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Chapter 1
Introduction
1.1 Description of Problem
Diabetes mellitus is a chronic metabolic disease characterized by the inability of the
human body to maintain blood glucose concentrations within the physiological limits.
Efforts to compensate for this inability include diet control and intravenous insulin
injections. These efforts reflect hypotheses about what is required for the body to
maintain homeostasis, which in this case means maintaining blood glucose
concentrations within the safe physiological limits. The two principal diagnostic tests are
the Intravenous Glucose Tolerance Test (IVGTT) and the Oral Glucose Tolerance test
(OGGT). These tests require the patient to submit to clinical conditions very different
from those encountered in every day living. Current differential equation models are
directed at explaining the results of IVGTT and OGGT tests.
The models are not
applicable, as formulated, to a patient wishing to control her or her disease.
Currently, work is ongoing to create an artificial pancreas capable of supporting,
or replacing, the human pancreas. Such an artificial pancreas must have three parts
working synchronously: (1) a glucose sensor, (2) an insulin infusion pump, and (3) a
computer program based on a mathematical algorithm that predicts healthy infusion rates
[Laino].
The object of this project is to propose and discuss a more comprehensive
mathematical model that will facilitate the development of control algorithms for an
artificial pancreas. The model proposed is the first to include recent research concerning
the existence of insulin pools inside the pancreas. Chapter 2, describes the nature of the
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disease, diabetes.
Chapter 3 contains an overview of compartmental modeling of
diabetes, and discusses previous IVGTT, and OGTT models. Chapter 4 discusses recent
understandings of the body’s insulin production and release mechanism. The pancreas,
shown in figure 1.1.1, is located near to the stomach produces insulin, and delivers it
directly to the portal vein.
Figure 1.1.1 The gastrointestinal system
Although, early models of the IVGTT took the rate of insulin release to be linear in time
with a slope dependent upon the glucose load, experiments have shown that insulin
release occurs in two phases. This is illustrated in figure 1.1.2 and referred to as biphasic
insulin release.
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Figure 1.1.2 The biphasic pattern of glucose stimulated insulin secretion from the ß-cell. (Reproduced with
permission from Williams G & Pickup JC, eds. Handbook of Diabetes, 2nd ed: Blackwell Science, Oxford,
1999, 30.)
A previously proposed biphasic insulin secretion model is introduced, and then
improvements are made. The improved secretion model is then inserted into an existing
model of glucose-insulin dynamics. The result is the Insulin Response Model (IRM). In
Chapter 5 looks at issues involved in system identification. Then, the IRM is fit to
IVGTT and to OGGT mean data sets that represent average healthy insulin-glucose
responses. Graphs of predicted insulin secretion rates are presented. The appendix
presents additional topics, including other OGTT models, equilibrium, and stability,
along with source code for the MATLAB programs.
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Chapter 2
Overview of Diabetes
2.1 What is Diabetes Mellitus?
Diabetes mellitus, commonly referred to as “diabetes”, meaning “sweet urine”, is a
chronic medical condition of great concern in the United States because of its growing
occurrence and high cost of treatment.
In its simplest description, blood presents
abnormally high levels of glucose (sugar). These elevated levels of blood glucose (a state
called hyperglycemia) lead to spillage of glucose into the urine, thus the name “diabetes”.
Normally, blood glucose levels are tightly controlled by insulin, a hormone produced by
the pancreas. Insulin functions to lower the blood glucose level: When blood glucose
elevates (after eating food, for example) the pancreas releases the hormone insulin to
normalize the glucose level. When this process does not function correctly the resultant
chronic medical condition may be treatable but is (at the present time) not only incurable,
but also progressive.
Over time, diabetes leads to several acute and chronic
complications including blindness, kidney failure, and nerve damage. In the United
States, diabetes mellitus affects 12 million people (six percent of the population) that we
know of with an additional undetected 10 million cases. Because diabetes is the third
leading cause of death in the United States (after heart disease and cancer) it is not
surprising that the direct and indirect costs of diabetes mellitus are estimated at
approximately $40 billion per year.
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2.2 Types of Diabetes Mellitus.
There are two related but distinct forms of diabetes mellitus. Type I, also called insulin
dependent diabetes, is the most severe. Type II, is the most common; it is controlled
by diet and exercise. Approximately 10% of diabetes mellitus patients have type I; the
remaining 90% exhibit type II.
In type I diabetes mellitus, the pancreas undergoes an autoimmune attack by the body
itself, and is rendered incapable of making insulin. The patient with type I diabetes must
rely on insulin medication for survival. Abnormal antibodies have been found in patients
with type I diabetes.
In contrast to the total lack of insulin production in type I diabetes, in type II the pancreas
can still produce insulin, but does so incorrectly.
In many cases, the pancreas is
producing larger than normal quantities of insulin but without effect. The major feature
of type II diabetes is a lack of sensitivity to insulin by the cells of the body and a
defective mechanism for the release of insulin by the pancreas.
2.3 Causes of Diabetes Mellitus.
Diabetes Mellitus is a disorder of the glycemic system. In addition to the particular
disorder in type I diabetes mellitus caused by a total lack of insulin production, the other
possible conditions causing the disease are insufficient production of insulin, production
of defective insulin (uncommon), or inability of cells to use insulin. All of these
conditions are included as causes because the term ‘diabetes’ is inclusive, describing
glycemic system dysfunctions caused by insufficiency of insulin secretion. Thus the
terms type I and type II are essential delineators when discussing causes and interpreting
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research and possible treatments of this disease, as in the case in current work being done
on the condition of insensitivity to insulin.
2.4 Relationship between Glucose and Insulin
Glucose is an essential nutrient that provides energy for the proper functioning of the
body cells. Glucose enters the liver from the gastrointestinal tract; it then enters the
bloodstream and is carried by blood to all the cells in the body. However, glucose cannot
enter the cells without the aid of insulin, which enables its transport into and utilization
within the cells. Without insulin, cells become energy starved despite the presence of
abundant glucose in the blood. The abundant, unutilized glucose is excreted in the urine.
Insulin is a hormone produced by specialized cells called beta cells located in the
pancreas. When the blood glucose levels are lowered, the insulin release from the
pancreas is turned off.
2.5 Symptoms of Diabetes Mellitus
The early symptoms of untreated diabetes mellitus are elevated blood sugar levels, and
loss of glucose in the urine. High amounts of glucose in the urine can cause increased
urine output and lead to dehydration. Dehydration causes increased thirst and water
consumption. The inability to utilize glucose energy eventually leads to weight loss
despite an increase in appetite.
Some untreated diabetes patients also complain of
fatigue, nausea, blurred vision, vomiting, and infections of the bladder, skin, and vaginal
areas.
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2.6 Diagnosis
Diabetes is the result of many abnormalities in insulin metabolism, from absolute
deficiency to a combination of both deficiency and resistance. These abnormalities affect
the ability of the body to dispose of glucose from the blood. Three factors are used to
describe, or quantify, a patient’s disease.

