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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
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
hb
a
b
aig
eba
daf
b
cab
fD
2
2
22
2
00
0
0
)0(
Steady States
Shifting the 1st steady state to the origin and
linearizing, we obtain
hb
a
b
aig
eba
daf
b
cab
fD
2
2
22
2
00
0
0
)0(
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
• 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.
• 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.
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
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