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

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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

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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

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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

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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

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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
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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

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ßIG Model

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

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ßIG Model

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

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ßIG Model
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ßIG Model
Change in Glucose
Change in Insulin
Change in Beta-cell Mass
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ßIG Model
Same as Minimal Model
Change in Glucose
Variant of Minimal Model
Change in Insulin
ß-cell mass changes as a parabolic function of
Glucose
Change in Beta-cell Mass
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ßIG Model
Fast dynamics
Fast dynamics
Slow dynamics
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Steady States
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3 Steady States
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Steady States
Shifting the 1st steady state to the origin and
linearizing, we obtain
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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
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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
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Parameter h decreasing
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• The saddle-node bifurcation describes a scenario
  in which ß -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)

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

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Parameter h decreasing
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(Butler, 2003)
63 Reduction in ß -cell MassBetween largest
glucose changes
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• Hence, it appears that for these individuals, the
  deficit in ß -cells is not extreme enough to
  encounter the saddle-node bifurcation, and
  approach the s.s with ß -cell mass 0
• Yet, there is a fast jump in glucose values when
  approaching the diabetic level

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

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• Although the ß -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.

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

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ßIG Model Revision 1
Change in Glucose
Change in Insulin
Change in Insulin Resistance
Change in Beta-cell Mass
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ßIG Model Revision 2

• The model is formulated to describe a slow moving
  fluctuation of beta-cells due to glucotoxicity
• However, the ßIG model has beta cell mass
  dynamics that fluctuate rapidly, as ß-cell level
  is modeled as a function of glucose level rather
  than steady state glucose level

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ßIG Model without correction
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ßIG Model Revision 2
A correction can be made by substituting in the
glucose steady state value
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ßIG 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
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(Sturis, 1991)
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ßIG Model Revision 2
Additionally, we may add, a glucose forcing term
to simulate daily feeding cycles
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ßIG Model Revision 2
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• Using the revised model, we may compare the
  glucose profiles obtained over a long time course
  with actual data from longitudinal studies

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Hypothetical Overlay of Revised ßIG Model and
Actual Long Term Data
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Works Cited

• Bergman RN, Ider YZ, Bowden CR, Cobelli C.
  Quantitative estimation of insulin sensitivity.
  Am J Physiol. 1979 Jun236(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 Jan52(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 ß-cell mass,
  insulin, and glucose kinetics pathways to
  diabetes, J. Theor. Biol. 2000 206, 605-619.
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