02. When the typical prediction was close to the LLOQ and the proportion of BLOQ samples was around 50%, the LLOQ/2 method underestimated the geometric mean (GMR = 0.85) whereas the M1 method overestimated it (median GMR = 1.4). In contrast, the M3 method allowed us to estimate the geometric mean without bias (GMR = 1.03). Even when typical predictions were so low that 93% of the simulated predictions were BLOQ, the M3 method still reasonably estimated the geometric mean (GMR = 1.34).
Conclusions: The M3 method is frequently encountered in population PK and PK/PD modeling with both NONMEM and Monolix. In this work, we have shown that the method can easily be adapted to situations such as data-exploratory analysis and NCA which usually precede population PK modeling. The implementation in R is straight forward, fast, and converges reliably. Based on our results, the method is especially helpful in interpreting the slope of concentration-time profiles at low concentrations, leading to a more accurate estimate of the terminal slope, and supporting the development of an appropriate population PK model. Similar benefits arise for population PK/PD modeling.
References:
[1] Ahn JE, Karlsson MO, Dunne A, Ludden TM. Likelihood based approaches to handling data below the quantification limit using NONMEM VI. Journal of pharmacokinetics and pharmacodynamics. 2008 Aug;35:401-21.
[2] Beal SL. Ways to fit a PK model with some data below the quantification limit. Journal of pharmacokinetics and pharmacodynamics. 2001 Oct;28:481-504.
[3] Monolix 2023R1, Lixoft SAS, a Simulations Plus company
Reference: PAGE 32 (2024) Abstr 10839 [www.page-meeting.org/?abstract=10839]
Poster: Methodology - Other topics