Dimensional Analysis of the Bergman Minimal Model for Glucose-Insulin Dynamics in Normal and Diabetic Individuals - Norazaliza Jamil; Chai Jian Tay (Jambura Journal of Biomathematics (JJBM), 2026)
The Bergman minimal model is commonly utilized to model glucose-insulin dynamics. However, the dimensional analysis of this model is somewhat limited. Dimensional analysis is an essential mathematical tool in fields such as engineering, physics, and applied sciences. Its main purposes include simplifying mathematical models, improving scalability and generalization, identifying key influencing factors, and enhancing computational efficiency. This study aimed to conduct a formal dimensional analysis of the Bergman minimal model, investigate the dimensionless model, simulate it, and investigate the dynamic of glucose-insulin to compare a normal and diabetic individual. The Bergman minimal model was transformed into a dimensionless format by reexpressing all state variables and parameters regarding their reference scales. The dimensionless model was solved and simulated using the fourth-order Runge-Kutta (RK4) numerical method, implemented in MATLAB software, to generate the dynamics of glucose and insulin in individuals with normal and diabetic conditions. In the dimensionless formulation, the number of parameters is reduced from eight to five and five new composite parameters are revealed, such as glucose clearance rate, insulin sensitivity, the rate of insulin degradation, insulin release response and basal glucose level. Results reveal that glucose levels decrease efficiently in normal individuals, while diabetic patients experience a slow decline in glucose level, not normalize (not return to 1 dimensionless) after a meal. A simulation study for the case where no insulin production occurs in response to glucose reveals that the glucose level increases over time rather than decreases, with no sign of stabilization. Hence, without insulin, cells cannot take up glucose, leading to uncontrolled blood sugar accumulation. These findings underscore the role of dimensional analysis in streamlining the Bergman model; it simplified the governing equations, enabling the direct identification of key composite factors and resulting in a more computationally efficient simulation framework for characterizing glucose metabolic control.