Modern Control System
for
Artificial Pancreatic A Cell
by
CH
KEVIN PAUL
B.S.,
Massachusetts Institute
(1976)
of
Technology
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF
June,
TECHNOLOGY
1981
Kevin Paul Koch
The author hereby grants to the Massachusetts Institute
of
Technology
permission to
reproduce and distribute
copies of
this thesis
document in
whole, or in part.
Signature of Author
Department of Mechanical
./
4
A
Engineering
June
1, 1981
Certified by
J. Karl Hedrick
Thesis Supervisor
Accepted by
Chairman, Departmental
Archives
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUL 31 1981
UIBRARIES
~
Warren M. Rohsenow
Graduate Committee
ABSTRACT
Modern Control System
for
8 Cell
Pancreatic
Artificial
by
KEVIX PAUL KOCH
Submitted to the Department of Mechanical Engineering
on 1 June 1981 in partial fulfillment of the
requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
of diabetes has been
Progress toward closed-loop control
years.
In spite of
the increasing
accelerating in recent
attention being devoted to this problem, apparently all work
has been based solely on empirical observations.
an engineering
thesis is
to
take
goal* of
this
The
A simple linear
approach to the glucose regulation problem.
be derived from
and insulin dynamics will
model of glucose
using straightforward
an existing complex, nonlinear model,
The simplified model will serve
linear analysis techniques.
as the basis for a modern control system.
is 'good enough'
Whether the simplified characterization
of the control law derived
is determined by the performance
Simulations indicate that the control law works
therefrom.
including
In simulations
as a
normal pancreas.
as well
insulin pump
hardware constraints, (glucose sensing delays,
the modern control system provides regulation
quantization)
artificial
than existing
as or
better
which
is as
good
glucose controllers.
Thesis Supervisor:
Title:
Dr. J. Karl Hedrick
Associate Professor
-
ii
-
ACKNOWLEDGEMENTS
My deepest thanks to Albert Hopkins and Basil
were
Smith,
able to let me devote my time to this project,
who
and who
provided encouragement and valuable editorial comments.
To Steve
Hall, for his
graphics package and many hours of
fruitful conversations.
To
Mario
computer
Santarelli
and
Dave
Haugar
for
unlimited
time and resources.
To Professor Clark Colton and
their support, encouragement,
To those
who helped
Dr.
iii
little ways:
Allison Brown,
and Jeanne Bueche.
-
Stuart Soeldner for
and enthusiasm.
in countless
Youcef-Toumi, Allan O'Connor,
J.
-
Kamal
John Sorensen,
-
iv -
TABLE OF CONTENTS
.
.
.
.
.
.
.
.
.
ACKNOWLEDGEMENTS
.
.
.
.
.
.
.
.
.
. . . .
TABLE OF CONTENTS
.
.
.
.
.
.
.
.
.
.
.
.
.
.
-
-
. . . . -
-
. . . . -
.
ABSTRACT
-
. . . . .
. ..
..
-
-
. . . viii
. ..
LIST OF TABLES
. . . . . . . . . . . . ..
LIST OF FIGURES
. . . . . . . . . . . . . . .. . . .
ix
. .
page
Chapter
.
.
.
.
.
.
1.
INTRODUCTION
2.
APPLICABLE CONTROL THEORY
.... . . . . . .
.
. .
.
.
THE PROBLEM OF DIABETES
1
.
. . . . . . . . . . . . . .
. . . . . . .
Results of Modern Control Theory .
Modern Control System Formulation . . . . . .
..
.. . .
Controllability . . . . . . . .. .
2uadratic Performance Index . . . . . . . . ..
. . . . . . . . . . . . .. .
State Estimation
. .. .
.
Kalman Filter . . . . . . . *. ....
Error Coordinate Transformation . . . . . . .
Integral Control/Intelligent Integrator . . .
An Example of the Modern Control Formulation
Classical Control as a Subset of the Modern
. . . . . . . . . . . .
Formulation
3.
V
.
.
.
.
.
.
.
.
.. .
.. .
.
.
.
.
5
6
6
8
9
10
12
12
13
14
. 16
17
17
.. .
Glucose - Insulin Physiology . . . . . .. .
18
.
.
.
.
.
.
. . . . . . . . . .
Diabetes Mellitus
20
.
.
.
.
.
.
. . . . . . . . . . . . . .
Treatment
4.
GLUCOSE/INSULIN MODELLING
. . . . . . . . . . . . . . 22
Existing Gluocse and Insulin Dynamics Models . .
. . . . . . . . . . . . . . .
The FLOWMOD Model
.. .
Physical Derivation . . . . . . . . . . .
. . . . . . . . . . . . . .
Glucose Space
. . . . . . . . . . . . . .
Insulin Space
. . . . . . . . . .
The Pancreatic 4 Cell
........
.
Value
Validation/Predictive
....
.
Cell
B
FLOWMOD
the
of
Irrelevance
. . .
.
.
.
.
.
FLOWMOD
Using
Sample Results
-
v -
. 23
. 24
24
. 26
. 29
. 30
30
31
. 31
5.
LINEAR
CHARACTERIZATION
Insulin
Dynamics
.
.
.
.
.
.
.
.
.
.
.
Controllability
. . . . . . . .
Transfer Functions
. . . . . .
Error Coordinate Transformation
Glucose Dynamics . . . . . . . . .
Frequency Response Analysis .
Impulse Response
. . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
49
OBSERVER DESIGN
.
.
.
.
.
.
.
.
.
.
.
.
.
52
RESULTS
.
.
.
.
.
CONCLUSIONS
.
.
.
.48
Observer
52
53
54
55
58
.
Controller
Design
. . . . . . . . . .
Comparison to Empirical Algorithms
Intelligent
Integrator
Design
.
Hardware Limitations Simulation
Insulin
Pump quantization.....
Glucose Measurement Delays
.
Peripheral versus Portal Delivery
Comparison with Other Results
8.
.
.
.
.
.
.
.
.
.
.
Summary
. . . . . . . . . . . . .
Future Work
. . . . . . . . . . .
. .
Discrete-Time Kalman Filter
Model Enhancement/Verification
Experimentation with Controller
Partial
3 Cell Function . . . .
Fault Tolerance . . . . . . . .
Conclusion . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . .
. . . . .
. . . . .
. . . . .
Parameters
. . . . .
. . . . .
. . . . .
.
.
.
.
.
.
.
.
.
58
61
61
65
65
6
. 71
. 74
76
76
.
.
78
78
78
.
.
79,
79
80
p.80
.
.
.
page
Appendix
A.
38
39
40
43
44
44
.
Parameter Readjustment Using Open Loop
Insulin States
. . . . . . . . . .
Glucose States
. . . . . . . . . .
Observer Implementation
. . . . . . .
7.
.
36
.
.
Summary
6.
.
MODEL
OF THE
COMPUTER PROGRAMS
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Simulation Data Definitions
/PCOM/
Database Generator
Named Common Definition
Observer Program . . .
Simulation Program . .
DYSYS Common Block
Modified FLOWMOD
.
-
.
.
.
.
.
.
.
.
.
vi -
.
.
.
.
.
.
.
.
.
.
82
83
86
87
89
89
89
REFERENCES
. .
. .
. . .
.
. .
.
-
vii
-
.
.
.
. .
.
.
. .
.
.
102
LIST
OF TABLES
Table
page
. . . . . . . . . 41
1.
Insulin Subsystem Poles and Zeroes
2.
Insulin
3.
Insulin Dynamics Transfer Functions
4.
Simplified Glucose
5.
Insulin Subsystem Final Parameter Settings
6.
Glucose Subsystem Final Parameter
7.
Observer
8.
Controller Gains
Subsystem
Gains
Fraction Numerators
Partial
Dynamics
.
.
.
.
.
. . . . . . . . .
Values
.
.
.
.
.
41
.
.
.
.
42
. . . . 49
.
.
.
.
53
. . . . . . 55
. . . . . . . . . . . . . . . . . . . 56
. . . . . . . . . . . . . . . . . . 59
-
viii
-
LIST OF FIGURES
page
Figure
Windup
14
.
. . . . . . . . . . . . .. . .
1.
Integrator
2.
Intelligent Integrator
3.
Glucose Compartments and Blood Circulation
4.
Glucose Uptake by Liver as a Function of Insulin
Concentration . . . . . . . . . . . . . . . . . . 28
5.
Glucose Uptake by Liver as
Concentration . . . . .
. . . . . . . . . . . . . . . 15
.
.
.
.
.
26
a Function of Glucose
.
..
. . . . . . ..
.
.
28
.
.
.
29
and Blood Circulation
.
.
6.
Insulin Compartments
7.
FLOWNOD Gut Absorption Assumption During OGTT
.
.
.
33
8.
FLOWMOD Pancreatic Insulin Secretion During OGTT
.
.
33
9.
FLOWMOD Peripheral Plasma Insulin Concentration
..
. ..
During OGTT . . . . . . . . . . . ..
.
34
10.
FLOWMOD Peripheral Blood Glucose Concentration
During OGTT . . . . . . . . . . . . . . . . . . . 34
11.
Simplified System Block Diagram
12.
Phase
13.
Error Coordinate Transformation of Canonical Second
. . . . . . . 43
. . . . . . . .. . .
order System
14.
Bode Plot of Peripheral Blood Glucose Concentration
Response When Driven By Liver Tissue Insulin
. . . . . . . . . . . . . . . . . . . .
Sinusoid
45
Bode Plot of Peripheral Blood Glucose Concentration
Response When Driven By Peripheral Tissue Insulin
. . . .
..
. . . . . . . . . . . .. .
Sinusoid
46
15.
. . . . . . . . . . 39
Variable Form for Second Order System
.
.
.
.
43
. . . . . . . . . . . . . 51
16.
Complete Linearized System
17.
Peripheral Plasma Insulin Concentration - Modern
. . . . . . . . . 60
Controller Compared to FLOWMOD
-
ix
-
18.
Peripheral Blood Glucose Concentration - Modern
Controller Compared to FLOWMOD
. . . . . . . . . 60
19.
Proposed Intelligent Integrator
. . . . . .
. . . . 63
20.
Peripheral Plasma Insulin Concentration Intelligent Integrator
. . . . . . . . .
. . . . 64
21.
Peripheral Blood Glucose Concentration - Intelligent
. . . . . . . . . . . . . . . . . . . 65
Integrator
22.
Peripheral Plasma Insulin Concentration - Pump
Constraints and Intelligent Integrator
. .
.
.
.
66
Peripheral Blood Glucose Concentration - Pump
Constraints and Intelligent Integrator
. .
.
.
.
67
23.
24.
Pump Rate with Pump Constraints and Intelligent
. . . . . . . . . . . . . . . . . . . 67
Integrator
25.
Synthetic Measurement for Discrete-Time
26.
Peripheral Plasma Insulin Concentration Measurement Delays of .5, 3, and 5 Minutes
Case
.
.
.
.
.
69
. 70
27.
Peripheral Blood Glucose Concentration - Measurement
. . . . . . . . . 70
Delays of .5, 3, and 5 Minutes
28.
Pump Rate with 3 Minute Measurement Delay
29.
Peripheral Plasma Insulin Concentration - Peripheral
Infusion
. . . . . . . . . . . . . . . . . . . . 72
30.
Peripheral Blood Glucose Concentration - Peripheral
Infusion
. . . . . . . . . . . . . . . . . . . . 73
31.
Peripheral Plasma Insulin Concentration - Peripheral
Infusion
with 3 Minute Measurement Delay
.
.
.
.
73
32.
Peripheral
Infusion
.
.
.
.
. 71
Blood Glucose Concentration - Peripheral
with 3 Minute Measurement Delay
. . . .
74
I
Chapter
1
INTRODUCTION
Steady
improvements in the
price, size,
of digital electronics bring
as
more
and capabilities
applications within reach
time goes on.
Controlling the blood sugar
time
has
in the
come.
Diabetes is
United States
associated with
of a diabetic is
the third leading
[281.
as atherosclerosis,
failure,
and gangrene
believe
that
precise
significantly
improve
cause
of death
The abnormal metabolic processes
diabetes heighten the risk
complications such
a task whose
[16,28,37].
degenerative
blindness,
kidney
Hence there is reason to
regulation of
the
of
quality
blood
of
glucose
life many
would
diabetics
could expect.
Conventional treatment
injections
state
of
of
function of
units,
blood
insulin once
the art
programmed
to
the
concentration,
is a
infuse
time
used for
from
for
severe
or twice
unit worn
varying
[36).
diabetes consists
daily.
externally which
amounts
of
to
determine
use
this
information
[33,35,45).
- 1
-
current
can be
as
a
nonportable
continuously withdraw
patient
insulin infusion rate
insulin
Substantially larger
short term research,
and
The
of
the
to
blood
sugar
compute
an
Analytic techniques and
control
to biological systems as to
biological
modelling
any others.
practices
have
theoretical considerations
In
the glucose-insulin
extensive review
theory are as applicable
been reconciled
of modern
realm,
of current
'algorithm'
for
literature incorporates other
[111.
[241 provides
approaches to
blood
with the
control theory
Hillman
algorithms. Numerous references are
No
Idiosyncracies of
glucose
also contained
glucose
an
control
herein.
control
in
the
than empirical observations
of
glucose or insulin dynamics, or any consideration of control
theory.
It is
body appears
and
its
determined
to be a
by trial
the
differentiating
Coefficients
and error.
the
goal
of this work will
law
derived
will
unit [381
[12,273,
each
term
knowledge
be
of the
obtained
A cubic function
and
are
by
of the
reported
as
a
[7].
glucose regulator.
sensor
only
empirically found
blood
implantable
can
measurement.
has been
of
Without any
derivative
superior algorithm
The
insulin secretion in a normal
function of both glucose concentration
derivative.
dynamics,
derivative
observed that
to methodically
It is envisioned that the
ultimately
which
be
be
incorporated
design a
control
into
an
includes a miniaturized 'glucose
an insulin reservoir,
power supply.
-2-
pump,
controller,
and
The
point
starting
investigation
of
of
glucose
this
design
and insulin
established physiological model [241.
is
similar
consists
to others described in
particular model may be
be used can be
it may
Insulin dynamics are,
ignore
possible to
be
predictive value.
of
portions
nonlinearities
glucose
-
essence
of
shown to
the
insulin
model is
the same
these
It is assumed
realistic enough
The
A
to
severities
captures
which
Some of the
form as
of
the
highly simplified linear
derived
the actual dynamics.
be of
(some of)
characterization is made of
model.
are elucidated.
for engineering
here.
used are
A linear
in
which does not yet appear
been addressed, is investigated
that the portions of the model
those
of a
acknowledge that glucose
However,
This possibility,
nonlinearities.
have
the merits
While
dynamics
probability, also nonlinear.
purposes
connected
applied to other models with equal ease.
dynamics have many nonlinearities.
to have
in that it
the analytic techniques to
argued,
All studies of glucose
all
an
The model to be used
the literature
(2,3,21,42,43,441.
an
using
dynamics
of lumped parameter organ representations
by circulation
be
will
those
the
results are
obtained
by other
investigators using experimental methods.
After deriving a
insulin dynamics,
linear model
simplified
principles of
-3
-
glucose and
control theory are applied
One of
to design a glucose controller.
of
the many results
of
control theory which
is exploited is
are never differentiated.
to
be
radically
that
the measurements
The derived control law
different
is shown
from
empirical
been
evaluated in
computer
it into
the model in
place of
A standard oral glucose
tolerance
algorithms
currently in use.
The
derived controller
simulations by
the pancreatic
substituting
beta cell.
test was then simulated.
as
the
model's
experimental
has
The
pancreas,
data.
controller performed
and
Corruption
implementational restrictions was
in the
glucose measurement and
finite bandwidth of the
compared
of
the
favorably
control
discrete output
that
either
superior or parts of
the
hardware
suggested
is
by
Delays
levels and
insulin pump were simulated.
the
controller
controllers
designed
herein
the mathematical model must be
Substituting
with
law
then considered.
Comparisons with existing closed loop glucose
indicate
as well
modern control
as
a
question.
- 4-
algorithm
method
of
into
is
refined.
existing
answering
this
Chapter
2
APPLICABLE CONTROL THEORY
The function of a control system can be
modification of the dynamics of the
controlled.
have
The most productive
been in
the analysis
system is one in which, if
a
multiplicative factor,
factor.
be described
virtually
anything
analytically.
linear
the output
of this
in
plant to be
systems.
an
is
A
linear
changed by the same
quality
linear
about
is
or
results in control theory
of linear
are vast.
differential
the
The cumulative
systems
system,
the
the input is scaled up or down by
The implications
system can
thought of as
system
byproduct
impressively
If a
equations,
can
of
be
found
the work
large
and
on
mature
collection of analytic tools.
Non-linear systems,
intractable.
on the
Analyses of those
of
one
applicable to another.
The
the
solution
are
generally
nonlinear systems
for which
are highly individual --
an analytic solution exists
of
other hand,
nonlinear
analysis
system
are
of nonlinear
results
seldom
systems
is
typically accomplished by application of the
first principle
to one
already solved.
of engineering:
Nonlinear
suitable
systems
reduce the problem
are
often
dealt
linear representation
of the
-
5
-
with
by
same system.
finding
a
This
is
the
method
system,
of
choice when
confronted
since the wealth of
with
a
nonlinear
experience with linear systems
can then be exploited.
Control theory
and
'modern.'
single
is
divided into two
multiple
representing
Because
'classical'
Classical control deals only with single
output systems.
input
camps:
and
of
Modern
output
control deals
systems,
and
these
characteristics,
applicable to a much wider
with multiple
is
states
controlling internal
input
capable
of
modern
of
a system.
control
is
range of problems.
RESULTS OF MODERN CONTROL THEORY
2.1
This
section summarizes
the results
to
the
of modern
theory which
are applicable
controller.
The capabilities of modern control
provide
a
strong
to
incentive
design of
represent
control
the
glucose
theory will
the
glucose
-
insulin dynamics linearly.
2.1.1
Modern Control
System Formulation
Modern control theory is based on a representation
system as a set of
equations.
the states
ordinary
linear
Each equation defines the
of the system.
It
classical transfer function
directly expressed in this
represented
first-order
differential
derivative
can easily
of
be shown
description of a system
form.
by
-
6 -
of the
one of
that the
can be
The modern formulation is
k2
= a 1 1 X1
= az 1 X 1
+ a1 2 X2 +
+ a 22 X 2 +
n
= ai1X1
+
R1
at) 2 X 2 +
.
