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THE DYNAMICS OF GLUCOSE-INSULIN ENDOCRINE METABOLIC

REGULATORY SYSTEM

by

Jiaxu Li

A Dissertation Presented in Partial Fulfillment

of the Requirements for the DegreeDoctor of Philosophy

ARIZONA STATE UNIVERSITY

December 2004


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THE DYNAMICS OF GLUCOSE-INSULIN ENDOCRINE METABOLIC

REGULATORY SYSTEM

by

Jiaxu Li

has been approved

December 2004

APPROVED:

, Chair

Supervisory Committee

ACCEPTED:

Department Chair

Dean, Division of Graduate Studies


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ABSTRACT

A model with two time delays is presented for modeling the insulin secretion ul-

tradian oscillations in the glucose-insulin metabolic system. One delay is for the insulin

response time delay (around 6 minutes) to the glucose concentration level increase, and

the other is for the hepatic glucose production time delay (around 36 minutes). The

results of the analysis of this model are in agreement with the experimental observations

and exhibit intrinsic insulin secretion ultradian oscillations. The results show that both

these time delays are necessary for the insulin secretion ultradian oscillation sustain-

ment and only the relative moderate glucose infusion rate and insulin degradation rate

can sustain the oscillations. The numerical simulations demonstrate that the insulin

concentration level peaks after the glucose concentration level. These results also indi-

cate that the hepatic glucose production and its time delay are insignificant in modeling

intravenous glucose tolerance tests (IVGTT).

A generic dynamic IVGTT model and two models for special cases are devel-

oped to simulate the short time (30-120 minutes) dynamics. As expected, such models

frequently produce globally asymptotically stable steady state dynamics. The easy-to-

check conditions, which guarantee the steady state to be stable, are provided.

In the last model, we take the active-cell mass into consideration and study the

effects of the-cells in the glucose-insulin regulatory system. The numerical simulations

show that the insulin concentration peaks after the active -cell mass peaks, which peaks

after the glucose concentration peaks. Other results are also in agreement with reported

data.

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In Memory of My Mother

To My Father

To My Wife and Daughters

To My Sisters

iv


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ACKNOWLEDGMENTS

I would first of all like to thank Dr. Yang Kuang for his guidance during my doc-

toral study at Arizona State University. I am forever indebted to his advise, suggestions,

support, encouragement, understanding and, in particular, patience. I would like to

thank Dr. Steven Baer, Dr. Carlos Castillo-Chaves, Dr. Hal Smith and Dr. Horst Thieme

for their interest, carefully reading the manuscript, valuable input and suggestions for

improving this dissertation. It is my great pleasure to work with them and I feel so

lucky and proud that I have such a wonderful supervisory committee, one of the best

in the world. I would also like to thank the external reviewer for the valuable input.

My special thanks go to my master thesis advisor Prof. Xiudong Chen.

I would also like to extend my gratitude to Dr. Bingtuan Li for the various

broad discussions, to Dr. Athena Makroglou for her initiating the collaborate paper

[59] and providing references (for example, [65] and [64]), to Prof. Edoardo Beretta for

his providing the manuscript of [23], to Mr. Clint Mason for his providing reference [9]

and [85], to Ms. Debbie Olson and Ms. Joan Person for their administrative support,

to Dr. Jialong He for his IT support, and to Mr. Rafael Mendez for the proof-reading

of the most of this dissertation.

Last, but not the least, I would like to thank my wife, Dr. Guihua Li, for her

long lasting love and support.

Jiaxu Li

December 11, 2004

Arizona State University, Tempe, Arizona USA

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TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

CHAPTER 1 Introduction and Physiological Background . . . . . . . . . . . . . 1

1. Diabetes Mellitus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Glucose-Insulin Endocrine Metabolic Regulatory System . . . . . . . . . 4

3. The pancreas and Its Endocrine Hormones . . . . . . . . . . . . . . . . . 6

3.1. The pancreas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2. Glucose Transporters . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3. Secretion and Actions of Insulin . . . . . . . . . . . . . . . . . . . 10

3.4. Insulin Receptors . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5. Insulin Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.6. Insulin Degradation and Clearance . . . . . . . . . . . . . . . . . 16

3.7. Production and Consumption of Glucose . . . . . . . . . . . . . . 17

4. Glucose Tolerance Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5. The Organization of This Dissertation . . . . . . . . . . . . . . . . . . . 21

CHAPTER 2 The Ultradian Oscillations of Insulin Secretion . . . . . . . . . . . 23

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2. Sturis-Tolic ODE Model and Current Research Status . . . . . . . . . . . 25

3. Two Time Delay DDE Model . . . . . . . . . . . . . . . . . . . . . . . . 34

4. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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5. Global Stability of Steady State . . . . . . . . . . . . . . . . . . . . . . . 43

6. Linearization and Local Analysis . . . . . . . . . . . . . . . . . . . . . . 44

7. Numerical Analysis of Stability Switches and Bifurcations . . . . . . . . . 56

7.1. Insulin Response Time Delay1 . . . . . . . . . . . . . . . . . . . 59

7.2. Glucose Infusion RateGin . . . . . . . . . . . . . . . . . . . . . . 60

7.3. Insulin Degradation Rate di . . . . . . . . . . . . . . . . . . . . . 63

7.4. Hepatic Glucose Production 2. . . . . . . . . . . . . . . . . . . . 63

7.5. Parameter1 vs. Gin . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.6. Parameter1 vs. di . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.7. ParameterGin vs. di . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.8. Insulin Concentration Peaks after Glucose Concentration Peaks . 68

8. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

CHAPTER 3 Modeling Intra-Venus Glucose Tolerance Test . . . . . . . . . . . 75

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2. Current Research Status . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3. More Generic IVGTT Model . . . . . . . . . . . . . . . . . . . . . . . . . 80

4. Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5. Global Stability of Steady State . . . . . . . . . . . . . . . . . . . . . . . 87

6. Local Stability of Steady State and Stability Switch . . . . . . . . . . . . 93

7. Delay Independent Stability Results for Discrete Delay Model . . . . . . 95

8. Delay Dependent Stability Conditions . . . . . . . . . . . . . . . . . . . . 99

8.1. The case of discrete delay . . . . . . . . . . . . . . . . . . . . . . 100

8.2. The case of distributed delay . . . . . . . . . . . . . . . . . . . . . 101

8.3. Expression ofH() . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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9. Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

10. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

CHAPTER 4 The Effects of Active-Cells: A Preliminary Study . . . . . . . .108

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2. Current Research Status . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3. Active -Cell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4. Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.1. Insulin Response Delay and Hepatic Glucose Production Are Crit-

ical for Sustain Insulin Secretion Oscillations . . . . . . . . . . . . 115

4.2. Insulin Response Time Delay1 as a Bifurcation Parameter . . . 115

4.3. Glucose Infusion RateGin as a Bifurcation Parameter . . . . . . . 116

4.4. Peaks of Oscillations in One Cycle . . . . . . . . . . . . . . . . . 118

4.5. -cell Deactivation Rate k[0.01, 2] as a Bifurcation Parameter . 119

4.6. Parameter as a Bifurcation Parameter . . . . . . . . . . . . . . 120

4.7. The Changes of Insulin Degradation Ratedi[0.025, 0.1] Do Not

Affect the Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 122

5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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LIST OF TABLES

Table Page

1.4.1. Fasting Glucose Tolerance Test . . . . . . . . . . . . . . . . . . . . . 20

1.4.2. Oral Glucose Tolerance Test . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.3. Gestational Diabetes Glucose Tolerance Test . . . . . . . . . . . . . . 21

2.2.1. Parameters in the Sturis-Tolic Model (2.2.1). . . . . . . . . . . . . . . 28

2.2.2. Parameters of the functions in the Sturis-Tolic Model (2.2.1). . . . . . 28

2.7.1. Parameters of the functions in Two Time Delay Model (2.3.1). . . . . 57

3.9.1. Parameters for subjects 6 and 7 in IVGTT Models (b5 = 23min.) . . . 104

4.2.1. Parameters of the Model 4.2.1 . . . . . . . . . . . . . . . . . . . . . . 111

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LIST OF FIGURES

Figure Page

1.2.1. Glucose-Insulin Regulatory System. . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1. Langerhans islets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2. The cells secrete insulin when glucose concentration level elevated . . . . . . . 12

1.3.3. Insulin signals cells to utilize glucose . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1. Insulin Secretion Ultradian Oscillations. . . . . . . . . . . . . . . . . . . . . 24

2.2.1. Physiological Glucose-Insulin Regulatory System . . . . . . . . . . . . . . . . 26

2.2.2. Functions fi(I), i= 1, 2, 4, 5. . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.1. Two Time Delay Glucose-Insulin Regulatory Model . . . . . . . . . . . . . . . 35

2.7.1. Bifurcation diagram of1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.7.2. Periods of periodic solutions when1[0, 20] and bifurcation diagram ofdi . . . 60

