Stochastic and Optimization Programming for Blood

P. Tirupathi Rao / International Journal of Engineering Science and Technology (IJEST)

Stochastic and Optimization Programming for Blood Glucose & Insulin Regulatory Systems of Type-2 Diabetes P. Tirupathi Rao Dept. of Statistics, Pondicherry University, Puducherry – 605 014, India, Email:[email protected] ABSTRACT In this paper a stochastic modeling and optimization problem is developed for blood glucose and insulin regulatory system suitable for management of type-2 diabetes patients. The rates of arrival and consumption of glucose and insulin in the blood are assumed as Poisson parameters and developed a model through stochastic processes. A stochastic programming problem for maintaining the optimized glucose and insulin levels of the blood is developed. An objective function for maximizing the energy release subject to the constraints on the consumption of glucose is formulated. Sensitivity analysis was carried out and observed the model behaviour through influencing parameters. Enhancing the developed programming problem as desk top automation system will make this work more accessible to the decision makers of Health care industry. Key words: Diabetes mellitus, Stochastic Programming, Glucose regulating system, Hypoglycemia, Hyperglycemia, 1. INTRODUCTION: Diabetes mellitus is a syndrome characterized by chronic hyperglycemia and disturbances of carbohydrate, fat, and protein metabolism associated with absolute or relative deficiencies in insulin secretion/action. Type II diabetes is associated with a deficit in the mass of β-cells resulting in reduced insulin secretion, so as to the development of a resistance to the action of insulin. Many individuals with the fully developed syndrome show impairments of both insulin secretion and insulin mediated glucose disposal. Insulin is secreted primarily in response to elevated blood glucose concentrations. It is an in charge of facilitating glucose entry into cells and also playing a key role in the control of intermediary metabolism. It has profound effects on both carbohydrate and lipid metabolism, and significant influences on protein and mineral metabolism. Diabetes mellitus is a metabolic disorder of human beings due to insulin deficiency. Non-insulindependent diabetes mellitus begins as a syndrome of insulin resistance in which targeted tissues fail to respond appropriately to insulin. Excessive insulin secretion is usually the result of insulin-secreting tumors. The high levels of insulin resulting from this condition or from an overdose of insulin causes a precipitous drop in blood glucose concentrations. The brain becomes starved for energy, leading to the syndrome of insulin shock, which is acutely life threatening. A major effect of insulin is to promote the entrance of glucose and amino acids in the cells of muscle tissue, adipose tissue and connective tissues. Glucose enters the cell by facilitated diffusion as inward gradient created by low intra cellular free glucose and by the availability of special carrier namely transporters. In the presence of insulin, the rate of movement of glucose in to the cell is greatly stimulated in a selective fashion. The regulation and functioning of insulin is more related to the mass of beta cells and the concentration of insulin receptors. There is a need of considering the contributions of mass of beta cells and the concentration of insulin receptors in understanding the dynamics of glucose and insulin. There is good evidence on the works of modeling the dynamics of glucose and insulin through mathematical models in the literature. Himsworth and Ker (1939) introduced the approach of measuring the insulin sensitivity in vivo. Bolie (1961) developed the model to estimate the glucose disappearance and insulinglucose dynamics by ordinary differential equations. Kapur (1981) considered the compartmental models to develop the glucose and insulin concentrations in diabetes. Mahan et.al (1987) developed a Mathematical model on early diabetes mellitus by considering the manifestation of the disease as a function of time. Volker Tresp etal (1999) developed a model on the blood glucose metabolism of diabetic by considering linear and non linear compartment models. Robert etal. (1999) developed a model based on predictive control algorithm to maintain normoglycemia in the Type –1 diabetic patients using compartment models. De Gaetano and Arino ( 2000) proposed an aggregated delay differential model called a dynamical model for measuring glucose levels in the blood. Jiaxu Li (2004) has developed the models on the dynamics of Glucose – Insulin endocrine metabolic regulatory system. P.T.Rao etal. (2011) developed a Stochastic Model on blood Glucose Levels in Type-2 Diabetes Mellitus. In all these papers, much emphasis was given on the general dynamics of glucose and insulin using the compartment models.

