**4. Two-tissue irreversible compartment model of [18***F***]***FDG*

[ <sup>18</sup>*F*]*FDG* is a glucose analogue used to evaluate brain's metabolic activity in vivo through positron emission tomography with computed tomography (PET/CT). The irreversible two-compartment model for [18*F*]*FDG* is used for description of this tracer, which is first entering a free compartment, C1, and is then metabolized irreversibly in the second compartment C2, [16].

*Biomechanical Model Improving Alzheimer's Disease DOI: http://dx.doi.org/10.5772/intechopen.92047*

$$\begin{cases} \frac{d}{dt} \mathbf{C}\_1(t) = K\_1 \mathbf{C}\_a(t) - (k\_2 + k\_3) \mathbf{C}\_1(t) \\\\ \frac{d}{dt} \mathbf{C}\_2(t) = k\_3 \mathbf{C}\_1(t) \\\\ \mathbf{C}\_1(\mathbf{0}) = \mathbf{0}, \mathbf{C}\_2(\mathbf{0}) = \mathbf{0} \end{cases} \tag{17}$$

where *Ca*ð Þ*t* is the input function and is considered to be known, *C*1ð Þ*t* and *C*2ð Þ*t* are the concentration in C1 and C2 compartments, respectively, and *K*1, *k*2, *k*<sup>3</sup> are positives proportionality rates describing, the tracer influx into and the tracer outflow from the compartment (transport constants).

Similarly to that developed in the previous section, considering *k*<sup>4</sup> ¼ 0, applying the Laplace transform with respect to *t* in Eq. (17), appear the algebraic system

$$\begin{cases} (s + k\_2 + k\_3)\overline{C}\_1(s) = K\_1 \overline{C}\_a(s) \\ -k\_3 \overline{C}\_1(s) + s \overline{C}\_2(s) = \mathbf{0} \end{cases} . \tag{18}$$

Eq. (18) is represented matrically

$$
\begin{bmatrix}
\overline{\mathbf{C}}\_1(s) \\
\overline{\mathbf{C}}\_2(s)
\end{bmatrix} = \begin{bmatrix}
s + k\_2 + k\_3 & \mathbf{0} \\
\end{bmatrix}^{-1} \begin{bmatrix}
K\_1 \overline{\mathbf{C}}\_d(s) \\
\mathbf{0}
\end{bmatrix}.
\tag{19}
$$

$$
\begin{bmatrix}
\overline{\mathbf{C}}\_1(\boldsymbol{\varepsilon}) \\
\overline{\mathbf{C}}\_2(\boldsymbol{\varepsilon})
\end{bmatrix} = \frac{1}{\mathfrak{s}(\boldsymbol{s} + k\_2 + k\_3)} \begin{bmatrix}
\boldsymbol{s} & \mathbf{0} \\
k\_3 & \mathfrak{s} + k\_2 + k\_3
\end{bmatrix} \begin{bmatrix}
K\_1 \overline{\mathbf{C}}\_\mathfrak{a}(\boldsymbol{\varepsilon}) \\
\mathbf{0}
\end{bmatrix}.\tag{20}
$$

Then,

*<sup>C</sup>*1ðÞ¼ *<sup>t</sup> £*�<sup>1</sup> ð Þ *<sup>s</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>4</sup>

*Recent Advances in Biomechanics*

*<sup>C</sup>*2ðÞ¼ *<sup>t</sup> £*�<sup>1</sup> *<sup>k</sup>*<sup>3</sup>

In Eq. (13) *s*<sup>1</sup> and *s*<sup>2</sup> are the roots of *s*

*<sup>C</sup>*1ðÞ¼ *<sup>t</sup> <sup>K</sup>*<sup>1</sup> *Ae<sup>s</sup>*1*<sup>t</sup>*

*<sup>C</sup>*1ðÞ¼ *<sup>t</sup> <sup>K</sup>*<sup>1</sup> *Ces*1*<sup>t</sup>*

[

**82**

order to make it possible to calculate the integral

irreversibly in the second compartment C2, [16].

