Fraunhofer-Chalmers Centre, Chalmers Science Park, SE-41288, Gothenburg, Sweden,
AAPS J. 2015 May;17(3):586-96. doi: 10.1208/s12248-015-9718-8. Epub 2015 Feb 19.
Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.
在混合效应模型中纳入随机微分方程为量化和区分数据中的三个变异性来源提供了手段。除了常见的两个来源，即测量误差和个体间变异性，我们还考虑了动态模型本身的不确定性。为此，我们将非线性混合效应模型中使用的常微分方程设置扩展到包括随机微分方程。使用一阶条件估计与交互方法和扩展卡尔曼滤波来推导近似总体似然。为了说明随机微分混合效应模型的应用，我们考虑了两个药代动力学模型。首先，我们使用具有一阶输入和非线性消除的随机单室模型在模拟研究中生成合成数据。我们表明，通过使用所提出的方法，可以成功地分离三个变异性来源。如果忽略随机部分，则参数估计会产生偏差，并且测量误差方差会被显著高估。其次，我们考虑了肥胖 Zucker 大鼠烟酸动力学的临床前研究中随机药代动力学模型的扩展。分别在确定性和随机 NiAc 处置模型之间比较参数估计。模型预测和观察之间的差异，以前仅描述为测量噪声，现在可以分为相对较低水平的测量噪声和模型动态的显著不确定性。这些示例表明，随机微分混合效应模型是识别不完整或不准确模型动态的有用工具，并可以减少由于模型缺陷导致的参数估计中的潜在偏差。
