Cellular and Whole Body Models
of Glucose Control on Insulin Secretion
Gianna Maria Toffolo
Morten Gram Pedersen
Department of Information Engineering
University of Padova
Padova, Italy
REx Workshop, February 17-18, 2009
Whole Body Models
Models to measure
• 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
Cellular Models
Models to understand
IVGTT: Young vs Elderly Subjects
N = 59 vs 145 (Dr. Rizza & Basu, Mayo Clinic)
[mg/dl]
350
GLUCOSE
250
Elderly
150
Young
50
0
0
60
[pmol/l]
900
120
180
t [min]
240
INSULIN
700
500
300
100
0
t [min]
0
60
[pmol/l]
2000
120
180
240
C-PEPTIDE
1600
1200
800
400
0
t [min]
0
60
120
180
240
IVGTT:C-peptide Minimal Model (Toffolo et al, 1995)
CP1(t)= - (k01 +
k21) =
CP
+m
k12X(t)
CP2(t) + SR(t)
SR(t)
SR
1(t)
b +
CP2(t)= k21 CP1(t) - k12 CP2(t)
SRb X(0)=X0
X(t) = - m X(t) + Y(t)
Basal Responsivity Φb=
Gb
X0
1
Φ1=
Y(t) = 1st Phase
[Y(t)Responsivity
– Φ2 (G-h)]
Glucose G
ΔG
T
SECRETION
Delay
Delay
Y
k21
2nd Phase
Releasable
Insulin
m
k12
1st Phase
k01
Glucose ΔG
CP2
CP1
X
Rate of Increase of
2nd Phase Responsivity
[pmol/min]
IVGTT: Insulin Secretion Phases
4000
ISR
3000
2000
1000
t [min]
0
-30 0
4000
BASAL
150
1st
3000
120
180
240
1000
PHASE
2000
500
50
1000
250
0
-30
0
0
0
60
120
t [min]
180
240
2nd PHASE
750
100
-30
Φb: Basal ISR / Basal G
0
60
120
t [min]
180
240
-30
0
60
120
t [min]
180
240
Φ2: Over Basal
Φ1: 1st Phase ISR / DG
2nd Phase ISR/ Over
Basal G
T: Delay between 2nd Phase ISR and G
400
[mg/dl]
[pmol/min]
200
60
Glucose
300
200
100
0
-30
t [min]
0
60
120
180
240
IVGTT: β-Cell Responsivity Indices
N=59Y vs 145E
Φ1
Φb
250
[10-9]
[10-9 min-1]
8
6
4
2
50
0
0
Y
*
150
E
Y
E
* p<0.05
Φ2
T
20
*
15
10
[min]
[10-9 min-1]
15
10
5
5
0
0
Y
E
Y
E
From IVGTT to
more physiological protocols
(OGTT or meal)
Meal: Young vs Elderly Subjects
N = 59 vs 145 (Dr. Rizza & Basu, Mayo Clinic)
GLUCOSE
[mg/dl]
200
160
Elderly
120
Young
t [min]
80
0
240
360
420
INSULIN
500
[pmol/l]
120
300
100
t [min]
0
[pmol/l]
3000
120
240
360
420
C-PEPTIDE
2000
1000
t [min]
Meal: C-peptide Minimal Model
(Toffolo et al, 2001; Breda et al, 2001, 2002)
Glucose
SR(t) = SRb + SRd(t) + SRs(t)
CP1(t)= - (k01 + k21) CP1(t)SR
+ k12CP2(t) + SR(t)
b phase SR
Static
dGs = Y
BasalkDynamic
Responsivity
= (t)
bCP
CP2(t)=
- kΦ
Phase
1d
21 CP1(t)
12SR
2(t)= Φ
dG
b
dt – Φ2 (G-h)]
Y(t) =
[Y(t)
Dynamic T
Responsivity
SECRETION
Static Responsivity
Delay
Delay
k21
Static Phase
CP1
CP2
k12
Dynamic Phase
k01
Rate of Increase of Glucose
(first 50-60 minutes)
[pmol/min]
Meal: Insulin Secretion Phases
4000
ISR
3000
2000
1000
t [min]
0
-30 0
BASAL
150
4000
100
3000
50
2000
0
-30
120
180
240
1000
0
60
120
t [min]
180
240
500
250
0
0
-30
-30
Φb: Basal ISR / Basal G
STATIC
750
DYNAMIC
1000
0
60
120
t [min]
180
240
Increase
0
60
120
t [min]
T: Delay between Static ISR and
400
300
GLUCOSE
100
0
240
static ISR /
over basal G
G
200
180
Φs: over basal
Φd: Dynamic ISR / Glucose Rate of
[mg/dl]
[pmol/min]
200
60
t [min]
Meal: β-Cell Responsivity Indices
N=59Y vs 145E
8
800
6
600
[10-9]
[10-9 min-1]
Φb
4
*
400
2
200
0
0
Y
Φd
E
Y
E
* p<0.05
Φs
20
30
[min]
[10-9 min-1]
40
T
20
15
10
10
5
0
0
Y
E
*
Y
E
To quantify the efficiency of the
glucose-insulin regulatory system…..
