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Control strategies for the regulation of blood glucose

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Control strategies for the regulation of blood glucose

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CONTROL STRATEGIES FOR THE
REGULATION OF BLOOD GLUCOSE

by

Michail G. Markakis

A Thesis Presented to the
FACULTY OF THE USC VITERBI SCHOOL OF ENGINEERING
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(ELECTRICAL ENGINEERING)

May 2008

Copyright 2008 Michail G. Markakis
ii
Acknowledgements

I would like to thank a number of people that contributed in this thesis, in one way or another.
First of all my academic adviser and mentor, Dr. Vasilis Marmarelis: I consider myself very
fortunate that I had the chance to work with him since my first steps in research. His experience,
enthusiasm, guidance and support were the foundations on which this thesis was built. I wish
our collaboration will be continued, even though our paths may split for a while… I would also
like to express my gratitude to the “control’s people” here at USC: Dr. Petros Ioannou that
believed in me and played an important role for me coming here. His analytical thinking and
professionalism have been a big influence for me; and Dr. Michael Safonov that introduced me to
the concepts of optimality and robustness, which became eventually the center of my research
interests. Being his student and teaching assistant has been a great benefit and pleasure. Last but
not least, I would like to thank my former adviser, Dr. George Papavassilopoulos, for guiding my
first steps in academia and for remaining my academic role-model ever since.
I am also greatly indebted to the Myronis Foundation that generously provided me a graduate
scholarship for the academic year 2007-2008, when this thesis was written.
Above all I need to thank my family, whose continuous love and support have always been
my major strength and inspiration.

iii
Table of Contents
Acknowledgements ii
List of Tables iv
List of Figures v
Abbreviations vi
Abstract vii
Chapter 1: Introduction 1
Chapter 2: Insulin-Glucose Dynamics 5
Chapter 3: Closed-Loop System 11
Glucose Disturbance 13
Glucose Measurement 16
Chapter 4: Proportional-Derivative Control 17
Chapter 5: Model Predictive Control 20
Hybrid Control Strategy 23
Switching Control Strategy 24
Chapter 6: Attenuation of Meal Disturbances 26
Attenuation of Stochastic Disturbances 27
Insulin Dependent Diabetes Mellitus 32
Mild Diabetic / ICU Case 33
Robustness to Variations in Endogenous Insulin Production 35
Chapter 7: Conclusions 36
Bibliography 39
Appendix A: Sorensen’s Model of Glucose Metabolism 44
Appendix B: Proof of Stability of the Augmented Minimal Model 53

iv
List of Tables
Table 1: Average behavior of control strategies for nominal diabetic patients 32
Table 2: Average behavior of control strategies for IDDM patients 33
Table 3: Average behavior of control strategies for mild diabetic / ICU patients 34
Table 4: Average behavior of control strategies for uncertain endogenous insulin production 35
Table 5: Constants in Sorensen’s model of glucose metabolism 51

v
List of Figures
Figure 1: Schematic of Closed-Loop Vs. Open-Loop 8
Identification of the Augmented Minimal Model
Figure 2: Standard Responses of the Augmented Minimal Model 10
Figure 3: Schematic of the closed-loop control system for blood glucose regulation 12
Figure 4: Stochastic Glucose Disturbance 15
Figure 5: Proportional Vs. Proportional-Derivative Control 19
Figure 6: Schematic of the Model Predictive Controller used in this study 21
Figure 7: Estimation of Meal Disturbances 22
Figure 8: Attenuation of Meal Disturbances 27
Figure 9: Attenuation of Stochastic Glucose Disturbances with PDC 29
Figure 10: Attenuation of Stochastic Glucose Disturbances with HCS 30
Figure 11: Attenuation of Stochastic Glucose Disturbances with SCS 31
Figure 12: Sorensen’s model of glucose metabolism 52

vi
Abbreviations
IDDM: Insulin Dependent Diabetes Mellitus (Type I diabetes)
NIDDM: Non-Insulin Dependent Diabetes Mellitus (Type II diabetes)
IGT: Impaired Glucose Tolerance
ICU: Intensive Care Unit
DMM: Diabetic Minimal Model
AMM: Augmented Minimal Model
OGTT: Oral Glucose Tolerance Test
PDC: Proportional Derivative Control
MPC: Model Predictive Control
HCS: Hybrid Control Strategy
SCS: Switching Control Strategy

vii
Abstract
In this computational study, we develop two nonlinear control algorithms (termed the
“Hybrid Control Strategy” and the “Switching Control Strategy”) and evaluate their performance
relative to the widely studied Proportional-Derivative Controller for the regulation of blood
glucose with intravenous insulin infusions in diabetic patients. For the requisite simulations of
the causal relationship between infused insulin and blood glucose we introduce a new model
(termed the “Augmented Minimal Model”) and a glucose “rate disturbance” signal that
represents the aggregate effects of the various neuro-hormonal, behavioral and metabolic factors
affecting blood glucose. The obtained results show that the Switching Control Strategy can
regulate blood glucose better than the other two controllers (based on widely accepted metrics of
performance), while eliminating the risk of inadvertent hypoglycaemia and remaining
remarkably robust in the event of unknown variations in critical parameters of the model.
However, the other two controllers exhibit a satisfactory performance too.

1
Chapter 1
Introduction
Diabetes represents a major threat to public health with alarmingly rising trends of incidence
and severity in recent years, with numerous detrimental consequences for public health. The
mean level of concentration of blood glucose in normal human subjects is about 90 mg/dl and the
zone from 70 mg/dl to 110 mg/dl is usually defined as the desired state of “normoglycaemia”.
Significant and prolonged deviations from this zone may give rise to numerous pathologies with
serious and extensive clinical impact that is increasingly recognized by current medical practice.
When blood glucose concentration falls under 60 mg/dl, we have the acute and very dangerous
state of hypoglycaemia that may lead to brain damage or even death if prolonged. On the other
hand, when blood glucose concentration rises above 120 mg/dl for prolonged periods of time, we
are faced with the detrimental state of hyperglycaemia that may cause a host of long-term health
problems (e.g. neuropathies, kidney failure, loss of vision etc.). The severity of the latter clinical
effects is increasingly recognized as medical science advances and diabetes is revealed as a major
lurking threat to public health with long-term repercussions. Prolonged hyperglycaemia is
usually caused by defects in insulin production (Insulin Dependent Diabetes Mellitus - IDDM),
insulin action (Non-Insulin Dependent Diabetes Mellitus - NIDDM) or other, more vague
genetically-based reasons (Impaired Glucose Tolerance - IGT). The goal of this paper is to
examine the efficacy of various control strategies for glucose regulation in diabetic patients by
use of appropriate intravenous insulin infusions at judiciously chosen times.
Numerous studies have been conducted over the last 40 years to examine the feasibility of
continuous control of blood glucose concentration with insulin infusions, starting with the
2
visionary works of Kadish in 1964 [27], Pfeiffer et al. on the “artificial beta cell” in 1974 [38], of
Albisser et al. on the “artificial pancreas” in 1974 [1] and of Clemens et al. on the “biostator” in
1977 [10]. These studies have had the common objective of regulating blood glucose levels in
diabetics with appropriate insulin infusions, with the ultimate goal of an automated closed-loop
glucose regulation (the “artificial pancreas”). Due to the inevitable difficulties introduced by the
complexity of the problem and the limitations of proper instrumentation or methodology, the
original grand goal of the “artificial pancreas” gradually yielded its place to the more modest
goal of “diabetes management” [2, 13, 22, 41, 42]. Much effort has been focused on advances in
the development of better mathematical and computational models of the relationship between
infused insulin and blood glucose [3, 11, 26, 45, 51, 55] that are critical for this purpose and enable
the prospect of closed-loop control or man-in-the-loop control with partial subject participation
(e.g. meal announcements) [5, 7, 12, 18, 21, 23, 43]. However, no such model has emerged yet that
is accepted by the peer community as universally valid or capable of describing the insulin-
glucose dynamics with the requisite accuracy for closed-loop control under all realistic conditions
of interest. In parallel, there have been many studies of control strategies suitable for the problem
of on-line regulation of blood glucose, starting with relatively simple linear control methods [8,
16, 24, 40] and extending through time to more sophisticated approaches including optimal
control [9, 19, 34, 47], adaptive control [6, 17], robust control [28, 37], run-to-run control [35] or
artificial neural networks [39, 50]. The majority of publications on the glucose regulation problem
to date have concentrated on applying Proportional-Derivative Control (PDC) or Model
Predictive Control (MPC) [14, 25, 32, 33, 36, 41, 54], due to their relative simplicity.
In spite of the considerable effort and resources that have been dedicated to this task to date,
the results have been mixed. Although many of these studies have contributed significantly to
3
our better understanding of the problem, no method or approach has been demonstrated yet to
produce an effective solution with potential clinical utility and wide applicability. In our opinion,
this is due primarily to the following reasons:
• Most of the computational studies of glucose regulation/control restrict themselves in the
Insulin Dependent Diabetes Mellitus (IDDM) case and therefore employ the Diabetic Minimal
Model (DMM) [18, 20] to represent the real system in their simulations. Very briefly, the DMM is
described by the following set of differential equations:

4
( )
( )
ex
ex
I
dI U t
p I t
dt V
=− +
2 3
( ) ( )
ex
dX
p X t pI t
dt
=− +
1
( ) ( )[ ( )] ( )
b
dG
pG t X t G G t D t
dt
=− − + +
( ) ( )
b
Y t G G t = +

where ( ) U t stands for the rate of exogenously infused insulin (input) and ( ) Y t for the value of
blood glucose (output). The DMM is an elaboration of the “minimal model” [3], which has gained
wide acceptability because of its simple structure and its ability to capture some of the basic
dynamics of the insulin-glucose system. However, closer examination of this model indicates that
certain major functional aspects of the system are not represented (e.g. the processes of
endogenous insulin and glucagon production) and consequently the DMM is not suitable for
cases other than the IDDM. The evidence to date suggests that the functional characteristics of the
4
glucose metabolism system in healthy and most of the diabetic subjects (i.e. the causal effect of
intravenously injected insulin on blood glucose concentration) are far more complex. Obviously,
the difference between the simulated and the real system is expected to affect the performance of
the various controllers and their ability to regulate blood glucose in practice. In order to
(partially) redress this problem, the present thesis introduces an “Augmented Minimal Model”
(AMM) that extends the DMM to include the process of endogenous insulin secretion from the
pancreatic beta cells.
• Most of the computational studies to date check a controller’s ability to regulate blood
glucose in the event of a glucose “disturbance” caused by a single meal or a sequence of meals
[14, 35]. However, the form of the actual glucose “disturbance” that the controller must face is
likely to be more complicated, since it will include the aggregate insulin-independent effects of
the endocrine system (e.g. systemic secretions of glucagon, cortisol, epinephrine, norepinefrine
etc.) and of many external factors (e.g. exercise, stress, mental activity etc.) on blood glucose
concentration. Although the exact quantitative effect of these numerous factors on blood glucose
concentration remains poorly understood, their significant qualitative effect is an indisputable
fact. In the context of this study, we choose to represent the aggregate effects of all these factors
(for modeling purposes) as a stochastic (and unknown) disturbance on the rate of change of
blood glucose concentration with a postulated form that reflects our best understanding of the
respective mechanisms at the present time. The postulated form of the stochastic disturbances
must be confirmed with experimental observations in the future. In the context of this thesis, it
simply offers a plausible and challenging paradigm for testing various control strategies.
5
Chapter 2
Insulin – Glucose Dynamics
As mentioned above, most of the computational studies of glucose regulation in diabetic
patients adopt the DMM. The implicit assumption of this model is that no endogenous insulin is
produced and secreted in plasma. Although this is true for IDDM cases, it is certainly not true for
other diabetic patients (NIDDM – IGT) or generally people in need of continuous insulin therapy
(e.g. patients in the ICU – see [53]). In order to incorporate this important fact in the model, the
DMM is augmented with an additional differential equation describing the endogenous insulin
secretion dynamics. Of several equations that have been proposed for modeling insulin secretion
[46, 48, 49], we select the one that utilizes a threshold function – see Equation (5) of [48]. The
resulting AMM operates in “closed-loop”, through the feedback exerted by the endogenously
secreted insulin, and is described by the following set of ordinary differential equations:
4
( )
( )
ex
ex
I
dI U t
p I t
dt V
=− +
( ) max[ ( ) ,0]
en
en b
dI
I t G G t
dt
α β θ =− ⋅ + ⋅ + −
2 3
( ) [ ( ) ( )]
en ex
dX
p X t p I t I t
dt
=− + +
1
( ) ( )[ ( )] ( )
b
dG
pG t X t G G t D t
dt
=− − + +
( ) ( ) ( )
b
Y t G G t N t = + +
where the implicated variables are:
G : the deviation of blood glucose concentration from its basal value of 90
b
G = mg/dl
6
X : the internal variable of “insulin action” (min
-1
)
D : the stochastic glucose rate disturbance caused by internal and external factors (mg/dl/min)
U : the rate of intravenously infused insulin (deviation from the basal rate of
4
b
b I ex
u p V I = ⋅ ⋅
mg/dl/min)
ex
I : the concentration of plasma insulin due to exogenous infusion (deviation from the basal
value which is denoted as
b
ex
I mU/L)
en
I : the concentration of plasma insulin due to endogenous production and secretion (deviation
from the basal value which is denoted as
b
en
I mU/L)
Y : the continuous glucose measurements (in mg/dL)
N : the measurement noise/errors (in mg/dL)
It is evident that the incorporation of endogenous insulin secretion gives the AMM far greater
modeling capabilities. A note on the basal values of exogenous and endogenous insulin: in
healthy subjects the basal value of plasma insulin is 15 mU/L and comes from endogenous insulin
secretion only. In contrast, in IDDM cases this basal value of plasma insulin must come solely
from exogenous insulin secretion (in IDDM the function of endogenous insulin production and
secretion is completely disabled). However, in all other cases the basal value of endogenous
insulin is between 0 and 15 mU/L, so the rest has to be complemented by exogenous insulin (until
their sum reaches the normal value of 15 mU/L). We assume in this computational study that
b
en
I
is a linear function of parameterβ , normalized by its nominal value in healthy subjects
0
β ( =
0.085 as we will see later on). The validity of this assumption has to be tested but plays no role in
the design and application of our control strategies.
7

0
15
b
en
I
β
β
=
15
b b
ex en
I I = −

To estimate the parameters of the AMM, input-output data and a “fitting” procedure of the
model to that data are needed. Since real data are very difficult to obtain we generate data sets
using the comprehensive model proposed in John Sorensen’s PhD Thesis in 1985 [45]. Sorensen’s
model of glucose metabolism is probably the most accurate model existing to date and is used as
the “real system” in many computational studies. An overview of this model can be found in the
thesis’ Appendix A. Since the AMM operates in “closed-loop”, we utilize the technique of
artificially “opening-the-loop”, which determines that every loop of the system (feed-forward or
feedback) has to be identified separately. A schematic of the “opening-the-loop” methodology
can be seen in Figure 1. All input signals that we use during the system identification procedure
are broadband (Gaussian White Noise) with carefully selected dynamic range, in order to excite
all the dynamics of each open loop.

