Conditional Universal Differential Equations Simultaneously Capture Population Level Dynamics and Inter-individual Variation in Human C-peptide Production

Universal differential equations (UDEs) are an emerging approach in biomedical systems biology, integrating physiology-driven mathematical models with machine learning for data-driven model discovery in areas where knowledge of the underlying physiology is limited. However, current approaches to training UDEs do not directly accommodate heterogeneity in the underlying data. As a data-driven approach, UDEs are also vulnerable to overfitting and consequently cannot sufficiently generalise to heterogeneous populations. We propose a conditional UDE (cUDE) where we assume that the structure and weights of the embedded neural network are common across individuals, and introduce a conditioning parameter that is allowed to vary between individuals. In this way, the cUDE architecture can accommodate inter-individual variation in data while learning a generalisable network representation. We demonstrate the effectiveness of the cUDE as an extension of the UDE framework by training a cUDE model of c-peptide production. We show that our cUDE model can accurately describe postprandial c-peptide levels in individuals with normal glucose tolerance, impaired glucose tolerance, and type 2 diabetes mellitus. Furthermore, we show that the conditional parameter captures relevant inter-individual variation. Subsequently, we use symbolic regression to derive a generalisable analytical expression for c-peptide production.