**Abstract**

This chapter is devoted to studying the physiology of the pituitary-thyroid axis and thyroid size in autoimmune thyroiditis via modeling. The pituitary-thyroid axis consists of a feed forward and backward loop in humans, which is responsible for maintaining the body's metabolism. Under a disease situation, the dynamics of the axis becomes more complex and unique among patients. Hashimoto's autoimmune thyroiditis disrupts the normal operation of the axis by slowly destroying the thyroid follicle cells through complex immune mechanisms. So, the size of thyroid and the axis operation are fully, partly, or totally not functional in this disease. Basically, the patient situation in the disease process is unique in describing the diffused goiter and/or a clinical symptom of hashitoxicosis, euthyroidism, subclinical hypothyroidism, or overt hypothyroidism. Using patient-specific modeling, we can predict the hidden dynamics of the natural history of autoimmune thyroiditis and test hypothesis on the operation of axis. In addition, we unfold case studies of three patients from the thyroid literature through the modeling viewpoint and describe their hidden dynamics.

**Keywords:** Hashimoto's thyroiditis, chronic lymphocytic thyroiditis, goiter, hypothyroidism, patient-specific modeling, pituitary-thyroid axis

#### **1. Introduction**

The normal operation of the pituitary-thyroid axis depends on the levels of thyroid stimulating hormone (TSH) and thyroid hormones, triiodothyronine (T3) and thyroxine (T4) [1]. Serum TSH is produced and released by the pituitary gland in response to the low levels of free thyroid hormones in the serum. Circulating TSH in turn stimulates the thyroid to produce and secrete T3 and T4 into the serum. When thyroid hormones reach their highest levels, it inhibits the production of TSH, which describes the normal day to day operation of the pituitary-thyroid axis. The axis is commonly referred as an important negative feedback loop in the endocrine system. The normal function of this loop is essential for the body's metabolic rate—it affects how quickly the cells in our body use the energy stored within itself [2, 3]. For the purpose of this modeling work, we use only serum free T4 for the levels of thyroid hormones as frequently asked to measure in the thyroid clinics for all patients with Hashimoto's autoimmune thyroiditis (see **Figure 1**).

The normal operation of the axis can be tested and verified clinically via one measurement of TSH and free T4 from the blood serum. Laboratory tests are important in diagnosing conditions of the thyroid gland [2]. The result of the blood test is determined approximately based on the normal reference range of TSH and free T4 is (0.4–4) mU/L and (7–18) pg/mL, respectively as recommended by the American Thyroid Association [2]. As the physiology of the axis is governed by TSH and free T4, the clinical state of the axis has been defined based on these values [3]. Suppose TSH and free T4 levels falls within the normal reference range; the state of the axis is said to be clinically normal. Suppose TSH levels falls above 4 mU/L but free T4 levels falls within the reference range; the state of the axis is said to be clinically subclinical hypothyroidism. Suppose TSH levels are above 4 mU/L and free T4 levels below 7 pg/mL; the state of the axis is said to be clinically hypothyroidism (an underactive thyroid gland). Keeping free T4 levels within the normal reference range is very important [3]. Lower levels of free T4 can cause hypothyroidism that results in several health problems including obesity, joint pain, infertility, slow metabolism, puffy face, constipation, stiffness, dry skin, depression, fatigue and higher heart rate [4].

through a simple blood test for anti-thyroid antibody biomarkers. More specifically,

(TPOAb) and/or thyroglobulin antibodies (TGAb) may indicate the malfunction of the immune system [8]. For simplicity in patient-specific modeling, we choose TPOAb as a biomarker representing the presence of the Hashimoto's autoimmune

In 1912, Hakaru Hashimoto published the description of four cases of diffused goiter which were showing the signs of infiltration of immune cells and antibodies in the thyroid gland [9]. All these goiters appeared differently from the colloid goiters that occur due to insufficient iodine in-take from the diet. In his time, the idea of autoimmune disease had not been established in the medical literature. He isolated these cases and asked the future clinicians to explore the mechanism of this new form of goiter [10, 11]. In 1936, the description of goiter due to lymphocytic thyroiditis was rediscovered in the United States and was labeled as Hashimoto's thyroiditis. It has been characterized as an organ-specific autoimmune disease and swelling might be the result of the chronic stimulation of the thyroid gland by the serum TSH [12, 13]. Now, scientists have described autoimmune thyroiditis with

the elevated titers of anti-thyroid antibodies, thyroid peroxidase antibodies

*Mathematical Modeling of Thyroid Size and Hypothyroidism in Hashimoto's Thyroiditis*

*DOI: http://dx.doi.org/10.5772/intechopen.90481*

atrophy, a variance from the original description of Hakaru Hashimoto.

mation is not available in the clinical setting.

with the normal production of hormones.

