Noninvasive k3 estimation method for slow dissociation PET ligands: application to [11C]Pittsburgh compound B

Background Recently, we reported an information density theory and an analysis of three-parameter plus shorter scan than conventional method (3P+) for the amyloid-binding ligand [11C]Pittsburgh compound B (PIB) as an example of a non-highly reversible positron emission tomography (PET) ligand. This article describes an extension of 3P + analysis to noninvasive ‘3P++’ analysis (3P + plus use of a reference tissue for input function). Methods In 3P++ analysis for [11C]PIB, the cerebellum was used as a reference tissue (negligible specific binding). Fifteen healthy subjects (NC) and fifteen Alzheimer's disease (AD) patients participated. The k3 (index of receptor density) values were estimated with 40-min PET data and three-parameter reference tissue model and were compared with that in 40-min 3P + analysis as well as standard 90-min four-parameter (4P) analysis with arterial input function. Simulation studies were performed to explain k3 biases observed in 3P++ analysis. Results Good model fits of 40-min PET data were observed in both reference and target regions-of-interest (ROIs). High linear intra-subject (inter-15 ROI) correlations of k3 between 3P++ (Y-axis) and 3P + (X-axis) analyses were shown in one NC (r2 = 0.972 and slope = 0.845) and in one AD (r2 = 0.982, slope = 0.655), whereas inter-subject k3 correlations in a target region (left lateral temporal cortex) from 30 subjects (15 NC + 15 AD) were somewhat lower (r2 = 0.739 and slope = 0.461). Similar results were shown between 3P++ and 4P analyses: r2 = 0.953 for intra-subject k3 in NC, r2 = 0.907 for that in AD and r2 = 0.711 for inter-30 subject k3. Simulation studies showed that such lower inter-subject k3 correlations and significant negative k3 biases were not due to unstableness of 3P++ analysis but rather to inter-subject variation of both k2 (index of brain-to-blood transport) and k3 (not completely negligible) in the reference region. Conclusions In [11C]PIB, the applicability of 3P++ analysis may be restricted to intra-subject comparison such as follow-up studies. The 3P++ method itself is thought to be robust and may be more applicable to other non-highly reversible PET ligands with ideal reference tissue.


Background
Various reversible-type radioligands have been developed for in vivo neuroreceptor study with positron emission tomography (PET). Both arterial blood sampling and long dynamic PET scan, up to 120 min, are required for standard nonlinear least-squares (NLS) analysis to estimate K 1 to k 4 in the two-tissue compartment four-parameter model (4P model): K 1 represents the blood-to-brain transport constant, k 2 represents the brain-to-blood transport constant, k 3 represents the first-order association rate constant for specific binding, and k 4 represents the dissociation rate constant for specific binding. The k 3 represents B max •k on , where B max is maximum receptor density and k on is the in vivo association rate constant. Since k 3 represents available receptors for the PET ligand, it is the target parameter of major interest in most PET studies. However, quantification of k 3 in the 4P model is often difficult because of uncertainty of the k 4 estimate and high correlation between the k 3 and k 4 estimates. As surrogate parameters for B max , binding potential and distribution volume have been widely used [1][2][3][4]. Several reference tissue methods have also been developed [5][6][7][8][9][10].
Irreversible (enzyme-substrate type) radiotracers [ 11 C] methylpiperidin-4-yl acetate and propionate have been developed for the measurement of cerebral acetylcholine esterase activity using PET [11,12]. In this case the twotissue compartment three-parameter (K 1 to k 3 ) model (3P model) was used to estimate k 3 , which is an index of acetylcholine esterase activity. In the 3P model, the precision of k 3 estimate is usually higher than in the 4P model, in spite of shorter PET scan time (40 to 60 min), since there is no need of k 4 estimation in the 3P model.
We have previously defined two mathematical functions, the information density function and information function, which are useful for model selection and optimization of scan time in PET [13]. Based on simulations using both functions, we proposed a new method (3P + method) for quantification of k 3 for moderately reversible ligands. '3P+' means three-parameter model plus short PET scan. In this method, the 3P model (k 4 = 0 model) was applied to the early-phase PET data (up to 30 to 40 min) from reversible ligands with moderate k 4 (moderately reversible ligands). Although the 3P + method was not always developed for a specific ligand, the amyloid-binding radiotracer [ 11 C]Pittsburgh compound B (PIB) was used as an example for the moderately reversible ligands (k 4 = 0.018/min). The 3P + method afforded a more stable k 3 estimate than the standard 90-min 4P analysis. However, there is still the drawback of the necessity for arterial blood sampling and radiometabolite analysis, which may restrict the widespread use of this method in daily clinical practice.
In this article, we propose a noninvasive 3P++ analysis using [ 11 C]PIB. 3P++ means 3P + analysis plus use of a reference tissue for input function. To validate the proposed method, the linear correlations of k 3 estimates were evaluated between 40-min 3P++ and 3P + analyses, as well as between 3P++ and 90-min 4P analyses in clinical PET studies. In addition, simulation studies were performed to explain k 3 biases observed in the 3P++ analysis.