+
+
ajnX
a2 nX
+ b11 u1
+ b21u1
+
+
birur
bzrur
+
annX
+
bn 1 ui
+
b
notation as
which is compactly expressed in matrix
=
X
AX
+ BU
coefficients
,
ai
vector U,
and
the control
inputs
the matrix B contains
of A and
The coefficients
B are
contains the
the system are the
to
the
determined
The characteristic equation is
coefficients bii.
by the plant.
the determinant of
(sI-A).
where
the characteristic equation is an equation in s,
Thus
the dimension
of s
is inverse
equation can be written either
transfer
factors
function
(i.e.
time.
as
are the
the
roots
are
system.
can be
interpreted physically as
are
the
complex
plant.
characteristic
'poles'
rather
they
For
equation
than time
represent
of
The roots
of
-
behavior
real and negative,
damped
referred
7
of s).
a
product
natural
the
to
frequencies
roots
as
they
When the
time constants.
constants.
-
a
(i.e.
poles of the system, and
generality,
are
characteristic
in determining the
most significant factor
of the
roots
or as
factored into the roots
When the
The
a polynomial in s
representation),
the characteristic equation are
of
matrix A
the
vector X,
are a
The states
of . the
'roots'
or
2.1.2
Controllability
The
control
combinations
signals
of
the
U
states
x2
= -clix
-
c1
U2
=-C21X1
-
CZZX.
-
-
cr2X2
-
cr1x1
Ur=
defined
to
be
linear
X via
U1
2
are
...
-
-
C1jrX
C2
X
x
-
...
which is expressed in matrix form as
U =
-CX
Since U
system
is
as
function
a
of the
states
X,
the
description may be rewritten as
X =
The
(A-BC)X
characteristic equation
(s_-(A-BC)).
as
defined
the
The
ability
controllability
of
the
control
characteristic equation of the
determined
by
controllable,
positioned
the
it
is
A and
can
arbitrarily
then,
if all the states
used
to form
control
control of the plant.
B
be
now
of
the determinant
the system
gains
system.
C
to
If
shown that
the
by the choice of
the
alter
system
poles
C.
the
may
is
be
Theoretically
are known at all times,
which
defined
Controllability is
matrices.
signals
is
of
will allow
they can be
arbitrary
Quadratic Performance Index
2.1.3
applied and how precisely the states can
C
can
to define
successful approach is
the
the
X and
states
The goal is
and Qo.
index
A well developed technique
or optimized.
minimized,
index which
performance
the
that
such
C
gains
Xo
A very
inputs?
a performance
controls U from their nominal values
find
all
satisfy
of the
the variation
of
a measure
to
How
on the values of the states and
constraints
is
be controlled.
simultaneously
to
chosen
be
can be
much control
on how
practical limits
There are
is
is to
define the quadratic performance index
t2
T
T
J =
LX
[
+
12][X]
[U
]RHUI
dt
ti
2 and
nominal
2
Lo.
state Ko,
with deviations
associated
penalties
R are
and
from the
be symmetric
R must
and
positive definite.
This
the
multiplies
integrates
the products.
performance
squares
2
Thus, in a simple case where
there
the weights
(.0 and R
identical
to
minimi-.ing
performing
formulation, it can be shown that
-09-
and
diagonal),
fit.
Using this
states,
and
terms
index is
of the
products
R,
by
products
are no cross-product
this
forms
performance index
a least
T
=
U(t)
-R-1B
where
It
S
is
Typically
yields
by
T
S + A S
+ SA
that
the
work
the
steady
for solving
equation
SBR-"B S + .0 = 0
-
is
a
function
state
value
gains
o-F
A,
of
C.
S
Many
control
2,
B,
is
and
desired,
have
equation.
utility
program
R.
which
techniques
the matrix-Ricatti
optimal
been
In this
OPTSYS
[231
used.
A and
the designer's
must
B are
choice
be stressed
matter
of 2
that
the
by
State
The
the
penalty
choice
of
2 and
Typically
the
control
is
U=-CX
unmeasurable
estimated by a
state
=
It
strictly a
choice
is
based
trial and error
performance.
(A-BC)X
,
10 -
on
To solve
reconstructed,
for the
is now
The
plant,
U
the
It is rarely
observer.
the control
-
X.
measured.
are
states
form postulated
predicated
vector
state estimator or
the
X
their
by
Estimation
of the complete
precisely
matrices.
R is
and adjusted by
case that all the states can be
problem
S is determined
desired characteristics and
optimal
availability
the plant,
and R,
and experience,
to achieve the
2.1.4
fixed
of engineering art.
on intuition
has
the matrix-Ricatti
feedback
linear
Since
the
---
T
S
constant
developed
is
defined
seen
is
S(t)X(t)
t)
'---C
=
or
observer
viz.
-CX
this
If A and
the
initial conditions are known exactly,
B and the
is never actually
This
perfectly.
error
observer errors,
the
predict
will
observer
of
measurements Z which can be made
as
of
combinations
linear
matrix
measurement
the
correct
The
are devised.
plant are described
according
states
the
To
possible.
signals
feedback
states
unmeasureable
a
to
M:
Z = MX
The identical measurements
and
is
measurements
X =
The error
is
fed back
-
observability) is
chosen to
and
estimated
observer
X -
and
X
=
(A-KM)X
parallel to the
controllability
to
and the
the error
zero
dynamics
the error
dependent on A and M,
force
the estimated states
^
L
X
controllability of
the
actual
dynamics
A
In a development
states and
to the
between the actual states
error X =
be
the
(A-BC)X + K(Z-Z)
shown to have the
plant,
between
difference
the
states:
of the observer
MX
=
Z
are made
observer states
the
plant states
converge
to form
U=-CX.
-
1 1 -
(termed
gains K can
plant
rapidly.
Thus
to the plant
are available
the
between the
the observer states arbitrarily
the
of
the
states,
and all
optimal control
A
successful
sufficiently
A
is
too
observer
accurate.
large
the
It
depends
on
can be shown
-
plant
A
and
B
being
that if the error in
observer
system can
become
unstable.
2.1.5
Kalman Filter
When the
plant states
with noise,
it
the observer
feedback
or measurements
because
it contains noise,
Kalman
filter
The
gains
to minimize
(noisy)
chosen
plant
states.
characteristics
Estimating
matter
2.1.6
of
the
on the
was
to
states
i.e.
A and
the
drive
and
of
whose
determined by
measurement
the
part of the
the
of the
problem was
states and
are to be regulated
states which
plant
noise
the
the
noise.
is
often
a
designer.
C via the quadratic
on deviations
To solve
observer
Coordinate Transformation
The derivation of
values;
an
which -is
the expected variance of
characteristics
judgement
was based
is
The choice of K is
of
the
Error
contaminated
is no longer possible to arbitrarily amplify
also amplified.
K are
are
this problem,
nominal values
and new
states from
their
formulated such that
controls
to zero.
nominal
the goal
Often the
about nonzero values.
the states are
are nominally zero.
B change,
performance index
During
transformed to new
the transformation
terms which reflect
are added.
-
12 -
the
original
2.1.7
Integral Control/Intelligent Integrator
A number of
factors
converge to zero.
plant
or
measurements,
One solution to
the
states.
cause
the error
states never
These include unknown disturbances
estimated setpoints
are
can
and
in the
incompatible
of
in the
incorrectly
error coordinate transformation.
this problem is to
integrals
or
to
the
define
important
It can easily be demonstrated
of interest will be driven to
nero
new
(error
states which
transformed)
that the error
with the use
states
of integral
control.
A
pitfall of
error can get
digestion
integral.
long
integral control
'wound up.'
of
glucose)
The error
enough for
A
large
produces
is
that the
disturbance
a
large
must swing in the
its integral
to
go to
integrated
(such as the
error
-
time
opposite direction
zero.
When
the
integral reaches zero the system may have enough momentum to
carry it through another
cycle.
Figure
1 illustrates
this
phenomenon.
To prevent integrator windup,
limits the
buildup of the
technique which
integral.
'throws away'
when the integral exceeds
some scheme is
Figure
-
13
2 illustrates a
additional error
a threshold value.
-
chosen which
state input
At t=a, Y(t) has returned to
zero, but in order for the
integral to also return to
zero, Y(t) must go negative
long enough for
the integral to
reach zero.
/
Y (t)
B2
i
If the B
B
trajectoiy
is
taken, when the
I
integral reaches
zero, Y(t) may have
enough momentum to
carry it through
another cycle.
s. \
Y (t)
a
MW
t=a
Figure
2.1.8
The
An Example
equation
considered.
be
Its
denoted as X,
of
of
1:
Integrator Windup
the Modern Control
motion
of a
single
position, velocity,
V,
and
A,
Formulation
14 -
be
and acceleration will
respectively.
-
particle will
The
equation
of
+
error
integral
error
state
Intelligent Integrator
Figure 2:
and external forces
as a function of the position, velocity,
general
In the
F.
case,
A = kX + bV
formulation)
motion
equation of
is
+F/m
a set
as
(the
equations
differential
linear
the
be expressed
equation can
This
of the particle
the acceleration
an expression for
motion is
of first
modern
order
control
as
X =V
A = kX + bV
V
+ F/m
which, in matrix notation is
X
0
1
It I
0
X
+F
L
k
LV
Note
that
are
differentiations
is
structured
and the
are
such
is
positio
successive
by
obtained
V
derivative
the second
the equation of motion
b
When
performed.
that
highest
the
-
15
-
1/m
obtained
directly from
n and first derivative
integrations.
No
the set of equations
order
derivative
is
solved
for
directly
obtained by
variable
and
integration,
the
Classical Control
Formulation
Positional control
All the
expressed
of the
formulation is
with the
the
as a Subset
of
modern
classical
control
control input is
are
called ohase
of the Modern
control
control
input
can
formulation.
a linear
be
From
combination
states.
only the
position X is included in F.
proportional
control.
proportional
plus
a new state
I is defined such
integral of X),
to
order derivatives
is achieved via the
variations
section 2.1.2,
If
lower
form.
2.1.8.1
F.
all
derivative
order
other
physical
control
I can be
form proportional
Higher
X and
If both
that I=X
included
plus
is
integrals or derivatives,
states
of
the
system
included.
-
'16
-
V are
result is
included,
implemented.
(i.e.
in the
integral,
the
I
If
is the
control signal
or PID
or
control.
functions of
could
also
be
Chapter
3
THE PROBLEM OF DIABETES
3.1
GLUCOSE
-
INSULIN PHYSIOLOGY
Virtually all
are
powered
physiological mechanisms
by
the
triphosphate (ATP)
energy
part by
exothermic
breakdown
into adenosine
required to reconstitute
requiring energy
of
diphosphate
ATP
adenosine
(ADP).
The
from ADP is provided in
oxidizing the monosaccharides
glucose, fructose,
and
galactose.
These sugars
are the
principal product
digestion.
Usually most of the
Further, the
liver rapidly converts
that glucose
sugar
is essentially the
of
carbohydrate
absorbed is glucose.
galactose to glucose, so
only monosacharide
in the
blood stream [20).
The
to
molecular weight
of the monosacharides is
permit diffusion across
actively transported.
transport.
rate
the
cell membrane;
In the absence of insulin
transport rate can be increased to
across
the
they must be
The hormone insulin facilitates
is approximately one quarter of
at elevated
too great
the glucose transport
its normal value.
cell membrane is bidirectional
17 -
The
four or five times normal
concentrations of insulin.
-
this
Glucose transport
[20].
When a cell
has
assimilated
converts the excess to
When the cell
has
converts excess
glycogen and
tissue,
glucose into
for example,
tissue types
their
liver plays
between
a major
of glucose and
also
elevated levels
the
the
normal
it
amount of
Nervous
glycogen or
store
no glycogen.
fat
very
Many
to
meet
regulation of blood
presence of elevated levels
liver
takes
up large
amounts
glycogen and fat.
glucose to
fatty
the blood.
The liver
acids,
In
which
released into the
mechanism for
raising
are
the presence
of the hormone glucagon, glycogen is
converted back to glucose and
is
glucose.
storage capability
In the
transported to fat tissue by
it
meals.
converts it to
converts excess
can,
tissue.
tissue can
role in the
and insulin
maximum
no
but virtually
glucose concentration.
of both glucose
The
In contrast, fat
needs
it
as possible,
the type of
have insufficient
metabolic
The
fat.
has virtually
of fat,
as
a stored form of
as much glycogen
fat depends on
amounts
much glucose
glycogen,
stored
storage capability.
large
as
the
rapidly
blood.
blood
of
This
glucose
concentration when it falls too low.
3.2
DIABETES MELLITUS
The
glucose
the
organ which
secretes
concentration is
islets
of
the
hormones
the pancreas.
Langerhans
respectively, as functions
secrete
of
-
The
-
a and
glucagon
blood glucose
18
which
and
regulate
S cells in
insulin,
concentration.
diabetes mellitus
The
insufficient insulin
in
the blood.
The
resulting
in urine
output
dump some of
(diabetes.)
the excess
and
of
osmotic differential
a corresponding increase
Simultaneously,
into
glucose
an
levels.
is an excessive concentration
dehydration of the cells
causes
elevated glucose
response to
The primary result of this
glucose
characterized by
syndrome is
the kidneys
the urine
(mellitus.)
1201.
The dehydration
effect.
The body's
Because
Depleted protein
oxidation
the
it makes
effectively,
tissues is the
first ill
inability to utilize glucose results in
alterations in glucose
processes.
on the
stress
body
and
intermediate metabolic
is unable
to utilize
increased use of lipids
stores
results in
a general
sugar
and proteins.
weakness and
susceptibility to other problems.
Increased
acids
lipid
and acidosis.
can result
in coma
complications
processes.
including
and
renal
are
Their
metabolism
leads
(Acidosis
and
death
caused
formation
stresses
the entire
[281.)
In
by
primary
to
these
forms
tissue
keto-
body tnd
the long
abnormal
are
of
term,
metabolic
changes,
atherosclerosis, an increased risk of heart attack
peripheral vascular disease, retinopathy,
failure.
-
19
-
blindness, and
3.3
TREATMENT
Some
diabetics with
partial B
cell
function
subnormal insulin response to glucose)
are
their
however
diabetes with diet and
require injections
of insulin once
the injections contain an
acting insulin,
exercise,
(i.e.
a
able to control
many others
or twice a day.
Although
optimized mix of fast-
and
the resultant glucose control is
slow-
still very
poor.
Because so
many people
control is a big,
DNA
Many
They
insulin
range from
delivery
have
insulin delivery
also
used
in
Existing closed-loop
of
pellets
based
The
clinical settings
20 -
on
most
BIOSTATOR,
controllers are large
-
are
insulin pump,
systems
been designed.
a variety
systems
being
113]
to
[3,37]
to
1361.
mentioned in the literature is the
been
Human insulin would
subcutaneous
sophisticated programmable pumps
sensors
the
insulin currently used for
simple systems based 'on an external
Closed-loop
have made
[40].
open-loop
developed.
being pursued.
technology
presumably be better than porcine
its
Several different
human insulin possible.
daily injections
diabetes,
control are currently
in recombinant
manufacture of
by
competitive business.
approaches to diabetes
Advances
are affected
glucose
frequently
[30]
which has
(33,35,451.
external units.
A significant advance would
that
they could
explored
The
be
to miniaturize
be implanted.
This area
the devices
is also
so
being
[29,381.
most exotic
culture
diabetics
and potentially
human 4
cells
[401.
Many
in
vitro
obstacles
before this idea reaches fruition.
-
21
-
effective
and
idea is
transplant them
must
still
be
to
into
overcome
Chapter 4
GLUCOSE/INSULIN MODELLING
a vehicle for understanding the
goes
Usually
the
bloodstream if
all the
phenomena being
Modelling
dispersed.
uniformly
it will
model is good,
behavior of the system under
of the
considerations
perform experiments on
expensive,
or
be able to
is
that
the model which would
do
to
impossible
in
simulations might cast doubt on the model
knowledge
will
about
the
it is
physiology)
or
real
in
22
-
time
for
[9,19).
predict the
One
possible to
be dangerous,
Such
life.
(i.e. the
state of
suggest experiments
provide insights which might not otherwise be
-
the
injection to
a variety of conditions.
a good model
values of
in
studied have
biological systems have been considered in detail
If the
injection
homogenized
than the time required for the
constants longer
be
instantaneously
being
as
restrictions,
an intravenous
example,
For
judgement.
of
relevant to the purpose the
and approximations
can be modelled
a matter
reasonable
embodies
model
to serve.
is
model
system.
is
model
into the
assumptions,
knowledge
the
the greater the value of the model as
incorporated into it,
What
better
The
system.
about that
knowledge
to describe
system is
of modelling a
the goals
One of
which
gained.
4.1
EXISTING GLUCOSE AND INSULIN
Much
of
insulin
what has
dynamics'
observations
confirm
or
of value
A
much
to
been
more
a hypothesis
the system.
those who
of
models
because
the
is
models
of
are
of parts
designed
knowledge
investigators
and
to
and
however,
has
a
body
proposed
insulin
are not of
design
the
vivo.
design models.
composed
of
in
data so obtained,
[5,211
to
than
on
and
experimental
in vitro or
rather
glucose
pancreatic
goal
The
glucose
than
based
actually
group
models
approximations
no
and pancreas
Proposed
circulation
'modelling of
are investigative
smaller
mathematical
called
liver
refute
understanding of
are
has
of the
Such experiments
been
DYNAMICS MODELS
dynamics.
interest here
pancreas.
of
Insulin
lumped
connected
parameter
by circulation
[21,24,431.
Models
lowest
so
the
glucose
dynamics exist
are descriptions
little
models
models
of
understanding
of the
there
equations
are
[21,24,42,431 which
dynamics
this
Next are
There
level
is
that
organ-level
models
into
describing
organs
[8].
and
connect the
organs
system.
23
at
At the
molecular-level
organs
-
levels.
interactions.
ability.
the entire
(usually nonlinear)
Finally
of molecular
have no predictive
which lump
at three
-
the
circulation
with the
models
circulatory
4.2
THE FLOWMOD MODEL
The
study
glucose -
model of
was
written
formulated by
in
the
Systems Dynamics
where
Foster
Group at MIT.
[21).
Hillman
[241
as
In
more
conventional
Simulation)
[15).
at
Since
simulated
a set
of
with
it
form the
the DYSYS
at the
differential
to the
point
recapitulated
by
equations.
behavior
of
facility
by West
the model has
(DYnamic
Computer
equation
the
by
differential
Joint
at
reasonably well
has been
(nonlinear)
originally
DYNAMO
refined
this
SYstem
Facility
[461
at
MIT
and Sorensen
form.
