**3. Model analysis**

As a first step in the process of analyzing the model, we solve for steady states of the reduced model (Eqs. (3)–(5)) from the following equations:

$$\frac{k\_1 k\_4 k\_d (k\_d + x)}{k\_4 k\_a k\_d + k\_4 k\_d \varkappa + k\_3 \varpi \varkappa} - k\_2 \varkappa = 0$$

$$k\_5 (\frac{\varkappa}{z} - N) - k\_6 \varpi \nu = 0$$

$$k\_7 \varpi \nu - k\_8 \varpi = 0$$

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 *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:

$$a\_1 \varkappa\_1^{\; 3} + a\_2 \varkappa\_1^{\; 2} + a\_3 \varkappa\_1 + a\_4 = 0 \tag{7}$$

where

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

**Name Normal value Normal range Source Unit** *k*<sup>1</sup> 5000 >4000 Literature [27] mU/L day *k*<sup>2</sup> 16*:*6 N/A Literature [3] 1/day

*k*<sup>3</sup> 90*:*11 4 � 257 Simulation Pg/mL L day *k*<sup>4</sup> 0*:*099021 N/A Literature [3] 1/day *k*<sup>5</sup> 1 N/A Simulation *L*3/mU day

*k*<sup>6</sup> 1 N/A Simulation mL/U day *k*<sup>8</sup> 0.035 N/A Literature [28] 1/day

*ka* 0*:*043 0*:*02 � 0*:*06 Calculation pg/mL *kd* 0*:*05 N/A Simulation mU/*L*<sup>2</sup>

*<sup>z</sup>* <sup>¼</sup> *<sup>k</sup>*4*y k*ð Þ *<sup>d</sup>* <sup>þ</sup> *<sup>x</sup>*

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

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

*<sup>k</sup>*4*kakd* <sup>þ</sup> *<sup>k</sup>*4*kax t*ðÞþ *<sup>k</sup>*3*z t*ð Þ*x 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> (5)

*<sup>k</sup>*3*<sup>x</sup>* <sup>¼</sup> *f x*ð Þ , *<sup>y</sup>* (6)

one algebraic equation:

**Table 2.**

**Figure 2.**

*free T4.* **32**

*dx*

*Goiter - Causes and Treatment*

*dt* <sup>¼</sup> *<sup>k</sup>*1*k*4*ka*ð Þ *kd* <sup>þ</sup> *x t*ð Þ

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

$$a\_1 = \frac{k\_2 k\_3}{N} > 0$$

$$a\_2 = k\_2 k\_4 k\_a > 0$$

$$a\_3 = k\_4 k\_a (k\_d k\_2 - k\_1) < 0 \text{ since } k\_1 > k\_d k\_2$$

$$a\_4 = -k\_4 k\_d k\_1 k\_d < 0$$

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

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

$$z\_2 = \frac{k\_8}{k\_7}$$

$$w\_2 = \frac{k\_7 k\_5}{k\_6 k\_8} \left(\frac{k\_7 \varkappa\_2}{k\_8} - N\right)$$

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

$$b\_1 \mathfrak{x}\_2^2 + b\_2 \mathfrak{x}\_2 + b\_3 = \mathbf{0} \tag{8}$$

where

$$b\_1 = \left(k\_2 k\_4 k\_a + \frac{k\_2 k\_3 k\_8}{k\_7}\right) > 0$$

$$b\_2 = k\_4 k\_a (k\_d k\_2 - k\_1) < 0 \text{ since } k\_1 > k\_d k\_2$$

*Goiter - Causes and Treatment*

$$b\_3 = -k\_4 k\_a k\_1 k\_d < 0$$

*d* ¼ *ax*<sup>0</sup> þ *by*<sup>0</sup> þ *cz*<sup>0</sup>

Next, we take the implicit differentiation of Eq. (9) with respect to time *t* and

*∂g ∂z* ∙ *dz*

þ

. This threshold value can be uniquely determined for every patient

*∂g ∂y* ∙ *dy dt* þ

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

<sup>0</sup> *ka* þ *y*<sup>0</sup>

*dt*

Notice that *N* is independent of the parameters *k*<sup>7</sup> and *k*8. So, we can use this

based on their initial blood test results and if we know other parameters *k*1, *k*2, *k*3, *k*4, *k*5, *k*6, *ka* and *kd*. We have estimated the values of all other parameters from the

As the bifurcation parameter *N* varies from the threshold value *N*� ¼ *k*7*x*0*=k*8, the number of the steady states changes, and its behavior near the steady states (linear stability) changes as well. Moreover, as the bifurcation parameter varies, the appearance and disappearance of steady states can be seen on the tangent plane at the initial state of the function *z* ¼ *f x*ð Þ , *y* . For this system (Eqs. (3)–(5)), there are

, *N* ¼ *N*� and *N* > *N*�

, the system (Eqs. (3)–(5)) has two steady states, which undergoes

overview of how the solutions become stable and unstable as parameter *N* changes

a saddle-node bifurcation—disappearance of the saddle diseased state. One can

*The bifurcation diagram shows the steady state values of TSH as the parameter N varies from 0 to 300. BP means the branch or bifurcation point where another equilibrium curve passes through and the system switches its stability. The parameter value of N at BP is 66.7. H is a neutral saddle, but not the bifurcation point for the*

*equilibrium curve. At H, there is a homoclinic orbit for the system.*

, which results in the following equa-

*x*0 *z*0

. **Figure 3** illustrates a nice

� ð Þ *<sup>k</sup>*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*6*w*<sup>0</sup> *<sup>z</sup>*<sup>0</sup> *k*5

(10)

*P* ¼ 0

substituting the initial point *P x*0, *y*0, *z*0, *w*<sup>0</sup>

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

literature or through calculation (see **Table 2**).

