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

Jacob Leander,Mats Jirstrand,Christine Ahlström,Johan Gabrielsson,Joachim Almquist

doi:10.1208/s12248-015-9718-8

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.

Highlights

In pharmacokinetic and pharmacodynamic modeling, the physical system is often assumed to be described by a system of ordinary differential equations (ODEs)
We introduce the concept of the extended Kalman filter (EKF), which serves as a state estimator for the stochastic model [29]
The relative standard errors (RSEs) in % are included in parentheses

Summary

Introduction

In pharmacokinetic and pharmacodynamic modeling, the physical system is often assumed to be described by a system of ordinary differential equations (ODEs). Since it is up to the modeler to define a model describing the drug administration and its effect, there is an uncertainty in the model itself. This uncertainty is not explicitly accounted for when considering a deterministic model, for example when using ODEs to describe the dynamics together with a measurement model to incorporate the error. This can lead to model deficiencies, such as correlated residuals, overestimated measurement noise, and incorrect inference [2]

Methods

Results

Conclusion