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**Harvard**

Almquist, J., Leander, J. och Jirstrand, M. (2015) *Using sensitivity equations for computing gradients of the FOCE and FOCEI approximations to the population likelihood*.

** BibTeX **

@article{

Almquist2015,

author={Almquist, Joachim and Leander, Jacob and Jirstrand, Mats},

title={Using sensitivity equations for computing gradients of the FOCE and FOCEI approximations to the population likelihood},

journal={Journal of Pharmacokinetics and Pharmacodynamics},

issn={1567-567X},

volume={42},

issue={3},

pages={191-209},

abstract={The first order conditional estimation (FOCE) method is still one of the parameter estimation workhorses for nonlinear mixed effects (NLME) modeling used in population pharmacokinetics and pharmacodynamics. However, because this method involves two nested levels of optimizations, with respect to the empirical Bayes estimates and the population parameters, FOCE may be numerically unstable and have long run times, issues which are most apparent for models requiring numerical integration of differential equations. We propose an alternative implementation of the FOCE method, and the related FOCEI, for parameter estimation in NLME models. Instead of obtaining the gradients needed for the two levels of quasi-Newton optimizations from the standard finite difference approximation, gradients are computed using so called sensitivity equations. The advantages of this approach were demonstrated using different versions of a pharmacokinetic model defined by nonlinear differential equations. We show that both the accuracy and precision of gradients can be improved extensively, which will increase the chances of a successfully converging parameter estimation. We also show that the proposed approach can lead to markedly reduced computational times. The accumulated effect of the novel gradient computations ranged from a 10-fold decrease in run times for the least complex model when comparing to forward finite differences, to a substantial 100-fold decrease for the most complex model when comparing to central finite differences. Considering the use of finite differences in for instance NONMEM and Phoenix NLME, our results suggests that significant improvements in the execution of FOCE are possible and that the approach of sensitivity equations should be carefully considered for both levels of optimization.},

year={2015},

keywords={Nonlinear mixed effects modeling, First order conditional estimation (FOCE), Sensitivity equations},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 218071

A1 Almquist, Joachim

A1 Leander, Jacob

A1 Jirstrand, Mats

T1 Using sensitivity equations for computing gradients of the FOCE and FOCEI approximations to the population likelihood

YR 2015

JF Journal of Pharmacokinetics and Pharmacodynamics

SN 1567-567X

VO 42

IS 3

SP 191

OP 209

AB The first order conditional estimation (FOCE) method is still one of the parameter estimation workhorses for nonlinear mixed effects (NLME) modeling used in population pharmacokinetics and pharmacodynamics. However, because this method involves two nested levels of optimizations, with respect to the empirical Bayes estimates and the population parameters, FOCE may be numerically unstable and have long run times, issues which are most apparent for models requiring numerical integration of differential equations. We propose an alternative implementation of the FOCE method, and the related FOCEI, for parameter estimation in NLME models. Instead of obtaining the gradients needed for the two levels of quasi-Newton optimizations from the standard finite difference approximation, gradients are computed using so called sensitivity equations. The advantages of this approach were demonstrated using different versions of a pharmacokinetic model defined by nonlinear differential equations. We show that both the accuracy and precision of gradients can be improved extensively, which will increase the chances of a successfully converging parameter estimation. We also show that the proposed approach can lead to markedly reduced computational times. The accumulated effect of the novel gradient computations ranged from a 10-fold decrease in run times for the least complex model when comparing to forward finite differences, to a substantial 100-fold decrease for the most complex model when comparing to central finite differences. Considering the use of finite differences in for instance NONMEM and Phoenix NLME, our results suggests that significant improvements in the execution of FOCE are possible and that the approach of sensitivity equations should be carefully considered for both levels of optimization.

LA eng

DO 10.1007/s10928-015-9409-1

LK http://dx.doi.org/10.1007/s10928-015-9409-1

OL 30