8
Figure 1: Schematic of Closed-Loop Vs. Open-Loop identification of the Augmented Minimal Model

The procedure above resulted in the following values for the parameters of our AMM:
1
0.016 p = min
-1

2
0.06 p = min
-1

5
3
1.2 10 p
−
= ⋅ L/ min
2
mU
90 θ = mg/dl
4
0.26 p = min
-1

9
3.1
I
V = L
As for the parameters that determine the production of endogenous insulin they were found
to be: 0.32 α = min
-1
, 0.085 β = (mU/L) (mg/dl)
-1
min
-1
. However, Sorensen’s model (with the
parameter values given in his PhD Thesis) corresponds to a healthy and not a diabetic subject, so
some of the parameters of the AMM need to be properly modified. According to Sorensen, the
primary physiological difference between diabetic and healthy subjects is the delayed and
attenuated pancreatic responsiveness to hyperglycaemia. In the context of the AMM this means
reducing the value of parameters α and β , so the following values were used as nominal:

0.16 α = min
-1

0.0085 β = (mU/L) (mg/dl)
-1
min
-1

We can gain some insight into the model’s dynamics by looking at the impulse and step
response of the AMM in Figure 2. When the parameter β is set to zero, then the AMM becomes
the DMM (after the effect of initial conditions ceases). Note that a closed-loop model similar to
the AMM has been used recently for glucose regulation studies in the ICU [54] with some
promising clinical results.

10
Figure 2: Standard Responses of the Augmented Minimal Model
0 200 400 600
89.986
89.988
89.99
89.992
89.994
89.996
89.998
90
Impulse Response
mg/dl
Time (min)
0 200 400 600
88.5
89
89.5
90
Step Response
Time (min)

Figure Description: Transient responses of the Augmented Minimal Model to an impulsive insulin
input (left panel) and to a step insulin input (right panel). The peak time of the system is approximately 35
min and the settling time 400 min. Disturbance and noise are assumed to be zero.
11
Chapter 3
Closed - Loop System
In this computational study we assume that the closed-loop system for blood glucose
regulation has the general structure of the block diagram shown in Figure 3, where the additional
implicated variables are:
R : the reference level, which is the target value for the controlled blood glucose (in mg/dl)
V : the measured error signal, i.e. the difference of Y from the constant level R (in mg/dl)
T : the sampling interval determined by the glucose sensor (in min)
The insulin – glucose AMM developed above plays the role of the real system in our
simulations. The glucose sensor generates simulated samples of the output Y with a sampling
interval T (discretized output of the AMM), upon which measurement noise N is added to
emulate a realistic situation. A digital controller is used to compute the control input U to the
system based on the measured error signal V . The use of an insulin micropump is simulated
with a zero-order hold (digital-to-analog conversion) and the imposition of certain constraints
that U has to satisfy in practice, i.e. to be non-negative and stay below 150 mU/min (following
the “conservative” approach of [37]).

12
Figure 3: Schematic of the closed-loop control system for blood glucose regulation

Theorem: The origin is an asymptotically stable equilibrium point of the Augmented Minimal
Model.
Proof: the proof can be found in Appendix B.
It is obvious that the objective of the digital controller of Figure 3 is not to stabilize the system
since it is already pretty stable (a physiological system is expected to be stable, at least in the
Bounded Input – Bounded Output sense). In contrast the objective is to attenuate the effects of the
disturbance signal ( ) D t and keep the error signal ( ) V t within bounds, defined as the
normoglycaemic region. Usually the targeted mean value of blood glucose concentration is set
equal to the basal value
b
G of 90 mg/dl, and a conservative definition of the normoglycaemic
region is from 70 to 110 mg/dl – i.e. | ( ) | 20 V t ≤ mg/dl. Since it is easier to derive the optimal
control strategy by minimizing the mean-square error, we may define our control objective as
achieving a Gaussian distribution of errors ( ) V t with zero mean and standard deviation smaller
than 8 mg/dl.
13
Glucose Disturbance
A critical set of assumptions in this computational study concerns the design of the glucose
rate disturbance signal ( ) D t that is used in the simulations. These assumptions must be based on
our best understanding of the physiological factors that affect the blood glucose concentration,
independently of the infused insulin. For this purpose, we have postulated various deterministic
and stochastic components that seek to capture the main factors influencing the blood glucose
concentration. Specifically, the following additive terms have been incorporated into the
disturbance signal ( ) D t :
• Terms of the exponential form exp( 0.05 ) t γ ⋅ − ⋅ , which represent the standard Fisher meals
[18]. Depending on the meal, the parameter γ takes values within the range [0.63 , 1.04] for
breakfast (corresponding to 15 - 25g of an OGTT), within the range [1.67 , 2.51] for lunch
(corresponding to 40 - 60g of an OGTT), and within the range [1.25 , 2.09] for dinner
(corresponding to 30 - 50g of an OGTT). The values of γ are selected randomly from a uniform
distribution defined over the respective range of values for each meal. The timing of each meal is
also selected randomly within a two-hour interval specified for breakfast, lunch and dinner. The
effect of this rate disturbance on glucose concentration has the form of a negative gamma-like
function that resembles the convolution of this exponential form with the exponential function
1
exp( ) p t ⋅ . This approximation improves as the parameter
3
p decreases and diminishes the
effect of the bilinear term in the glucose-rate equation. The peak-time of this gamma-like curve is
around 30 minutes and its peak amplitude varies from 70 to 285 mg/dl.
• Terms of the exponential form exp( 0.08 ) t δ ⋅ − ⋅ , which represent small-scale neurohormonal
effects caused by stress, mental exertion and other external factors affecting the nervous system.
14
The appearance of these terms follows a Poisson distribution with parameter 0.033 λ = min
-1
.
The parameter δ is uniformly distributed within the range [-0.06 , 0.12] in mg/dl/min. The effect
of this rate disturbance term on glucose concentration has the form of a gamma-like function that
resembles the convolution of this exponential form with the exponential function
1
exp( ) p t ⋅ .
The peak-time of this gamma-like curve is approximately 25 minutes and its peak amplitude
varies from -5 to 10 mg/dl.
• Terms of the exponential form exp( 0.015 ) t ε⋅ − ⋅ , which represent larger random effects due
to factors such as exercise or strong emotions. The appearance of these terms follows a Poisson
distribution with parameter 0.016 λ = min
-1
. The parameter ε is uniformly distributed within
the range [-0.042 , 0.087] in mg/dl/min. The effect of this rate disturbance term on glucose
concentration has the form of a negative gamma-like function that resembles the convolution of
this exponential form with the exponential function
1
exp( ) p t ⋅ . The peak-time of this gamma-
like curve is approximately 60 minutes and its peak amplitude varies from -10 to 20 mg/dl.
• Two sinusoidal terms of the form sin( )
i i i
t α ω φ ⋅ ⋅ + with specified amplitudes and
frequencies (
i
α and
i
ω ) and random phase
i
φ uniformly distributed within the range [-π/2 ,
π/2]. These terms represent circadian rhythms [30, 52] (endocrine cycles) with periods 8 and 24
hours and amplitudes around 10 mg/dl. The effect of this rate disturbance terms on glucose
concentration is periodic (with minor harmonics generated by the bilinear term) and their
combined amplitude ranges up to 20 mg/dl depending on the randomly selected phases and the
secretion of endogenous insulin.
• A constant term B which is uniformly distributed within the range [0.035 , 0.107] and
represents a random bias of the subject-specific basal glucose from the nominal value of 90
b
G =
15
mg/dl. The effect of this rate disturbance bias is a steady-state offset of the basal glucose value
from 90 mg/dl that is uniformly distributed from 20 to 60 mg/dl.
Specific references regarding the qualitative aspects of the terms included in our rate
disturbance signal can be found dispersed in the diabetes literature and some basic aspects in
physiology textbooks. Our goal in this paper is to model the disturbance in a way that is
consistent with the accumulated qualitative knowledge to date in a realistic context and similar to
actual observations in clinical trials (e.g. see the patterns of glucose fluctuations shown in [7, 15,
25]). An illustrative example of the combined effect of these rate disturbance factors on glucose
fluctuations is shown in Figure 4.
Figure 4: Stochastic Glucose Disturbance
0 500 1000 1500 2000 2500
-0.5
0
0.5
1
1.5
2
2.5
mg/dl/min
Stochastic Glucose Disturbance D(t)
0 500 1000 1500 2000 2500
0
100
200
300
400
mg/dl
Corresponding Blood Glucose Y(t)
Time (min)

