Mixed-Effects Modeling Using Stochastic Differential Equations – Application to Pharmacokinetic Modeling

J. Leander, J. Almquist, C. Ahlström, J. Gabrielsson, M. Jirstrand, 23rd PAGE meeting, Alicante, Spain, June 2014.


The model dynamics is often assumed to be deterministic in traditional mixed-effects modeling. We want to extend the non-linear mixed-effects model to a so called stochastic differential mixed-effects model, to account for model deficiencies and uncertainty in the dynamics [1-4]. In extension to previous results, interactions between the output covariance and the random effects, together with correlation between random effects are considered. Moreover, we aim for a robust calculation of the gradient of the objective function by using sensitivity equations. Methods: The ordinary non-linear mixed-effects modeling framework is extended by considering stochastic differential equations. The population likelihood is approximated using Laplace’s approximation together with the First Order Conditional Estimation with Interaction (FOCEI) method. The state variables of system (e.g., drug concentration) is estimated using the extended Kalman filter on an individual level. In contrast to the commonly used finite difference approximation of the gradient we utilize the so called sensitivity equations. These equations provide a robust and efficient evaluation of the objective function and its gradient. They are obtained by differentiating the update and prediction equations in the extended Kalman filter.


An algorithm for parameter estimation in stochastic differential mixed-effects models has been developed. It features sensitivity equations for a robust and efficient calculation of the gradient in both the outer and inner optimization problem. The stochastic differential mixed-effects framework is illustrated by using a pharmacokinetic model of nicotinic acid (NiAc) turnover in obese rats [5-7]. The analysis shows that the total error consists of pure measurement error together with a significant uncertainty in model dynamics. The smoothed state variables estimates are used to provide a visualization of uncertainty in variables after the parameter estimation has been completed.


We account for three sources of variability by considering stochastic differential mixed-effects models. We are able to account for uncertainty in the dynamics, in addition to measurement noise and interindividual variability. The new model structure is able to handle interaction effects and correlation between random parameters. The uncertainty plots derived from smoothing serve as an illustrative way to understand output variability.

Photo credits: Nic McPhee