Nonlinear Mixed Effects models (NLME) models are widely used in pharmacometrics and related fields to analyze hierarchical and longitudinal data. However, as the number of parameters and random effects increases, traditional methods for maximizing the marginal likelihood become computationally expensive. This paper explores the Variational Expectation Maximization (VEM) algorithm, a scalable alternative for fitting NLME models. Originally introduced in the context of probabilistic graphical models and later popularized through variational autoencoders, VEM has not been extensively applied to NLME modeling. By leveraging flexible variational families and reverse-mode automatic differentiation, VEM can efficiently maximize the marginal likelihood, scaling to NLME models with over 15,000 population parameters. This work provides a detailed description of VEM, compares it to other NLME fitting algorithms, and highlights its scalability through computational experiments. Using the Pumas statistical software, we fit two test models: 1) a standard warfarin model, and 2) a DeepNLME Friberg model with 15,410 population parameters and 16 random effects. The warfarin model was fitted to completion to demonstrate the correctness of VEM, while the DeepNLME Friberg model was fitted for a limited number of iterations to measure the time per iteration and demonstrate VEM's scalability.

翻译：非线性混合效应模型（NLME）广泛应用于药物计量学及相关领域，用于分析层次结构和纵向数据。然而，随着参数和随机效应数量的增加，最大化边际似然的传统方法计算成本急剧上升。本文探讨了变分期望最大化（VEM）算法，这是一种可扩展的NLME模型拟合替代方案。该算法最初在概率图模型中引入，随后通过变分自编码器推广，但在NLME建模中尚未得到广泛应用。通过利用灵活的变分族和反向模式自动微分，VEM能够高效最大化边际似然，可扩展至包含超过15,000个群体参数的NLME模型。本研究详细阐述了VEM算法，将其与其他NLME拟合算法进行比较，并通过计算实验突出其可扩展性。使用Pumas统计软件，我们拟合了两个测试模型：1）标准华法林模型，2）具有15,410个群体参数和16个随机效应的DeepNLME Friberg模型。华法林模型被完整拟合以验证VEM的正确性，而DeepNLME Friberg模型则通过有限次迭代评估每次迭代时间，以展示VEM的可扩展性。