2.7.3. Bifurcation diagram ofGin[0, 2.16] . . . . . . . . . . . . . . . . . . . . . 61

2.7.4. Limit cycles in (Gin,G,I )-space when Gin[0, 2.16] . . . . . . . . . . . . . . 62

2.7.5. Periods of periodic solutions whenGin[0, 2.16] . . . . . . . . . . . . . . . . 62

2.7.6. Periods and peak time differences whendi changes in [0.001, 0.7] . . . . . . . . 63

2.7.7. Hepatic production delay has no impact to sustained oscillations. . . . . . . . . 64

2.7.8. Stability Region in (1, Gin)-plane . . . . . . . . . . . . . . . . . . . . . . . 65

2.7.9. Bifurcation diagrams and stability regions in (1, Gin)-space . . . . . . . . . . . 66

2.7.10. Stability Regions in (1, di)-plane and (Gin, di)-plane . . . . . . . . . . . . . . 67

2.7.11. Glucose concentrations peak before insulin does. . . . . . . . . . . . . . . . . 69

3.9.1. Periodic solutions for the discrete delay model (3.3.2) for subject 6 and 7. . . . . 105

4.3.1. Glucose-Insulin with Active-cell Interaction Diagram . . . . . . . . . . . . . 113

4.3.2. Function g(G) in GI-Model . . . . . . . . . . . . . . . . . . . . . . . . . 115

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4.4.1. Orbits of (G,I, ) ofGI model . . . . . . . . . . . . . . . . . . . . . . . . 116

4.4.2. Bifurcation diagram of1[0, 20] . . . . . . . . . . . . . . . . . . . . . . . 117

4.4.3. Bifurcation diagram ofGin[0, 3.0] . . . . . . . . . . . . . . . . . . . . . . 118

4.4.4. Periodic solutions and periods when Gin[0, 3.0] . . . . . . . . . . . . . . . . 119

4.4.5. Peaks of Oscillations in One Cycle . . . . . . . . . . . . . . . . . . . . . . . 120

4.4.6. Bifurcation diagram ofk[0.01, 2]. . . . . . . . . . . . . . . . . . . . . . . 121

4.4.7. Bifurcation diagram of[0.0001, 0.1] . . . . . . . . . . . . . . . . . . . . . 121

4.4.8. Limit Cycles when [0.0001, 0.1]. . . . . . . . . . . . . . . . . . . . . . . 122

4.4.9. There is no bifurcation whendi[0.005, 0.01] . . . . . . . . . . . . . . . . . 122

4.5.1. Possible -cell pulsatile oscillation? . . . . . . . . . . . . . . . . . . . . . . 124

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2

diabetes. The direct and indirect cost of diabetes in 2002 was $132 billions. The world

wide diabetics population is much higher, especially in underdeveloped countries.

Diabetes mellitus is currently classified as type 1 diabetes or type 2 diabetes

([9], [85]). Type 1 diabetes was previously called insulin-dependent diabetes mellitus

(IDDM) or juvenile-onset diabetes. It develops when the bodys immune system de-

stroys pancreatic beta cells, the only cells in the body that make the hormone insulin,

which regulates blood glucose. This form of diabetes usually strikes children and young

adults, although disease onset can occur at any age. Type 1 diabetes may account

for 5% to 10% of all diagnosed cases of diabetes. Risk factors for type 1 diabetes in-

clude autoimmune, genetic, and environmental factors. Type 2 diabetes is adult onset

or non-insulin-dependent diabetes mellitus (NIDDM) as this is due to a deficit in the

mass ofcells, reduced insulin secretion [53], and resistance to the action of insulin

[32]. The relative contribution and interaction of these defects in the pathogenesis of

this disease remains to be clarified [17]. About 90% to 95% of all diabetics diagnose

type 2 diabetes. Type 2 diabetes is associated with older age, obesity, family history

of diabetes, prior history of gestational diabetes, impaired glucose tolerance, physical

inactivity, and race/ethnicity. African Americans, Hispanic/Latino Americans, Native

Americans, some Asian Americans, Native Hawaiian, and other Pacific Islanders are

at particularly high risk for type 2 diabetes. Type 2 diabetes is increasingly being

diagnosed in children and adolescents ([93]).

In addition to Type 1 and Type 2 diabetes, gestational diabetes is a form of

glucose intolerance that is diagnosed in some women during pregnancy ([9], [85], [97]).

Gestational diabetes occurs more frequently among African Americans, Hispanic/Latino

Americans, and Native Americans. It is also more common among obese women and


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3

women with a family history of diabetes. During pregnancy, gestational diabetes re-

quires treatment to normalize maternal blood glucose levels to avoid complications in

the infant. After pregnancy, 5% to 10% of women with gestational diabetes are found

to have type 2 diabetes. Women who have had gestational diabetes have a 20% to

50% chance of developing diabetes in the next 5-10 years. Other specific types of dia-

betes result from specific genetic conditions (such as maturity-onset diabetes of youth),

surgery, drugs, malnutrition, infections, and other illnesses. Such types of diabetes may

account for 1% to 5% of all diagnosed cases of diabetes ([97]).

The relative contribution and interaction of these defects in the pathogenesis of

this disease remains to be clarified ([17]).

Due to the large population of diabetes patients in the world and the big health

expenses, many researchers are motivated to study the glucose-insulin endocrine metabolic

regulatory system so that we can better understand how the mechanism functions ([79],

[84], [85], [67], [74], [31], [85], [4] and their references), what cause the dysfunctions of

the system ([9] and its rich references), how to detect the onset of the either type of

diabetes including the so called prediabetes ([10], [83], [8], [97], [23], [57], [6], [63] and

their references), and eventually provide more reasonable, more effective, more efficient

and more economic treatments to diabetics. For example, according to Bergman ([6],

2002), there are now approximately 50 major studies published per year and more than

500 can be found in literature related to the so called minimal model([10], [83], [8]) for

modeling the intra-venous glucose tolerance test.


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4

2. Glucose-Insulin Endocrine Metabolic Regulatory System

Metabolism is the process of extracting useful energy from chemical bounds. A

metabolic pathway is a sequence of enzymatic reactions that take place in order to

transfer chemical energy from one form to another. The chemical adenosine triphos-

phate (ATP) is a common carrier of energy in a cell. There are two different ways to

form ATP:

1. adding one inorganic phosphate group (HP O24 ) to the adenosine diphosphate

(ADP), or

2. adding two inorganic phosphate groups to the adenosine monophosphate (AMP).

The process of inorganic phosphate group addition is referred to phosphorylation. Due

to the fact that the three phosphate groups in ATP carry negative charges, it requires

lots of energy to overcome the natural repulsion of like-charged phosphates when addi-

tional groups are added to AMP. So considerable amount of energy is released during

the hydrolysis of ATP to ADP ([51], [89] and [91]).

In the glucose-insulin endocrine metabolic regulatory system, the two pancre-

atic endocrine hormones, insulin and glucagon, are the primary dynamic factors that

regulate the system.

When the plasma glucose concentration rises, the elevation in the ratio of ATP/ADP

in a cell in the pancreas causes ATP-sensitive K+ channels (KATP channels) in the

plasma membrane to close. The decreased K+ permeability leads to membrane depo-

larization, opening of voltage-dependent Ca2+ channels, Ca2+ influx, and eventual rise

of the cytosolic Ca2+ concentration ([Ca2+]c) that triggers exocytosis ([91]).


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When the serum insulin concentration increases, more insulin receptors of cells

are bound by insulin. The binding of insulin to its receptors on the surfaces of cell

membranes leads to an increase in glucose transporter (GLUT4) molecules in the outer

membrane of muscle cells and adipocytes, and therefore to an increase in the uptake

of glucose from blood into muscle and adipose tissue. Thus, the intracellular glucose is

consumed and energy is released ([91]).

After some amount of the plasma glucose is utilized by the cells and the concen-

tration level is low, the cells are signaled not to release insulin. Then the amount of

extracellular glucose transported into intracellular by the glucose transporters is signifi-

cantly reduced or even stopped due to the decreased number of insulin receptors bound

by insulin. Therefore, the consumption of glucose is tremendously decreased.

When the glucose concentration level is low, the cells in the pancreas will

release glucagon to the liver and the liver will convert glucagon into glucose. The liver

also converts glycogen into glucose.

In short, when humans the plasma glucose concentration level is high, the fol-

lowing processes will occur:

1. the pancreas is signaled to release insulin from cells;

2. serum insulin (including newly secreted insulin) binds to the cells insulin recep-

tors,

3. the insulin receptors bound by insulin cause the glucose transporters (GLUT4)

transport glucose molecules into the cells;

4. the cells consume the glucose and convert to energy.


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6

These processes decrease the glucose concentrations in the plasma. Almost all the cells

in human body have insulin receptors, including fat cells and muscle cells. Glucose

is also utilized by other cells without insulin involvement. The brain cell is a typical

example.

When a humans the plasma glucose concentration level is low, a different series

of processes will occur:

1. the pancreas is signaled to release glucagon from cells;

2. glucagon is transported to the liver;

3. the liver converts the glucagon to glucose.

These processes increase the glucose concentration level in human plasma.

Exogenous glucose infusion also increases glucose concentration. The typical ex-

ogenous glucose infusions include meal ingestion, oral glucose intake, continuous enteral

nutrition, and constant glucose infusion.

The liver plays a key role in keeping the glucose and insulin amount in human

plasma oscillating smoothly ([96]). Figure 1.2.1, which is adapted from [96], illustrates

the plasma glucose-insulin endocrine metabolic regulatory system.