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Vol. 4 No.11 November 2012

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Ryan D.Heradez et.all (2001) has made an attempt in modeling of Glucose and Insulin by considering the mass of Beta Cells and the concentrations of Insulin Receptors. The rates of arrivals/ consumption of the said components are assumed as constants/deterministic. He has applied this model for diabetic patients. But when we studied the levels of the Glucose and Insulin, influenced by the components like mass of Beta Cells and density of Insulin Receptors, they act as random variables rather than constants. More over the conditions of metabolic phenomena are non deterministic and effected by many uncertainties. Therefore assuming the stochastic environment for modeling the levels of Glucose, Insulin, Mass of Beta cells and the concentrations of Insulin receptors is more appropriate. These things motivated us for developing the stochastic model to study the glucose and insulin regulatory mechanism. Our work has explored a programming problem for finding the optimal regulating methods of diabetes treatment. 2. ASSUMPTIONS & NOTATIONS: In this model we assume that the arrivals of glucose to the blood plasma through different modes are random and follows Poisson processes with parameters a1,a2,a3,a4 and a5 representing arrival rates of glucose through food intake, liver supply( when the glucose levels are reached to the minimum levels), Gluconeogenesis, Lipogenolosis process, and in other modes respectively. a2, a3, a4 are the parameters that are 4

dependent one another and also influenced by a1 and at a1=0.

Let a =

 i 2

glucose in the blood through various means, where μi  0,

a  a1  a5 ; be the total

i 1 i

3

 i 1

i

= 1 ; ai  0, 1, 2, 3 are the constants.

Various amounts of glucose clearance/consumption by body are also random and follows Poisson processes with parameters b1, b2, b3, b4,b5 and b6, Representing the rates of consumption of glucose through GLUT1 to GLUT4 ( Insulin dependent consumption), through GLUT-5( Insulin independent), through methodology of protein synthesis, through conversion of glucose to fats and fatty acids, through renal passage, and through liver 5

respectively. i.e. while conversion of glucose in to glycogen( Insulin dependent). And let b =

b

i

i 1

be the

total clearance of glucose through various means. Let c1, c2 be the rates of induced insulin per unit time through pancreas and through Drugs respectively. It is assumed that the arrival of the insulin to the blood 2

stream through physiological and external modes are random and follow the Poisson process. Let c=

c  i

i 1

λi  0 ,

2

 i 1

i

i

,

 1 , is the total induced insulin to the blood plasma. Let d1, d2, d3, and d4 be the rates of insulin

clearance for conversion of glucose in to glycogen, for facilitating glucose consumption by Muscle/tissue cells, for excreting through renal passage (kidney) when the patient is insulin resistant, and for converting as the 4

enzyme of insulin receptors respectively. All those parameters follow the Poisson process. Let d =

d i 1

i

, is

the total clearance of insulin through the above modes. Let ‘e’ be the inflexion point of sigmoid function, ‘  ’is the mass of beta cells in the pancreas, ‘m’ is the rate of death of beta cells, ‘h1’ is the beta cell glucose tolerance range , ‘h2’ is the beta cell glucose tolerance range , on the surface of muscle cells. Let j be the recycling rate of insulin receptors, l1 be the endocytosis rate of insulin dependent receptor, l2 be the endocytosis rate of insulin independent receptor. Let G(t) be the blood glucose concentrations (measured in mg/dl), ‘I(t)’ be the insulin concentration in the blood (measured in μU/ml),  (t) be the beta cell mass (measured in mg), and R(t) be the fraction of insulin receptors on the surface of the muscle cells, at time t. The fraction of insulin receptors decreases by both natural endocytosis and insulin-induced down regulation, and increases due to a natural recycling of the internalized receptors. The system of the linear differential equation are in the form of the consumption of glucose is influenced by the quantum of insulin, mass of beta cells, rate of receptors of insulin and the existing glucose. Insulin is secreted by the β-cells and cleared by various parts of the body like the liver, kidney, muscle cells, and insulin receptors. The relationship between the extra cellular glucose concentration and the rate of insulin secretion has been shown to follow a sigmoid function in plasma glucose concentration. It also depends on the β-cell mass and fraction of receptors on the cell surfaces, as they relate to insulin resistance. It is assumed that the normal rate of insulin clearance at the muscle cell receptors to be proportional to the rate of clearance at liver and kidneys. Normal glucose levels can be maintained in individuals with insulin resistance via

