Mathematical Model on Glucose, Insulin and Β-cells Mass Dynamics in Type 2 Diabetes
Chapter One
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
- Obtain numerical solution of the model
- Perform Equilibrium, Stability Analysis and Simulations on the model.
CHAPTER TWO
LITERATURE REVIEW
Attempts have been made to solve problems posed by diabetes disease, through the use of mathematical models. The literature on mathematical modelling in different aspects of diabetes including: diabetes progression, tools and computational study of diabetes models, plasma and insulin level in diabetes mellitus, homeostatic assessment, insulin sensitivity/resistance, plasma and blood glucose level regulation and human glucose-insulin regulatory system is very vast.
WHO (1999), through the expert committee released a report on Diabetes mellitus, which includes its classifications, types, symptoms, urination, diagnoses, amount of glucose in blood and many other ways, as well as treatment of diabetes.
Wild et al. (2004), estimated the incidence of diabetes and the number of diabetic people of all ages for years 2000 to 2030. The data collected for all age groups having diabetes mellitus worldwide was estimated to be 2.8% of the world’s population in 2000 and 4.4% in 2030. The incidence of diabetes is higher in men than women, but as for the record there are more diabetic women as compared to men. These findings indicate that the “diabetes occurrence” will continue even if levels of obesity remain constant.
In a review, Gisela (2005) provided an overview of insulin, its history, structure, synthesis, secretion, actions and interactions. Specifically, the work focused on insulin and manifestation of insulin resistance in specific organs and tissues; physiological, environmental and pharmacological influences on insulin action and insulin resistance. In conclusion, he stated that rapid globalization, urbanization and industrialization spawned epidemics of obesity, diabetes and their attendant co-morbidities. However, the approach to the work was non-mathematical.
Topp et al. (2000), developed amodel on Glucose (G), Insulin (I) and -cells mass( )given as follows
The model showed three equilibrium points: a stable physiological equilibrium point ( = 100, = 20, = 6000), a stable trivial pathological equilibrium point ( = 600, = 0, = 0) and a saddle point ( = 250, = 9.8, = 129.36) for = 1. In the case of = 0 (absence of predisposition to diabetes), the model has an unstable pathological equilibrium point ( = 600, = 0, = 0) and a stable physiological equilibrium point ( = 82.6, = 23, = 900). The model was further subjected to simulations to find the effects of obesity, physical exercise and other factors that affect insulin sensitivity.
De Gaetano et al. (2008), presented mathematical models of diabetes progression. A model of the pancreatic islet compensation was formulated, its physiological assumption were presented and some fundamental qualitative characteristics of its solutions established. Numerical values were assigned to model parameters. The performance over the lifetime span was simulated under certain conditions including worsening insulin resistance and primary replication defects. Equally, differences in the models as compared to previously proposed models of diabetes progression were highlighted. The model tries to give a descriptive evolution to the compensation of the glucose-insulin system in healthy and diabetic individuals.
Lin (2011), developed a glucose-insulin kinetics model with the development of type 2 diabetes in offspring of diabetic parents. Investigation using a population – based Bayesian nonlinear hierarchical model of pharmacokinetics/pharmacodynamics (PK/PD) risk factors preceding the onset of T2D was done. Also, a methodology describing disease development of T2D was developed based on fasting blood glucose (FBG), fasting serum insulin (FSI), homeostatic model assessment of insulin resistance (HOMA-IR) and body mass index (BMI) of T2D.
Delay Differential Equation (DDE) models on diabetes were reviewed by Makroglou, Karaous, Li and Kuang (2011). They stated that DDE models can generate rich dynamics using minimum number of parameters, which makes the models to play important roles in a growing number in areas of diabetes study. They identified such areas to include insulin/glucose regulatory system, Intravenous Glucose Tolerance Test (IVGT), and insulin therapies. The models were presented with some computational results and theoretical results for models on ultradian oscillations of insulin and models for diagnostic tests.
CHAPTER THREE
TYPE 2 DIABETES(T2D) DISEASE
Type2 Diabetes (T2D) or Non-Insulin Dependent Diabetes Mellitus (NIDDM) Diabetes mellitus type 2 is a long term metabolic disorder that is characterized by high blood sugar, insulin resistance, and relative lack of (Wikipedia, 2011a). This type of diabetes is slow to develop, occurring mainly in older, obese individuals as a result of resistance to insulin action combined with a relative deficiency insulin secretion. Although insulin is still produced by the β-cells, it is insufficient to overcome this resistance arising from the defectiveness in some features of insulin-response system (Nelson & Cox, 2005).