**29**

The pituitary gland has no clue about the undergoing immune process in the thyroid gland and does not know how to adapt to the changing environment and the abnormal function of the thyroid [14, 15]. It simply tells the thyroid to keep up with the release of thyroid hormones for its signal. By responding through bulging its size, the poor thyroid gland needs to accomplish the task of producing enough levels of hormones if it could. What initiates the immune process against the thyroid gland? The answer is still largely unknown to the researchers, but they all speculate the immune process might be initiated through a complex combination of the genetic and environmental pollution [15]. Herein, we care about the functional size of thyroid gland as opposed to the exact size of the thyroid gland, which is treated as a hidden compartment [16]. The exact size of the thyroid gland includes the size of the parathyroid glands, the isthmus, capillaries, blood flow and so on whereas the functional size includes the follicular cells that contains thyroglobulin molecules in which hormones are stored. Basically, the functional size of the gland counts the part of the gland that is able to make and secrete hormones; obviously this infor-

A measurement of TPOAb titers is required from the blood test to diagnose autoimmune thyroiditis and afterward, TPOAb titers is not usually measured as its levels not helpful in the treatment. As mentioned above, the axis function is determined from one-time measurement of TSH and free T4, respectively. However, the functional size of the thyroid gland cannot be measured through the laboratory experiments or with any medical tools, which does not exist in practice at least so far. Even if it exists, then it would be very expensive in terms of cost and time. However, the mathematical model can help by telling us the functional size from the clinical measurements of specific patient and from the physiological point of view. Moreover, the functional size of the gland is different for everyone, such as boy's thyroid size versus men's thyroid size or girl's thyroid size versus women's thyroid size and so on. Using the model and the clinical measurements of TSH and free T4, we will determine the initial functional size of the gland for a given patient. Goiter due to this disease occurs more frequently in women than men [17]. Using the model, we describe the diffused goiter as the functional size increases to keep up

In reality, it is hard to perform experiments on patients in the laboratory setting and collect data to understand the underlying dynamics of the pituitary-thyroid axis

thyroiditis.

Hyperthyroidism is another clinical state of the axis, in which free T4 levels stay above 18 pg/mL while the levels of TSH stay below the reference range 0.4 mU/L [5]. This state of the axis occurs when the thyroid gland over produces and secretes the thyroid hormones either in response to chronic TSH stimulation or due to Graves' disease. Transient hyperthyroidism is a phenomenon that refers to the leakage of stored thyroid hormones from the gland, typically called Hashitoxicosis or thyroid burst [6]. It happens due to the Hashimoto's autoimmune thyroiditis. The bursting of thyroid gland can happen at any clinical stage of the patients.

Hashimoto's autoimmune thyroiditis is one of the immune disorders hosted by the thyroid gland [6, 7]. Under this disease process, the thyroid follicular tissue undergoes the slow destruction by the immune system and thereby the thyroid struggles to produce enough hormones for the body's requirement. Currently, the incidence rate of Hashimoto's autoimmune thyroiditis is estimated to be 300–500 cases per 100,000 individuals per year. In this disease, the interaction of the immune system with the thyroid is highly complex involving the cellular and humoral mechanisms. The involvement of humoral mechanism can be verified

#### **Figure 1.**

*The pituitary-thyroid axis is shown in this picture. It is commonly referred as a negative-feedback control.*

#### *Mathematical Modeling of Thyroid Size and Hypothyroidism in Hashimoto's Thyroiditis DOI: http://dx.doi.org/10.5772/intechopen.90481*

through a simple blood test for anti-thyroid antibody biomarkers. More specifically, the elevated titers of anti-thyroid antibodies, thyroid peroxidase antibodies (TPOAb) and/or thyroglobulin antibodies (TGAb) may indicate the malfunction of the immune system [8]. For simplicity in patient-specific modeling, we choose TPOAb as a biomarker representing the presence of the Hashimoto's autoimmune thyroiditis.

