ABSTRACT
A mathematical model, describing glucose, insulin and β-cells mass dynamics of a type 2
diabetic patient was developed in the form of a system of ordinary differential equation,
considering insulin resistance, the body inability to overcome the resistance and the fact that
glucose production from food intake is not constant. Numerical solution of the model using
RungeKutta code in MATLAB, graphically shows rise in blood glucose concentration and
further decline over time in glucose concentration below fasting glucose level as a result of
low storage of glucose by the liver from food intake; rise in insulin level and fall in β-cells
mass. Results from the equilibrium and stability analysis are interesting and compare
favourably with available medical literature. Simulation of insulin resistance parameters
showed that glucose concentration in the body is proportional to insulin resistance rate.Also,
insulin resistance in glucose conversion to glycogen has a significant impact on glucose
concentration in the blood.
TABLE OF CONTENTS
Certification i
Declaration ii
Dedication iii
Acknowledgement iv
Abstract v
1. INTRODUCTION 1
1.1.1 Diabetes 1
1.1.2 Insulin 2
1.1.3 Mechanism of Insulin Action on Glucose Level in Blood 5
1.1.4 Blood Glucose Concentration 6
1.2 Objectives of the Study 7
1.3 Scope of the Study 7
1.4 Limitations of the Study 7
1.5 Significance of the Study 8
2. LITERATURE REVIEW 9
3. TYPE 2 DIABETES DISEASE (T2D)
3.1 Type 2 Diabetes (T2D)/Non-Insulin Dependent Diabetes Mellitus (NIDDM) 18
3.2 The Aetiology of Type 2 Diabetes 18
3.2.1 Causes of T2D 18
3.2.2 Risk Factors for the Development of T2D 19
3.3 Prevention 20
3.4 Management 20
3.5 Medication 20
3.6 Epidemiology of Diabetes (T2D) 21
3.7 Signs and Symptoms of T2D 22
3.8 Diagnoses of Diabetes 23
3.9 Insulin Resistance and Development of T2D 26
4. MODEL PRESENTATION AND ANALYSIS
4.1 Model Presentation 28
4.2 Definition of Model Parameters 30
4.3 Model Solution 31
4.4 Equilibrium Analysis of the Model 33
4.5 Stability Analysis of the Model 34
4.6 Model Simulation with Varying Degrees of Insulin Resistance 35
5. DISCUSSION AND RECOMMENDATIONS
5.1 Discussion of Results 40
5.2 Conclusion 41
CHAPTER ONE
INTRODUCTION
Mankind in search for happiness has sought a disease free earth, trying to find long lasting
solutions to the diseases threatening the existence of man through various means including
mathematical modelling. The wide speculations that some diseases, including diabetes have
no cure poses serious challenge to man. International Diabetes Federation (IDF) in 2013
reported that over 387 million people live with diabetes in the world. World Health
Organization (WHO) puts it that diabetes led to approximately 3.4 million deaths worldwide
in 2004 with 90 percent of the people suffering from diabetes suffer from type 2 diabetes. In
2010, American Diabetes Association (ADA) in their account listed diabetes as the seventh
disease leading to death in United States of America. Hence, the call for the study of diabetes,
types of diabetes, development, treatment and factors associated to it through the use of
mathematical modelling.
1.1.1 Diabetes
Diabetes is a disorder of metabolism that results in excessive thirst and production of large
volume of urine. It occurs in two forms; diabetes insipidus and diabetes mellitus. Although in
most cases, diabetes is referred to as diabetes mellitus(Oxford Concise Medical Dictionary,
2003).
· Diabetes Insipidus
Diabetes Insipidus is a metabolic disorder caused by deficiency of vasopressin (antidiuretic),
a pituitary hormone that regulates reabsorption of water by the kidney. It is characterized by
the production of large volume of diluted urine and constant thirst, although it is a rare kind
of disease (Oxford Concise Medical Dictionary, 2003).