BRAIN
PRODUCTION
LIVER
-
GLUCOSE
UTILIZATION
+
MUSCLE
Insulin Sensitivity
β-CELLS
+
SECRETION
TISSUES
INSULIN
B-cell Responsivity
DEGRADATION
LIVER
• Measurement of insulin secretion alone provides
limited insight
•
It is important to determine whether b-cell secretion
is appropriate for the degree of insulin resistance
Glucose Minimal Model
IVGTT
MEAL
(Dalla Man & Cobelli, 2002)
(Bergman & Cobelli, 1979)
OGTT/MEAL
Gastrointestinal Tract
IVGTT
k5
k1
GLUCOSE
LIVER
k5
k1
TISSUES
GLUCOSE
LIVER
TISSUES
k4
SI
k6
PLASMA
INSULIN
SI
k6
k2
REMOTE
INSULIN
REMOTE
INSULIN
INSULIN
k2
k3
k4
k3
12
59 Y vs 145 E
*
8
4
0
Y
E
* p<0.05
[10-4 dl/kg/min per mU/ml]
[10-4 dl/kg/min per mU/ml]
Insulin Sensitivity
20
*
15
10
5
0
Y
E
Efficiency of the Control: Disposition Index
(Bergman & Cobelli, 1981, Cobelli et at, 2007)
Beta-Cell
Responsivity
Insulin Sensitivity x Beta-Cell Function= Constant
Increased
II
Normal
2
I
Normal Tolerance
Impaired Tolerance
Reduced
Normal
Insulin Sensitivity
IVGTT
Disposition Indices
59 Y vs 145 E
[10-14 dl/kg/min per pmol/l]
[10-14 dl/kg/min per pmol/l]
DI1
3000
*
2000
1000
0
E
DI2
[10-14 dl/kg/min2per pmol/l]
[10-14 dl/kg/min2 per pmol/l]
Y
160
*
120
80
40
0
Y
Meal
E
* p<0.05
DId
20000
15000
*
10000
5000
0
Y
E
DIs
1200
*
800
400
0
Y
E
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)
Efficiency of Anti-aging Drugs
(Nair et al, New England Journal of Medicine
2006)
Type 2 Diabetes ( Basu et al, Diabetes Care, 2009)
Children and Adolescent (Sunehag et al, Obesity 2008)
Diurnal Variation of Glucose Tolerance
(Dr. E. Van Cauter, University of
Chicago, Chicago, IL)
Reduced OGTT & Meal Protocols (Dalla Man et al, Diabetes ,2006)
Whole body models vs cellular events
Glucose
Delay
k21
Static Phase
CP1
CP2
k12
Dynamic Phase
k01
Rate of Increase of
Glucose
Mechanistic intepretation of
minimal model parameters
Cellular Model
Dynamics of β-Cell Turnover in Rats
(Manesso et al, EASD 2008)
Pancreatic islet - himmunohystochemistry
Replication Rate
(RR)
(Dr P.Butler, UCLA)
Other Sources
of β-Cells
(OSB)
β-Cell Mass
M
?