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increases in blood insulin levels. Glucose homeostasis can be maintained despite significant loss of beta cell mass, it indicates that greater beta cell mass is required in the presence of insulin resistance. As per experimental observations, insulin receptors internalization is the major mechanism by which cell surface insulin receptors are down regulated. The internalized and subsequent recycling of the insulin requires insulin binding. This insulin-induced regulation mediated by internalization decreases the concentration of insulin receptors on the cell surface. 3. STOCHASTIC MODEL: With the above assumptions and the mentioned structure of Glucose, Insulin, Mass of Beta Cells, The size of Insulin Receptors, and the aspects of Metabolism , The differential equation of the glucose at time ‘t’ is 5 d G (t )  a  ( bi  I (t ).b6 .R(t )).G (t ) = a – E(t).G(t) dt i 1

( 3.1 )

Where, E(t).G(t) is the total body glucose up take rate and proportional to G(t) and E(t)= I (t ).b6 .R (t ) is the effectiveness of glucose or the ability of the body to consume the glucose from blood. Usually is measured in 1/d. The differential equation for insulin dynamics is

(c1  (t )  c 2 ) d I (t )  G 2 (t )  d .I (t )  d .R (t ).I (t ) 2 dt (e  G (t )).(1  R(t ))

(3. 2 )

Where c1/ (1+R(t)) is the rate of insulin secretion per one  cell. (in units of µU / ml mg d). It is assumed that β-cells reach their maximal secretion capacity when R(t) = 0. The differential equation for beta cell mass is

d G (t )  (t )  h1 .G (t ).(1  )  (t )  m (t ) dt h1 / h2 Where h1 =

( 3.3 )

dl 2 dl , is the beta cell glucose tolerance range and h2 = , is the beta cell glucose mg .d mg 2 .d

tolerance range on the surface of muscle cells and m is the rate of death of beta cells. The differential equation for insulin receptor dynamics is

d R(t )  j.(1  R(t ))  l1 .I (t ).R(t )  l 2 .R(t ) dt

( 3.4 )

Where j is the recycling rate of insulin receptors; l1 is the endocytosis rate of insulin dependent receptor; and l2 is the endocytosis rate of insulin independent receptor. Solving the differential equations (3.1) to (3.4) simultaneously, we can obtain the functions of Glucose level, Insulin Level, The mass level of Beta cells, The size of the Insulin Receptors. The function of the Glucose level at a point of time is G(t) =

a (1  e  k1t )  G0 .e  k1t k1

Where k1= c + I(t).b.R(t); The initial conditions are G(0) = 0, G(t) =

( 3.5 )

a (1  e k1t ) , for t ≠ 0 k1

The function of the Insulin level at a point of time is I(t) = (1  e

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 kt

h ).  I 0 .e kt k


( 3.6 )

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(c1  (t )  c 2 )G 2 (t ) Where k= d+k3 , k3 = d.R(t), h= ; the initial conditions for Insulin levels are I(0) = (e  G 2 (t ))(1  R(t )) h  kt 0, I(t) = (1  e ). , for t ≠ 0 k The function of the Mass of the Beta cells level at a point of time is

 (t )   (t ).[(k 4  m).t  1] ,

( 3.7 )

G (t ) ), The initial conditions for the levels of Mass of Beta cells are  (t )   0 . (h1 / h2 ) for t=0;  (t )  ( k 4  m). 0 .t , for t ≠ 0

Where k4 = h1.G(t).(1-

The function of the Insulin Receptors level at a point of time is R(t) = (1- e

 h.t

).

k5  R0 .e  h.t , h

( 3.8 )

Where k5 = j, h= k5 + k6 + k1, k6= l1.I, k7=l2; and the initial conditions are R(t) = R0, at t=0, R(t)=(1- e

 h .t

).

k5 , h

for t ≠ 0 4. THE ENERGY FUNCTION: As Glucose is the prime source of energy and to get energy to the cell, the metabolism part of the glucose will be taken place as an act of intracellular activity. The usual mechanism is the combination of glucose + water + oxygen will give us Energy + Corbondioxide + water, observed as chemical reaction C6H12O6 + 6H2O + 6O2 = 38ATP + 6CO2 + 12H2O. The metabolism of glucose is an interesting feature, which involves compartment functions. The induced food has changed in to many formats and finally it will be absorbed in to blood in the form of glucose. Insulin will act as the facilitator in consuming the glucose for getting energy. The combination of glucose and insulin together will play a role in the blood metabolism. This model considers the concept of chemical reactions and law of mass actions. It is considered that some part of the glucose will be consumed for producing the insulin in the pancreas. The rate of glucose transformed in to ATP is proportional to the amount of glucose at time t. Let A(t) represents the number of units of ATP and OE present at time t with A(t) = 0. the differential equation for A(t) at time t is

