One Computation Method of the Constant K3/K4 in Three-Compartment Modeling of 13N-NH3 PET Images Tractor

This study aims to quantitatively analyze and compute the velocity constant of three-compartment modeling, which is based on 13NNH3 PET images of human brain tumors. We selected the parietal lobe, cerebellum, frontal lobe, and the average of these three reference regions to analyze the transfer constant ratio, K3/K4, in three-compartment modeling. The study was based on the results of threecompartment modeling. Data sampling was performed for a left frontal lobe tumor, and simultaneously, data sampling was performed for three reference regions (including the parietal lobe, cerebellum, and right frontal lobe). The dynamic frames were 4 × 10 s, 7 × 20 s, 4 × 60 s, and 1 × 480 s. The parietal lobe, cerebellum, frontal lobe and the average of the three reference regions, as determined by the slopes of the fitted curves, are 1.6207, 1.5931, 1.5293, and 1.5803, respectively. The F-test values are 5552.4, 2943.6, 3756.8, and 5650.2, respectively; the average F-test value is the largest. And we have experiment with 11 ROIs with REF in this way, they are the line relative. All the R2 and P for all the fitted curves are almost 1, and all the P of all the fitted curves are almost 0. Moreover, the 95% confidence interval, which is based on the variance test, are enough short respectively. The comparison results show that the deviation and relative deviation for mean of REF are within the acceptable level. Thus, we have thought the K3/K4 as 0.5803; therefore, the transfer constant K4 is about 1.72 times that of K3 for the clinical use of N-NH3 PET tracer for brain tumors. The N-NH3 PET tracer is feasible for clinical use with brain tumors. The transfer constant, K4, is approximately two times of K3 in the three-compartment modeling of the parietal lobe, cerebellum, frontal lobe and the average of the three reference regions. This method is feasible and effective.


Introduction
Positron emission tomography (PET) is a medical imaging technique that is used to study tissue functions in vivo by a tracer, which is labeled with positron-emitting radionuclides. Arterial sampling is considered to be the accuracy standard for obtaining the input functions. However, arterial sampling is invasive, laborious and sensitive to errors, and it has a minor risk of adverse effects [1]. In a previous study [2], a new method was developed that combine's partial volume correction (PVC) during reconstruction with a simple automatic procedure for extracting the IDIF from the internal carotid arteries.
No particular model structure is assumed, which can be an advantage in many cases because one model may not describe equally well all data sets from the same region of interest (ROI). It is assumed that the plasma concentrations of unchanged tracer are monitored following tracer injection. The requirement of plasma measurements can be eliminated in some cases when a reference region is available.
The kinetic modeling of data that are obtained from PET can provide quantitative information regarding the spatial distribution of radiopharmaceuticals [1]. This modeling often requires knowledge of the input function that is traditionally obtained by arterial sampling, which is a burdensome and potentially risky procedure. Imagebased time activity curves, which are obtained by placing regions of interest (ROIs) over vascular structures on PET dynamic studies, are an appealing way to obtain individual input functions, concurrently reducing or obviating the requirement for blood samples. For brain scans, several studies have demonstrated the possibility of obtaining a reliable input function using the smaller intracranial vessels, such as the internal carotids [2] or venous sinuses [3]. Positron emission tomography is widely used to investigate tumors in the brain [4].
Several PET tracers have been applied to the investigation of microvascular structures in proof of-principle experiments. Partial volume effects that are caused by overlapping signals from other tissues can be significant when the tumor size is less than twice the scanner resolution [5]. The measurement of the tumor blood volume can be achieved by the inhalation of 11 C-CO, which then forms 11 C-CO-hemoglobin. Unlike 15 O-H 2 O, 11 C-CO-hemoglobin remains entirely intravascular and enables the calculation of blood volume from the ratio of tracer concentrations in tissue to that in blood [6]. 13 N (half-life: 9.965 min, 100% decay) is one of the most important of all positron emitters and has been primarily used in nuclear medicine in the chemical form 13 N-NH 3 [7] or as enzymatically synthesized 13 N-labeled amino acids [8]. Cardiac 13 N-ammonia ( 13 N-NH 3 ) PET is frequently used to assess myocardial blood flow and the coronary flow reserve [9]. However, gated cardiac 13 N-NH 3 imaging is not currently used for the estimation of left ventricular function or for brain tumors. In addition, in clinical applications, 13 N is of limited use compared with other positron emitters, such as 11 C and 18 F, which is primarily due to its short half-life of 10 minutes. However, the short half-life is also convenient to both shorten the time in the clinic and to shorten the pain of the patient [10]. 13 N may come to be used more widely if more 13 N-labeled compounds are made available. High specific activity may also increase the applicability of 13 N-labeled compounds for receptor studies using PET [11].
In this paper, we analyzed kinetic modeling that uses the 13 N-NH 3 PET tracer for brain tumors in a clinically used reference region method. Because the parietal lobe, cerebellum and right frontal lobe have close ties with the tumor regions (left frontal lobe) of the brain, whether structurally or functionally, we selected the three reference regions to compute the relative of the constant K 3 /K 4 for threecompartment modeling. In addition, we analyzed and statistically tested for the transfer constant of the three-compartment modeling.