*∂g ∂x* ∙ *dx dt* þ

*<sup>N</sup>* <sup>¼</sup> �*k*4*y*<sup>0</sup> �*k*1*kakd* � *<sup>x</sup>*0ð Þ �*k*2*kd* <sup>þ</sup> *<sup>k</sup>*4ð Þ *kd* <sup>þ</sup> *<sup>x</sup>*<sup>0</sup> *ka* <sup>þ</sup> *<sup>y</sup>*<sup>0</sup>

*k*5*k*3*x*<sup>2</sup>

tion for *N*:

value of *N* as *N*�

**3.3 Linear stability**

from 0 to 300. **Case 1:** *N* < *N*� When *N* < *N*�

**Figure 3.**

**35**

three cases, namely when *N* < *N*�

By Descarte's rule of signs, Eq. (8) has one positive real solution when *x*<sup>2</sup> > *Nk*<sup>8</sup> *<sup>k</sup>*<sup>7</sup> so the reduced model has diseased state in the positive octant for the system parameters.

**Remark:** When *N* ¼ *k*7*x*2*=k*8, the second steady state (diseased state) of the system (Eqs. (3)–(5)) has *w*<sup>2</sup> ¼ 0. So, it must coincide with the first steady state (diseased-free state) in the positive quadrant if

$$N = \frac{k\_7 \varkappa\_2}{k\_8} = \frac{k\_7 \varkappa\_1}{k\_8}$$

#### **3.1 Definition: bifurcation parameter**

We let *N*� be the unique value of *N*, where

$$N^\* = \frac{k\gamma\varkappa\_0}{k\_8} = \frac{k\gamma\varkappa\_2}{k\_8} = \frac{k\gamma\varkappa\_1}{k\_8}.$$

the system undergoes a bifurcation. We call *N*� is a bifurcation value of the system (Eqs. (3)–(5)) and *N* is the bifurcation parameter. We can take the initial state of the system at where the diseased-free and diseased steady states merge together. Using this definition, we can also calculate the parameter *k*<sup>7</sup> (say *k*<sup>7</sup> � ) provided the values of *k*<sup>8</sup> and *x*<sup>0</sup> is known.

#### **3.2 Equation of tangent plane**

We first define the level surface (*S*) through a function of three variables from Eq. (6) as

$$\mathbf{g}(\mathbf{x}(t), \mathbf{y}(t), \mathbf{z}(t)) = \frac{k\_4 \mathbf{y}(t)(k\_d + \mathbf{x}(t))}{k\_3 \mathbf{x}(t)} - \mathbf{z}(t) = \mathbf{0} \tag{9}$$

The normal surface vector *n* ! can be calculated from the gradient of this function

$$\overrightarrow{n} = \langle \frac{\partial \mathbf{g}}{\partial \mathbf{x}}, \frac{\partial \mathbf{g}}{\partial y}, \frac{\partial \mathbf{g}}{\partial \mathbf{z}} \rangle$$

In particular, the normal vector of the surface *S* at initial state *P x*0, *y*0, *z*<sup>0</sup> is a vector perpendicular to the tangent plane of *S* at *P* is given by

$$\overrightarrow{n} = \langle \frac{-k\_4 k\_d \wp\_0}{k\_3 \varkappa\_0^2}, \left(\frac{k\_4 k\_d}{k\_3 \varkappa\_0} + \frac{k\_4}{k\_3}\right), -1 \rangle.$$

The equation of the tangent plane at point *P x*0, *y*0, *z*<sup>0</sup> is given by

$$a\infty + by + cz = d$$

where

$$a = \frac{-k\_4 k\_d y\_0}{k\_3 \varkappa\_0 r^2}, b = \left(\frac{k\_4 k\_d}{k\_3 \varkappa\_0} + \frac{k\_4}{k\_3}\right), c = -\mathbf{1} \wedge \mathbf{1}$$

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

*d* ¼ *ax*<sup>0</sup> þ *by*<sup>0</sup> þ *cz*<sup>0</sup>

Next, we take the implicit differentiation of Eq. (9) with respect to time *t* and substituting the initial point *P x*0, *y*0, *z*0, *w*<sup>0</sup> , which results in the following equation for *N*:

$$\left(\frac{\partial \mathbf{g}}{\partial \mathbf{x}} \bullet \frac{d\mathbf{x}}{dt} + \frac{\partial \mathbf{g}}{\partial \mathbf{y}} \bullet \frac{d\mathbf{y}}{dt} + \frac{\partial \mathbf{g}}{\partial \mathbf{z}} \bullet \frac{d\mathbf{z}}{dt}\right)\_P = \mathbf{0}$$

$$N = \frac{-k\_4 \mathbf{y}\_0 \left(-k\_1 k\_d k\_d - \mathbf{x}\_0 (-k\_2 k\_d + k\_4 (k\_d + \mathbf{x}\_0)) \left(k\_d + \mathbf{y}\_0\right)\right)}{k\_5 k\_3 \mathbf{x}\_0^2 \left(k\_d + \mathbf{y}\_0\right)} + \frac{\mathbf{x}\_0}{\mathbf{z}\_0} - \frac{(k\_4 + k\_6 \mathbf{z}\_0) \mathbf{z}\_0}{k\_5} \tag{10}$$

