**2.4 Biological terms commonly used in diabetes**

**Insulin:** An anabolic hormone, produced by the beta cells of the islets of Langerhans of pancreas in response of elevated blood sugar level in the body. It helps to control the blood sugar level in the desirable range.

**Glucose:** Glucose is a simple sugar present in everyone's body. It is an essential nutrient that provides energy for the proper functioning of the body cells. After meals, food is digested in the stomach and intestines. The glucose in digested food is absorbed by the intestinal cells into the blood stream and is carried by the blood to all the cells in the body. Glucose needs insulin to enter into the body as it can not get into the cells alone.

**Glucagon:** Glucagon is a hormone synthesized and secreted from alpha cells of the pancreatic islets used for carbohydrate metabolism. Its secretion increases rapidly when the sugar level is too low in the body. It maintains the level of glucose in the blood by binding to specific receptors on hepatocytes causing the liver to release its intracellular stores of glucose. As these stores become depleted, glucagon then encourages the liver to synthesize glucose by gluconeogenesis which will be released to prevent the development of hypoglycemia, low sugar level.

**Insulin Resistance**: Sometimes the cells throughout the body become resistant to the insulin produced by the pancreas due to which it becomes difficult for the sugar to enter the cells. This condition is known as insulin resistance.

**Diabetic Ketoacidosis:** It is a condition in which the cells of muscle, liver and other body parts are unable to take up glucose for producing energy due to the absence of insulin. It is a

Therapeutic Modelling of Type 1 Diabetes 467

The insulin input I(t) will be given through injection at subcutaneous level at periodic intervals, which leaks its contents into the system over a period of time. Therefore, I(t) may

> 0 *<sup>t</sup> It b t t*

> > *I t*

Food input source term, F(t), is the source for food input to the plasma glucose level, the contents of which are reduced in a simple exponential manner. Therefore, F(t) may be

<sup>0</sup>

 

For 0 *t t* , in non – diabetic case, *F t*() 0 and *I t*() 0 and for diabetic case, *F t*() 0 , *I t*() 0 . A mathematical model for the dynamics of glucose concentration in patients with type 1 diabetes using CSII [15] therapy as an external source of insulin has been developed by us. We attempt to model the effect of an external source of insulin release, as a prescribed function of time, on glucose levels. The model is then used to assess the optimal insulin release profile, and the threshold amount required to bring the level of glucose to within a

To model the pump's delivery of insulin, we take into account three major factors: (i) the total amount of insulin released over a specific period; (ii) the time profile of insulin release, f(t); and (iii) the glucose threshold concentration Gc, below which the pump stops releasing insulin. The amount of insulin (TDD) is proportional to the total amount of glucose, whose concentration is assessed by the sensor in the pump's controller. This amount is released by the pump in a dual wave shaped insulin bolus which allows the patient to combine both normal and square wave techniques. The body characteristics of the patient determine how much insulin is needed to maintain the glucose level within the normal physiological range after each meal. The dual wave shape also provides a rapid increase in insulin plasma concentration, and sustained high circulating insulin levels while a meal is being consumed. Here, we extend the minimal model to incorporate the above factors, which leads to the

*dG XG l G G*

,

<sup>0</sup>

*t t* 

0 *t t*

, - quantity of injection, t0 – time of injection, *t* -time lag to

0 0

*t t*

 

(4)

(5)

*t* (3)

<sup>0</sup>

<sup>0</sup> *t t* 

where, S - quantity constant of meal, - delay parameter.

1( ), *<sup>b</sup>*

*dt*

*<sup>t</sup> <sup>b</sup> t t* 

0

0, *t t Se t t F t*

be defined as

At t = t0, I(t) = 0

where, <sup>0</sup>

maximum.

modeled as

0 *t t t* 

,

normal physiological range.

following differential equations:

state of absolute or relative **insulin deficiency** aggravated by hyperglycemia, dehydration, and acidosis-producing derangements in intermediary metabolism. To avoid starvation the body begins to break down fat for energy. Fatty acids and ketone bodies are released due to the break down of fat causing chemical imbalance (metabolic imbalance) called Diabetic Ketoacidosis. Moderate or large amounts of **ketones** in urine are dangerous. They upset the chemical balance of the blood.