ð*t* 0 *e*

ð*t* 0 *e*

> *I* ¼ ð*t* 0 *e*

**4. Two-tissue irreversible compartment model of [18***F***]***FDG*

inverse Laplace transform is obtained

Then,

*s*<sup>2</sup> þ ð Þ *k*<sup>2</sup> þ *k*<sup>3</sup> þ *k*<sup>4</sup> *s* þ *k*2*k*<sup>4</sup> � �

*s*<sup>2</sup> þ ð Þ *k*<sup>2</sup> þ *k*<sup>3</sup> þ *k*<sup>4</sup> *s* þ *k*2*k*<sup>4</sup> � �

> *s*<sup>2</sup> þ ð Þ *k*<sup>2</sup> þ *k*<sup>3</sup> þ *k*<sup>4</sup> *s* þ *k*2*k*<sup>4</sup> � �

*s*<sup>2</sup> þ ð Þ *k*<sup>2</sup> þ *k*<sup>3</sup> þ *k*<sup>4</sup> *s* þ *k*2*k*<sup>4</sup> � �

on transport constants *k*2, *k*3, and *k*4. The parameters *A*, *B*, *C*, and *D* are obtained by partial fraction decomposition technique. Then, because that the inverse Laplace transforms are simply linear combinations of exponential functions with the exponents *s*<sup>1</sup> and *s*<sup>2</sup> depending on *k*2, *k*3, and *k*4. Applying the linearity property of the

*<sup>C</sup>*1ðÞ¼ *<sup>t</sup> <sup>K</sup>*<sup>1</sup> *Ae<sup>s</sup>*1*<sup>t</sup>* <sup>þ</sup> *B e<sup>s</sup>*2*<sup>t</sup>* f g <sup>∗</sup>*Ca*ð Þ*<sup>t</sup>*

The analytical solution of the reversible two-compartment model for [11*C*]*PIB* (4) is

�*s*1*uCa*ð Þ *<sup>u</sup> du* <sup>þ</sup> *Be<sup>s</sup>*2*<sup>t</sup>*

�*s*1*uCa*ð Þ *<sup>u</sup> du* <sup>þ</sup> *Des*2*<sup>t</sup>*

In Eq. (15), it is visible the importance of construction of input function *Ca*ð Þ*t* in

<sup>18</sup>*F*]*FDG* is a glucose analogue used to evaluate brain's metabolic activity in vivo through positron emission tomography with computed tomography (PET/CT). The irreversible two-compartment model for [18*F*]*FDG* is used for description of this tracer, which is first entering a free compartment, C1, and is then metabolized

� �

<sup>¼</sup> *<sup>A</sup> s* � *s*<sup>1</sup> þ *B s* � *s*<sup>2</sup>

<sup>¼</sup> *<sup>C</sup> s* � *s*<sup>1</sup> þ *D s* � *s*<sup>2</sup>

*<sup>C</sup>*2ðÞ¼ *<sup>t</sup> <sup>K</sup>*<sup>1</sup> *Ces*1*<sup>t</sup>* <sup>þ</sup> *D es*2*<sup>t</sup>* f g <sup>∗</sup>*Ca*ð Þ*<sup>t</sup> :* (14)

ð*t* 0 *e*

ð*t* 0 *e*

� � *:* (15)

*<sup>C</sup>*1ðÞ¼ *<sup>t</sup> <sup>K</sup>*<sup>1</sup> *£*�<sup>1</sup> *<sup>s</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>4</sup>

*<sup>C</sup>*2ðÞ¼ *<sup>t</sup> <sup>K</sup>*<sup>1</sup> *£*�<sup>1</sup> *<sup>k</sup>*<sup>3</sup>

*s* þ *k*<sup>4</sup> *s*<sup>2</sup> þ ð Þ *k*<sup>2</sup> þ *k*<sup>3</sup> þ *k*<sup>4</sup> *s* þ *k*2*k*<sup>4</sup>

*k*3 *s*<sup>2</sup> þ ð Þ *k*<sup>2</sup> þ *k*<sup>3</sup> þ *k*<sup>4</sup> *s* þ *k*2*k*<sup>4</sup> <sup>∗</sup>*K*<sup>1</sup> *£*�<sup>1</sup> *Ca*ð Þ*<sup>s</sup>* � �

<sup>∗</sup>*K*<sup>1</sup> *£*�<sup>1</sup> *Ca*ð Þ*<sup>s</sup>* � �*:*

∗*Ca*ð Þ*t*

∗*Ca*ð Þ*t :*

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>3</sup> <sup>þ</sup> *<sup>k</sup>*<sup>4</sup> *<sup>s</sup>* <sup>þ</sup> *<sup>k</sup>*2*k*<sup>4</sup> <sup>¼</sup> 0, dependent

�*s*2*uCa*ð Þ *<sup>u</sup> du*

�*s*2*uCa*ð Þ *<sup>u</sup> du*

*si uCa*ð Þ *<sup>u</sup> du:* (16)

(11)

(12)

(13)

$$\begin{aligned} \overline{\mathbf{C}}\_{1}(s) &= \frac{K \mathbf{1} \mathbf{C}\_{a}(s)}{s + k\_{2} + k\_{3}} \\ \overline{\mathbf{C}}\_{2}(s) &= \frac{k\_{3} K\_{1} \overline{\mathbf{C}}\_{a}(s)}{(s + k\_{2} + k\_{3})s} = \frac{k\_{3} \mathbf{C}\_{1}(s)}{s} \end{aligned} \tag{21}$$

$$\begin{split} \mathbf{C}\_{1}(t) &= \boldsymbol{\varepsilon}^{-1} \left\{ \frac{K\_{1} \overline{\mathbf{C}}\_{a}(s)}{(s + k\_{2} + k\_{3})} \right\} = K\_{1} \boldsymbol{\varepsilon}^{-1} \left\{ \frac{1}{(s + k\_{2} + k\_{3})} \right\} \* \boldsymbol{\varepsilon}^{-1} \{ \overline{\mathbf{C}}\_{a}(s) \} \\ \mathbf{C}\_{2}(t) &= \boldsymbol{\varepsilon}^{-1} \left\{ \frac{k\_{3} \overline{\mathbf{C}}\_{1}(s)}{s} \right\} = k\_{3} \* \boldsymbol{\varepsilon}^{-1} \{ \overline{\mathbf{C}}\_{1}(s) \} . \end{split} \tag{22}$$

The representation Eq. (22) implies that

$$\begin{split} \mathbf{C}\_{1}(t) &= \mathbf{K}\_{1} \mathbf{e}^{-(k\_{2}+k\_{3})t} \ast \mathbf{C}\_{a}(t) = \mathbf{K}\_{1} \int\_{0}^{t} \mathbf{e}^{-(k\_{2}+k\_{3})(t-u)} \mathbf{C}\_{a}(u) du \\ \mathbf{C}\_{2}(t) &= k\_{3} \ast \mathbf{C}\_{1}(t) = k\_{3} \int\_{0}^{t} \mathbf{C}\_{1}(u) du. \end{split} \tag{23}$$

Then, with *k*<sup>2</sup> þ *k*<sup>3</sup> >0, the analytical solution of the irreversible two compartment model for [18*F*]*FDG* Eq. (17) is

$$\begin{aligned} \mathbf{C}\_{1}(t) &= \mathbf{K}\_{1} e^{-(k\_{2}+k\_{3})t} \int\_{0}^{t} e^{(k\_{2}+k\_{3})u} \mathbf{C}\_{a}(u) du \\ \mathbf{C}\_{2}(t) &= k\_{3} \int\_{0}^{t} \mathbf{C}\_{1}(u) du. \end{aligned} \tag{24}$$

It is important now to choose a suitable model to represent the input function *Ca*ð Þ*<sup>t</sup>* , which makes it possible to calculate the integral <sup>Ð</sup>*<sup>t</sup>* <sup>0</sup> *<sup>e</sup>*ð Þ *<sup>k</sup>*2þ*k*<sup>3</sup> *<sup>u</sup> <sup>K</sup>*1*Ca*ð Þ *<sup>u</sup> du*.

#### **4.1 The input concentration**

The knowledge of the input function is mandatory in quantifying by compartmental kinetic modeling. The radioactivity concentration of arterial blood can be measured during the course of the scan collecting blood samples.

