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. 4 th, 2006

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

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

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

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

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

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 3 rd dynamical variable – β - cell mass

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

βIG Model

Change in Glucose Change in Insulin Change in Beta-cell Mass

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

βIG Model Fast dynamics Slow dynamics

Steady States

3 Steady States

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

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

Steady States The 2 nd steady state is a saddle point, and the 3 rd steady state is a locally stable spiral This 3 rd 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 2 nd unstable steady state, and upon crossing this threshold, a saddle node bifurcation occurs, leaving only the 3 rd steady state - approached rapidly by all trajectories

Parameter h decreasing

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)

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.

Parameter h decreasing

(Butler, 2003) 63 % Reduction in β -cell Mass Between largest glucose changes

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

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 resistance which causes insulin secretion levels to rise

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.

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

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

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 βIG Model – Revision 2

βIG Model – without correction

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

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

(Sturis, 1991)

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

βIG Model – Revision 2

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

Hypothetical Overlay of Revised βIG Model and Actual Long Term Data Time (years) Overlay of Long-term Glucose Dynamics and Longitudinal Data

Works Cited Bergman RN, Ider YZ, Bowden CR, Cobelli C. Quantitative estimation of insulin sensitivity. Am J Physiol Jun;236(6):E 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 Jan;52(1): 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, Background image modified from