Assumptions in 3P++ analysis
The following are assumptions used in 3P++ analysis: Assumption 1 (on the nature of radioligand used): We apply 3P++ analysis only to moderately reversible or nearly irreversible radioligands (k 4 ≤ 0.03/min), but exclude highly reversible ligands. [ 11 C]PIB is an example of moderately reversible ligands (k 4 = 0.018/min). Assumption 2 (on the duration time of PET scan): We use early-phase PET data in the curve fitting. In [ 11 C]PIB, dynamic PET data during 0 to 40 min was described well with the 3P model, since the effect of the k 4 process on PET data was negligible within these early-phase kinetics [13]. Assumption 3 (on the specific binding in the reference tissue, k 3r ): Specific binding of radioligand is negligible in the reference tissue (k 3r = 0). In [ 11 C]PIB, the gray matter of the cerebellum is usually used as a reference tissue for input function [14]. We apply the one-tissue compartment two-parameter (K 1 , k 2 ) model (2P model) to the reference tissue.

Working equation for 3P++ analysis
The working equation for the 3P++ analysis has been reported [15]: where C t (t) is the radioactivity concentration in the target tissue and C r (t) is that in the reference tissue; k 2r is the k 2 in the reference tissue and ⊗ is the convolution integral. The rate of tracer penetration into the target tissue is obtained as the relative value R 1 , which is the ratio of target K 1 to reference K 1 .

Clinical PET study Human subjects
Two groups of subjects, a normal control (NC) group and an Alzheimer's disease (AD) group, participated in the current study with written informed consent. The NC group consisted of 15 healthy subjects (age ranging from 48 to 90 years, 66.7 ± 11.5 years (mean ± SD); eight males and seven females) without a history of central nervous system diseases or psychiatric disorders, and the AD group consisted of 15 patients (ages 55 to 85, 68.9 ± 9.6 years; four males and 11 females) diagnosed as probable AD according to the criteria of the National Institute of Neurological and Communication Disorders, Alzheimer's Disease and Related Disorders Association [16]. The study was approved by the Institutional Review Board of the National Institute of Radiological Sciences.

PET scan protocol
PET images were acquired with a Siemens ECAT EXACT HR + scanner (CTI PET systems, Inc., Knoxville, TN, USA) with an axial field of view of 155 mm, providing 63 contiguous 2.46-mm slices with 5.6-mm transaxial and 5.4-mm axial resolution. After a 10-min transmission scan for tissue attenuation correction, infusion of [ 11 C]PIB (about 370 MBq in 5 mL for 1 min) began. A PET scan in 3D mode was started after the arrival of tracer to the brain (approximately 30 s after the beginning of tracer infusion). The dynamic scans consisted of 19 frames (3 × 20 s, 3 × 40 s, 1 × 1 min, 2 × 3 min, 5 × 6 min, and 5 × 10 min) with the total scan duration of 90 min. All data processing and image reconstruction were performed using standard Siemens software, which included scatter correction, randoms, and dead time correction.

Region-of-interest delineation
Region-of-interest (ROI) analysis was performed using the PMOD software package (PMOD version 3.2; Technologies Ltd., Adliswil, Switzerland). The [ 11 C]PIB PET images were co-registered to T 1 weighted images in each subject. The following 15 ROIs were drawn manually on T 1 weighted images: frontal, mesial temporal, lateral temporal, parietal, occipital, anterior cingulate, and posterior cingulate cortices in both hemispheres as well as the reference tissue (gray matter of cerebellum). ROIs were transferred to co-registered [ 11 C]PIB PET images, and time-activity curves (TACs) were obtained in those brain regions.

