Johannes Schropp 1, Freya Bachmann 1, Britta Steffens 2, Mona Kasper 1, Gilbert Koch 2, Jan Rohleff 1
1 University Of Konstanz (Konstanz, Germany), 2 University Children's Hospital (Basel, Switzerland)
Objectives
Pharmacometrics (PMX) modeling is essential for optimizing drug therapy and drug development. Nonlinear Mixed-Effects (NLME) modeling is used to account for inter-individual variability, and to quantify uncertainty on the population level. However, NLME model building remains compu-tationally demanding, structurally constrained, and heavily dependent on expert-driven decisions. The recently developed NLME Variational Autoencoder (VAE) [1] is combined with Neural Ordi-nary Differential Equations (NODEs) to automate and generalize NLME modeling tasks. This framework allows NODEs to be embedded anywhere in the NLME structure, enabling automated modeling of complex dynamic processes such as time-varying covariates. Our objective is to demonstrate how generative VAEs can unify parameter estimation, uncertainty quantification, and Empirical Bayesian Estimation (EBE) while providing a flexible framework for NLME modeling.
Methods
We developed a VAE framework specifically adapted to PMX NLME models. The encoder learns distributions of individual parameters directly from the data, while the decoder reconstructs predictions using structural or NODE-augmented models. NODEs can capture complex temporal patterns, allowing dynamic individual or population characteristics to be learned automatically without manual structural adjustments.
Following the PMX gold standard, parameter uncertainty is quantified using a Fisher Information Matrix-based variance estimator (see Lavielle [2]). The solution accuracy is measured in terms of Log-Likelihood (LL) and corrected Bayesian Information Criterion (BICc). Beyond parameter estimation, the generative VAE NLME framework enables accurate EBEs within the training range.
Results
To assess the performance of the generative VAE NLME framework, we investigated levothyrox-ine treatment in newborns and infants with congenital hypothyroidism (see [3] for details). Here, body weight changed substantially during the first two years of life. Two versions of a structural model are applied. Model 1 incorporates weight as time-varying covariate, whereas Model 2 takes weight as a second output modeled by an expert-based growth model. For both structural models, we compared the VAE results with those of Monolix 2024.
Using the expert-based growth model (Model 2) results in a standard NLME formulation with 6 individual parameters (3 for the thyroid model and 3 for the weight model). The solution can be computed either with the VAE or Monolix yielding nearly identical parameter estimates and Log-Likelihood (-2LL = 4227 (VAE), -2LL = 4226 (Monolix)) values.
Time-varying covariates are treated differently in the VAE NLME framework than in Monolix. Model 1 requires only two individual parameters. The VAE automatically computes weight trajectories using a NODE-based encoder, whereas Monolix uses weight as a regressor with Last Observation Carried Forward interpolation. Here, the VAE achieved better Log-Likelihood and BICc (-2LL = 3047, BICc = 3087 vs. -2LL = 3361, BICc = 3394) and a lower proportional error model parameter (b = 0.228 vs. 0.271).
In a direct comparison between the two models, the VAE with NODE not only matched the expert-defined growth model, it successfully recovered biologically plausible weight-age trajectories closely resembling World Health Organization standards [4].
Quite interesting is the comparison with respect to uncertainty quantification, i.e., relative standard error (RSE). Parameters linked to the covariate weight, i.e., volume V, exhibited better RSEs (%RSE (V) = 0.35 (VAE) vs. 3.25 (Monolix), and %RSE (omega_V) = 3.59 (VAE) vs. 12.7 (Monolix)), while parameters not linked to weight show similar RSE values. In addition, the trained VAE model provides excellent thyroid and weight predictions.
Conclusion
VAEs with integrated NODEs provide a flexible and scalable NLME modeling framework that reproduces classical NLME results while enabling automated handling of complex dynamic modeling tasks, including time-varying covariates. The framework achieves accurate parameter estimation, stable uncertainty quantification, and biologically plausible predictions, highlighting generative AI VAEs as a foundational tool to extend NLME methodology and facilitate previously challenging modeling tasks in pharmacometrics.
References:
[1] Rohleff, J., Bachmann, F., Nahum, U., Bräm, D., Steffens, B., Pfister, M., Koch, G. and Schropp, J. (2025), Redefining Parameter Estimation and Covariate Selection via Variational Autoencoders: One Run Is All You Need. CPT Pharmacometrics Syst Pharmacol, 14: 2232-2243. https://doi.org/10.1002/psp4.70129
[2] Lavielle M. (2014), Mixed Effects Models for the Population Approach, 1st ed., Chapman and Hall, CRC.
[3] Koch G, Steffens B, Leroux S, Gotta V, Schropp J, Gächter P, Bachmann F, Welzel T, Janner M, L’Allemand D, Konrad D, Szinnai G, Pfister M. Modeling of levothyroxine in newborns and infants with congenital hypothyroidism: challenges and opportunities of a rare disease multi-center study. J Pharmacokinet Pharmacodyn. 2021 Oct;48(5):711-723. doi: 10.1007/s10928-021-09765-w. Epub 2021 Jun 11. PMID: 34117565; PMCID: PMC8405503.
[4] World Health Organization. who.int website. https://www.who.int/tools/child-growth-standards/standards/weight-for-age. Accessed February 26, 2026.
Reference: PAGE 34 (2026) Abstr 12142 [www.page-meeting.org/?abstract=12142]
Poster: Methodology - New Modelling Approaches