Cellular and Whole Body Models of Glucose Control on Insulin Secretion

Gianna Maria ToffoloMorten Gram Pedersen

Department of Information EngineeringUniversity of Padova

Padova, Italy

REx Workshop, February 17-18, 2009

Whole Body Models

• Simple models, with a few parameters to be identified on data of an individual

• Measure glucose control on insulin secretion

• Usable in clinical practice

• Based on plasma measurements (C-peptide and glucose) during a minimally invasive protocol

From i.v. and oral tests

Models to measure

Cellular Models Models to understand

IVGTT: Young vs Elderly SubjectsN = 59 vs 145 (Dr. Rizza & Basu, Mayo Clinic)

Young0

50

150

250

350

0 60 120 180 240t [min]

GLUCOSE

Elderly[mg

/dl]

0

400

800

1200

1600

2000

0 60 120 180 240t [min]

C-PEPTIDE

[pm

ol/

l]

t [min]0100

300

500

700

900

0 60 120 180 240

INSULIN

[pm

ol/

l]

CP1(t)= - (k01 + k21) CP1(t) + k12CP2(t) + SR(t)

CP2(t)= k21 CP1(t) - k12 CP2(t)

SR(t) = SRb + m X(t)

Basal Responsivity Φb=SRb

Gb

Y(t) = [Y(t) – Φ2 (G-h)]1

T

Delay2nd Phase Responsivity

IVGTT:C-peptide Minimal Model (Toffolo et al, 1995)

CP2

Rate of Increase of

Glucose ΔG

k01

k21

k12

CP1

Delay

1st Phase

2nd Phase

Glucose

ReleasableInsulin

SECRETION

m

X

Y

G1st Phase Responsivity Φ1=

X0

ΔG

X(t) = - m X(t) + Y(t) X(0)=X0

0

100

200

300

400

-30 0 60 120 180 240t [min]

[mg

/dl]

Glucose

IVGTT: Insulin Secretion Phases

-30 0 60 120 180 2400

1000

2000

3000

4000

t [min]

[pm

ol/

min

]

ISR

0

50

100

150

200

-30 0 60 120 180 240t [min]

BASAL

[pm

ol/

min

]

Φb: Basal ISR / Basal G

0

250

500

750

1000

-30 0 60 120 180 240t [min]

2nd PHASE

Φ2: Over Basal 2nd Phase ISR/ Over

Basal G

T: Delay between 2nd Phase ISR and G

t [min]

0

1000

2000

3000

4000

-30 0 60 120 180 240

1st PHASE

Φ1: 1st Phase ISR / G

IVGTT: β-Cell Responsivity Indices N=59Y vs 145E

EY

*

Φ1

[10

-9]

[10-9

min

-1]

EY

Φb

0

2

4

6

8

50

150

250

EY

[10-9

min

-1]

Φ2[m

in]

0

5

10

15 *T

EY0

5

10

15

20

0

* p<0.05

(OGTT or meal)

From IVGTT to more physiological protocols

GLUCOSE

INSULIN

100

300

500

0 120 240 360 420t [min]

C-PEPTIDE

1000

2000

3000

0 120 240 360 420t [min]

Meal: Young vs Elderly SubjectsN = 59 vs 145 (Dr. Rizza & Basu, Mayo Clinic)

80

120

160

200

0 120 240 360 420t [min]

Young

Elderly

[mg

/dl]

[pm

ol/

l][p

mo

l/l]

Rate of Increase of Glucose

(first 50-60 minutes)

k01

k21

k12

CP1 CP2

Delay

Dynamic Phase

Static Phase

GlucoseSECRETION

Meal: C-peptide Minimal Model (Toffolo et al, 2001; Breda et al, 2001, 2002)

CP1(t)= - (k01 + k21) CP1(t) + k12CP2(t) + SR(t)

CP2(t)= k21 CP1(t) - k12 CP2(t)

SR(t) = SRb + SRd(t) + SRs(t)

Basal Responsivity Φb=SRb

GbY(t) = [Y(t) – Φ2 (G-h)]1

T

Delay Static Responsivity

Static phase SRs = Y

Dynamic Responsivity

Dynamic Phase SRd(t) = Φd

dG

dt

Meal: Insulin Secretion PhasesISR

0

1000

2000

3000

4000

-30 0 60 120 180 240

t [min]

[pm

ol/

min

]