Materials and Methods
The institutional review board of our hospital approved this retrospective study. A 47-year-old man had been diagnosed with a brain tumor on the left frontal lobe, detailed information regarding the study purpose and imaging procedure were explained to the patient, and informed consents were obtained from the patient.
To perform the quantitative analysis of 13 N-NH 3 PET, we have used three-compartment modeling and have selected three reference regions (including the parietal lobe, cerebellum, and right frontal lobe) that are close to the tumor region (left frontal lobe). Based on the relation between the region of interest (ROI) and the region of reference (REF), we have computed the constant K 3 /K 4 of the threecompartment modeling, which is based on the 13 N-NH 3 PET imaging of a human brain tumor.

PET imaging
Tracers were produced at our center by applying standard techniques and commercially available systems for isotope generation (Ion Beam Applications, Cyclone-10, Belgium). PET/CT imaging was performed with a Gemini GXL 16 scanner (Philips, Netherlands) in the 3-dimensional acquisition mode. The specific imaging protocol for brain was selected with a FOV of 180 mm, and the Trans axial spatial resolution was 2 mm full width at half maximum at the center of the FOV for 13 N-ammonia. PET images were reconstructed by the line-ofresponse RAMLA algorithm, with low-dose CT images for attenuation correction, which resulted in 3-dimensional images that consisted of 128 × 128 × 90 voxels of 2.0 × 2.0 × 2.0 mm 3 .
Because the half-life of 13 N is 9.965 min, there is 100% decay [11]. For the patient, the tracers were examined with a time interval of at least 24 hours between the scans, and fasting for more than 8 hours was required before 13 N-ammonia imaging. At 5 minutes after the injection of 13 N-ammonia (370-740 MBq), a 15-minute PET acquisition began ( Figure 1).
The study was based on the results of three-compartment modeling. We have obtained data sampling from a left frontal lobe tumor; simultaneously, we have obtained data from the three reference regions (including the parietal lobe, cerebellum, and right frontal lobe) and the average of the three reference regions. The dynamic frames consisted of 4 × 10 s, 7 × 20 s, 4 × 60 s, and 1 × 480 s.

