Long Term Verification Long Term Verification of Glucose-Insulin of Glucose-Insulin Regulatory System Regulatory System

Model DynamicsModel DynamicsTHE 26th ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE

ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY

J. Lin, J. G. Chase, G. M. Shaw, T. F. Lotz,C. E. Hann, C. V. Doran, D. S. Lee

Department of Mechanical EngineeringUniversity of Canterbury

Christchurch, New Zealand

Hyperglycemia in the ICUHyperglycemia in the ICU

• Stress-induced hyperglycemia

• Insulin resistance or

deficiency enhanced

• High dextrose feeds don’t suppress glucagon release or gluconeogenesis

• Drug therapy

Source: www.endocrine.com

There is a need for validatedModels to aid treatment

Physiological ModelPhysiological Model

Blood PlasmaThe utilisation of insulin and the removal of glucose over time

LiverProduces endogenous glucose (GE)

Exogenous Glucose

Food intake etc. (P(t))

Exogenous InsulinInsulin injection etc. (uex(t))

PancreasProduces endogenous insulin I(t)

GE P(t)

G

)(1

)( tPQ

QGGSGpG

GEIG

t tk deIkQ

0

)()(

I

Btu

II V

Ie

V

tu

I

nII

)()(

1

Glucose DynamicsGlucose Dynamics

The ability to regulate blood glucose level

Tissue sensitivity to insulin

Blood PlasmaThe utilisation of insulin and the removal of glucose over time

)(1

)( tPQ

QGGSGpG

GEIG

t tk deIkQ

0

)()(

Saturated effect of insulin over time

Q(t)

time

Parameter Fitting Parameter Fitting RequirementsRequirements

• Very low computation time required if fitting over long periods of several days or using for control

• High accuracy for tracking changes in time varying patient specific parameters pG and SI

• Physiologically realistic values of optimised parameters

• Convex and not starting point dependent, like the commonly used non-linear recursive least squares (NRLS) method

Parameter ValuesParameter Values

Parameter

Controls Value

VIInsulin Volume of

Distribution 12 L

n 1st Order Plasma Insulin Decay 0.16 min-1

k Delay in Interstitial Transfer 0.0099 min-1

αGInsulin Receptor

Saturation0.015

L∙min∙mU-1

αIInsulin Transport

Saturation1.7 × 10-3

L∙min∙mU-1

Generic Parameters found from an extensive literature review

Integration-Based Integration-Based OptimizationOptimization

ET GGG

)1( QQQ G

t

t

t

tTI

t

t ETGTT PdtdtQGSdtGGptGtG0 00

)()()( 0

N

iiiGiG ttHttHpp

1)1( ))()((

N

iiiIiI ttHttHSS

1)1( ))()((

t

t EIG

t

tdttPQGGSGpdtG

00

))()((

Use different values of t and t0 to develop a number of linear equations, where pG and SI at different times are the only unknowns

bS

pA

Ii

Gi

Approximate glucose curve between data points as linear

Error AnalysisError Analysis)()()( ttGtG approxTrealT )(0 t small

)()()()()()()(

)()()())(()()(

000

0 00

00

0

tEdttPdttQtGSdttGpGttptG

dttPdttQtGSdtGtGptGtG

t

t

t

t approxTI

t

t approxTGEGapproxT

t

t

t

trealTI

t

t ErealTGrealTrealT

)(

)()(

)(|)(||)(|

|)()(||)(||)(|

|)()()()(||)(|

0

00

00

00

0

0

0

O

dttQSttp

dttQtSdttp

dttQtSdttpt

dttQtSdttpttE

t

tIG

t

tI

t

tG

t

tI

t

tG

t

tI

t

tG

t

t

t

t realTG

G

dt

tt

dttGp

ttp

00

1

)(

)(

)( 00

t

t

t

tt

t realTI

t

tI

dttQ

dttQ

dttQtGS

dttQS

0

0

0

0

)(

)(

)()(

)(

t

t

t

tTI

t

t ETGTT PdtdtQGSdtGGptGtG0 00

)()()( 0

Approximate glucose curve does not compromise the fitting quality

AdvantagesAdvantages

• Least squares problem (constrained)

