INTEGRAL-BASED IDENTIFICATION OF A
PHYSIOLOGICAL INSULIN AND GLUCOSE
MODEL ON EUGLYCAEMIC CLAMP AND
IVGTT TRIALS
T Lotz1, J G Chase1, K A McAuley2, J Lin1, J Wong1, C E Hann1
and S Andreassen3
1Centre
for Bioengineering, University of Canterbury, Christchurch, New Zealand
2Edgar National Centre for Diabetes Research, University of Otago, Dunedin, New Zealand
3Centre for Model-based Medical Decision Support, Aalborg University, Denmark
Why model glucose and insulin
kinetics?
• Glycaemic control from critically ill to diabetic individuals
– Tight glycaemic control in ICU reduces mortality by up to 45%
– Type 1 and insulin dependent Type 2 diabetes growing rapidly
• Diagnosis of insulin resistance
– Requires knowledge of glucose and insulin kinetics
– Currently, diagnosis occurs ~7 years after initial occurrence
• Current models not physiological, difficult to identify, or do not
provide high resolution in clinical validation!
ID - Goals
1. Physiologically accurate model identification
• Higher predictive power and resolution
2. Simple application in a clinical setting
• Simple identification without the need of complicated tests
(minimal data required)
• Use population parameters where possible, fit critical
parameters
• Computationally efficient
2-compartment insulin kinetics model
+ glucose pharmacodynamics
PANCREAS
x·uen
PLASMA
I
nK
nL
KIDNEYS
LIVER
GLUCOSE
nI
diffusion
uex
INTERSTITIAL
FLUID
Q
nC
G   pG G  S I (G  GE )
CELLS
Q
1  GQ

P(t )
VG
n
Q  nC Q  I ( I  Q)
VQ
u (t ) x  uen (t )
n I
n
I  nK I  L  I ( I  Q)  ex 
1   I I VP
VP
VP
ID - problems
• 2-exponential insulin model but 8 parameters
• Physiological solution required
Try to identify a priori as many parameters as possible
Fit only the most critical parameters!
Critical parameters:
– Hepatic clearance nL
– First pass extraction of endogenous insulin x (if enough resolution in data)
– Insulin sensitivity SI
– Insulin independent glucose clearance pG
– Distribution volumes (if enough resolution in data)
A priori ID - Similarities with
C-peptide
PANCREAS
uen
C-peptide
(Van Cauter et al 1992)
PLASMA
VP
nI
INTERSTITIAL
FLUID
VQ
nK
KIDNEYS
Equimolar
secretion
PANCREAS
x·uen
Insulin
PLASMA
VP
nK
KIDNEYS
nI
INTERSTITIAL
FLUID
nC
CELLS
VQ
nL
LIVER
Additional losses
A priori ID – insulin model
u (t ) x  uen (t )
n I
n
I  nK I  L  I ( I  Q)  ex 
1   I I VP
VP
VP
n
Q  nC Q  I ( I  Q)
VQ
•
Distribution volumes (VP, VQ), transcapillary diffusion (nI), kidney clearance (nK)
assumed to match values for C-peptide (similar molecular size, equimolar
secretion)
•
Parameters taken from well validated population model for C-peptide kinetics
(Van Cauter et al. 1992)
•
Saturation of hepatic clearance (αI) fixed from published literature
•
Clearance by the cells (nC) fixed to achieve ss-concentration gradient between
the compartments (Iss/Qss=5/3) (Sjostrand et al 2005)
1 (2) key insulin parameters to be estimated, liver clearance nL (+ first pass hepatic
extraction x if data available)
A priori ID – glucose model
G   pG G  S I (G  GE )
Q
1  GQ

P(t )
VG
•
Glucose clearance saturation αG= 1/65 (from literature mean, validated
in glycemic control trials)
•
Equilibrium glucose concentration GE= fasting glucose level
•
Glucose distribution volume VG= 0.19 x body weight (can be estimated
if data allows)
•
Estimate pG, SI, (VG)
Integral-based fitting method
•
•
•
Convex, not starting point dependent
Reduces ID to solving a set of very well known linear equations
2 steps, first insulin, then glucose
•
Integrate insulin model between [t0,t1]:
t1
t
t
t
t
1
I (t )
nI 1
nI 1
1 1
I (t1 )  I (t0 )  nL 
dt  (nK  )  I (t )dt   Q(t )dt   uex (t )dt  x  uen (t )dt
1