· Diabetes Mellitus
Diabetes Mellitus, commonly referred to as diabetes is defined as chronic metabolic disorder
of carbohydrates, proteins and fats occurring in the endocrine system (Jarald, Balakrishnan&
2
John; 2008), due to an absolute or relative deficiency of insulin secretion with/without
varying degree of insulin resistance (Barar, 2000; Delvin 2006). This is a disease
characterized by chronic high blood plasma glucose concentration (hyperglycaemia), as a
result of sugar not being oxidized in the body to produce energy (Oxford Concise Medical
Dictionary, 2003). The disease can cause the body to do any of the following; produce little
insulin, cease to produce insulin at all or become progressively resistant to insulin action
(Ranjan&Ramanujam, 2002).
Diabetes mellitus is classified into subclasses: Insulin-Dependent Diabetes Mellitus (IDDM)
or Type 1 Diabetes (T1D), Non-Insulin Dependent Diabetes Mellitus (NIDDM) or Type 2
Diabetes (T2D), Gestational Diabetes Mellitus (GDM) etc. T1D is a case where there is
production of little or no insulin because of destruction of the β – pancreatic cells or may be
caused by factors such as genetic infections etc. (Oxford Concise Medical Dictionary, 2003).
In this work, the emphasis is on Type 2 Diabetes (T2D) disease in man. This type of disease
is slow to develop occurring in older, obese individuals due to resistance to insulin action
combined with a relative deficiency to overcome this resistance arising from the
defectiveness in some features of insulin-response system (Nelson & Cox, 2005).
NIDDM/T2D is a heterogeneous disease, ranging from insulin resistance to insulin deficiency
(Lokesh& Amit, 2006). It is caused by both genetic and non-genetic factors (Torben, 2002).
It accounts for 90-95% of most adults who develop diabetes (Centres for Disease Control and
Prevention, 2008). Development entails the process of growth or directed change. Hence,
development of disease has to do with the process of growth of such a disease.
1.1.2 Insulin
Insulin is small peptide hormone that consist of 51amino acids and has a molecular mass of
5808 Daltons (Wikipedia, 2011g). In mammals, it is produced by the β-cells of the islets of
3
Langerhans in the pancreas in the form of pre-proinsulin, a single-chain precursor composed
of 110 amino acids. Other issues associated with insulin in diabetes include:
· Synthesis and Release of insulin
Insulin is coded on the short arm of chromosome II and synthesized in the β-cells of the
pancreatic islets of Langerhans as its precursor, proinsulin. Proinsulin is synthesized in the
ribosomes of the rough endoplasmic reticulum (RER) from mRNA as pre-proinsulin. Preproinsulin
is formed by sequential synthesis of a signal peptide, the β chain, the connecting
(C) peptide and then the A chain comprising a single chain of 100 amino acids. Removal of
the signal peptide forms proinsulin, which acquires its characteristics 3 dimensional structure
in the endoplasmic reticulum. Secretory vesicles transfer proinsulin from the RER to the
Golgi apparatus, whose aqueous zinc and calcium rich environment favours formation of
soluble zinc-containing proinsulinhexamers. As immature storage vesicles form from the
Golgi, enzymes acting outside the Golgi convert proinsulin to insulin and C-peptide, Gisela
(2005).
Insulin secretion from the islet cells into the portal veins is characteristically pulsatile,
reflecting the summation of co-ordinate secretory bursts from millions of islet cells. In
response to stimuli such as glucose, insulin secretion is characteristically biphasic, with an
initial rapid phase of insulin secretion, followed by a less intense but more sustained release
of the hormone, Gisela (2005).