In 1912, Hakaru Hashimoto published the description of four cases of diffused goiter which were showing the signs of infiltration of immune cells and antibodies in the thyroid gland [9]. All these goiters appeared differently from the colloid goiters that occur due to insufficient iodine in-take from the diet. In his time, the idea of autoimmune disease had not been established in the medical literature. He isolated these cases and asked the future clinicians to explore the mechanism of this new form of goiter [10, 11]. In 1936, the description of goiter due to lymphocytic thyroiditis was rediscovered in the United States and was labeled as Hashimoto's thyroiditis. It has been characterized as an organ-specific autoimmune disease and swelling might be the result of the chronic stimulation of the thyroid gland by the serum TSH [12, 13]. Now, scientists have described autoimmune thyroiditis with atrophy, a variance from the original description of Hakaru Hashimoto.

The pituitary gland has no clue about the undergoing immune process in the thyroid gland and does not know how to adapt to the changing environment and the abnormal function of the thyroid [14, 15]. It simply tells the thyroid to keep up with the release of thyroid hormones for its signal. By responding through bulging its size, the poor thyroid gland needs to accomplish the task of producing enough levels of hormones if it could. What initiates the immune process against the thyroid gland? The answer is still largely unknown to the researchers, but they all speculate the immune process might be initiated through a complex combination of the genetic and environmental pollution [15]. Herein, we care about the functional size of thyroid gland as opposed to the exact size of the thyroid gland, which is treated as a hidden compartment [16]. The exact size of the thyroid gland includes the size of the parathyroid glands, the isthmus, capillaries, blood flow and so on whereas the functional size includes the follicular cells that contains thyroglobulin molecules in which hormones are stored. Basically, the functional size of the gland counts the part of the gland that is able to make and secrete hormones; obviously this information is not available in the clinical setting.

A measurement of TPOAb titers is required from the blood test to diagnose autoimmune thyroiditis and afterward, TPOAb titers is not usually measured as its levels not helpful in the treatment. As mentioned above, the axis function is determined from one-time measurement of TSH and free T4, respectively. However, the functional size of the thyroid gland cannot be measured through the laboratory experiments or with any medical tools, which does not exist in practice at least so far. Even if it exists, then it would be very expensive in terms of cost and time. However, the mathematical model can help by telling us the functional size from the clinical measurements of specific patient and from the physiological point of view. Moreover, the functional size of the gland is different for everyone, such as boy's thyroid size versus men's thyroid size or girl's thyroid size versus women's thyroid size and so on. Using the model and the clinical measurements of TSH and free T4, we will determine the initial functional size of the gland for a given patient. Goiter due to this disease occurs more frequently in women than men [17]. Using the model, we describe the diffused goiter as the functional size increases to keep up with the normal production of hormones.

In reality, it is hard to perform experiments on patients in the laboratory setting and collect data to understand the underlying dynamics of the pituitary-thyroid axis

The normal operation of the axis can be tested and verified clinically via one measurement of TSH and free T4 from the blood serum. Laboratory tests are important in diagnosing conditions of the thyroid gland [2]. The result of the blood test is determined approximately based on the normal reference range of TSH and free T4 is (0.4–4) mU/L and (7–18) pg/mL, respectively as recommended by the American Thyroid Association [2]. As the physiology of the axis is governed by TSH and free T4, the clinical state of the axis has been defined based on these values [3]. Suppose TSH and free T4 levels falls within the normal reference range; the state of the axis is said to be clinically normal. Suppose TSH levels falls above 4 mU/L but free T4 levels falls within the reference range; the state of the axis is said to be clinically subclinical hypothyroidism. Suppose TSH levels are above 4 mU/L and free T4 levels below 7 pg/mL; the state of the axis is said to be clinically hypothyroidism (an underactive thyroid gland). Keeping free T4 levels within the normal reference range is very important [3]. Lower levels of free T4 can cause hypothyroidism that results in several health problems including obesity, joint pain, infertility, slow metabolism, puffy face, constipation, stiffness, dry skin, depression,

Hyperthyroidism is another clinical state of the axis, in which free T4 levels stay above 18 pg/mL while the levels of TSH stay below the reference range 0.4 mU/L [5]. This state of the axis occurs when the thyroid gland over produces and secretes the thyroid hormones either in response to chronic TSH stimulation or due to Graves' disease. Transient hyperthyroidism is a phenomenon that refers to the leakage of stored thyroid hormones from the gland, typically called Hashitoxicosis or thyroid burst [6]. It happens due to the Hashimoto's autoimmune thyroiditis. The