Notice that *N* is independent of the parameters *k*<sup>7</sup> and *k*8. So, we can use this value of *N* as *N*� . This threshold value can be uniquely determined for every patient based on their initial blood test results and if we know other parameters *k*1, *k*2, *k*3, *k*4, *k*5, *k*6, *ka* and *kd*. We have estimated the values of all other parameters from the literature or through calculation (see **Table 2**).

#### **3.3 Linear stability**

*b*<sup>3</sup> ¼ �*k*4*kak*1*kd* < 0

By Descarte's rule of signs, Eq. (8) has one positive real solution when *x*<sup>2</sup> > *Nk*<sup>8</sup>

**Remark:** When *N* ¼ *k*7*x*2*=k*8, the second steady state (diseased state) of the system (Eqs. (3)–(5)) has *w*<sup>2</sup> ¼ 0. So, it must coincide with the first steady state

> <sup>¼</sup> *<sup>k</sup>*7*x*<sup>1</sup> *k*8

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

the system undergoes a bifurcation. We call *N*� is a bifurcation value of the system (Eqs. (3)–(5)) and *N* is the bifurcation parameter. We can take the initial state of the system at where the diseased-free and diseased steady states merge together. Using this definition, we can also calculate the parameter *k*<sup>7</sup> (say *k*<sup>7</sup>

We first define the level surface (*S*) through a function of three variables from

*gxt* <sup>ð</sup> ð Þ, *y t*ð Þ, *z t*ð ÞÞ ¼ *<sup>k</sup>*4*y t*ð Þð Þ *kd* <sup>þ</sup> *x t*ð Þ

*n* ! <sup>¼</sup> ⟨ *∂g ∂x* , *∂g ∂y* , *∂g ∂z* ⟩

�*k*4*kdy*<sup>0</sup>

*<sup>k</sup>*3*x*0<sup>2</sup> , *<sup>b</sup>* <sup>¼</sup> *<sup>k</sup>*4*kd*

vector perpendicular to the tangent plane of *S* at *P* is given by

The equation of the tangent plane at point *P x*0, *y*0, *z*<sup>0</sup>

*<sup>a</sup>* <sup>¼</sup> �*k*4*kdy*<sup>0</sup>

*n* ! ¼ ⟨

In particular, the normal vector of the surface *S* at initial state *P x*0, *y*0, *z*<sup>0</sup>

*<sup>k</sup>*3*x*0<sup>2</sup> , *<sup>k</sup>*4*kd*

*ax* þ *by* þ *cz* ¼ *d*

*k*3*x*<sup>0</sup> þ *k*4 *k*3

*k*3*x*<sup>0</sup> þ *k*4 *k*3

<sup>¼</sup> *<sup>k</sup>*7*x*<sup>1</sup> *k*8

*<sup>k</sup>*3*x t*ð Þ � *z t*ðÞ¼ <sup>0</sup> (9)

! can be calculated from the gradient of this function

, �1⟩

is given by

,*c* ¼ �1∧

the reduced model has diseased state in the positive octant for the system

*<sup>N</sup>* <sup>¼</sup> *<sup>k</sup>*7*x*<sup>2</sup> *k*8

*<sup>N</sup>*� <sup>¼</sup> *<sup>k</sup>*7*x*<sup>0</sup> *k*8

(diseased-free state) in the positive quadrant if

We let *N*� be the unique value of *N*, where

**3.1 Definition: bifurcation parameter**

provided the values of *k*<sup>8</sup> and *x*<sup>0</sup> is known.

**3.2 Equation of tangent plane**

The normal surface vector *n*

Eq. (6) as

where

**34**

parameters.

*Goiter - Causes and Treatment*

*<sup>k</sup>*<sup>7</sup> so

� )

is a

As the bifurcation parameter *N* varies from the threshold value *N*� ¼ *k*7*x*0*=k*8, the number of the steady states changes, and its behavior near the steady states (linear stability) changes as well. Moreover, as the bifurcation parameter varies, the appearance and disappearance of steady states can be seen on the tangent plane at the initial state of the function *z* ¼ *f x*ð Þ , *y* . For this system (Eqs. (3)–(5)), there are three cases, namely when *N* < *N*� , *N* ¼ *N*� and *N* > *N*� . **Figure 3** illustrates a nice overview of how the solutions become stable and unstable as parameter *N* changes from 0 to 300.

