Mixed Effects Modeling Using Stochastic Differential Equations: Illustrated by Pharmacokinetic Data of Nicotinic Acid in Obese Zucker Rats

Abstract

Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.

Asymptotic Theory of Bayes Factor in Stochastic Differential Equations with Increasing Number of Individuals

“…In these time-dependent situations it is only natural that the available covariates are also continuously varying with time. Examples of statistical applications of SDE-based models with time-dependent covariates are Oravecz et al (2011), Overgaard et al (2005), Leander et al (2015), the first one also considering the hierarchical Bayesian paradigm.…”

“…Apart from the covariates there may also be random effects associated with the individuals, which may be useful in modeling variabilities between the individuals. SDE-based models with time-dependent covariates are considered in Oravecz et al (2011), Overgaard et al (2005), Leander et al (2015); moreover, Oravecz et al (2011) analyse their covariate-based SDE model in the hierarchical Bayesian paradigm. In the literature, random effects SDE models without covariates seem to be more popular than those based on covariates.…”

Parametric inference for discrete observations of diffusion processes with mixed effects

“…Multidimensional SDEMEs with linear random effects are often encountered in applications (see e.g. Leander et al (2015), Berglund et al (2001)) and could be studied by the same approach .…”

“…Hence, SDEMEs provide a good framework to study population characteristics when longitudinal data are collected on multiple individuals ruled by the same intraindividual mechanisms (see e.g. Overgaard et al (2005), Ditlevsen and De Gaetano (2005b), Picchini et al (2008), Møller et al (2010), Donnet et al (2010), Berglund et al (2001), Leander et al (2015) for discussions and applications on real data sets).…”

Bayesian inference for stochastic differential equation mixed effects models

“…SDEMEMs are emerging as a useful class of models for biomedical and pharmacokinetic/pharmacodynamic data (Donnet et al, 2010;Donnet and Samson, 2013a;Leander et al, 2015). They have also been applied in psychology (Oravecz et al, 2011) and spatiotemporal modelling (Duan et al, 2009).…”