Physical Derivation
the model
independent
the
which is
body
'spaces';
spaces are divided
is
imagined to
insulin space and
into a number of
with
rules
creates glucose
Each compartment
the
body.
is
compartments
transfer
for how the
compartment
The
3
the glucose
and
compartment
-
2L4
-
The
in the same
the
consumes or
to define
the
describe
the
and insulin spaces.
an idealination
Each
6
two
each of
between
first task is
Figures
compartmentalization of
glucose space.
affecting
or insulin.
compartmentalization.
consist of
compartments,
interconnected with other
compartments and rules
part of
was
It has been further refined
4.2.1
space,
then
developed
the JCF in
In
language
It
used in
It was
experimental data
Guyton
been
[181.
system dynamics
it reproduced
this
insulin dynamics
of
a real
is assumed
organ or
to be
a
continuously stirred tank reactor,
i.e.
is not
reasonable approximation.
strictly true,
For example,
as being
but
is
instantly dispersed
through the
In fact,
required for this to occur.
in the model are
phenomena being
affect
the
rules,
each
a definite amouat of time
As
long as
shorter
investigated,
the
than time constants of the
of compartments in
or
bodily
dynamics.
constituitive
should not
the model is
relationships
subsystem
Parts
affects
of
the
relationships are
describing
glucose
body
lumped
determined by
on
of
results
experiments
The
the
flow between them.
glucose
Thus the
tissue,
compartments
In the
heart,
diffuses
between
liver,
glucose
concentrations
and
the venous
as
BLOOD compartment
flow-rate
are in units
body
of
the
are based
investigative
mentioned previously.
various BLOOD
kidneys,
large
similar
together into
The constituitive relationships
how
and/or
with
same compartment.
the
is
time constants
such assumptions
or constituitive
organ
insulin
compartment into
accuracy of the model.
The number
the
This
an intravenous injection of insulin is modelled
which it is injected.
assumed
a
homogeneous.
mixing
connected by
periphery,
the blood
liver,
and
and kidney compartments
of their respective
blood mixes,
changes.
The
equations and
of inverse
are
time.
-
25 -
the
and
tissues.
change the
blood supplies,
glucose
in
coefficients of
tissue
blood
transport
the HEART
both the
equations
A
<<-2
BLOOD
HEADiT
B
HEAD TISSUE
2->
H
VENOUS
CIRCULATION
HEART
BLOOD
HEART
TISSUE
C F
IARTERIAL
I
LIVER
CIRCULATION
<--2
L
KIDNEY
I
>
PERIPHERAL
<--9
P"
BLOOD <-2
P
TISSUE
Figure
4.2.1.1
those
Compartments
compartment represents
most part
of the
its tissue
brain.
concentration.
The
HEART
cells
(RBCs).
Though
the
TISSUE
also consume
constitui'iVe
-
26 -
For
correspond
glucose
of glucose
BLOOD compartment
HEART
RBCs
supply
tissue absorbs
independent
cardiovascular system.
rate.
Circulation
the nervous system.
and blood
Nerve
constant rate, essentially
blood
and Blood
Glucose Space
The HEAD
the
Glucose
3:
or
at
glucose
relations
at
to
a
insulin
represents
corresponds
to
the
the
red
a constant
±or
these
of the
compartments are
constants are different.
transmembrane equilibration time
glucose
Peripheral
PERIPHERY.
(the derivative
peripheral uptake
in
the
functions
of
level
derivative of
partial
a
is
the
of
means
the
Further,
the
to
function with respect
the
That
product
system
the
the
peripheral glucose
of the
is nonlinear.
relationship
constituitive
in
states
two
and insulin.
equation)
differential
on
depends
uptake
of glucose
relative concentrations
together under
lumped
body tissues are
The remainder of
masses and
the tissue
form,
same
insulin
concentration is also nonlinear.
The
constituitive
kidneys'
until the
At this
point
dump glucose
the kidneys
the
to
amount
by
also
negligible
excretion is
glucose concentration exceeds
proportional
rate
glucose
kidneys'
The
nonlinear.
is
relationship
threshold value.
a
urine
into the
which
the
at a
glucose
concentration exceeds the threshold.
The liver's
chapter,
introduced when the
and fat stores.
uptake
off
A saturation
liver has
glucose uptake.
The
concentration on
insulin concentrations,
insulin concentration
zero.
shown in-figures
full glycogen
in the
saturates for high
level to
the preceding
in
The most significant nonlinearity is
of insulin
when the
elucidated
have several nonlinearities.
also
nonlinearity is
effect
as
'functions,
The uptake
4 and
5
from its
functions used in
(from Guyton
-
drops
27 -
(211).
and
cuts
basal
FLOWMOD are
x2.
-
e
05
4,,
0
c1
LUc
NL
x2
x4
x3
x5
x6 x7
A G. < 40 mg %
X 8 X 9 X 10
Insulin in liver space
Figure 4:
by Liver
Glucose Uptake
Concentration
as
a Function of Insulin
x2
C')
4
-- ;
Ln
xl
0
U
NL
x2
x3
x4
Glucose in liver space
Figure
5:
Glucose Uptake
Concentration
by Liver
-
28
-
as
x5
a Function
of Glucose
I
p
---
>
LYtVEART
DY
TLASZ!A
HEART
TISSUE
LIVER
PLASMA
H
<
<-2
L
LIVER
<
TISSUE
KIDNEY
PLASMA
KIDNEY
TISSUE
<-K
-<
PERIPHERAL
PLASMA
<-P
TISSUE
Figure
4.2.1.2
The
6:
Insulin Compartments
and Blood
Insulin Space
relations for
constituitive
diffuse between the
blood
tissue
assumed to
and interstitial fluid
at a rate
concentration.
Thus
the insulin
of insulin in
degraded in all
proportional
rate
a
at
concentrations
insulin is
Further,
compartments
compartments in
Insulin is
dependent only on the relative
compartments.
the
all
insulin space are identical in form.
the
Circulation
dynamics are
to
the
its
completely
linear.
As in
insulin
glucose space,
between
the
the BLOOD
various
-
tissue
29 -
compartments transport
compartments.
The
insulin concentration in the
are
those
used
compartments
dynamics,
tissue,
to
in
determine
glucose
nerve tissue
so the HEAD
liver,
kidneys,
glucose
space.
In
of
periphery
uptake
terms
is indistinguishable
compartments
and
glucose
in
of
those
insulin
from peripheral
space
are
lumped
into PERIPHERY in insulin space.
4.2.1.3
The Pancreatic .8 Cell
The pancreatic R cell is
to insulin space.
and
The
secretes insulin
6).
0
FLOWMOD
for
nonexistent.
postulated,
A
of the /3
pertinent
mechanism
and its
senses glucose concentration
(2(L)
cell is the
in Figure
cell are
The
one
experimental
describing
parameters
closely approximated
vein
differential equations.
workings
which
glucose space
relationships for the /3
set of highly nonlinear
internal
cell
into the portal
The constituitive
of the
the coupling from
the
/3
'tuned' until its
the performance
of
an
a
model
part of
data
is
cell
was
performance
actual pancreas
(21,221.
4.2.2
Most
Validation/Predictive
physiological
Value
models are
difficult
to
validate.
Often there is so little experimental data available
is all used to help construct the
cannot
be
used
to
then
model
verify
correctly.
-
30
-
[9].
that
the
The
that it
same data
model
works
constants
The
transmembrane
blood
time
equilibration
FLOWMOD model
volumes,
for
glucose
constants
of
these
laboratory
numerous
were
have
sorts
incorporated
4.2.2.1
into
FLOWMOD
of the
derivation of
insulin producing
model are
value of
the
the
control system to be designed.
intolerance is
OGTT is
procedure used to
containing
100
the
sugar stresses
are taken
glucose
at
15
or
pancreatic part of
the
(i.e.
with
for comparison with
test
twelve
then
subject
grams
the
for
tolerance test
by an eight to
The
overnight.
Therefore
Using FLOWMOD
the oral glucose
preceded
assumed to have no
is
will be used only
Sample Results
The most common
weak point in
The unmodified FLOWNOD
FLOWMOD
4.2.3
cell)
the
this analysis.
the
,
and
FLOWMOD 13 Cell
function in
of no concern.
in
[24].
the pancreas
validity and predictive
obtained
researchers,
by other
the 4 cell model is
Fortunately,
FLOWMOD.
by the liver.
experimentally
experiments
Irrelevance
The
been
the effect
and
insulin concentration on glucose absorption
Data
with
and tissue
insulin degradation time constants,
uptake,
of
concerned are
thesis is
which this
of the
the parts
in
drinks
of glucose.
minute
a
water
This large
intervals
and insulin concentration.
-
31
-
(OGTT).
hour fast,
The
usually
solution
ingestion of
Blood samples
glucoregulatory system.
30
carbohydrate
and
analyzed
for
control system
by which the
will be the basis
The OGTT
and
clinical data,
designed herein is compared to FLOWMOD,
existing glucose regulators.
an intravenous glucose
FLOWMOD originally simulated only
function of
time is known because
it is controlled
a
by the
tolerance
simulate an oral glucose
To
experimenter.
input as
glucose
case,
the intravenous
In
infusion.
test
with FLOWMOD required modifications in two areas.
the digestive system.
absorption of
effect
the
gastric
of
This
is
due
glucose
The
second was
to model
hormones
on
insulin
secretion.
larger
than to an
digested glucose is
(Insulin response to
intravenous
the
model
to
was
modification
first
The
injection of
amount of
an identical
of hormones secreted by
to the effect
glucose.
the gut
These modifications were made by Hillman
during digestion.)
[24).
OGTT,
9,
8,
Figures 7,
and
10
show a FLOWMOD simulation of
compared with a pool of clinical data
in the results
of the simulation are
and
modelling of
gastric
glucagon and
Embedded
FLOWMOD's assumptions
4 cell dynamics,
of alucose,
about the gut absorption rate
[391.
an
hormone effect
on
insulin secretion.
Although FLOWMOD does not reproduce clinical OGTT results
perfectly,
this
is
thought
-
to be
32 -
more
a
function
of
0;
-p.
)
r"
Z-D
)
On 0.
UCN )
.1
E-A
-,D
50
100
150
200
T IME (MINUT ES)
Figure
7:
FLOWMOD
Absorption Assumption
Gut
250
3 00
During OGTT
Cuj
E-4
II
-
Ur>
U)
H
.-
z
9
qJ'
50
200
150
100
250
300
TIME (MINUTES)
Figure
8:
FLOWMOD Pancreatic
-
Insulin
33
Secretion During OGTT
120
MEAN t SEM
100
-
z
%%T
80
Ln
MODEL
EXPERIMENTAL
(n=145)
so0
z
0
U
0
40
2.
(L
20r
(L
0
-50
f
0
I
50
£00
150
200
250
300
TIME CMINUTES)
Figure
FLOWMOD Peripheral
During OGTT
9:
Plasma
Insulin
Concentration
130
MEAN: SEM
L2
z
0
1.20
-MO-MODEL
\
£10
----
-
EXPERIMENTAL
LI
0
-
(n=145)
100
L3
0
0
0
4j
4
wj
IU
90
s0
70
w
0r
s0
-50
0
50
00
£50
200
250
300
I
TIME CMINUTES)
Figure
10:
FLOWMOD Peripheral Blood Glucose
During OGTT
-
Concentration
34 I
imperfect modelling
effect
of gut
than a fundamental
absorption and
weakness
model works well for intravenous
response could be
gastric
gastric hormone
of the model,
glucose
because
infusion.
altered by changing the
the
The OGTT
gut absorption and
hormone assumptions.
The model
is still
of great value
insulin dynamics.
It can
algorithm
herein
designed
be
used
to
BIOSTATOR algorithms).
-
35 -
for
its
to compare
existing
glucose
the
algorithms
and
control
(e.g.
Chapter 5
LINEAR CHARACTERIZATION OF THE MODEL
Before launching into a controller design,
to
have
some
controlled.
idea
of
In fact,
the dynamics
of
the
the
system.
be exploited.
a
simplified
It
dynamics.
can
never
is
model
realized
capture
only capture the essence
assumptions
and
ex'ample,
the kidney
dumping
Any
chapter
glucose
and
of
actual
not necessary.
It must
of the actual dynamics.
nonlinearity
is
insulin
the
used
to be
to
can simply
good
enough
reasonable
simplify
of the glucoregulatory
control
is to
representation
behavior
is
are
of control
this
simplified
approximations
representation
glucose
this
results
describes what are believed
dynamical
the
the
the
but
of
There
representation of
the
both
of
that
all
physiological system,
This chapter
The goal
be
impossible
plant.
a linear
With a linear model, all
theory can
derive
for finding
prudent
plant to
a modern control design is
without a mathematical description of the
strong motivations
it is
be
so that
the
system.
For
discarded
the
if
kidney
threshold is never reached.
glucose -
insulin model
sites at which glucose
has many
consumption
-
36
-
occurs.
states
and
many
Hotwever,
in a
represented as
simplified model the dynamics can be
-
input
single
system.
output
concentration controls
the
site
which dominates all others.
One of
This
in FLOWMOD.
one
is
the
series
Taylor's
would be extremely tedious
series
are table
because the nonlinearities
necessary to evaluate more than
and it might be
functions,
liver is-that site in
ways to linearize a nonlinear
common is
the most
The Taylor's
expansion.
The
one
system.
There are several different
system.
insulin
there is
justified if
can be
gross simplification
'global'
glucose concentration.
'global'
This
the glucoregulatory
The
a single
operating point.
the insulin
used to simplify
The method
This
partial fraction expansion.
technique can be used when
The transfer function can
the transfer function is known.
as
be expressed
number
a sum
the
in
denominator
of
fractions,
and an
numerator
1321.
some
If
of
each
has a
of which
term
(s-root)
in
are
the numerators
those terms can be
smaller than others,
is a
subsystem
the
much
discarded without
sacrificing much accuracy.
The
The
glucose
goal
subsystem
of analyzing
simplified transfer
how
much
nonlinear
it
transfer function
is not
known.
is to
find a
glucose subsystem
the
function describing it,
varies with
initial
it is).
-
37 -
conditions
and
determine
(i.e.
how
The method
frequency
used to
response
correspondence
in s
in the
phase
shift
In
in the
estimate
the
a
unique
function as a polynomial
and
as an
attenuation and
time domain.
system,
The
a
is
the steady
a sinusoid of
state
Bode plot
poles and
is
response to
the same frequency
a
as the
phase relative to
an analytic
tool used
zeroes of a linear system by
to
from the
domain transfer function [14,321.
In
a truly linear linear system, the transfer function is
independent of
Taking
the
the initial conditions and driving
frequency
response
amplitudes will reveal
A
the transfer
exists
but with a different amplitude and
input.
time
There
frequency domain,
sinusoidal input is
the
the glucose subsystem
analysis.
between
a linear
input,
simplify
Bode
analysis
with many
will
implicitly
it will
be
poles
and zeroes which are very close
5.1
INSULIN DYNAMICS
secretion.
insulin
to
insulin
This is the
space
acts
control input and the
periphery,
which are
controlled.
The
first
different driving
how linear or nonlinear a system is.
characterization because
The only input
amplitude.
as
space
control
a
is
yeild
a
simplified
impossible
to resolve
to each other.
the
pancreatic
input to the system.
filter
interposed
insulin concentration in
where
task
the
is
-
glucose
to characterize
38
insulin
-
between
the
the
liver and
concentration
this
The
is
filter.
Insulin
Insulin
Liver Insulin
Glucose
Glucose
Secretion
Dynamics
Concentration
Dynamics
Conc.
Figure
Since
linear,
the
11:
Simplified System Block Diagram
insulin
dynamics are
time invariant,
linear analysis
eight states,
first
techniques
represented
to
The control
PLASMA compartment, and the
as
order differential equations,
are easily applied.
corresponding
concentrations.
already
There
the four PLASMA
input is applied
outputs are
are
and TISSUE
to
the LIVER
the LIVER TISSUE and
PERIPHERAL TISSUE insulin concentrations.
After
determining
equations from
the coefficients
of the
constants in the model, the
differential
matrices of
X = AX + BU
have been specified.
A matrix
analysis utility program is
used to examine the system described
5.1.1
by these matrices
Controllability
First,
the
examined.
The
indicating
that
controllability
of
the
there
are
the
states
in the
-
39
modes,
insulin dynamics.
(the state
three uncontrollable
-
is
a rank of 5,
three uncontrollable
system is transformed to canonical form
uncoupled),
output
controllability matrix only has
combinations of states,
are
[1].
modes
or
When the
variables
can
-be
identified, but their physical
grasp.
However,
it is intuitive
hence liver tissue
the
peripheral
significance is impossible
that the liver
is not
as
be assumed that that state is
5.1.2
Transfer
Next,
to
two
outputs
characteristic
liver in
not uncontrollable,
it
controllable.
Functions
the transfer functions
the
Because
important as the
glucose regulation, and by itself is
will
plasma and
concentrations are controllable.
tissue
to
are
equation of
relating the control
found
from
the system
input
X=AX+BU.
The
has eight
negative
real roots.
This is consiste-nt with intuition, as
there are
no
storage modes.
cannot be
"energy"
stored and
parallel
problem.
dashpot
later
to a
released;
mass -
system
can
poles
cancellation of
of half
diffuse.
This is
a heat
transfer
A mass -
spring -
system or
Neither system can oscillate.
system
function.
(insulin)
it can only
dashpot
store energy
oscillations are possible
The
"Energy"
one
Four of
a minute
and
of
and
later
release
in such a system.
zeroes are given
the
poles
the remaining
or less.
The
in Table
poles have
time constants
fastest time
to two minutes,
may
so these faster poles
remaining
three
2.
-
40 -
A
each transfer
on the
of the
1.
occurs in
physiological interest is assumed to be
The residues
it;
poles
constant
order of
of
one
be disregarded.
are shown
in Table
TABLE
1
Insulin Subsystem Poles
System Poles
Location
Time
(s)
The
smaller
Zeroes
LIVER
Constant
of
TISSUE
-12.18
-11.67
-3.784
-2.533
-1.876
-. 4453
-. 07689
.08
min.
.264
.395
.53
2.25
13.0
min.
min.
min.
min.
min.
-. 03371
29.7
min.
residues
of
the
and Zeroes
Zeroes of
PERIPHERAL
TISSUE
-11.67
-3.469
-2.045
-11.67
-8.75
-2.469
-. 5609
-. 08117
-. 10
-. 04041
pole at s=-.07689
than those of the other
partial fraction expansion are
poles.
are
significantly
These terms
in the
also discarded.
TABLE 2
Insulin Subsystem Partial
Pole
Fraction Numerators
Residue of
Residue of
LIVER
-. 4453
-. 07689
-. 03371
The
now
simplified
contains
only
PERIPHERY
.004646
.00009395
.0003107
-. 3046
-. 08929
.3775
representation of
two
poles.
measuring insulin disappearance show
has
the
Laboratory
that the
the form of two decaying exponentials
-
41
-
insulin
[17].
subsystem
experiments
disappearance
To
check
the
accuracy of
impulse response of
and compared
approximations,
the complete insulin model
to the
response predicted
transfer functions.
is shown in Table
these
The simplified
3 in factored
the
is measured
by the
simplified
characteristic equation
(values
of the roots)
form.