I
(
t
)
VP t0
VP t0
VP t0
I
t0
t0
•
I(t) estimated by interpolating between discrete data
•
 ( nC  )( t  )
nI
VQ
Q(t) known from analytical solution: Q(t )   I ( )e
d
VQ 0
•
Inputs u(t) known (endogenous insulin estimated from C-Peptide)
t
nI
Integral-based fitting method
t1
t
t
t
t
1
I (t )
nI 1
nI 1
1 1
I (t1 )  I (t0 )  nL 
dt  (nK  )  I (t )dt   Q(t )dt   uex (t )dt  x  uen (t )dt
1   I I (t )
VP t0
VP t0
VP t0
t0
t0
known
known
identify
•
Repeat for different time-steps [t0,t1] ... [tn-1,tn]:
solve
C 0 , n L C 0 , x 
d 0 

 n L   
       0
 
x
Cn ,n Cn , x    d n 
 L

known
identify
Integral-based fitting method
•
ID glucose model – same approach as shown on insulin
t1
t1
t
1 1
G (t1 )  G(t0 )   pG  G (t )dt  S I  (GE  G (t ))dt 
P(t )dt

VG t0
t0
t0
solve
C0, pG C0, S I 
d 0 

  pG   
       0
 
S
Cn , p Cn , S   I  d n 
G
I 

Example of result accuracy
• Estimation of two parameters in insulin model, nL and x
2D error grid
RMSE
100
Identified values in 1 iteration!
nL= 0.21
50
x= 0.3
0
0
0.3
0.5
nL
1
0.5
0.2
0.3 0.21
0.4
x
0.1
nL
0.1 ± 0.024 min-1
pG
0.01 ± 0.002 min-1
SI
12 ± 3.8 x 10-4 l/mU/min
VP
4.49 ± 0.37 l
VQ
5.6 ± 0.56 l
VG
12.1 ± 1.07 l
nK
0.021 ± 0.003
min-1
nI
0.272 ± 0.028 l/min
nC
0.032 ± 0.0004 min-1
GE
4.85 ± 0.59 mmol/l
Glucose [mmol/l]
0.4
100
glucose
insulin
0.2
0
50
0
20
40
60
80
100
0
120
100
120
Insulin [mU/l/min]
Euglycaemic clamp trials (N=146)
VG=0.19xbw
uen(t) assumed suppressed
Fitting errors within measurement noise:
eG=5.9±6.6% SD; eI=6.2±6.4% SD
Insulin [mU/l]
•
•
•
•
Glucose [mmol/l/min]
Validation on clamps
8
G(t)
6
4
2
0
20
40
60
80
200
I(t)
100
Q(t)
I
0
0
20
40
60
t [min]
80
100
Q
120
Validation on IVGTT
nL
0.13 min-1
x
0.39
20
18
16
Blood Glucose [mmol/l]
Data taken from Mari (Diabetologia 1998)
N=5 normal subjects
22g glucose, 2.2U insulin (5min IV infusion)
Errors in area under curve: eAG=1.6%; eAI=6.7%
14
12
G(t)
10
8
6
4
2
pG
0.023
min-1
0
0
50
100
150
t [min]
SI
8.4 x 10-4 l/mU/min
400
VG
10.7 l
350
VP
4.22 l
300
VQ
4.37 l
nK
0.06 min-1
nI
0.22 l/min
nC
0.033 min-1
GE
5.2 mmol/l
Plasma Insulin [mU/l]
•
•
•
•
250
I(t)
200
150
100
Q(t)
50
0
0
50
100
t [min]
150
Clinical validation: Dose response test
at low and high dosing
Same subject on 2 different visits
10g glucose/ 1U insulin
20g glucose/ 2U insulin
14
9
0.23 min-1
0.23
x
0.34
0.34
6
pG
0.011 G(t)
min-1
-1
0.01 min
8
5
SI
12.3 x 10-4 l/mU/min
16.2 x6 10-4 l/mU/min
4
VG
13.6 l
15.4 l 4
VP 30
4.54
40 l
7
3
0
10
20
50
Blood Glucose [mmol/l]
nL
8
Blood Glucose [mmol/l]
min-1
12
10
4.54 l 2 0
G(t)
10
20
250
VQ
5.69 l
5.69400
l
200
nK
0.06 min-1
0.06 min-1
150
nI
I(t)0.28 l/min
0.28 l/min
100
0.033 min-1
200min-1
0.033
GE
4.1 mmol/l
4.7 mmol/l
Q(t)
0
10
20
30
t [min]
40
50
30
t [min]
40
0
60
Q(t)
100
0
50
I(t)
300
nC
50
0
Plasma Insulin [mU/l]
Plasma Insulin [mU/l]
t [min]
10
20
30
t [min]
40
50
60
Conclusions
• Physiological insulin kinetics model
• Easy a-priori identification with C-peptide population model
• Additional fitting of key parameters (1(2) for insulin, 2(3) for
glucose)
• Integral-based fitting method convex, accurate and not starting
point dependent
• Great potential for use in clinical applications
Acknowledgements – Questions?
Jessica Lin
Geoff Shaw
Kirsten
McAuley
Jason Wong
Chris Hann
Geoff Chase
Dominic Lee
Jim Mann
Steen
Andreassen