Long Term Verification of
Glucose-Insulin Regulatory
System Model Dynamics
THE 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 Engineering
University of Canterbury
Christchurch, New Zealand
Hyperglycemia in the ICU
• Stress-induced
hyperglycemia
• Insulin resistance or
deficiency enhanced
• High dextrose feeds
don’t suppress
glucagon release or
gluconeogenesis
• Drug therapy
There is a need for validated
Models to aid treatment
Source:
www.endocrine.com
Physiological Model

G   pG G  S I (G  GE )
Q
1   GQ
 P (t )
t
Q  k  I ( )e k ( t  ) d
0
Pancreas
Produces
endogenous insulin
Exogenous Insulin
Insulin injection etc.
(uex(t))
I(t)
Blood Plasma
The utilisation of
insulin and the
removal of
glucose over time
GE
u(t )
nI
u
(
t
)
e
IB
I  


1   I I VI
VI
Liver
Produces
endogenous
glucose (GE)
P(t)
Exogenous
Glucose
Food intake
etc. (P(t))
G
Glucose Dynamics
The ability to regulate
blood glucose level
Tissue sensitivity
to insulin
Saturated effect of
insulin over time
Q(t)

G   pG G  S I (G  GE )
t
Q  k  I ( )e k ( t  ) d
0
Blood Plasma
The utilisation of
insulin and the
removal of
glucose over time
Q
1   GQ
 P (t )
time
Parameter Fitting Requirements
• 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 Values
Parameter
Controls
Value
VI
Insulin Volume of
Distribution
12 L
n
1st Order Plasma
Insulin Decay
0.16 min-1
k
Delay in Interstitial
Transfer
0.0099 min-1
αG
Insulin Receptor
Saturation
0.015
L∙min∙mU-1
αI
Insulin Transport
Saturation
1.7 × 10-3
L∙min∙mU-1
Generic Parameters found from an extensive literature review
Integration-Based Optimization
t 
t
t0
t0
 G dt   ( p G  S (G  G
G
I
E
GT  G  GE
) Q  P(t )) dt
Q  Q (1  GQ )
t
t
t
t0
t0
t0
GT (t )  GT (t0 )    pG (GT  GE ) dt   S I GT Q dt   Pdt
N
pG   pGi ( H (t  t( i 1) )  H (t  ti ))
i 1
N
S I   S Ii ( H (t  t( i 1) )  H (t  ti ))
i 1
Approximate
glucose curve
between data
points as linear
 pGi 
A  b
 S Ii 
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
Error Analysis
GT real (t )  GT approx (t )   (t )
0   (t )  
t
t
t0
t0

  small
t
t
t0
t0


t
(GTSI GGE T)dt
 S GT Q dt
GT real (t )  GT real (t0G
) T(pt )G 
t ) GEp)Gdt
0 ) (
real (t ) Q (It ) dt
 G(GT T(treal
 P(t ) dtPdt
t
t0
t0
t
t
t
t0
t0
t0
 GT approx (t0 )  pG (t  t0 ) GE  pG  GT approx (t ) dt  S I  GT approx (t ) Q (t ) dt   P(t ) dt  E (t )
t
t
t0
t0
| E (t ) |  |  (t 0 )  p G   (t ) dt  S I   (t ) Q (t ) dt |
t
t
t0
t0
 |  (t 0 ) |  p G |   (t ) dt |  S I |   (t ) Q (t ) dt |
pG (t  t0 )
t
pG  GT  real (t )dt
t
t0
t0
S I   Q (t ) dt
t
t
   pG  (t  t 0 )  S I   Q (t ) dt
t0
 (t  t0 )
t
 1dt
t0
t
   pG  |  (t ) | dt  S I  |  (t ) |Q (t ) dt

t0
t
t0
S I  GT  real (t )Q (t ) dt
t0

t

  Q (t ) dt

t0
t
t0

Q (t ) dt
 O ( )
Approximate glucose curve does
not compromise the fitting quality
Advantages
 pGi 
A  b
 S Ii 
•
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
Patient 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
• Mean Error =
0.87 %
• Standard
Deviation =
pG (1/min)
-1
Blood Glucose Level (mmol L )
Results – Patient 1090
glucose versus time
20
10
0
0
0.2
0.4
0.6
0.8
1
t (days)
pG versus time
1.2
1.4
1.6
0.2
0.4
0.6
0.8
1
t (days)
SI versus time
1.2
1.4
1.6
0.2
0.4
0.6
1.2
1.4
1.6
0.05
0.80 %
0
0
SI (L/(mU  min))
-3
3
x 10
2
1
0
0
0.8
t (days)
1
• Mean Error =
2.35 %
• Standard
Deviation =
pG (1/min)
-1
Blood Glucose Level (mmol L )
Results – Patient 87
glucose versus time
20
10
0
0
1
2
3
4
t (days)
pG versus time
5
6
1
2
3
4
t (days)
SI versus time
5
6
1
2
5
6
0.05
2.69 %
0
0
SI (L/(mU  min))
-3
3
x 10
2
1
0
0
3
t (days)
4
Fitting Error
• Absolute Error Metric
G fit (ti )  Gdata (ti )
ei 
 100%
Gdata (ti )
• 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 Error
• “Chi-square” quantity
– Value used in non-linear, recursive, least-squares fitting
 G fit  Gdata
  

si
i 1

N
2




2
2

• Expected value exp    N  M
– (Number of Measurements – Number of Variables)
•
si = 4.79 % matches model across all patients
– Within measurement Error of 3.5 - 7 % (Arkray Inc, 2001)
Predictive Ability Verification
One hour predictions
8 hour window of modelling
Error
standard
deviation
[%]
Patient
No. of
predictions
Average
prediction
error e [%]
24
22
5.86
4.00
87
41
4.71
5.21
• Using previous 8 hours of
measured data
130
18
10.12
9.55
519
76
5.25
5.98
• Hold pG and SI constant over
the next hours
554
24
10.90
8.89
666
13
4.66
3.01
• Compare with measured data
1016
13
7.01
6.27
1025
14
5.09
4.54
1090
13
1.86
0.87
1125
14
6.83
4.78
G
e
time
• 1 hour predictions have an
average absolute error of 211%
Conclusions
• 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
Acknowledgements
Engineers and Docs
Dr Geoff Shaw
Maths and Stats Gurus
Dr Dom Lee
Dr Bob
Broughton
Prof
Graeme Wake Dr Chris Hann
Dr Geoff Chase
The Danes
Students
Maxim
Bloomfield
Steen
Andreassen
AIC2, Kate, Carmen and Nick
AIC3, Pat, Jess, and Mike
Thomas Lotz
Questions?