Input function measurement
During PET scan, arterial blood was collected from radial artery, starting 6 s (transit delay at the blood sampling site) after the beginning of PET scan to 85 min post injection (10 × 10 s, 1 × 30 s, 9 × 2 min, 6 × 10 min, and 1 × 5 min; 27 samples). Radioactive metabolites were analyzed by a radio-thin layer chromatography (TLC) method [12], with a TLC-developing solvent (ethyl acetate/n-hexane = 2:1 vols). The metabolite-corrected radioactivity as well as total radioactivity in blood plasma was fitted to a monoexponential saturation function during infusion (0 to 1 min) and the sum of three-exponential functions after the end of infusion (1 to 85 min) [12].
4P and 3P + analyses (arterial-plasma input) Brain regional TACs were analyzed by the weighted NLS method under positive constraint of all k i with metabolitecorrected input function to afford K 1 to k 4 estimates in 4P analysis (scan time of 90 min) and K 1 to k 3 estimates in 3P + analysis (40 min). Correction was made for bloodpool (5%) radioactivity in brain tissue [14]. Custom software operating in IDL software (version 6.0; Jicoux Datasystems, Inc., Tokyo, Japan) environment was used for the compartment model analysis.

3P++ analysis (reference tissue input)
For successful convergence in NLS optimization using Equation 1, we fixed k 2r to 0.178/min (mean cerebellar k 2 value by 40-min 3P + analysis; N = 30; SD = 0.034). Based on Equation 1 and cerebellar TAC with a fixed k 2r value, the time-integral of C r (t) (the second term on the right side of Equation 1) and the convolution integral (the third term) were calculated numerically without data interpolation for each scan mid-times during 0 to 40 min, and the three parameters R 1 , k 2 , and k 3 were estimated.

Simulation study Generation of error-added TACs for Monte Carlo simulation
The error-free, baseline TACs (19 frames/90 min) simulating the target ROI of the NC and AD subjects were generated by using the 4P model with parameter set (K 1 = 0.180 mL/g/min, k 2 = 0.180/min, k 3 = 0.018 and 0.036/min for the NC and AD subjects, respectively, and k 4 = 0.018/min; typical values for [ 11 C]PIB) and averaged (N = 20) input function of [ 11 C]PIB. The reference ROI was the same between NC and AD subjects and was generated by using the 2P model with parameter set (K 1 = 0.180 mL/g/min, k 2 = 0.180/min) and the same input function as above. The error-added TACs for simulation were generated according to the following formula [18]: where C i is noise-free simulated radioactivity concentration at frame number i, Rand is a random number from a Gaussian distribution with a mean 0 and variance 1, ε is a scaling factor that determines the noise level, Δt i is scan duration of frame number i, t i is mid-scan time of frame number i, and λ is 11 C decay constant. In all Monte Carlo simulations, a data set of 100 noise-added TACs was analyzed with weighted NLS, using a relative weight w i : Effects of PET noise on 4P, 3P+, and 3P++ analyses , relative values empirically determined) were added to the baseline TACs of the target ROI of the NC subjects. From 100 error-added TACs for each PET noise level, 100 k 3 values were estimated using 90-min 4P, 40min 3P+, and 3P++ analyses. Coefficient-of-variation (CV) of k 3 was calculated as CV (%) = (SD/mean) × 100. In the following simulations, the PET noise was fixed at 0.1.

