A Hybrid Physiological and Neural ODE Model with Probabilistic Outputs for Personalized Glucose Dynamics in Type 2 Diabetes

Introduction:
Type 2 Diabetes Mellitus (T2DM) is a heterogeneous metabolic disorder characterized by insulin resistance, impaired insulin secretion, and altered hepatic glucose regulation, resulting in complex and patient-specific glucose dynamics. Accurate mathematical modeling of these processes is essential for understanding disease mechanisms and supporting personalized therapeutic strategies.

Methods:
We present a hybrid modeling framework that integrates a physiologically interpretable system of ordinary differential equations (ODEs) with Neural ODE residual dynamics and a probabilistic output mechanism. The physiological component explicitly models key glucose–insulin processes relevant to T2DM, including gastric emptying, intestinal glucose absorption, plasma glucose and insulin dynamics, and insulin action. Each state variable corresponds to a measurable or identifiable biological process, ensuring physiological plausibility. A Neural ODE component is introduced to learn residual dynamics from patient-specific glucose time-series data. A Gaussian Mixture Model (GMM) output layer enables probabilistic prediction.

Results:
The proposed model admits a well-defined and stable fasting equilibrium and generates physiologically consistent postprandial glucose trajectories. The Neural ODE residual component enhances adaptability to inter-patient variability, while the GMM output captures uncertainty and multimodal glucose responses.

Conclusions:
This hybrid Physiological–Neural ODE framework combines mechanistic interpretability, data-driven adaptability, and probabilistic forecasting, providing a robust mathematical foundation for personalized glucose dynamics modeling in Type 2 Diabetes.