**Case 1:** *N* < *N*�

When *N* < *N*� , the system (Eqs. (3)–(5)) has two steady states, which undergoes a saddle-node bifurcation—disappearance of the saddle diseased state. One can

#### **Figure 3.**

*The bifurcation diagram shows the steady state values of TSH as the parameter N varies from 0 to 300. BP means the branch or bifurcation point where another equilibrium curve passes through and the system switches its stability. The parameter value of N at BP is 66.7. H is a neutral saddle, but not the bifurcation point for the equilibrium curve. At H, there is a homoclinic orbit for the system.*

imagine if we start the system from the initial steady state, which splits into the diseased-free and diseased state. The disease state is asymptotically stable which means all its eigenvalues are purely negative real. However, the diseased-free state is saddle because two of its eigenvalues are positive real and one of its eigenvalues is a negative real. See [22] for proof of the stability of the diseased-free and diseased steady state.

antibodies can be tested. The numerical simulation of the model shows TPOAb

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

When *N* > *N*�, the system (Eqs. (3)–(5)) has one steady state (diseased-free), which is asymptotically stable. As the value of *N* increases beyond the threshold value, the diseased free state is moving on the function *z* ¼ *f x*ð Þ , *y* describing the progression of thyroid size and the function of the axis. One can imagine if we start the system from the initial state of normal thyroid size, then the system may approach goiter or atrophy depending upon the parameter value of *N* while keeping other parameters fixed. On the other hand, varying the parameter *k*7, the clinical

*The parameter value is set to N* ¼ *66:7*, *k7* ¼ *2:3345 and the initial state is at euthyroidism (1, 0.015, 10). The levels of anti-thyroid peroxidase decrease slowly whereas TSH and the functional thyroid size remains unchanged over time, which indicates the normal operation of the axis with fully functional thyroid size.*

*Clinical progression from euthyroidism to mild subclinical hypothyroidism when N* ¼ *80*, *k7* ¼ *2:3345 with*

slowly approaches zero as time increases over months (see **Figure 6**).

**Case 3:** *N* > *N*�

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

**Figure 6.**

**Figure 7.**

**37**

*initial state (1, 0.015, 10).*

As the value of *N* decreases from the threshold value and when it is close to zero, there exists a homoclinic orbit associated at the initial state which forms the boundary for the asymptomatically stable interior diseased state. The orbit captures the hidden dynamics of the diffused goiter and hashitoxicosis (see **Figures 4** and **5**).

**Case 2:** *N* ¼ *N*�

When *N* ¼ *N*� , the system (Eqs. (3)–(5)) has one steady state (diseased-free), which is asymptotically stable. Since there is only one steady state in the system, the operation of the pituitary-thyroid axis in the presence of anti-thyroid peroxidase

#### **Figure 4.**

*When the value of N = 10 and k7* ¼ *2:3345, the hidden dynamics of the axis and the thyroid size reveal hashitoxicosis and goiter. The model simulation is performed for 2 years from the initial state: TSH = 1 mU/L, free T4 = 13 pg/mL, and thyroid size = 0.015 mL and TPOAb = 10 U/mL.*

#### **Figure 5.**

*When N = 0.01 and k7* ¼ *2:3345, the hidden dynamics of the thyroid size capture the development of goiter and return to normal size.*

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

antibodies can be tested. The numerical simulation of the model shows TPOAb slowly approaches zero as time increases over months (see **Figure 6**).

**Case 3:** *N* > *N*�

imagine if we start the system from the initial steady state, which splits into the diseased-free and diseased state. The disease state is asymptotically stable which means all its eigenvalues are purely negative real. However, the diseased-free state is saddle because two of its eigenvalues are positive real and one of its eigenvalues is a negative real. See [22] for proof of the stability of the diseased-free and diseased

there exists a homoclinic orbit associated at the initial state which forms the

As the value of *N* decreases from the threshold value and when it is close to zero,

, the system (Eqs. (3)–(5)) has one steady state (diseased-free),

boundary for the asymptomatically stable interior diseased state. The orbit captures the hidden dynamics of the diffused goiter and hashitoxicosis (see **Figures 4** and **5**).

which is asymptotically stable. Since there is only one steady state in the system, the operation of the pituitary-thyroid axis in the presence of anti-thyroid peroxidase

*When the value of N = 10 and k7* ¼ *2:3345, the hidden dynamics of the axis and the thyroid size reveal hashitoxicosis and goiter. The model simulation is performed for 2 years from the initial state: TSH = 1 mU/L,*

*When N = 0.01 and k7* ¼ *2:3345, the hidden dynamics of the thyroid size capture the development of goiter and*

*free T4 = 13 pg/mL, and thyroid size = 0.015 mL and TPOAb = 10 U/mL.*

steady state.

**Figure 4.**

**Figure 5.**

**36**

*return to normal size.*

**Case 2:** *N* ¼ *N*� When *N* ¼ *N*�

*Goiter - Causes and Treatment*

When *N* > *N*�, the system (Eqs. (3)–(5)) has one steady state (diseased-free), which is asymptotically stable. As the value of *N* increases beyond the threshold value, the diseased free state is moving on the function *z* ¼ *f x*ð Þ , *y* describing the progression of thyroid size and the function of the axis. One can imagine if we start the system from the initial state of normal thyroid size, then the system may approach goiter or atrophy depending upon the parameter value of *N* while keeping other parameters fixed. On the other hand, varying the parameter *k*7, the clinical

#### **Figure 6.**

*The parameter value is set to N* ¼ *66:7*, *k7* ¼ *2:3345 and the initial state is at euthyroidism (1, 0.015, 10). The levels of anti-thyroid peroxidase decrease slowly whereas TSH and the functional thyroid size remains unchanged over time, which indicates the normal operation of the axis with fully functional thyroid size.*

#### **Figure 7.**

*Clinical progression from euthyroidism to mild subclinical hypothyroidism when N* ¼ *80*, *k7* ¼ *2:3345 with initial state (1, 0.015, 10).*

mechanisms euthyroidism ! subclinical hypothyroidism ! overt hypothyroidism can be explained while keeping *N* fixed [20]. See **Figure 7** that shows the clinical progression of subclinical hypothyroidism and thyroid size.