**Chronic hyperglycemia:** Chronic hyperglycemia means elevated blood sugar level in the blood.

#### **2.5 Treatment therapies for diabetes**

Type 1 Diabetes is very serious, with a sudden and dramatic onset, usually in youth. Type 1 diabetes is an autoimmune condition, where the body attacks its own insulin producing cells. The body's immune cells, or white blood cells, include B cells and T cells. B cells make antibodies and present 'antigens' to T cells, allowing them to recognize, and kill invaders. People with Type 1 diabetes must maintain an insulin-monitoring and insulin-injecting regimen for the rest of their lives as the islets of Langerhans are destroyed in this type of diabetes. Treatment for type 1 diabetes includes taking insulin shots or insulin pump to deliver insulin in the body, making wise food choices, exercising regularly and controlling blood pressure and cholesterol.

Type 2 diabetes can be treated successfully with diet, physical activity and medication, if necessary.[23] Physical activity can help to control blood sugar levels and increases body's sensitivity to insulin.[6] Also, it helps delays or stop heart diseases, a leading complication of diabetes. Diet plays an extremely important role in controlling this type of diabetes. Being overweight can increase the chances of developing type 2 diabetes. Usually GDM in pregnant women disappears itself after delivery.

#### **2.6 Mathematical model**

The first approach to measure the insulin sensitivity *in vivo* was introduced by Himsworth and Ker [24] and the first mathematical model to estimate the glucose disappearance and insulin sensitivity was proposed by Bolie. In this model, he assumed that glucose disappearance is a linear function of both glucose and insulin. The insulin secretion and disappearance is proportional to glucose and plasma insulin concentration respectively.

The main objective here is to prescribe a more accurate, but less simple, method of arranging the palatable composition of a diabetic diet.

The modified coupled differential equations for the plasma glucose and insulin concentration [1-14, 16-22], when the normal fasting level of plasma glucose is 70 - 120 mg/dl, are given as follows

$$\frac{d\mathbf{g}}{dt} = -l\_1 h \overline{\mathbf{g}} + l\_2 \left(\mathbf{g}\_0 - \mathbf{g}\right) \mathcal{U}\left(\mathbf{g}\_0 - \mathbf{g}\right) + l\_3 F\left(t\right) \tag{1}$$

$$\frac{d\mathbf{h}}{dt} = l\_4 \left(\mathbf{g} - \mathbf{g}\_0\right) \mathcal{U} \left(\mathbf{g} - \mathbf{g}\_0\right) - l\_5 h\_0 + l\_6 I \left(t\right) \tag{2}$$

where, g(t) - plasma glucose concentration, h(t) - insulin concentration, li - sensitivity constants, i = 1,2,3,4,5,6, F(t) - food source input for plasma glucose, I(t) – insulin input and U(g0 - g) is unit step function.

The insulin input I(t) will be given through injection at subcutaneous level at periodic intervals, which leaks its contents into the system over a period of time. Therefore, I(t) may be defined as

$$I(t) = \frac{\rho t}{\overline{t} - t\_0} + b$$

At t = t0, I(t) = 0

466 Type 1 Diabetes – Complications, Pathogenesis, and Alternative Treatments

state of absolute or relative **insulin deficiency** aggravated by hyperglycemia, dehydration, and acidosis-producing derangements in intermediary metabolism. To avoid starvation the body begins to break down fat for energy. Fatty acids and ketone bodies are released due to the break down of fat causing chemical imbalance (metabolic imbalance) called Diabetic Ketoacidosis. Moderate or large amounts of **ketones** in urine are dangerous. They upset the

