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 ([11C]PIB) and a two-tissue irreversible compartment model for [18F]2-fluoro-2-deoxy-D-glucose ([18F]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.
- noninvasive input function
- Laplace transform
- kinetic modeling
- positron emission tomography (PET)
- reference region
- region of interest
- time activity curve
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
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
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, 9, 10, 11, 12, 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 , 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 [11
1.1 Image analysis and data generation
Data used in this work was obtained with [18
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 , the effective dose injected can be calculated as:
where is the dose measured before injection at time
The radioactivity of [11
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:
where is the concentration of radioactive tracer in compartment
Figure 3 illustrates a reversible compartment model, that is be used to investigated the [11
The irreversible two compartment model (Figure 2 with ) is used for description of tracer [18
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.
2.1 Estimation of rate constants
In order to estimate the parameters , , 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 ,
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.
3. Two-tissue reversible compartment model of [11
where is the arterial input function (AIF) considered to be known, and are, respectively, the concentration within the non-displaceable and displaceable compartments and and are kinetic rate constants which have to be determined.
The Laplace transform with respect to
An algebraic system is obtained
that can be written in matrix form as
The solution of the algebraic system (6) is
The inverse matrix is
Using the inverse Laplace in Eq. (9), results
Now, the proprieties inverse Laplace transform are used, considering * to denote the convolution1.
In Eq. (13)
The analytical solution of the reversible two-compartment model for [11
In Eq. (15), it is visible the importance of construction of input function in order to make it possible to calculate the integral
4. Two-tissue irreversible compartment model of [18
where is the input function and is considered to be known, and are the concentration in C1 and C2 compartments, respectively, and , , 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 , applying the Laplace transform with respect to
Eq. (18) is represented matrically
The representation Eq. (22) implies that
Then, with , the analytical solution of the irreversible two compartment model for [18
It is important now to choose a suitable model to represent the input function , which makes it possible to calculate the integral .
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.  present five different forms to measure this data and  eight methods for the estimation image input function in dynamic [18
4.2 Input function derived of PET image
where is the concentration of the radiotracer in the arterial blood, is the concentration of the radiotracer in the reference region and and are the proportionality rates describing, respectively, the tracer influx into and the outflow from the reference tissue.
is constructed from a TAC of a reference region .
After this, deriving we obtain , which is the AIF, using
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 in a few stages as piecewise function, . This is defined for three stages in the equation
where , and are the concentration of the radiotracer on the reference region, respectively, for the fast, intermediate and slow stage. is the Heaviside function defined by
5. Results and discussion
In order to obtain the analytical solution of two-compartment model, Eq. (15) for [11
5.1 for [11
Nonlinear regression is applied to determine the parameters of the model chosen to approximate , from a discrete TAC curve.
The technique was applied for 7 patients considering the activity for the reference regions: right cerebellum, left cerebellum, and also for the mean of both (total of 24 simulations). The minor
5.2 for [18
In order to obtain for [18
Then, it may be appropriate to estimate the considering the fast and slow stage.
After this, we apply regression techniques, and in two stages of the time, a good option that came up was the piecewise function logistical to describe the behavior of the mean of the discrete TACs of four patients (considering left volume), Figure 8, with correlation coefficient of 0.9947 (at least) is
It may be convenient in the diagnosis of Alzheimer’s disease to consider the specific time interval seconds, [22, 23]. In this time interval, the graphs in Figure 8 show the comparison between the values estimated by the function and the concentration of the FDG radiotracer in the left VOI. As we can see, the estimated values between 1170 and 2970 s were close to the original values, with the lowest average relative error is 0.0582 and the highest is 0.1096.
The aim this study was described the algorithms of kinetic modeling to analyze the pattern of deposition of amyloid plaques and glucose metabolism in Alzheimer’s dementia, obtaining the exact solution of the [11
Longitudinal studies, without arterial blood samples, can assist in the calculation of the dose of medicine, providing the stabilization of cognitive impairment, behavior and the performance of activities of daily living. The technique here described can be used to analyze the pattern of deposition of amyloid plaques, glucose metabolism, the cortical and functional structure of the brain of SuperAgers in relation to cognitively normal elderly and individuals with Alzheimer’s dementia. Older adults with exceptional memory ability are coined SuperAgers. Their preserved cognitive capacities with aging may help uncover neuromechanisms of dementia. These individuals showed whole-brain volume similar to middle-aged individuals and some areas thicker than usual agers. Intriguingly, they also exhibited decreased atrophy rate when compared to normal older adults. To our knowledge, their brain functional integrity is yet to be uncovered.
This study was made possible by a team work from all members of the SuperAgers project: Lucas Porcello Schilling, Louise Mross Hartmann, Ana Maria Marques da Silva, Cristina Sebastiao Matushita, Mirna Wetters Portuguez, Alexandre Rosa Franco, and Ricardo Bernardi Soder. This research was partially supported by CNPq, project number 403029/2016-3 FAPERGS, project number 27971.414.15498.22062017.
|AIF||arterial input function|
|EDI||effective dose injected|
|MIP||maximum intensity projection|
|PET||positron emission tomography|
|ROI||region of interest|
|TAC||time activity curve|
|VOI||volume of Interest|
- The property of commutativity is valid in convolution operation for Laplace transform of ft and gt functions, defined by ft∗gt=∫0tfugt−udu=∫0tft−ugudu.