Handling Below Limit of Quantification Data in Optimal Trial Design
Camille Vong*, Sebastian Ueckert*, Joakim Nyberg and Andrew C. Hooker
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: The analysis of clinical trial data with classical statistical methods is often influenced by data below the limit of quantification (LOQ). Nonlinear mixed effect models provide methods for using the information present in that data [13]. The aim of this work was to evaluate different methods of handling LOQs in Optimal Design (OD).
Methods: Six different methods were implemented in PopED [4]:

D1: Ignore LOQ.

D2: Noninformative Fisher information matrix (FIM) for median response below LOQ (FO) i.e. set the contribution to the FIM to zero if a design point gives a median response below LOQ.

D3: Noninformative FOCE linearized FIM for individual response below LOQ i.e. set the individual contribution to the FIM to zero if a design point gives an individual response below LOQ.

D4: Simulation & Rescaling i.e. Scale FIM with the probability of BLQ predicted from simulation.

D5: Integration & Rescaling i.e. Scale FIM with the probability of BLQ calculated from the FO approximated joint density.

D6: Calculation of FIM by integrating over simulated data with a joint likelihood for data above (normal likelihood) and below LOQ (M3 method) using the Laplace approximation.
Comparisons were performed using a 1cmp IV bolus model with a standard design of 50 patients and 4 sample times per individual. Performance of D1D6 was assessed for 5 LOQs (39, 42, 47, 51, 63% < LOQ). Predicted parameter relative standard errors (RSE) were compared to empirical RSEs obtained from multiple stochastic simulations and estimations (SSE) in NONMEM using the M3 method [2]. Optimizations using a 2cmp IV bolus model with standard design of 200 patients and 5 sample times per individual were performed using the fastest methods. Resulting designs were assessed in terms of bias and precision from SSEs using the M3 method.
Results: Evaluated and SSEderived RSEs for the 6 methods were in good agreement. Determinants of the FIM derived from Method D4D6 in general were the closest to the empirical covariance obtained from SSEs. FIM calculation times relative to D1 were D2=1.27, D3=21115, D4=137, D5=7.99 and D6=37904. While optimizing with methods D1, D2, D4 and D5 for LOQs up to 70% censored data, D5 provided the most accurate and precise parameter estimates. Method D2 resulted in the least robust designs for estimation.
Conclusion: The use of OD methods anticipating BLQ data in planned designs allows better parameter estimations. For the scenarios investigated, method D5 showed the best compromise in terms of speed and accuracy.
Acknowledgement: This work was part of the DDMoRe project.
References:
[1] Beal, S.L., Ways to fit a PK model with some data below the quantification limit. J Pharmacokinet Pharmacodyn, 2001. 28(5): p. 481504.
[2] Ahn, J.E., et al., Likelihood based approaches to handling data below the quantification limit using NONMEM VI. J Pharmacokinet Pharmacodyn, 2008. 35(4): p. 40121.
[3] Bergstrand, M. and M.O. Karlsson, Handling data below the limit of quantification in mixed effect models. AAPS J, 2009. 11(2): p. 37180.
[4] Foracchia, M., et al., POPED, a software for optimal experiment design in population kinetics. Comput Methods Programs Biomed, 2004. 74(1): p. 2946.
* Both authors contributed equally to this work.