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Introduction

Universal Differential Equations (UDEs)¹ integrate knowledge-driven mechanistic models with data-driven machine learning methods by incorporating a universal approximator, such as a neural network (NN), into differential equations. In this way, UDEs enable data-driven model discovery, especially in areas where biological knowledge is limited.² However, conventional UDEs learn a single representation of the data and consequently struggle with variability in data due to inter-individual variation. To address this, we introduce the conditional UDE (cUDE), which introduces a trainable condition-specific input parameter in the NN. The conditional input parameter can account for differences in system behaviour arising from diverse experimental conditions or heterogeneity within a subject population. In this work, we apply the cUDE framework, based on an existing model of c-peptide kinetics, on human time-series glucose and c-peptide measurements after an oral glucose tolerance test (OGTT). The model is applied on measurements in individuals with normal glucose tolerance, impaired glucose tolerance, and type 2 diabetes.

Approach

An existing model of plasma c-peptide kinetics³ was extended with a neural network term to account for pancreatic production, forming a UDE. The cUDE was developed by incorporating a trainable, person-specific parameter into the neural network to account for inter-individual differences in c-peptide secretion. During training, the neural network’s weights and biases were estimated on the population-level, representing the general model behaviour. Both the neural network parameters and the subject-specific conditional parameter were trained simultaneously. Model selection was performed by training on a subset of the data and validating with a separate dataset. During validation, the neural network parameters are fixed while only the conditional parameter was fitted to the unseen data. The final model was selected using a separate test subset.

Results

The trained cUDE model was able to accurately fit to the plasma c-peptide measurements in individuals of each group. Furthermore, the derived conditional parameters exhibit a strong correlation with independent ground-truth measurements of c-peptide production capacity.

Discussion

In this work, we demonstrate the conditional UDE (cUDE) as an extension to the UDE framework. We showcase its ability to derive a function of c-peptide production capacity from OGTT data, Moreover the individual conditioning parameters capture inter-individual variation in beta-cell function which correlates substantially with gold-standard measurements. As demonstrated on the c-peptide production model, the cUDE’s dual ability to derive general biological dynamics and capture individual or group-specific differences positions the cUDE as a valuable tool for understanding the heterogeneity observed in health and disease.

References