Biomechanical Model Improving Alzheimer's Disease

*Eliete Biasotto Hauser, Wyllians Vendramini Borelli and Jaderson Costa da Costa*

### **Abstract**

The aim this study is to describe the algorithms of kinetic modeling to analyze the pattern of deposition of amyloid plaques and glucose metabolism in Alzheimer's dementia. A two-tissue reversible compartment model for Pittsburgh Compound-B ([11*C*]*PIB*) and a two-tissue irreversible compartment model for [18*F*]2-fluoro-2-deoxy-D-glucose ([18*F*]*FDG*) are solved applying the Laplace transform method in a system of two first-order differential equations. After calculating a convolution integral, the analytical solutions are completely described. In order to determine the parameters of the model, information on the tracer delivery is needed. A noninvasive reverse engineer technique is described to determine the input function from a reference region (carotids and cerebellum) in PET image processing, without arterial blood samples.

**Keywords:** noninvasive input function, Laplace transform, kinetic modeling, radiotracer, positron emission tomography (PET), reference region, region of interest, time activity curve

#### **1. Introduction**

Positron emission tomography (PET) [1, 2], is a functional imaging technology that visualizes physiological changes through the administration of radiopharmaceutical molecular tracers into living systems. PET with measures the local concentration of a tracer in the region of interest (ROI) or target tissue.

PET with [11*C*]*PIB* and [18*F*]*FDG* radiotracers are widely used in the clinical setting for patients with neurodegenerative diseases like Alzheimer's disease. The [ <sup>18</sup>*F*]*FDG*-PET indirectly measures neuronal metabolism, subsequently allowing the identification of brain regions with increased or decreased activity. Individuals with progressive amnestic dementia show a specific pattern of FDG uptake that distinguishes their brains from other types of pathologies. Thus, this technique directly impacts the treatment selected for this patient. However, this technique is still under study to improve its accuracy power and to decrease patient discomfort undergoing this diagnostic tool.

Cognitive aging is also a subject of interest of PET studies. This technique can be used to investigate abnormal binding occurs in clinically normal individuals, prior to the development of cognitive changes. Higher binding in nondemented subjects suggests that [11*C*]*PIB* amyloid imaging may be sensitive for detection of a preclinical Alzheimer's disease state. Age-related cognitive changes impact the brain functioning and subsequently neuronal activity. Frontal and medial temporal regions are particularly vulnerable for the aging process. Nonetheless, a group of elderly named SuperAgers exhibit exceptional memory ability and a specific brain signature [3–7]. SuperAgers appear to maintain neuronal activity throughout the aging process, showing stable neuronal activity in the frontal lobe when compared with normal agers.

**1.1 Image analysis and data generation**

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

ethics committee with *PET*/*CT* imaging.

*FDG* with a half-life of 109.7 min.

**2. Models compartments**

where *C<sup>i</sup>*

balance equations.

specific compartment:

ment *j* to compartment *i*.

**79**

*d*

*dtCi*ðÞ¼ *<sup>t</sup>* <sup>X</sup>

*N*

*j*¼1, *j*6¼*i*

and since it does not find amyloid plaques to bind it leaves rapidly.

and is then metabolized irreversibly in the second compartment C2.

the time-course of tracer concentration in arterial blood or plasma.

decay, attenuation, scatter and dead time corrections.

**1.2 Effective dose injected (EDI) and half-life**

Data used in this work was obtained with [18*F*]*FDG* and [11*C*]*PIB* synthesized by the Cyclotron at the Instituto do Cérebro (InsCer/BraIns) at the Pontifical Catholic University of Rio Grande do Sul PUCRS, in studies (**Figure 2**) approved by medical

Using software PMOD, the 3D Gaussian pre-processing tool is used to make

� *<sup>C</sup><sup>e</sup> ae* � ln 2 *t* 1*=*2 ð Þ *te*�*t*<sup>0</sup>

The radioactivity of [11*C*]*PIB* decays with a half-life of *t*1/2 = 20 min and of [18*F*]

