The topic of this thesis is nonlinear mixed-effects models. A nonlinear mixed-effects model is a hierarchical regression model used to analyze measurements from several individuals simultaneously, which allows sharing of information between similar individuals. In many applications, the response is described by an ordinary differential equation together with an equation describing the measurement process. To allow for uncertainty in the underlying model as well, we consider extension of the ordinary differential equation to a stochastic differential equation. Moreover, since the parameter estimation problem requires solution of an optimization problem, we derive a novel method for calculating the gradient of the objective function in nonlinear mixed-effects models. Instead of utilizing a finite difference approximation of the gradient, we instead derive an expression for the gradient of the objective function using sensitivity equations.
In Paper I, stochastic differential equations are introduced in the single individual case. In this paper we show the ability of stochastic differential equations to regularize the parameter estimation problem for continuous time dynamical systems given discrete time measurements. We also introduce the concept of the extended Kalman filter, which serves as a state estimator in the stochastic models. In Paper II we propose a combination of the nonlinear mixed-effects model and stochastic differential equations. By utilizing stochastic differential equations in the nonlinear mixed-effects model three different sources of variability in data can be modeled. In contrast to the commonly used measurement error and parameter variability we also consider uncertainty in the underlying dynamics. The stochastic mixed-effects model is applied to two different pharmacokinetic models describing drug interaction, where we show the ability of the proposed method to separate the three sources of variability in the stochastic differential mixed-effects model. In Paper III we present the theory for efficient gradient calculation in nonlinear mixed-effects models. Instead of utilizing the common finite difference approximation of the gradient, we consider explicit derivation of the objective function using sensitivity equations. It is shown that the novel method significantly decreases the time needed for gradient calculation and at the same time increases the precision.