Hashimoto's autoimmune thyroiditis is one of the immune disorders hosted by the thyroid gland [6, 7]. Under this disease process, the thyroid follicular tissue undergoes the slow destruction by the immune system and thereby the thyroid struggles to produce enough hormones for the body's requirement. Currently, the incidence rate of Hashimoto's autoimmune thyroiditis is estimated to be 300–500 cases per 100,000 individuals per year. In this disease, the interaction of the immune system with the thyroid is highly complex involving the cellular and humoral mechanisms. The involvement of humoral mechanism can be verified

*The pituitary-thyroid axis is shown in this picture. It is commonly referred as a negative-feedback control.*

bursting of thyroid gland can happen at any clinical stage of the patients.

fatigue and higher heart rate [4].

*Goiter - Causes and Treatment*

**Figure 1.**

**28**

in autoimmune problem. Hypothetically speaking, if the experiment is possible, then one needs to consider patients' time, cost, safety, and variability. Two patients with similar characteristics from societies might react to an experiment in an unusual way [18, 19]. The inter and intra variability among patients causes a major problem for treatment and challenges the obtained information from labs. Using the model, we can carry out an experiment for a specific patient and the hidden dynamics of the axis can be investigated at the microscopic level for shorter or longer time-period. In general, the model is implemented through a computer program and the parameters are tuned to a specific patient and then the experiments can be performed with the test hypothesis [20]. Also, we can identify sensitive and insensitive parameters that are responsible for the physiology of the axis under autoimmune thyroiditis, that causes the hypothyroidism and goiter. This is an effective and modern way to explore the complex interaction of the immune system to the thyroid and its consequence on the negative feedback loop [21, 22].

4.The humoral immune system uses serum TPOAb to attack the thyroid and those titers can be used as a biomarker for the level of the anti-thyroid immune

*Mathematical Modeling of Thyroid Size and Hypothyroidism in Hashimoto's Thyroiditis*

5.The patient does not demonstrate central or peripheral resistance to thyroid

6. Serum free T4 have much slower dynamics in the disease process compared to

We denote *x t*ð Þ, *y t*ð Þ and *w t*ð Þ to represent the concentrations of serum TSH (mU/L), serum free T4 (pg/mL) and serum TPOAb (U/mL), respectively. We let *z t*ð Þ represent the functional size of the gland. According to the fundamental principle of rate laws, the rate of change of concentration of TSH over time *t* is the secretion rate from the pituitary gland minus the elimination rate of TSH via kidney through the unspecified mechanism. The rate of change of concentration of free T4 over time *t* is the secretion rate of free T4 from the thyroid minus the elimination rate of free T4 via kidney through unspecified mechanism. Similarly, the rate of change of concentration of TPOAb over time *t* is the production rate due to the abnormal interaction thyroid gland and immune system minus the elimination rate of TPOAb via kidney through unspecified mechanism. Finally, the rate of change of

serum TSH, serum TPOAb and the functional size of the gland.

the functional size of thyroid gland over time *t* is the growth rate minus the

Next, we write a coupled model that represents the interaction of two subsystems: the negative feedback loop and the humoral immune system.

*ka* <sup>þ</sup> *y t*ð Þ � *<sup>k</sup>*2*x t*ð Þ, *<sup>x</sup>*ð Þ¼ <sup>0</sup> *<sup>x</sup>*<sup>0</sup> (1)

� *k*6*z t*ð Þ*w t*ð Þ *z*ð Þ¼ 0 *z*<sup>0</sup> (3)

*kd* <sup>þ</sup> *x t*ð Þ � *<sup>k</sup>*4*y t*ð Þ, *<sup>y</sup>*ð Þ¼ <sup>0</sup> *<sup>y</sup>*<sup>0</sup> (2)

*dt* <sup>¼</sup> *<sup>k</sup>*7*z t*ð Þ*w t*ð Þ� *<sup>k</sup>*8*w t*ð Þ *<sup>w</sup>*ð Þ¼ <sup>0</sup> *<sup>w</sup>*<sup>0</sup> (4)

where *x t*ð Þ≥0, *y t*ð Þ≥0, *z t*ð Þ> 0 and *w t*ð Þ≥0. The normal average value for each state variable is given in **Table 1**. The model has 11 positive parameters unchanged over time, in which each parameter has a unique physiological meaning and their corresponding values listed in **Table 2**. The model (1)–(4) captured the dynamics of two subsystems coupled through the functional size of the thyroid gland. Using assumption (8) that serum free T4 has much slower dynamics compared to serum

**Name Normal value Normal range Source Unit** *x t*ð Þ 1 0*:*4 � ð Þ 2*:*5 � 4 Literature [2] mU/L *y t*ð Þ 13 7 � 18 Literature [2] pg/mL *z t*ð Þ 0.015 0*:*005 � 0*:*125 Literature [16] L *w t*ð Þ 0 0 � <200 Dataset U/mL

activity.

