Manuel Huth 1,2, Jonas Arruda 1,2, Nina Schmid 1,2, Clemens Peiter 1,2, Jan Hasenauer 1,2
1 Bonn Center for Mathematical Life Sciences (Bonn, Germany), 2 Life and Medical Sciences (LIMES) Institute (Bonn, Germany)
Introduction/Objectives
Hidden Markov models (HMMs) are increasingly applied in pharmacometric disease progression modeling to represent latent disease states and transitions [1]. In parallel, neural ordinary differential equations (NODEs) have been proposed as flexible structural models within nonlinear mixed-effects (NLME) frameworks [2,3]. However, integration of HMM structures, neural components, and flexible inter-individual variability within a unified open NLME environment remains limited. We introduce NoLimits.jl (NOn LInear MIxed effecTS), an open-source julia framework enabling estimation of NLME models with neural components, hidden Markov structures, and flexible random-effect distributions parameterized via normalizing flows [4]. The framework enhances modeling flexibility by supporting a comprehensive set of frequentist and Bayesian estimation and inference approaches. Its documentation and code are openly available [5].
The objective was to assess the feasibility and estimation performance of the framework across established NODE benchmarks, to compare neural network and SoftTree parameterizations, and to evaluate the stable estimation of a large-scale disease progression model using an HMM with subject-specific transition dynamics.
Methods
NODEs models were implemented for the Theophylline and Warfarin benchmark datasets, using our built-in Neural Network and Soft Tree support. These datasets have previously been in Monolix and NONMEM using manually specified neural network structures [2]. Estimation was performed using a stochastic approximation expectation-maximization (SAEM) algorithm [6]. Neural network parameterizations were compared with SoftTree structures within the same estimation framework. Model performance was evaluated using mean squared error (MSE) and median relative absolute error (MRAE).
Additionally, a 7-state hidden Markov NLME model was fitted to a real-world disease progression dataset comprising approximately 5094 patients with 9 partially observed binary outcome indicators measured at 5 time points [1]. Inter-individual variability was modeled by specifying the transition rates as random-effects, and estimation was conducted using SAEM [6]. Flexible random-effect distributions via normalizing flows were implemented as an extension of the classical Gaussian assumption [4,7].
Results
For both Theophylline and Warfarin datasets, neural NLME models reproduced benchmark performance reported in the literature [2] with respect to MSE and MRAE. SoftTree parameterizations achieved comparable predictive accuracy to neural networks while providing structured and stable model representations.
The 7-state HMM-NLME model converged successfully for the disease progression dataset. Estimated transition probabilities were stable and clinically interpretable. The model accommodated partially missing binary outcomes without numerical instability. Inter-individual variability parameters were estimable under both Gaussian and flow-based parameterizations. These findings demonstrate the practical feasibility of combining latent state modeling, neural structural components, and flexible variability modeling within a unified NLME framework.
Conclusions
These results highlight the capabilities of NoLimits.jl to integrate neural ordinary differential equations, hidden Markov structures, and flexible random-effect distributions within a unified nonlinear mixed-effects modeling framework. By combining latent state modeling, neural structural components, and extensible variability parameterizations in an open environment, NoLimits.jl expands the range of complex pharmacometric models that can be specified and estimated in practice.
The package is under active development and steadily maturing into a robust and extensible platform for advanced pharmacometric methodology and collaborative research.
References:
[1] Gusinow R, Górska A, Canziani LM, et al. (2026). Latent transition analysis for longitudinal studies of post-acute infection syndromes. Nature Communications.
[2] Bräm, D. S., Steiert, B., Pfister, M., Steffens, B., & Koch, G. (2025). Low‐dimensional neural ordinary differential equations accounting for inter‐individual variability implemented in Monolix and NONMEM. CPT: Pharmacometrics & Systems Pharmacology, 14(1), 5-16.
[3] Bräm, D. S., Nahum, U., Schropp, J., Pfister, M., & Koch, G. (2024). Low-dimensional neural ODEs and their application in pharmacokinetics. Journal of Pharmacokinetics and Pharmacodynamics, 51(2), 123-140.
[4] Rezende, D., & Mohamed, S. (2015). Variational inference with normalizing flows. International conference on machine learning (pp. 1530-1538). PMLR.
[5] NoLimits.jl documentation. https://manuhuth.github.io/NoLimits.jl/dev/
[6] Kuhn, E., & Lavielle, M. (2005). Maximum likelihood estimation in nonlinear mixed effects models. Computational statistics & data analysis, 49(4), 1020-1038.
Reference: PAGE 34 (2026) Abstr 12098 [www.page-meeting.org/?abstract=12098]
Poster: Methodology – AI/Machine Learning