3. The pancreas and Its Endocrine Hormones

3.1. The pancreas. The pancreas lies interior to a humans stomach, in the

size of a humans fist and is in the bend of the duodenum. Scattered through out

inside of the pancreas, there are about a million Langerhans islets. Each Langerhans

islet contains about three hundred cells and each cell contains about one thousand

granules. Approximately 5% of the total pancreatic mass is comprised of endocrine


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7

Glucose Infusion,

meal, enteral,

oral intake

and others

Exercises,

fasting

and others

-cells release

glucagon -cells release insulin

Liver converts partial

glucagon released

from -cells and partial

glycogen stored in liver

to glucose

Low

Plasma Glucose

Level

High

Plasma Glucose

Level

Normal

Plasma Glucose

Level

Pancreas

Insulin

Liver

Glucagon

Insulin helps

to consume

plasma glucose

Figure 1.2.1. Glucose-Insulin Regulatory System

The dashed lines indicate that exercises and fasting consume glucose and lower the glucose concentra-

tion, which signals the pancreas to release glucagon and the liver converts the glucagon and glycogen

to glucose. The solid lines indicate that the glucose infusion elevate the plasma glucose concentration

level which signals the pancreas to secrete insulin and consume the glucose. (This figure is adapted

from [96].)

cells. These endocrine cells are clustered in groups within the pancreas, which look

like little islands of cells when examined under a microscope. The pancreas is both an

endocrine and an exocrine gland. The exocrine functions are concerned with digestion.

The endocrine function consists primarily for the secretion of the two major hormones,

insulin and glucagon, which participate in the regulation of carbohydrate metabolism.

Five types of cells in a Langerhans islet are identified: cells, which occupy

65-80% of the islet and make insulin; cells, which occupy 15-20% and make glucagon;

cells, which occupy 3-10% and make somatostatin ([87]); and pancreatic polypeptide-

containingP Pcells and D1 cells comprise 1% ([2]), about which little is known. Figure


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9

new perspective ([13], [14], [34], [82] and [49]).

The cells release glucagon, a protein hormone that has important effects in the

regulation of carbohydrate metabolism. Glucagon is a catabolic hormone, that is, it

mobilizes glucose, fatty acids and amino acids from storage into the blood. When the

glucose concentration level in the plasma is low, the liver will convert the glucagon to

glucose.

Both insulin and glucagon are important in the regulation of carbohydrate, pro-

tein and lipid metabolism.

Somatostatin is secreted from the cells in the Langerhans islets in the pan-

creas and is a hormone inhibiting the secretion of many other hormones. Somatostatin

acts through both endocrine and paracrine pathways to affect its target cells. In the

pancreas, somatostatin appears to act primarily in a paracrine manner to inhibit the se-

cretion of both insulin and glucagon. In the brain (hypothalamus) and the spinal cord

it may act as a neurohormone and neurotransmitter. The effects of somatostatin to

glucose-insulin regulatory system is small, indirect and negligible. Its paracrine manner

makes the secretion of insulin and glucagon smoother.

3.2. Glucose Transporters. Glucose is transported by its transporters. There

are total five transporters in the family, that is, GLUT1 to GLUT5 ([91]).

GLUT1 is ubiquitously distributed in various tissues.

GLUT2 is found primarily in intestine, kidney and liver.

GLUT3 is found in the intestine.

GLUT4 is primarily contained in insulin-sensitive tissues such as skeletal muscle

and adipose tissue.


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10

GLUT5 is found in the brain and testis. GLUT5 is also the major glucose trans-

porter present in the membrane of the endoplasmic reticulum (ER) and serves the

function of transporting glucose to the cytosol following its dephosphorylation by

the ER enzyme glucose-6-phosphatase.

When the concentration of blood glucose increases in response to food intake,

pancreatic GLUT2 molecules mediate an increase in glucose uptake which leads to

increased insulin secretion. Recent evidence has shown that the cell surface receptor

for the human T cell leukemia virus (HTLV) is the ubiquitous GLUT1. ([91])

3.3. Secretion and Actions of Insulin. Insulin secretion is pulsatile and is

regulated primarily by the glucose metabolism ([67], [74]). Numerous in-vivo and in-

vitroexperiments have shown that insulin concentration oscillates in two different time

scales: rapid oscillation with a period of 5-15 minutes and ultradian oscillation with a

range of 50-140 minutes ([67], [74] and their cited references). The rapid oscillations

are caused by coordinating periodic secretory bursting of insulin from cells contained

in millions of the Langerhans islets in the pancreas. These bursts are the dominant

mechanism of insulin release at basal level ([67]). Ultradian oscillations of insulin con-

centration are believed to be mainly due to glucose interaction in the plasma ([79], [84],

[74]). These ultradian oscillations are best seen after meal ingestion, oral glucose intake,

continuous enteral nutrition or intravenous glucose infusion ([79]). In addition, muscle,

the brain, nerve and others utilize the plasma glucose to complete the regulatory system

feedback loop. So, insulin production, glucose infusion and production (for example,

meal and continuous enteral nutrition in daily life) and glucose utilization (for example,

in daily life, exercise) are the three major variables of this intricate regulatory system

([74], [79]).


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11

P. Gilon, M. A. Ravier, J.-C. Jonas, and J.-C. Henquin summarized the mech-

anism of insulin secretion control in 2002 ([39]). Glucose stimulates insulin secretion

from-cells by activating two pathways that require metabolism of the sugar as follows

([47]).

Triggering Pathway The GLUT2 transports the glucose into the cell. It

causes the rise in the ratio of ATP/ADP which causes ATP-sensitive K+ channels

(KATP

channels) in the plasma membrane to close. The decreased K

+

permeability

leads to membrane depolarization, opening of voltage-dependent Ca2+ channels,

Ca2+ influx, and the eventual rise of the cytosolic Ca2+ concentration ([Ca2+]c)

that triggers exocytosis. This pathway is also called KATP channel-dependent

pathway. See Figure 1.3.2 for an illustration.

Amplifying Pathway The KATPchannel-independent pathway simply increases

the efficiency of the Ca2+ on exocytosis when the concentration of Ca2+ has been

elevated.

The pulsatility of insulin secretion might result from oscillations in either of these

transduction pathways. Because metabolism and [Ca2+]c play key roles in the control

of insulin secretion and have been reported to oscillate, many efforts have been spent

to investigate which of these two mechanisms is the primary factor of pulsatile insulin

secretion ([39]). The essential role of Ca2+ influx in the generation of [Ca2+]coscillations

by glucose, in either whole islets or single -cells, is demonstrated by their abrogation

upon omission of extracellular Ca2+ ([44], [38]) or blockade of voltage-dependent Ca2+

channels ([26]). [Ca2+]c oscillations are linked to oscillations of the membrane potential

in -cells ([72], [38]), and it is assumed that mixed [Ca2+]c oscillations result from an

irregular (so-called periodic) electrical activity ([3], [46], [20]). Synchronization of the


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12

-cell electrical activity ([62]) by gap junctions is likely to underlie the synchronization

of [Ca2+

]c oscillations between -cells within the islet ([44], [50] and [43]). See Figure

1.3.2 for an illustration.

Elevate

K+

Close K+channels

Open Ca2+channeles

Ca2+influx

Granules

Insulin

Elevated Ca2+

NAD(P)HH+

Glucokinase

Glu

cose

Cell Depolarization

Protein

Phosphorylations

Glucose-6-phosphate

ATP

ADP

GLUT2

GlucoseMetabolism

Figure 1.3.2. The cells secrete insulin when glucose concentration level elevated

The facilitated GLUT2 transport the glucose into the cell and the glucose is phosphorylated by

glucokinase. The ratio of ATP:ADP is elevated. The glucose metabolism causes ATP-sensitive K+

channels to close, the membrane to depolarize and the Ca2+ channels to open. This triggers a cascade

of protein phosphorylations and leads to insulin exocytosis [68]. (The figure is partially adapted from

[68].)

The insulin has five major actions. These include:

facilitation of glucose transport through certain membranes (e.g. adipose and

muscle cells);

stimulation of the enzyme system for conversion of glucose to glycogen (liver and

muscle cells);

slow-down of gluconeogenesis (liver and muscle cells);


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13

regulation of lipogenesis (liver and adipose cells); and

promotion of protein synthesis and growth (general effect).

These actions of insulin are mediated by the binding of the hormone to membrane re-

ceptors to trigger several simultaneous actions. A major effect of insulin is to promote

the entrance of glucose and amino acids in cells of muscle tissues, adipose tissue and

connective tissue. Glucose enters the cell by facilitated diffusion along an inward gradi-

ent created by low intracellular free glucose and by the availability of a specific carrier

called transporter. In the presence of insulin, the rate of movement of glucose into the

cell is greatly stimulated in a selective fashion. ([89].)

In the liver, insulin does not affect the movement of glucose across membranes

directly but facilitates glycogen deposition and decreases glucose output. Consequently,

there is a net increase in glucose uptake. Insulin induces or represses the activity of

many enzymes; however, it is not known whether these actions are direct or indirect. For

example, insulin suppresses the synthesis of key gluconeogenic enzymes and induces the

synthesis of key glycolytic enzymes such as glucokinase. Glycogen synthetase activity is

also increased. Insulin likewise increases the activity of enzymes involved in lipogenesis

.

3.4. Insulin Receptors. In molecular biology, the insulin receptor is a trans-

membrane glycoprotein that is activated by insulin. It belongs to the large class of

tyrosine kinase receptors. Two subunits and two subunits make up the insulin re-

ceptor. Thesubunits pass through the cellular membrane and are linked by disulfide

bonds ([90]).

The insulin receptors are embedded in the plasma membrane of hepatocytes

and myocytes. The binding of insulin to the receptors is the initial step in a signal


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14

transduction pathway, triggering the consumption and metabolism of glucose ([89], [86]).