Figure Description: The top panel shows a typical pattern of the glucose rate disturbance signal
employed in this study. The bottom panel shows the corresponding blood glucose fluctuations due to this
rate disturbance. The large “impulsive” events are caused by meals.

16
Glucose Measurements
Several studies have shown that most existing glucose sensors tend to underestimate the
extreme hyperglycaemic and hypoglycaemic events, as well as introduce a short time lag. A
deviation of less than 15% from the venous blood measurements is usually acceptable. It has also
been observed that successive measurement errors are not statistically independent (i.e. the
glucose measurement error is not a white-noise signal), although they follow roughly a Gaussian
distribution. Therefore we construct a measurement noise signal for our simulations using a low-
pass filtered Gaussian white-noise measurement signal (added to the blood glucose signal) that
has a bandwidth of approximately 0.45 min
-1
and a variation coefficient of 6% (which
corresponds to an approximate range of measurement errors of ±12%), consistent with most
observations. For a thorough review of the existing commercial glucose sensors and their tested
performance see [29].
17
Chapter 4
Proportional-Derivative Control
One of the simplest forms of control action is through a linear Proportional-Derivative
Controller (PDC) that has been discussed extensively in connection with regulating blood
glucose, e.g. see [24]. For this reason, we use it as a reference control strategy against which we
can evaluate the performance of the later proposed Hybrid Control Strategy (HCS) and Switching
Control Strategy (SCS). The discrete-time PDC utilizes a control signal of the form:

( ) ( 1)
( ) ( )
p d
V n V n
U n K V n K
T
− −
=− −

where n denotes the discrete-time index for samples obtained every 3 T = min, and the values of
the controller parameters used in our simulations are 5
p
K = (mU dl)/(mg min) and 100
d
K =
(mU dl)/mg. The control input is clipped when it exceeds the imposed lower and upper bounds
of 0 and 150 mU/min respectively (including the basal value). These values of
p
K and
d
K
correspond to the best PD controller as determined by successive trials, using the joint criterion of
attenuation of the effects of meal disturbances and avoidance of hypoglycaemic events. The z-
transfer function of this PDC for the selected parameter values is:

38.33 33.33
( )
z
K z
z
− +
=

18
In order to examine the limits of performance that PDC can achieve in theory, we linearize the
given nonlinear model around the origin and we discretize it with sampling period of 3 T = min.
This yields the following z-transfer function for the linearized system:

2
3 2
( ) 0.00123 0.003863 0.0007436
( )
( ) 2.247 1.616 0.3649
G z z z
H z
H z z z z
− − −
= =
− + −

The closed-loop sensitivity transfer function ( ) S z is defined as:

1
( ) [1 ( ) ( )] S z K z H z
−
= +

In order to achieve good controller performance (i.e. disturbance attenuation), the magnitude
of ( ) S z must be small within the frequency range where the disturbance signal has significant
power. Figure 5 shows the closed-loop sensitivity of the PDC compared to the one achieved with
the best Proportional Controller having fixed gain of -2, as determined also by successive trials
and using the aforementioned joint criterion of performance. The utility of including the
derivative term in the linear controller (PDC instead of PC) is evident, since the PDC suppresses
the disturbance signal at least 6.5 dB more than the PC within the frequency range of interest.
19
Figure 5: Proportional Vs. Proportional-Derivative Control

Figure Description: Top panel shows the closed-loop sensitivity for the cases of the Proportional
Controller (blue line) and Proportional-Derivative controller (green line). Bottom panel presents the
spectrum of a typical disturbance signal.
20
Chapter 5
Model Predictive Control
We present first the concept of Model Predictive Control which constitutes the core of the
model-based strategies that we develop afterwards. A schematic of the closed-loop system
implementing the MPC in this study is shown in Figure 6. At time instant n, we have knowledge
of the nonlinear model and of all the past input-output pairs. The goal of MPC is to determine the
control input value ( ) U n , so that the following cost function be minimized:

2 2
( ) [ ( | ) ] ( )
y
J n Y n p n R U n = Γ + − +

where ( | ) Y n p n + is the vector of predicted output values over a future horizon of p steps
using the model and the input past values, R is the target reference value for the output, and
y
Γ
is a diagonal matrix of weighting coefficients that assign greater importance to the near-future
predictions. Since simulations have shown that only the immediate control input value ( ) U n
ought to be decided at each discrete time instant, the procedure is repeated at each time step to
compute ( 1) U n+ and so on. More details on MPC and relevant control issues can be found in
[4].
The selection of the parameter p in the definition of the prediction horizon - i.e. the size of the
vector of predicted output values ( | ) Y n p n + - is important and affects the performance of the
MPC algorithm. It generally depends on the severity of the glucose disturbances in a given case
and on the system dynamics (i.e. how far into the future do the effects of an impulsive input last).
21
This selection is made empirically in each specific case. Since it is difficult to estimate the future
values of the stochastic glucose disturbance, we are likely to keep the prediction horizon short. In
our simulations we use a prediction horizon of 60 min (20 samples) and an exponential weighting
vector
y
Γ that is not “aggressive” in order to minimize the risk of hypoglycaemic events. As
another measure of precaution against hypoglycaemia we may use asymmetric weighting of the
predicted output vector, as in [24], whereby we penalize 10 times more the deviations of this
vector ( | ) Y n p n + that are below 70 mg/dl.