Insulin sensitivity: The capability of insulin to increase glucose disposal to
muscles, liver and adipose tissue.

Glucose effectiveness: The ability of glucose to enhance its own disposal at basal
insulin level.

Pancreatic responsiveness: The ability of the pancreas to secrete insulin in
response to glucose stimuli [Anderson].
There are two types of tests used to ‘measure’ an individual’s disease: the Intra-Venous
Glucose Tolerance Test (IVGTT) and the Oral Glucose Tolerance Test (OGTT).
The IVGTT test:
Fig 2.6.1 Typical IVGTT results
The standard IVGTT consists in injecting, say at time 0, glucose over a period of 30-60
seconds and measuring in plasma the resultant glucose and insulin concentrations. The
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sampling schedule of the standard IVGTT usually consists of 3 pretest samples taken at 15, -5 and 0 (immediately before the injection) and 25 test samples taken at 2, 3, 4, 5, 6,
8, 10, 12, 15, 18, 20, 2 5, 30, 35, 40, 60, 70, 80, 100, 120, 140, 160, 180, 210, 240
minutes.
The OGTT test:
Fig 2.6.2 Typical OGTT results
The most common glucose tolerance test is the oral glucose tolerance test (OGTT). After
an overnight fast, a patient drinks a solution containing a known amount of glucose.
Blood and urine are obtained before the patient drinks the glucose solution, and blood is
drawn again every hour after the glucose is consumed for up to three hours. Blood
glucose levels above normal limits at the times measured can diagnose Type 2 ("noninsulin-dependent") diabetes or gestational diabetes (high blood glucose during
pregnancy).
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2.7 Treatment of Diabetes
The major goal in treating diabetes mellitus is controlling elevated blood sugars (glucose)
without causing abnormally low levels of blood sugar. Type I diabetes mellitus is treated
with insulin, exercise, and a diabetic diet. Type II diabetes mellitus is first treated with
weight reduction, diet, and exercise. When these measures fail to control the elevated
blood sugars, oral medications are used. Weight reduction and exercise are important
treatments of diabetes. Weight reduction and exercise increase the body's sensitivity to
insulin, thus helping to control blood sugar elevations. Insulin is the mainstay of
treatment for patients with type I diabetes mellitus. Ideally, insulin medication should be
administered in a manner that mimics, as closely as possible, the natural pattern of insulin
secretion by a healthy pancreas [Shiel].