Bound by insulin, the insulin receptor phosphorylates from ATP to several proteins

in the cytoplasm, including insulin receptor substrates (IRS-1 and IRS-2) containing

signaling molecules, activates Phosphatidylinositol 3-kinase (PI-3-K) and leads to an

increase in glucose transporter (GLUT4) molecules ([98]) in the outer membrane of

muscle cells and adipocytes, and therefore to an increase in the uptake of glucose from

blood into muscle and adipose tissue ([89]). GLUT4 will transport the glucose to the

cells efficiently. Figure 1.3.3 elucidates this signaling pathway.

Intracellular phosphorylation of glucose is rapid and efficient and therefore the

glucose concentration is low. Thus, a certain amount of glucose moves into the cell

regardless of the existence of insulin. With insulin, however, the rate of glucose entry is

much increased due to the facilitated diffusion as mediated by the glucose transporters

([89]). Refer to Figure 1.3.3.

However, the kinetics of insulin receptor binding are complex. The number of

insulin receptors of each cell changes opposite to the circulating insulin concentration

level. Increased insulin circulating level reduces the number of insulin receptors per cell

and the decreased circulating level of insulin triggers the number of receptors to increase.

The number of receptors is increased during starvation and decreased in obesity and

acromegaly. But, the receptor affinity is decreased by excess glucocorticoids. The

affinity of the receptor for the second insulin molecule is significantly lower than for the

first bound molecule. This may explain the negative cooperative interactions observed

at high insulin concentrations. That is, as the concentration of insulin increases and

more receptors become occupied, the affinity of the receptors for insulin decreases.

Conversely, at low insulin concentrations, positive cooperation has been recorded. That


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ATP

Phosphorylations

IRS-1IRS-2

PI-3 Kinase GLUT4

Insulin

Cell membrane

G

Other activities

Glucose

GG

G GG

G

I

G

I

I

G

G

G

G

Insulin receptor

G

-Unit

-Unit

-Unit

-Unit

-S-S-

-S-S-

-S-S-G

G

Figure 1.3.3. Insulin signals cells to utilize glucose

Insulin binds to its receptors on the membrane of the cells and phosphorylates several proteins in the

cytoplasm, including insulin receptor substrates (IRS-1 and IRS-2) containing signaling molecules, ac-

tivates Phosphatidylinositol 3-kinase (PI-3-K) and leads to an increase in glucose transporter (GLUT4)

molecules. This leads to an increase in glucose transporter (GLUT4) molecules. GLUT4 will transport

the glucose to the cells efficiently.

is, the binding of insulin to its receptor at low insulin concentrations seems to enhance

further binding (([89]), [86]).

3.5. Insulin Resistance. Insulin resistance is defined as when insulin is inef-

ficient in causing the plasma glucose to enter the cells of a body and to be utilized by

the cells for energy, even if there is enough insulin in serum. That is, the cells resist

the insulin. In addition, the liver may continue to secrete glucose into the bloodstream

even when the glucose is not needed.

The reasons for insulin resistance occurring are still uncertain. Certain genes

predispose certain people to develop insulin resistance. Some factors are, for example,

lack of exercise, obesity, and chronically high blood sugar levels may cause insulin

resistance in susceptible individuals. [95]


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16

Previously, the perspective was that the abnormal binding to the insulin receptors

of the cells was the major reason of insulin resistance. This is no longer believed to be

the case. [95]

Currently, many researchers are active in determining the cause of insulin resis-

tance at the cellular and molecular levels. Postbinding abnormalities, believed by

most researchers, is the cause of insulin resistance. Several chemical pathways and

genes causing the abnormalities have been identified. A typical example is that the

glucose transporter GLUT4 is deficient in some individuals showing insulin resistance.

The activity of GLUT4 is to transport the glucose into the body cells after the insulin

is bound to the insulin receptors. [95]

3.6. Insulin Degradation and Clearance. Insulin degradation is a broad

and rich research area and this is not the major focus of this dissertation. We will only

discuss this briefly. (For more information, refer to [5], [27], [33], [42] and their cited

references.)

Insulin is cleared mainly by the liver and kidney, but most other tissues also

degrade the hormone ([33]). Insulin-degrading enzyme (IDE) is the major enzyme in

the proteolysis of insulin in addition to several peptides ([27]). It resides in a region

of chromosome 10q that is linked to Type 2 diabetes ([42]). IDE is the major enzyme

responsible for insulin degradation in vitro, but the extent to which it mediates insulin

catabolism in vivo has been controversial, with doubts expressed that IDE has any

physiological role in insulin catabolism ([33] and cited references). Insulin is degraded

by enzymes in the subcutaneous tissue ([64]) and interstitial fluid as well ([7]). The

insulin is degraded by insulin receptors as well as when the insulin is bound to its

receptors ([85]).


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17

3.7. Production and Consumption of Glucose. Glucose is liberated from

dietary carbohydrates such as starch or sucrose by hydrolysis within the small intestine,

and then is absorbed into the blood. The most often ways of glucose infusion are through

meal ingestion; oral glucose intake; continuous enteral nutrition; and constant glucose

infusion ([79] and [84]).

Insulin controls the hepatic glucose production (conversion from glucagon) and

release rate by the liver ([89]). When the blood glucose level drops, the liver converts

glycogen to glucose and releases it into the bloodstream. When there is enough glucose

in the bloodstream, insulin secreted by the pancreas signals the liver to shut down

glucose production. In healthy people, the pancreas continually measures blood glucose

levels and responds by secreting just the right amount of insulin. The liver converts the

glycogen to glucose as well as when the plasma glucose concentration level is low.

The insulin receptor leads that the glucose molecules go into the muscle cells,

fat cells and others. These cells utilize the glucose. Elevated concentrations of glucose

in the blood stimulate the release of insulin. Insulin acts on cells throughout the body

to stimulate uptake, utilization and storage of glucose. Within seconds to minutes the

rate of glucose entry into tissue cells increases 15 to 20 times. Once glucose enters

the tissue cells, insulin enhances its oxidation, stimulates its conversion to glycogen,

activates transport of amino acids into cells, promotes protein synthesis and inhibits

virtually all liver enzymes that promote gluconeogenesis. The effects of insulin on

glucose metabolism vary depending on the target tissue. Two important effects are

([89]) (see also Figure 1.3.3 for an illustration.):

Higher Insulin Concentration Leads to More Glucose Uptake Insulin

facilitates entry of glucose into muscle, adipose and several other tissues. The only


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18

mechanism by which cells can take up glucose is by facilitated diffusion through a

family of hexose transporters. In many tissues, e.g., muscle, the major transporter

used for uptake of glucose (GLUT4) is made available in the plasma membrane

through the action of insulin.

Lower Insulin Concentration Leads to Less Glucose Uptake In the

absence of insulin, GLUT4 glucose transporters are present in cytoplasmic vesicles,

where they are useless for transporting glucose. Binding of insulin to receptors

on such cells leads rapidly to fusion of those vesicles with the plasma membrane

and insertion of the glucose transporters, thereby giving the cell the ability to

efficiently take up glucose. When blood levels of insulin decrease and insulin

receptors are no longer occupied, the glucose transporters are recycled back into

the cytoplasm. Therefore, the glucose uptake is significantly decreased.

Insulin stimulates the liver to store glucose in the form of glycogen. A large

fraction (50%) of glucose absorbed from the small intestine is immediately taken up by

hepatocytes, which convert it into the storage polymer glycogen ([89]).

Insulin has several effects in the liver that stimulate glycogen synthesis. First, it

activates the enzyme hexokinase, which phosphorylates glucose, trapping it within the

cell. Coincidentally, insulin acts to inhibit the activity of glucose-6-phosphatase. Insulin

also activates several of the enzymes that are directly involved in glycogen synthesis,

including phosphofructokinase and glycogen synthase. The net effect is clear: when the

supply of glucose is abundant, insulin signals the liver to store as much of it as possible

for use later ([89]).

Many cells consume the glucose without involvement of the insulin receptor effect.

The brain and the liver do not use GLUT4 to transport glucose. Instead, a type of


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19

insulin-independent transport is used. This constitutes the insulin-independent glucose

utilizations ([89]).

4. Glucose Tolerance Test

A series of glucose tolerance tests have been developed over the year and applied

in clinics and experiments ([93], [10], [8], [41], [76], [16] and [61]). Each of the glucose

tolerance tests is to diagnose if an individual has diabetes or has potential to have

diabetes. The basic idea is to test ones glucose-insulin endocrine metabolic system

after a large amount of glucose infusion.

The glucose tolerance tests include Fasting Glucose Tolerance Test (FGTT),

Oral Glucose Tolerance Test (OGTT), Intra Venous Glucose Tolerance Test (IVGTT),

frequently sampled Intra Venous Glucose Tolerance Test (fsIVGTT) ([93], [60] and [59]).

The Fasting Glucose Tolerance Test (FGTT) needs the individual to fast for 8-10 hours

before his/her the plasma glucose is sampled. The meanings of the test results are

summarized in Table 1.4.1. The Oral Glucose Tolerance Test (OGTT) is another type

of glucose tolerance test. The individual is given a glass of glucose liquid (75mg) to

intake and his/her the plasma glucose level will be sampled. The test result meanings

are defined in Table 1.4.2. To diagnose gestational diabetes, a pregnant woman is

required to drink a glass of glucose water containing 50mg glucose. Her the plasma

glucose is sampled one hour later. The meanings of the test results are listed in Table

1.4.3. The American Diabetes Association suggests two tests need to be performed to

determine if an individual has diabetes or pre-diabetes ([93]).