Mathematical modeling seeks to describe the processes of distribution and elimination through compartments, which represent different regions (for example, the

Transferring rate from one compartment to another, is proportional to concentration in the compartment of origin. Compartmental model is an important kinetic modeling technique used for quantification of PET. Each compartment is characterized by the concentration within it as a function of time. The physiological and metabolic transport processes are described mathematically by the analysis of mass

A compartment model is represented by a system of differential equations, where each equation represents the sum of all the transfer rates to and from a

where *Ci*ð Þ*t* is the concentration of radioactive tracer in compartment *i*, *N* is the number of sections of the model, *Kji* is the rate constant for transfer from compart-

**Figure 3** illustrates a reversible compartment model, that is be used to investigated the [11*C*]*PIB* metabolism, [15], because this tracer enters a reference region

The irreversible two compartment model (**Figure 2** with *k*<sup>4</sup> ¼ 0) is used for description of tracer [18*F*]*FDG*, [16, 17], which first enter a free compartment, C1,

In order to determine the parameters of the model, it is necessary to have information about the tracer delivery in the form of an input function representing

*KijCj*ð Þ�*<sup>t</sup> KjiCi*ð Þ*<sup>t</sup>* � �, (2)

(1)

*<sup>a</sup>* is the rest dose after

According [1], the effective dose injected can be calculated as:

*ae* � ln 2 *t* 1*=*2 ð Þ *t*0�*ti*

injection measured at time *te* and *t*1/2 is the half-life of the tracer.

vascular space, interstitial, intracellular) o different chemical stages.

*<sup>a</sup>* is the dose measured before injection at time *ti*, *C<sup>i</sup>*

*EDI* <sup>¼</sup> *Ci*

Mathematical modeling seeks to describe the processes of distribution and elimination through compartments, which represent different regions (for example, the vascular space, interstitial, intracellular) or different chemical stages. Noninvasive methods have been used successfully in PET image studies [8–13].

In order to determine the parameters of the model, information on the tracer delivery is needed in the form of the input function that represents the time-course of tracer concentration in the arterial blood or plasma is non-invasively obtained by non-linear regression [14], from the time-activity curve in a reference region (carotids and cerebellum).

The Laplace transform is used to generate the exact solution solutions of the [ <sup>11</sup>*C*]*PIB* two-tissue reversible compartment model, [15], and [18*F*]*FDG* two-tissue irreversible compartment model, [16, 17], applying the Laplace transform method in a system of two first order differential equations. From a reference region (carotids and cerebellum) the technique allows to estimate the concentration in each compartment of the region of interest, illustrated in **Figure 1**.

**Figure 1.** *Region of interest outlined in temporal areas bilaterally in an axial slice.*

**Figure 2.** *SuperAgers project study protocol.*

#### **1.1 Image analysis and data generation**

and subsequently neuronal activity. Frontal and medial temporal regions are particu-

In order to determine the parameters of the model, information on the tracer delivery is needed in the form of the input function that represents the time-course of tracer concentration in the arterial blood or plasma is non-invasively obtained by non-linear regression [14], from the time-activity curve in a reference region

The Laplace transform is used to generate the exact solution solutions of the

<sup>11</sup>*C*]*PIB* two-tissue reversible compartment model, [15], and [18*F*]*FDG* two-tissue irreversible compartment model, [16, 17], applying the Laplace transform method in a system of two first order differential equations. From a reference region (carotids and cerebellum) the technique allows to estimate the concentration in

larly vulnerable for the aging process. Nonetheless, a group of elderly named SuperAgers exhibit exceptional memory ability and a specific brain signature [3–7]. SuperAgers appear to maintain neuronal activity throughout the aging process, showing stable neuronal activity in the frontal lobe when compared with normal agers. Mathematical modeling seeks to describe the processes of distribution and elimination through compartments, which represent different regions (for example, the vascular space, interstitial, intracellular) or different chemical stages. Noninvasive

methods have been used successfully in PET image studies [8–13].

each compartment of the region of interest, illustrated in **Figure 1**.

