T Lotz 1 , J G Chase 1 , K A McAuley 2 , J Lin 1 , J Wong 1 , C E Hann 1 and S Andreassen 3

INTEGRAL-BASED IDENTIFICATION OF A INTEGRAL-BASED IDENTIFICATION OF A PHYSIOLOGICAL INSULIN AND GLUCOSE PHYSIOLOGICAL INSULIN AND GLUCOSE MODEL ON EUGLYCAEMIC CLAMP AND MODEL ON EUGLYCAEMIC CLAMP AND

IVGTT TRIALS 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 Zealand2Edgar 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 Why model glucose and insulin kinetics?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 - GoalsID - 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 2-compartment insulin kinetics model + glucose pharmacodynamics+ glucose pharmacodynamics

PLASMAINTERSTITIAL

FLUID

KIDNEYS

LIVER

diff

usio

nCELLS

PANCREAS

nC

nK nL

nI

x·uen

uexGLUCOSE

GGEIG V

tP

Q

QGGSGpG

)(

1)(

P

en

P

ex

P

I

I

LK V

tux

V

tuQI

V

n

I

InInI

)()()(

1

)( QIV

nQnQ

Q

IC

I Q

ID - problemsID - 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 A priori ID - Similarities with C-peptideC-peptide

PLASMAVP

INTERSTITIALFLUID

VQ

KIDNEYS

PANCREAS

nK

nI

uen

PLASMAVP

INTERSTITIALFLUID

VQ

KIDNEYSLIVER

CELLS

PANCREAS

nC

nK

nL

nI

x·uen

Additional losses

C-peptide(Van Cauter et al 1992)

Insulin

Equimolarsecretion

A priori ID – insulin modelA priori ID – insulin model

• 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)

P

en

P

ex

P

I

I

LK V

tux

V

tuQI

V

n

I

InInI

)()()(

1

)( QI

V

nQnQ

Q

IC

A priori ID – glucose modelA priori ID – glucose model

• 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)

GGEIG V

tP

Q

QGGSGpG

)(

1)(

Integral-based fitting methodIntegral-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]:

• I(t) estimated by interpolating between discrete data

• Q(t) known from analytical solution:

• Inputs u(t) known (endogenous insulin estimated from C-Peptide)

1

0

1

0

1

0

1

0

1

0

)()(1

)()()()(1

)()()( 01

t

t

en

t

t

ex

t

t P

t

t P

I

P

IK

t

t IL dttuxdttu

VdttQ

V

ndttI

V

nndt

tI

tIntItI

t t

V

nn

Q

I deIV

ntQ Q

IC

0

))((

)()(

Integral-based fitting methodIntegral-based fitting method

• Repeat for different time-steps [t0,t1] ... [tn-1,tn]:

1

0

1

0

1

0

1

0

1

0

)()(1

)()()()(1

)()()( 01

t

t

t

t

enex

t

t P

t

t P

I

P

IK

t

t IL dttuxdttu

VdttQ

V

ndttI

V

nndt

tI

tIntItI

knownknownidentify

00

,

,0

,

,0

n

L

xn

x

nn

n

d

d

x

n

C

C

C

C

L

L

solve

identifyknown

Integral-based fitting methodIntegral-based fitting method

• ID glucose model – same approach as shown on insulin

dttPV

dttGGSdttGptGtGt

tG

t

t

EI

t

t

G 1

0

1

0

1

0

)(1

))(()()()( 01

00

,

,0

,

,0

nI

G

Sn

S

pn

p

d

d

S

p

C

C

C

C

I

I

G

G

solve

Example of result accuracyExample of result accuracy

• Estimation of two parameters in insulin model, nL and x

0.10.2

0.30.4

0.5

0

0.5

1

0

50

100

xnL

RM

SE

2D error grid

Identified values in 1 iteration!

nL= 0.21

x= 0.3

0.3

0.21

Validation on clampsValidation on clamps

• 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 0 60 1200

0.2

0.4

Glu

cose

[mm

ol/l/

min

]

0 20 40 60 80 100 1200

50

100

Insu

lin [m

U/l/m

in]

0 20 40 60 80 100 1202

4

6

8

Glu

cose

[mm

ol/l]

0 20 40 60 80 100 1200

100

200

t [min]

Insu

lin [m

U/l]

insulin

I Q

glucose

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

G(t)

I(t)

Q(t)

Validation on IVGTTValidation on IVGTT

• 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%

0 50 100 1500

2

4

6

8

10

12

14

16

18

20

t [min]

Blo

od G

lucose [

mm

ol/l]

0 50 100 1500

50

100

150

200

250

300

350

400

t [min]

Pla

sm

a I

nsulin

[m

U/l]

nL 0.13 min-1

x 0.39

pG 0.023 min-1

SI 8.4 x 10-4 l/mU/min

VG 10.7 l

VP 4.22 l

VQ 4.37 l

nK 0.06 min-1

nI 0.22 l/min

nC 0.033 min-1

GE 5.2 mmol/l

I(t)

Q(t)

G(t)

0 10 20 30 40 50 602

4

6

8

10

12

14

t [min]

Blo

od G

luco

se

[mm

ol/l]

0 10 20 30 40 50 600

100

200

300

400

t [min]

Pla

sma

Insu

lin [

mU

/l]

Clinical validation: Dose response test Clinical validation: Dose response test at low and high dosingat low and high dosing

0 10 20 30 40 500

50

100

150

200

250

t [min]

Pla

sma

Insu

lin [

mU

/l]

0 10 20 30 40 503

4

5

6

7

8

9

t [min]

Blo

od

Glu

co

se

[m

mo

l/l]

I(t)

Q(t)

G(t)

10g glucose/ 1U insulin 20g glucose/ 2U insulin

I(t)

Q(t)

G(t)

nL 0.23 min-1 0.23 min-1

x 0.34 0.34

pG 0.011 min-1 0.01 min-1

SI 12.3 x 10-4 l/mU/min 16.2 x 10-4 l/mU/min

VG 13.6 l 15.4 l

VP 4.54 l 4.54 l

VQ 5.69 l 5.69 l

nK 0.06 min-1 0.06 min-1

nI 0.28 l/min 0.28 l/min

nC 0.033 min-1 0.033 min-1

GE 4.1 mmol/l 4.7 mmol/l

Same subject on 2 different visits

ConclusionsConclusions

• 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?Acknowledgements – Questions?

Geoff ChaseGeoff Shaw

Dominic LeeSteen Andreassen

Jim MannKirsten McAuley

Jessica Lin Chris HannJason Wong