The Intra-venous Glucose Tolerance Test (IVGTT) and the frequently sampled

Intra-venous Glucose Tolerance Test (fsIVGTT) are to test the insulin sensitivity or


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Table 1.4.1. Fasting Glucose Tolerance TestThe plasma Glucose Meaning

70-99 mg/dl (3.9-5.4 mmol/l) normal glucose tolerance100-125 mg/dl (5.5-6.9 mmol/l) impaired fasting glucose (pre-diabetes)

Over 126 mg/dl (7.0 mmol/l) and above probable diabetes

Table 1.4.2. Oral Glucose Tolerance Test

The plasma Glucose MeaningBelow 140 mg/dl (7.8 mmol/l) normal glucose tolerance

140-200 mg/dl (7.8-11.1 mmol/l) impaired fasting glucose (pre-diabetes)Over 200 mg/dl (11.1 mmol/l) probable diabetes

response to high the plasma glucose concentration. The procedure of IVGTT is similar

to other glucose tolerance tests but the plasma glucose and serum insulin are sampled

more frequently. In the test, the individual to be tested needs to fast 8-10 hours and

is then given a bolus of glucose infusion, for example, 0.33 g/kg body weight [23]

or 0.5 g/kg body weight of a 50% solution and is administered into an antecubital

vein in approximately 2.5 minutes. Within the next 180 minutes, the individuals

the plasma glucose and serum insulin are sampled frequently. According to the rich

information in the sampled data, the insulin sensitivity can be accurately determined.

Many models study the Intravenous Glucose Tolerance Test (IVGTT), which focuses

on the metabolism of glucose in a short time period starting from the infusion of big

bolus (0.33 g/kg) of glucose at time t = 0. As pointed out in Chapter 2, due to the

large amount of intravenous glucose infusion, the insulin response time delay of the

small amount of hepatic glucose production is insignificant and thus negligible and

furthermore is assumed at a small constant infusion rate in the models ([10], [8], [23],

[57] and [63]). The most noticeable model is the so called Minimal Model which


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Table 1.4.3. Gestational Diabetes Glucose Tolerance TestThe plasma Glucose Meaning

Below 140 mg/dl (7.8 mmol/l) normal glucose toleranceOver 140 mg/dl (7.8 mmol/l) abnormal, needs oral glucose tolerance test

contains minimal number of parameters ([10], [8]) and it is widely used in physiological

research work to estimate metabolic indices of glucose effectiveness (SG) and insulin

sensitivity (SI) from the intravenous glucose tolerance test (IVGTT) data by sampling

over certain periods (usually 180 minutes) ([41]). Also a few are on the control through

meals and exercise ([25]). See also a review paper by Mari ([60]) for a classification of

models.

5. The Organization of This Dissertation

In this dissertation, we propose a more realistic DDE model for the insulin secre-

tion ultradian oscillations in Chapter 2. This model (Model (2.3.1)) contains two time

delays: the first mimic the hepatic glucose production time delay and the other reflects

the insulin response time delay to increased glucose concentration. Both analytical and

numerical analysis are performed. The results obtained include global and local sta-

bility analysis of steady state, persistence of solutions and numerical simulation with

insightful results.

In Chapter 3, we propose three models (Model (3.3.1), (3.3.2) and (3.3.3)) for

modeling the effective and powerful intravenous glucose tolerance test. We performed

global and local stability analysis of the steady state and numerical simulations based

on clinic data from diabetics.


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22

In Chapter 4, we present another DDE model to investigate the effects of the

mass of the active cells. Our numerical analysis shows that we simulated the glucose-

insulin endocrine metabolic system taking active cell mass into account. Due to the

fact that this area is relatively new, our study is still preliminary. More thorough studies

are needed.


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CHAPTER 2

The Ultradian Oscillations of Insulin Secretion

1. Introduction

Endocrine systems often secrete hormones in pulses [21] [56]. Examples include

the release of growth hormone and gonadotropins, and also the secretion of insulin from

the pancreas, which are secreted over intervals of 1-3 hours and 80-150 minutes, respec-

tively. It has been suggested that relative to constant or stochastic signals, oscillatory

signals are more effective at producing a sustained response in the target cells [40] [58].

Numerousin-vivoandin-vitroexperiments have shown that insulin concentration

oscillates in two different time scales: rapid oscillation with a period of 5-15 minutes

and ultradian oscillation with a range of 80-150 minutes ([79], [67], [74] and [73]).

The mechanisms underlying both types of oscillations are not fully understood.

The rapid oscillations may arise from an intra-pancreatic pacemaker mechanism [77]

and caused by coordinating periodic secretory bursting of insulin from cells contained

in the millions of the Langerhans islets in the pancreas. These bursts are the domi-

nant mechanism of insulin release at basal level ([67]). Often, the rapid oscillation is

superimposed on the slow (ultradian) oscillation ([79]).

Ultradian oscillations of insulin concentration are believed to be mainly due to

glucose interaction in the plasma and an instability in the insulin-glucose feedback sys-


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Glucose(gm/dl)

In

sulin(U/ml)

80

100

120

140

240 480 720 960 1200 1440

30

A

10

20

40

0

50

Glucose(gm/dl)

60

100

120

160

60 120 160 240

140

80

0

40

60

100

80

20In

sulin(U/ml)

B

Glucose

(gm/dl)

Insulin(U/ml)

240 480 720 960 1200 1440

30

C

10

20

40

0

50

Glucose(gm/dl)

100

120

140

180

240 480 720 1200

160

10

20

30

40

Insulin(U/ml)

D

40

80

100

140

120

60

840

Figure 2.1.1. Insulin Secretion Ultradian Oscillations

These figures illustrate the insulin secretion ultradian oscillations. The glucose infusion rate are A.

meal ingestion; B. oral glucose intake; C. continuous enteral nutrition; D. constant glucose infusion,

respectively. (The figures are adapted from [79].)

tem ([79], [84], [74] and [60]). These ultradian oscillations are best seen after meal

ingestion, oral glucose intake, continuous enteral nutrition or intravenous glucose in-

fusion (Figure 2.1.1). In addition, muscles, the brain, nerves and others utilize the

plasma glucose to complete the regulatory system feedback loop ([79], [84]). So, insulin

production, glucose infusion and production (for example, meal and continuous enteral

nutrition in daily life) and glucose utilization (for example, in daily life, exercise) are

the three major factors of this intricate regulatory system ([74], [79] and [59]).

The hypothesis that the ultradian insulin secretion is an instability in the insulin-

glucose feedback system has been the subject of a number of studies, including some

which have developed a mathematical model of the insulin-glucose feedback system

([51], [79], [84], [31] and [4]).

This chapter is organized as follows. Section 2 summarizes the current study

status with focus on the Sturis-Tolic Model. Section 3 presents our two time delay


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Insulin

Glucose

Glucose

ProductionGlucose

Utilization

Insulin

Secretion

(-) (-)

(-) (-)

Figure 2.2.1. Physiological Glucose-Insulin Regulatory System

These four negative feedback loop show the glucose stimulating pancreatic beta cells to secrete insulin,

insulin stimulating glucose uptake and inhibiting hepatic glucose production, and also positive feedback

as glucose enhances its own uptake ([79]). (This figure is adapted from [79].)

dG(t)

dt =G

=Gin f2(G(t)) f3(G(t))f4(Ii(t)) +f5(x3),

dIp(t)

dt =Ip=f1(G(t)) E(

Ip(t)

VpIi(t)

Vi) Ip(t)

tp,

dIi(t)

dt =Ii =E(

Ip(t)

Vp Ii(t)

Vi) Ii(t)

ti,

dx1(t)

dt =x1=

3

td(Ip x1),

dx2(t)

dt =x2=

3

td(x1 x2),

dx3(t)

dt =x3=

3

td

(x2 x3),

(2.2.1)


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27

where G(t) is the amount of glucose, Ip(t) and Ii(t) are the amount of insulin in the

plasma and the intercellular space, respectively, Vp is the plasma insulin distribution

volume,Vi is the effective volume of the intercellular space, E is the diffusion transfer

rate, tp and ti are insulin degradation time constants in the plasma and intercellu-

lar space, respectively, Gin indicates (exogenous) glucose supply rate to plasma, and

x1(t), x2(t) and x3(t) are three auxiliary variables associated with certain delays of the

insulin effect on the hepatic glucose production with total time td. f1(G) is a function

modeling the pancreatic insulin production as controlled by the glucose concentration,

f2(G) and f3(G)f4(Ii) are functions, respectively, for insulin-independent and insulin-

dependent glucose utilization by various body parts (for example, brain and nerves (f2),

and muscle and fat cells (f3f4)) and f5(x3) is a function modeling hepatic glucose pro-

duction with time delay td collaborated with auxiliary variables x1, x2 and x3. Based

on experimental results ([79], [84]), all the parameters in the model are given in Table

(2.2.1) andfi,i = 1, 2, 3, 4, 5, take following forms and the parameters listed in Table

2.2.2.

f1(G) = Rm

1 + exp((C1 G/Vg)/a1) , (2.2.2)

f2(G) =Ub(1 exp(G/(C2Vg))), (2.2.3)

f3(G) = G

C3Vg, (2.2.4)

f4(Ii) =U0+

0.1(Um

U0)

1 + exp(ln(Ii/C4(1/Vi+ 1/Eti))) , (2.2.5)


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Table 2.2.1. Parameters in the Sturis-Tolic Model (2.2.1).

Parameters Values UnitsVp 3 lVi 11 lE 0.2 l min1tp 6 minti 100 min

Table 2.2.2. Parameters of the functions in the Sturis-Tolic Model (2.2.1).