Apoptosis Rate
(RA)
OSB
[mg/month]
dM
= RR – RA + OSB
dt
16
8
0
0
5
age [month]
10
Thanks
CLAUDIO COBELLI
ROBERT RIZZA
Elena Breda
Rita Basu
Marco Campioni
Ananda Basu
Chiara Dalla Man
F. John Service
Erica Manesso
(Rochester, MN)
Paolo Denti
(Padua, Italy)
PETER BUTLER
(UCLA, Los Angeles)
Andrea Caumo
(Milan, Italy)
Whole body models vs cellular events
Glucose
Delay
k21
Static Phase
CP1
CP2
k12
Dynamic Phase
k01
Rate of Increase of
Glucose
Mechanistic intepretation of
minimal model parameters
Cellular Model
Model Assessment
Insulin Secretion: Model vs Deconvolution
Insulin Secretion
[pmol/min]
1600
Deconvolution
1200
800
C-Peptide Minimal Model
400
t [min]
0
0
100
200
300
Need of All the MM Ingredients
e.g. Dynamic Glucose Control
[pmol/min]
1600
Insulin Secretion
Deconvolution
1200
Model without Φd
800
C-Peptide Minimal Model
400
t [min]
0
0
100
200
300
MM Indices vs HGC Counterparts
IVGTT vs Hyperglycemic Clamp
1st Phase β-cell Responsivity
Meal vs Hyperglycemic Clamp
Static β-cell Responsivity
Φ1IVGTT
(103 pmol/l/min)
(nmol/min per mmol/l)
Φsmeal
(103 pmol/l/min)
Φ1HGC
(Steil et al, 2004)
ΦsHGC
(pmol/min per mmol/l)
Φ1: Correlation with Other Indexes
N=204
R=0.72, p<0.001
10000
Φ1 (10-9)
8000
6000
4000
2000
0
0
100
200
300
AIR (pmol/l∙ min)
400
500
600
MM Indices vs AV Measurements
VALIDATION PROBLEMS
1. Simple net balance equations are not appropriate out of steady state
2. The transit time of the substance needs to be explicitly considered
C-peptide AV Model
CPA: C-peptide
Femoral Artery
Glucose
Glucose
Increase
Φs
Φd
Delay
SR g(t)
F
CPV: C-peptide
Hepatic Vein
Distribution of
C-peptide intransit times from
Femoral femoral artery to
hepatic vein
Artery Secretion
t
CPV(t) =
CPA(s)+
SR(s)
g(t - s) ds
F
0
C-peptide in
Hepatic Vein
Blood Flow
Meal vs AV (N=12)
1000
[pmol min-1]
750
Insulin Secretion
500
Meal
AV
250
0
0
60
120
180
300
240
360
Time [min]
Φs
Φd
3000
0
100
0
[l 109]
0
[l 109 min-1]
200
1500
0
Meal
AV
Meal
AV
Protocols, Attributes and Information Content
Is it Simple?
Is it
Physiological?
BASAL STATE
Can Assess Insulin
Sensitivity?
Can Assess Beta-Cell
Function?
Yes
Yes
Yes, but limited
Hyperglycemic Clamp
No
No
Yes, but requires
a model
Euglycemic Clamp
No
No
Yes
IVGTT
No
No
Yes, but limited
without a model
Yes, but limited
without a model
Yes, but no
nutrients
Yes
Yes, but requires
a model
Yes, but limited
without a model
Yes
Yes
Yes, but requires
a model
Yes, but limited
without a model
Yes, but limited
INTRAVENOUS PERTURBATION
Yes, but limited
without a model
No
ORAL PERTURBATION
OGTT
Meal
Reproducibility: IVGTT
β - Cell Responsivity
Φ2
Φ1
∆ = 4%
∆ = 17%
[10-9 min-1]
25
250
15
5
50
0
0
Day A
Day B
Day A
Insulin Sensitivity
[10-4 dl/kg/min per mU/ml]
[10-9]
500
8
SI
∆ = 12%
6
4
2
0
Day A
Day B
Day B
Reproducibility: Meal
β - Cell Responsivity
[10-9 min-1]
1200
600
0
Φs
60
∆ = 1%
∆ = 7%
40
20
0
Day A
Day B
Day A
Insulin Sensitivity
[10-4 dl/kg/min per mU/ml]
[10-9]
1800
Φd
16
SI
∆ = 7%
12
8
4
0
Day A
Day B
Day B
b-cell Function: IVGTT vs Meal
IVGTT vs Meal: β-Cell Responsivity
N=204
Φ1
Φd
+ 251%
*
800
600
Φ2
+ 253%
*
40
30
400
20
200
10
0
0
IVGTT
Φs
Meal
IVGTT
Meal
• Incretin hormones?
