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Jacob Leander1,2,3, Joachim Almquist1,3, Helga Kristín Ólafsdóttir2, Anna Johnning1,2, and Mats Jirstrand1
1Fraunhofer-Chalmers Centre, Gothenburg, Sweden, 2Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg,
Gothenburg, Sweden, 3Quantitative Clinical Pharmacology, Early Clinical Development, IMED Biotech Unit, AstraZeneca, Gothenburg, Sweden
Parameter Estimation for Nonlinear Mixed
Effects Models Implemented in Mathematica
Background
In many applications within biology and
medicine, measurements are gathered from
several entities in the same experiment. This
could for example be patients exposed to a
treatment or cells measured after stimuli.
To characterize the variability in response
between entities, the nonlinear mixed effects
(NLME) model is a suitable statistical model. An
NLME model enables quantification of both
within- and between subject variability.
The parameter estimation in NLME models is
not straightforward, due to the intractable
expression of the likelihood function.
In this work we present a Mathematica package
for parameter estimation in NLME models
where the longitudinal model is defined by
differential equations. The parameter
estimation problem is solved by the first-order
conditional estimation (FOCE) method with
exact gradients. The package is demonstrated
using data from a simulated drug concentration
model.
Statistical model
The dynamical model for an individual is
defined by a system of ODEs
together with an observation model
The individual parameters are linked to
population parameters by a functional
relationship with the random
effects .
Extension to a longitudinal model described by
stochastic differential equations (SDEs) is also
supported.
Parameter estimation
The aim is to estimate the model parameters
from a set of observations
.
Since the random effects are unobserved, the
joint probability distribution is marginalized over the
unobserved quantities to obtain the likelihood
function.
Due to the normality assumptions in the model we
have
with residual
Since the integral over is problematic, the
integral is approximated using a second order Taylor
expansion of , which yields the objective function
where the point
() is the value maximizing
(for a fixed ). This leads to a nested optimization
problem which is computationally demanding.
The Hessian can further be simplified to give the
so called first order conditional estimation (FOCE)
approximation.
Exact gradients
•A quasi-Newton method with a finite difference
approximation of the gradient has traditionally
been used to compute the maximum likelihood
estimate.
•In this work, we use sensitivity equations to
compute exact gradients for the optimization of
and
.
•The ODE system is differentiated with respect to
the model parameters to obtain the sensitivity
equations [1,2].
•Exact gradients enable faster and more robust
optimization compared to finite differences, and
have been implemented in the NLME software
NONMEM 7.4 [3].
•The package has previously been used in several
applications, see [4,5,6].
Modeling workflow
The measurements are collected as a list of time-
value pairs with easy-to-use plotting tools
available.
The NLME model is defined by an ODE system
and an observation model.
The estimation requires dataset, model and initial
guesses for the population parameters:
Several options are available:
The optimization returns an object which
contains the estimated model, including
parameter estimates and optimization history.
The model object can be used for easy plotting of
predictions
Acknowledgements
This project has been supported by the Swedish Foundation for Strategic Research,
which is gratefully acknowledged.
References
[1] Almquist, J., Leander, J. & Jirstrand, M. J Pharmacokinet Pharmacodyn (2015) 42:
191. https://doi.org/10.1007/s10928-015-9409-1
[2] Olafsdottir, H.K., Leander, J., Almquist, J. et al. AAPS J (2018) 20: 88.
https://doi.org/10.1208/s12248-018-0232-7
[3] Beal, S., Sheiner, L.B., Boekmann, A. & Bauer, R.J. NONMEM’s User’s Guides (ICON
Development Solutions, Ellicott City, MD, USA, 2009)
[4] Leander, J., Almquist, J., Ahlström, C. et al. AAPS J (2015) 17: 586.
https://doi.org/10.1208/s12248-015-9718-8
[5] Cardilin, T., Almquist, J., Jirstrand, M. et al. Cancer Chemother Pharmacol (2019).
https://doi.org/10.1007/s00280-019-03829-y
[6] Andersson, R., Jirstrand, M., Almquist, J,. Gabrielsson, J. European Journal of
Pharmaceutical Sciences (2019) 128:1. https://doi.org/10.1016/j.ejps.2018.11.015
Summary and conclusions
•A Mathematica package for parameter
estimation in nonlinear mixed effects
models has been implemented and
demonstrated.
•The package enables easy-to-use NLME
modeling, is free, and can be further
demonstrated upon request.