*DOI: http://dx.doi.org/10.5772/intechopen.90481*

hormone.

destruction rate of the gland.

*dx*

*dy*

*dz dt* <sup>¼</sup> *<sup>k</sup>*<sup>5</sup>

*dw*

*Variable names, normal values, ranges, sources and units.*

**Table 1.**

**31**

*dt* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup> � *<sup>k</sup>*1*y t*ð Þ

*dt* <sup>¼</sup> *<sup>k</sup>*3*z t*ð Þ*x t*ð Þ

*x t*ð Þ *z t*ð Þ � *<sup>N</sup>* 

The hormone TSH has a half-life of 1 hour in serum on the average scale. Similarly, the hormone free T4 has a half-life of 7 days in serum on average, the thyroid per-oxidase antibodies (TPOAb) has a half-life of 24 hours in serum on average and the half-life of functional size of the thyroid gland is not known and probably varies among individuals. Several time scales have been involved in the disease process. Based on the fundamental principle of rate laws, the model will be developed here to predict the history of patient-specific dynamics of the axis and thyroid size in autoimmune thyroiditis [23–25]. The model can unfold the consequences of the presence of antibodies titers in the blood. Basically, it can replace humans but mimics the system that provides convenience and flexibility for scientists to run experiments on the computer. Information obtained through the dynamics might be useful in treating patients and improving the accuracy of the levothyroxine treatment [26]. For instance, the levothyroxine drug can be targeted and administered to the specific patient so that the levels of TSH and free T4 maintained within the normal reference range, which in turn can avoid other health problems.

The remaining of this chapter is divided into three main sections. Section 2 introduces the development and construction of a coupled model describing the interactions of two subsystems (the axis and humoral immune system) and reduces the coupled model based on the physiological assumptions. Section 3 analyzes the solutions of the reduced model qualitatively and numerically for various parameter values. Section 4 describes the case studies of three patients, validity of the model and provides the conclusion of this work.

### **2. Model outline**

The model is constructed as a system of four ordinary differential equations based on the following known physiological assumptions [20, 21].

#### **2.1 Assumptions**


*Mathematical Modeling of Thyroid Size and Hypothyroidism in Hashimoto's Thyroiditis DOI: http://dx.doi.org/10.5772/intechopen.90481*


We denote *x t*ð Þ, *y t*ð Þ and *w t*ð Þ to represent the concentrations of serum TSH (mU/L), serum free T4 (pg/mL) and serum TPOAb (U/mL), respectively. We let *z t*ð Þ represent the functional size of the gland. According to the fundamental principle of rate laws, the rate of change of concentration of TSH over time *t* is the secretion rate from the pituitary gland minus the elimination rate of TSH via kidney through the unspecified mechanism. The rate of change of concentration of free T4 over time *t* is the secretion rate of free T4 from the thyroid minus the elimination rate of free T4 via kidney through unspecified mechanism. Similarly, the rate of change of concentration of TPOAb over time *t* is the production rate due to the abnormal interaction thyroid gland and immune system minus the elimination rate of TPOAb via kidney through unspecified mechanism. Finally, the rate of change of the functional size of thyroid gland over time *t* is the growth rate minus the destruction rate of the gland.