Figure 6: Schematic of the Model Predictive Controller used in this study

Figure Description: the Nonlinear Model Predictive Controller (NMPC) computes the optimal insulin
infusion, having an internal description of the Augmented Minimal Model (AMM) and the input-output
data.

The MPC relies on the ability to predict the future values of the output vector, including the
effects of the rate disturbance. However, to predict these future values accurately, we must have
an accurate estimate of the future values of the rate disturbance (in addition to an accurate model
and set of parameters). This is not feasible in the presence of stochastic disturbances, although it
is feasible in the case of large disturbances caused by meals, when their specific exponential
22
structure (e.g. standardized Fischer meals) can be assumed -- without further assumptions about
their magnitude or timing. This implies that MPC can be successfully applied during the time
periods of large disturbances due to meals, but not during time periods of small stochastic
disturbances. Figure 7 illustrates the estimates of meal-related glucose rate disturbances when
their exponential structure is assumed to be known.

Figure 7: Estimation of Meal Disturbances
0 200 400 600 800 1000 1200 1400 1600
0
0.5
1
1.5
2
2.5
Time (min)
mg/dl/min

Figure Description: Estimates of the meal-related glucose rate disturbances of assumed known
exponential form (green line) and the “actual” glucose rate disturbances (blue line) in a simulated example.

23
Hybrid Control Strategy
One of the contributions of this computational study is the incorporation of random small-
scale disturbances (random bias, sinusoids and small-scale exponentials) in the rate disturbance
signal. Their stochastic nature, combined with the random measurement errors, degrade the
performance of the MPC and lead to the introduction of hybrid and switching control strategies
that can mitigate these weaknesses of the MPC but utilize its strengths.
The Hybrid Control Strategy (HCS) applies to the case where the main aggregate effects of the
small-scale stochastic disturbances amount to a slowly varying random bias. This gives rise to the
idea of including, apart from the control input determined by the MPC (to reject meal-related
disturbances), an additional insulin infusion of slowly varying magnitude proportional to a
moving average ( ) A n of the observed glucose deviations from the reference level (i.e. the
glucose residuals of the model prediction) – i.e. a form of “baseline control” action. The moving
average may be computed over the last one hour of glucose residuals after meals have been
detected and their effects computationally removed from the glucose measurements. Then the
insulin infusion rate
0
( ) U n to achieve this baseline control may be determined by the steady-
state relation between a constant insulin infusion rate
0
U and the corresponding steady-state
glucose value
0
G :

0 1 2 4
0
3 0
I
b
G p p p V
U
p G G
⋅ ⋅ ⋅
=
+

This steady-state relation is obtained from the AMM equations after setting all the derivatives
and the disturbance signal equal to zero and defines the baseline control action (i.e. the insulin
24
infusion rate
0
U ) at each time-step n by setting
0
( ) G A n = . Since the steady-state value requires
several hours to be established and the effects of endogenous insulin have been neglected, this
approach is only an approximation. A heuristic constant has been included to help the HCS
perform better, so the actual “baseline infusion” is
0 0
kU (were
0
k is about 0.8). The results of
simulations have shown that considerable improvement of performance can be achieved with
this additional “baseline control” action.

Switching Control Strategy
As mentioned above, the control strategies of PDC and MPC have dominated the blood
glucose regulation literature. Since the results of our computational study with stochastic glucose
rate disturbances indicate relative strengths and weaknesses of the two approaches depending on
the level of disturbance, we may consider a “mixed strategy” that combines their strengths and
mitigates their respective weaknesses. Specifically, our results lead us to the following
conclusions:
• An “aggressive” PDC performs very well in the presence of small-scale disturbances (where
the nonlinear effects are limited and disturbance is difficult to predict) and is computationally
efficient. However, the same “aggressive” PDC would cause hypoglycaemias if applied to large-
scale or meal disturbances.
• The MPC performs very well in the presence of large disturbances (where the nonlinear
effects are significant) and carefully avoids hypoglycaemias. Similar performance can be achieved
with a “conservative” PDC like the one presented in Chapter 4, but the MPC guarantees optimal
performance under any (known) parameter variation.
25
These conclusions suggest a possible “switching control strategy” (SCS) between MPC and
PDC that can be applied as follows: use MPC when we observe large disturbances (e.g. effects of
meals) and PDC when we observe small disturbances. Thus, the SCS can combine the strengths
of PDC and MPC, as well as mitigate their respective weaknesses. The implementation of this
algorithm is easy and the switching points can be clearly identified, as demonstrated in the
presented examples below.
26
Chapter 6
Attenuation of Meal Disturbances
We begin with a study, common in the blood glucose regulation literature, of the attenuation
of meal disturbances, which are usually represented in the literature as Fisher meals [18] or
Lehmann-Deutsch meals [31] (the former are used in this study). Figure 8 shows the performance
achieved by the application of PDC and MPC separately on three meal disturbances (emulating
breakfast, lunch and dinner). The conclusion is that both PDC and MPC are capable of rejecting
meal disturbances (as efficiently as possible) and avoid hypoglycaemias at the same time. The
fundamental difference is at the method the two algorithms follow: PDC achieves this
performance with its gains having been tuned empirically after a large number of simulations
(they are also parameter-specific). On the other hand the MPC guarantees by definition an
optimal (or close to optimal) behavior and this fact by itself justifies the additional computational
burden.
27
Figure 8: Attenuation of Meal Disturbances
0 200 400 600 800 1000 1200 1400 1600
0
50
100
150
200
250
300
350
Time (min)
mg/dl
No Control
PDC
MPC

Figure Description: Performance of PDC and MPC for three meal disturbances (emulating breakfast,
lunch and dinner): blood glucose with only the basal insulin infusion (blue line), blood glucose with PDC
action (green line) and blood glucose with MPC action (red line).

Attenuation of Stochastic Disturbances
We now examine the case of stochastic glucose disturbances in the general form described in
Chapter 3. Illustrative results of regulating the total glucose signal using the PDC, HCS and SCS
are shown in Figures 9, 10 and 11, respectively, for typical simulated runs over 48 hours, along
with the required amounts of insulin infusions in each case. In these figures we also report the
mean value (MV) and the standard deviation (SD) of the glucose signal, as well as the percentage
of time that the glucose level is found outside the target (normoglycaemic) region of 70 to 110
mg/dl (termed the “PTO” parameter) in the absence and presence of the respective controller.
28
The average daily insulin usage (IU) in U/day is also reported for each controller (in addition to
the basal value).
Table 1 reports the averages of the MV, SD, PTO values of the regulated glucose fluctuations
and the average daily IU value for the required insulin infusions for the three types of control
action, along with the unregulated values (no control action), for 10 independent runs over 48
hours each. The number of hypoglycaemic events (blood glucose under 60 mg/dl) that occurred
in each case is also reported. The results presented in Figures 9-11 and Table 1 lead to the
conclusion that all three control strategies are able to achieve a satisfactory level of glucose
regulation in the presence of stochastic disturbances. The SCS seems to perform better compared
to the other two strategies, in terms of MV and PTO (which are probably the two most important
criteria). Moreover, the hypoglycaemic event reported, combined with the significantly lower
MV, leaves some questions about the safety of the patient when the HCS is employed.
29

Figure 9: Attenuation of Stochastic Glucose Disturbances with PDC
500 1000 1500 2000 2500
0
100
200
300
400
Blood Glucose with and without Control
MV: 180.8 -> 109.1 SD: 50.6 -> 27.7 PTO: 100% -> 23% IU: 42.6
mg/dl
500 1000 1500 2000 2500
0
50
100
Insulin
Time (min)
mU/L*min

Figure Description: The top panel shows the blood glucose levels with basal insulin infusion only (blue
line) and after PDC action (green line). The mean value (MV), standard deviation (SD) and the percentage
of time that the glucose is found outside the normoglycemic region of 70-110 mg/dl (PTO) are reported
between the panels for PDC (right values) and without control action (left values). The bottom panel shows
the amounts of infused insulin used by the PDC action (the average insulin usage is 42.6 U/day in this
case).