0

100

200

300

400

-30 0 60 120 180 240t [min]

GLUCOSE [mg

/dl]

0

50

100

150

200

-30 0 60 120 180 240t [min]

BASAL

[pm

ol/

min

]

Φb: Basal ISR / Basal G

t [min]

0

1000

2000

3000

4000

-30 0 60 120 180 240

DYNAMIC

Φd: Dynamic ISR / Glucose Rate of

Increase

0

250

500

750

1000

-30 0 60 120 180 240t [min]

STATIC

Φs: over basal static ISR /

over basal G

T: Delay between Static ISR and G

EY

Φb

[10-9

min

-1]

Meal: β-Cell Responsivity IndicesN=59Y vs 145E

*

Φd

[10

-9]

EY* p<0.05

0

2

4

6

8

0

200

400

600

800

Φs

[10-9

min

-1]

EY0

10

20

30

40 *

EY

T

[min

]

0

5

10

15

20

• Measurement of insulin secretion alone provides limited insight

• It is important to determine whether -cell secretion

is appropriate for the degree of insulin resistance

To quantify the efficiency of the glucose-insulin regulatory system…..

β-CELLS

LIVER GLUCOSE

INSULIN

BRAIN

MUSCLE

TISSUESDEGRADATION

SECRETION

LIVER

PRODUCTION UTILIZATION

+

B-cell Responsivity

-+

Insulin Sensitivity

Glucose Minimal Model

k3

k5

GLUCOSE TISSUES

k1

k4

k6

REMOTE INSULIN

k2

PLASMA INSULIN

LIVER

IVGTT

SI

IVGTT (Bergman & Cobelli, 1979)

OGTT/MEAL

k3

k5

GLUCOSE TISSUES

k1

k4

k6

REMOTE INSULIN k2

INSULIN

LIVER

Gastrointestinal Tract

SI

MEAL (Dalla Man & Cobelli, 2002)

Insulin Sensitivity 59 Y vs 145 E

*

[10-4

dl/k

g/m

in p

er

U/m

l]

EY0

4

8

12

EY

*

[10-4

dl/k

g/m

in p

er

U/m

l]

0

5

10

15

20

* p<0.05

Beta-Cell Responsivity

Beta-Cell Responsivity

Insulin Sensitivity Insulin Sensitivity

Normal ToleranceNormal

NormalReduced

I

II

2

Impaired Tolerance

Increased

Efficiency of the Control: Disposition Index(Bergman & Cobelli, 1981, Cobelli et at, 2007)

Insulin Sensitivity x Beta-Cell Function= Constant

Disposition Indices59 Y vs 145 E

* p<0.05

*

DI1

[10

-14 d

l/k

g/m

in p

er

pm

ol/

l]

EY0

1000

2000

3000

*

DI2

[10-1

4 d

l/k

g/m

in2

pe

r p

mo

l/l]

EY0

40

80

120

160

Meal IVGTT

[10

-14 d

l/k

g/m

in p

er

pm

ol/

l]

EY

*

DId

0

5000

10000

15000

20000

EY

*

DIs

[10

-14 d

l/k

g/m

in2 p

er

pm

ol/

l]

0

400

800

1200

Use in Pathophysiology

Role of age and gender (Basu et al, Diabetes 2006)

Pathogenesis of Prediabetes (Bock et al, Diabetes 2006)

Role of Race (Petersen et al, Proceedings of the National Academy of Science 2006)

Diurnal Variation of Glucose Tolerance (Dr. E. Van Cauter, University of Chicago, Chicago, IL)

Efficiency of Anti-aging Drugs (Nair et al, New England Journal of Medicine 2006)

Reduced OGTT & Meal Protocols (Dalla Man et al, Diabetes ,2006)

Type 2 Diabetes ( Basu et al, Diabetes Care, 2009)

Children and Adolescent (Sunehag et al, Obesity 2008)

Whole body models vs cellular events

Rate of Increase of Glucose

k01

k21

k12

CP1 CP2

Delay

Dynamic Phase

Static Phase

Glucose

Mechanistic intepretation of minimal model parameters Cellular Model

Dynamics of β-Cell Turnover in Rats(Manesso et al, EASD 2008)

β-Cell MassM

Replication Rate (RR)

Apoptosis Rate (RA)

Other Sources of β-Cells

(OSB)

?

dMdt = RR – RA + OSB

Pancreatic islet - himmunohystochemistry

(Dr P.Butler, UCLA)