Parameters Units ValuesVg l 10

Rm Umin1 210

a1 mg l1 300C1 mg l1 2000Ub mg min1 72C2 mg l1 144

C3 mg l1

1000

Parameters Units ValuesU0 mgmin

1 40Um mgmin

1 940 1.77

C4 Ul1 80

Rg mgmin1 180

lU1 0.29

a1 Ul

1

26

f5(x) = Rg

1 + exp((x/Vp C5)) , (2.2.6)

Figure (2.2.2) display the graphs of the above functions, fi, i= 1, 2, 3, 4, 5. The

importance of these functions is their shapes rather than their forms [51].

This model comprised of two major negative feedback loops describing the effects

of insulin on glucose utilization and glucose production, respectively, and both loops

include the stimulatory effect of glucose on insulin secretion. The authors of [84] hoped

to identify a possible mechanism behind the efficiency of oscillatory insulin secretions.

Analysis of the original model revealed that the slow oscillations of insulin secretion

could arise from a Hopf bifurcation in the insulin-glucose feedback mechanism. The

model included several feedback loops (see Figure 2.2.1), including: glucose stimulating


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29

G

400003000020000100000

70

60

50

40

30

20

10

0

800

600

400

200

I

4003002001000

f2(G) f4(I)

160

120

80

40

0

x

200150100500

200

150

100

50

0

G

400003000020000100000

f5(I) f1(G)

Figure 2.2.2. Functions fi(I),i= 1, 2, 4, 5.

pancreatic beta cells to secrete insulin, insulin stimulating glucose uptake and inhibiting

hepatic glucose production, and also positive feedback as glucose enhances its own

uptake.

The model includes two significant delays. One, 5-15 min., is sluggish effect of

insulin on glucose utilization, reflecting that the effect is dependent on the concentra-

tion of insulin in a slowly equilibrating intercellular compartment as opposed to the

concentration of the plasma insulin. The other delay, 25-50 min., is due to the time

lag between the appearance of insulin in the plasma and its inhibitory effect on hepatic


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30

glucose production. This delay is simulated by introducing three auxiliary variables

x1, x2 and x3, which is called the third order delay. We demonstrate how the auxiliary

variables simulate time delay as follows. For simplicity, assume the first order delay,

that is,x1(t) = (Ip(t) x1(t))/td, where td>0 is the time delay. Then

Ip(t td) =x1(t td) +x1(t td)td

Observe the Taylors expansion ofx1(t) at t td,

x1(t) =x1(t td) +x1(t td)td+o(td).

So x1(t)Ip(t td). The occurrence of sustained insulin and glucose oscillations was

found numerically to be dependent on these two time delays.

Model simulations suggested that the interaction of the oscillatory insulin supply

with the glucose receptors of the glucose utilizing cells was of minimal importance. This

was because the oscillations in the concentration of the intercellular insulin were small,

and changes in the average glucose utilization only depend weakly on amplitude. How-

ever, with their model they were able to resolve conflicting results from clinical studies.

Different experimental conditions will influence hepatic glucose release. If hepatic glu-

cose release is occurring near its maximum limit, an oscillatory insulin supply will be

more effective at lowering the blood glucose level than a constant supply. However, if

the insulin level is sufficiently high to cause the hepatic release of glucose to virtually

disappear, the opposite is observed. For insulin concentrations close to the point of

inflection of the insulin-glucose curves (f1 andf5), an oscillatory and a constant insulin

secretion produce similar effects. Under the assumption of constant glucose infusion,

the authors observed following numerical observations.

ST1 The ultradian insulin secretion oscillation is critically dependent on hepatic glu-

cose production, that is, if there is no hepatic glucose production, then there is


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31

no insulin secretion oscillation.

ST2 When the hepatic glucose production time delay2(25, 50), the period of the

periodic solutions of both insulin and glucose is in interval (95, 140) (min.), that

is,(95, 140).

ST3 To obtain the ultradian oscillation (periodic solutions), it is necessary to break

the insulin into two separate compartments, the plasma and interstitial tissues.

ST4 The ultradian oscillation is sensitive to both the speed of insulin reaction to the

increased plasma glucose concentration level and the speed of the hepatic glucose

production triggered by insulin. Specifically, if the slope in the reflexive points of

function f1 and f5 is reduced by 10 20%, the oscillation becomes damped.

K. Engelborghs, V. Lemaire, J. Belair and D. Roose ([31], 2001) introduced a

single time delay in the Negative Feedback Loop Model and proposed following DDE

model.

G(t) =Eg f2(G(t)) f3(G(t))f4(I(t)) + f5(I(t )),

I

(t) =f1(G(t)) I(t)

t1 ,

(2.2.7)

where the functions, fi, i = 1, 2, 3, 4, 5, and their parameters are assumed to be the

same as those in the Model (2.2.1). Eg stands for the glucose infusion rate and the

term 1/t1 is the insulin degradation rate. The positive constant delay mimics the

hepatic glucose production delay (5-15 min.). This model ignores the glucose stimulat-

ing insulin secretion time delay. Due to the complex chemical reactions on the cells,


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32

the insulin secretion occurs a few minutes after the plasma glucose concentration rises.

This significant time delay (5-15 min.) is not negligible in physiology.

The other DDE model proposed by K. Engelborghs, V. Lemaire, J. Belair and

D. Roose ([31], 2001) is trying to model the exogenous insulin infusion. The authors

assumed that the exogenous insulin infusion function takes the same form as internal

insulin production, which is, as the authors admitted, too artificial.

G(t) =Eg f2(G(t)) f3(G(t))f4(I(t)) +f5(I(t 2)),

I(t) =f1(G(t)) I(t)t1

+ (1 )f1(G(t 1)).

(2.2.8)

Nevertheless, a noticeable addition to the work of [31] is the usage of DDE-

BifTool software package ([30]) to analyze and simulate the bifurcation diagram and

other numerical analysis.

Due to the lack of physiological meanings, we would not summarize the analytical

and numerical results presented in [31].

In 2004, D. L. Bennett and S. A. Gourley ([4]) modified the Sturis-Tolic ODE

Model ([79] and [84]) by removing the three auxiliary linear chain equations and their

associated artificial parameters and introducing a time delay into the model explicitly.

This time delay stands for the hepatic glucose production, which is the same as

proposed in [31]. Unlike [31] in which the sluggish effect of glucose on insulin is ignored,

D. L. Bennett and S. A. Gourley ([4]) kept the idea in [79] and [84] of breaking the

insulin in two compartments to simulate the time delay of insulin secretion triggered

by rising glucose concentration level. The DDE model takes following form. All the


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parameters and functions are the same as that in model (2.2.1) given in (2.2.2) to (2.2.6)

and Table 2.2.1 and 2.2.2.

G(t) =Gin f2(G(t)) f3(G(t))f4(Ii(t)) +f5(Ip(t )),

Ip(t) =f1(G(t)) E(Ip(t)

VpIi(t)

Vi) Ip(t)

tp,

Ii(t) =E(Ip(t)

VpIi(t)

Vi) Ii(t)

ti,

(2.2.9)

Their major analytical results are a sufficient condition of global asymptotical

stability induced by a Liapunov function for the case that the hepatic glucose production

time delay = 0 and one for the case > 0. This analytical result shows that if the

hepatic glucose production time delayand the insulin degradation time delay between

the plasma and interstitial compartmentsti andtd are sufficiently small, then solutions

converge globally to the steady state or the basel levels of glucose and insulin. In other

words, there are no sustained oscillations. For larger delay, whose range is not given in

[4], oscillatory solutions become possible and under these circumstances it seems that

likely candidates for having sustainable oscillatory insulin and glucose levels are those

subjects with low degradation rates of the two insulin compartments.

Two other observations in [4] are that large glucose infusion rate could cause

insulin secretion oscillations, and the insulin oscillations are sensitive to the values of

|f1(C1Vg)| = Rm/(4a1Vg) or|f5(C5Vp)| = Rg/(4Vp). This means if the cells do not

release enough insulin into the bloodstream, or glucose production is not sensitive to

insulin and keeps at a constant moderate rate (Rg/2), then the insulin oscillation will


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not sustain. Similarly, if the hepatic glucose production rate Rg is too small, regardless

of sensitivity to insulin, the oscillations of insulin and glucose disappear.

3. Two Time Delay DDE Model

Glucose molecules are in the bloodstream or the plasma. When the concentration

level rises, electronic signals are sent to the pancreas and the cells secrete insulin.

The liver delivers the insulin into the plasma. This process takes about 5-15 minutes

depending different individuals. So, to more intuitively and precisely model the glucose-

insulin ultradian oscillations, we introduce two time delay parameters in to the glucose

and insulin regulatory system. The model diagram is shown in Figure 2.3.1. We remove

the insulin compartment split in the Sturis-Tolic Model ([79], [84]). The two time

delays are the hepatic glucose production time delay2 as in [4] and [31] and the effect

of glucose concentration level on insulin secretion time delay 1 due to the complex

electro-chemical reactions when the rising glucose concentration level triggers the

cells to release insulin. The delay 1 can be referred as insulin response time delay. The

two time delay DDE model we propose is as follows.

dG(t)

dt =Gin f2(G(t)) f3(G(t))f4(I(t)) +f5(I(t 2)),

dI(t)

dt =f1(G(t 1)) diI(t),

(2.3.1)

where the initial condition I(0) =I0 >0, G(0) =G0 > 0, G(t)G0 for all t[1, 0]

and I(t)I0 for t[2, 0] with 1, 2>0. In addition,


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Glucose

utilization

Glucose

production

Liver converts

glucagon and

glycogen to

glucose

Ins

ulinproduction

Insulinclearance

Insulin independent:

brain cells, and

others

Insulin dependent:

fat cells, and

others

Insulin degradation:

receptor, enzyme, and

others

Delay

Delay

Glucose Infusion:

meal ingenstion,

oral intake,

enteral nutrition,constant infusion

Glucagon

secrete

Glucose Controls

insulin secretion

Glucose Controls

glucagon secretion

Insulin helps cells consume glucose

Insulin secretion

Insulin Controls

Hepatic

glucose production

Glucose

Insulin

Pancreas

Liver

cells -cells

Figure 2.3.1. Two Time Delay Glucose-Insulin Regulatory Model

The divide lines (dash-dot-dot) indicate insulin controlled hepatic glucose production with time delay;

the dash-dot lines indicate the insulin secretion from the -cells stimulated by elevated glucose concen-

tration level with time delay; the dashed lines indicate low glucose concentration level triggers -cells

in pancreas to release glucagon; and the dot line indicates the insulin accelerates glucose utilization in

cells.