· Factors Influencing Insulin Biosynthesis and Release
Insulin secretion may be influenced by alteration in synthesis at level of gene transcription,
translation and post-translational modification in the Golgi as well as by factors influencing
insulin release from secretory granules. Considering the insulin’s role in glucose utilization
4
and metabolism, glucose has multiple influences on insulin biosynthesis and secretion. Other
factors also influence these process such as amino acids, fatty acids, acetyleholine, pituitary
adenylate cyclase-activating polypeptide (PACAP), glucose-dependent insulinotropic
polypeptide (GIP), glucagon – like peptide – 1 (GIP -1), in combination with other agonists,
Gisela (2005).
On physiology of insulin secretion, glucose is the principal stimulus though other
macronutrients, hormones, humoral factors and neural input may modify this response.
Insulin, together with hormone glucagon, regulates blood glucose concentrations. Pancreatic
β cells secrete 0.025 – 1.5 units of insulin per hour during fasting (or basal) state, sufficient to
enable glucose insulin – dependent entry into cells, which consequently, maintain normal
fasting blood glucose levels, Gisela (2005)
· Distribution of Insulin
In the blood, insulin circulates in its free monomer form, where its volume of distribution
almost equals the volume of extracellular fluid. During fasting periods, the pancreas secretes
about 40ıı (1unit) of insulin per hour into the portal vein, in order to obtain an insulin
concentration in the portal blood of 2 – 4ıı/ı ı(50 to 100ı ııııı/ı ı) and 0.5 ıı/ıı(12
units/ml) or about 0.1 ıı in the peripheral circulation.
The ingestion of a meal leads to rapid rise in the concentration of insulin in the portal blood,
followed by a parallel but a smaller rise in the peripheral circulation. At digestion, insulin
release from the pancreas is not continuous rather it oscillates with a period of 3 – 6 minutes,
changing from generating a blood insulin concentration greater than 800pmol/l to less than
100pmol/l. Hellman, Gyife, Dansk andSalehi (2007).
· Physiological Effects of Insulin
The effects of insulin include the triggering notable array of biological responses. The most
important target tissues for the regulation of glucose homeostasis by insulin are liver muscle
5
and adipocyte. Although insulin exerts potent regulatory effects on other cell types as it is
basically the hormone responsible for controlling the uptake, use and storage of cellular,
nutrients (Laurence, Lazo&Parker 2006). It also increases DNA replication and protein
synthesis through the control of amino acid uptake with modification of the activity of several
enzymes, Wikipedia (2011g). The anabolic action of insulin occurs mainly at high
concentrations and it includes promotion of fatty acid synthesis. Enhancing amino acid
uptake in order to synthesize proteins, promotion of protein translocation between cellular
compartments, increases glycogen synthesis whereby insulin induces glucose storage in the
lever and muscle cells in the form of glycogen. Insulin promotes the uptake of serum
potassium thereby reducing potassium in the blood. Insulin also, increases the secretion of
hydrochloric acid by the parietal cells in the stomach and induces arterial muscle relaxation
by enhancing the flow of blood in micro arteries, Wikipedia (2011e).
· Pancreas
This is a gland organ of the digestive and endocrine system of vertebrates. The pancreas is an
endocrine gland, which produces several important hormones such as insulin, glucagon and
somatostatin. It equally secretes pancreatic juice containing digestive enzyme, Wikipedia
(2011e). Islets of Langerhans is a portion of the pancreas which is made up a cluster of cells
of about a million in number, which is about 1-2% of the mass of pancreas and compose of
four main cell types namely: Alpha (ı) cells that secretes glucagon, Beta (β) cells that
produces insulin and amylin, Delta (ı) cells that produce somatostatin, Epsilon (ı) cells that
secretes Ghrelin, and pancreatic polypeptide (ıı) cells that produces pancreatic polypeptide.
(Oxford Concise Medical Dictionary, 2003).
1.1.3 Mechanism of Insulin Action on Glucose Level in Blood
The increase in the plasma glucose concentration induces insulin release, which in turn aids
reduction and maintaining the level of glucose in the same plasma.