TABLE 3
Insulin Dynamics Transfer Functions
LIVER
S + .0595
(s+.4453)(s+.03771)
Predicted by
Simplified Model:
Estimated from
Impulse Response
of Complete Model:
Both poles
reflect the
The values
in the
The
influence
obtained
of all
from the
the
s + 2.385
(s+.5)(s+.05)
than predicted,
poles
impulse
which may
which were
response
ignored.
will be used
following chapter.
system
is
in the
further
term
liver.
reduced
system.
The
by
amd assuming all
Thus
the
discarding
glucose
the
uptake
the eighth order insulin system
has been simplified to a second order,
output,
s + 2.165
(s+.4453)(s+.03771)
s + .0866
(s+.5)(s+.05)
are slightly faster
peripheral uptake
occurs
PERIPHERY
accuracy
-
42
o±
-
single input,
these
assumptions
single
and
the
performance
in section 5.2.1
5.1.3
and in the following chapter.
Error Coordinate Transformation
The insulin subsystem can be
form directly from
the
below
of the second order system are examined
system
the
expressed in phase
transfer function.
can be expressed in
and controllable.
The
Note
this form it
variable
that when
is observable
states must then be
transformed to
error coordinates.
=
X1
-a
X2
is the
0
1
-b
X1
+
0 U
;Y
canonical representation of
Y =
U
=
2 a
X1
X 2
X 2.
the
transfer function
as + z
s2 + bs + a
Figure
12:
Phase Variable Form for Second Order System
1-
Xi
aa2
S
+
-aa
X,
as
+
-a
X2
I:
U :
Figure
-b+aa
X2
Io
-a
+
U
ZB
Basal
Insulin
L iver Tissue Insulin Concentration
Secretion Rate
13:
Error Coordinate Transformation of
Canonical Second Order System
-
43
-
GLUCOSE DYNAMICS
5.2
Many of the
of
FLOWNOD
use nonlinear
dynamics.
Because
table
the
a linear
can still
be
the
model
functions
glucose
analytic determination of poles
However,
in the glucose
equations
differential
dynamics
and
:eroes
characterization
of
attempted.
Advantage
has
predictive
good
to
is
the
nonlinear,
is not
possible.
glucose
ability
the
describe
are
taken of
section
subsystem
the
fact
to
that
the
do
physiologically impossible.
5.2.1
All
Frequency
Response Analysis
parts of
the
disabled.
In
model except
terms
of figure
examining the right hand box
exist.
The glucose
as
the
this
11
if
glucose
space
are
to
corresponds
the left hand box didn't
space is driven with
Bode
tissue insulin concentration inputs.
sinusoidal
plots
liver
are
made
of
each time domain transfer function.
The severity of
variation
in the
amplitude and
relating
the nonlinearities will be
Bode
diagram as
initial conditions.
peripheral blood
liver
tissue
shown
in figures
insulin,
14 and
a
and peripheral
observed.
414
transfer
-
by
driving
functions
the control
inputs,
tissue insulin
Nonlinearities
15.
-
function of
The
glucose to
indicated
are
are
indeed
10
-~~-
114
-~z~
-
---
-
7
-- - -
-
t Z
-
----
.1
N,1
I;
1 11
._-
-a
.
..
- .....
4.
.....
-,
4
'
.---
-4
-
+
----
--
-t--+ ----..-.---.
.0/
~I47~
(Ae 0 0I
-
-
-
--
~
____
-
~1
I
-
--6-
- ----
tAVE
--
--
,--.--A
--
-
~
*
-
-
-
--
r
41
-z8r -
-30
-2e
IZiZ7IZi7iZZ~~
P#9Sf/9A4LE
-
/
ii
660606)
.(nt
Figure
14:
a).
(A)~.OI
(4>):
1.
Bode Plot of Peripheral Blood Glucose
Concentration Response When Driven By Liver
Tissue Insulin Sinusoid
-
45
-
Cuqo,
L2
/i
NI
------
IF IT
--------------
-77 -:-
-~
-I
! N
~\TL.2 2~\.~
r..u:\.:mY
A- 1A
-
~--
-
A
7-7
\Wx
/NJ =
ID~7AbP~/
..
.
.
\
0/
7:t
....
.... -
....-
...
.-.
,-
...-
~
t..ff
- -li
.I:!
/SD
-94
FL-.-r-
-3w
ic.
&7 ,I /
Figure
15:
Bode Plot of Peripheral Blood Glucose
Concentration Respons.E When Priven By
Sinusaid
Tissue Insulin
-
46
-
Peripheral
Valuable information may be
obtained
nonetheless.
First, it is noted that
for
control
the
two
shape.
Further,
function
is
function.
the
the
ten
have
inputs
Thus
of
the transfer functions
substantially
magnitude
times that
from the Bode plots
of
the
the
transfer
peripheral
transfer
peripheral
insulin characterization was
justified.
Bode plots
three poles and no zeroes.
of how
complex the
This
contribution in
indicate the presence of
is a significant indication
simplified glucose
Because there are only three
poles,
subsystem must
the
important to
'more
linear'
One
two.
than
of the
system.)
example
liver
can store
This is
given
poles are
than the other
or
"energy"
complex.
storage mode.
later return
The poles
-
The
spring
could be real
it to
-
the
dashpot
or complex
It is guessed that the
complex.
phase portion
additional
be real
glucose and
is
function.
significantly slower
amount of damping.
From the plot it is
The
transfer
represents an
It
transfer function is
parallel to the mass
above.
depending on the
liver
two poles could
The other
liver
that the
the peripheral
poles is
glycogen in the
(The
note
be.
simplified glucose
subsystem can be represented with only three states.
also
same
liver
discarding the
Most importantly, the
the
poles
of
with
seen that these poles
the
indicates
plot
time
-
constants
47
-
of
are overdamped.
that there
are
less
one
than
minute.
that these poles
Note
simulation.
are fast enough that they
they
In any event,
of the
could be artifacts
can be ignored.
Whether the three poles are real
and complex, or all real
will be investigated in the next chapter.
5.2.2
Impulse Response
The impulse response
to check the
control
of the glucose subsystem
theory
that
shows
of the
transformation from time domain to frequency domain)
impulse
a
response of
Thus by applying
transfer'function.
insulin,
identically
system is
linear
(i.e.
transform
Laplace
the
Linear
plot.
Bode
predictions made from the
taken
was
an impulse
the
of liver
the transfer function of the glucose subsystem can
be obtained.
Identifying the
than for the
proved to be more difficult
This was
glucose subsystem
impulse response of the
because -the
real pole
insulin subsystem.
is easily
identified when
plotted on semi-log paper, but the decaying sinewave is best
plotted
on
linear
the
When
paper.
two
signals
are
superimposed, neither plotting technique is suitable.
The method used
was to guess
on estimates made from the Bode
search on the coefficients of the
an
analytic function based
plot and perform a gradient
function to find the
response.
fit to the measured impulse
-
48 -
Higher
best
order terms,
amplitudes
an order of magnitude less
have been
discarded.
compared in Table
4.
one minute,
less than
time constants of
which either had
or
than the dominant ones
predicted
The
validity of
are
fitted roots
and
The
these assumptions
is
investigated in the following chapter.
4
TABLE
Simplified
Glucose
Dynamics
Complex Poles
Real Pole
Predicted
.002
.707
.707
Estimated from
.00534
.953
.1692
Impulse
Response
1
(s+.00534)((s+.1613)2+.051z)
Transfer Function from
Impulse Response
magnitude
The
strength
space.
of the
of
The -coupling term
to
insulin space
from
coupling
indicates
response
the impulse
r is
the
glucose
be
.1 from the
applied
to both the
estimated to
impulse response.
5.3
SUMMARY
Linear analysis
glucose and
techniques have
The eighth
insulin subsystems.
a more tractable
dynamics have been reduced to
representation.
The
assumption
-
been
49
-
that
the
order insulin
second
order
peripheral
contribution to glucose uptake
was negligible in
to
the
the
liver was supported
by
analysis of
comparison
the
glucose
dynamics.
Frequency
response
demonstrated
However
that
analysis of
the
glucose
the
dynamics
the nonlinearities are well
an engineering
Figure
glucose
are
nonlinear.
enough behaved
16
depicts
left hand corner
of A
the
insulin
and
glucose
is
The upper
the insulin subsystem and the
glucose subsystem.
subsystems
each
are isolated
r.
term
secretion rate.
glucose
respectively.
I
dynamics
into one matrix equation.
right hand corner contains the
and
that for
approximation they can be ignored.
descriptions assembled
coupling
subsystem
from
The
control
and
Go
are the
concentrations
in the
Numerical values are
-
50 -
The
for
other except
input is
basal
U,
two
the
the
insulin
(nominal)
insulin
liver
given
lower
and
periphery,
in Appendix A.
I1
z
61
dz
3a
X
c II
012
c01
0
0
-r
0
0
0
=
0
0
0
0
-a
0
0
1
0
-b
0
0
0
I1
+
G3
-c
+
X
A
b1
b
oU
0
0
0
0
-1
+
0 I0
0
0
+
0
0
-a
G0
B U + error coordinate
terms
States:
Liver Tissue Insulin Mass
I1
12
Linear combination of I1 and its first derivative
G1
Peripheral Blood Glucose Concentration
First
Derivative of G 1
G2
G3
Second Derivative of G 1
Figure
16:
Complete
-
51
Linearized
-
System
Chapter
6
OBSERVER DESIGN
blood glucose concentration,
In the body, only one state,
the states
The rest of
is measurable.
the simplified
of
model must be reconstructed, in order to be able to form the
Because the noise characteristics for the
feedback U = -Ct.
be designed.
free)
observer will
6.1
PARAMETER READJUSTMENT USING
chapter the
In the previous
OPEN LOOP OBSERVER
parameters of the simplified
The ultimate
insulin and glucose subsystems were estimated.
is how well
these parameters
test of
the plant
conditions of
should
observer
Chapter
from
Recall
observer.
(noise
a deterministic
glucose sensor are unknown,
pump and
-track
and observer
the
if
initial
the
are identical,
without
perfectly
plant
the
criterion will be used
This
feedback correction.
that
2
in the
they perform
any
to make
final adjustments to the parameters.
The
observer must
simplified
reconstruct
Two
system.
all
those
of
insulin and peripheral blood glucose,
The performance
with respect
of
five
states,
states of
liver
two
states.
-'4
tissue
also exist in FLOWMOD.
the open-loop observer can
to these
the
The control
be evaluated
input,
the
pancreatic
also
is
insulin secretion.
the
accessible in
model.
Insulin States
6.1.1
control
performance
FLOWMOD and
of the
are
index is
of the
square
equal
input
phase
to minimize
then adjusted
is discovered
This
response
to
given in
been
may have not
because only
z,
a,
detected
the first 70 minutes
in
The
5.
The
from table
previously
the impulse
of the
were plotted.
TABLE
5
Insulin Subsystem Final Parameter Settings
a =
b =
z =
a
UO
I
.025
.8
.07
=
=
=
1.76
21.9
4.7
1,76(s+,04)
(s+.768)(s+.0325)
Transfer
function
-
53
-
12)
index.
table
than
in
and /3
(figure
(s=-.03771
more important
be
b.
the performance
subsystem
insulin
a,
A
the
tissue insulin
variable form
the parameters are
slowest pole of the
thought.
liver
The parameters
insulin subsystem in
integral of
the
to be
the
error between
the observer.
final values of
1)
defined
output.
pancreatic
FLOWMOD's
to
and
(in error coordinates)
initial conditions equal to zero
the
setting the
b.y
insulin observer is tested
The open loop
response
6.1.2
Glucose States
To
test the
glucose
parameters,
closed-loop until t=40. minutes.
time
during the
peaked
and
its
approximately
insulin
During
c,
zero.
the
and
remaining
the
observer
260
minutes
index is
the
r of
and
to minimize
the
nonlinearity
subsystem.
F
was
concentration
to be
the
below
concentration
is
basal
concentration
increases.
instead of
different
---
it
downward.
The
observer runs
open-
the square
the parameters
figure
in
on one
fasting
or
the
16)
the
b,
are
glucose
of two
was
level
below
is
glucose
two
a,
index.
and
r
insulin
glucose
'negative
concentration
of
or
glucagon
effectively
values
values
above
basal,
subsystem parameter
slowest,
accurately,
zero.
upward
represent the
of the two hormones.
The final glucose
6.
its
The
effectivities
set to
concentration
Glucagon
forces
are
absorption,
Physiologically, when the
falls
insulin'
are
admitted
insulin
value.
derivatives
(see
to take
allowed
depending on whether
below its nominal
performance
had
has
the integral of
subsystem
the glucose
run
concentration
glucose
gains
is
approximately the
second
the glucose concentration estimate;
One
fairly
and
observer
is
glucose
At t=40.
The performance
adjusted
table
This
which
first
secretion,
loop.
of
OGTT at
the
but
values are
most significant
the other
-
54
two poles
-
pole
were
shown in
was estimated
real
instead
of complex.
One of them (s=-.2554)
location of the complex poles
a =
b =
c =
Subsystem
6
Final
1.888 X 10-4
.03
.37
the estimated
(s=-.1613).
TABLE
Glucose
was near
Parameter Values
r(insulin)
F(glucagon)
=
=
.0302
.00695
1
(s+.00686)(s+.0108)(s+.2554)
OBSERVER IMPLEMENTATION
6.2
When all the parameters
determined,
of the
the observer gains can
the observer error is described
X =
From figure
system A matrix have been
be chosen.
Recall
that
by
(A-KM)X
16 it is seen
that if
the measurement is the
glucose concentration, the measurement matrix M is
=
0
0
1 0 0
Because the system
many
zero
is only fifth order and
the determinant
entries,
evaluated analytically.
The
of
+
coefficients
a 3 s 3 + azs 2 + a 1 s'
-
55
-
and M have
(sI-(A-_K1))
of the
equation
assS + a 4s 4
A
+
ao
is
resulting
are
functions
13,
12,
those
of
Ks,
of
the parameters
and Ks.
stability
equations
The
Butterworth
of
the
in five
is a pole
observer
unknowns,
observer gains
for
gains
The coefficients
a fifth order
Butterworth pattern
the
insulin
and the unknown
There
which may
are
shown in
states
are
weak
coupling
glucose space via F.
To check
The
the
insulin
gains
are
be
table
are
solved
7.
now
Note
insulin
vary
Further,
five
that
this
hypothesis,
to
The
for the K's.
Physically,
this
be
maximizes the
orders of magnitude
between
found
verifying the hypothesis.
with w=10.
pattern which
than those of the glucose states.
represent
are selected to
pattern
[34).
gains KI,
greater
should
space
r is
and
varied.
inversely
the glucose
the
with
F,
gains
are
OGTT
and
be independent of r.
-ound to
TABLE
7
Observer Gains
K1-=
K2
The
observer
examining
become
and
very
the
is
-943011.
950273.
tested
reconstructed
negative
oscillate wildly
as
by
for several
56G -
31.1907
486.763
4655.57
simulating
states.
soon as
-
=
=
=
13
K'
Ks
the
an
The insulin
states
gut absorption
begins,
minutes before
dying away.
The insulin
glucose
oscillations induce
states.
large
The performance
oscillations
of the
in the
observer is totally
unacceptable.
The
explanation may be
Insulin causes
negative
makes
glucose
gut absorption begins,
Because
unknown
to the
insulin
the
it
as
through how
are
Therefore they should
correction
term.
to
insulin
corresponding
the
insulin states
The reconstructed
those of
independent of
very
conclude that
obtained again by reasoning
rapidly to the
poles
When the
very negative.
accurate without feedback.
not respond
rise.
can only
has become
system works.
observer
concentration
Therefore
glucose concentration rises
observer,
solution is
reasonably
the
to fall.
the gut absorption is a state disturbance
concentration
The
by physiological reasoning.
glucose concentration
insulin
rapidly.
found
the
the glucose
states,
Because
the
states
are
they
can
be
adjusted.
In the final observer
are
moved
by
scaling
determined factor of
five
orders
of
implementation,
K1
and
10".
the
ten times
gut absorption than during the
The
observer
the estimated
now
states
down
When the
magnitude
observation is still
K2
are
in
during
reduced by
the
insulin
the
onset of
the OGTT simulation.
the
well enough
to
signal.
57
error
empirically
rest of
used
-
insulin poles
by an
gains
greater
reconstructs
can be
the
-
states
form the
optimal
that
control
Chapter
7
RESULTS
7.1
CONTROLLER
Designing
the
design
a
DESIGN
good
observer
process.
If
reconstruct the states
be
operating with
Once the
designed
and
by
R.
choosing
Only
concentration,
the
values
insulin
of the
square
of the
method).
The
system
is
controller
and
controller
figures
trial
and
17
concentration in the
same as in
the
then
matrices
penalized.
set
value
simulated with the
and R adjusted
2
glucose
are
are
is
to
(a
the
common
artificial
to actually
desired values.
are
18.
so
rate
maximum desired
values of 2
FLOWMOD
and error
to
controller
penalty
R elements
gains C are given
and
unable
concentration,
and
*
the
the
infusion
inverse
The control
of
insulin
of those
the maximum
is
controller will never
been designed,
values
achieve
observer
has
Initial
the
the
part of
information.
the
and
the most critical
accurately, the
correct
observer
is
in table 8.
compared
The penalties
that
the
modern
FLOWMOD
in
an OGTT simulation
have been
maximum
controller
simulation.
-
58
-
The modern
adjusted by
peripheral
is
The
in
insulin
approximately the
control
gains
and
performance
R.
selection is a matter
concentration-time
insulin
the
modern
the
with
hypoglycemic undershoot in the
and there is no
controller,
lower
is
integral
that
in
strongly
and
judgement.
of the designer's
differ
results
The
.0
Their
R.
changes in 0 and
results will vary with
The
of
designer's choice
are determined by the
concentration.
glucose
8
TABLE
Controller Gains
State
(from figure
Gain
16)
-10.612
37 .091
15. 319
96.41
Insulin Conc.
I1
Liver
G1
Periph Glucose Conc.
Gz
First
Derivative
of
G1
183.9
G 3 Second Derivative of G 1
below its
basal
control gains
neither
For this reason,
(nominal) level.
gains are
are used;
determined by r(insulin)
only the
calculated nor used for r(glucagon).
three elements
Only
and R
of .
have
There are
13 other elements which can
obtain
different
results.
investigating the
C.
has
been made
P efficiently
-
1471.
59
time
be varied to
could
-
on
be
spent
the
five
a methodology
for
16 parameters on
effect of all
Progress
choosing 2 and
Much
manipulated
been
here.
gains
hardly goes
that the insulin concentration
noted
It is
Light: Modern controller
FLOWNOD
Dark:
0
__
_
Or,
__
_
zc____
I
F.