**Chronic hyperglycemia:** Chronic hyperglycemia means elevated blood sugar level in the

Type 1 Diabetes is very serious, with a sudden and dramatic onset, usually in youth. Type 1 diabetes is an autoimmune condition, where the body attacks its own insulin producing cells. The body's immune cells, or white blood cells, include B cells and T cells. B cells make antibodies and present 'antigens' to T cells, allowing them to recognize, and kill invaders. People with Type 1 diabetes must maintain an insulin-monitoring and insulin-injecting regimen for the rest of their lives as the islets of Langerhans are destroyed in this type of diabetes. Treatment for type 1 diabetes includes taking insulin shots or insulin pump to deliver insulin in the body, making wise food choices, exercising regularly and controlling

Type 2 diabetes can be treated successfully with diet, physical activity and medication, if necessary.[23] Physical activity can help to control blood sugar levels and increases body's sensitivity to insulin.[6] Also, it helps delays or stop heart diseases, a leading complication of diabetes. Diet plays an extremely important role in controlling this type of diabetes. Being overweight can increase the chances of developing type 2 diabetes. Usually GDM in

The first approach to measure the insulin sensitivity *in vivo* was introduced by Himsworth and Ker [24] and the first mathematical model to estimate the glucose disappearance and insulin sensitivity was proposed by Bolie. In this model, he assumed that glucose disappearance is a linear function of both glucose and insulin. The insulin secretion and disappearance is proportional to glucose and plasma insulin concentration respectively. The main objective here is to prescribe a more accurate, but less simple, method of arranging

The modified coupled differential equations for the plasma glucose and insulin concentration [1-14, 16-22], when the normal fasting level of plasma glucose is 70 - 120

1 20 0 3 *dg l hg l g g U g g l F t dt*

4 0 0 50 6 *dh <sup>l</sup> g g <sup>U</sup> g g lh lI t*

where, g(t) - plasma glucose concentration, h(t) - insulin concentration, li - sensitivity constants, i = 1,2,3,4,5,6, F(t) - food source input for plasma glucose, I(t) – insulin input and

(1)

(2)

chemical balance of the blood.

blood pressure and cholesterol.

**2.6 Mathematical model** 

mg/dl, are given as follows

U(g0 - g) is unit step function.

pregnant women disappears itself after delivery.

the palatable composition of a diabetic diet.

*dt*

**2.5 Treatment therapies for diabetes** 

blood.

$$\implies b = -\frac{\rho \cdot t\_0}{\overline{t} - t\_0} \qquad\qquad\therefore \; I(t) = \frac{\rho \left(t - t\_0\right)}{\overline{t} - t\_0} \qquad\qquad\Rightarrow \lambda + \mu \text{ t}\tag{3}$$

where, <sup>0</sup> 0 *t t t* , <sup>0</sup> *t t* , - quantity of injection, t0 – time of injection, *t* -time lag to

maximum.

Food input source term, F(t), is the source for food input to the plasma glucose level, the contents of which are reduced in a simple exponential manner. Therefore, F(t) may be modeled as

$$F(t) = \begin{cases} \mathcal{S}e^{-\alpha \left(t - t\_0\right)}, & t > t\_0 \\ 0, & t \le t\_0 \end{cases} \tag{4}$$

where, S - quantity constant of meal, - delay parameter.

For 0 *t t* , in non – diabetic case, *F t*() 0 and *I t*() 0 and for diabetic case, *F t*() 0 , *I t*() 0 .

A mathematical model for the dynamics of glucose concentration in patients with type 1 diabetes using CSII [15] therapy as an external source of insulin has been developed by us. We attempt to model the effect of an external source of insulin release, as a prescribed function of time, on glucose levels. The model is then used to assess the optimal insulin release profile, and the threshold amount required to bring the level of glucose to within a normal physiological range.

To model the pump's delivery of insulin, we take into account three major factors: (i) the total amount of insulin released over a specific period; (ii) the time profile of insulin release, f(t); and (iii) the glucose threshold concentration Gc, below which the pump stops releasing insulin. The amount of insulin (TDD) is proportional to the total amount of glucose, whose concentration is assessed by the sensor in the pump's controller. This amount is released by the pump in a dual wave shaped insulin bolus which allows the patient to combine both normal and square wave techniques. The body characteristics of the patient determine how much insulin is needed to maintain the glucose level within the normal physiological range after each meal. The dual wave shape also provides a rapid increase in insulin plasma concentration, and sustained high circulating insulin levels while a meal is being consumed. Here, we extend the minimal model to incorporate the above factors, which leads to the following differential equations:

