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‐
4]. Several reference tissue methods have also been developed [
5‐
10].
Irreversible (enzyme-substrate type) radiotracers [
11C]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 two-tissue 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 [
11C]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 [11C]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.
Discussion
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 [
11C]PIB (
k
4 = 0.018/min) as a moderately reversible ligand, only a poor model fit was obtained [
19]. Previous simulation studies on [
11C]PIB using information 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 [
11C]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.
K1 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 k3 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 [
11C]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 [
11C]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 k3 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 (
r2; 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
r2 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 r2 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 [
11C]PIB as an example of moderately reversible ligands, a somewhat lower inter-subject
k
3 correlation (
r2 = 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 [11C]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. post-comparison 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.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
KS participated in clinical PET study and the simulation study, and drafted the manuscript. KF conceived of the study, participated in the simulation study, and helped to draft the manuscript. HS (Shinotoh), HS (Shimada), SH, and NT participated in clinical PET study and contributed to the discussion. TS, TI, and HI supervised the design and coordination of the study. All authors read and approved the final manuscript.