5b
1'o
1'50
TIME
Figure
17:
2bo
250
300
(MIN)
Peripheral Plasma Insulin Concentration Controller Compared to FLOWIMOD
Modern
Light: Modern Cqntroller
FLOWMOD
Dark:
C:
z
C
(n
Cj
-4
-LD
CIO
0.0
50
150
100
TIME
Figure
18:
(MIN)
200
250
Peripheral Blood Glucose Concentration Controller Compared to FLOWMOD
-
60 -
3300
Modern
7.1.1
Comparison to Empirical
Existing empirical
Algorithms
algorithms
consider only
concentration and its first derivative
the
power
states
8
of the modern control
are
it is
heavily weighted
derivatives
not
the
state
Part of
formulation is that all
glucose
in the
signal.
acceleration
of the glucose concentration are
From table
is the
control law.
the
most
Although two
used,
they are
obtained by differentiation.
The
insulin
empirical
which
states,
are
This
is probably
absence of hypoglycemic
the
a significant factor
in the
undershoot in the modern controller.
derivative were known because
Insulin concentration and its
insulin
in
unavailable
heavily as the glucose
algorithms, are weighted as
concentration.
the
glucose
(velocity).
incorporated into the control
seen that
the
infusion rate was
known,
and the
dynamics of
insulin diffusion and degradation were incorporated into the
simplified model.
INTELLIGENT INTEGRATOR
7.2
In some
simulations not
concentration
insulin
setpoint were
setpoints
which
are
DESIGN
shown here,
setpoint
chosen incorrectly.
depend on
many factors,
weight and
individuals and as
sex.
Thus
a particular
-
61
and
In
liver
the
tissue
insulin
secretion
the real
'world the
the most
significant of
they will
vary
among
subject's weight changes.
-
The result of the setpoint errors
in
glucose concentration.
state,
the
integral of the
defined to drive
integral to
As
the
is a steady
explained in Chapter
2,
glucose error state,
steady state error to
not grow,
state error
the glucose
a new
must be
zero.
For the
error state must
go to
zero.
The intelligent integrator described in
usable here.
Further,
In the
enough to offset them.
in the
it always integrated the entire
glucoregulatory system,
the
large enough,
are
errors
There
if the
setpoint
intearal may never
build up
are
glucose error state which
During
ignored.
are normal
for
OGTT,
an
also large perturbations
40 mg./dl.
For these
figure
much
19.
and ought to
the
example,
concentration rises from 80 mg./dl.
zero to
2 is not
In that design the integral could never exceed
a certain value.
error.
Chapter
to
120 mg./dl.
be
glucose
(Or
from
in error coordinates).
reasons,
an alternate
This -design allows the
as necessary,
but ignores
design
is
integral
large
to
proposed
in
build up
as
transients in
the
glucose error state.
When the control
system,
the gains for
glucose acceleration
the gain
gains are recalculated for
on the
smaller than the
the
glucose,
double
integral
glucose
from their
state
gain.
-
glucose
62
is
The
-
an
a six state
velocity,
previous
order
optimal
of
values
and
and
magnitude
control does not
Error
Signal
Output to
Controller
Gain
Integrator
Signal Limiter
Saturation
Threshold
Figure
'know'
about
integral by
19:
the
letting
the
what
are
two
The
the
It
parameters
intelligent
threshold,
is
integral
existing controller.
integrator
it build up
controller's
integral control,
provided by
Intelligent
controller
desired
state,
of the
integrated
value
of
parameters:
is
the
selected by
63
added
to
onto the
adjust the
There
the saturation
The
never
-
undershoots
that there
will be
magnitude
of
the
related
to
the
directly
overshoot.
-
without
integrated value.
integrator.
the
good
controller.
recognized
(hypoglycemia)
is
desirable
but it is
undershoot
first place.
is
integrator
the
minimizes the
'correction term,'
without the
with
can be
a
which
independent
undershoot
hypoglycemia
in the
is
would be
integrator
state,
it
performance
and the gain on the
glucose
Integrator
the intelligent integrator;
never
Because
Proposed
Thus
saturation
the
degree
threshold
of
of
4
the
integrator.
determined
Based
by trial
on
trial
The
the
the
and error
and
gain
intelligent
are both
integrator is compared
concentration
anticipated,
but
is
by
set
integrator
to
1 and
the
The controller performance
to the
the integrator and with FLOWMOD in
glucose
state
4
control gains unchanged.
with the
integrated
and error.
saturation threshold
other
gain on
controller without
figures
undershoots
less than
its
20 and
21.
basal
The
level
FLOWMOD's
does
4
as
4
'normal'
pancreas.
4
Light: Modern + Integrator
di
akW
dd
UIL
ll-
tA
fC,
1 DL
FILOWMOD
____hark:
Cj
F
U
z
0c-_
_
_
_
_
_
_
_
_
_
_
_
_
z
Nz_
53
a.
150
100
TIME
Figure
20:
200
250
MIN)
300
4
Peripheral Plasma Insulin Concentration
Intelligent Integrator
4
-
614 -
4
Light: Modern + Ihtegrator
Medium:Modern Controller
F:OWMOD
!Dark:
-j
D.
M
0._
_
_
_
_
__
_
_
_
_
M
LJ U-
_
_
U-)
:D
5b
e.
10o
TIME
Figure
7.3
21:
the
300
Two
redesigned.
of the
Simulations
hardware
constraints
include
will
implementation
the
-
LIMITATIONS SIMULATION
hardware.
whether
250
200
(MIN)
Peripheral Blood Glucose Concentration
Intelligent Integrator
HARDWARE
Any
150
control
or
of
types
hardware
imposed
can
algorithm
indicate
need
to
limitations
practical
by
be
are
considered here.
7.3.1
Insulin Pump Ruantization
An actual
nor
insulin pump is neither
The
instantly adjustable.
discrete output rates which
small time
be an
intervals.
may
frequency of
is
pump
only
be
of
2/min.
-
65 -
assumed
changed
to
have
at finitely
were
estimated to
20mu/min and
an infusion
Typical parameters
infusion rate step size
rate change
continuously variable
and pump quantization are
Simulations with the integrator
shown in figures 22,
identical to
plots
indistinguishable
that
the
(i.e.
constrained
pump
are
so nearly
pump constraints
that the
The results
24.
simulations without
are
indicates
23, and
when
be
can
less expensive)
This
superimposed.
even
more
without
severely
compromising
performance.
0)
-J
C
0
iuu
.r'
rigure
7.3.2
The
22:
TIME
(MIN)
i2uu
250
Peripheral Plasma Insulin Concentration Constraints and Intelligent Integrator
30
Pump
Glucose Measurement Delays
other,
the glucose
hardware limitation is that
more significant
measurements
are
-
not continuous
66
-
and are delayed.
-
LO.I-
t
V
__
C:)
U)
V
U-)
-i
-j
0.
Figure
_______
_______
_______
0. 0
5b
23:
100
T I ME
250
2b0
150
(MI.N)
300
Peripheral Blood Glucose Concentration Constraints and Intelligent Integrator
Pump
0
Cu
0-
U'L0
_
_
_
ICD
__
_
_
_
_
_
_
__
Figure
Ua
24:
1OU
TIME
150
(MIN)
200
Pump Rate with Pump Constraints
Integrator
-
67 -
_
_
i
____
U. U
_
_
_
_____
250
300
and Intelligent
That is,
every time
period A 1 t a measurement is taken.
measurement is not available until
The
results
of
control theory
throughout this work
are for
measurement systems.
powerful in
theory
the
in its
outside the
have
continuous-time,
discrete-time domain as
domain.
is
been
used
theory is
as
continuous-time
control theory
Discrete-time
However,
thesis.
later.
continuous-
is
a crude estimate
indication of how well the
can be made to get an
controller
stressed that the methods to be used
It is
not the correct
which
Discrete-time control
scope of this
might work.
A2 t
some time
That
approach to the problem, as
are
is apparent from
the results they yield.
The goal
case
'look
of this estimate
like'
problem to
a
The
measurements.
Figure
measurement.
It
From
is
delayed
must again
correct.
25
this
by
figure
it
approximately
Continuous-time
is
-
that this
introduced.
68
-
the
is
a
discrete-time
the
synthetic
seen
that
the
d,t-aet-
controllers
are
from
measurement
two
illustrates
be stressed
unstable when time delays
last
the
reduce
accomplished by
measurement
synthetic
between the
discrete-time
(i.e
This is
'continuous-time'
interpolation
the
case
the continuous-time
discrete-time measurement.
measurement
to make
one already solved).
synthesizing
linear
is
technique is
typically
not
become
*
o
Measurement taken
Measurement
available
If the
synthetic
0-~
measurement
is
the sample and
'A'
hold signal 'A'
the measurement
is too choppy for
---- the continuous time
'B', a
observer.
linear interpolation
of the previous two
measurements, is much
smoother.
In 'B', the synthetic
measurement is approx'B'
imately the signal,
delayed by Alt + A 2 t.
---
Figure 25:
The
Synthetic
pump rate
reason
Discrete-Time Case
is observed to become
gut absorption onset
this
Measurement for
as the time
the pump
rate
is
t
a
unstable during the
delay
is increased.
limited between
For
zero
and
350mu./min. in the simulations.
Simulations of
seconds,
and
3 minutes, and 5 minutes
27.
It can
concentration
increases.
will
minutes.
with time
control of
deteriorate as
part
suboptimal
of
The glucose
the
the
controller
the pump output when the
-
poorer
30
be
better
glucose
delay
control
performance.
output remained at 350mu/min instead of
69
the
measurement
measurements
control would
-
delays of
are compared in figures 26
be expected that
However,
attributable to
28 shows
portal delivery
is
Figure
are delayed
3
if the pump
oscillating.
0
Light:
Medium:
O.\_Dar:
! Minute delay
Minute delay
.5 Minute delay
-J
N.
U
ote: A1 t
=
A2t =
.5, 3, 5 mih.
z
0
z
LO
z U)
U.
DL
Figure 26:
0
1
U
U
I
300
Peripheral Plasma Insulin Concentration Measurement Delays of .5, 3, and 5 Minutes
Minute Delay
Light:
Medium:!3 Minute delay
DMk:1
Minu3te- Malay
C)
Note: A
LO
: =
. 5,
z
2 l
2)U
(MIN)
TIME
3,
A2 t
5 min.
W-
U
Li
U
CD0
,0.
Figure
so
27:
1 30
150
TIME
(MIN)
200
2Z,7
Peripheral Blood Glucose Concentr ation Measurement Delays of .5, 3, and 5 Minutes
-
7vf
-
3300
0
t--
C.
Cr1
I~
0~
-L(n
CL
Fi.gu
C)1'
Figure 28:
7.4
100
50
TIME
DELIVERY
controllers
portal
into the
whereas
be recognized
the
changes
equilibrium.
occurs
insulin
A
in the
that
greater percentage
peripheral tissue
be an
long
not
In the
dynamics
acceptable
linear
model,
can change
term delivery
when the
-
infusion
71
-
regulation
of glucose
when
insulin is
is
of
infused
infusion may
technique
numerator
site
glucose
hence
and
that peripheral
only the
was
insulin infusion
equilibrium
It is believed
pancreas
comparison.
peripheral
peripherally.
the
Peripheral infusion
vein.
simulated with the modern controller for
It must
BIOSTATOR [301)
(e.g.
vein,
insulin into a peripheral
secretes
300
Measurement Delay
Pump Rate with 3 Minute
Existing closed-loop
_
250
(MIN)
PERIPHERAL VERSUS PORTAL
infuse
t
200
150
[2,3].
the insulin
changed.
The
two parameters
adjusted
in the numerator of
for peripheral
observer technique
In
Chapter 5
modes
than when it
is
problems
is
infusion with
was determined
when the
portal.
insulin
Thus
the same
(figures 29,
30)
the
glucose
control was
the controller becare unstable.
the glucose
are
measurements
nearly
as
When the glucose measurement
insulin
when
peripheral
was simulated with peripheral
was delayed,
results
is
fewer
infusion.
portal infusion.
show the
are
of instability
good as with
32
open-loop
there
possibility
with peripheral
greater
that
infusion
the
When the modern controller
infusion
transfer function are
described previously.
it
controllable
the
is
infused
delazyed
three
Figures
31
and
and
peripherally
minutes.
C0
mr
Light: Peripheral ';Infusion
Portal Ifso
0IarLk:
-J
CD
_
_
_
__
_
_
_
_
_
_
__
_
_
_
_
_
_
CD
U)
Z:
I
50
100
Figure
29:
200
150
Peripheral
Plasma Insulin
Peripheral
Infusion
-
250
Soo
(M N
T I ME
7
'4
Concentration
I
-
-
I
C
bight: Peripheral Infusion
-CZ
Dark:
I
o
_
Po
tal Infusion
__
_
_
_
-LJ
0.
Figure
50
30:
100
TIME
150
250
-
I300
(MIN)
Peripheral Blood Glucose
Peripheral Infusion
Light:
C)
200
tL
r
Concentration
Peripheral 'Inf us ion
3 Minute dE lay
rjhtrAL Infs-in
no delay
C)
zhIn
a. u
Figure
53
31:
130
TIME
150
(MIN)
200
250
300
Peripheral Plasma Insulin Concentration Peripheral Infusion with 3 Minute Measurement
Delay
- 73 -
C ru 1
ight: Peripheral Infusion
3 Minute d lay
)ark: pe ripherallInfusion
nc delay
[0
n
L.
V7
C
rl0Fg. n
32
TIME
(MIN)
25
-
30
Peripheral Blood Glucose Concentration Peripheral Infusion with 3 Minute Measurement
Delay
Figure 32:
7.4.1
2u
IOU
.3L
Comparison with Other Results
Results of the
reported in
in these
[10).
tests
BIOSTATOR controlled subjects are
OGTT on
concentration reached
The maximum glucose
is- 160mg/dl.
With the
modern controller,
peripheral infusion, and measurements delayed three minutes,
the maximum
30).
is
(see figure
135mg/dl
This is the same maximum concentration as with portal
and a three minute
infusion
It might
the
glucose concentration
appear that
BIOSTATOR's
verified,
simulated
the modern
(empirical)
however,
measurement
until
delay.
controller
algorithm.
the
on the same dynamic model.
74 -
BIOSTATOR
outperforms
This cannot
be
algorithm
is
Hillman's results
also
(241
of
simulated measurement
indicate that sometimes the
-
75 -
delays
controller is unstable.
8
Chapter
CONCLUSIONS
8.1
SUMMARY
Analytic
techniques
simplified
derive a
linear
The
insulin dynamics.
intractable
current
sucessfully
glucose
all the
a linear
to not be
made
model
tools of modern control
algorithms,
empirical
to
glucose and
found
dynamics were
Having
applied
of
characterization
nonlinear.
possible to exploit
Unlike
have been
measurements
it
theory.
were
never
differentiated.
modern control
The
state
estimator,
solution
controller,
and
The modern
control formulation
unavailable
to empirical
and
its
derivative
simplified
model,
factor
information,
concentration
not
provided state
Insulin
obtained
well
as
were
in
the
including
and
its
control
exhibit hypoglycemic
with empirical algorithms.
concentration
from
glucose
law.
information
time derivative,
information
directly
Glucose acceleration was found
acceleration.
important
as
intelligent integrator.
algorithms.
were
a
of designing
consisted
velocity
undershoot common
and
to be the most
With
about
the
the
better
insulin
controller
did
in nature
and
Steady
glucose
state
errors
error-integral state.
to not exhibit
'windup' and
the
law.
control
The
to be
Adding
hypoglycemic undershoot,
controllable
by
were eliminated
integrator was
adjustable
the
but the
through
the
addition of
undershoot
of
designed
independent of
integrator
choice
a
introduced
was
the
entirely
integrator
parameters.
The
in
effect
an
of hardware
of
implementation
investigated.
much smaller
constraints
which would
an
artificial
/3
be
imposed
cell
were
Insulin pump limitations were found
effect
on
performance
than glucose
to have a
measurement
delays.
When
connected to
the
body at
the
controller provided glucose regulation
Glucose
control
was unchanged
portally
glucose
connected
instability.
This
discrete-time
system being
is
nature's.
infusion site
were
remained
controller
peripherally
hoc
when the
measurement delays
connected
as good as
the
was
vein.
changed to a peripheral
When
portal vein,
controller
suspected to be
controller
77
exhibited
the
-
the
but
the
signs
of
stable
result of the ad
design rather
inherently uncontrollable.
-
simulated,
than
of
the
FUTURE WORK
8.2
has
This thesis
theory
of control
complex nonlinear
biological
biological system.
of
design
the
then
and
a
of
the analysis
in
system
demonstrated the application
a
regulator
for
that
It is really only a proof of concept ---
an even larger amount of analytic work is necessary before a
is
work
further
areas where
the
Some
of the art artificial organ can be realized.
state
described
are
required
of
below.
Discrete-Time Kalman Filter
8.2.1
A
was
filter
Kalman
charateristics of the sensor,
because
designed
not
actuator,
the
and plant noise were
not known nor could they be guessed at.
It
by good
was only
observer worked at
and
delayed,
properly,
fortune
that
all when the measurements
infusion may
The
disappear when
to operate
(observer)
with
instabilities associated
a discrete-time
time
were sampled,
For the controller
manipulated.
a discrete-time Kalman filter
designed.
the continuous
must be
peripheral
controller is
used.
8.2.2
Model Enhancement/Verification
Simulations with the artifical
peripheral
portal
vein were
infusion.
almost
This
itentical
differee.
-
controller connected to a
78 -
to simulations
strongly
from
of
clinical
with
results
[10).
PIOSTATOR
closed-loop
peripheral
Either
evaluate
the
first
BIOSTATOR algorithm
other
needed to
of FLOWMOD is
experiments
Two
possibility.
suggested to resolve the
represent
or a mistake was
the field,
in depth investigation
An
not
are
simulate the
possibilities:
and implement
in FLOWMOD;
the
law derived here is
control
the
superior to algorithms in use in
made.
does
FLOWIMOD
circulatory dynamics well,
using
control
the modern
control law in the BIOSTATOR.
8.2.3
Only
Experimentation with Controller Parameters
out
five
of
parameters were modified in
the
parameters on
the
penalty
eighteen
this study.
and
still possible.
Much experimentation with the parameters is
demonstrated.
and
is
required
to determine how
measurement delays were
of both of
More investigation
effects of all
The
performance are unknown.
controller's
Modelling of pump limitations
integrator
pump
much
these variables
discretization and
measurement delay is tolerable.
8.2.4
This,
Partial B
Cell Function
and other work
no . cell function.
