Prediction of precision of individual parameter estimates and of shrinkage via the Bayesian Fisher information matrix in non-linear mixed-effects models with application in pharmacokinetics
F. Combes (1,2,3), S. Retout (2), N. Frey (2) and F. Mentré (1)
(1) INSERM, UMR 738, Univ Paris Diderot, Sorbonne Paris cité, Paris, France; (2) Pharma Research and Early Development, Translational Research Sciences, Modeling and Simulation, F. Hoffmann-La Roche ltd, Basel, Switzerland; (3) Institut Roche de Recherche et Médecine Translationnelle, Boulogne-Billancourt, France
Objectives: In population pharmacokinetics (PK), precision of population parameter estimates depends on design and are evaluated using Fisher information matrix [1]. Individual parameters are usually estimated by the Maximum A Posteriori (MAP) and precision of individual estimates can be evaluated using the Bayesian Fisher information Matrix (BMF) [2]. Shrinkage of individual parameters towards the mean occurs when information is sparse and can be quantified as a reduction of variance of the estimated Random Effects (RE) [3]. This study aims at 1) exploring the relationship between BMF and shrinkage in order to propose a prediction of shrinkage and 2) evaluating by simulation the prediction of individual parameter precision and shrinkage.
Methods: We first derived the BMF for additive RE and constant residual error and then extended it for exponential RE and/or combined residual error. From the formula of shrinkage in linear mixed effects models, we derived the normalized Estimation Variance (nVE) from the expected BMF as a prediction of shrinkage. Regarding the evaluation by simulation, we simulated data from sparse and rich design for two PK examples: a simple one (one compartment) with six different scenarios (additive or exponential RE, with low and high variabilities, additive or combined residual error); a more complex example derived from a real case study [4] (two compartment, dual linear and non-linear elimination). We used NONMEM 7.2 and MONOLIX 4.0 to perform individual estimation via MAP assuming known population parameters and fixed to their exact value. We also recorded individual standard errors (SE). We then compared predicted and estimated SE for each scenario and example as well as the predicted and estimated shrinkage, evaluated using the formula with ratio of variances.
Results: For the simple example, considering all scenarios and designs, predicted SE of the two parameters using BMF were close to the estimated SE with both software and varied as expected with the richness of the design and the variabilities. There were also a very good agreement (almost identity line) between estimated shrinkage (which varies from 0 to 70%) and predicted shrinkage. Similar results were observed for all the parameters of the real example.
Conclusion: The Bayesian Information Matrix allows to evaluate impact of design on precision of individual parameters and to predict shrinkage. This can be used for design optimization and will be implemented in PFIM.
References:
[1] Retout S, Duffull S, Mentré F. Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs. Comput Methods Programs Biomed, 2001;65(2):141-151
[2] Merlé Y, Mentré F. Bayesian design criteria: computation, comparison and application to a pharmacokinetic and a pharmacodynamic model. J Pharmacokinet Biopharm, 1995;23(1):101-25
[3] Savic R, Karlsson M. Importance of shrinkage in empirical bayes estimates for diagnostics: problems and solutions. The AAPS J, 2009;11(3):558-69
[4] Frey N, Grange S, Woodworth T. Population pharmacokinetics analysis of tocilizumab in patients with rheumatoid arthritis. J Clin Pharmacol, 2010 Jul;50(7):754-66