Effects of k 2 or k 3 change in reference ROI on 3P++ analysis
In 3P++ analysis, k 3r was assumed to be 0 and k 2r was fixed as an empirical constant. The effects of k 2r or k 3r change were investigated as follows. The error-added target TACs were generated by 4P model with two different k 3 values (0.018/min for NC and 0.036/min for AD); other parameters were the same as the baseline target TAC. The error-added reference TACs were generated by 2P model with five different k 2 (0.12, 0.15, 0.18, 0.21, and 0.24/min) and fixed K 1 values (0.18 mL/g/min). Another set of simulated reference TACs was generated by 3P model (not 2P model) with five different k 3 (0, 0.002, 0.004, 0.006, and 0.008/min) and fixed K 1 (0.18 mL/g/min) and k 2 (0.18/ min). The k 3 bias in 3P++ analysis was expressed relative to 3P + analysis as bias (%) = (3P++ k 3 /3P + k 3 − 1) × 100. Although k 3r was assumed to be 0 in Equation 1, each subject may have different k 3r values that deviated from 0. In simulations to investigate the effect of the individual k 3r variation on 3P++ analysis, we defined the k 3 value empirically corrected for nonzero k 3r as follows: where k 3 is the k 3 estimate of target ROI by 3P++ analysis and k 3r is the k 3 estimate of reference ROI by 3P + analysis (true reference k 3 ). Bias in 3P++ k 3 ′ relative to 3P + k 3 was compared with the bias in 3P++ k 3 to 3P + k 3 . Figure 1A shows an example of the curve fitting of [ 11 C] PIB cerebellar TAC data to the 2P model, where a good fit is seen during 0 to 40 min after tracer injection. Figure 1B shows the fits of cerebral cortical TAC data (0 to 40 min) to the 3P + and 3P++ models. The goodness-of-fit by 3P++ model (reference tissue input) is almost indistinguishable from that by 3P + model (arterial-plasma input). Kinetic parameters (K 1 = 0.161 mL/g/min, k 2 = 0.167/min and   Figure 2A is an example of the intra-subject k 3 correlation between 40-min 3P + (X-axis) and 3P++ (Y-axis) analyses, where the k 3 values of 15 ROIs, including the cerebellum (reference tissue in 3P++ analysis) from one particular NC subject or one particular AD subject, are shown. The regression lines and the coefficients of determination are Y = 0.845X − 0.006 (r 2 = 0.972) for the NC subject and Y = 0.655X − 0.004 (r 2 = 0.982) for the AD subject. Cerebellar k 3 values for both subjects are naturally calculated to be 0 in the 3P++ analysis. The slopes of the regression lines indicate the presence of negative bias in the 3P++ against the 3P + analysis. Figure 2B shows the k 3 correlation between 90-min 4P (X-axis) and 40-min 3P++ (Y-axis) analyses in the same subjects. The regression lines are Y = 0.590X − 0.005 (r 2 = 0.953) for the NC subject and Y = 0.338X + 0.000 (r 2 = 0.907) for the AD subject. When the cerebellar data (X = 0.008, Y = 0.000) was removed from calculation for the AD subject, the regression line became Y = 0.295X − 0.002 with slightly larger r 2 (0.935; not shown in the figure). The slopes of the regression lines show that k 3 bias in 3P++ against 4P analysis is larger than that against 3P + analysis. for the NC group alone, and Y = 0.286X + 0.003 (r 2 = 0.411) for the AD group alone; the latter two lines are not shown in the figure. The slopes of the regression lines also indicate the presence of negative biases in 3P++ against 3P + analysis. Figure 3B shows the inter-subject correlation of left lateral temporal k 3 between 90-min 4P (X-axis) and 40min 3P++ (Y-axis) analyses, where the regression line is Y = 0.225X + 0.000 (r 2 = 0.711) for all subjects. The lines of Y = 0.090X + 0.001 (r 2 = 0.122) for the NC group alone and Y = 0.135X + 0.005 (r 2 = 0.513) for the AD group alone were also calculated. The slopes of the regression lines show larger negative k 3 biases in 3P++ against 4P analysis than that shown in Figure 3A. The results in other cerebral regions were essentially the same as those in the left lateral temporal cortex.

Inter-subject k 3 correlation
Simulation on the effects of PET noise on k 3 CV Figure 4 compares the noise sensitivity of k 3 estimates among the 90-min 4P, 40-min 3P+, and 3P++ analyses. In all three analyses, the k 3 CVs increased as the PET error became larger. The k 3 CV in 3P++ analysis was comparable to that in 3P + analysis and lower than that in 4P analysis; for example, k 3 CVs at 0.1 of noise level were 6.6% in 3P++, 7.0% in 3P+, and 11.4% in 4P analyses.
Simulation on the effects of target K 1 change on k 3 bias Figure 5 shows the effects of K 1 change in the target ROI on the k 3 biases in the 90-min 4P, 40-min 3P+, and 3P++ analyses. The 4P analysis remained almost biasfree (+0.6%) within K 1 from 0.12 to 0.24 mL/g/min. 3P + and 3P++ analyses showed larger negative biases (−33% to −34% bias in 3P + and −33% to −35% bias in 3P++) compared with 4P analysis. Although 3P++ analysis showed slightly larger k 3 bias than 3P + analysis when K 1 was low (0.12 mL/g/min), k 3 bias in 3P++ analysis was almost the same as 3P + analysis.

