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
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 A-etiology 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
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.
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 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 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).
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.
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 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 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
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).
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.
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. Signaling 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,
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 (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 a mathematical 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.
1. Obtain numerical solution of the model.
2. 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
It is known that there are two major types of diabetes mellitus caused by different 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.