#### **3.4 Exploration of parameter curve**

Suppose that an individual is diagnosed with Hashimoto's autoimmune thyroiditis; we can use the reduced model to explain the physical and clinical symptoms occurring for this individual in the course of this disease. Having this disease means everyone has a unique behavior and knowing that may be very helpful in managing the course. In fact, certain parameter values in the reduced model are responsible for the uniqueness in patient behavior. More specifically, we have identified through the stability analysis that the parameter *N* can explain the size of the thyroid (goiter, normal or atrophy) whereas *k*<sup>7</sup> can explain the clinical progression of the disease [21, 22]. Putting these two parameters together, one can test the hypothesis on the patients'symptoms. See **Figure 8** that shows the parameter curve in terms of *N* and *k*<sup>7</sup> that is responsible for different thyroid size and clinical conditions.

Suppose a patient's information is given; then a threshold *k*� 7, *<sup>N</sup>*� can be found for the patient using Eq. (10) and the definition of the bifurcation parameter. If another information is available in the future, then another threshold can be obtained. To be exact, the threshold is an ordered pair found on the curve given by the bifurcation definition of parameter (see **Figure 8**). To simulate physical and clinical symptoms, we take two test ordered pairs from the parameter curve below and above the threshold *k*� <sup>7</sup> , *<sup>N</sup>*� <sup>¼</sup> <sup>2</sup>*:*3345, 66*:*7) while other parameters came from **Table 2**. With the first test point for ð Þ¼ *k*7, *N* ð Þ 0*:*9904, 12*:*65 , we simulated the model for a day and found the symptoms of the disease are hashitoxicosis and goiter (see **Figure 9**). Similarly, by keeping the second test point as (5.8, 397.9), the one-day simulation showed mild atrophy and overt hypothyroidism (see **Figure 10**). Finally, by keeping the third test point away from the curve as (4.5, 150), the one-day simulation showed subclinical hypothyroidism and normal size of the thyroid (**Figure 11**).

**4. Case studies of three patients**

*N* ¼ *397:9 and k7* ¼ *5:8 are taken from the curve.*

**Figure 10.**

**39**

**Figure 9.**

*0:9904 are taken from the parameter curve.*

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

Using patients' information from the peer-reviewed published article, we will predict patients' natural history of the disease. More precisely, the reduced model can describe thyroid size and clinical progression from euthyroidism to subclinical or overt hypothyroidism for each patient given below. These patients' information was provided by Salvatore Benvenga originally and were already published in the article [21]. The data consists of TSH, free T4 and TPOAb information whose normal reference ranges are (0.4–2.5) mU/L, (7–18) pg./mL and (0–200) U/mL, respectively. Using Eq. (6) and data, we have computed the functional size of the

*The result of physical and clinical symptoms is mild atrophy and overt hypothyroidism, when test values*

*The result of physical and clinical symptoms is goiter and hashitoxicosis when test values N* ¼ *12:65 and k7* ¼

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

#### **Figure 8.**

*The sample parameter curve of a Hashimoto's patient is shown here. The hypothesis testing can be done for different k7 and N values from the curve.*

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

#### **Figure 9.**

mechanisms euthyroidism ! subclinical hypothyroidism ! overt hypothyroidism can be explained while keeping *N* fixed [20]. See **Figure 7** that shows the clinical

Suppose that an individual is diagnosed with Hashimoto's autoimmune thyroiditis; we can use the reduced model to explain the physical and clinical symptoms occurring for this individual in the course of this disease. Having this disease means everyone has a unique behavior and knowing that may be very helpful in managing the course. In fact, certain parameter values in the reduced model are responsible for the uniqueness in patient behavior. More specifically, we have identified through the stability analysis that the parameter *N* can explain the size of the thyroid (goiter, normal or atrophy) whereas *k*<sup>7</sup> can explain the clinical progression of the disease [21, 22]. Putting these two parameters together, one can test the hypothesis on the patients'symptoms. See **Figure 8** that shows the parameter curve in terms of *N* and *k*<sup>7</sup> that is responsible for different thyroid size and clinical

7, *<sup>N</sup>*� can be

progression of subclinical hypothyroidism and thyroid size.

