Leveraging Neural ODEs for Population Pharmacokinetics of Dalbavancin in Sparse Clinical Data

Abstract

This study investigates the use of Neural Ordinary Differential Equations (NODEs) as an alternative to traditional compartmental models and Nonlinear Mixed-Effects (NLME) models for drug concentration prediction in pharmacokinetics. Unlike standard models that rely on strong assumptions and often struggle with high-dimensional covariate relationships, NODEs offer a data-driven approach, learning differential equations directly from data while integrating covariates. To evaluate their performance, NODEs were applied to a real-world Dalbavancin pharmacokinetic dataset comprising 218 patients and compared against a two-compartment model and an NLME within a cross-validation framework, which ensures an evaluation of robustness. Given the challenge of limited data availability, a data augmentation strategy was employed to pre-train NODEs. Their predictive performance was assessed both with and without covariates, while model explainability was analyzed using Shapley additive explanations (SHAP) values. Results show that, in the absence of covariates, NODEs performed comparably to state-of-the-art NLME models. However, when covariates were incorporated, NODEs demonstrated superior predictive accuracy. SHAP analyses further revealed how NODEs leverage covariates in their predictions. These results establish NODEs as a promising alternative for pharmacokinetic modeling, particularly in capturing complex covariate interactions, even when dealing with sparse and small datasets, thus paving the way for improved drug concentration predictions and personalized treatment strategies in precision medicine.

Keywords: deep learning; neural ODE; population pharmacokinetics; precision medicine.

Figure A1

Loss function profiles for training with different number of parameters.

Figure A2

Comparison of residuals of the best model fitted in Monolix with the one without covariates.

Figure A3

Comparison of distributions from a kernel density estimations of the residuals of the NLME (a) and the residuals of the NODE (b).

Figure 1

Sketch of the integration process. t represents time, C the concentration, D the dose, n the number of previous administrations and final t means the time of the last concentration measure (during training) or the total time for the simulation during inference.

Figure 2

Scheme of a two-compartment model. The antibiotic is administered in the central compartment, which is the only compartment that can eliminate the antibiotic at rate Cl, the clearance. Q is the intercompartmental clearance, while V1 and V2 are the compartment volumes.

Figure 3

Comparison of the concentration prediction of NODE with respect to the two-compartment model and NLME for a test patient (for a fixed cross-validation iteration). The red dashed line represents the drug efficacy threshold, fixed to $t=8.04\mathrm{mg}/\mathrm{L}$, while red dots are real data, corresponding to drug concentration measurements, and the spikes correspond to dose administrations.

Figure 4

Absolute residuals (left) and relative residuals (right) as a function of time for the two-compartment model (a), the NLME model (b) and the NODE (c). Each administration resets the time.

Figure 5

Distributions (via kernel density estimation) of the absolute residuals (a) and the relative residuals (b) for the two-compartment model, the NLME and the NODE.

Figure 6

Residuals as a function of time (a) and the distributions (via kernel density estimation) of the absolute residuals (b) and relative residuals (c) for the two-compartment model, the NLME and the NODE trained, including the patients’ covariates.

Figure 7

(a) Beeswarm plot of SHAP values for each input feature in the NODE model without covariates. (b) Beeswarm plot of input features in the model with covariates, indicating their minimal impact on the output. The color maps give an insight on the relation between SHAP values and feature values.

Figure 8

Violin plot of the SHAP values used in the computation of $V_{\mathrm{NODE}}$ for the model with covariates, where volume is treated as an individual-specific parameter, illustrating a comparable contribution from each covariate.