Several techniques have been proposed for obtaining input function. [9] present five different forms to measure this data and [8] eight methods for the estimation image input function in dynamic [18*F*]*FDG* PET human brain. The image arterial input function provides data that are similar to arterial blood input methods and can be used to quantify, noninvasively, in PET studies, according to previous studies [8, 10, 13, 15, 20]. This technique calculate the input function using linear and nonlinear regression applied in a applied to a discrete set of data, discrete time activity curve (TAC) of reference region [11].

#### **4.2 Input function derived of PET image**

The dynamics of the radiotracer, [11, 17], on the reference region is governed by the differential equation

$$\frac{d\mathbf{C}\_r}{dt} = K\_1'\mathbf{C}\_a(t) - k\_2'\mathbf{C}\_r(t) \tag{25}$$

**5. Results and discussion**

*Biomechanical Model Improving Alzheimer's Disease DOI: http://dx.doi.org/10.5772/intechopen.92047*

**5.1** *Cr*ð Þ*<sup>t</sup>* **for [11***C***]***PIB* **radiotracer**

**Figure 4.**

**Figure 5.**

**85**

*Discrete and fitted rational cerebellum TAC (Cr(t)): the best model.*

In order to obtain the analytical solution of two-compartment model, Eq. (15) for [11*C*]*PIB* and Eq. (24) for [18*F*]*FDG* radiotracer, the important step is to determine *Cr*ð Þ*t* that will allow you to calculate the input function (Eq. (26)). In the reference region, *Cr*ð Þ*t* , is approximated by means of linear and nonlinear regression of the data obtained from a discrete TAC curve on a positron emission tomography

For [11*C*]*PIB* tracer is chosen as reference region the left and right cerebellum, known to be amyloid free illustrated in **Figure 4**. The left and right cerebellum are

*Region of reference outlined in both cerebellar gray matter in a sagittal (left) and coronal (right) slices.*

(PET) image, using PMOD, a biomedical image quantification software.

where *Ca*ð Þ*t* is the concentration of the radiotracer in the arterial blood, *Cr*ð Þ*t* is the concentration of the radiotracer in the reference region and *K*<sup>0</sup> <sup>1</sup> > 0 and *k*<sup>0</sup> <sup>2</sup> >0 are the proportionality rates describing, respectively, the tracer influx into and the outflow from the reference tissue.

*Cr*ð Þ*t* is constructed from a TAC of a reference region [11].

After this, deriving *Cr*ð Þ*t* we obtain *Ca*ð Þ*t* , which is the AIF, using

$$C\_a(t) = \frac{1}{K\_1'} \frac{dC\_r}{dt} + \frac{k\_2'}{K\_1'} C\_r(t) \tag{26}$$

The transport of the radiotracer across of arterial blood is very fast in the first few minutes and then decreases slowly. Then, it may be appropriate to estimate the *Cr*ð Þ*t* in a few stages as piecewise function, [16]. This is defined for three stages in the equation

$$\begin{aligned} \mathbf{C}\_{r}(t) &= (H(t-t\_{0}) - H(t-t\_{1}))\mathbf{C}\_{\eta'}(t) + (H(t-t\_{1}) - H(t-t\_{2}))\mathbf{C}\_{rl}(t) \\ &+ H(t-t\_{2})\mathbf{C}\_{n}(t), \end{aligned} \tag{27}$$

where *Crf*ð Þ*t* , *CrI*ð Þ*t* and *Crs*ð Þ*t* are the concentration of the radiotracer on the reference region, respectively, for the fast, intermediate and slow stage. *H t*ð Þ is the Heaviside function defined by

$$H(t - a) = \begin{cases} 0, t < a, \\ 1, t \ge a. \end{cases} \tag{28}$$

$$H(t - a) - H(t - b) = \begin{cases} 0, t < a \text{ and } t \ge b, \\ 1, a \le t < b. \end{cases} \tag{29}$$

*Biomechanical Model Improving Alzheimer's Disease DOI: http://dx.doi.org/10.5772/intechopen.92047*