OSB

0

8

16

0 5 10age [month]

[mg

/mo

nth

]

Thanks

CLAUDIO COBELLI

Elena Breda

Marco Campioni

Chiara Dalla Man

Erica Manesso

Paolo Denti

(Padua, Italy)

Andrea Caumo

(Milan, Italy)

ROBERT RIZZA

Rita Basu

Ananda Basu

F. John Service

(Rochester, MN)

PETER BUTLER

(UCLA, Los Angeles)

Whole body models vs cellular events

Rate of Increase of Glucose

k01

k21

k12

CP1 CP2

Delay

Dynamic Phase

Static Phase

Glucose

Mechanistic intepretation of minimal model parameters Cellular Model

Model Assessment

Insulin Secretion: Model vs Deconvolution

t [min]

[pm

ol/

min

]

C-Peptide Minimal Model

Deconvolution

0

400

800

1200

1600

0 100 200 300

Insulin Secretion

C-Peptide Minimal Model

t [min]

[pm

ol/

min

] Deconvolution

Model without Φd

0

400

800

1200

1600

0 100 200 300

Need of All the MM Ingredients

Insulin Secretion

e.g. Dynamic Glucose Control

(Steil et al, 2004)

Φ1IVGTT

(103 pmol/l/min)

Φ1H

GC

(1

03 p

mo

l/l/

min

)

IVGTT vs Hyperglycemic Clamp 1st Phase β-cell Responsivity

Meal vs Hyperglycemic Clamp Static β-cell Responsivity

Φsm

eal

(n

mo

l/m

in p

er

mm

ol/

l)

ΦsHGC

(pmol/min per mmol/l)

MM Indices vs HGC Counterparts

N=204

R=0.72, p<0.001

AIR (pmol/l∙ min)

Φ1

(10-9

)

Φ1: Correlation with Other Indexes

0

2000

4000

6000

8000

10000

0 100 200 300 400 500 600

VALIDATION PROBLEMS1. Simple net balance equations are not appropriate out of steady state

2. The transit time of the substance needs to be explicitly considered

MM Indices vs AV Measurements

GlucoseIncrease

DelayGlucose SRΦs

g(t)

F

CPA: C-peptide Femoral Artery

CPV: C-peptideHepatic Vein

Φd

C-peptide AV Model

0

SR(s)

FCPV(t) = CPA(s)+ g(t - s) ds

t

Distribution of transit times from femoral artery to

hepatic vein

C-peptide in Hepatic Vein

C-peptide in Femoral Artery Secretion

Blood Flow

Meal vs AV (N=12)

Φs

Meal AV

0

200

100

0

[l

109

min

-1]

Time [min]

0

250

500

750

1000

0 60 120 180 240 300 360

Meal

AV

[pm

ol

min

-1]

Insulin Secretion

Φd

Meal AV

0

3000

1500

0

[l

109

]

Protocols, Attributes and Information Content

Can Assess Beta-Cell Function?

Yes, but limited

Yes, but limited without a model

No

Yes, but limited without a model

Yes, but limited without a model

Yes, but limited without a model

BASAL STATE

Is it Physiological?

Yes

Is it Simple?

Yes

Can Assess Insulin Sensitivity?

Yes, but limited

INTRAVENOUS PERTURBATION

Hyperglycemic Clamp

Euglycemic Clamp

IVGTT

No

No

No

No

No

No

Yes, but requires a model

Yes

Yes, but limited without a model

ORAL PERTURBATION

OGTT

Meal

Yes, but no nutrients

Yes

Yes

Yes

Yes, but requires a model

Yes, but requires a model

Reproducibility: IVGTT

SI

Day A Day B

β - Cell Responsivity

Insulin Sensitivity

[10-4

dl/

kg

/min

p

er U

/ml]

[10-9

min

-1]

Φ2

0

50

250

500

Φ1

[10

-9]

Day A Day B Day A Day B

∆ = 4% ∆ = 17%

∆ = 12%

0

5

15

25

0

2

4

6

8

Reproducibility: Meal

SI

Day A Day B

β - Cell Responsivity

Insulin Sensitivity

Φd

[10

-9]

Day A Day B

Φs

[10

-9 m

in-1]

Day A Day B

[10-4

dl/

kg

/min

p

er U

/ml]

0

600

1200

1800

0

20

40

60∆ = 1% ∆ = 7%

∆ = 7%

0

4

8

12

16

-cell Function: IVGTT vs Meal

IVGTT vs Meal: β-Cell ResponsivityN=204

MealIVGTT

ΦdΦ1 ΦsΦ2

0

200

400

600

800

MealIVGTT0

10

20

30

40+ 251%*

• Incretin hormones?

