Universal differential equations (UDEs) are an emerging approach combining physiology based mathematical models (PBMMs) with artificial neural networks (ANNs). However, in many biological problems, there is considerable inter-sample variation. This variation may arise from measurement noise, or due to differences in genetic background or acquired exposures that are indicative of pathological conditions in the development of a disease state of interest. For this reason, it is important that modelling approaches can adequately capture and quantify this data heterogeneity, thereby improving our understanding of the underlying disease. Conventional UDEs cannot accommodate data heterogeneity, limiting direct application to biological modelling problems. We propose a conditional UDE (cUDE) framework which introduces a tunable conditioning parameter to the ANN in the model to account for individual differences. In this way, the biases and weights of the ANN learn the population level response while the conditioning parameter captures inter-individual variation.
We applied cUDEs to learn a model of c-peptide production, a marker of insulin secretion, in individuals with varying glucose tolerance status. The trained cUDE model accurately describes postprandial c-peptide levels across the whole population, outperforming a conventional UDE. Furthermore, the tunable individual parameter produced a strong correlation with independently measured hyperglycemic clamp indices, the gold standard measure of beta-cell capacity (rho = -0.81). We demonstrated that this mathematical model can reliably quantify insulin production capacity from oral glucose tolerance tests data, providing a valuable surrogate index for use in precision healthcare in diabetes and metabolic diseases.