Hoai-Thu Thai (1), France Mentré (1), Nicholas H.G. Holford (3), Christine Veyrat-Follet (2), Emmanuelle Comets (1)
(1) INSERM, UMR 738, F-75018 Paris, France; Univ Paris Diderot, Sorbonne Paris Cité, UMR 738, F-75018 Paris, France; (2) Drug Disposition Department, Sanofi, Paris, France; (3) Department of Pharmacology and
Clinical Pharmacology, University of Auckland, Auckland, New Zealand
Objectives: Nonparametric case bootstrap is frequently used in PK/PD for estimating standard error (SE) and confidence interval (CI) of parameters [1-2]. Residual bootstraps resampling both random effects and residuals are an alternative approach to case bootstrap which resamples entire individuals [3-4]. These methods have not been well studied in mixed-effects models (MEM). We aimed to study and propose appropriate bootstrap methods in MEM and to evaluate their performance by simulation using examples of disease progression model in Parkinson's disease [5] (for linear MEM) and PK model of aflibercept (Zaltrap®) [6], a novel anti-VEGF drug (for nonlinear MEM).
Methods: Different bootstraps accounting for between-subject and residual variabilities were implemented in R 2.14. Corrections of random effects and residuals for variance underestimation were investigated [7].
The bootstrap performances were first assessed in LMEM with homoscedastic error by a simulation (k=1000 replicates and B=1000 bootstrap samples/replicate) with 3 balanced designs (rich, sparse, large error). The best bootstraps in LMEM were then evaluated in the NLMEM with heteroscedastic error by a simulation (k=100/B=1000) with 2 balanced (frequent/sparse) and 1 unbalanced designs. Bootstraps were compared in terms of bias of parameters, SE and coverage rate of 95% CI. R 2.14 and MONOLIX 4.1 were used to fit the data in LMEM and NLMEM, respectively.
Results: Our simulations showed a good performance of the case bootstrap and the
nonparametric/parametric residual bootstraps with a correction for variance underestimation in LMEM [8].
In NLMEM, these methods performed well in the balanced designs, except for the sparse design where they greatly overestimated SE of a parameter estimate having a skewed distribution. In the unbalanced design, the case bootstrap overestimated the SE of this parameter and the nonparametric residual bootstrap overestimated the SE of variances even with stratification on frequent/sparse sampling. The asymptotic method performed well in most cases, except for low coverage rates of highly nonlinear parameters.
Conclusion: The bootstraps only provide better estimates of uncertainty in NLMEM with high nonlinearity compared to the asymptotic method. The nonparametric residual bootstrap works as well as the case bootstrap. However, they may face practical problems, e.g skewed distributions in parameter estimates and unbalanced designs where stratification may be insufficient.
References:
[1] Ette EI. Stability and performance of a population pharmacokinetic model. J Clin Pharmacol 1997;
37(6):486-495.
[2] Parke J, Holford NHG, Charles BG. A procedure for generating bootstrap samples for the validation of nonlinear mixed-effects population models. Comput Methods Programs Biomed. 1999; 59:19-29.
[3] Das S, Krishen A. Some bootstrap methods in nonlinear mixed-effects models. J Stat Plan Inference 1999;
75: 237-245.
[4] Ocana J, El Halimi R, Ruiz de Villa MC, Sanchez JA. Bootstrapping repeated measures data in a nonlinear
mixed-models context. Mathematics Preprint Series 2005.
[5] Holford NHG, Chan PLS, Nutt JG, Kieburtz K, Shoulson I and Parkinson Study Group. Disease progression and pharmacodynamics in Parkinson disease-evidence for functional protection with Levodopa and other treatments. J Pharmacokinet Pharmacodyn. 2006; 33 (3): 281-311.
[6] Thai H-T, Veyrat-Follet C, Vivier N, Dubruc C, Sanderink G, Mentré F, Comets E. A mechanism-based model for the population pharmacokinetics of free and bound aflibercept in healthy subjects. Br J Clin Pharmacol 2011, 72(3): 402-414.
[7] Carpenter JR, Goldstein H, Rasbash J. A novel bootstrap procedure for assessing the relationship between class size and achievement. Appl Statist 2003; 52:431-443. 26.
[8] Thai H-T, Veyrat-Follet C, Mentré F and Comets E. A comparison of bootstrap approaches for estimating standard errors of parameters in linear mixed effects models. Pharm Stat 2013, doi :10.1002/pst.1561.
Oral: Methodology - New Tools
Celia Barthelemy A-42 New methods for complex models defined by a large number of ODEs. Application to a Glucose/Insulin model
Celia Barthelemy and Marc Lavielle INRIA Saclay and University Paris-Sud
Objectives: Modelers are increasingly faced with complex physiological models represented by a large number of ordinary differential equations (ODEs). Widely used modeling algorithms need to evaluate the structural model a large number of times, and for instance SAEM, MCMC and Monte-Carlo algorithms can be extremely time-consuming when the model is defined by a large system of ODEs. There is therefore a need for new efficient numerical tools to help the modeler deal with such complex models. We propose an extension of these algorithms that limits the total number of times the ODEs need to be solved.
Methods: This new approach consists in first evaluating the structural model on a well-defined grid of parameters. Then, the structural model is approximated by interpolating these isolated values. The number of points of the grid defines the quality of the approximation. “Exact methods” are obtained when this number tends to infinity.
The proposed method has been evaluated using simulations. Performance was assessed by computing the root mean square error (RMSE) and the computing time required for several tasks: estimation of the population and individual parameters, evaluation of the Fisher information matrix, evaluation of the log-likelihood, and creation of VPCs.
Results: We illustrate the method on simulated datasets from the glucose/insulin model proposed by Alvehag [1]. This model is composed of 29 EDOs, and 5 parameters are estimated. For a grid of 11^5 points, the elapsed time for each task is approximately divided by 7. For example, the times for the original and extended SAEM algorithms are respectively 3 minutes and 42 seconds, and the increase in the RMSE does not exceed 5%.
Conclusions: Encouraging results have been obtained with models defined by a large system of ODEs and a relatively small number of unknown parameters. In particular, the population parameters are estimated with little bias whereas the estimation is significantly faster compared to standard SAEM. This method makes feasible the use of more and more realistic physiological models.
Attempts can now be made to extend such approaches to models defined with a larger number of parameters. Application to spatial models defined by Partial Differential Equations (PDEs) could also be considered.
References:
[1] Karin Alvehag. Glucose regulation, a mathematical model (2006).
[2] Donnet and Samson. Estimation of parameters in missing data models defined by differential equations.
ESAIM: Probability and Statistics (2005)
Acknowledgments: This work was supported by the DDMoRe project (www.ddmore.eu).
Poster session I: Wednesday morning 10:05-11:40 Poster: New Modelling Approaches