$$\frac{d\mathbf{G}}{dt} = -\mathbf{X}\,\mathbf{G} + l\_1 \{\mathbf{G}\_b - \mathbf{G}\}^+,\tag{5}$$

$$\frac{dX}{dt} = -p\_1 X + p\_2 (I - I\_b)\_{\prime} \tag{6}$$

Therapeutic Modelling of Type 1 Diabetes 469

plasma 118 mg dl-1

plasma 7 µU ml-1

uptake 10 Min-1

uptake 0.0107 min-1

0.007 min-2 µU ml-1

**S No Parameter Description Value Unit** 

2 Gc Glucose threshold concentration in plasma 100-107 mg dl-1

5 l2 Scaling factor determining TDD of insulin Variable min-1 µU mg-1 6 l3`` The rate of decay for insulin in plasma 0.264 min-1

Base line value of glucose concentration in

Baseline value of insulin concentration in

The insulin dependent rate of tissue glucose

The rate of spontaneous decrease of glucose

The rate of insulin – dependent increase in tissue glucose uptake due to insulin concentration excess over its baseline

Table 1. Description and values of the model parameters obtained from the published

This particular work is published in Applied Mathematics and computation, 2007, pages 1476 – 1483 and has been cited by Kato, R, Munkhjargal, M and Takahashi, D "An autonomous drug release system based on chemo- mechanical energy conversion "Organic Engine" for feedback control of blood glucose", Biosensors and Bioelecetronics in 2010 Vol

More advanced mathematical models can be formulated to explain the effects of obesity on diabetes, effects of exercise on management of type 2 diabetes. Parameters involving glucose sensors can be added to the insulin pump model for a better programmed insulin delivery

[1] D. Araujo-Vilar, C.A. Rega-Liste, D.A. Garcia-Estevez, F. Sarmiento-Escalona, V.

[2] R.N. Bergman, L.S. Phillips, C. Cobelli, Physiologic evaluation of factors controlling

[3] B.W. Bode, R.D. Steed, P.C. Davidson, Reduction in severe hypoglycemia with long term

[4] A. Boutayeb, E.H. Twizell, K. Achouayb, A. Chetouani, A mathematical model for the

[5] A. De Gaetano, O. Arino, Mathematical modelling of the intravenous glucose tolerance

subject, Diabetes Res. Clin. Pract. 39 (1998) 129–141.

glucose tolerance in man, J. Clin. Invest. 68 (1981) 1456–1467.

Mosquera-Tallon, J. Cabezas-Cerrato, Minimal model of glucose metabolism: modified equations and its application in the study of insulin sensitivity in obese

continuous subcutaneous insulin infusion in type 1 diabetes, Diabetes Care 19

burden of diabetes and its complications, BioMed. Eng. (2004), doi:10.1186/1475-

1 Gb

3 Ib

4 l1

7 p1

8 p2

26(4), pages 1455 - 1459.

**2.7 Future work** 

by insulin pump.

**3. References** 

(1996) 324–327.

925X-3-20, Online.

test, J. Math. Biol. 40 (2000) 136–168.

literature

$$\frac{d\mathcal{I}}{dt} = -l\_2(\mathcal{G} - \mathcal{G}\_c)^+ f(t) - l\_3(I - I\_b)\_\prime \tag{7}$$

where G is the blood glucose concentration, X is an auxiliary function representing remote insulin action, and l is the insulin plasma concentration. A description of the model parameters and their values are given in Table 1.

The important part of this extension is the first term of (7) which models all three factors mentioned above. This term contributes to the insulin plasma when the glucose concentration exceeds the threshold Gc, and is defined as

$$\mathbf{l}\_2(\mathbf{G} - \mathbf{G}\_c)^+ = \begin{cases} \mathbf{l}\_2(\mathbf{G}(t) - \mathbf{G}\_c) \, f(t) & \text{if } \mathbf{G}(t) > \mathbf{G}\_c \\ 0 & \text{if } \mathbf{G}(t) < \mathbf{G}\_c \end{cases} \tag{8}$$

The function models the profile of insulin release from the pump, and the coefficient represents a scaling factor determining TDD of insulin released by the pump. In the next section, we discuss different profiles of insulin release and compare their effects on the optimal control of glucose concentration. The newer generation of pumps can be programmed to release insulin using three different bolus techniques.