B
cell function
control.
but
artificial B cells have
on
There are
still
many diabetics
do
not
To properly address the
amount of partial /3 cell function
-
79
-
have
who have
adequate
possibility of
is the
assumed
some
glucose
an unknown
domain of
adaptive
field.
still developing
Fault Tolerance
8.2.5
-
have
will
-
organ
artificial
truly
pancreas
artificial
implanted
the
meet
to
the life or well-being of the
warn that service will
reliability
highest
require
requirements
It must be able to
patient.
there is
be required well before
And it
a catastrophic failure.
seldom
an
It must not fail in any way which would endanger
criteria.
of
itself a
[26]),
(for example, see
taken in adaptive control
A
approaches which can be
There are many different
control.
access
surgical
must be considered
robust enough to
must be
These
maintenance.
for
risk
in the design
of each
aspect
of the implantable unit.
approach to reliability that
Fault-tolerant design is an
enables the system
components have
failed
faults
tolerate
objective.
to continue functioning
There
system is
the
and
(failures),
are many
fault-tolerant system
8.3
--
after some of its
designed
still
considerations
carry
so it
can
out
its
in
designing a
linear
analytic
[251.
CONCLUSION
This
thesis
techniques
has
and modern
demonstrated
control
that
theory
can
be
applied
to
I
nonlinear
results
which
natural
glucose
contained
herein will
will further
advance
and
dynamics.
and insulin
artifical glucose
-
provoke
additional
state
the
regulation.
81
-
Hopefully
the
research
of knowledge
about
Appendix A
COMPUTER PROGRAMS
This appendix
used in this
the
All
study.
modified FLOWMOD
For details
DYSYS.
A.1
contains the computer programs
except
is a
one
are
subroutine which
on this
interface,
see
is
were
programs;
called
by
1 15].
SIMULATION DATA DEFINITIONS
The first program is used to write
all the
the
standalone
which
constants
data file
definition of
a data
which FLOWMOD uses.
into the
the /PCOM/
program which writes the
named common
common block
file
FLOWMOD
block
containing
then
reads
/PCOM/.
is given
after
The
the
data file.
-
82
-
4
A.1.1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
/PCOM/ Database Generator
PURPOSE --THIS PROGRAM CREATES THE FIRST GENERATION OF PARAMETER
FILES
NOTES --THE BASELINE PARAMETER CONSTANTS ARE DECLARED IN DATA
ALL THE PARAMETERS ARE
STATEMENTS CONTAINED HEREIN.
DECLARED TO BE IN THE NAMED COMMON /PCOM/.
THE ARRAY PCOM IS E2UIVALENCED TO THE FIRST ELEMENT OF
/PCOM/; THE ARRAY IS WHAT IS WRITTEN OUT TO THE FILE
AND READ IN BY FLOWMOD.
INCLUDE
C
C
C
C
C
C
C
C
C
C
C
C
'PCOM.FOR'
LOCAL VARIABLE DECLARATIONS --LOGICAL*1 FILMAM(40)
BASELINE PARAMETER VALUE DECLARATIONS --INPUT FOR SIMULATION CONSTANTS:
DATA BFHDBVHD,BFL,TVLBFK,TVK/.72 5,.2,1.45,2.,1.16,
1 .6/
DATA BFP,BVP,GTETH,BFH,BVH,FVHT/2.465,2.4,1.,5.8,2.2,
13.5/
DATA TBVFVHD,GTETHD,BVK,BVL,FVP/6.,.2,. 2,.4,.8,7./
DATA GTETP,ITETH,PFK,PVK,PFL,PVL/5.,10., .696,.24,.87,
1 .'48/
DATA PFP,PVP,PVH,FVHI,ITETK,FVK/1.479,1. 44,1.44,3.7,
1.2,.2/
DATA FIKTKT,FVL,ITETL,FILTLT,ITETP,FIPTPT/2., .4, .2,
13.75,20.,.0125/
INPUT FOR IVGTT SIMULATION:
DATA ISTARTISTOP,IRATE/-3.,0., 11666.67/
INPUT FOR OGTT SIMULATION
DATA EOGISF/0.,1.,1.,7.273,5.714,3.636,2.677,2.,1.6,
11.333,
1.143,1., 1./
DATA EOGRHS/0.,0.,1..,1.,1.667,1.429,1.282,1.190,1.124,
11.087,
*1.053,1.020,1./
INPUT FOR GLUCOSE DISTRIBUTION SECTOR
DATA ILSNF,GLYBTN,GLYBNF,GLNF/4.702756,3.,100.,
11846.437/
545
0./
DATA GLYSN,GNEMTM,GNEOF,GPSNF/100.,3.,60.,
IPSNF,MAGMTI,MAGUF,pBCU/84.,3.,30.,10./
DATA
DATA HUG,CNSU,GLYSTfl/20.,100.,3./
C
-
83
C
INPUT
FOR BETA
CELL
SECTOR:
DATA GMET,IRRF,ISPF,:FTSF/.465025,21 .86756,200000.,
144.40491/
DATA
DISTM,IRTM,GDC,FGTIrIM/.2,.4,.01,2./
DATA FiTGM,F IMPTl, MRNASNH, RTI N/.5,1.,.05,20./
DATA PISNF,PITIME,IGT!ME,RHSDTM/21.86756,20.,20.,45./
DATA DEGRTM,HSN,RSN/45.,2206.589,116. 1372/
C
C
NONLINEAR TABLE FUNCT I O - 1so.,0.,
DATA TB1 /0.,
0.,
I
720.,690.,
1
900. ,920./
1
10.
DATA TB2 /0., 0.,
.2,
50., 1./
1
.,
DATA TB3 /.5, V
6, 0
360.,230.,
20.,.8,
3 0., 1.,
.8 ,.3
,0.,
.
540.,460.,
,.9,
.8
40.
,1.,
,
1.,
11./
DATA TB4
1
/0.,
1 .5,
1.
,
2., .64,
1.,
3.
.45,
4
5.,.35,
6.,
32,
DATA TB5 /20. 4.5,
12.4,
70. 1.2,
1
0
/0.,
TB6
DATA
5.,2.0/
1
DATA TB7 /0., 2 .25,
11.4,
.5, 1 .00,
1
DATA TBS /0.0 2.8,
10.8,
1
2.3 0.7,
DATA TB9 /0.0 0.0,
12.0,
5.0
1
2.0/
DATA TB10/0.0 0.0,
16.0,
1
2.5
7.7,
110.4,
5., 10.8,
1
1
.,1
17
30
9 .,.23,
29,
8.
.26,
10
,4.3,
40., 3 .9,
50., 3.4,
80. ,1.0/
1. 0,
2., 1.5,
3.
1.75
1 90
.2,2. 2 5,
.3,
.6, 0.60,
0.5
.7,0. 3 0,
1.0,1 0,
.3, 0
3.0 ,0. 1
1.0
1.0,
3.5,0
2.0,1
5/
4,
1. 0,
3.0
/
,2.0,
4
.1, 2.25,
,.2
60
,
.4,
0/
2.0,
0 .9
3.0,1 .7 5,
4.0,
.0,
1.5
.0,
2 0,
3. 5,
.5,
4.0
0.,
4 5,
5.5
6.
.5,
6.5
1.8,
7 0,
60.
40.
70
80
.0,
.0,
80.
120
2.,
240
28 0.
5.4,
320
5.7,
160.,
1.,
240.
,1.3,
80.
1 .0,
120.
,1.
280
4.,
0.5
0.5,
112.0/
DATA TB11/50. 0.2,
DATA TB12/00. 0.0,
1
160.,3.,
200 ,4.
1
1 360.,5.9,
400 , 6.0/
1
DATA TB13/.0.0 1.0,
1 320.,1.6,
400 ,1.8,
1
DATA TB14/O.0 0.0,
1 160.,1.4,
1
200 ,2.2,
1
360.,6.0,
400.,7.,
1
DATA TB15/61.0345,0
80.
480
40.
,03.8,
,2 .
,0.
240
I
/
440. ,8.,
480.
7/
i.206
1,.9,
. ,o
P, 4
.0/
,
320. ,5.,
1,
I
9./
31.3793,1./
I
DATA TB16/0.0,0.0,
1 160.,.6,
1
200.,.7,
1 360.,.92,
2
400.,.95,
C
C
C
40.,0.0,
80.,.05,
120.,.43,
240.,.78,
280.,.84,
320.,.89,
440.,.98,
480.,1.0/
START OF EXECUTABLE CODE ---
10
20
30
40
50
60
WRITE(5,10)
FORMAT('[CREATE HOW MANY FILES>')
READ(5,20) {F
FORHAT(14)
WRITE(5,30)
FORMAT('IENTER FILENAME>')
READ(5,40) LENF,FILNAM
FORMAT(2,40Al)
IF(LENF.E2.0) STOP
FILNAM(LENF+1)=0
DO 60 I1=1,NF
OPEN(UNIT=1,,HAME=FILEAr,TYPE='NEW',FORM='FORMATTED'
CARRIAGECONTROL='LIST')
1
WRITE(1,50) PCOM
FORMAT(8F10.3)
CLOSE(UNIT=1)
END
-
&5
-
A.1.2
C
C
Named Common Definition
PCOM.FOR --PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
PARAMETER
/PCOM/
DECLARATIONS
!NONLINEAR TABLE FUNCTIONS
NT1=6
DIMENSION DECLARATIONS.
NT2=6
NT3=6
NT4= 11
NT5=7
NT6=6
NT7=9
NT8=8
NT9=6
NT10=15
NT1 1=4
NT12=1 1
NT13=7
NT14=13
NT15=3
NT16= 13
PSIZE=358
C
REAL
REAL
REAL
REAL
REAL
REAL
REAL
REAL
C
C
C
MAGUFILSNF,IPSNF,MAGMTM,ITETH,ITETK,ITETLITETP
MRNASN,MRTIME,IGTIME,IRTM,ILSN,IPSN,IRRF,ISPF
EOGISF(NT1 4),EOGRHS(NT14), IFTSF TB5(2, NT5)
TB1(2,NT1) ,TB2(2,NT2),TB3( 2 ,NT3 ) ,TB4(2,NT4)
TB6(2,NT6) ,TB7(2,NT7),TB8( 2 ,NT8 ) ,TB9(2,NT9)
TB11(2,NT1 1) ,TB 12(2,NT 12), TB 13 ( 2 ,NT13)
TB10(2,NT1 0),TB14(2,NT14), TB15( 2 ,NT15)
TB16(2,NT1 6),PCOM(PSIZE)
/PCOM/ DECLARATIONS --COMMON /PCOM/ BFP,BVP,GTETH,BFH,BVH,FVHT,BFHD,BVHD,BFL
COMMON /PCOM/ GTETP,ITETH,PF,PVIK,PFL,PVL,TBV,GTETHD
COMMON /PCOM/ FIKTT,FVL,ITETL,FILTLT,ITETP,FIPTPT,PFP
COMMON /PCOM/ ILSNF,GLYBTM,GLYBNF,GLNF,EOGISF,EOGRHS
COMMON /PCOM/ IPSNF,MAGMTM,MAGUF,GLYSN,GNEMTM,GNEOF
COMMON /PCOM/ GMET,IRRF,ISPF,IFTSF,HUG,CNSU,GLYSTM
COMMON /PCOM/ FMTGM,FMMPM,MRIASN,MRTIME,DISTM,IRTM,GDC
COMMON /PCOM/ DEGRTM,HSN.RS13,PISNF,PITIME,IGTIME,FV
COMMON /PCOM/ FVP,TVfFV'I.TVL,BFK,FVHD,BV,BVL,PVP
COMMON /PCOM/ PVH,RBCU,GPSNF,ITETK,FGTrPM,RHSDTM
COMMON /PCOM/ TB1,TB2,TB3,TB4,TB5,TB6,TB7,TBS,TB9
COMMON /PCOM/ TB10,TB11,TB1 2,TB13,TB14,TB15,TB16
EQUIVALENCE (PCOM,BFP)
-
86 -
A.2
OBSERVER PROGRAM
The
will
observer
the
program calculates
produce a
fifth
observer poles.
order
This is
observer
gains
Butterworth pattern
done
by
which
the
for
solving the determinant of
A-KM directly.
C
C
C
OBSERVER.FOR
WRITTEN:
C
C
C
C
PURPOSE --CALCULATE
C
C
C
C
C
C
11APR81
STARK DRAPER
LABORATORY
OF
COEFFICIENTS
BUTTERWORTH
POLYNOMIAL AND
LOCAL VARIABLE DECLARATIONS --REAL*8 AK(5,5),AINV(5,5),DUMMY(5,5),V(5),DUM2(5,5)
SYSTEM
FORMULATION:
C
C
E
G
0
0
-D
F
H
0
0
0
0
0
C
C
C
C
C
C
C
C
C
C
C
0
0
0
0
0
0
C
-D
0
C
THE CHARLES
THEN FIND K MATRIX TO YEILD BUTTERWORTH POLES IN A-KM
C
C
C
C
C
AT
BY KEVIN KOCH
REVISED:16APR81 KPK CORRECT A MATRIX!
C
M
=
DH
0
0
1
0
-B
0
0
0
1
-C
1
0
0
-A
1
C-P
0
1
C-P
R-CP
BR
CR
DATA A/1.888E-4/
DATA B/.03/
DATA C/.37/
DATA D/.0302/
DATA E/-.62857/
DATA F/-3.31020/
DATA G/-.025/
DATA H/-.17143/
NEXT DATA STATEMENTS
MEASUREMENT
OF
+
IS:
! K1
! K2
1
!
!
K3
K4
R
!
K5
BUTTERWORTH
ARE
-
87
-
MATRIX
(A-KM)
IS SET EQUAL TO
A2*SS + A1*S + A0
0
0
-P
VARIABLES:
R=(EH-FG)
P=E+H
EQUATION
B+R-CP
CR-BP
IS
INTERMEDIATE
EQUATION
+ A3*SSS
RESULTING MATRIX
-DF
A MATRIX
0
0
0
0
THE CHARACTERISTIC
SSSSS + A4*SSSS
THE
THE
=
A4
A3
A2
Al
A0
+ P
+CP
+BP
+AP
-C
- R -B
-CR -A
-BR
-AR
COEFFICIENTS
BEFORE
C
C
C
MULTIPLYING BY POWERS OF W
DATA AO/1/
DATA A1/3.236068/
DATA A2/5.236068/
DATA A3/5.236068/
DATA A4/3.236068/
START OF EXECUTABLE CODE --DO 10 11=1,5
DO 10 12=1,5
10 AK(I1,I2)=O
C
C
C
P=E+H
R=E*H-F*G
BUILD MATRIX OBTAINED BY TAKING DETERMINANT OF A-KM AND
ISOLATING THE EXPRESSIONS FOR EACH POWER OF S.
AK(1,3)=1
AK(2,3)=C-P
AK(2,4)=1.
AK( 3,3)=B+R-C*P
AK( 3,4)=C-P
AK(3,5)=1.
AKC4,1)=-D
AK(4,3)=C*R-B*P
AK(4,4)=R-C*P
AK(4,5)=-P
AK(5, 1 )=D*H
AK(5,2)=-D*F
AK(5,3)=B*R
AK(5,4)=C*R
AK (5, 5)= R
C
WRITE(5,20)
20 FORMAT('IENTER
READ(5,30) W
30 FORMATCF10.6)
C
C
MAKE
OMEGA>')
VECTOR WHICH AK*K EQUALS
+ P
-C
2)=A3*W*W
R -B
+C*P 3)=A2*W*DW*W
+B*P -C* R -A
4)=A1*W*W*W*
+A*P -B* R
5)=A0*W*W*W*W *W
-A*.
V( 1)=A4*W
V(
V(
V(
V(
C
CALL MPRINT(AK,5,5,5)
CALL MPRINT(V,5,1,5)
CALL MATINV1(AKAINV,5,DUMMY,DUMZ,IER)
WRITE(5,40) IER
40 FORMAT(' MATINV1 IER=',I5)
CALL DMATMUL(AK,AINV,DUMZ,5,5,5)
CALL MPRINT(DUM2,5,5,5)
CALL DMATMUL(AINV,V,DUMMY,5,5,1)
CALL MPRINT(DUMMY,5,1,5)
END
-
P, I
A.3
SIMULATION PROGRAM
The
±or
all
investigations described
the
simulation
calculates
variables.
It
is
subroutine,
A.3.1
C
of
details
see
DYSYS
of
FLOWMOD modified
in this
derivatives
time
The
thesis.
and
subroutine by.
as a
called
which integrates the
For
is a version
simulation program
auxiliary
DYSYS
115],
derivatives supplied by the subroutine.
the
interface
between
DYSYS
and
its
[15].
Common
Block
EQSIMDECL.FOR
PARAMETER MAX=98
REAL*4 Y(MAX),Y_IHITIAL(MAX),F(MAX),PRNTC(3),PLOTC(3)
REAL*4 CONC(3)
COMMON TIME,TIME_STEP,Y,F,STARTTIME,FIAL_TIE,NEWDT
COMMON IREAD,NSYS,IDUMMY(3),TBREAK,IPLOT_FLAG,TBACK
COMMON CONSTANT(30)
EQUIVALENCE (DT,TIMESTEP),(STARTTIME,STIME)
E2UIVALENCE (FTIME,FINALTIME)
A.3.2
Modified FLOWMOD
FLOWMOD IS A FLOW LIMITED MODEL OF GLUCOSE - INSULIN
C
METABOLISM ORIGINALLY DEVELOPED BY JOHN GUYTON AT HARVORD
C
THE PROGRAM WAS LATER MODIFIED BY ROBERT
C
MEDICAL SCHOOL.
THE PROGRAM AS
C
HILLMAN AT MIT TO PERMIT OGTT SIMULATION.
PROGRAM
HILLMAN'S
ADAPTATION OF
HERE IS AN
C
IT APPEARS
THE NEWEST VERSION
TO ACCOMODATE
WHICH HAS BEEN UPDATED
C
ALSO,
FACILITY.
COMPUTER
JOINT
AT THE MIT
C
OF DYSYS
PROGRAM CONSTANTS FOR THE SIMULATION HAVE BEEN CHANGED TO
C
C
REFLECT THEIR PHYSIOLOGICAL ORIGIN.