d A(t )  kA(t ), k >0 . The chemical reaction for this type is denoted by a first dt

order reaction. Consider that the combination of two substances say Glucose G(t), and Insulin I(t) will release Adenosinetripaspate[ATP] and Other Emissions[OE]. G0 units of glucose combines with I0 units of insulin will form G0+I0 units of ATP and OE. Where G0, I0 are the initial amounts of glucose and insulin respectively. The rate at which A(t) changes is governed by the chemical and physical reactions of human body and ultimately it can be attributed as the law of mass action. In the usual mechanism of human physiology, the rate at which the two substances say glucose and insulin react to form the substance ATP+ OE. The total emission of ATP + OE is proportional to the consumption of glucose and insulin at time t. The amounts of glucose and insulin are combine in the ratio G(t) and I(t).

I (t ) G (t ) A(t ) units of glucose and A(t ) units of insulin. I (t )  G (t ) I (t )  G (t ) G (t ) Hence the glucose and insulin remain untransformed at time t are G0  A(t ) and I (t )  G (t ) I (t ) I0  A(t ) respectively. I (t )  G (t )

It implies that A(t) consists of

The differential equation of A(t) with the chemical and mass action is

d G (t ) I (t ) A(t )  K. ( G0  A(t ) ).( I 0  A(t ) ) dt I (t )  G (t ) I (t )  G (t )

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( 4.1 )

4658


where k>0 is the constant of proportion. The kind of reaction in (4.1) is called as a second order reaction and

d A (t) is a quadratic function of dt

A(t). The usual mechanism behind this is the rate at which the reaction proceeds is proportional to the product of the remaining concentration of glucose and insulin. This implies the equation (4.1) can be written as

G (t ).I (t ) d G (t )  I (t ) G (t )  I (t ) ( G0  A(t )).( I 0  A(t )) = L.(D-A).(B-A) A(t ) = K. 2 I (t ) G (t ) dt (G (t )  I (t )) If D=B; i.e

G0 I 0 G (t ) 1 (G0  I 0 ) 2 G (t )  I (t ) G0  , then A(t) = G I G (t ) K .I 0 .G0 .t  (G0  I 0 )

D.B[1  e  L ( D  B ) t ] G (t )  I (t ) If D  B, and Let D>B, then A(t) = I0 , as t   , A  B =  L ( D  B ).t I (t ) D  B.e where L= K.

G (t ).I (t ) G (t )  I (t ) G (t )  I (t ) G0 , B = I0 , D= 2 G (t ) I (t ) (G (t )  I (t ))

(4.2)

The limiting value of A(t) depends on the amount of the insulin present initially and on the ratio

I (t ) . I (t )  G (t )

The laws of chemical mass action can also be applicable to more complicated reactions involving more than two compounds is described by the differential equation n d Q(t ) = k.  ( xi  ri .Q(t )) dt i 1

(4.3)

5. STOCHASTIC PROGRAMMING: Glucose metabolism results in release of energy, usually taken place in the inner parts of cell where the process of energy release is required. The classical mechanism of glucose entry to the internal parts of the cell will be possible when the insulin receptors can accommodate the entry of glucose molecules to inner parts of the cell. Diet, Physical Exercise and Medication are the three ways of managing the diabetes problem for effective health administration. Physical exercise is more useful than the remaining two because of its impact on the general health of the patient. This study is on optimal design and monitoring of healthy glucose levels as well as insulin levels in the blood for a diabetic patient. High density glucose levels without enough insulin put the patient in Hyperglycemia; on the other hand Low density levels of glucose with more insulin in the blood leads to the condition of Hypoglycemia. Both the situations are unwarranted. There is an absolute need to maintain the glucose levels and insulin levels are in healthy threshold limits. Providing reasonable size of physical exercise to the body is always good for health. Usually the endocrinology system of Diabetic patients is in passive mode and hence the size of insulin produced by pancreas may not sufficient for wanted metabolism levels. The considerable decrement in mass of beta cells is one of the reasons for less secretion of insulin. Obesity conditions of the body due to lack of exercise may also one of the causes for low levels of beta mass cells. The act of more physical exercise leads to healthy maintenance of glucose and insulin levels. In other way, increasing the intensity of physical exercise will make the body system to produce the required size of energy. The optimal levels of glucose in a healthy person usually lies in between 70 to 110 mg/dl in fasting, where as they are in between 80 to 140 after having the meal. These limits may not be considered as it is to the case of diabetic patient due to their disturbed endocrinology system. These limits are subject to alter form patient to patient and even for the same patient form time to time. Keeping these points in mind we develop a stochastic programming problem with an objective of maximizing the energy release subject to the constraints of maintaining optimal threshold limits of Glucose and Insulin through suitable health administration strategies. G ( t ). I ( t )

