Department of Pharmacy, National and Kapodistrian University of Athens, Panepistimiopolis, 15771, Athens, Greece.
J Pharmacokinet Pharmacodyn. 2023 Aug;50(4):283-295. doi: 10.1007/s10928-023-09851-1. Epub 2023 Mar 21.
Fractional differential equations (FDEs), i.e. differential equations with derivatives of non-integer order, can describe certain experimental datasets more accurately than classic models and have found application in pharmacokinetics (PKs), but wider applicability has been hindered by the lack of appropriate software. In the present work an extension of NONMEM software is introduced, as a FORTRAN subroutine, that allows the definition of nonlinear mixed effects (NLME) models with FDEs. The new subroutine can handle arbitrary user defined linear and nonlinear models with multiple equations, and multiple doses and can be integrated in NONMEM workflows seamlessly, working well with third party packages. The performance of the subroutine in parameter estimation exercises, with simple linear and nonlinear (Michaelis-Menten) fractional PK models has been evaluated by simulations and an application to a real clinical dataset of diazepam is presented. In the simulation study, model parameters were estimated for each of 100 simulated datasets for the two models. The relative mean bias (RMB) and relative root mean square error (RRMSE) were calculated in order to assess the bias and precision of the methodology. In all cases both RMB and RRMSE were below 20% showing high accuracy and precision for the estimates. For the diazepam application the fractional model that best described the drug kinetics was a one-compartment linear model which had similar performance, according to diagnostic plots and Visual Predictive Check, to a three-compartment classic model, but including four less parameters than the latter. To the best of our knowledge, it is the first attempt to use FDE systems in an NLME framework, so the approach could be of interest to other disciplines apart from PKs.
分数阶微分方程（FDE），即具有非整数阶导数的微分方程，可以比经典模型更准确地描述某些实验数据集，并在药代动力学（PK）中得到应用，但由于缺乏合适的软件，其更广泛的适用性受到了阻碍。本工作引入了 NONMEM 软件的扩展，作为一个 FORTRAN 子程序，该子程序允许用 FDE 定义非线性混合效应（NLME）模型。新的子程序可以处理任意用户定义的具有多个方程、多个剂量的线性和非线性模型，并且可以无缝集成到 NONMEM 工作流程中，与第三方软件包配合良好。通过模拟和对安定的真实临床数据集的应用，评估了子程序在参数估计练习中的性能，包括简单的线性和非线性（米氏）分数 PK 模型。在模拟研究中，对两个模型的 100 个模拟数据集的每个数据集进行了模型参数估计。为了评估方法的偏差和精度，计算了相对平均偏差（RMB）和相对均方根误差（RRMSE）。在所有情况下，RMB 和 RRMSE 均低于 20%，表明估计值具有很高的准确性和精度。对于安定的应用，最能描述药物动力学的分数模型是一个单室线性模型，根据诊断图和视觉预测检查，该模型与三腔经典模型的性能相似，但比后者少包含四个参数。据我们所知，这是首次尝试在 NLME 框架中使用 FDE 系统，因此该方法可能除 PK 以外的其他学科也感兴趣。