Suppose a patient's information is given; then a threshold *k*�

found for the patient using Eq. (10) and the definition of the bifurcation

parameter. If another information is available in the future, then another threshold can be obtained. To be exact, the threshold is an ordered pair found on the curve given by the bifurcation definition of parameter (see **Figure 8**). To simulate physical and clinical symptoms, we take two test ordered pairs from the parameter curve

came from **Table 2**. With the first test point for ð Þ¼ *k*7, *N* ð Þ 0*:*9904, 12*:*65 , we simulated the model for a day and found the symptoms of the disease are

hashitoxicosis and goiter (see **Figure 9**). Similarly, by keeping the second test point as (5.8, 397.9), the one-day simulation showed mild atrophy and overt hypothyroidism (see **Figure 10**). Finally, by keeping the third test point away from the curve as (4.5, 150), the one-day simulation showed subclinical hypothyroidism and

*The sample parameter curve of a Hashimoto's patient is shown here. The hypothesis testing can be done for*

<sup>7</sup> , *<sup>N</sup>*� <sup>¼</sup> <sup>2</sup>*:*3345, 66*:*7) while other parameters

**3.4 Exploration of parameter curve**

*Goiter - Causes and Treatment*

below and above the threshold *k*�

normal size of the thyroid (**Figure 11**).

conditions.

**Figure 8.**

**38**

*different k7 and N values from the curve.*

*The result of physical and clinical symptoms is goiter and hashitoxicosis when test values N* ¼ *12:65 and k7* ¼ *0:9904 are taken from the parameter curve.*

#### **Figure 10.**

*The result of physical and clinical symptoms is mild atrophy and overt hypothyroidism, when test values N* ¼ *397:9 and k7* ¼ *5:8 are taken from the curve.*

#### **4. Case studies of three patients**

Using patients' information from the peer-reviewed published article, we will predict patients' natural history of the disease. More precisely, the reduced model can describe thyroid size and clinical progression from euthyroidism to subclinical or overt hypothyroidism for each patient given below. These patients' information was provided by Salvatore Benvenga originally and were already published in the article [21]. The data consists of TSH, free T4 and TPOAb information whose normal reference ranges are (0.4–2.5) mU/L, (7–18) pg./mL and (0–200) U/mL, respectively. Using Eq. (6) and data, we have computed the functional size of the

#### **Figure 11.**

*The result of physical and clinical symptoms are normal thyroid size and subclinical hypothyroidism, when test values of N* ¼ *4:5 and k7* ¼ *150 are not taken from the curve.*


#### **Table 3.**

*Patient 1 information.*


thyroid gland. Using Eq. (10), data and the bifurcation definition, we have

*Three courses of the natural history of the disease is shown here for patient 1. Patient 1 visited the clinic three times and was not required any treatment based on their TSH and free T4 values. The model is simulated for 10 years in which the first course of disease shows thyroid size started with mild goiter and then returned to*

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

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

Assuming all patients had diseased-free state (1, 13, 0.015, 0) at some point, the data provides us the diseased state from which one can back trace the physical and clinical conditions caused by the disease and experienced by the patients. We have taken three untreated patients with a known biomarker for the presence of the autoimmune thyroiditis. Patients 1 and 2 visited the clinic three times whereas patient 3 visited the clinic six times due to mild goiter or other symptoms. We do

*Three courses of the disease is shown here for the second patient. This patient visited the clinic three times and was not required treatment based on their TSH and free T4 measurements. The model is simulated for 1 year to observe the hidden dynamics. The course of the disease showed patient had developed subclinical hypothyroidism*

In **Table 3**, the clinical measurements, the functional thyroid size and parameters are listed for patient 1 which form the diseased states. The model simulation started from each diseased state for 10 years to back trace the natural symptoms of patient 1 due to autoimmune thyroiditis. The course of disease revealed thyroid size changes from mild goiter to normal while the function of axis remained normal (see **Figure 12**). In **Table 4**, the clinical measurements, the functional thyroid size and parameters are listed for patient 2. The model simulation is done for 1 year to back trace the symptoms from the course of the disease. The first disease course revealed a development of subclinical hypothyroidism with normal functional thyroid size (see **Figure 13**). Similarly, patient 3 diseased states are found in **Table 5**. The model

simulation is done only for 2 days to capture the early course of the disease.

computed *N* and *k*<sup>7</sup> for all patients given below (see **Tables 3**–**5**).

**Figure 12.**

*normal.*

**Figure 13.**

**41**

*with normal thyroid size.*

not have any information about their thyroid sizes in the data.

#### **Table 4.**

*Patient 2 information.*


#### **Table 5.** *Patient 3 information.*

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

#### **Figure 12.**

*Three courses of the natural history of the disease is shown here for patient 1. Patient 1 visited the clinic three times and was not required any treatment based on their TSH and free T4 values. The model is simulated for 10 years in which the first course of disease shows thyroid size started with mild goiter and then returned to normal.*

#### **Figure 13.**

**Time (months)**

**Figure 11.**

*Goiter - Causes and Treatment*

**Table 3.**

**Time (months)**

**Time (months)**

*Patient 2 information.*

**Table 4.**

**Table 5.**

**40**

*Patient 3 information.*

*Patient 1 information.*

**TSH (mU/L)**

*values of N* ¼ *4:5 and k7* ¼ *150 are not taken from the curve.*

**TSH (mU/L)**

**TSH (mU/L)**

**FT4 (pg/mL)**

**FT4 (pg/mL)**

**FT4 (pg/mL)** **TPOAb (U/mL)**

**TPOAb (U/mL)**

0 1.54 11.78 3810 13.37 64.301 1.4614 31 1.57 12.36 1310 14.02 93.662 2.088 39 1.54 11.97 1480 13.58 93.294 2.1203