Maximize A (t) =

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( K. 2 G (t )  I (t ) G (t )  I (t ) G0 . I 0 (1  e ( G (t )  I ( t )) G (t ) I (t ) G ( t ). I ( t )

( K. 2 G (t )  I (t ) G (t )  I (t ) G0  I 0 .e ( G (t )  I (t )) G (t ) I (t )


G (t ) I (t ) G (t )  I (t ) G0  I 0 )t G (t ) I (t )

G (t )  I (t ) G (t )  I (t ) G0  I 0 )t G (t ) I (t )

) (5.1)

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Where G(t) =

a (1  e k1t ) k1

; I(t) = (1  e

 kt

).

h k

Subject to the constraints

a (1  e  k1t )  G0 .e  k1t ≤ GU k1

(5.2)

a (1  e  k1t )  G0 .e  k1t ≥ GL k1

(5.3)

G L ≤ GU

(5.4)

4

Where a =

 i 2

a  a1  a5 , μi  0,

i 1 i

3

 i 1

i

= 1. , ai  0, 1, 2, 3 are the constants and ‘a’ is the total

glucose in the blood through various means; GL, GU are the optimal and required Lower and Upper limits of Glucose level for diabetic patients.; k1= c + I(t).b.R(t); b =

k3 = d.R(t),

e  h.t ).

(c1  (t )  c 2 )G 2 (t ) h= ; d = (e  G 2 (t ))(1  R(t ))

5

2

i 1

i 1

 bi ; c=  ci i , λi  0 ,

4

d i 1

i

;

2

 i 1

i

 (t )   (t ).[(k 4  m).t  1] ;

 1 ; k= d+k3 , R(t) = (1-

k5  R0 .e  h.t ; k5 = j, h= k5 + k6 + k1, k6= l1.I, k7=l2 h

The Decision Variables are (stochastic Parameters under study) such as μi  0, ai  0, bi  0, ci  0, λi  0 ,

(5.5)

As the control of Insulin, Beta cells mass and the Insulin receptor enzyme is not in our control, The Constraints is considered only on Glucose. We have obtained the simulated values of the objective function for varying values of Glucose inputs and Glucose consumptions on different accounts. Also developed the data set for calculating the energy released functions on hypothetical environments of Insulin inputs and their consumptions on different counts. All these calculation were carried out with the software MATHCAD 6.0 6. NUMERICAL ILLUSTRATION AND FINDINGS: We obtain the values of G(t), I(t), and A(t) for different values of the parameters using the equations (3.5), (3.6) and (4.3) and presented in Table 1 and Table-2. Table-1: The values of the parameters and the glucose, insulin and energy released

a1 2 5 8 13 20

For b1 = 1, b2 = 3, b3 = 5, b4= 3, b5= 8, C1= 3, C2= 6, e=2, are a 2 a 3 a 4 a 5 I0 R  T G(T) I(T) A(T) 1

5

3

4

4 6 8 12 7 10 13 16 5 6 9 11 6 12 14

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6

16

6

10

0.006 0.007 0.008 0.010 0.012 0.007 0.007 0.008 0.010 0.006 0.007 0.009 0.010 0.006 0.007 0.009 0.010 0.011 0.013 0.014


0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024

7.381 7.657 7.933 8.393 9.038 7.657 7.841 8.025 8.393 7.565 7.841 8.117 8.393 7.565 7.841 8.117 8.393 8.669 9.222 9.406

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a1

a2

a3

a4

a5

I0

R



T

G(T)

I(T)

A(T)

0.01 0.04 0.06 0.09

0.015 0.009 0.005 0.003 0.003 0.012 0.009 0.007 0.006 0.015 0.015 0.015 0.015 0.015 0.006 0.006 0.006 0.006

0.024 0.024 0.024 0.024 0.024 0.014 0.009 0.005 0.004 0.024 0.029 0.037 0.042 0.048 3.052 0.418 0.125 0.037

9.774 13.376 22.278 33.378 35.779 10.817 12.071 13.743 14.789 9.974 9.108 8.457 8.157 7.921 4.794 6.069 6.266 6.895

