Neural ODE structures based on pharmacokinetic principles
Dominic Stefan Bräm (1), Uri Nahum (1), Marc Pfister (1), Johannes Schropp (2), and Gilbert Koch (1)
(1) Pediatric Pharmacology and Pharmacometric, University Children’s Hospital Basel, Switzerland (2) Department of Mathematics and Statistics, University of Konstanz, Germany
Objectives:
Neural ordinary differential equations (NODEs) are a machine learning (ML) approach that can fit data from dynamic systems [1], such as pharmacokinetic (PK) and pharmacodynamic (PD) data. It was shown in few published examples, that NODEs are able to describe PK data well [2–5]. However, despite some similarities compared to classical ODEs in pharmacometrics (PMX), there are substantial differences that might have impede the broad application of NODEs in the PMX community.
In this research work, we developed NODEs that are motivated by general PK principles, such as distribution and elimination processes, to improve application to PK problems [6]. We also investigated common ML challenges of NODEs, such as overfitting and extrapolation, and developed concepts to improve their applicability in PMX. With this presentation, we aim at improving the understanding of NODEs in the PMX community and illustrate, how they could be applied efficiently and successfully to various PK problems.
Methods:
In this work, NODEs were developed that can be generally applied to PK scenarios. This basic NODE structure was developed for different administration routes and it was applied to different PK datasets simulated with a variety of PK models, including an IV two-compartment model, an IV target-mediated drug disposition (TMDD) model, delayed absorption with different numbers of transit compartments and an IV infusion one-compartment model.
In a further development, a combination of mechanistic parts and NODEs was developed. This structure was tested on a dataset simulated with a PO TMDD model.
Additionally, the challenge of extrapolation in ML was investigated, i.e. the fact, that ML approaches generally perform well on data that they were trained on but are unable to make predictions for unseen data ranges.
Results:
Applied NODE structures were able to fit all simulated PK datasets well. Investigation of the right-hand side of trained NODEs versus concentration plots showed that trained NODEs were able to learn dynamics of ODEs, i.e. leverage underlying mechanism utilized to simulate data.
Remarkably, in the PO case with delayed absorption, the NODE could fit data from both PK models with four and eight transit compartments, respectively. As such time-consuming discussion about how many transit compartments are necessary to describe a delayed absorption process is obsolete when applying developed NODE structure.
We could overcome the challenge of overfitting by pooled training of the NODEs. With this, the NODEs did not fit residual errors but underlying dynamics was learned. Further, we could demonstrate that NODEs trained on PK data with different dose levels were able to perform appropriate predictions inside trained dose range.
Conclusions:
This research work reveals that NODEs are able to describe PK concentration-time data well, including PK data usually described with intricated models such as TMDD and delayed absorption. Incorporating PK principles, such as route of administration, into structures of NODEs allows to reduce size of NODE structures mitigating the risk of overfitting. As demonstrated, the risk of overfitting can further be minimized by pooled. Plotting the learned right-hand side of an NODE versus the concentration allows an assessment and consequently understanding of learned underlying mechanism. Hence, an insight into the “blackbox” NODE can be gained. In addition, it could be demonstrated that NODEs provide accurate predictions for new doses, as long as they are within the dose range used in the training data.
Based on these results, we conclude that application of NODEs might differ from that of classical PMX modeling. Lack of explicitly interpretable PK model parameters, such as drug clearance, is a clear difference compared to mechanistic modeling. However, the option to apply one NODE structure and make accurate predictions for various PK scenarios might be an advantage in terms of efficiency. Hence, NODEs may facilitate implementation of automated modeling approaches in situations, where interpretation of individual PK parameters is less important than the ability to efficiently characterize PK data, even with intricated profiles such as TMDD, that may drive a certain PD response. Our research work enhances understanding of how NODEs can be applied in PK analyses and illustrates the potential for NODEs in the field of pharmacometric PK/PD modeling and clinical pharmacology.
References:
[1] Chen RTQ, Rubanova Y, Bettencourt J, Duvenaud D (2018) Neural ordinary differential equations. arXiv [2] Lu J, Bender B, Jin JY, Guan Y (2021) Deep learning prediction of patient response time course from early data via neural-pharmacokinetic/pharmacodynamic modelling. Nat Mach Intell. https://doi.org/10.1038/s42256-021-00357-4
[2] Lu J, Bender B, Jin JY, Guan Y (2021) Deep learning prediction of patient response time course from early data via neural-pharmacokinetic/pharmacodynamic modelling. Nat Mach Intell. https://doi.org/10.1038/s42256-021-00357-4
[3] Lu J, Deng K, Zhang X, Liu G, Guan Y (2021) Neural-ODE for pharmacokinetics modeling and its advantage to alternative machine learning models in predicting new dosing regimens. iScience. https://doi.org/10.1016/j.isci.2021.102804
[4] Janssen A, Leebeek FWG, Cnossen MH, Mathôt RAA, for the OPTI-CLOT study group and SYMPHONY consortium (2022) Deep compartment models: A deep learning approach for the reliable prediction of time-series data in pharmacokinetic modeling. CPT Pharmacometrics Syst Pharmacol 11:934–945. https://doi.org/https://doi.org/10.1002/psp4.12808
[5] Janssen A, Bennis FC, Mathôt RAA (2022) Adoption of Machine Learning in Pharmacometrics: An Overview of Recent Implementations and Their Considerations. Pharmaceutics 14:. https://doi.org/10.3390/pharmaceutics14091814 [6] Bräm DS, Nahum U, Schropp J, Pfister M, Koch G (2023) Neural ODEs in Pharmacokinetics: Concepts and Applications. Prepr (Version 1) available Res Sq. https://doi.org/https://doi.org/10.21203/rs.3.rs-2428689/v1 [7] Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Networks. https://doi.org/10.1016/0893-6080(89)90020-8
[6] Bräm DS, Nahum U, Schropp J, Pfister M, Koch G (2023) Neural ODEs in Pharmacokinetics: Concepts and Applications. Prepr (Version 1) available Res Sq. https://doi.org/https://doi.org/10.21203/rs.3.rs-2428689/v1