Steady-State Optimal Insulin Infusion for
Hyperglycemic ICU Patients
J G Chase, G C Wake, Z-H Lam, J-Y Lee, K-S Hwang and G. Shaw
University of Canterbury
Dept of Mechanical Engineering
Christchurch
New Zealand
ICARCV 2002, Singapore
Diabetes – A Brief Overview
• Diabetes – A disorder of the metabolism
– Type I: Body produces little or no insulin.
– Type II: Insulin resistance or impaired glucose tolerance.
• Complications – kidney failure, blindness, nerve damage, amputation, heart
attack, stroke.
• High annual costs growing exponentially with number of cases
• Estimated cost to NZ is $1B per year in 2020 – A growing epidemic!
– Similar numbers hold true throughout most of the world, including Singapore.
Diabetes in the ICU
• Elevated blood glucose levels or Hyperglycaemia is very common
among the critically ill in the ICU
– Stress of the disease
– Many older patients are Type II diabetic individuals
– Direct result of disease
• Current Treatment
–
–
–
–
Sliding scale protocols based on magnitude with very coarse resolution
Feeding 1-2x daily in slow infusion
Generally poor control (<8 mmol/L is considered “very good”)
Often overlooked because of severity of other issues and disease
• Why bother? 45 reasons for every 100!
– Vandenberghe et al (2001) showed that tight glucose regulation in the ICU
(levels < 6mmol/L) resulted in up to a 45% decrease in mortality
3 Elements of Control Systems in an ICU
• Sensing
– Typically done with GlucoCard™ or similar
arterial blood measurement
– Modern methods of automatic measuring being
developed (Trajanoski et al, 1994)
• Computation
– Sliding scale protocol could be replaced by an
algorithm implemented on DSP
• Actuation
– Standard systems such as a Graseby 3500
• Necessary technologies emerging very rapidly
to close the loop!
2-Compartment Glucose-Insulin System Model
d
G   p1G  X (G  Gb )  P(t )
dt
d
X   p2 X  p3 I
dt
d
u (t )
I   n( I  I b ) 
dt
V
• Model derived and validated in “Bergman et al.” [1985]
– More amenable for real-time control analysis than many models
• G and I are variations from basal levels of Glucose and Insulin.
• Coefficients p1, p2, p3 vary for Type I, Type II, Normal, and n varys
for insulin type.
• System simulated with time step of 1 minute, actuation and sensor
bandwidth are varied to determine trade-offs and diminishing returns.
Optimal Steady State Infusion Rate
• Equations for I(t) and X(t) solved analytically and the optimal solution
for u(t) obtained for G = d/dt(G) = 0 – no excursion or slope
V
u (t ) 
p3Gb
 d2

d
d
P
(
t
)

P
(
0
)

(
n

p
)(
P
(
t
)

P
(
0
))

np
P
(
t
)
2
2
 2
  u0
dt
dt
 dt

u0  nVI b
• Solution depends on 1st and 2nd derivatives of exogenous glucose input
P(t) as well as its initial conditions. I.e. you must know P(t) very well.
• If P(t)=0 for all t then the optimal steady state rate is simply u0 as
expected for G=d/dt(G)=0 status
Solution of Steady State Optimal Infusion I
• First solve for I(t) insulin level in first compartment in terms of infusion u(t)
 nt
e
I (t )  I B (e nt  1)
VI

t
0
e n (t  )u ( )d
• Use I(t) solution to obtain remote compartment analytical solution for X(t) in terms
of the input u(t) from the solution for I(t).
X (t ) 
p3 I B  nt  p2t
pI
(e  e )  3 B (e  p2t  1)
p2  n
p2
t
p3
 p2 ( t  )
 n ( t  )

(
e

e
)u ( )d

0
VI ( p2  n)
Solution of Steady State Optimal Infusion II
• Insulin utilization equation if dG/dt = G = 0 for a Type 1 diabetic  the steady state
P (t )  X (t )G (t )
p1  0
• Inserting solutions for X(t) and using Laplace transforms to simplify the convolution
integrals and algebra the steady state optimal infusion u(t) can be obtained from the
inverse Laplace transform of the above equation solved for U(s)
V
u (t ) 
p3Gb
 d2

d
d
 2 P(t )  P(0)  (n  p2 )( P(t )  P(0))  np2 P(t )  u0
dt
dt
 dt

The algebra is “ugly” but fairly direct and much easier if the initial conditions for P(t)
are equal to zero, which should be true for a slow, smooth infusion.
Optimal Control of a Glucose Slow Infusion
• Infusion will “follow” the normal
response shown
• Optimal response essentially flat
because P(t) is very well known,
smooth and continuous
• This input profile is not unlike a
typical ICU night feeding via IV.
• Infusion occurs over ~3hours for
500kcals of feeding
The optimal controller handles this case very well
Optimal Infusion for Slow Infusion
• Glucose Response is ~ flat with
small errors due to numerical time
step size. At infinitely small size
the response is almost perfectly flat.
• Small negative infusion or glucose
input is due to numerical issues. The
solution is not very stable on Matlab
• Much more like an injection than the
normal modeled response.
A Difficult Test
• 1000 calories in 4 hours over five “meal” inputs of glucose which is
rapidly absorbed
• Inputs vary in magnitude from 50 – 400 calories
• Inputs occur in two groups of rapid succession at t = 0, 10, 30 minutes
and at t = 210 and 300 minutes
– The last meal is 40 calories from 980 – 1020 calories so the full absorption
of about 1000 calories occurs by 4 hours quite easily.
• Controller has no knowledge of glucose input except in optimal case
– Input knowledge is not currently practicable in any way for this system in
general
The goal is to “hammer” the system and see if it breaks!
Comparison with other Controllers
• Relative proportional controller (RPC).

G 
,
u t   u0 1 
 Gb 
u0  nVI b
• Optimal steady state infusion rate by solving analytically with
V
u (t ) 
p3Gb
 d2

d
d
P
(
t
)

P
(
0
)

(
n

p
)(
P
(
t
)

P
(
0
))

np
P
(
t
)
2
2
 2
  u0
dt
dt
 dt

• PD controller – controls slopes of incresing/decreasing blood sugar
level rather than actual glucose concentration
dG 

u t   u0 1  k p G  k d

dt


Control of Glucose Inputs
Optimal control very nearly flat as desired and much lower than other forms of control
Insulin Infusion Rates for Glucose Inputs
• Insulin rates are sharper and nearer injections expected
• Lower insulin rates less effective control as might be expected
• Total insulin used is very similar for each case  better usage w/ optimal
Summary & Conclusions
• A steady state optimal infusion solution is developed for a
physiologically verified 3 compartment model of the glucose
regulatory system
• Solution is shown to provide the desired ~flat glucose response to
steady, slow inputs as well as more significant challenges
• Optimal solution does require knowledge of the glucose absorption
function P(t) which is unlikely to be known outside of a controlled
setting such as the ICU. Hence, it’s limited application clinically.
• Optimal insulin infusions mimic the injection solutions which have
been hand optimized for care over the prior 50+ years
Acknowledgements…
Lipids and Diabetes
Research Group
Questions, Comments, “Complements”, ….
“Failure is not an option (but it is
much more interesting).”
-- G. Shaw, MD
“No, no, no… (explicit adjective(s))”
-- G. Chase, PhD