> **TPOAb (U/mL)**

 1.46 15.16 164 17.23 81.908 1.9635 1.56 13.74 153 15.58 97.724 2.1925 1.85 13.09 191 14.77 122.4 2.3157 2.06 11.67 482 13.14 150.5 2.557 4.61 10.47 773 11.63 387.39 2.9411 5.04 7.63 537 8.467 590.65 4.1018

0 0.8282 13.5 50 15.73 51.865 2.1918 8 0.93 13 55 15.1 60.952 2.2939 30 1.178 11.8 159 13.5 84.998 2.5254

*The result of physical and clinical symptoms are normal thyroid size and subclinical hypothyroidism, when test*

**Thyroid size (mL)**

**Thyroid size (mL)**

**Thyroid size (mL)**

*N* **<sup>∗</sup>** *k***<sup>7</sup> ∗**

*N* **<sup>∗</sup>** *k***<sup>7</sup> ∗**

*N* **<sup>∗</sup>** *k***<sup>7</sup> ∗**

*Three courses of the disease is shown here for the second patient. This patient visited the clinic three times and was not required treatment based on their TSH and free T4 measurements. The model is simulated for 1 year to observe the hidden dynamics. The course of the disease showed patient had developed subclinical hypothyroidism with normal thyroid size.*

thyroid gland. Using Eq. (10), data and the bifurcation definition, we have computed *N* and *k*<sup>7</sup> for all patients given below (see **Tables 3**–**5**).

Assuming all patients had diseased-free state (1, 13, 0.015, 0) at some point, the data provides us the diseased state from which one can back trace the physical and clinical conditions caused by the disease and experienced by the patients. We have taken three untreated patients with a known biomarker for the presence of the autoimmune thyroiditis. Patients 1 and 2 visited the clinic three times whereas patient 3 visited the clinic six times due to mild goiter or other symptoms. We do not have any information about their thyroid sizes in the data.

In **Table 3**, the clinical measurements, the functional thyroid size and parameters are listed for patient 1 which form the diseased states. The model simulation started from each diseased state for 10 years to back trace the natural symptoms of patient 1 due to autoimmune thyroiditis. The course of disease revealed thyroid size changes from mild goiter to normal while the function of axis remained normal (see **Figure 12**). In **Table 4**, the clinical measurements, the functional thyroid size and parameters are listed for patient 2. The model simulation is done for 1 year to back trace the symptoms from the course of the disease. The first disease course revealed a development of subclinical hypothyroidism with normal functional thyroid size (see **Figure 13**). Similarly, patient 3 diseased states are found in **Table 5**. The model simulation is done only for 2 days to capture the early course of the disease.

The literature description of Hashimoto disease begins with a gradual swelling of

the thyroid gland and development of mild clinical condition, euthyroidism or subclinical hypothyroidism and subsequent gradual progression of overt hypothyroidism. Small goiter and hashitoxicosis are the very early stages of the disease and typically go untreated and hidden in the view of patients and physicians. Overt hypothyroidism is the irreversible end clinical state (where levothyroxine treatment is needed) whereas atrophy is irreversible end physical state of the disease. Basically, the mechanism involved in the progression of the disease is unique and sequential. For instance, some patients may have the disease course of goiter and euthyroidism, some may have goiter and the clinical progression euthyroidism ! subclinical hypothyroidism and some patients may have gradual progression from

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

normal thyroid size to atrophy and euthyroidism to overt hypothyroidism.

data by describing their natural history of the disease.

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

mathematical modeling of Hashimoto's autoimmune thyroiditis.

We declare there is no conflict of interest on this chapter.

**Acknowledgements**

**Conflict of interest**

**Notes/thanks/other declarations**

University of Wisconsin, Whitewater, WI, USA

provided the original work is properly cited.

\*Address all correspondence to: pandiyab@uww.edu

data for the modeling work.

**Author details**

**43**

Balamurugan Pandiyan

Herein, we have developed and used patient-specific model to describe all possible mechanisms involved in the autoimmune thyroiditis. This can be achieved using two parameters *N* and *k***7**. To be precise, the parameter *N* and *k***<sup>7</sup>** can trace the thyroid size and clinical progression. We validated the model with three patients'

I would like to dedicate this chapter to Dr. Stephen J. Merrill for his guidance on

I would like to thank Dr. Salvatore Benvenga for providing Hashimoto's patients

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Figure 14.**

*Six courses of the disease has been shown here for the third patient. This patient visited the clinic six times and was not required any treatment based on their TSH and free T4 measurement. The model is intentionally simulated for 2 days to capture the early dynamics of the patient. The model simulation shows this patient had developed both clinical and physical symptoms at different points in time.*

Surprising results are seen in the early course of the disease (see **Figure 14**). This patient had developed both clinical and physical symptoms at different points in time. In particular, the first course shows mild goiter and mild hashitoxicosis whereas the sixth course shows the overt hypothyroidism and mild atrophy.

#### **5. Conclusion**

The human body is made up of so many subsystems such as the pituitarythyroid axis and the immune system. These subsystems do not disrupt the function of each other in healthy people. A function of the pituitary-thyroid axis is to secrete the appropriate levels of thyroid hormones and take care of body's metabolism whereas the function of the immune system is to protect the body's organs such as the thyroid gland and remove foreign substances that try to invade the organs. Hashimoto discovered the abnormal interaction of the immune system to the thyroid gland, which resulted in the disruption of the physiology of the axis. Later this diseased condition has been named as Hashimoto's autoimmune thyroiditis. Hashimoto noticed the destruction of thyroid follicular cells through physical and clinical symptoms in four middle aged female patients.

