Improved Modeling of the Glucose-Insulin Dynamical System Leading to a Diabetic State

1. Improved Modeling of the Glucose-Insulin Dynamical System Leading to a Diabetic State Clinton C. Mason Arizona State University National Institutes of Diabetes and Digestive and Kidney Diseases Feb. 4th, 2006

2. Diabetes Overview • The cells in the body rely primarily on glucose as their chief energy supply • This glucose is mostly a by-product of the food we eat • After digestion, glucose is secreted into the bloodstream for transport to the various cells of the body

3. Diabetes Overview • Glucose is not able to enter most cells directly – insulin is required for the cells to uptake glucose • Insulin is secreted by the pancreas, at an amount regulated by the current glucose level – a feedback loop • If the steady state level of glucose in the bloodstream gets too high (200 mg/dl) – Type 2 Diabetes is diagnosed

4. Glucose-Insulin Modeling • Various models have been proposed to describe the short-term glucose-insulin dynamics • The Minimal Model (Bergman, 1979) has been widely accepted

5. Minimal Model Net Glucose Uptake & Product of Remote Insulin and Glucose Change in Glucose Const. times Plasma Insulin minus Const. times Remote Insulin Change in Remote Insulin 2nd Phase Insulin Secretion depends on Glucose excess of threshold (e) minus amount of 1st Phase Secretion Change in Plasma Insulin

9. Minimal Model • The model describes quite well the short-term dynamics of glucose and insulin • Drawbacks: • No Long-term simulations possible • Describes return to a normal glucose steady state level only • Provides no pathway for diabetes development

10. βIG Model • The first model to describe long-term glucose-insulin dynamics was the βIG model (Topp, 2000) • This model provided a pathway for diabetes development through the introduction of a 3rd dynamical variable – β - cell mass

11. βIG Model • The βIG model combines the fast dynamics of the minimal model, with the slower changes in β-cell mass due to glucotoxicity • This effect was modeled from data gathered from studies on Zucker diabetic fatty rats

12. βIG Model

13. βIG Model Change in Glucose Change in Insulin Change in Beta-cell Mass

14. βIG Model Same as Minimal Model Change in Glucose Variant of Minimal Model Change in Insulin β-cell mass changes as a parabolic function of Glucose Change in Beta-cell Mass

15. βIG Model Fast dynamics Fast dynamics Slow dynamics

16. Steady States

17. 3 Steady States

18. Steady States Shifting the 1st steady state to the origin and linearizing, we obtain

19. Steady States As the diagonal elements (eigenvalues) are negative for all normal parameter ranges, we find the steady state to be a locally stable node

20. Steady States The 2nd steady state is a saddle point, and the 3rd steady state is a locally stable spiral This 3rd s.s. represents a normal physiological steady state. The change of a given parameter can move this steady state closer and closer to the glucose level of the 2nd unstable steady state, and upon crossing this threshold, a saddle node bifurcation occurs, leaving only the 3rd steady state - approached rapidly by all trajectories

21. Parameter h decreasing

22. The saddle-node bifurcation describes a scenario in which β -cell mass goes to zero, and the glucose level rises greatly. • This is typical of what happens in Type 1 diabetes (usually only occurs in youth)

23. In Type 2 diabetes, the β-cell level is sometimes decreased, but the zero level of B-cell mass is never reached. • In fact, in some Type 2 diabetics, the β -cell level is completely normal.

24. Parameter h decreasing

25. (Butler, 2003) 63 % Reduction in β -cell MassBetween largest glucose changes

26. Hence, it appears that for these individuals, the deficit in β -cells is not extreme enough to encounter the saddle-node bifurcation, and approach the s.s with β -cell mass = 0 • Yet, there is a fast jump in glucose values when approaching the diabetic level

27. We will explore a different pathway for diabetes development that is independent of the β -cell level (i.e. let β’ = 0) • The pathway involves an increase in insulin resistancewhich causes insulin secretion levels to rise

28. Although the β -cells can increase their capacity to secrete insulin, there is a maximal level, and once reached, further increases in IR will cause the glucose steady state value to rise • Such a scenario may be sufficient to explain this pathway to diabetes.

29. This scenario is possible by merely looking at the 2-dimensional glucose-insulin dynamics.

30. βIG Model –Revision 1 Change in Glucose Change in Insulin Change in Insulin Resistance Change in Beta-cell Mass

32. βIG Model – Revision 2 • The model is formulated to describe a slow moving fluctuation of beta-cells due to glucotoxicity • However, the βIG model has beta cell mass dynamics that fluctuate rapidly, as β-cell level is modeled as a function of glucose level rather than steady state glucose level

33. βIG Model – without correction

34. βIG Model – Revision 2 A correction can be made by substituting in the glucose steady state value

35. βIG Model – Revision 2 Additionally, regular perturbations to the glucose system occur as often as we eat While these perturbations have usually decayed by the time of the next feeding, they may be modeled to give a more realistic profile

36. (Sturis, 1991)

38. βIG Model – Revision 2 Additionally, we may add, a glucose forcing term to simulate daily feeding cycles

39. βIG Model – Revision 2

40. Using the revised model, we may compare the glucose profiles obtained over a long time course with actual data from longitudinal studies

41. Overlay of Long-term Glucose Dynamics and Longitudinal Data 450 400 350 300 250 200 150 100 0 5 10 15 20 25 30 35 40 Time (years) Hypothetical Overlay of Revised βIG Model and Actual Long Term Data

42. Works Cited • Bergman RN, Ider YZ, Bowden CR, Cobelli C. Quantitative estimation of insulin sensitivity. Am J Physiol. 1979 Jun;236(6):E667-77. • Butler AE, Janson J, Bonner-Weir S, Ritzel R, Rizza RA, Butler PC. Beta-cell deficit and increased beta-cell apoptosis in humans with type 2 diabetes. Diabetes. 2003 Jan;52(1):102-10. • Sturis, J., Polonsky, K. S., Mosekilde, E., Van Cauter, E. Computer model for mechanisms underlying ultradian oscillations of insulin and glucose. Am. J. Physiol. 1991; 260, E801-E809. • Topp, B., Promislow, K., De Vries, G., Miura, R. M., Finegood, D. T. A Model of β-cell mass, insulin, and glucose kinetics: pathways to diabetes, J. Theor. Biol. 2000; 206, 605-619. • Background image modified from http://www.fraktalstudio.de/index.htm