Simulation on the effects of k 2r change on 3P++ analysis
In 3P++ analysis (Equation 1), k 2r was fixed at 0.178/ min, though k 2r was not always the same among subjects (CV = 19%). Figure 6 shows the effects of individual k 2r change in 40-min 3P++ analysis. When k 2r was equal to the fixed value (0.18/min), 3P++ analysis was bias-free, relative to 3P + analysis. However, when k 2r was different from the fixed value, 3P++ analysis showed a negative k 3 bias relative to 3P + k 3 . The k 2r effects were similar between NC ROI (k 3 = 0.018/min) and AD ROI (k 3 = 0.036/min); for example, the biases were −14.1% for NC and −12.1% for AD at k 2r = 0.12/min and −14.1% for NC and −11.3% for AD at k 2r = 0.24/min.

Simulation on the effects of k 3r change on 3P++ analysis
In 3P++ analysis we assume that k 3r = 0, that is, specific binding is negligible in the reference tissue. However, in all subjects examined, this assumption did not hold: the k 3r values in 40-min 3P + analysis were 0.008 ± 0.004/min in the AD group, 0.007 ± 0.002/min in the NC group, and 0.007 ± 0.003/min in the AD + NC group.    Figure 7 shows the effects of individual k 3r change (0 to 0.008/min) on 40-min 3P++ analysis. When k 3r was 0, 3P++ analysis was bias-free, relative to 3P + analysis. The k 3 biases (negative biases) increased as k 3r increased: −38% for NC and −27% for AD at k 3r = 0.004/min and −70% for NC and −48% for AD at k 3r = 0.008/min. The NC ROI (k 3 = 0.018/min) showed larger biases than the AD ROI (k 3 = 0.036/min). Figure 7 also shows the results of the simulation study on the relationship between 3P++ k 3 ′ and 3P + k 3 , where 3P++ k 3 was empirically corrected with individual k 3r . In this case, negative bias in 3P++ k 3 ′ was significantly decreased compared to that in 3P++ k 3 ; for example, bias was decreased from −70% to −7% for NC, and from −48% to −15% for AD at k 3r = 0.008/min. Figure 8 shows the correlation between 3P++ k 3 ′ and 3P + k 3 using the same data as in Figure 3A, where 3P++ k 3 in Figure 3A was replaced by 3P++ k 3 ′. The regression line was Y = 0.678X + 0.003 (r 2 = 0.975) for all subjects, where X = 3P + k 3 and Y = 3P++ k 3 ′. The lines of Y = 0.798X + 0.002 (r 2 = 0.897) for the NC group alone and Y = 0.620X + 0.004 (r 2 = 0.960) for the AD group alone were also calculated. The determination coefficient was increased by this correction from 0.739 to 0.975. The slope of the regression line was also increased from 0.461 ( Figure 3A) to 0.678 (Figure 8), which showed the reduction of negative bias in 3P++ analysis.