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(i) Gin is due to glucose infusion, e.g., by meal ingestion, oral glucose intake, contin-

uous enteral nutrition or intravenous glucose infusion;

(ii) f2(G(t)) stands for insulin independent glucose consumption by the brain, nerve

cells and others. f2(0) = 0, f2(x) > 0 and f2(x) > 0 are bounded for x > 0.

Denote M2:= sup{f2(x) :x0}0} 0 is a constant. f4(0) > 0, for

x > 0, f4(x) > 0 and f4(x) > 0 are bounded above. f4(I(t)) is in sigmoidal

shape. Denote M3 := sup{f3(x) : x > 0} 0,

M4 := sup{f4(x) :x0}0}0. The time delay 2 > 0 reflects that the liver

does not convert the stored glucose and glycogen into glucose immediate when the

insulin concentration level decreases. When insulin concentration level increases,

the liver converts glucagon and glycogen to glucose decreasingly. f5(0) > 0 and,

forx >0, f5(x)> 0 andf5(x)< 0. f5(x) and |f5(x)| are bounded above forx >0.

Denote M5 := sup{

f5(x) : x

0}

0

} 0 and, for

x > 0, f1(x) > 0, f

1(x) > 0, f

1(x) > 0 and bounded. DenoteM1 := sup{f1(x) :x0}0}0, (2.4.1)

and

I =d1i f1(G). (2.4.2)

ProofAll we have to show is that equation (2.4.1) has a unique root in (0 , ). In fact,

observe thatf1(x)> 0, f2(x)> 0, f

4(x)> 0, f

3(x)> 0, andf

5(x)< 0, then H

(x)< 0.

Notice that

H(0) = Gin

f2(0)

f3(0)f4(d

1i f1(0)) + f5(d

1i f1(0))

= Gin+f5(d1i f1(0))> 0,


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and

limx H(x) = Gin limx f2(x) limx f3(x)f4(d1i limx f1(x))

+f5(d1i limx

f1(x))

= Gin M2 f4(d1i M1) limx(k3x) +f5(d1i M1)

< 0.

In addition,f1(x) is strictly monotone increasing, so the proof is completed.

We show the positiveness and boundedness of the solutions of the model (2.3.1).

Proposition 2.4.2 All solutions of model (2.3.1) exist for all t > 0, are positive and

bounded. Furthermore,

lim supt

G(t)MG := Gin+M5m4k3

(2.4.3)

and

lim supt

I(t)MI :=d1i f1(MG). (2.4.4)

Proof. Observe that the|fi(x)|, i = 1, 2, 3, 4, 5, are bounded, thus fi(x), i = 2, 3, 4,

and fj(xt), j = 1, 5, are Lipschitz and completely continuous in x 0 and xt

C[ max{1, 2}, 0], respectively. Then by Theorem 2.1, 2.2 and 2.4 on page 19 and 20

in [54], the solution of equation (2.3.1) with given initial condition exists and unique

for all t0. If there exists a t0 >0 such that G(t0) = 0 and G(t)> 0, for 0 < t < t0,

then G(t0)0. So

0 G(t0)

= Gin f2(G(t0)) f3(G(t0))f4(I(t0)) +f5(I(t0 2))

= Gin f2(0) f3(0)f4(I(t0)) +f5(I(t0 2))

= Gin+f5(I(t 2))> 0


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This implies that G(t) > 0, for all t > 0. Ift0 > 0 such that I(t0) = 0 and I(t) > 0

for all 0 < t < t

0, then I(t

0) < 0. Therefore, 0 > I(t

0) = f1(G(t

0)diI(t

01)f1(G(t

0))> 0 implies that I(t)> 0 for all t >0.

Notice that m4f4(x)M4 and f5(x)M5 and f3(x) =k3x, for x >0. Thus

G(t) = Gin f2(G(t)) f3(G(t))f4(I(t)) +f5(I(t 2))

Gin m4k3G(t) +M5.

Therefore, for any given t >0, ift >t, we have

d

dt(em4k3tG(t))(Gin+M5)em4k3t

em4k3tG(t)G(t) + tt

(Gin+M5)em4k3sds

G(t) G(t)em4k3t + tt

em4k3sds

= G(t)em4k3t +Gin+M5m4k3

(em4k3t em4k3t)

Thus

lim supt

G(t) Gin+M5m4k3

:=MG

Since|f1(x)| M1, given >0, I(t)f1(MG+ ) diI(t) for sufficiently large t >0.

Then we have

lim supt

I(t)d1i f1(MG+ ).

Notice that >0 is arbitrary, so

lim supt

I(t)d1i f1(MG) :=MI.

The following lemma is elementary. See [48] for a proof.


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Lemma A Let f : R R be a differentiable function. If l = liminft f(t) 0. So,

f1(G) diI(sk)f1(G(sk 1)) diI(sk) for k= 1, 2, 3,...

Thus,

f1(G) diI0.

Now we show (2.4.6) holds. Again, due to Proposition 2.4.2 and Fluctuation

Lemma, there exists a sequence{tk} as k such that limk G(tk) =G and

0 = G(t

k)

= Gin

f2(G(t

k))

f3(G(t

k))f4(I(t

k)) +f5(I(t

k

2)), k= 1, 2, 3, ....

Then, notice that f4 and f50,

0 = Gin f2(G(tk)) f3(G(t

k))f4(I(t

k)) +f5(I(t

k 2))

Gin f2(G(tk)) f3(G(t

k))f4(I) +f5(I), k= 1, 2, 3,...

and therefore

Gin f2(G) f3(G)f4(I) +f5(I)0.

Similarly we can show (2.4.7) is true. According to Proposition 2.4.2 and Fluctu-

ation Lemma, there exists a sequence{sk} as k such that limk G(sk) =G

and

0 = G(s

k)

= Gin f2(G(sk)) f3(G(s

k))f4(I(s

k)) +f5(I(s

k 2)), k= 1, 2, 3, ....


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On the other hand side, (2.4.3) and (2.4.5) imply thatI(t) andG(t) are bounded

above.

5. Global Stability of Steady State

In this section, we will give one result of globally asymptotically stable equilib-

rium of this model using Lemma 2.4.1.

Theorem 2.5.1 Let

F(x, y) =f3(x)f4(d1i f1(y)) +f5(d

1i f1(x)), x, y0. (2.5.1)

If

F(x, y)

F(y, x), x

y

0, (2.5.2)

then the steady state(G, I) of (2.3.1) is globally asymptotically stable.

Proof Let (G(t), I(t)) be a solution of (2.3.1). Due to Lemma 2.4.1, we have

Gin f2(G) f3(G)f4(I) +f5(I)Gin f2(G) f3(G)f4(I) +f5(I)

that is,

0 [f2(G) +f3(G)f4(I) f5(I)] [f2(G) +f3(G)f4(I) f5(I)]

= [f2(G) +f3(G)f4(I) + f5(I)] [f2(G) +f3(G)f4(I) +f5(I)]

[f2(G) f2(G)] + [(f3(G)f4(d1i f1(G)) +f5(d1i f1(G)))

(f3(G)f4(d1i f1(G)) +f5(d1i f1(G)))]

= [f2(G) f2(G)] + [F(G, G) F(G, G)]

f2(G) f2(G)


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due to (2.5.2). Thus G= G.

Remark Notice that f5(d1i f1(x)) f5(d1i f1(y)) for x y 0 means higher hep-atic production of glucose helps to make oscillations happen (the case that (G, I) is

unstable).

Remark Notice thatf3(G) can be linear and f4 is bounded. If the glucose concentra-

tionG is big enough and there is no hepatic production (f50), then the steady state

(G, I) will be globally stable and thus there is no oscillation.

6. Linearization and Local Analysis

We need following theorem for two special cases, where one of the two time

delays equals to zero. When both delays equal to zero, the linearized system of the

model (2.3.1) becomes a trivial 2-dimensional ODE. Now we state theorem here without

proof. For a proof, see Kuang ([54], 1993)(Theorem 3.1, page 77).

Theorem B In the following second order real scalar linear neutral delay equation

x(t) +x(t ) +ax(t) +bx(t ) + cx(t) +dx(t ) = 0, (2.6.1)

where 0. Assume|| < 1, c+ d= 0 and a2 +b2 + (d c)2 = 0. Consider the

characteristic equation of (2.6.2)

2 + 2e +a+be +c+de = 0. (2.6.2)

The number of different imaginary roots with positive (negative) imaginary parts of

(2.6.2) can be zero, one, or two only.

(I) If there are no such roots, then the stability of the zero solution does not

change for any >0.