*Region of interest outlined in temporal areas bilaterally in an axial slice.*

(carotids and cerebellum).

*Recent Advances in Biomechanics*

[

**Figure 1.**

**Figure 2.**

**78**

*SuperAgers project study protocol.*

Data used in this work was obtained with [18*F*]*FDG* and [11*C*]*PIB* synthesized by the Cyclotron at the Instituto do Cérebro (InsCer/BraIns) at the Pontifical Catholic University of Rio Grande do Sul PUCRS, in studies (**Figure 2**) approved by medical ethics committee with *PET*/*CT* imaging.

Using software PMOD, the 3D Gaussian pre-processing tool is used to make decay, attenuation, scatter and dead time corrections.

#### **1.2 Effective dose injected (EDI) and half-life**

According [1], the effective dose injected can be calculated as:

$$EDI = \mathbf{C}\_a^i e^{-\frac{\ln 2}{t\_{1/2}}(t\_0 - t\_i)} - \mathbf{C}\_a^t e^{-\frac{\ln 2}{t\_{1/2}}(t\_e - t\_0)} \tag{1}$$

where *C<sup>i</sup> <sup>a</sup>* is the dose measured before injection at time *ti*, *C<sup>i</sup> <sup>a</sup>* is the rest dose after injection measured at time *te* and *t*1/2 is the half-life of the tracer.

The radioactivity of [11*C*]*PIB* decays with a half-life of *t*1/2 = 20 min and of [18*F*] *FDG* with a half-life of 109.7 min.

### **2. Models compartments**

Mathematical modeling seeks to describe the processes of distribution and elimination through compartments, which represent different regions (for example, the vascular space, interstitial, intracellular) o different chemical stages.

Transferring rate from one compartment to another, is proportional to concentration in the compartment of origin. Compartmental model is an important kinetic modeling technique used for quantification of PET. Each compartment is characterized by the concentration within it as a function of time. The physiological and metabolic transport processes are described mathematically by the analysis of mass balance equations.

A compartment model is represented by a system of differential equations, where each equation represents the sum of all the transfer rates to and from a specific compartment:

$$\frac{d}{dt}\mathbf{C}\_i(t) = \sum\_{j=1, j\neq i}^{N} \left[ K\_{ij}\mathbf{C}\_j(t) - K\_{ji}\mathbf{C}\_i(t) \right],\tag{2}$$

where *Ci*ð Þ*t* is the concentration of radioactive tracer in compartment *i*, *N* is the number of sections of the model, *Kji* is the rate constant for transfer from compartment *j* to compartment *i*.

**Figure 3** illustrates a reversible compartment model, that is be used to investigated the [11*C*]*PIB* metabolism, [15], because this tracer enters a reference region and since it does not find amyloid plaques to bind it leaves rapidly.

The irreversible two compartment model (**Figure 2** with *k*<sup>4</sup> ¼ 0) is used for description of tracer [18*F*]*FDG*, [16, 17], which first enter a free compartment, C1, and is then metabolized irreversibly in the second compartment C2.

In order to determine the parameters of the model, it is necessary to have information about the tracer delivery in the form of an input function representing the time-course of tracer concentration in arterial blood or plasma.

*£ C*f g *<sup>i</sup>*ð Þ*t* ¼ *Ci*ðÞ¼ *s*

*dCk*ð Þ*t*

*£*

*s* þ *k*<sup>2</sup> þ *k*<sup>3</sup> �*k*<sup>4</sup> �*k*<sup>3</sup> *s* þ *k*<sup>4</sup> � � *<sup>C</sup>*1ð Þ*<sup>s</sup>*

*C*2ðÞ¼ *s*

Using the inverse Laplace in Eq. (9), results

An algebraic system is obtained

(

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

that can be written in matrix form as

The solution of the algebraic system (6) is

*C*1ð Þ*s C*2ð Þ*s*

The inverse matrix is

*s* þ *k*<sup>2</sup> þ *k*<sup>3</sup> �*k*<sup>4</sup> �*k*<sup>3</sup> *s* þ *k*<sup>4</sup> � ��<sup>1</sup>