A keystone of the functional thyroid gland is the follicular cells, which die due to the aggressive and destructive attack by the immune system. Hashimoto patients discovered the disease sometimes by themselves due to discomfort in the neck (small goiter) or accidentally during annual checkup by the family physicians. Consequences of Hashimoto disease can be classified into physical and clinical symptoms at various stages of the disease. Both types of symptoms occur in a sequential manner from one extreme to another. More precisely, the physical symptom runs from goiter ! atrophy whereas the clinical symptom runs from hashitoxicosis ! overt hypothyroidism. In order to describe these symptoms mathematically, we chose four state variables namely the size of functional thyroid gland to describe the physical symptom, TSH and free T4 to describe the clinical symptoms and thyroid peroxidase antibodies as a representative for the presence of Hashimoto disease. Modeling the disruption of the axis through a coupled model is the key to unlock the hidden dynamics experienced by the patients. The hidden dynamics can be seen in patients through physical and clinical symptoms.

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

The literature description of Hashimoto disease begins with a gradual swelling of the thyroid gland and development of mild clinical condition, euthyroidism or subclinical hypothyroidism and subsequent gradual progression of overt hypothyroidism. Small goiter and hashitoxicosis are the very early stages of the disease and typically go untreated and hidden in the view of patients and physicians. Overt hypothyroidism is the irreversible end clinical state (where levothyroxine treatment is needed) whereas atrophy is irreversible end physical state of the disease. Basically, the mechanism involved in the progression of the disease is unique and sequential. For instance, some patients may have the disease course of goiter and euthyroidism, some may have goiter and the clinical progression euthyroidism ! subclinical hypothyroidism and some patients may have gradual progression from normal thyroid size to atrophy and euthyroidism to overt hypothyroidism.

Herein, we have developed and used patient-specific model to describe all possible mechanisms involved in the autoimmune thyroiditis. This can be achieved using two parameters *N* and *k***7**. To be precise, the parameter *N* and *k***<sup>7</sup>** can trace the thyroid size and clinical progression. We validated the model with three patients' data by describing their natural history of the disease.

#### **Acknowledgements**

Surprising results are seen in the early course of the disease (see **Figure 14**). This patient had developed both clinical and physical symptoms at different points in time. In particular, the first course shows mild goiter and mild hashitoxicosis whereas the sixth course shows the overt hypothyroidism and mild atrophy.

*developed both clinical and physical symptoms at different points in time.*

*Six courses of the disease has been shown here for the third patient. This patient visited the clinic six times and was not required any treatment based on their TSH and free T4 measurement. The model is intentionally simulated for 2 days to capture the early dynamics of the patient. The model simulation shows this patient had*

The human body is made up of so many subsystems such as the pituitarythyroid axis and the immune system. These subsystems do not disrupt the function of each other in healthy people. A function of the pituitary-thyroid axis is to secrete the appropriate levels of thyroid hormones and take care of body's metabolism whereas the function of the immune system is to protect the body's organs such as the thyroid gland and remove foreign substances that try to invade the organs. Hashimoto discovered the abnormal interaction of the immune system to the thyroid gland, which resulted in the disruption of the physiology of the axis. Later this diseased condition has been named as Hashimoto's autoimmune thyroiditis. Hashimoto noticed the destruction of thyroid follicular cells through physical and clinical

A keystone of the functional thyroid gland is the follicular cells, which die due to the aggressive and destructive attack by the immune system. Hashimoto patients discovered the disease sometimes by themselves due to discomfort in the neck (small goiter) or accidentally during annual checkup by the family physicians. Consequences of Hashimoto disease can be classified into physical and clinical symptoms at various stages of the disease. Both types of symptoms occur in a sequential manner from one extreme to another. More precisely, the physical symptom runs from goiter ! atrophy whereas the clinical symptom runs from hashitoxicosis ! overt hypothyroidism. In order to describe these symptoms mathematically, we chose four state variables namely the size of functional thyroid gland to describe the physical symptom, TSH and free T4 to describe the clinical symptoms and thyroid peroxidase antibodies as a representative for the presence of Hashimoto disease. Modeling the disruption of the axis through a coupled model is the key to unlock the hidden dynamics experienced by the patients. The hidden dynamics can be seen in patients through physical and clinical symptoms.

symptoms in four middle aged female patients.

**5. Conclusion**

**42**

**Figure 14.**

*Goiter - Causes and Treatment*

I would like to dedicate this chapter to Dr. Stephen J. Merrill for his guidance on mathematical modeling of Hashimoto's autoimmune thyroiditis.

#### **Conflict of interest**

We declare there is no conflict of interest on this chapter.

#### **Notes/thanks/other declarations**

I would like to thank Dr. Salvatore Benvenga for providing Hashimoto's patients data for the modeling work.

#### **Author details**

Balamurugan Pandiyan University of Wisconsin, Whitewater, WI, USA

\*Address all correspondence to: pandiyab@uww.edu

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