A normal bolus can be used if small amounts of carbohydrates are consumed or if a correction to the blood glucose level outside the physiological range needs to be made. A square wave profile is helpful when eating foods that are high in both fat/protein and carbohydrate (fat and protein delay the absorption of carbohydrates). If a normal bolus is given for a meal high in protein and fat concentrations, circulating insulin levels rise rapidly and may peak before the carbohydrates are absorbed. This mismatch in insulin and blood glucose levels can result in postprandial hypoglycemia. Therefore, a dual wave bolus, as a combination of the normal and square wave bolus techniques, can be introduced. Using this technique, half of the insulin dose is given (over a short period of time) at the onset of the meal, and the remainder over a 2–4 h period. The profile of a dual wave bolus is modeled as a function of time, f(t), in Eq. (4) over a period of 3 h (Fig. 1(a)–(c)).

Fig. 1. Profile of insulin release by the pump *f*(*t*), for 3h: HLL release; (b)LHL release; (c) LLH release, where *H* stands for high amount release of insulin and *L* stands for its low amount per hour *f*(*t*) is normalized so that *H*=*L*

*<sup>p</sup> X pI I dt*

*dI l G G ft l I I dt*

where G is the blood glucose concentration, X is an auxiliary function representing remote insulin action, and l is the insulin plasma concentration. A description of the model

The important part of this extension is the first term of (7) which models all three factors mentioned above. This term contributes to the insulin plasma when the glucose

2

( () ) () () ( ) <sup>0</sup> ( )

*l G t G f t if G t G lG G*

The function models the profile of insulin release from the pump, and the coefficient represents a scaling factor determining TDD of insulin released by the pump. In the next section, we discuss different profiles of insulin release and compare their effects on the optimal control of glucose concentration. The newer generation of pumps can be

A normal bolus can be used if small amounts of carbohydrates are consumed or if a correction to the blood glucose level outside the physiological range needs to be made. A square wave profile is helpful when eating foods that are high in both fat/protein and carbohydrate (fat and protein delay the absorption of carbohydrates). If a normal bolus is given for a meal high in protein and fat concentrations, circulating insulin levels rise rapidly and may peak before the carbohydrates are absorbed. This mismatch in insulin and blood glucose levels can result in postprandial hypoglycemia. Therefore, a dual wave bolus, as a combination of the normal and square wave bolus techniques, can be introduced. Using this technique, half of the insulin dose is given (over a short period of time) at the onset of the meal, and the remainder over a 2–4 h period. The profile of a dual wave bolus is modeled as

Fig. 1. Profile of insulin release by the pump *f*(*t*), for 3h: HLL release; (b)LHL release; (c) LLH release, where *H* stands for high amount release of insulin and *L* stands for its low

*dX*

parameters and their values are given in Table 1.

2

concentration exceeds the threshold Gc, and is defined as

*c*

programmed to release insulin using three different bolus techniques.

a function of time, f(t), in Eq. (4) over a period of 3 h (Fig. 1(a)–(c)).

amount per hour *f*(*t*) is normalized so that *H*=*L*

1 2 ( ), *<sup>b</sup>*

2 3 ( ) ( ) ( ), *c b*

(6)

*c*

(8)

(7)

*c c*

*if G t G*


Table 1. Description and values of the model parameters obtained from the published literature

This particular work is published in Applied Mathematics and computation, 2007, pages 1476 – 1483 and has been cited by Kato, R, Munkhjargal, M and Takahashi, D "An autonomous drug release system based on chemo- mechanical energy conversion "Organic Engine" for feedback control of blood glucose", Biosensors and Bioelecetronics in 2010 Vol 26(4), pages 1455 - 1459.
