Steady-State Optimal Insulin Infusion for Hyperglycemic ICU Patients

1
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

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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.

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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 (lt8 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 lt 6mmol/L)
  resulted in up to a 45 decrease in mortality

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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!

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2-Compartment Glucose-Insulin System Model

• 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.

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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

• 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 Gd/dt(G)0
  status

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Solution of Steady State Optimal Infusion I

• First solve for I(t) insulin level in first
  compartment in terms of infusion u(t)

• 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).

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Solution of Steady State Optimal Infusion II

• Insulin utilization equation if dG/dt G 0
  for a Type 1 diabetic ? the steady state

• 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)

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.
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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
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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.

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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!
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Comparison with other Controllers

• Relative proportional controller (RPC).

• Optimal steady state infusion rate by solving
  analytically with

• PD controller controls slopes of
  incresing/decreasing blood sugar level rather
  than actual glucose concentration

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Control of Glucose Inputs
Optimal control very nearly flat as desired and
much lower than other forms of control
14
Insulin Infusion Rates for Glucose Inputs
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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, its limited
  application clinically.
• Optimal insulin infusions mimic the injection
  solutions which have been hand optimized for care
  over the prior 50 years

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Acknowledgements
Lipids and Diabetes Research Group
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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