C
SEPT 1980
JOHN T. SORENSEN
C
C
C ---------C
*****
CONSTANT DEFINITIONS *****
C
C
DEFINIT101N
C
CONSTANT
C
BLOOD FLOW THROUGH HEART (LITERS/MINUTE)
C
BFH
BLOOD FLOW THROUGH HEAD (LITERS/MIN)
C
BFHD
-
89
-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
BFK
BFL
BFP
BVH
BVHD
BVK
BVL
BVP
CNSU
DEGRTM
DISTM
EOGISF
EOGRHS
FGTMTM
FIKTKT
FILTLT
FIPTPT
FMMPM
FMTGM
FVHD
FVHI
FVHT
FVK
FVL
FVP
GDC
GLNF
GLYBNF
GLYBTM
GLYSN
GLYSTM
GMET
GNEMTM
GNEOF
GPSNF
GTETH
GTETHD
GTETP-
BLOOD FLOW THROUGH KIDNEY (LITERS/MINUTE)
BLOOD FLOW THROUGH LIVER (LITERS/MIM)
BLOOD FLOW THROUGH PERIPHERY (LITERS/MINUTE)
BLOOD VOLUME OF HEART (LITERS)
BLOOD VOLUME OF HEAD (LITERS)
BLOOD VOLUME OF KIDNEY (LITERS)
BLOOD VOLUME OF LIVER (LITERS)
BLOOD VOLUME OF PERIPHERY (LITERS)
RATE OF CENTRAL NERVOUS SYSTEM UPTAKE OF
GLUCOSE (MG/MINUTE)
DELAYED EFFECT OF GLUCOSE ON INSULIN RELEASE
(MINUTES)
RELEASING-HOLDING SITES DISTRIBUTION TIME
(MINUTES)
EFFECT OF OGTT ON INSULIN TRANSFER
EFFECT OF OGTT ON DISTRIBUTION OF
RELEASING-HOLDING SITES
FRACTION OF GLUCOSE METABOLIZED
(FRACTION/MINUTE)
FRACTION OF INSULIN IN KIDNEY PLASMA TO
KIDNEY TISSUE (FRACTION/MINUTE)
FRACTION OF INSULIN I
LIVER PLASMA TO LIVER
TISSUE (FRACTION/MINUTE)
FRACTION OF INSULIN IN PER TISSUE TO PER
PLASMA (FRAC/MINUTE)
FRACTION OF GLUCOSE METABOLITE METABOLIZED
(FRACTION/MIN)
FRACTION OF METABOLITE TO GLUCOSE
(FRACTION/MINUTE)
FLUID VOLUME OF HEAD TISSUE (LITERS)
FLUID VOLUME OF HEART AND HEAD TISSUE(LITERS)
FLUID VOLUME OF HEART TISSUE (LITERS)
FLUID VOLUME OF KIDNEY TISSUE (LITERS)
FLUID VOLUME OF LIVER TISSUE (LITERS)
FLUID VOLUME OF PERIPHERAL TISSUE (LITERS)
GLUCOSE DIFFUSION COEFFICIENT FOR BETA CELL
(MG/MG%/MINUTE)
FASTING LIVER GLYCOGEN STORE (GRAMS)
FASTING RATE OF GLYCOGEN BREAKDOWN (MG/MINUTE)
TRANSPORT DELAY IN ACTION OF GLYCOGEN
BREAKDOWN (MINUTES)
FASTING RATE OF GLYCOGEN SYNTHESIS(MG/MINUTE)
TRANSPORT DELAY IN ACTION OF GLYCOGEN
SYNTHESIS (MINUTES)
FASTING GLUCOSE METABOLITE (MG)
TRANSPORT DELAY IN ACTION OF GLUCONEOGENESIS
(MINUTES)
FASTING RATE OF GLUCONEOGENESIS (MG/MINUTE)
FASTING PERIPHERAL GLUCOSE (MG)
GLUCOSE TRANSCAPILLwtY EQUILIBRIUM TIME OF
HEART (MINUTES)
GLUCOSE TRANSCAPILLARY EQUILIBRIUM TIME OF
HEAD (MINUTES)
GLUCOSE TRANSCAPILLARY EQUILIBRIUM TIME OF
-
90 -
PERIPHERY
(MINUTES)
C
C
HUG
RATE OF
C
HSN
C
IFTSF
FASTING TOTAL HOLDING SITES (MU EQUIVALENTS)
INSULIN TRANSFER FAST TO SLOW POOL FASTING
((MU/MINUT
E)
IGTIME
INSULIN IN
ILSHF
(MINUTES)
FASTING LIVER TISSUE INSULIN (MG)
FASTING PERIPHERAL TISSUE INSULII
c
C
C
C
C
IPSNF
HEART
UPTAKE
GOLGI
OF
GLUCOSE
COMPLEX
TIME
(MG/MINUTE)
CONSTANT
(MU)
C
C
C
C
IRATE
IRRF
IRTM
IVGTT GLUCOSE INFUSION RATE (MG/MINUTE)
FASTING RATE OF INSULIN RELEASE FROM BETA
CELL (MU/MINUTE)
INSULIN RELEASE TIME CONSTANT (MINUTES)
C
ISPF
FASTING
IN SLOW
POOL
(MU)
C
ISTART
IVGTT
STARTING
TIME
(MINUTES)
C
ISTOP
IVGTT INFUSION STOPPING
TIME
(MINUTES)
C
C
ITETH
INSULIN TRANSCAPILLARY
HEART (MINUTES)
C
C
ITETK
INSULIN TRANSCAPILLARY EQUILIBRIUM
KIDNEY (MINUTES)
TIME FOR
C
C
C
C
ITETL
INSULIN TRANSCAPILLARY
LIVER (MINUTES)
INSULIN TRANSCAPILLARY
PERIPHERY (MINUTES)
TIME FOR
C
MAGMTM
ITETP
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
INSULIN
INFUSION
TRANSPORT
UPTAKE
DELAY
EQUILIBRIUM
EQUILIBRIUM
TIME OF
EQUILILIBRIUM
IN ACTION
TIME
FOR
OF PERIPHERAL
(MINUTES)
PITIME
PVH
PVK
PVP
RA
FASTING RATE OF PERIPHERAL GLUCOSE UPTAKE
(MG/MINUTE)
FASTING RATE OF MRNA SYNTHESIS
(MOLECULES/MINUTE)
MRNA TIME CONSTANT (MINUTES)
PLASMA FLOW THROUGH KIDNEY (LITERS/MINUTE)
PLASMA FLOW THROUGH LIVER (LITERS/MINUTE)
PLASMA FLOW THROUGH PERIPHERY (LITERS/MINUTE)
FASTING RATE OF PROINSULIN SYNTHESIS
(MU/MINUTE)
PROINSULIN TIME CONSTANT (MINUTES)
PLASMA VOLUME OF HEART AND HEAD (LITERS)
PLASMA VOLUME OF KIDNEY (LITERS)
PLASMA VOLUME OF PERIPHERY (LITERS)
MAXIMUM RATE OF GUT GLUCOSE ABSORPTION
RBCU
(MG/MINUTE)
RATE OF RED BLOOD
MAGUF
MRNASN
MRTIME
PFK
PFL
PFP
PISNF
C
CELL
UPTAKE
SITE
DECAY
OF
GLUCOSE
(MG/MINUTE)
TIME
C
RHSDTM
RELEASING-HOLDING
C
C
RSN
TBV
FASTING TOTAL RELEASING SITES(MU
TOTAL BLOOD VOLUME (LITERS)
C
TEND
END
C
TVK
TOTAL VOLUME
OF GUT
GLUCOSE
ABSORPTION
OF KIDNEY
(MINUTES)
EQUIVALENTS)
(MINUTES)
(LITERS)
TOTAL VOLUME OF LIVER (LITERS)
C
TVL
C
C------------------------------------------------------------
C
C
-
91
-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
INDEX DEFINITIONS:
DEFINITION
INDEX
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
HEART, LUNG, AND CENTRAL VASCULAR BLOOD
GLUCOSE (MG)
HEART, LUNG, AND CENTRAL VASCULAR TISSUE
GLUCOSE (MG)
HEAD BLOOD GLUCOSE (MG)
HEAD TISSUE GLUCOSE (MG)
KIDNEY BLOOD GLUCOSE (MG)
LIVER BLOOD GLUCOSE (MG)
PERIPHERAL TISSUE GLUCOSE (MG)
PERIPHERAL BLOOD GLUCOSE (MG)
LIVER GLYCOGEN (GRAMS)
GLUCOSE IN BETA CELL (MG)
GLUCOSE METABOLITE (MG)
MRNA (MOLECULES)
PROINSULIN (MU)
INSULIN IN GOLGI COMPLEX (MU)
INSULIN IN SLOW POOL (MU)
HOLDING SITES EMPTY (MU EQUIVALENTS)
HOLDING SITES OCCUPIED (MU)
RELEASING SITES EMPTY (MU EQUIVALENTS)
RELEASING SITES OCCUPIED (MU)
DELAYED EFFECT OF GLUCOSE ON INSULIN RELEASE
HEART, LUNG, CENTRAL VASCULAR, AND HEAD
PLASMA INSULIN (MU)
HEART, LUNG, CENTRAL VASCULAR, AND HEAD
TISSUE INSULIN (MU)
KIDNEY PLASMA INSULIN (MU)
KIDNEY TISSUE INSULIN (MU)
LIVER PLASMA INSULIN (MU)
LIVER TISSUE INSULIN (MU)
PERIPHERAL PLASMA INSULIN (MU)
PERIPHERAL TISSUE INSULIN (MU)
MEDIATOR OF GLYCOGEN BREAKDOWN
MEDIATOR OF GLYCOGEM SYNTHESIS
MEDIATOR OF GLUCONEOGENESIS
MEDIATOR OF PERIPHERAL GLUCOSE UPTAKE
NON-DERIVATIVE
90
89
88
87
86
85
84
83
82
80
VARIABLES
OGTT GUT GLUCOSE ABSORPTION RATE (MG/MINUTE)
IVGTT GLUCOSE INFUSION RATE (MG/MINUTE)
RATE OF KIDNEY GLUCOSE EXCRETION (MG/MINUTE)
EFFECT OF GLYCOGEN LEVELS ON GLYCOGEN
BREAKDOWN
EFFECT OF INSULIN ON MEDLGM
EFFECT OF ARTERIAL GLUCOSE ON MEDLGM
MEDIATOR OF LIVER GLUCOSE METABOLISM <MEDLGM>
EFFECT OF MEDLGM CM GLYCOGEN BREAKDOWN
MEDIATOR OF GLYCOGEN EREAV"DNWN-BEFORE
TRANSPORT DELAY
RATE OF GLYCOGEN ERAKDOWN (MG/MINUTE)
-
92
-
C
C
79
C
C
78
77
EFFECT OF LIVER GLUCOSE CONC.
ON GLYCOGEN
SYNTHESIS
EFFECT OF MEDLGM ON RATE OF GLYCOGEN SYNTHESIS
MEDIATOR OF GLYCOGEN SYNTHESIS-BEFORE
TRANSPORT
C
DELAY
C
C
C
C
75
74
73
RATE OF GLYCOGEN SYNTHESIS (MG/MINUTE)
EFFECT OF MEDLGM ON GLUCONEOGENESIS
MEDIATOR OF GLUCONEOGENESIS-BEFORE TRANSPORT
DELAY
C
71
RATE
C
70
EFFECT OF GLUCOSE ON PERIPHERAL GLUCOSE
C
C
(MG/MINUTE)
UPTAKE
69
EFFECT OF INSULIN ON PERIPHERAL GLUCOSE
68
EFFECT OF ARTERIAL GLUCOSE
C
C
OF GLUCONEOGENESIS
UPTAKE
C
GLUCOSE
ON PERIPHERAL
UPTAKE
C
67
MEDIATOR OF PERIPHERAL GLUCOSE UPTAKE-BEFORE
C
C
65
TRANSPORT DELAY
RATE OF PERIPHERAL
GLUCOSE
UPTAKE
(MG/MINUTE)
C
C
C
C
C
C
C
C
C
C
C
63
62
58
57
56
EFFECT OF GLUCOSE ON MRNA
EFFECT OF GLUCOSE ON INSULIN SLOW TO FAST
POOL
EFFECT OF ARTERIAL GLUCOSE ON INSULIN
TRANSFER SLOW TO FAST POOL
EFFECT OF GLUCOSE ON INSULIN RELEASE
RATE OF INSULIN TRANSFER SLOW TO FAST POOL
(MU/MINUTE)
SITES EMPTY TO SITES OCCUPIED (MU/MINUTE)
SITES HOLDING TO SITES RELEASING (MU/MINUTE)
ERROR FOR RELEASING SITES (MU)
C
55
SITES
HOLDING
(MU/MINUTE)
C
54
SITES HOLDING TO SITES RELEASING
(MU/MINUTE)
C
C
53
TOTAL PANCREATIC INSULIN
(MU/MINUTE)
C
C
C
C
50
49
48
HEART, LUNG, CENTRAL VASCULAR BLOOD GLUCOSE
CONCENTRATION
PERIPHERAL BLOOD GLUCOSE CONCENTRATION(MG/DL)
PERIPHERAL TISSUE GLUCOSE CONCENTRATION
C
47
KIDNEY
C
C
46
45
LIVER (BLOOD) GLUCOSE CONCENTRATION
HEART, LUNG, CENTRAL VASCULAR, HEAD
C
C
C
44
43
INSULIN CONC.
PERIPHERAL PLASMA INSULIN CONCENTRATION(MU/L)
KIDNEY PLASMA INSULIN CONCENTRATION (MU/L)
61
60
59
RELEASING
TO
SITES
RELEASE RATE
(MG/DL)
C
(BLOOD) GLUCOSE CONCENTRATION
(MG/DL)
(MG/DL)
PLASMA
C
42
LIVER PLASMA INSULIN CONCENTRATION (MU/L)
C
C----------------------------------------------------------C
C
GIPORTAL.FOR
C
EXCERPTED:08DEC80
C
C
MASSACHUSETTS INSTITUTE OF
TECHNOLOGY
FROM FLOWMOD.FOR
AT THE
-
93
-
C
C
C
C
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C
C
C
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REVISED:08DEC80
REVISED:15DEC80
REVISED:12APR81
REVISED:28APR81
.REVISED:
BY KEVIN KOCH
KPK REMOVE MOST COMMENTS (DESCRIPTIONS
OF VARIABLES) FOR DEVELOPMENT EASE.
KPK READ PARAMETERS FROM FILE RATHER
THAN FROM DATA STATEMENTS
KPK BASIC FLOWNMOD PLUS OBSERVER
KPK ADD MEASUREMENT SAMPLE-AND-HOLD;
INSULIN PUMP GRANULARITY
KPK ADD INTEGRATOR; FIX PUMP INTERVAL
METHOD
PURPOSE --SIMULATE GLUCOSE-INSULIN METABOLISM/INTERACTION
ADDITIONAL DERIVATIVE VARIABLES --INTEGRAL OF G3 WITH INTEGRATOR LIMITING
10
STATE
I
33
12 STATE
34
G1 STATE
35
G2 STATE
36
G3 STATE'
37
CONTROL OUTPUT INTEGRAL FOR PUMP GRANULARITY
38
SYNTHETIC MEASUREMENT
39
START OF FORTRAN STATEMENTS *
!DEFINE SUBROUTINE CALLED BY DYSYS
SUBROUTINE E2SIM
C
!DYSYS COMMON
INCLUDE 'E2SIMDECL.FOR'
!SIMULATION CONSTANTS
INCLUDE 'PCOM.FOR'
REAL*8 GIAGIB,GIC,GIDP,GIDM,GIAE2,GICEM1,GIDME1
C
C
LOCAL DECLARATIONS --LOGICAL*l FILNAM(40)
REAL IV89TB(2,4),K1,K2,K3,K4,K5
-3.,116 66.67, -. 001,11666.67,
DATA IV89TB/-3.001,0.,
1 0.0,0./
!INSU LIN VARIABLES IN
/.025/
DATA CANIA
!PHp.SE VARIABLE CANONICAL
/.8/
DATA CANIB
!FORM
/.07/
DATA CANZ
DATA CANALPHA /1.76/
!SETP OINTS FOR:LIV TIS INS
/4.7/
DATA SETI26
PER BLO GLU
/80./
DATA SET149
RATE
SECRETION
INS
/21.9/
DATA SETI53
C
!COEFFICIENTS OF GLUCOSE
!SUBSYSTEM REPRESENTATION
/1.888E-4/
/ .03/
.37/
/
/ .0302/
/ .00695/
DATA
DATA
DATA
DATA
DATA
GIA
GIB
GIC
GIDP
GIDM
DATA
DATA
DATA
DATA
K1/-9.43/
K2/ 9.502/
K3/31.1907/
C
!OBSERVER
!OBSERVER
K4/486.763/
-
94 -
GAINS FROM
PROGRAM
DATA K5/4655.57/
C
DATA
DATA
DATA
DATA
DATA
DATA
C
C
C
C
G1 /-12.483/
G2 / 36.727/
G3 / 17.585/
G4 /100.98/
G5 /181.5/
/0/
IPASS
!CONTROL GAINS FROM
!OPTSYS
START OF EXECUTABLE CODE --COMPUTE THE SIMULATION CONSTANTS
IF(NEWDT.NE.-J) GO TO 90
!BEGINNING OF S IM
IF(IPASS.NE.0) GO TO 20
!FIRST SIM
OPEN(UNIT=1,NAME='PCOM.DAT',TYPE='OLD')
READ(1,10) PCOM
!READ CONSTANTS MADE BY PCOM PGM
CLOSE(UNIT=1)
10 FORMAT(8F10.3)
GET INITIAL CONDITIONS
20 OPEN(UNIT=1,AME='GISYSTEM.ICS',TYPE='OLD ')
READ(1,10) (Y(I1),I1=1,39)
CLOSE(UNIT=1)
21=BFHD/BVHD
22=BFL/TVL
Q3=BFK/TVK
24=BFP/BVP
25=1./GTETH
26=-BFH/BVH-FVHT/(BVH*GTETH)
27=-BVH/TBV
28=FVHT/(BVH*GTETH)
29=-1./GTETH
210=BFHD/BVH
211=-BFHD/BVHD-FVHD/(BVHD*GTETHD)
212=1 ./GTETHD
913=-BVHD/TBV
214=FVHD/(BVHD*GTETHD)
215=-1. /GTETHD
216=BFK/BVH
217=-BFK/TVK
218=-BVK/TBV
919=BFL/BVH
220=-BFL/TVL
221=-BVL/TBV
222=FVP/(BVP*GTETP)
223=-i./GTETP
224=BFP/BVH
225=-BFP/BVP-FVP/(BVP*GTETP)
226=-BVP/TBV
227=1 ./GTETP
228=1 ./ITETH
229=PFK/PVK
230=PFL/PVL
231=PFP/PVP
232=(-1./PVH)*(PFP+PFL+PFK+FVHI/ITETH)
233=FVHI/(PVH*ITETH)
-
95
C
C
C
C
C
C
C
234=-i./ITETH
235=PFK/PVH
236=-PFK/PVK-FVK/ ( PVK*ITET )
Q37=1./ITETK
238=FVK/( PVK*ITETK)
239=-i./ITETK-FIKTKT
240=PFL/PVH
241=-PFL/PVL-FVL/(PVL*ITETL)
242=1./ITETL
243=FVL/(PVL*ITETL)
244=-i./ITETL-FILTLT
245=PFP/PVH
246=-PFP/PVP-FVP/ ( PVP*ITETP)
247= 1 ./ITETP
248=FVP/(PVP*ITETP)
249=-i./ITETP-FIPTPT
IVGTT=1
THIS SECTION ONLY EXECUTED FOR OGTT
IF(IPASS.NE.0) GO TO 40
!FIRST SIM
ADJUST SOME OF THE PANCREAS TABLE FUNCTIONS TO REFLECT
THE EFFECT OF GUT ABSORPTION ON INSULIN SECRETION
DO 30 I1=1,NT14
TB14(2,I1)=TB14(2,I1)*EOGISF(Ii)
30 TB16(2,I1)=TB16(2,I1)*EOGRHS(Ii)
40 IPASS=1
GET PARAMETERS FOR THIS PARTICULAR SIMULATION
WRITE(5,50)
50 FORMATC'[ENTER MEASUREMENT DELAY TIME>')
READ(5,60) DELAYTIME
60 FORMATC5F10.4)
Y49HOLD=SET149
Y49DELAYED=SETI49
NDELAYINT=DELAYTIME/DT
NDELAYCOUNT=0
Y53DELAYED=Y(35)
Y(39)=SETI49
WRITE(5,70)
70 FORMAT('[ENTER DT, DH FOR PUMP GRANULARITY>')
READ(5,60) PUMPDTPUMPDH
NPUMPINT=PUrlTDT/DT+1
NPUMPCOUNT=0
WRITE(5,80)
80 FORMAT('IENTER INTEGRATOR GAIN AND SATURATION LIMIT>')
READ(5,60) G6,Y35LIM
Y(39)=Y(49)
!SET INTEGRATOR BASE
TRANSFORM INSULIN SUBSYSTEM COEFFICIENTS FROM CANONICAL
FORM TO ERROR COORDINATE FORM
CANBETA=SETI26*CANIA/ (CANZSETI53)
C11=
-CANIA*CANALPHA/CAEZ
C12=1 -CAXIB*CAIALPHA/CANZ+CAIIA*(CANALPHA**2)/(CANZ**2)
-CANIA
Cz1=
-CANIB -Cli
C22=
V. 1=K 1
-
96 -
90
C
C
K2=K2
CONTINUE
FACTOR=1.0
IF(TIME.GT.40.)
