Stochastic nonlinear mixed effects: a metformin case study

Abstract

In nonlinear mixed effect (NLME) modeling, the intra-individual variability is a collection of errors due to assay sensitivity, dosing, sampling, as well as model misspecification. Utilizing stochastic differential equations (SDE) within the NLME framework allows the decoupling of the measurement errors from the model misspecification. This leads the SDE approach to be a novel tool for model refinement. Using Metformin clinical pharmacokinetic (PK) data, the process of model development through the use of SDEs in population PK modeling was done to study the dynamics of absorption rate. A base model was constructed and then refined by using the system noise terms of the SDEs to track model parameters and model misspecification. This provides the unique advantage of making no underlying assumptions about the structural model for the absorption process while quantifying insufficiencies in the current model. This article focuses on implementing the extended Kalman filter and unscented Kalman filter in an NLME framework for parameter estimation and model development, comparing the methodologies, and illustrating their challenges and utility. The Kalman filter algorithms were successfully implemented in NLME models using MATLAB with run time differences between the ODE and SDE methods comparable to the differences found by Kakhi for their stochastic deconvolution.

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

“…First, additional applications of SDE-NLME models are needed to further demonstrate the benefits of PK/PD models with uncertain dynamics. Some applications have already been mentioned in this work (6,16,17,19). Second, for novel methods to be widely used, they must be implemented in software with large user-bases, such as NONMEM, Phoenix, and Monolix (22,38,39), or open-source initiatives such as nlmixr (40).…”

“…Nevertheless, some attempts to develop methods for parameter estimation in NLME models with stochastic dynamics have been successful, using for example (i) Bayesian inference (9,10), (ii) expectation maximization (EM) methods (11)(12)(13), and (iii) by expanding the traditional gradient-based estimation methods using Kalman filters (14,15). These methods have been used for several PK/PD applications (16)(17)(18)(19). This paper focuses on gradient-based methods.…”

Beyond Deterministic Models in Drug Discovery and Development

“…NLME models based on SDEs extend the first-stage model of the hierarchical structure by decomposing the residual variability into measurement noise accounting for uncorrelated errors (assay sensitivity, sampling errors, etc.) and system noise accounting for model misspecifications or true random physiological fluctuations [ 58 , 59 ]. Thus, here three levels of random effects are included.…”

“…Thus, here three levels of random effects are included. One example of the application of SDEs into PK/PD modeling is work by Mazuka et al [ 58 ], who evaluated the use of SDEs to characterize the absorption properties of single oral dose of metformin. The authors suggested a state equation for the absorption rate parameter of the drug that fluctuates randomly, which managed to better capture the absorption phase of the concentration profiles.…”

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

“…Typically, the underlying system of interest is described by a system of ordinary differential equations in combination with an observation model. In recent years there has been an increasing interest in extending the NLME framework to incorporate stochastic differential equations, leading to a class of models called stochastic differential equations mixed effects models (SDEMEMs) [1,2,3,4,5,6]. There are several software options available for parameter estimation in NLME models with ordinary differential equations, including both commercial software such as NONMEM [7,8,9], Monolix [10], and Phoenix, and open-source such as nlmixr [11,12] and Stan [13].…”