Parameter Estimation for Nonlinear Mixed Effects Models Implemented in Mathematica

J. Leander, J. Almquist, H. K. Ólafsdóttir, A. Johnning, M. Jirstrand. Workshop on Modelling in Biology and Medicine, Gothenburg, Sweden, 9-10 May 2019.


In many applications within biology and medicine, measurements are gathered from several entities in the same experiment (e.g. patients, cells, tissue samples) with the aim of characterizing for example response to a specific drug treatment or other stimuli. To characterize the variability in response between entities, the nonlinear mixed effects model is a suitable statistical model. A nonlinear mixed effects model enables quantification of both within- and between subject variability. Parameter estimation in a nonlinear mixed effects model is not straightforward, due to the intractable expression of the likelihood function. Approximations of the likelihood function or other MCMC-based approaches are typically used to infer parameters from data. Several parameter estimation software are available, but few have reached the academic community. In this work, we present a framework for parameter estimation in nonlinear mixed effects models with a longitudinal response model described by ordinary differential equations. The framework is implemented in the computational environment Mathematica. The estimation algorithm is based on the FOCE approximation of the likelihood, and the symbolic capabilities of Mathematica is used to enable a gradient-based optimization method using sensitivity equations (denoted S-FOCE). Moreover, extension to stochastic differential equations is also supported. We exemplify the framework using data from a simulated pharmacokinetic (drug concentration) model. The implementation enables a user-friendly framework for parameter estimation in nonlinear mixed effects model with ordinary or stochastic differential equations. Moreover, the proposed method has been shown to reduce computational time and has lately been implemented in the industry-standard software NONMEM 7.4.

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