Abstract
Nonlinear mixed effects (NLME) modeling based on stochastic differential equations (SDEs) have evolved into a promising approach for analysis of PK/PD data. SDE-NLME models go beyond the realm of standard population modeling as they consider stochastic dynamics, thereby introducing a probabilistic perspective on the state variables. This article presents a summary of the main contributions to SDE-NLME models found in the literature. The aims of this work were to develop an exact gradient version of the first-order conditional estimation (FOCE) method for SDE-NLME models and to investigate whether it enabled faster estimation and better gradient precision/accuracy compared to the use of gradients approximated by finite differences. A simulation-estimation study was set up whereby finite difference approximations of the gradients of each level were interchanged with the exact gradients. Following previous work, the uncertainty of the state variables was accounted for using the extended Kalman filter (EKF). The exact gradient FOCE method was implemented in Mathematica 11 and evaluated on SDE versions of three common PK/PD models. When finite difference gradients were replaced by exact gradients at both FOCE levels, relative runtimes improved between 6- and 32-fold, depending on model complexity. Additionally, gradient precision/accuracy was significantly better in the exact gradient case. We conclude that parameter estimation using FOCE with exact gradients can successfully be applied to SDE-NLME models.
Highlights
Nonlinear mixed effects (NLME) models are used to describe populations of individuals that behave qualitatively similar, but where every individual has its own quantitative characteristics. These models have been highly applicable in pharmacokinetics (PK) and pharmacodynamics (PD) [1,2,3,4]
This paper extends the S-first-order conditional estimation (FOCE) algorithm to a general NLME model based on Stochastic differential equations (SDEs) using the extended Kalman filter (EKF) approach
As the models become more complex in terms of number of fixed effects and number of state variables, the relative benefit of using sensitivities instead of finite differences increases
Summary
Nonlinear mixed effects (NLME) models are used to describe populations of individuals that behave qualitatively similar, but where every individual has its own quantitative characteristics. Since the mechanisms of the physiological systems of interest are often not completely understood, a natural extension is to account for the uncertainty in the dynamics This is the motivation behind extending the NLME models with yet a stochastic part [5]. An ODE model can be turned into an SDE model by the addition of possibly non-linearly scaled noise terms These additional terms can be thought of as a type of slack-variable introduced to pick-up potential discrepancies between the mechanistically modeled dynamics and unknown but indirectly observed dynamical effects. In other words, these noise terms in the dynamical system equations represent the lumped effect of all not explicitly mechanistically modeled effects in the system. This includes turning model parameters into state variables with random dynamics, which for instance could be used to model inter occasion variability [6] or time-dependent changes in parameters where the trend is unknown a priori, and for the estimation of an unknown input signal [7]
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