• Integration based approach to fitting reduces noise

• Effectively low-pass filter noise with numerical integration

• Not starting point dependent like typical methods

• Convex, easily solved, single global minima

bS

pA

Ii

Gi

Patient Data and MethodsPatient Data and Methods• Patients selected from retrospective study were those

with glucose measurement intervals < 3 hours– 17 out of 201 patients– Good general cross-section of ICU population

• Details from patient charts used in the fitting process– Glucose Measurements– Insulin Infusions– Feed Details

• 1.4 – 12.3 days were fit to the model (average is 3.1 days)– Not always entire length of stay

• Resulting patient specific parameters, pG and SI, were smoothed to reduce noise, and the overall fit was compared to measured glucose data

Results – Patient 1090Results – Patient 1090

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

10

20glucose versus time

t (days)

Blo

od G

luco

se L

evel

(m

mol

L-1)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.05

pG versus time

t (days)

p G (

1/m

in)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3x 10

-3 SI versus time

t (days)

SI (

L/(m

U

min

))• Mean Error

= 0.87 %

• StandardDeviation = 0.80 %

Results – Patient 87Results – Patient 87

0 1 2 3 4 5 60

10

20glucose versus time

t (days)

Blo

od G

luco

se L

evel

(m

mol

L-1)

0 1 2 3 4 5 60

0.05

pG versus time

t (days)

p G (

1/m

in)

0 1 2 3 4 5 60

1

2

3x 10

-3 SI versus time

t (days)

SI (

L/(m

U

min

))• Mean Error

= 2.35 %

• StandardDeviation = 2.69 %

• Absolute Error Metric

• Mean Absolute Error → 4.39 %– Mean Error Range across 17 patients → 1.03 – 7.62 %– Measurement Error is 3.5 – 7 % (Arkray Inc, 2001)

• Standard Deviation → 4.45 %– SD Range across 17 patients → 0.93 - 9.75 %

Fitting ErrorFitting Error

%100)(

)()(

idata

idataifiti tG

tGtGe

• “Chi-square” quantity– Value used in non-linear, recursive, least-squares fitting

• Expected value – (Number of Measurements – Number of Variables)

• i = 4.79 % matches model across all patients– Within measurement Error of 3.5 - 7 % (Arkray Inc,

2001)

Fitting ErrorFitting Error

N

i i

datafit GG

1

2

2

MN 2exp

Predictive Ability Predictive Ability VerificationVerification

• Using previous 8 hours of measured data

• Hold pG and SI constant over the next hours

• Compare with measured data

• 1 hour predictions have an average absolute error of 2-11%

8 hour window of modelling

eG

time

PatientNo. of

predictions

Average prediction error e [%]

Error standard deviation

[%]

24 22 5.86 4.00

87 41 4.71 5.21

130 18 10.12 9.55

519 76 5.25 5.98

554 24 10.90 8.89

666 13 4.66 3.01

1016 13 7.01 6.27

1025 14 5.09 4.54

1090 13 1.86 0.87

1125 14 6.83 4.78

One hour One hour predictionspredictions

ConclusionsConclusions• Minimal computation and rapid identification of

time-varying parameters pG and SI using the integral-based fitting method presented

• Long term validation of the physiological model

• Accurate results and significant computational speed compared to traditional NRLS method

• Forward prediction error ranging 2-11% as further validation

AcknowledgementsAcknowledgementsEngineers and Docs

Dr Geoff Chase

Dr Geoff Shaw

Students

Maxim Bloomfield

AIC2, Kate, Carmen and Nick AIC3, Pat, Jess, and Mike Thomas Lotz

Maths and Stats Gurus

Dr Dom LeeDr Bob Broughton Dr Chris Hann

Prof Graeme Wake

QuestionsQuestions??

The Danes

Steen Andreassen