6
Insulin is important in hormone regulating cellular energy supply and macronutrient balance,
directing anabolic processes of the fed state. It is essential for the intra-cellular transport of
glucose into insulin-dependent tissues such as muscle and adipose tissue. Signalling
abundance of exogenous energy, adipose tissue and fat breakdown is suppressed and its
synthesis and stored in muscle cells. Glucose entry enables glycogen synthetize in muscle,
Gisela (2005).
Sequel to glucose concentration in the blood plasma, the pancreas β-cell releases insulin
amylin. The insulin is released into the extra cellular fluid in the pancreas by exocytosis and
it diffuses out into the blood stream, which are taken along to fluids surrounding the liver
cells where insulin binds with appropriate protein on the plasma membrane and clear the
proteins inhibiting glucose entry into the cells. Insulin binds with the receptors in the
phosphorylated form or the unphosphorylated form, Quon and Campfield (1991).
In a case of diabetic patient (T2D), there is an insulin resistance or deficiency, the pancreas
fails to produce adequate or effective insulin to overcome the insulin resistance which makes
it impossible for glucose entry into the cells for body use and storage. Hence, the glucose is
not oxidized and this leads to excess glucose in the blood while the cells are starved of
glucose.
1.1.4 Blood Glucose Concentration
Glucose is the main source of energy for body cells and blood lipids (in the form of fats and
oils which are primary compact energy store). It is transported from intestines or liver to body
cells through the bloods stream and presented to the cells for absorption via the hormone
insulin. Blood sugar concentration or blood glucose level is the amount of glucose (sugar)
present in the blood of a human or animal, Wikipedia (2011a).
The blood glucose concentration is measured based on international standards with respect to
molar concentration. It is measured in units such as millimoles per litre(mm/l) or millimolar
7
(mM). In some countries such as the United States, blood glucose concentration is expressed
as mass concentration of glucose in the blood with unit as mg/dl (milligrams per deciliter).
Both units can be used interchangeably, since the molecular mass of glucose (C6H12O6) is
approximately 180g/ml. Thus the measurement of glucose is the difference between these
two scales, which is the factor of 18. Therefore 1mmol/l of glucose is equal to 18 mg/dl,
Wikipedia (2011a). Glucose concentrations vary prior to and after each meal consumption, so
that with a substantial carbohydrate load the human blood glucose levels usually remain
within the normal range, although shortly after eating, the blood glucose concentration could
temporarily increase a bit in non-diabetics. Several factors influence an individual’s blood
glucose concentration, such as homeostasis, that functions to restore blood glucose
concentration back to normal. Normal blood glucose level/concentration varies among
Medical professionals, with the values of blood use concentration at 4.4 – 6.1mmol/l (82-
110mg/dl).
1.2 Objectives of the Study
The study aims at developing amathematical model on glucose production, insulin and β-cells
mass dynamics in development of type 2 diabetes. Specifically, the objectives are to:
· Develop mathematical model on glucose, insulin and β-cells mass dynamics,
describing development of T2D in a case of insulin resistance individuals.
· Obtain numerical solution of the model.
· Perform Equilibrium, Stability Analysis and Simulations on the model.
1.3 Scope of the Study
This study is restricted to T2D without loss of generality to other types of diabetes.
Furthermore, all formulations are based on the results of the practical work of medical
literature.
1.4 Limitations of the Study
8
It is known that there are two major types of diabetes mellitus caused by diferent factors. The
study did not perform any physical experiment but relied solely on literature and limited itself
to development of a mathematical model on glucose, insulin, β-cells mass dynamics in an
individual having insulin resistance and genetically predisposed to development of type 2
diabetes. Without considering other risk factors leading to development of type 2 diabetes,
effects of α-cells in glucose production. Also, the case of type 1 diabetes and development of
diabetes in other animals were not covered.
1.5 Significance of the Study
A mathematical formulation to assist in the quest for curative solutions to contemporary
diseases cannot be over emphasized. This study presents T2D development pattern using
mathematical biology contributing to the literature in the effort for better management and
possible cure for T2D.
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