Low-dimensional neural ordinary differential equations accounting for inter-individual variability implemented in Monolix and NONMEM.

Neural ordinary differential equations (NODEs) are an emerging machine learning (ML) method to model pharmacometric (PMX) data. Combining mechanism-based components to describe "known parts" and neural networks to learn "unknown parts" is a promising ML-based PMX approach. In this work, the implementation of low-dimensional NODEs in two widely applied PMX software packages (Monolix and NONMEM) is explained. Inter-individual variability is introduced to NODEs and proposals for the practical implementation of NODEs in such software are presented. The potential of such implementations is shown on various demonstrational datasets available in the Monolix model library, including pharmacokinetic (PK), pharmacodynamic (PD), target-mediated drug disposition (TMDD), and survival analyses. All datasets were fitted with NODEs in Monolix and NONMEM and showed comparable results to classical modeling approaches. Model codes for demonstrated PK, PKPD, TMDD applications are made available, allowing a reproducible and straight-forward implementation of NODEs in available PMX software packages.