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(II) If there are any imaginary roots with positive imaginary part, an unstable

zero solution never becomes stable for any 0. If the zero solution is asymptoticallystable for = 0, then it is asymptotically stable for < 0, and it becomes unstable

for > 0 where 0 >0 is a constant. It undergoes a supercritical Hopf bifurcation at

=0.

(III) If there are two imaginary roots with positive imaginary part,i+ andi,

such that+ > >0, then the stability of the zero solution can change (when changes

from stable to unstable, the zero solution undergoes a supercritical Hopf bifurcation) a

finite number of times at most as is increased, and eventually it becomes unstable.

The number of such roots are determined by the following conditions.

Ifc2 d2, then there is only one such root.

Ifc2 > d2, then there are two such roots provided that

(A) b2 + 2c a2 2d >0, and

(B) (b2 + 2c a2 2d)2 >4(1 2)(c2 d2).

Otherwise, there is no such solution.

Now we try to linearize the model (2.3.1). Let G(t) = G1(t) +G and I(t) =

I1(t) +I. Then system (2.3.1) becomes

G1(t) = Gin f2(G1(t) +G) f3(G1(t) +G)f4(I1(t) +I) + f5(I1(t 2) +I)

= [f2(G) +f3(G)f4(I)]G1(t) f3(G)f4(I)I1(t) +f5(I)I1(t 2)

I1(t) = f1(G1(t 1) +G) di(I1(t) +I)

= f1(G)G1(t

1)

diI1(t).

We still use G(t) and I(t) to denote G1(t) and I1(t), respectively. Thus the linearized


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system of (2.3.1) can be written as

dG(t)dt

=AG(t) BI(t) CI(t 2)

dI(t)

dt =DG(t 1) diI1(t)

(2.6.3)

where

A := f2(G) +f3(G

)f4(I)> 0,

B := f3(G)f4(I

)> 0,

C := f5(I)> 0,

D := f1(G)> 0.

Let

G(t)

I(t)

=etG

0

I0

, G0, I0>0, C, t >0

be a solution of (2.6.3). Then

et

G0

I0

=

AG0et BI0et CI0e(t2)

DG0e(t1) diI0et

= et A B Ce

2

De1 di

G0

I0

.

So the characteristic equation of (2.6.3) is given as

det

0

0

A B Ce2

De1 di

= det +A B+Ce

2

De1 +di


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= (+ A)(+di) +De1(B+Ce2)

= 2

+ (A+di)+ diA+DBe1

+DC e(1+2)

= 0.

We denote the characteristic equation as

() =2 + (A+di)+diA+DBe1 +DC e(1+2) = 0. (2.6.4)

Note (0) = diA+ DB+ DC > 0. So = 0 is not a solution of the characteristic

equation (2.6.4). So if there is any stability switch of the trivial solution of the linearized

system (2.6.3), there must exist a pair of pure imaginary roots of the characteristic

equation (2.6.4).

If1 = 0 and 2 = 0, the original model (2.3.1) is an ODE model. The charac-

teristic equation of its linearized equation is given by

() =2 + (A+di)+diA+DB +DC= 0.

Then due to A +di > 0 and diA+ DB+DC >0, the steady state (G, I) is stable.

If2 = 0 but 1 > 0, the characteristic equation of the linearized system takes

the following form.

() =2 + (A+ di)+diA+ (DB+DC)e1 = 0. (2.6.5)

Then due to Theorem B ([54]), di D(B+C)A means there exists only one positive root

of (2.6.5). That is, there exists an 10>0 such that the trivial solution of the linearized

system (2.6.3) is stable when 1(0, 10) and unstable when 110.

Similarly if1 = 0 and 2 >0, then d2i2DB A2 implies the trivial solution

of the linearized system (2.6.3) is stable. If d2i < 2DBA2 and 2DB + D2C2 >

A2 + d2i + (diA + DB)2, then the trivial solution of the linearized system (2.6.3) has at

most finite number of stability switches and eventually is unstable.


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(c.1) if d2i 2f1(G)B (f2(G) + f3(G)f4(I))2, then the steady state(G, I)

is stable.

(c.2) if

d2i (f2(G) + f3(G

)f4(I))2 +d2i +

+(di(f2(G

) + f3(G)f4(I

)) +f1(G)(f3(G

)f4(I)))2

then there are at most a finite number of stability switch and eventually steady

state(G, I) is unstable.

(d) When1 > 0 and2 > 0, if the insulin degradation rate

dif1(G

)(f3(G)f4(I

) f5(I))f2(G

) +f3(G)f4(I)

:=d0, (2.6.7)

the steady state(G, I) is stable.

Remark If the parameters and functionsfi, i= 1, 2, 3, 4, 5, take the values in (2.7.1)

to (2.7.5) and Table (2.2.1) and (2.2.2), then the threshold value d0 = 0.6669 when

Gin = 0.54. So when di = 1/26 = 0.03849 < d0. So, (2.6.7) does not hold. In fact, the

insulin and glucose oscillation is sustained provided that 2 = 36 and 1 is sufficiently

large (greater than 5.2).

To further analyze the stability of the steady state of the model (2.3.1) and the

cases of the oscillations to be sustained, we will apply Rouches Theorem to analyze


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when 1 > 0 and 2 > 0 that the steady state (G, I) is unstable. Recall following

Rouche

s Theorem([19], p.125-126).

Rouches TheoremGiven two functions f(z) andg(z) analytic in a simple connected

regionA C with boundary , a simple loop homotopic to a point inA. If|f(z)|>

|g(z)| on , then f(z) and f(z) + g(z) have the same number of roots inA.

We start from a more generic equation and leave the system (2.6.3) as a special

case.

Let

S1 ={ 2m2n 1 :m, nZ

+, m , n1}

and

S2={2m 12n

:m, nZ+, m , n1}.

Clearly Q+ =S1 S2 and S1 S2=. Further we have

Lemma 2.6.1 S1 andS2 are dense inQ+ thus inR+.

Proof. rQ+ \ S1, p, qZ+ such that r= 2p12q . Thus

rk = 2p 1 2

2k

2q 12k

=(4kp 2k 2)/2k

(4kq 1)/2k=

2(2kp 2k 1)2(2kq) 1 S1 k= 1, 2, 3,...

and limk rk= (2p 1)/2q= r. That is,S1=Q+. Similarly, S2 = Q+.

Proposition 2.6.2 For characteristic equation

k +k1j=1

ajj +b+ce1 + de2 = 0, k2, 1, 2>0, (2.6.8)

whereb,c,d >0, aj R, j = 1, 2, 3,...,k,, ifb < d c orb < c d, then10 >0 and

20 >0 such that the characteristic equation (2.6.8) has at least one root with positive

real part for1 > 10 and2 > 20 and1/2S1 or1/2S2.


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We need following lemmas to prove Proposition 2.6.2.

Lemma 2.6.2 For the equation

kzk +k1j=1

ajjzj +b+cep1z +dep2z = 0, k2, p1, p2>0, zC (2.6.9)

whereb,c,d >0, aj R, j = 1, 2, 3,...,k, assume

(i) b < d c, andp1/p2S1, or

(ii) b < c d, andp1/p2S2.

Then,0 >0 such that for all, 0< < 0, the equation (2.6.9) has at least one root

with positive real part.

Proof. Let

f(z) =b+cep1z +dep2z.

We show that f(z) has a zero with positive real part. Since p1 and p2 are S1 related

in case (i) or S2 related in case (ii), there exist integer m, n 1 such that p1p2 = 2m2n1for case (i), or p1

p2= 2m1

2n for case (ii). Letz = x+ qi, where q = 2m/p1 = (2n

1)/p2 for case (i) or q= (2m 1)/p1= 2n/p2 for case (ii). Then

f(z) = b+cep1xep1qi + dep2xep2qi

= b+cep1x cosp1q +dep2x cosp2q i(cep1x sinp1q + dep2x sinp2q)

= b+cep1x cos2m+dep2x cos(2n 1)

(=b+cep1x cos(2m 1)+dep2x cos2n for case (ii))

= b+cep1x dep2x (=b cep1x +dep2x for case (ii))

:= H(x).


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Let z=yi,y[0, 2q], then

|f(z)| = |b+cep1yi +dep2yi|

d c b, for case (i),

c d b, for case (ii):= 0>0.

Let 0 := min{0, b/2}. Denote

:={z=x+yiC : z=x or z=x 2qi, x[0, Kx0]

or z=yi or z=K x0+yi y[0, 2q].

:={z=x+yiC : 0< x < Kx0, 0 < y

0 on . Choose r0 > 0 such that A:={zC :|z|< r0}. Denote A:={zC :

|z|= r0}. ThuszA,z=r0ei, [0, 2], we have

|g(z)|=|kzk +k1j=1

ajjzj| krk0+

k1j=1

|aj|jrj0. (2.6.11)

Obviously0>0 such that, 0< < 0,

|g(z)|< 0, zA.

z A, z=rei, then r < r0, and

|g(z)|=|kzk +k1j=1

ajjzj| krk +

k1j=1

|aj|jrj < krk0+k1j=1

|aj|jrj0.

Thus

|g(z)|< 0 for allz.


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Therefore|f(z)| > |g(z)| on . By Rouches Theorem ([19], p125-126), f(z) and

f(z) + g(z) have the same number of zeros in

. That is,f(z) + g(z) = 0 has at least

one root z.

Proof of Proposition 2.6.2. Assume b < dc, and 1/2 S1 (or b < cd,

and 1/2 S2). In Lemma 2.6.2, choose p

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