• Differences in the pattern of glucose stimulus?
• Effect of fat/protein in the meal?
IV vs Oral Glucose: Assessment of Incretin Effect
N=10 (Dr. Rizza & Service, Mayo Clinic)
GLUCOSE
[mg/dl]
180
IV-OGTT
140
OGTT
100
80
-30
0
120
180
240
180
240
t [min]
INSULIN
50
[uU/ml]
60
30
10
-30
0
120
t [min]
C-PEPTIDE
6
[ng/ml]
60
4
2
t [min]
-30
0
60
120
180
240
Dynamic Secretion
Static Secretion
p<0.05
400
[pmol/min]
300
OGTT
I-IVG
200
100
200
0
0
0
0
60
120
180
240
60
120
180
-200
t [min]
t [min]
30
Φs
+ 55%
*
600
[10-9]
[10-9 min-1]
[pmol/min]
400
20
10
Φd
+ 68%
*
400
200
IV-OGTT
OGTT
* p<0.05
IV-OGTT
OGTT
240
Reduced Oral Protocols
OGTT
(N=100)
REDUCED
120 min – 7 Samples
0 10
20 30
60
120
90
Φd
0
full
red
1500
00
1500
full
3000
red
red
600
120
60
R=0.98,
p<0.0001
(10-9 min-1)
3000
1200
(10-9 )
Φs
30
0
full
red
R=0.88,
p<0.0001
60
0
0
60
full
120
Hepatic Insulin Extraction
Rationale
C-PEPTIDE
ISR
LIVER
k2,1
CP1
CP2
k1,2
k0,1
ISR
b-CELLS
INSULIN
IDR
I
LIVER
n
ISR - IDR
HEPATIC EXTRACTION =
ISR
Estimation of Hepatic Insulin Extraction
(Toffolo et al, 2006)
C-peptide Model
IM-IVGTT
350
Glucose
200
Delay
Glucose
0
0
60
120
180
240
Releasable
C-peptide
Glucose
Increase
MEAL
KINETICS
SECRETION
ISR
Glucose
k21
CP1
k12
CP2
80
k01
900
0
500
60 120 180 240 300 360 420
Insulin
Insulin
Insulin Model
0
SECRETION
0
60
120
180
Delay
Glucose
C-peptide
Glucose
Increase
0
0
60
120
180
0
0
240
2000
KINETICS
Releasable
Insulin
IDR
60 120 180 240 300 360 420
VI
3000
C-peptide
I
n
240
Insulin Bolus
time [min]
0
0
60 120 180 240 300 360 420
time [min]
HEPATIC EXTRACTION =
ISR - IDR
ISR
Population Model
(Van Cauter, 1991)
From Insulin Bolus
C-peptide Kinetics
Population Model
Insulin Kinetics
Population Model
(Van Cauter, 1992)
IVGTT: Hepatic Extraction
N=59Y vs 145E
Profile
1
ELDERLY
0.8
ISR(t) - IDR(t)
(%)
HE(t) =
ISR(t)
0.6
0.4
YOUNG
0.2
0
0
60
120
t [min]
180
Index
1.00
T
HE =

0.80
ISR(t)dt – IDR(t)dt
0
0

0
T
ISR(t)dt
(%)

T
*
0.60
0.40
0.20
0.00
* p<0.05
Y
E
240
Meal: Hepatic Extraction
N=59Y vs 145E
Profile
1.00
ELDERLY
(%)
0.80
0.60
0.40
0.20
YOUNG
0.00
0
60
120
180
240
300
360
420
t [min]
Index
1.00
(%)
0.80
0.60
*
* p<0.05
0.40
0.20
0.00
Y
E
Disposition index
Open Questions
• 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?
