Improved Modeling of the Glucose-Insulin Dynamical

System Leading to a Diabetic State

Clinton C. MasonArizona State University

National Institutes of Diabetes and Digestive and Kidney Diseases

Feb. 4th, 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

fItI

cX

GXadt

dG

)e-d(G dt

d

bI dt

dX

)(Change in Glucose

Change in RemoteInsulin

Change in PlasmaInsulin

Net Glucose Uptake & Product of Remote Insulin and Glucose

Const. times Plasma Insulin minus Const. times Remote Insulin

2nd Phase Insulin Secretion depends on Glucose excess of threshold (e) minus amount of 1st 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 3rd 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

)h (-g dt

d

dt

dI

)(

2

2

2

iGG

fIGe

dG

GcIbadt

dG

βIG Model

)h (-g dt

d

dt

dI

)(

2

2

2

iGG

fIGe

dG

GcIbadt

dG

βIG Model

Change in Glucose

Change in Insulin

Change in Beta-cellMass

)h (-g dt

d

dt

dI

)(

2

2

2

iGG

fIGe

dG

GcIbadt

dG

βIG Model

Change in Glucose

Change in Insulin

Change in Beta-cellMass

Same as MinimalModel

Variant of MinimalModel

β-cell mass changesas a parabolic function of Glucose

)h (-g dt

d

dt

dI

)(

2

2

2

iGG

fIGe

dG

GcIbadt

dG

βIG Model

Fast dynamics

Fast dynamics

Slow dynamics

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0

)(0

2

2

2

iGG

fIGe

dG

GcIba

Steady States

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b

aIG

,,),,(

,,),,(

0,0,),,(

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3 Steady States

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a

b

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b

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Steady States

Shifting the 1st steady state to the origin and

linearizing, we obtain

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a

b

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b

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00

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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

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

Parameter hdecreasing

• 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 hdecreasing

(Butler, 2003)

63 % Reduction in β -cell MassBetween 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.

0 dt

d

gdt

BMId

dt

d

dt

dI

)(

2

2

R

fIGe

dG

GR

Iba

dt

dGβIG Model –Revision 1

Change in Glucose

Change in Insulin

Change in Beta-cellMass

Change in InsulinResistance

• 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

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2299.6

299.65

299.7

299.75

299.8

299.85

299.9

299.95

300

-C

ell M

ass

(mg)

Time (days)

Diabetes Model -Cell Mass vs. Time

βIG Model – Revision 2

statesteadyglucosetherepresentsGwhere

)h (-g dt

d

dt

dI

)(

2

2

2

iGG

fIGe

dG

GcIbadt

dG

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)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

120

140

160

180

200

220G

luco

se

Time

Diabetes Model Glucose vs. Time

βIG Model – Revision 2

Additionally, we may add, a glucose forcing term to simulate daily feeding cycles

)h (-g dt

d

dt

dI

)(

2

2

2

)6sin(8

iGG

fIGe

dG

jeGcIbadt

dG t

βIG Model – Revision 2

0 10 20 30 40 50 60 70 80 90 100100

150

200

250

300

350

Glu

cose

Time

Diabetes Model Glucose vs. Time

• 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

0 5 10 15 20 25 30 35 40100

150

200

250

300

350

400

450

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. 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