30
Figure 10: Attenuation of Stochastic Glucose Disturbances with HCS
500 1000 1500 2000 2500
0
100
200
300
400
Blood Glucose with and without Control
MV: 180.8 -> 99.4 SD: 50.6 -> 27.2 PTO: 100% -> 21% IU: 53.5
mg/dl
500 1000 1500 2000 2500
0
50
100
Insulin
Time (min)
mU/L*min

Figure Description: The top panel shows the blood glucose levels with basal insulin infusion only (blue
line) and after HCS action (green line). The mean value (MV), standard deviation (SD) and the percentage
of time that the glucose is found outside the normoglycemic region of 70-110 mg/dl (PTO) are reported
between the panels for HCS (right values) and without control action (left values). The bottom panel shows
the amounts of infused insulin used by the HCS action (the average insulin usage is 53.5 U/day in this
case).
31
Figure 11: Attenuation of Stochastic Glucose Disturbances with SCS
500 1000 1500 2000 2500
0
100
200
300
400
Blood Glucose with and without Control
MV: 180.8 -> 103.7 SD: 50.6 -> 27.9 PTO: 100% -> 18% IU: 48.7
mg/dl
500 1000 1500 2000 2500
0
50
100
Insulin
Time (min)
mU/L*min

Figure Description: The top panel shows the blood glucose levels with basal insulin infusion only (blue
line) and after SCS action (green line). The mean value (MV), standard deviation (SD) and the percentage
of time that the glucose is found outside the normoglycaemic region of 70-110 mg/dl (PTO) are reported
between the panels for SCS (right values) and without control action (left values). The bottom panel shows
the amounts of infused insulin used by the SCS action (the average insulin usage is 48.7 U/day in this
case).
32
Table 1: Average behavior of control strategies for nominal diabetic patients
NO CONTROL PDC HCS SCS
MV 184.3 110.8 98.6 104.6
SD 56.9 32.4 30.9 32.2
PTO 99 21.5 22.6 17.2
INSULIN/DAY 0 43.3 56.7 50.3
HYPOGLYCAEMIAS 0 0 1 0

Table Description: Averages of 10 independent runs over 48 hours each, when all parameters have their
nominal values. Presented are the mean value (MV ) and the standard deviation (SD) of glucose
fluctuations, the percentage of time glucose is found outside the normoglycemic region 70-110 mg/dl
(PTO), and the average insulin usage (IU) in U/day, as well as the number of hypoglycaemic events.

Insulin Dependent Diabetes Mellitus
In this section we consider the case that parameter β has a significantly reduced value (from
its nominal value of 0.0085) emulating cases of IDDM, while the rest of the parameters retain
their nominal values. This change is known to the controllers and we examine the performance of
the modified closed-loop system (i.e. when the parameter β is reduced but correctly estimated).
Note that a significant reduction in the value of parameter β will cause greater values and
fluctuations of blood glucose for a given rate disturbance signal or control action, that may
increase the likelihood of hypoglycaemic events. However the parameters of our controllers
remain unchanged.
33
Table 2 presents the results of 10 independent runs for the case of 0 β = , which indicate that
we can apply the control strategies developed above to IDDM cases (corresponding to a zero
value for parameter β ) without risking the safety of the patient by causing hypoglycaemic
events. Also, comparison with the results of Table 1 indicates that the PDC, HCS and SCS can be
applied to both IDDM and NIDDM patients without significant degradation in performance
.
Table 2: Average behavior of control strategies for IDDM patients
NO CONTROL PDC HCS SCS
MV 198.6 111.4 102.2 105.3
SD 61.7 33 31.5 33
PTO 100 22.8 25.4 17.6
INSULIN/DAY 0 44.3 53.7 51
HYPOGLYCAEMIAS 0 0 0 0

Table Description: Averages of 10 independent runs over 48 hours each, when 0 β = . Presented are the
mean value (MV) and standard deviation (SD) of glucose fluctuations, the percentage of time glucose is
found outside the normoglycemic region 70-110 mg/dl (PTO), the average insulin usage (IU) in U/day, as
well as the number of hypoglycaemic events.

Mild Diabetic / ICU Case
In this section we consider the case that parameter β has a significantly increased value,
emulating mild NIDDM cases or patients in the ICU with a need of continuous insulin therapy.
The rest of the parameters retain their nominal values. This change is again known to the
34
controllers and we examine the performance of the various controllers for the modified system
(i.e. when the parameter β is increased and correctly estimated).
Table 3 presents the results of 10 independent runs for the case of β=0.0425, which indicate that
we can apply the HCS and SCS approaches to mild NIDDM and ICU cases (corresponding to a
zero value for parameter β) without degradation in performance. Again some questions arise
from the fact that a hypoglycaemic event is reported in the HCS. In contrast the PDC and SCS
seem to guarantee the patient’s safety.

Table 3: Average behavior of control strategies for mild diabetic / ICU patients
NO CONTROL PDC HCS SCS
MV 157.8 109.3 96.5 103.5
SD 46.6 30.3 29.5 30.5
PTO 97.5 19.2 17.4 16.4
INSULIN/DAY 0 41.9 58.1 49.3
HYPOGLYCAEMIAS 0 0 1 0

Table Description: Averages of 10 independent runs over 48 hours each, when 0.0425 β = . Presented
are the mean value (MV) and standard deviation (SD) of glucose fluctuations, the percentage of time
glucose is found outside the normoglycemic region 70-110 mg/dl (PTO), the average insulin usage (IU) in
U/day, as well as the number of hypoglycaemic events.

35
Robustness to Variations in Endogenous Insulin Production
We now examine the robustness of performance in the face of parametric uncertainty due to
inter-patient variability, intra-patient variability, or simply mis-estimation of parameters α and
β , which determine the production of endogenous insulin. The range of random variations of
α and β is ± 50% from their nominal values. Unlike the previous case, the altered parameter
values are not known to the controllers. The effects of these parameter mis-estimations are very
limited as demonstrated by the results shown in Table 4, when compared with the results of
Table 1. This indicates that the control algorithms developed in this study are robust to
parametric uncertainty.