Next, we write a coupled model that represents the interaction of two subsystems: the negative feedback loop and the humoral immune system.

$$\frac{d\mathbf{x}}{dt} = k\_1 - \frac{k\_1 y(t)}{k\_a + y(t)} - k\_2 \mathbf{x}(t), \qquad \mathbf{x}(0) = \mathbf{x}\_0 \tag{1}$$

$$\frac{dy}{dt} = \frac{k\_3 x(t)x(t)}{k\_d + x(t)} - k\_4 y(t), \qquad \quad \quad \mathbf{y}(\mathbf{0}) = \mathbf{y}\_0 \tag{2}$$

$$\frac{dz}{dt} = k\varsigma \left(\frac{\varkappa(t)}{z(t)} - N\right) - k\varsigma z(t)w(t) \qquad z(\mathbf{0}) = z\_0 \tag{3}$$

$$\frac{dw}{dt} = k\_7 z(t)w(t) - k\_8 w(t) \qquad w(0) = w\_0 \tag{4}$$

where *x t*ð Þ≥0, *y t*ð Þ≥0, *z t*ð Þ> 0 and *w t*ð Þ≥0. The normal average value for each state variable is given in **Table 1**. The model has 11 positive parameters unchanged over time, in which each parameter has a unique physiological meaning and their corresponding values listed in **Table 2**. The model (1)–(4) captured the dynamics of two subsystems coupled through the functional size of the thyroid gland. Using assumption (8) that serum free T4 has much slower dynamics compared to serum


**Table 1.**

*Variable names, normal values, ranges, sources and units.*

in autoimmune problem. Hypothetically speaking, if the experiment is possible, then one needs to consider patients' time, cost, safety, and variability. Two patients with similar characteristics from societies might react to an experiment in an unusual way [18, 19]. The inter and intra variability among patients causes a major problem for treatment and challenges the obtained information from labs. Using the model, we can carry out an experiment for a specific patient and the hidden dynamics of the axis can be investigated at the microscopic level for shorter or longer time-period. In general, the model is implemented through a computer program and the parameters are tuned to a specific patient and then the experiments can be performed with the test hypothesis [20]. Also, we can identify sensitive and insensitive parameters that are responsible for the physiology of the axis under autoimmune thyroiditis, that causes the hypothyroidism and goiter. This is an effective and modern way to explore the complex interaction of the immune system to the thyroid and its consequence on the negative feedback loop [21, 22]. The hormone TSH has a half-life of 1 hour in serum on the average scale. Similarly, the hormone free T4 has a half-life of 7 days in serum on average, the thyroid per-oxidase antibodies (TPOAb) has a half-life of 24 hours in serum on average and the half-life of functional size of the thyroid gland is not known and probably varies among individuals. Several time scales have been involved in the disease process. Based on the fundamental principle of rate laws, the model will be developed here to predict the history of patient-specific dynamics of the axis and thyroid size in autoimmune thyroiditis [23–25]. The model can unfold the consequences of the presence of antibodies titers in the blood. Basically, it can replace humans but mimics the system that provides convenience and flexibility for scientists to run experiments on the computer. Information obtained through the dynamics might be useful in treating patients and improving the accuracy of the levothyroxine treatment [26]. For instance, the levothyroxine drug can be targeted and administered to the specific patient so that the levels of TSH and free T4 maintained within the normal reference range, which in turn can avoid other health

The remaining of this chapter is divided into three main sections. Section 2 introduces the development and construction of a coupled model describing the interactions of two subsystems (the axis and humoral immune system) and reduces the coupled model based on the physiological assumptions. Section 3 analyzes the solutions of the reduced model qualitatively and numerically for various parameter values. Section 4 describes the case studies of three patients, validity of the model

The model is constructed as a system of four ordinary differential equations

1.The pituitary gland is diseased free, so the feed forward loop is intact.

2.Total TSH receptors concentration does not change during Hashimoto's

3. Serum TSH stimulates the growth of functional thyroid and the production

based on the following known physiological assumptions [20, 21].

problems.

*Goiter - Causes and Treatment*

**2. Model outline**

**2.1 Assumptions**

**30**

and provides the conclusion of this work.

autoimmune thyroiditis.

and secretion of thyroid hormones.


#### **Table 2.**

*Parameter names, normal values, ranges, sources and units.*

TSH, serum TPOAb and the functional size of the thyroid gland, we can take *dy=dt* ¼ 0 and obtain a reduced model consisting of three differential equations and one algebraic equation:

$$\frac{d\mathbf{x}}{dt} = \frac{k\_1 k\_d k\_d (k\_d + \mathbf{x}(t))}{k\_4 k\_d k\_d + k\_4 k\_d \mathbf{x}(t) + k\_3 \mathbf{z}(t)\mathbf{x}(t)} - k\_2 \mathbf{x}(t), \qquad \mathbf{x}(0) = \mathbf{x}\_0 \tag{5}$$

$$z = \frac{k\_4 \wp(k\_d + \varkappa)}{k\_3 \varkappa} = f(\varkappa, \jmath) \tag{6}$$

model, we can describe the clinical progression of overt hypothyroidism and

*Mathematical Modeling of Thyroid Size and Hypothyroidism in Hashimoto's Thyroiditis*

the reduced model (Eqs. (3)–(5)) from the following equations:

*k*5 *x <sup>z</sup>* � *<sup>N</sup>*

*z*<sup>1</sup> ¼ *x*1*=N*, *w*<sup>1</sup> ¼ 0 and the value of *x*<sup>1</sup> is given by the cubic equation:

<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*x*<sup>1</sup>

*a*1*x*<sup>1</sup>

tional in the presence of diseased-free steady state solution.

and the value *x*<sup>2</sup> is given by the quadratic equation:

*k*1*k*4*ka*ð Þ *kd* þ *x*

As a first step in the process of analyzing the model, we solve for steady states of

*<sup>k</sup>*4*kakd* <sup>þ</sup> *<sup>k</sup>*4*kax* <sup>þ</sup> *<sup>k</sup>*3*zx* � *<sup>k</sup>*2*<sup>x</sup>* <sup>¼</sup> <sup>0</sup>

*k*7*zw* � *k*8*w* ¼ 0

which leads to diseased-free and diseased steady state solutions besides the initial condition. The first steady state (diseased-free) denoted as ð Þ *z*1, *w*1, *x*<sup>1</sup> where

*<sup>a</sup>*<sup>1</sup> <sup>¼</sup> *<sup>k</sup>*2*k*<sup>3</sup>

*a*<sup>2</sup> ¼ *k*2*k*4*ka* > 0 *a*<sup>3</sup> ¼ *k*4*ka*ð Þ *kdk*<sup>2</sup> � *k*<sup>1</sup> <0 since *k*<sup>1</sup> >*kdk*<sup>2</sup> *a*<sup>4</sup> ¼ �*k*4*kak*1*kd* <0

By Descarte's rule of signs, Eq. (7) has one positive real solution, so the reduced model has diseased-free state in the positive octant for the system parameters. In fact, this steady state solution lives on the surface of the function *z* ¼ *f x*ð Þ , *y* given by Eq. (6). The operation of the pituitary-thyroid axis is healthy and fully func-

Next, the second steady state is the diseased state solution denoted as ð Þ *z*2, *w*2, *x*<sup>2</sup>

*<sup>z</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>8</sup> *k*7

> *k*7*x*<sup>2</sup> *k*8

� *N* 

*k*2*k*3*k*<sup>8</sup> *k*7 <sup>&</sup>gt; <sup>0</sup>

*b*<sup>2</sup> ¼ *k*4*ka*ð Þ *kdk*<sup>2</sup> � *k*<sup>1</sup> < 0 since *k*<sup>1</sup> > *kdk*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*2*x*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>3</sup> <sup>¼</sup> <sup>0</sup> (8)

*<sup>w</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*7*k*<sup>5</sup> *k*6*k*<sup>8</sup>

*b*1*x*<sup>2</sup>

*b*<sup>1</sup> ¼ *k*2*k*4*ka* þ

*<sup>N</sup>* <sup>&</sup>gt;<sup>0</sup>

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*3*x*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*<sup>4</sup> <sup>¼</sup> <sup>0</sup> (7)

� *<sup>k</sup>*6*zw* <sup>¼</sup> <sup>0</sup>

thyroid size such as goiter and atrophy.

*DOI: http://dx.doi.org/10.5772/intechopen.90481*

**3. Model analysis**

where

where

where

**33**

Using (Eq. (6)), the functional size can be determined for patients if a result of TSH (*x*) and free T4 (*y*) value is known from the blood test, which in turn can be used as an initial condition on (Eq. (3)) for the model simulation. The graph of (Eq. (6)) is shown on the **Figure 2**, which illustrates the functional size of thyroid gland in terms of varying TSH and free T4 values. Using the reduced

#### **Figure 2.**

*The functional size of the thyroid gland can be calculated from the graph of this function of TSH and free T4.*

*Mathematical Modeling of Thyroid Size and Hypothyroidism in Hashimoto's Thyroiditis DOI: http://dx.doi.org/10.5772/intechopen.90481*

model, we can describe the clinical progression of overt hypothyroidism and thyroid size such as goiter and atrophy.