NIDDM/T2D is classified as a heterogeneous disease, ranging from insulin resistance to insulin deficiency (Lokesh& Amit, 2006), also as a multifactorial, with both genetic and monogenetic components/caused factors (Torben, 2002). It accounts for 90 – 95% of most adults who develop diabetes (Centres for Disease Control and Prevention, 2008) and it is associated also with family history of diabetes, history of gestational diabetes, physical inactivity impaired glucose metabolism and race/ethnicity.
The Aetiology of Type 2 Diabetes
This is the study of the causes or causative factors of type 2 diabetes.
Causes of T2D
The development of T2D is caused by a combination of lifestyle and genetic factors.
CHAPTER FOUR
MODEL PRESENTATION AND ANALYSIS
Model Presentation
Considering the fact that the major condition leading to the development of type 2 diabetes is insulin resistance, the models of Mbah (1998), Topp et al. (2000) modified by Hernandez (2001) and subsequently Boutayeb et al. (2014) were extended to reflect the case of insulin resistance on the cells receptors of an individual having insulin resistance predisposed to development of T2D. Also, given the fact that glucose production from food intake is not constant as assumed by Topp et al. and Boutayeb et al., glucose production is given by ( ) . The resistance in the rate at which glucose is converted into glycogen in the cells (w) and resistance rate of insulin on the cells receptors (r) were introduced. Hence, the proposed model on Glucose, Insulin and β-cells mass dynamics is given by the set of ordinary differential equations (4.1), (4.2) and (4.3) respectively.
CHAPTER FIVE
DISCUSSION AND RECOMMENDATIONS
Discussion of the Results
Mathematical model on glucose, insulin, and β-cells mass dynamics considering insulin resistance as a major factor leading to development of type 2 diabetes, was presented in the form of a system of ordinary differential equations. The numerical solution using RungeKutta code in MATLAB was obtained graphically for the model behaviour. This shows rise in blood glucose level and further decline over time of glucose production below fasting glucose level from food intakewhich can be as a result of low conversion for storage in the liver. There isrise in insulin level because of insulin resistance and consequently, a slight fall in β- cells mass to overcome insulin resistance. The equilibrium points and stability analysis over time under the varying rate of insulin resistance indicated three pathways to development of T2D of which two are asymptotically stable while one is unstable at different degrees of glucose production from food intake. Further the model simulation of the parameters reveals that insulin resistance, has a significant impact on glucose concentration in the body, where high rate of insulin resistance results in high blood glucose level leading to development of T2D and low storage of glucose.
The model developed which was an extension of Mbah (1998) model is similar to those of Topp et al. (2000), Hernandez et al. (2001) and Boutayeb et al. (2014). For glucose, insulin, and β-cells mass dynamics. However, it differs significantly by its peculiarity in the case of type 2 diabetes, for taking care of insulin resistance at the cells receptors and reduction in the rate of sugar oxidation in the cells as a result of insulin resistance in T2D. Also, the realistic nature on glucose production from food intake which is never constant, rather decreases with time over the period of digestion, contrary to the assumption that glucose production is constant by some of the works reviewed.
The result of equilibrium analysis undervarying degrees of insulin resistance in type 2 diabetes gave three pathways to the development of T2D over time, in line with the three pathway equilibrium points of the results of the general diabetes models obtained by Topp et al. (2000)and genetically predisposition consideration in Boutayeb et al. (2014) model.However, the work deviated with Boutayeb et al. (2014) in area of risk factor to the development of diabetes, where Boutayeb et al. (2014) considered genetic predisposition.
Also, the simulation of the model over the varying degrees of insulin resistance, upholds Topp et al. (2000)submission that when plasma glucose concentration is above 250mg. , it results in death rate of β – cells exceeding replication rates, which accounts for the depletions of β – cells mass.
The result also upholds the proposition of Samantha (2009) and Ibrahim et al. (2016) that lack of exercise in case of insulin resistance significantly increase risk of T2D.
Conclusion
We noticed a persistent increase in glucose concentration within the first two hours from the time of meal, which subsequently decreases after hours. This shows that glucose concentration is proportional to insulin resistance rates.
Furthermore, variations in insulin resistance in glucose conversion to glycogen has significant impact on glucose concentration in the blood.
Recommendations
In order to check the prevalence of T2D as a result of insulin resistance, physical exercise and diet control are good control measures for individuals having insulin resistance. This will help to cushion the effects of high glucose concentration in the body.
Suggestion for Further Studies
The following has been suggested for further studies;
- The extension of the model on glucose production, insulin and β-cells mass dynamics in a case of insulin resistance to include α-cells mass
- A mathematical model on body glucose concentration from glucose production, insulin and β-cells mass dynamics in a case of individuals having insulin resistance and genetically predisposed to development of diabetes is to be carried
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