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An amortized approach to non-linear mixed-effects modeling based on neural posterior estimation
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Proposes an amortized NLME inference method that trains a neural posterior estimator (conditional normalizing flow with summary network) on simulations to approximate individual-specific posteriors, then fits population parameters by maximizing a Monte Carlo marginal likelihood using those posterior samples.
Problem
Fitting non-linear mixed-effects models to large populations is computationally expensive because marginalizing over individual random effects requires per-individual integration, and existing methods (SAEM, FOCEI) are slow, biased, or cannot handle stochastic individual models.
Approach
Train a single global neural posterior estimator (conditional normalizing flow with LSTM summary network, via BayesFlow) using only simulations from the generative model, then reuse it amortized across datasets and population models; population parameters are obtained by minimizing a Monte Carlo approximation of the marginal likelihood using samples from the learned posteriors, with prior as impor
Key results
- On synthetic single-cell NLME data, the amortized approach achieved lower parameter recovery error than SAEM across data set sizes; the trained estimator is reused for all data sets of a model.
- Training (~6.5 hours) amortizes against MCMC posterior sampling (~20 minutes per cell) after around 20 cells; for 10,000 cells the approach beats SAEM for a single data set.
- On the pharmacokinetic cohort, the amortized approach had correlations 0.752 (sunitinib) and 0.746 (SU12662) between simulations and measurements vs 0.235 and 0.006 for FOCEI, due to FOCEI outliers.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use amortized neural posterior estimation instead of FOCEI for individual patient estimates | Pharmacokinetic NLME model · Patient-level sunitinib and SU12662 measurements | correlation of simulations and measurements: 0.235 -> 0.752 (sunitinib); 0.006 -> 0.746 (SU12662) | Helped |
| Use amortized neural posterior estimation instead of FOCEI for population-level predictions | Pharmacokinetic NLME model · Population-level sunitinib and SU12662 measurements | correlation of simulations and measurements: 0.806 -> 0.779 (sunitinib); 0.743 -> 0.748 (SU12662) | Mixed |
| Allow a full covariance matrix for random effects instead of a diagonal covariance matrix (ablation) | Amortized neural posterior estimation · Detailed single-cell mRNA transfection model | fit of population variance | Helped |
| Use a stochastic SDE model instead of the simple deterministic ODE model (ablation) | Single-cell mRNA transfection NLME model · Synthetic data generated by the SDE NLME model | population variance relative to true variance: 3 times larger -> 1.3 times larger | Helped |
| Use profile likelihood intervals from the amortized approach instead of SAEM-based intervals | Simple single-cell NLME model · Synthetic data, three variance parameters | 80% confidence interval coverage of true parameter | Helped |
| Use the detailed ODE model instead of the simple ODE model (ablation) | Single-cell mRNA transfection model · Real single-cell data | fit to initial fluorescence activity | Helped |
Surprising findings
- The SAEM confidence intervals for three variance parameters failed to cover the true values at 80%, while profile-likelihood CIs from the amortized approach did.
- The simple ODE model with fixed structural non-identifiability overestimated population variance about 3 times on SDE-generated data, while the SDE model captured it with about 1.3 times.
Limitations
- Simulation and training phases are computationally expensive and can exceed established methods for small data sets without model selection or uncertainty quantification.
- Erroneous estimates may arise from too-narrow priors or model misspecification, and the normalizing flow may be insufficiently expressive for multimodal posteriors.
- For the pharmacokinetic application, individual-level fits were less accurate than FOCEI in some respects and the computational advantage remains untested for larger cohorts.
Experimental setup
Datasets: synthetic single-cell mRNA transfection data (simple, detailed, SDE models) real single-cell eGFP/d2eGFP fluorescence data from Fröhlich et al. (5488 cells) sunitinib pharmacokinetic patient cohort (47 patients)
Benchmarks: synthetic single-cell NLME recovery vs SAEM real single-cell population mean fit pharmacokinetic NLME fit vs FOCEI
Metrics: mean squared error of parameters computation time simulation-based calibration confidence interval coverage correlation of simulations and measurements
Baselines: SAEM (Monolix) FOCEI (NONMEM) MCMC with adaptive parallel tempering (pyPESTO) Fröhlich et al. 2018 Laplacian approximation fit
Builds on: BayesFlow SAEM FOCEI simulation-based calibration
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