FACTOR=0.0
!ARTIFACT FROM OPEN-LOOP
!OBSERVER TESTS
IF (IVGTT.E2.1) GO TO 100
IF (NEWDT.E2.0) GO TO 100
Y(89)=R(TIME,Y(89),IV89TB,4)
100
!COMPUTE IVGTT
INPUT
CONTINUE
c
(IVGTT.E2.0) GO TO
IF (NEWDT.E2.0) GO TO
RA=800.
TEND= 240.
TO=30.
200
200
IF
!COMPUTE OGTT INPUT
IF
IF
IF
(TIME.LT.3.) GO TO 110
GO TO 130
(TIME.GE.3..AND.TIME.LE.10.)
(TIME.GT.10..AND.TIIME.LE.30.) GO TO 120
IF
(TIME.GT.TEND) GO TO
150
Y(90)=RA*(1.-SIH(((TIME-TO)/(TEND-To))*1.5707963)
110
120
130
140
150
160
170
180
190
200
GO TO 140
Y(90)=0.
GO TO 140
Y(90)=RA
GO TO 140
Y(90)=RA*SIN(((TIME-3.)/(10.-3.))*1.570796)
CONTINUE
GO TO 200
TO=330.
TEND=540.
IF (TIME.LT 303.) GO TO 160
IF (TIME.GE 303..AND.TIME.LE.310.) GO TO 180
IF (TIME.GT 310..AND.TIME.LE.330.) GO TO 170
IF (TIME.GT TEND) GO TO 160
Y(90)=RA*(1 -SIN(((TIME-TO)/(TEND- TO))*1.570796))
GO TO 190
Y(90)=0.
GO TO 190
Y(90)=RA
GO TO 190
Y(90)=RA*SIN(((TIME-30 3.)/(310.-303.))*1.570796)
CONTINUE
CONTINUE
C
IF (NEWDT.E2.0) GO TO 210
GKC=Y(5)/6.
Y(88)=R(GKC,Y(88),TB1,NT1)
Y(87)=R(Y(9),Y(87),TB2,NT2)
ILSN=Y(26)/ILSNF
Y(86)=R(ILSN,Y(86),TB4,NT4)
GLC=Y(1)/22.
Y(85)=R(GLC,Y(85),TB5,NT5)
Y(84)=Y(86)*Y(85)
Y(83)=R(Y(84),Y(83),TB3,NT3)
-
97
!COMPUTE GLUCOSE
!DISTRIBUTION
Y(82)=Y(84)*Y( 83)
Y(80)=Y(87)*Y( 29)*GLYBNF
GLN=Y( 6 )/GLNF
Y(79)=R(GLN,Y( 79),TB6,INT6)
Y(78)=R(Y(84), Y(78),TB7,lt'T7)
210
Y(77 )=Y(79)*Y( 78)
Y(75)=GLYSN.*Y( 30)
Y(74)=R(Y(84), Y(74),TB8,NT8)
Y(73)=Y(84)*Y( 74)
Y(71)=Y(31)*GN EOF
GPSN=Y(7)/GPSN F
Y(70)=R(GPSN,Y (70),TB9,NT9)
IP SN=Y(28)/IPS NF
Y(69)=R(IPSN,Y (69),TB10,NT10)
GHC=Y( 1)/22.
Y(68)=R(GHC,Y( 68),TB11,NT11)
Y(67)=Y(69 )*Y( 68)
7(65)=MAGUF*Y( 70)*Y(32)
CONTINUE
C
F( 1 )=26*Y( 1)+2 5*Y(2)+91*Y(3)+03*7 (5)+92*Y(6)+94*Y(8)+
127*RBCU+Y(89)
F(2)=28*Y(1)+29*Y(2) -HUG
F(3)=0210*Y(1)+211*Y( 3)+O12*Y(4)4+013*RBC
F(4)=214*7(3)+Q15*Y( 4) -CI{SU
F(5)=216*7 (1)+217*Y( 5)+218*RBCU-Y(88)
F(6)=219*Y(1)+220*Y( 6)+221*PBCU+Y (80)-Y (75)+Y(71)+Y(90)
1*FACTOR
7)=222*Y(8)+22 3*Y(7)-Y(65)
8)=224*Y(1)+22 5*Y(8)+226*RBCU+2Z7*Y(7)
9 )=0.001*(Y(75 )-Y(80) )
29)=(Y(82)-Y(2 9))/GLYBTN
30)=(Y(77)-Y(3 0))/GLYSTM
31)=(Y(73)-Y(3 1))/GNENTM
32)=(Y(67)-Y(3 2 )) /rAGHTtl
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
cC
BETA CELL SECTOR
IF (NEDT.E2.0)
GO TO 220
FLOWHOD BETA CELL IS NOT USED; VARIABLES 10-20 AND 53-63
ARE AVAILABLE.
GMIETC=80.*Y(11)/GMET
Y(63)
GMETC, Y( 63) ,TB12 ,NT 12 )
=R(
Y(62)
GIMETC, Y( 62) ,TB 14 ,NT14 )
=R(
Y(61)
GHC,Y(61 ) ,TB 15, N T15)
=R(
GMETC, Y ( 60) ,TB13 ,NT 13)
Y(60)
y(59)
( 15)*Y(6 2) *Y( 20) *IRRF/ISPF+IFTSF)*Y(61)
=R(
Y(58) =1. /(Y( 16)+ Y(18) )*Y (59)
GMETC, Y ( 57) ,TB 16 ,NT16)
Y 57)
Y(56)
( 18)+Y( 1 9))-(Y(1 8)+Y(19)+Y(16)+Y(17))*Y (57)
Y(55) =( 1 ./(Y( 18) +Y( 19))) *ALIMIT(Y(56),0.,1.E37, IST 1,
1NED T )/DI ST M
'Y(54 )=-(1. / (Y (16 )+Y (17))
L M T Y 56),
1 .E37, 0.,IST2,
1NE11DT )/DIST1
7(53)=Y(19)/IRTM
-
I
I
I
98 -
4
C
C
C
Y(53)=Y(53)*FACTOR
SET OUTPUT
PLUS THE
(IN ERROR COORDINATES)
TO CONTROL LAW RESULT
BASAL SECRETION RATE
Y(53)=Y(41)+SETI53
IF(Y(53).LT.0) Y(53)=0
IF(Y(53).GT.300.) Y(53)=300.
CONTINUE
220
!CLAMP IT
C
C
C
F(10)=GDC*GHC-Y(10)*(GDC*100.+FGTMPM)+FMTGM*Y(11)
C
F(12)=MRNASN*Y(63)-Y(12)/MRTIME
C
C
F(13)=Y(12)*PISNF-Y(13)/PITLIME
C
C
C
C
C
C
C
C
C
F(11)=Y(10)*FGTMlPM-Y(11)*(FrTGM+FMIPM)
F(14)=Y(13)/PITIjrE-Y(14)/IGTIME
F(15)=Y(14)/IGTIME-(Y(16)+Y(18))*Y(58)+(Y(17)+
1Y(19))/RHSDTI
F(20)=(Y(60)-Y(20))/DEGRTM
F(17)=Y(16)*Y(58)+Y(19)*Y(55)-Y(17)*(Y(54)+1./RHSDTM)
F(16)=Y(60)*HSI/RHSDTM+Y(18)*Y(55)-Y(16)*(Y(54)+Y(58)+
11./RHSDTM)
F(19)=Y(18)*Y(58)+Y(17)*Y(54)-Y(19)*(Y(55)+1./RHSDTM+
11./IRTM)
F(18)=Y(19)/IRTM+Y(60)*RSN/RHSDTri-Y(18)*(Y(58)+
11./RHSDTM+Y(55))
+Y(16)3Y(54)
C
C
C
C
C
C
C
C
COMPUTE INSULIN DISTRIBUTION
PERIPHERAL
.VS.
PORTAL
DELIVERY
PERIPHERAL
DETERMINED
BY
THE
INFUSION
(THE
PERIPHERAL
VEIN
LEADS
DIRECTLY TO THE HEART COMPARTMENT)
F (2 1) 22 8*Y(22)+229*Y(23)+230*Y(25)+231*Y(27)+232*Y(21)
F (22) 23 3*Y(21)+234*Y(22)
F 23) =23 5*Y(21)+236*Y(23)+237*Y(24)
F (24) =23 8*Y(23)+239*Y (24)
F (25) =24 0*Y(21)+241*Y(25)+242*Y(26)+Y(53)!PORTAL
F (26 )=24 3*Y(25)+244*Y(26)
F (27) =24 5*Y(21)+246*Y(27)+247*Y(28)
F (28) =24 8*Y( 27) +2493:Y(28)
C
AUXILIARY
VALUES
C
C
COMPUTE
C
GLUCOSE DISTRIBUTION IN MG/DL:
IF (NEWDT.E2.0) GO TO 230
Y(50)=Y(1)/22.
Y(49)=Y(8)/24.
Y(48)=Y(7)/70.
Y(47)=Y(5)/6.
C
IS
COMPARTMENT INTO WHICH THE INSULIN IS ADDED.
LIVER PLASMA=>PORTAL INFUSION; HEART PLASIA=>
Y(46)=Y(6)/20.
INSULIN DISTRIBUTION IN MU/L:
Y(45)=Y(21)/1.44
Y(44)=Y(27)/1.44
Y(43)=Y(23)/.24
-
99 -
Y(42)=Y(25)/.48
C
230 CONTINUE
C
C
C
C
C
C
C
C
C
MEASUREMENT DELAY
IF(NEWDT.E2.0) GO TO 250
IF(DELAYTIME.GT.0) GO TO 240
Y49DELAYED=Y(49)
GO TO 250
240 NDELAYCOUNT=MOD(NDELAYCOUNT+1,NDELAYINT)
TIME TO PROPOGATE THE MEASUREMENT
IF(NDELAYCOUNT.GT.0) GO TO 250
Y49DELAYED=Y49HOLD
Y49HOLD=Y(49)
Y35DELAYED=Y35HOLD
Y35HOLD=Y(35)
DY39=(Y49DELAYED-Y(39))/NDELAYINT
FULL ORDER GLUCOSE-INSULIN OBSERVER
250 CONTINUE
GID=GIDM
!POSITIVE/NEGATIVE GAMMA
IF(Y(33).GT.0) GID=GIDP
F(33)=C11*Y(33) +C12*Y(34) +CANALPHA*CANBETA*Y(53)
1 +C11*SETI26
F(34)=C21*Y(33) +C22*Y(34) +CANZ*CANBETA*Y(53)
1 +C21*SETI26
F(35)=
+Y(36)
F(36)=
Y( 37)
F(37)= -GID*Y(33)
-GIA*Y( 35)-GIB*Y(36)-GIC*Y(37)
F(37)=F(37)-GIA*SETI49
Y(40)=Y(35)+SETI49
INTEGRATOR
Y35=Y(35)
!GET GLUCOSE ERROR STATE
Y35=AMIN1(Y35, Y35LIM)
!SATURATION LIMIT
"T"v
Y35=AMAX1(Y35,-Y35LIM)
F(10)=Y35
!DO THE INTEGRATION
MEASUREMENT AND MEASUREMENT E RROR
IF(DELAYTIME.EQ.0) Y(39)=Y (49)
!IF NO MEAS DELAY
IF(NEWDT.EQ.0) GO TO 260
Y(39)=Y(39)+DY39
!SYNTHETIC MEASUREMENT
260 F(33)=F(33)-K1*Z*FACTOR
F(34)=F(34)-92*Z*FACTOR
F(35)=F(35)-K3*Z*FACTOR
F(36)=F(36)-K4*Z*FACTOR
F(37)=F(37)-K5*Z*FACTOR
CONTROL LAW:
IFCNEWDT.EQ.0) RETURN
Y41=G1*Y(33)+G2*Y(34)+G3*Y (35)+G4*Y(36)+G5*Y(37)
1+G6*Y(10)
IF(PUMPDH.GT.0) GO TO 270 !PUNP CONSTRAINTS
Y(41)=Y41
!NO PUMP CcNST; SET ACTUATOR SIGNAL
RETURN
:NTEGRATE CONTROL ACTION AND WAIT FOR IT TO BE BIG ENOUGH
TO TURN THE PUMP ON
270 F(38)=Y41
-
100
IF(NEWDT.E2.0)
RETURN
NFUMPCOUNT=MOD(NPUMPCOUNT+1
IF(NPUMPCOUNT.GT.0) RETURN
,NPUMPINT)
Y( 41)=IFIX(Y(38)/(PUMPDT*PUt1PDH))
!MULTIPLY BY PUMPING INCREMENT
Y(41)=Y(41)*PUNPDH
!CORRECT CONTROL INTEGRAL
Y(38)=Y(38)-Y(41)*PUMPDT
RETURN
END
FUNCTION R(X,Z,TAB,NT)
C
C
C
PURPOSE --LINEAR INTERPOLATION
FUNCTION
C
INPUT
C
C
C
C,
C
C
C
C
---
TAB(2,NT)?
COORDINATES
X
C
VARIABLE
INDEPENDENT
VALUE
INTERPOLATED
SAME
OF
FUNCTION
AS R
TAB(2,NT)
OF EXECUTABLE CODE
START
INPUT
---
OUTPUT
z
R
REAL
C
C
C
(INDEPENDENT,DEPENDENT)
OF
ORDERED PAIRS
---
IF(X.GT.TAB(1,1)) GO TO 10
Z=TAB(2,1)
GO TO 50
10 IF(X.GE.TAB(1,NT)) GO TO 30
SCAN FOR X<TAB(1,I1)
DO 20 I1=2,NT
GO
IF(X.LT.TAB(1,I1))
40
TO
20 CONTINUE
30 Z=TAB(Z,NT)
GOTO 50
40 DX=TAB(-1,I1)-TAB(1,Il-1)
DY=TAB(2,I1)-TAB(2,Il-1)
Z=TAB(2,Il-1)+(DY/DX)*(X-TAB(1,Il-1))
50
R=Z
RETURN
END
-
101
-
REFERENCES
1.
ACCESS User's Guide, Joint Computer Facility, MIT,
Cambridge.
2.
Albisser, A. M., Botz, C. K., Leibel, B. S., "Blood
Glucose Regulation Using an Open-Loop Insulin
Delivery System in Pancreatomized Dogs Given Glucose
Infusions", Diabetologia, Vol. 16, 1979.
3.
Albisser, A. M., et alii, "Studies With An Artificial
Endocrine Pancreas", Archives of Internal Medicine,
Vol. 137, May 1977.
4.
Bale, Gurunanjappa S., Entmacher, Paul S., "Estimated
Life Expectancy of Diabetics", Diabetes, Vol. 26 No.
5, May 1977.
5.
Bergman, R. N., Bucolo, R. J., "Nonlinear Metabolic
Dynamics of the Pancreas and Liver", Journal of
Dynamic Systems, Measurement, and Control, Sept.
1973.
6.
Bergman, Richard N., Refai, Mahmoud El, "Dynamic
Studies by
Control of Glucose Metabolism:
Experiment and Computer Simulation", Annals of
Biomedical Engineerinq, Vol. 3, 1975.
7.
Botz, Charles K, "An Improved Control Algorithm for an
Artificial iS Cell", IEFE Transactions on Biomedical
Enctineerinci, Vol. BME-23 No. 3, May, 1976.
8.
Carson, E. R., Cramp, Derek G., "A Systems Model of
Blood Glucose Control", International Journal of
Biomedical Computina, Vol. 7, 1976.
9.
Carson, E. R., et al., "Mathematical Modelling of
of
Metabolic and Endocrine Systems", Institute
1980.
July
U.1K.,
Measurement and Control, Guilford,
10.
"Development of an
H. et alii,
Clemens, A.
Electrochemical Glucose Analyzer and New Algorithms
for a Glucose Controlled Insulin Infusion System
(Artificial S Cell)", Proceedings, 9th Congress of
Diabetes Federation, Mew Delhi,
the International
1976. Excerpta Nedica, Princeton, 1976.
11.
Cobelli, Claudio, Romanin-Jacur, Giorgio,
"Controllability, Observability and Structural
Identifiability of Multi Input and Multi Output
Biological Compartmental Systems", IEEE Transactions
on Biomedical Engineering, Vol. BME-23 No. 2, Mar.
1976.
12.
Colton, Clark K., Giner, Jose, Lerner, Harry, Marincic,
Ljiljana, Soeldner, J. S., "Development of an
Implantable Electrochemical Glucose Sensor",
Vol.
Transplantation and Clinical Immunology,
Lyon, May 1978.
10,
13.
Creque, Halimena M., Langer, Robert, Folkman, Judah,
"One Month of Sustained Release of Insulin from a
Polymer Implant", Diabetes, Vol. 29, Jan. 1980.
14.
Dorf, Richard C., Modern Control Systems,
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