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