Sensitivity Equations Provide More Robust Gradients and Faster Computation of the FOCE Approximation to the Population Likelihood

J. Almquist, J. Leander, M. Jirstrand. Proceedings of the 24th Annual meeting of the Population Approach Group in Europe, 2-5 June 2015, Crete, Greece


Objectives: The first order conditional estimation (FOCE) method [1] is still one of the parameter estimation workhorses for nonlinear mixed effects (NLME) modeling used in population pharmacokinetics and pharmacodynamics [2]. 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.

Methods: We propose an alternative implementation of the FOCE method, and the related FOCEI, for parameter estimation in NLME models [3]. 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.

Results: The advantages of the approach are 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 [4]. 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.

Conclusions: Considering the use of finite differences in for instance NONMEM and Phoenix NLME, our results suggests that signicant improvements in the execution of FOCE are possible and that the approach of sensitivity equations should be carefully considered for both levels of optimization.


[1] Wang Y. Derivation of various NONMEM estimation methods. J of Pharmacokin Pharmacodyn (2007) 34(5): 575-593.

[2] Johansson ÅM, Ueckert S, Plan EL, Hooker AC, Karlsson MO. Evaluation of bias, precision, robustness and runtime for estimation methods in NONMEM 7. J of Pharmacokin Pharmacodyn (2014) 41(3):223-238.

[3] Almquist J, Leander J, Jirstrand M. Using sensitivity equations for computing gradients of the FOCE and FOCEI approximations to the population likelihood. In press J of Pharmacokin Pharmacodyn (2015).

[4] Tapani S, Almquist J, Leander J, Ahlström C, Peletier LA, Jirstrand M, Gabrielsson J. Joint Feedback Analysis Modeling of Nonesterified Fatty Acids in Obese Zucker Rats and Normal Sprague–Dawley Rats after Different Routes of Administration of Nicotinic Acid. J Pharmaceutical Sciences (2014), 103(8):2571–2584.

Photo credits: Nic McPhee