Theoretical basis and merits of 3P++ analysis
The previous 3P + analysis allowed for estimating k 3 of moderately reversible ligands, where the 3P model was applied to early-phase (up to 30 to 40 min) PET data with arterial input function [13]. It was reported that when the 3P model was applied to 60-min PET scan data from [ 11 C]PIB (k 4 = 0.018/min) as a moderately reversible ligand, only a poor model fit was obtained [19]. Previous simulation studies on [ 11 C]PIB using information  The k 3 bias in 3P++ analysis was expressed relative to 3P + analysis as bias (%) = (3P++ k 3 /3P + k 3 -1) × 100. Effects on bias in 3P++ k 3 ′ relative to 3P + k 3 are also shown (NC, open triangle; AD, closed triangle), where 3P++ k 3 ′ was calculated as (3P++ k 3 ′) = (3P++ k 3 ) + reference k 3 . density theory suggested that scan time reduction to 40 min would be necessary to obtain a good fit to the 3P model [13].
When 3P + or 3P++ analysis can be applied to a ligand, such ligand is specified as a moderately reversible ligand. This applicability is determined by the information function curves of k 3 and k 4 [13], and thus is dependent on the scan time as well as k 3 and k 4 values of the ligand in a ROI. Differentiation of a moderately reversible ligand from general reversible ligands is somewhat arbitrary, though we conveniently defined this with the k 4 value (≤0.03/min) in this study.
In the present study, the 3P + plasma input model was extended to the 3P++ reference tissue input model. The 3P++ analysis has three merits over previous methods. First, the PET scan time is short, usually less than 40 min, which may be important in PET studies with elderly or demented subjects. Secondly, the target parameter k 3 can be isolated from the other model parameters. Thirdly, neither arterial cannulation nor labor-intensive measurements of labeled metabolites are required.
One of the conventional models for the estimation of binding of [ 11 C]PIB is the Logan plot analysis [2], which employs data of long duration (more than 60 min). Noninvasive Logan analysis (distribution volume ratio) [6] requires late-phase (equilibrium-phase) PET data, whereas late-phase data are not necessary for 3P++ analysis. In the noninvasive Logan model or simplified reference tissue model [8], the K 1 -to-k 2 ratio in the target and reference tissues is assumed to be equal. 3P++ analysis does not require such an assumption. Since 3P++ analysis is a kind of irreversible-model analysis, K 1 (R 1 ) and k 3 can be independently estimated (k 2 must be fixed to a certain constant).

Noise sensitivity of 3P++ analysis
Loss of PET data in short-scan 3P++ and 3P + analyses might be considered to deteriorate the precision of the k 3 estimate. In the present simulation for noise sensitivity, k 3 CV values in 40-min 3P++ and 3P + analyses were lower than (almost three fifths of ) that in 90-min 4P analysis (Figure 4), which was in accordance with the previous report [13]. It is considered that the loss of PET data may be compensated for by the reduction in the number of free parameters from four in the 4P model to three in the 3P + and 3P++ models.

K 1 effect on 3P++ analysis
In the K 1 simulation, the stableness of k 3 estimation in changes of cerebral blood flow was investigated. The magnitudes of k 3 bias were independent of the K 1 change, ranging from 0.12 to 0.24 mL/g/min, in 3P++, 3P+, and 4P analyses ( Figure 5). The 3P++ as well as 3P + and 4P analyses were less affected by K 1 , which is owing to the capability of isolating the k 3 estimation. The 40-min 3P + analysis showed −33% k 3 bias relative to 90-min 4P analysis, which is in accordance with the previous report [13]. In this K 1 simulation, 3P++ k 3 showed negligible bias relative to 3P + k 3 . These results suggested that in 3P++ analysis, the effects of ignoring vascular volume as well as numerical integration error due to discrete time points were not significant.

Causes of negative k 3 bias in 3P++ analysis
Firstly, the k 3 bias in 3P++ analysis originates from 3P model approximation. Our previous simulation study [13] showed that the 3P + analysis with 28-min scan had large negative k 3 bias relative to 4P analysis with 90-min scan; for example, there was about −22% to −24% bias to true k 3 (4P k 3 ) ranging from 0.01 to 0.04/min including NC and AD k 3 . 3P++ analysis showed further negative k 3 bias relative to 3P + analysis due to the following two reasons.
Secondly, the bias is due to individual k 2r change from the fixed value in Equation 1. In 3P++ analysis, we also assumed that k 2 in the reference tissue was constant and was fixed at 0.178/min, which was the average k 2 value with the 3P + model. In simulation, negative k 3 bias was predicted when k 2r was larger or smaller than fixed k 2 ( Figure 6). Each subject in the NC and AD groups had different k 2 values in the reference tissue, and it is considered that such biological variance as for reference tissue may result in a negative k 3 bias in 3P++ analysis, relative to 3P + analysis for [ 11 C]PIB.
Thirdly, the bias is due to the discrepancy between the model assumption and the actual reference ROI. The basic assumption (assumption 3) in 3P++ analysis is k 3r = 0. The working equation of 3P++ analysis (Equation 1) is derived under this assumption, and reference k 3 is naturally calculated to be 0. However, in 3P + analysis with [ 11 C]PIB, the cerebellum showed nonzero k 3 (0.007 ± 0.003/min in all 30 subjects). Thus, 3P++ k 3 is expected to be underestimated. Simulation studies showed that 3P++ analysis was bias-free for ideal reference with zero k 3 and that k 3 bias became larger as k 3r increased (Figure 7). When k 3 was replaced by k 3 ′, negative bias was significantly decreased in the simulation (Figure 7), as well as the slope of the regression line between 3P++ and 3P + analyses being increased from 0.461 ( Figure 3A) to 0.678 (Figure 8), which also suggested that nonzero k 3r caused underestimation of 3P++ k 3 .