Methods
For reversible systems, the form of the graphical analysis equation can be derived from the compartmental equations that describe tracer accumulation in tissue. For the two-tissue compartment model, the compartment ordinary differential equations are shown [1,2].
In the equations, Cp (t), C1 (t) and C2 (t) are concentrations for each compartment at time t. The units of concentrations that are used in the examples that are presented in this paper are Ci/mL. The transfer constants, K 1 , K 2 , K 3 and K 4 , describe the tracer concentration efflux between tissue and plasma. K 1 describes the transfer from plasma to tissue and is a function of blood flow, capillary permeability, and plasma protein binding. K 2 describes the transfer from tissue to plasma and is a function of blood flow. K 3 and K 4 are, respectively, the product of a bimolecular rate constant and the concentration of free receptor/ enzyme, which is assumed to be constant. However, in experiments with changing neurotransmitter levels, K 3 and K 4 represent an average over the duration of the experiment.
The transfer constant and the distribution volume ratio (DVR) can be calculated directly using the graphical method with data from a reference region [REF (t)] and with an average tissue-to-plasma efflux constant. In fact, the DVR changes when we use a different conference region and a different tracer. Specifically, the DVR that we calculated is a relative value; thus, DVR is calculated using the following equation:
We have selected three reference regions (including the parietal lobe, cerebellum, and right frontal lobe) that are close to the tumor region (left frontal lobe) and the average gray value of the region. For fitting convenience, we have used the formulation y=bx+c, and the fitting results are shown in Table 1 (Figure 2).

Discussion
Single scan acquisitions in PET are the most commonly protocols that are used in clinical imaging, irrespective of the tracer or the organ to be imaged. However, the obtained single scan images are primarily used for qualitative observations only. In brain imaging with PET, when the input curve is not available, the simulation or reference images can be converted to parametric images, which can be archived. These reference images are also useful for patient follow-up, for database constitution, and particularly for quantitative diagnoses.
From Table 1, among the reference regions are the line relative. If the parietal lobe is the ROI, and we take the cerebellum, right frontal  Table 1, the reference region and the tumor region are relatively in line. In addition, the slopes of the fitted curves for the parietal lobe, cerebellum, right frontal lobe and the average of the three reference regions are 1.6207, 1.5931, 1.5293, and 1.5803, respectively. Thus, based on the method that is described in this paper, we can compute the relative constant K 3 /K 4 of the three-compartment modeling, which are 0.6207, 0.5931, 0.5293, and 0.5803, respectively. Based on the computer data and the resulting analysis of the clinical use of PET 13 N-NH 3 tracer kinetic modeling of a brain tumor using the reference region method, we have found that the transfer constant K 4 of the modeling is approximately two times the transfer constant of K 3 .
Moreover, the 95% confidence interval, which is based on the variance test, are [1.5943, 1.6471], [1.5644, 1.6218], [1.5006, 1.5580],  and [1.5548, 1.6059], respectively. The confidence interval of the average is the shortest interval. The F-test values are 5552.4, 2943.6, 3756.8, and 5650.2, respectively, with the average of the regions being the largest value, i.e., the average is much more prevalent than the others are in the four results. Thus, we have chosen the K 3 /K 4 value of 0.5803; therefore, the transfer constant K 4 is 1.7216 times the transfer constant K 3 for the clinical use of PET 13 N-NH 3 tracer for brain tumor imaging.
From Table 2, we have experiment with 11 ROIs with REF in this way, they are the line relative. All the R 2 and P for all the fitted curves are almost 1, and all the P of all the fitted curves are almost 0. Moreover, the 95% confidence interval, which is based on the variance test, are enough short respectively. The F-test values are meeting the requirement of the F-test table respectively. Thus, the transfer constant K 4 is about 1.7 times the transfer constant K 3 for the clinical use of PET 13 N-NH 3 tracer for brain tumor imaging. Therefore, this method is feasible and effective based on these results.
From Table 3, the mean value of K 4 /K 3 is 1.6907 with cerebellum REF, and the variance is 0.0254. The mean value of K 4 /K 3 is 1.8746 with frontal lobe REF, and the variance is 0.0405. The comparison results show that the deviation and relative deviation for mean of REF are within the acceptable level. From Table 4, The comparison results show that the deviations between this paper with method [12], method [13] and method [14] are within the acceptable level.
To avoid withdrawing blood samples from patients and handling radioactive samples, in this work, we proposed a new feasible and effective reference method to calculate parametric images for the analysis of 13 N-NH 3 tracer kinetic modeling. Additionally, we determined that the average of the last three conference regions (including the parietal lobe, cerebellum, and right frontal lobe) are optimal in the four results for the tumor (left frontal lobe). Therefore, this method is feasible and valid, with the average of the several related or associated regions being taken as the REF for the ROI.