18 10 19 30 32 21 27 35 40 6 8 11 13 15

Table-2: The values of the parameters and the glucose, insulin and energy released

b1

b2

For a1 = 2, a2 =1, a3=5, a4 = 3, a6 =4, I0= 6, R= 16,  = 6, T=10 are c1 b3 b4 b5 d1 d2 d3 G(T) I(T) c2

3 7 10 20 5 12 25 42 7 21 27 40 5 10 22 38 10 21 29 39 225 2000 5000 10000 900 4700 11000 24000 4 7 9 11 6 10 14 18 5 9 12 18

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0.014 0.013 0.012 0.009 0.009 0.008 0.006 0.005 0.014 0.010 0.009 0.007 0.014 0.012 0.009 0.007 0.014 0.010 0.009 0.007 0.015 0.012 0.009 0.007 0.015 0.012 0.009 0.007 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015

0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 1.176 10.386 25.952 51.896 0.797 4.083 9.532 20.774 0.014 0.010 0.008 0.007 0.012 0.008 0.006 0.005 0.012 0.008 0.006 0.005

A(T) 9.523 9.109 8.857 8.250 8.158 7.888 7.533 7.230 9.523 8.403 8.115 7.679 9.523 9.020 8.25 7.679 9.523 8.579 8.158 7.792 6.076 6.007 6.002 6.001 6.109 6.018 6.006 6.012 12.605 15.435 17.322 19.209 13.548 17.322 21.096 24.870 13.548 17.322 20.153 25.814

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7. FINDINGS & CONCLUSIONS: From Tables 1 and 2 it is observed that the glucose at time t is the increasing functions of the glucose arrival rate through food intake, liver, gluconeogenesis and lipogenolysis processes when all the remaining parameters are fixed. It is also observed that the released energy is an increasing function of glucose arrival rates due to the said formats. It is inferred that the release of energy is proportionate the quantum of the glucose transformed. This established the hypothesis that the mass and chemical reactions of substances Glucose and Insulin. The glucose at time t is a decreasing function of the glucose clearance rates through protein synthesis, conversion of glucose to fats and fatty acids, renal passage, and conversion of glucose in to glycogen etc. It is observed that the energy is a decreasing function of glucose clearance rates due to various mentioned means as the released energy is always proportional to the available quantum of glucose in the plasma. The glucose levels are decreasing functions of initial levels of Insulin and the energy is an increasing function of initial Insulin levels. It is observed that increasing levels of insulin acts on two ways as by decreasing the glucose levels and by consumption of glucose so that there will be increase in energy function. An insulin level in the plasma is a decreasing function of number of insulin receptors as the other parameters are constant. The glucose level is also decreasing function of the number of insulin receptors. The energy released is an increasing function of number of insulin receptors. It is observed that as the number of insulin receptors increasing it acts on the decreasing in the levels of glucose leads to increase in release of energy. Insulin level is an increasing function of beta cell mass and it has no impact on the glucose levels, leads to decrease in energy. Hence it is observed that the Beta cell mass will increase the insulin levels, leads to consumption of existing glucose and releases the energy proportionate to insulin consumption. It supports the hypothesis that increasing levels of insulin may leads to hypoglycemia as the existing glucose levels will come down. The Glucose level is a decreasing function, the Insulin levels is an increasing function and the energy release is a decreasing function of the arrival rate of Insulin as the other parameters are constants. This result is also establishing that the release of energy is directly proportional to the glucose consumption and inversely proportional to the insulin consumption. Hence the release of energy is influenced by the insulin sensitivity. If the insulin concentrations are more in the blood then there is a possibility of insulin resistance as the receptors do not respond properly. This phenomenon is due to the non requirement of glucose in the intracellular compartment. It is also observed that the insulin concentrations are increasing and the energy release levels are increasing functions of the rate of insulin clearance for conversion of glucose, as the other parameters are fixed. This leads to increase the effectiveness of insulin receptors so that the consumption of insulin is increased. Insulin-stimulated glucose disposal is reduced in individuals with type II diabetes as compared to non-diabetic controls insulin resistance of a similar magnitude has been observed in many non-diabetic individuals, including obese subjects, or during pregnancy, puberty, and aging. Therefore, normal glucose levels can be maintained in individuals with insulin resistance via increases in blood insulin levels. In addition, it has been suggested that glucose homeostasis can be maintained despite significant loss of β-cell mass function when an individual has normal insulin sensitivity. It may be suggested that a greater β-cell mass is required in the presence of insulin resistance.

8. REFERENCE: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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