Correlation of k 3 between 3P++ and 3P + analyses
Strong intra-subject k 3 correlation was shown between 3P++ and 3P + analyses, and the rank-order of k 3 was almost the same between the two analyses (Figure 2A), suggesting the stability of both 3P++ and 3P + analyses.
The inter-subject k 3 correlation (r 2 ; Figure 3A) was significantly lower than the intra-subject correlation (Figure 2A). Such a lower inter-subject k 3 correlation can be partly explained by the sample variance of cerebellar k 3 . In order to explain this, k 3 ′ was calculated for each subject. When k 3 was replaced by k 3 ′, the determination coefficient between 3P++ and 3P + analyses was increased from 0.739 ( Figure 3A) to 0.975 ( Figure 8); the latter is comparable to r 2 of the intra-subject k 3 correlation (0.982; Figure 2A).
Such an estimation of parameter k 3 ′ is not always practical, as 3P + analysis with arterial input function is necessary for individual cerebellar k 3 estimation. However, these results suggest that the lower r 2 in the inter-subject correlation compared with the intra-subject correlation is due to the sample variance of cerebellar k 3 and that 3P++ analysis itself is robust, as far as the reference is ideal.
Practically, the use of mean k 3r may be meaningful. When target k 3 is empirically corrected as corrected k 3 = estimated k 3 + mean cerebellar k 3 , the absolute bias in target k 3 would decrease. However, the precision of target k 3 would not necessarily be improved owing to the variance of individual k 3r .
In addition to the nonzero effect of k 3r , inter-subject variation of k 2r from the fixed value (k 2 = 0.178/min) may also produce individually different k 3 bias in 3P++ analysis, resulting in lower inter-subject k 3 correlation between 3P + and 3P++ analyses.

Limitations of 3P++ analysis
When 3P++ analysis was applied to [ 11 C]PIB as an example of moderately reversible ligands, a somewhat lower inter-subject k 3 correlation (r 2 = 0.739 or 0.711; Figure 3A or Figure 3B) was shown between the 3P++ and 3P + or 4P analyses, respectively, across a k 3 range including NC and AD (3P + k 3 , 0.004 to 0.040/min). The rank order of 3P++ k 3 also differed considerably from 3P + k 3 or 4P k 3 . These results were mainly due to nonzero k 3r and the sample variance of both k 2r and k 3r as described above. The negative k 3 bias (3P++ vs. 3P+) was larger in NC ROI (−70%) than in AD ROI (−48%) when k 3r = 0.008/min (Figure 7). The previous report showed that the difference in k 3 bias (28-min 3P + vs. 90-min 4P) was small between NC ROI (−23%) and AD ROI (−24%) [13]. Therefore, the k 3 value in 3P++ analysis may be somewhat underestimated in the ROI with lower amyloid deposition compared to 3P + or 4P analysis.
In [ 11 C]PIB PET, 3P++ analysis may be inadequate for inter-subject k 3 comparison and useful only for intra-subject (inter-ROI) comparison or pre-vs. postcomparison in the same subject. 3P++ analysis would be more suitable for such reversible ligands that have moderate k 4 and reference tissue without specific binding.

Conclusions
The 3P++ analysis is a k 3 estimation method for moderately reversible PET ligands with a short scan time such as 40 min and without arterial blood sampling. Although the applicability of 3P++ method to [ 11 C]PIB PET may be restricted to intra-subject comparison, 3P++ analysis itself is robust. The 3P++ method would be useful for PET study with non-highly reversible ligands, as far as the reference tissue without specific binding is available.