Table 4: Average behavior of control strategies for uncertain endogenous insulin production
NO CONTROL PDC HCS SCS
MV 194.3 111.4 103.5 105.4
SD 58.3 32.1 31.1 32.5
PTO 100 22 24.4 17.5
INSULIN/DAY 0 45.8 53.9 52.4
HYPOGLYCAEMIAS 0 0 0 0

Table Description: Averages of 10 independent runs over 48 hours each, when [0.08,0.24] α∈ and
[0.00425,0.01275] β∈ (random values, uniformly distributed). Presented are the mean value (MV)
and standard deviation (SD) of glucose fluctuations, the percentage of time glucose is found outside the
normoglycemic region 70-110 mg/dl (PTO), the average insulin usage (IU) in U/day, as well as the
number of hypoglycaemic events.
36
Chapter 7
Conclusions
The basic findings of our computational study are:
• The proposed Switching Control Strategy gives excellent results and performs better overall
than the other two control strategies considered. Without even changing its parameters, the SCS
can deal with mild NIDDM / ICU, nominal NIDDM and IDDM cases without degradation in
performance or risk of the patient’s safety. It is also significantly robust to a wide range of
variations of the crucial parameters determining the endogenous insulin production.
• The proposed Hybrid Control Strategy gives satisfactory results and achieves a closer to the
reference mean value of controlled blood glucose (compared to the PDC and SCS). However,
since the baseline control it employs during small scale disturbances does not reduce the glucose
fluctuations (only removes the bias), this increases the chances of hypoglycaemia (and indeed we
had 2 cases of mild hypoglycaemia in totally 40 trials). This could mean that another baseline
control (e.g. a PI) or a modification of the present may be more appropriate for the rejection of
small scale disturbances. Moreover a slight degradation in performance is noted in the cases of
IDDM and random variations of α and β .
• The overall performance of the Proportional-Derivative Controller is shown to be good (in
fact much better than we expected before this study). It gives satisfactory results in all cases we
examined and avoids hypoglycaemic events. At the same time it seems to be more efficient in
term of insulin usage.
37
A note regarding the average insulin daily usage: we observe that all the controllers
considered in this study utilize amounts of insulin from 45 - 55 U/day. This is in accordance with
results from clinical trials.
Another point comes up when one compares the average insulin usage in Tables 1 and 2,
corresponding to nominal and zero endogenous insulin secretion: the fact that the values
appearing in these Tables are very close might seem counter-intuitive. However endogenous
insulin secretion, which is based on a threshold nonlinearity, occurs only in cases when blood
glucose exceeds 90 θ = mg/dl and remains pretty small as long as blood glucose is in the
normoglycaemic region. Therefore the presence of an effective controller not only maintains
normoglycaemia but at the same time minimizes the effects of endogenous insulin on blood
glucose.
The main goal of this study has been to develop, test and evaluate alternative approaches for
blood glucose regulation under conditions that are deemed as realistic as possible, based on our
current knowledge. For this reason we introduced a minimal model which is appropriate for all
patients requiring continuous insulin treatment (IDDM, NIDDM, ICU), we estimated its
parameters and, more importantly, we suggested a blood glucose disturbance signal that
captures the aggregate effects of the known systemic and external factors affecting glucose
(including meals). Thus, we formulated a more difficult, but more realistic, control problem and
presented three different approaches (PDC, HCS, SCS) that produce satisfactory results and
remain computationally efficient (i.e. easily realizable).
Future work may extend these results in the non-stationary case (when the system
characteristics change through time) that requires adaptive or robust control strategies. We must
38
also examine the case of random variations in parameters determining insulin action (
2
p and
3
p ) which are very important too. Our future efforts should also address the issue of predicting
partially the stochastic components of the rate disturbance signal (not just meals) and, thus,
improving the performance of the HCS and SCS approaches.
39
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808, 1982.

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290, pp. 169-176, 2006.

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subcutaneous route”, IEEE Trans. Biomed. Eng., vol. 45, pp. 1122–1134, 1998.
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of a diabetic”, IEEE Trans. Neural Networks, vol. 10, pp. 1204-1213, 1999.

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44
Appendix A
Sorensen’s Model of Glucose Metabolism

Variables
G: glucose concentration (mg/dl)
I: insulin concentration (mU/l)
Γ: glucagon concentration (pg/ml)
U: (exogenous) insulin infusion rate (mU/min)
Q: vascular flow rate (dl/min for glucose – l/min for insulin)
S: metabolic source rate (mg/min for glucose – mU/min for insulin – pg/min for glucagon)
Σ: metabolic sink rate (mg/min for glucose – mU/min for insulin – pg/min for glucagon)
M: multiplier of basal metabolic rate (dimensionless)
T: time constant (min)
V: volume (dl for glucose – l for insulin - ml for glucagon)
t: time (min)

First Subscript (Physiologic Compartment)
B: brain
G: gut
H: heart and lungs
L: liver
K: kidney
45
P: periphery
(A: hepatic artery)
Second Subscript (Physiologic Sub-compartment)
I: interstitial fluid space
V: vascular space

First Superscript
G: glucose model
I: insulin model
Γ: glucagon model
B: basal value
N: normalized value (divided by basal value)

Second Superscript
0: initial value (normalized value as t→0
+
)
∞: asymptotic/steady state value (normalized)

Metabolic Rate Subscripts
BGU: brain glucose uptake
GGU: gut glucose utilization
HGP: hepatic glucose production
HGU: hepatic glucose uptake
KGE: kidney glucose excretion
46
PGU: peripheral glucose uptake
RBCU: red blood cell glucose uptake
KIC: kidney insulin clearance
LIC: liver insulin clearance
PIC: peripheral insulin clearance
PIR: pancreatic insulin release
PΓR: plasma glucagon release
PΓC: plasma glucagon clearance

47
GLUCOSE MODEL
( ) ( )
G
G G BV BI
BV B H BV BV BI
B
dG V
V Q G G G G
dt T
= − − −
( )
G
G BI BI
BI BV BI BGU
B
dG V
V G G
dt T
= − −Σ
G G G G G G H
H B BV L L K K P PV H H RBCU
dG
V Q G Q G Q G Q G Q G
dt
= + + + − −Σ
( )
G G G
G G H G GGU
dG
V Q G G
dt
= − −Σ
G G G G L
L A H G G L L HGP HGU
dG
V Q G Q G Q G S
dt
= + − + −Σ
( )
G G K
K K H K KGE
dG
V Q G G
dt
= − −Σ
( ) ( )
G
G G PV PI
PV P H PV PV PI G
P
dG V
V Q G G G G
dt T
= − − −
( )
G
G PI PI
PI PV PI PGU G
P
dG V
V G G
dt T
= − −Σ

70
BGU
Σ =
10
RBCU
Σ =
20
GGU
Σ =
[7.03 6.52 tanh(0.338 ( 5.82))]
B N N
PGU PGU PI PI
G I Σ =Σ ⋅ ⋅ + ⋅ ⋅ −
48
71 71 tanh[0.11 ( 460)], 0 460
330 0.872 , 460
K K
KGE
K K
G G
G G
+ ⋅ ⋅ − ≤ ≤ 
Σ =

− + ⋅ ≤


B I G
HGP HGP HGP HGP HGP
S S M M M
Γ
= ⋅ ⋅ ⋅
1
( )
25
I
I I HGP
HGP HGP
dM
M M
dt
∞
= −
1.21 1.14 tanh[1.66 ( 0.89)]
I N
HGP L
M I
∞
= − ⋅ ⋅ −
0
2 HGP HGP
M M f
Γ Γ
= −
0
2.7 tanh(0.39 )
N
HGP
M
Γ
= ⋅ ⋅Γ
0
2
2
1 1
( )
65 2
HGP
M df
f
dt
Γ
−
= −
1.42 1.41 tanh[0.62 ( 0.497)]
G N
HGP L
M G = − ⋅ ⋅ −