• Differences in the pattern of glucose stimulus?

• Effect of fat/protein in the meal?

+ 253%*

IV vs Oral Glucose: Assessment of Incretin EffectN=10 (Dr. Rizza & Service, Mayo Clinic)

GLUCOSE

80

100

140

180

-30 0 60 120 180 240t [min]

IV-OGTT

OGTT[m

g/d

l]

C-PEPTIDE

2

4

6

-30 0 60 120 180 240t [min]

[ng

/ml]

INSULIN

10

30

50

-30 0 60 120 180 240t [min]

[uU

/ml]

Dynamic Secretion Static Secretion

t [min]

[pm

ol/

min

]

-200

0

200

400

0 60 120 180 240

t [min]

[pm

ol/

min

]

0

100

200

300

400

0 60 120 180 240

OGTTI-IVG

p<0.05

+ 55%*

+ 68%*

Φs

[10-9

min

-1]

10

20

30

OGTTIV-OGTT

200

400

600

OGTTIV-OGTT

[10

-9]

Φd

* p<0.05

Reduced Oral Protocols

OGTT (N=100)

0

60

120

0 60 1203000full

R=0.88, p<0.0001

red

(10-9

m

in-1)

0

30

60

full red

red

R=0.98, p<0.0001

0

1500

3000

0 1500

(10

-9 )

full red0

600

1200

full

Φd Φs

REDUCED

0 90 120 10 20 30 60

120 min – 7 Samples

Hepatic Insulin Extraction

Rationale

ISR

C-PEPTIDE

INSULIN

ISR

IDR

LIVER

-CELLS

LIVER I

n

k0,1

k2,1

k1,2

CP1 CP2

HEPATIC EXTRACTION =ISR - IDR

ISR

HEPATIC EXTRACTION =ISR - IDR

ISR

C-peptide Kinetics

Insulin Kinetics

IM-IVGTT

0

350

0

900

0

2000

0 60 120 180

0 60 120 180

0 60 120 180 240

240

240

time [min]

Glucose

Insulin

C-peptide

MEAL200

500

0 120 240 360 420

800 120 240 360 420

3000

0 120 240 360 420

0

0

time [min]

60 180 300

60 180 300

60 180 300

Glucose

Insulin

C-peptide

Estimation of Hepatic Insulin Extraction(Toffolo et al, 2006)

C-peptide Model

CP2

IncreaseGlucose

k01

k21

k12

CP1

Delay

GlucoseReleasableC-peptide

SECRETION KINETICS

ISR

Delay

SECRETION

IncreaseGlucose

Glucose

n

I

KINETICS

VI

IDR

Insulin Model

ReleasableInsulin

Insulin Bolus

From Insulin Bolus

Population Model(Van Cauter, 1991)

Population Model(Van Cauter, 1992)

Population Model

IVGTT: Hepatic Extraction

Profile

t [min]

0

0.2

0.4

0.6

0.8

1

0 60 120 180 240

ELDERLY

YOUNG

(%)

ISR(t) - IDR(t)

ISR(t)HE(t) =

N=59Y vs 145E

Index

0.00

0.20

0.40

0.60

0.80

1.00

EY

*

(%)

T

0

T

0

ISR(t)dt

ISR(t)dt – IDR(t)dt

HE =T

0

* p<0.05

Meal: Hepatic Extraction

0.00

0.20

0.40

0.60

0.80

1.00

0 60 120 180 240 300 360 420

ELDERLY

YOUNG

t [min]

(%)

Profile

0.00

0.20

0.40

0.60

0.80

1.00

EY

*(%)

Index

* p<0.05

N=59Y vs 145E

Disposition index

• Does the hyperbolic relation DI=Φ x SI hold in a population or is it DI=Φ x SIα?

• Average individual DI or value estimated in the population?

• Linear regression on log-trasformed variables or nonlinear regression on original variables?

• Ordinary regression with errors in one variable or regression with errors in two variables?

Open Questions

Average individual DI or value estimated in the population?

