**5. Results and discussion**

It is important now to choose a suitable model to represent the input function

The knowledge of the input function is mandatory in quantifying by compartmental kinetic modeling. The radioactivity concentration of arterial blood can be

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

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

<sup>1</sup>*Ca*ð Þ�*t k*<sup>0</sup>

where *Ca*ð Þ*t* is the concentration of the radiotracer in the arterial blood, *Cr*ð Þ*t* is

are the proportionality rates describing, respectively, the tracer influx into and the

1 *K*0 1

*Cr*ðÞ¼ *t* ð Þ *H t*ð Þ� � *t*<sup>0</sup> *H t*ð Þ � *t*<sup>1</sup> *Crf*ð Þþ*t* ð Þ *H t*ð Þ� � *t*<sup>1</sup> *H t*ð Þ � *t*<sup>2</sup> *CrI*ð Þ*t*

*H t*ð Þ¼ � *a*

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

�

�

*H t*ð Þ� � *<sup>a</sup> H t*ð Þ¼ � *<sup>b</sup>* 0, *<sup>t</sup>*<*a and t*<sup>≥</sup> *<sup>b</sup>*,

*dCr dt* þ

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

*k*0 2 *K*0 1

þ *H t*ð Þ � *t*<sup>2</sup> *Crs*ð Þ*t* , (27)

0, *t*<*a*, 1, *t*≥ *a:*

1, *a*≤*t*< *b:*

<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*.

<sup>2</sup>*Cr*ð Þ*t* (25)

*Cr*ð Þ*t* (26)

<sup>1</sup> > 0 and *k*<sup>0</sup>

<sup>2</sup> >0

(28)

(29)

*Ca*ð Þ*<sup>t</sup>* , which makes it possible to calculate the integral <sup>Ð</sup>*<sup>t</sup>*

activity curve (TAC) of reference region [11].

**4.2 Input function derived of PET image**

the differential equation

the equation

**84**

outflow from the reference tissue.

Heaviside function defined by

measured during the course of the scan collecting blood samples.

*dCr dt* <sup>¼</sup> *<sup>K</sup>*<sup>0</sup>

the concentration of the radiotracer in the reference region and *K*<sup>0</sup>

*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

*Ca*ðÞ¼ *t*

**4.1 The input concentration**

*Recent Advances in Biomechanics*

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