B I G
HGU HGU HGU HGU
M M Σ =Σ ⋅ ⋅
1
( )
25
I
I I HGU
HGU HGU
dM
M M
dt
∞
= −
2 tanh(0.55 )
I N
HGU L
M I
∞
= ⋅ ⋅
5.66 5.66 tanh[2.44 ( 1.48)]
G N
HGU L
M G = + ⋅ ⋅ −
49
INSULIN MODEL
( )
I B
B H B
dI
V Q I I
dt
= −
I I I I I I H
H B B L L K K P P H H
dI
V Q I Q I Q I Q I Q I U
dt
= + + + − +
( )
I I G
G G H G
dI
V Q I I
dt
= −
I I I I L
L A H G G L L PIR LIC
dI
V Q I Q I Q I S
dt
= + − + −Σ
( )
I I K
K K H K KIC
dI
V Q I I
dt
= − −Σ
( ) ( )
I
I I PV PI
PV P H PV PV PI I
P
dI V
V Q I I I I
dt T
= − − −
( )
I
I PI PI
PI PV PI PIC I
P
dI V
V I I
dt T
= − −Σ

0.4 ( )
I I
LIC A H G G PIR
Q I Q I S Σ = ⋅ + +
0.3
I
KIC K K
Q I Σ = ⋅
0.85
0.15
PI
PIC I
P
I I
P PI
I
T
Q V
Σ =
−
⋅

4
4
( )
( )
B H
PIR PIR B
H
P G
S S
P G
=
50
3.27
3.27 3.02
( )
( )
(132) 5.93 ( )
H
H
H
G
X G
G
=
+ ⋅

1.11
( ) [ ( )]
H H
Y G X G =
1
1
( )
dP
Y P
dt
α = ⋅ −
2
2
( )
dP
X P
dt
β = ⋅ −
3
3 3 1 4
( )
B
dP
K P P P P
dt
γ = ⋅ − + ⋅ −
4 1 2 2 3
( ) [ ( ) max( ( ) ,0)]
H H H
P G MY G M X G P P = + − ⋅

GLUCAGON MODEL

P R P C
d
V S
dt
Γ
Γ Γ
Γ
= −Σ

9.1
P C Γ
Σ = ⋅Γ
P R
B G I
P R P R P R
S S M M
Γ
Γ Γ Γ
= ⋅ ⋅
2.93 2.1 tanh[4.18 ( 0.61)]
G N
P R H
M G
Γ
= − ⋅ ⋅ −
1.31 0.61 tanh[1.06 ( 0.47)]
I N
P R H
M I
Γ
= − ⋅ ⋅ −
51
CONSTANTS

Table 5: Constants in Sorensen’s model of glucose metabolism
3.5
G
BV
V = 5.9
G
B
Q = 2.1
B
T =
4.5
G
BI
V = 43.7
G
H
Q = 5
G
P
T =
13.8
G
H
V = 2.5
G
A
Q = 20
I
P
T =
25.1
G
L
V = 12.6
G
L
Q = 35
B
PGU
Σ =
11.2
G
G
V = 10.1
G
G
Q = 155
B
HGP
S =
6.6
G
K
V = 10.1
G
K
Q = 20
B
HGU
Σ =
10.4
G
PV
V = 15.1
G
P
Q = 17.09
B
PIR
S =
67.4
G
PI
V = 0.45
I
B
Q = 682.5
P R
B
S
Γ
=
0.26
I
B
V = 3.12
I
H
Q =
0.0482 α =
0.99
I
H
V = 0.18
I
A
Q =
0.931 β =
0.94
I
G
V = 0.72
I
G
Q =
0.575 γ =
1.14
I
L
V = 0.90
I
L
Q =
0.00747 K =
0.51
I
K
V = 0.72
I
K
Q =
1
0.0908 M =
0.74
I
PV
V = 1.05
I
P
Q =
2
0.0958 M =
6.74
I
PI
V =

3
6.33
B
P =
11310 V
Γ
=

52
The basal values resulting from Sorensen’s model for venous blood glucose, insulin and glucagon
concentrations are:
90
B
H
G = mg/dl
15
B
H
I = mU/l
75
B
Γ = pg/ml

Figure 12: Sorensen’s model of glucose metabolism

Figure Description: Block diagram of Sorensen’s comprehensive model of glucose metabolism
(Figure from Parker et al., 2000).
53
Appendix B
Proof of Stability of the Augmented Minimal Model
Theorem: The origin is an asymptotically stable equilibrium point of the Augmented Minimal
Model.
Proof: Let us consider two cases:
• Gb G θ + ≥ . Then the linearized around the origin AMM can be written as:
4
( )
( )
ex
ex
I
dI U t
p I t
dt V
=− +
( ) ( )
en
en
dI
I t G t
dt
α β =− ⋅ + ⋅
3 3 2
( ) ( ) ( )
ex en
dX
pI t pI t p X t
dt
= + −
1
( ) ( )
b
dG
G X t pG t
dt
=− −
• Gb G θ + ≤ . Then the linearized around the origin AMM can be written as:
4
( )
( )
ex
ex
I
dI U t
p I t
dt V
=− +
( )
en
en
dI
I t
dt
α =− ⋅
3 3 2
( ) ( ) ( )
ex en
dX
pI t pI t p X t
dt
= + −
1
( ) ( )
b
dG
G X t pG t
dt
=− −
54
In both cases it is easy to see that for the nominal values of the implicated parameters the
linearized AMM is strictly stable (in fact one can verify that the linearized AMM is strictly stable
for a wide range of variations of the parameters from their nominal values). According to
Lyapunov’s Linearization Method (e.g. see Theorem 3.1 of [44]) this implies that the origin is an
asymptotically stable equilibrium point of the actual (nonlinear) AMM.

Abstract (if available)

Abstract In this computational study, we develop two nonlinear control algorithms (termed the Hybrid Control Strategy and the Switching Control Strategy ) and evaluate their performance relative to the widely studied Proportional-Derivative Controller for the regulation of blood glucose with intravenous insulin infusions in diabetic patients. For the requisite simulations of the causal relationship between infused insulin and blood glucose we introduce a new model (termed the Augmented Minimal Model ) and a glucose rate disturbance signal that represents the aggregate effects of the various neuro-hormonal, behavioral and metabolic factors affecting blood glucose. The obtained results show that the Switching Control Strategy can regulate blood glucose better than the other two controllers (based on widely accepted metrics of performance), while eliminating the risk of inadvertent hypoglycaemia and remaining remarkably robust in the event of unknown variations in critical parameters of the model. However, the other two controllers exhibit a satisfactory performance too.

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Asset Metadata

Creator Markakis, Michail G. (author)

Core Title Control strategies for the regulation of blood glucose

School Viterbi School of Engineering

Degree Master of Science

Degree Program Electrical Engineering

Publication Date 03/27/2010

Defense Date 02/27/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag blood glucose,Diabetes,intravenous insulin,model predictive control,OAI-PMH Harvest,PID control

Language English

Advisor Marmarelis, Vasilis Z. (committee chair), Ordonez, Fernando (committee member), Safonov, Michael G.. (committee member)

Creator Email markakis@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1060

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Rights Markakis, Michail G.

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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