Φ1 [10-9]
Does
Average
Linear
Regression
theregression
hyperbolic
individual
with error
on
DI
relation
Disposition
log-trasnformed
orinvalue
one
DI=
variable
estimated
Φ
Index
x SIvariables
or
in in
a population
the
twoor
population?
variables?
nonlinear
or is it
regressionDI=
on original
Φ x Siα?variables?
Averaging
XY
Averaging
XY fit
fit
0.68=330
xxSISI=732
Φ
xx SI
0.68=330
Φ
SI=732
ΦΦ
logXY
Y fit fit
XY
0.68=
0.77 330
ΦΦ
SI
Φxx
xSI
SI0.19=578
=135
SIivgtt [10-5 min-1 per pmol/l]
From 2D to 3D?
1-HE [dimensionless]
The classical DI ignores the effect of hepatic insulin
extraction
Φ1 [10-9]
SIivgtt [10-5 min-1 per pmol/l]
From IVGTT to slower profiles
Glucose
SECRETION
Y
Delay
2nd Phase
Releasable
Insulin
1st Phase
m
k21
CP1
X
k12
CP2
k01
Rate of Increase of
Glucose ΔG
• 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)
Meal: Non Diabetics vs Type 2 Diabetics
N=14 vs 11 (Dr. A. Basu)
[mg/dl]
GLUCOSE
400
Diabetics
200
t [min]
Non Diabetics
0
240
360
INSULIN
120
[uU/ml]
120
80
40
t [min]
0
240
360
C-PEPTIDE
5000
[pmol/l]
120
3000
1000
t [min]
0
120
240
360
β-Cell Responsivity Indices
N=14 vs 11
Φb
4
400
0
*
0
ND
D
ND
Φs
60
D
T
140
*
100
40
*
[min]
[10-9 min-1]
Φd
* p<0.05
[10-9]
[10-9 min-1]
8
800
60
20
20
0
ND
D
ND
D
Type 2 Diabetes: Effect of Pioglitazone
(Basu A. et al submitted)
Glucose
Pre-pio
Post-pio
Nondiabetic
25
Insulin
C-peptide
Pre-pio
Post-pio
Nondiabetic
140
120
5.000
20
Pre-pio
Post-pio
Nondiabetic
4.000
15
nmol/L
uU/ml
mM
100
80
3.000
60
10
2.000
40
5
1.000
20
0
0
0
60
120
180
240
300
-60
360
0.000
0
60
30
15
240
300
360
-60
pre pio
post pio
Nondiabetic
700
600
60
50
40
400
300
30
20
10
200
5
0
60
10
100
0
0
* p<0.05 pre vs post treatment
^ p<0.05 post treatment vs
nondiabetic
120
180
s
pre pio
post pio
Nondiabetic
500
*
0
time (min)
800
10^-9
10^-5 dl/kg/min per pmol/L
20
180
d
SI
25
120
time (min)
Time (min)
10^-9 min^-1
-60
pre pio
post pio
Nondiabetic
240
300
360
Children and Adolescents
NGT_NFG
NGT_IFG
IGT_NFG
(Dr. Caprio)
INSULIN
GLUCOSE
450
160
400
140
350
120
300
100
250
C-PEPTIDE
7000
6000
5000
pmol/l
uU/ml
180
200
80
60
150
40
100
20
50
0
60
90
120
150
180
1000
0
0
30
min
5
4
3
3000
NGT_IFG
* * *
^
IGT_NFG
IGT_IFG
2500
2000
10-9
6
90
120
150
180
0
30
60
* *
^
1500
Φs
90
*
75
45
30
500
15
0
* p<0.05 vs NGT-NFG; ^ p<0.05 vs NGT-IFG; # p<0.05 vs IGT-NFG
*
#
60
1000
1
120
min
2
0
90
Φd
NGT_NFG
7
60
min
SI
8
3000
10-9 min-1
30
4000
2000
0
0
10-4 dl/kg/min per uU/ml
mg/dl
IGT_IFG
0
150
180
Whole Body Models
• Provide estimates of beta
cell function in an
individual
• Based on plasma
measurements (Cpeptide and glucose)
during a minimally
invasive protocol
INSULIN SYSTEM
Secretion+Kinetics
PLASMA C-peptide
PLASMA GLUCOSE
• Usable in clinical practice
From i.v. to oral tests
PLASMA INSULIN
GLUCOSE SYSTEM