Therefore,

the convolution<sup>1</sup>

**81**

.

functions, defined by *f t*ð Þ <sup>∗</sup> *g t*ðÞ¼ <sup>Ð</sup>*<sup>t</sup>*

" #

and

ð<sup>∞</sup> 0 *e*

*dt* � � <sup>¼</sup> *sCi*ðÞ�*<sup>s</sup> Ci*ð Þ <sup>0</sup> *:*

ð Þ *s* þ *k*<sup>2</sup> þ *k*<sup>3</sup> *C*1ðÞ�*s k*4*C*2ðÞ¼ *s K*1*Ca*ð Þ*s*

*C*2ð Þ*s*

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

" #

<sup>¼</sup> *<sup>K</sup>*1*Ca*ð Þ*<sup>s</sup>* 0 " #

> 0 " #

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

�*k*3*C*1ð Þþ*s* ð Þ *s* þ *k*<sup>4</sup> *C*2ðÞ¼ *s* 0

<sup>¼</sup> *<sup>s</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>3</sup> �*k*<sup>4</sup>

<sup>¼</sup> <sup>1</sup>

*<sup>C</sup>*1ðÞ¼ *<sup>s</sup>* ð Þ *<sup>s</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>4</sup> *<sup>K</sup>*1*Ca*ð Þ*<sup>s</sup>*

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

*<sup>C</sup>*2ðÞ¼ *<sup>t</sup> £*�<sup>1</sup> *<sup>k</sup>*3*K*1*Ca*ð Þ*<sup>s</sup>*

<sup>0</sup> *f u*ð Þ*g t*ð Þ � *<sup>u</sup> du* <sup>¼</sup> <sup>Ð</sup>*<sup>t</sup>*

�*k*<sup>3</sup> *s* þ *k*<sup>4</sup>

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

<sup>0</sup> *f t*ð Þ � *u g u*ð Þ*du*.

Now, the proprieties inverse Laplace transform are used, considering \* to denote

<sup>1</sup> The property of commutativity is valid in convolution operation for Laplace transform of *f t*ð Þ and *g t*ð Þ

�*stCi*ð Þ*<sup>t</sup> dt*

, (5)

*:* (6)

*:* (7)

(8)

(9)

(10)

#### **Figure 3.**

*A schematic diagram of a reversible two compartments model to illustrate the flux of tracer between blood (Ca) and and two tissues (C1 and C2).*

#### **2.1 Estimation of rate constants**

In order to estimate the parameters *kij*, *kji*, a nonlinear regression problem is solved using the Levenberg-Marquardt method [18, 19]. The sensitivity equations are generated partially deriving Eq. (2) with respect to the parameters *kij*, *kji*

$$\begin{cases} \frac{\partial}{\partial K\_{\vec{\boldsymbol{\eta}}}} \left( \frac{d}{dt} \mathbf{C}\_{i}(t) = \sum\_{j=1, j \neq i}^{N} \left[ K\_{\vec{\boldsymbol{\eta}}} \mathbf{C}\_{j}(t) - K\_{\vec{\boldsymbol{\mu}}} \mathbf{C}\_{i}(t) \right] \right) \\\ \frac{\partial}{\partial K\_{\vec{\boldsymbol{\mu}}}} \left( \frac{d}{dt} \mathbf{C}\_{i}(t) = \sum\_{j=1, j \neq i}^{N} \left[ K\_{\vec{\boldsymbol{\mu}}} \mathbf{C}\_{j}(t) - K\_{\vec{\boldsymbol{\mu}}} \mathbf{C}\_{i}(t) \right] \right) \end{cases} \tag{3}$$

Over which region of interest (ROI) is defined discrete TAC using the image processing. The Jacobian matrix it consists of the column vectors whose values resulting from the numerical integration of the sensitivity equations with respect to time.
