Exact Gradients Improve Parameter Estimation in Nonlinear Mixed Effects Models with Stochastic Dynamics

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.

A stochastic mixed effects model to assess treatment effects and fluctuations in home‐measured peak expiratory flow and the association with exacerbation risk in asthma

“…The model parameters were estimated using the first‐order conditional estimation with interaction method, as implemented in the open‐source Wolfram Mathematica package NLMEModeling 22,23 . Because of the stochastic dynamics of the underlying system, the extended Kalman filter (EKF) was used to estimate the state of the system 16,24 . Similar to other implementations, 25 NLMEModeling automatically generates the necessary equations for the EKF.…”

NLMEModeling: A Wolfram Mathematica Package for Nonlinear Mixed Effects Modeling of Dynamical Systems

“…Second, we describe how the estimation of model parameters is done according to the maximum likelihood approach and give a high-level description of the numerical methods used. A more detailed description of the methodology can be found in [26,27]. Instructions for retrieving and installing the package are also provided.…”

“…For ODE models, R ij is equal to the observation error covariance matrix Σ as the dynamical model is deterministic. For SDE models, on the other hand, NLMEModeling utilizes the extended Kalman filter [29] to estimate ŷij and R ij [23,27,30].…”

“…The derivation requires symbolic differentiation of the model with respect to the model parameters and this is achieved using the symbolic engine in Wolfram Mathematica. For derivation of the exact gradients in NLMEModeling, we refer the interested reader to [26,27].…”

Beyond Deterministic Models in Drug Discovery and Development

“…However, research efforts that incorporate stochastic phenomena at higher than the single-cell level are rarely found and there is almost no experience in the application of those models to guide clinical trial design or patient management. Some examples of the implementation of SDEs in NONMEM already exist where the absorption or elimination rate constant have been defined as stochastic processes and the extended Kalman filter has been coupled for parameter estimation [63][64][65]. This same method has also been implemented in R [66] and Matlab [67] for estimation of insulin secretion rates.…”