Mathematical model on glucose, insulin and β-cells mass dynamics in type 2 diabetes

Project Abstract

<p> A mathematical model, describing glucose, insulin and β-cells mass dynamics of a type 2<br>diabetic patient was developed in the form of a system of ordinary differential equation,<br>considering insulin resistance, the body inability to overcome the resistance and the fact that<br>glucose production from food intake is not constant. Numerical solution of the model using<br>RungeKutta code in MATLAB, graphically shows rise in blood glucose concentration and<br>further decline over time in glucose concentration below fasting glucose level as a result of<br>low storage of glucose by the liver from food intake; rise in insulin level and fall in β-cells<br>mass. Results from the equilibrium and stability analysis are interesting and compare<br>favourably with available medical literature. Simulation of insulin resistance parameters<br>showed that glucose concentration in the body is proportional to insulin resistance rate.Also,<br>insulin resistance in glucose conversion to glycogen has a significant impact on glucose<br>concentration in the blood. <br></p>

Project Overview

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