Averaging Φ x SI=732

XY fit

Φ x SI=732

Φ x SI0.68=330

Averaging

logXY fit Φ x SI0.77=578

XY fit Φ x SI0.68=330

Y fit Φ x SI0.19=135

XY fit Φ x SI0.68=330

SIivgtt [10-5 min-1 per pmol/l]

Φ1 [

10

-9]

Linear regression on log-trasnformed variables or nonlinear regression on original variables?

Does the hyperbolic relation DI= Φ x SI in a population or is it DI= Φ x Siα?

Regression with error in one variable or in two variables?

Disposition Index

The classical DI ignores the effect of hepatic insulin extraction

Φ1 [10-9]

SIivgtt [10-5 min-1 per pmol/l]

1-H

E [d

imen

sion

less

]

From 2D to 3D?

From IVGTT to slower profiles

CP2

Rate of Increase of

Glucose ΔG

k01

k21

k12

CP1

Delay

1st Phase

2nd Phase

Glucose

ReleasableInsulin

SECRETION

m

X

Y

• Since m is rapid (1/m=1-2 minutes) with respect to the slower (than IVGTT) time course of insulin and glucose concentration, m cannot be resolved from the data.

• 1st phase dynamic phase (related to dG/dt, active during the glucose rising phase)

• 2nd phase static phase (related to G)

GLUCOSE

Meal: Non Diabetics vs Type 2 Diabetics N=14 vs 11 (Dr. A. Basu)

t [min]

INSULIN

40

80

120

0 120 240 360

t [min]

200

400

0 120 240 360

Diabetics

Non Diabetics[m

g/d

l]

t [min]

C-PEPTIDE

1000

3000

5000

0 120 240 360

[pm

ol/

l][u

U/m

l]

Φb

β-Cell Responsivity Indices N=14 vs 11

* p<0.05

ND D0

4

8Φd

ND D

*

0

400

800

Φs

ND D

*

0

20

40

60

20

60

100

140

*

ND D

T

[10

-9 m

in-1]

[10

-9]

[10-9

min

-1]

[min

]

Type 2 Diabetes: Effect of Pioglitazone (Basu A. et al submitted)

Insulin

0

20

40

60

80

100

120

140

-60 0 60 120 180 240 300 360

time (min)

uU/m

l

Glucose

0

5

10

15

20

25

-60 0 60 120 180 240 300 360

Time (min)

mM

Pre-pioPost-pioNondiabetic

C-peptide

0.000

1.000

2.000

3.000

4.000

5.000

-60 0 60 120 180 240 300 360

time (min)

nmol

/L

Pre-pioPost-pioNondiabetic

Pre-pioPost-pioNondiabetic

* p<0.05 pre vs post treatment

^ p<0.05 post treatment vs nondiabetic

0

5

10

15

20

25

30

10^-

5 dl

/kg/

min

per

pm

ol/L

pre pio

post pio

Nondiabetic

*

0

100

200

300

400

500

600

700

800

10^-

9

pre pio

post pio

Nondiabetic

0

10

20

30

40

50

60

10^-

9 m

in^-

1

pre pio

post pio

Nondiabetic

SI ds

Children and Adolescents (Dr. Caprio)

GLUCOSE

0

20

40

60

80

100

120

140

160

180

0 30 60 90 120 150 180

min

mg/

dl

NGT_NFG

NGT_IFG

IGT_NFG

IGT_IFG

INSULIN

0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180

min

uU/m

l

C-PEPTIDE

0

1000

2000

3000

4000

5000

6000

7000

0 30 60 90 120 150 180

min

pmol

/l

SI

0

1

2

3

4

5

6

7

8

10-4 d

l/kg/

min

per

uU

/ml

NGT_NFG

NGT_IFG

IGT_NFG

IGT_IFG

* *^

*

* p<0.05 vs NGT-NFG; ^ p<0.05 vs NGT-IFG; # p<0.05 vs IGT-NFG

Φd

0

500

1000

1500

2000

2500

3000

10-9

*^

*

Φs

0

15

30

45

60

75

90

10-9 m

in-1

*#

*

Whole Body Models

• Provide estimates of beta cell function in an individual

• Based on plasma measurements (C-peptide and glucose) during a minimally invasive protocol

• Usable in clinical practice

From i.v. to oral tests

INSULIN SYSTEMSecretion+Kinetics

PLASMA GLUCOSE

PLASMA INSULIN

GLUCOSE SYSTEM

PLASMA C-peptide