Yasunori Aoki, Joakim Nyberg, Mats O. Karlsson
Department of Pharmaceutical Biosciences, Uppsala University, Sweden, National Institute of Informatics, Japan
Objectives:
Linearization of models with respect to the random effect variable has been proved to work well for continuous models and the FOCE(I) approximation [1]. The approach enables fast and sometimes more stable estimation of covariate and random effects models, once as a structural model has been identified. However, for likelihood-based models, first order mean and variance linearization is not feasible since the data are not implicitly assumed normally distributed. Instead, a second order Taylor expansion is needed. The aim of this work is to implement a second order Taylor expansion to improve speed and stability for likelihood-based models.
Methods:
The -2 log likelihood of the individual observation is approximated with second-order Taylor series expansion at the empirical Bayes estimate (EBE). The gradient vector and Hessian matrix required for this expansion are calculated using e.g. numerical differentiation of the individual log likelihood with respect to the EBEs (e.g., in NONMEM these values can be obtained as G(,) in verbatim code). The key motivation for this approach is that, since Laplace approximation uses the second order approximation of the log likelihood when using Laplace approximation, the above approach does not influence the final outcome of the likelihood calculation.
Once the approximate model is constructed, it can be modified to add and test covariate relationships or change the random effects. The key advantage of this approximation is that it does not require any additional computation of the structural model predictions. Hence extensive explorations of various functions for the covariates and random effects models become feasible, in fact using an analytical quadratic approximation, even for models with long computation time, e.g. for differential equations models. This methodology was tested using the following different models:
- pharmacokinetics model of Phénobarbital (PHENO) [2]
- pharmacokinetics model of Phénobarbital with lower limit of quantification treatment using M3 method (PHENO with M3) [2]
- minimal continuous-time Markov model for the Likert pain score (mCTMM) [3]
- first-order Markov model for adverse reactions (Fatigue) experienced by Sunitinib-treated patients (MARKOV) [4]
- bounded integer model for ADAS-cog score (BI) [5,6]
For MARKOV and BI, we have conducted extensive covariate search using the SCM method implemented in PsN [7,8].
Results:
The comparison of the computed -2 log likelihood (OFV) and computation time of the final model are tabulated below. (original model/approximated model using proposed method)
PHENO: 867.61 / 868.31 0.02 sec / 0.01 sec
PHENO with M3: 818.18 / 817.63 0.02 sec / 0.01 sec
mCTMM: 48902.16 / 48900.61 526 sec / 0.02 sec
MARKOV (base model): 6765.61 / 6763.99 3632sec / 59 sec
MARKOV (SCM): 6765.61 / 6763.99 9512.95 sec / 697.38 sec
BI (base model): 28122.99 / 28122.99 76.07 sec / 1.49 sec
BI (SCM): 27569.13 / 27569.10 14099.83 sec / 168.08 sec
As can be seen from the above table, some approximation error can be observed (e.g., OFV difference of 1.62 for MARKOV); however, a significant speed up in the computation was achieved (e.g., 4hr to 3min for SCM of BI).
For MARKOV, the SCM did not find any statistically significant covariates. For BI, the baseline Minimum Mental State Examination score was significant covariates for both baseline ADAS-cog score and disease progression. Both of these results were consistent between the original model and approximated model while it was significantly faster to run SCM using approximated model.
Conclusions:
A strategy to implement second-order Taylor expansion of the likelihood of the nonlinear mixed effect model was derived, which enables fast and stable assessments of covariates for likelihood-based models. The proposed method expands the use of the linearization technique [1,7,9] to also include likelihood-based models.
References:
[1] Khandelwal A, Harling K, Jonsson EN, Hooker AC, Karlsson MO. A Fast Method for Testing Covariates in Population PK/PD Models, AAPS J. 2011;13:464–472
[2] Grasela Jr. TH, Donn SM. Neonatal Population Pharmacokinetics of Phénobarbital Derived from Routine Clinical Data, Dev Pharmacol Ther. 1985;8:374–83
[3] Schindler E and Karlsson MO. PAGE 26 (2017) Abstr 6077 [www.page-meeting.org/?abstract=6077]
[4] Hansson EK, Ma G, Amantea MA, French J, Milligan PA, Friberg LE, and Karlsson MO. PKPD Modeling of Predictors for Adverse Effects and Overall Survival in Sunitinib-Treated Patients With GIST, CPT Pharmacometrics Syst Pharmacol. 2013;2(12):e85
[5] Karlsson MO and Wellhagen G. A bounded integer model for rating and composite scale data. PAGANZ 2018
[6] Wellhagen G and Karlsson MO. A bounded integer model for rating and composite scale data. PAGE 27 (2018)
[7] Jonsson EN, Karlsson MO. Automated covariate model building within NONMEM., Pharm Res. 1998 Sep;15(9):1463-8.
[8] Keizer RJ, Karlsson MO, Hooker A. Modeling and Simulation Workbench for NONMEM: Tutorial on Pirana, PsN, and Xpose. CPT Pharmacometrics Syst Pharmacol. 2013 Jun 26;2:e50. doi: 10.1038/psp.2013.24.
[9] Svensson EM, Karlsson MO. Use of a linearization approximation facilitating stochastic model building, J Pharmacokinet Pharmacodyn. 2014;41:153-158
Reference: PAGE 27 (2018) Abstr 8751 [www.page-meeting.org/?abstract=8751]
Poster: Methodology - Covariate/Variability Models