A Model of β -Cell Mass, Insulin, and Glucose Kinetics: Pathways to Diabetes

Abstract Diabetes is a disease of the glucose regulatory system that is associated with increased morbidity and early mortality. The primary variables of this system are β -cell mass, plasma insulin concentrations, and plasma glucose concentrations. Existing mathematical models of glucose regulation incorporate only glucose and/or insulin dynamics. Here we develop a novel model of β -cell mass, insulin, and glucose dynamics, which consists of a system of three nonlinear ordinary differential equations, where glucose and insulin dynamics are fast relative to β -cell mass dynamics. For normal parameter values, the model has two stable fixed points (representing physiological and pathological steady states), separated on a slow manifold by a saddle point. Mild hyperglycemia leads to the growth of the β -cell mass (negative feedback) while extreme hyperglycemia leads to the reduction of the β -cell mass (positive feedback). The model predicts that there are three pathways in prolonged hyperglycemia: (1) the physiological fixed point can be shifted to a hyperglycemic level (regulated hyperglycemia), (2) the physiological and saddle points can be eliminated (bifurcation), and (3) progressive defects in glucose and/or insulin dynamics can drive glucose levels up at a rate faster than the adaptation of the β -cell mass which can drive glucose levels down (dynamical hyperglycemia).