The present study investigated and characterized two graphical methods for *in vivo* noninvasive quantification of the binding levels of [^{18}F]FDDNP in human brain. The graphical method is widely used for analysis of dynamic PET data acquired with irreversible or reversible radiotracers because: 1) it is more computationally efficient and easier to implement than the conventional compartmental analysis that uses iterative model fitting, which is slow and sensitive to high noise levels for parametric imaging and 2) it allows derivation of DVR (and DV, if the time course of metabolite-corrected plasma input function is available) without assuming a specific compartmental model configuration for the acquired dynamic PET data. Improvement in scanner resolution and increase in slice number often come with increased noise level in the acquired data. A better quantitative method to process and analyze noisy data especially over short scan times is thus essential for clinical studies. Although both Logan graphical analysis and RE plot approaches assume somewhat similar equilibrium conditions in their formulation, our data show that variations in time window have different effects on the bias and variability of Logan DVR estimates obtained at voxel and ROI levels, but no significant effect was observed on the DVR estimates derived with the RE plot approach. While the two graphical approaches were evaluated with [^{18}F]FDDNP as a case in point and some of the issues may be specific to the kinetics of [^{18}F]FDDNP, it is worth mentioning that many of these issues have implications for other radiotracer studies as well.

As can be seen from

Eq. (3), the regression matrix of the independent variables (

**M**_{L}) in the Logan graphical analysis is dependent upon both

*C*_{R}(

*t*) and

*C*_{T}(

*t*). Its matrix inversion needs to be calculated for all voxel TACs included in the construction of parametric DVR images (

Eq. (7)) and could be poorly conditioned at the voxel level because of the high noise levels in

*C*_{T}(

*t*). For a large volume of dynamic data, calculation of matrix inversion on a voxel-by-voxel basis is computationally intensive and numerically unstable. On the contrary, the relatively stable parametric DVR image constructed by the RE plot approach is largely attributed to two reasons: 1) both

*C*_{R}(

*t*) and its time integral that are used as the regression independent variables are derived from averaging the voxel TACs within the reference region, whose TAC is expected to be less noisy than an individual voxel TAC; and 2) the regression matrix of the regression independent variables (

**M**_{E}) is independent of the target tissue activity and is common to all voxel calculations (

Eqs. (10) and

(12)). Moreover, the matrix inversion of

**M**_{E} needs to be calculated only once, followed by a matrix multiplication that is applied to all voxel TACs (

Eq. (13)). Therefore, the computational time is substantially reduced with the RE plot approach.

While both graphical approaches provide model-independent estimate of DVR, which is the slope of the linear portion of the graphical plots, it should be noted that the asymptotic linearity of

Eqs. (2) and

(9) (or

Eqs. (1) and

(8) in the original forms) holds only for

*t*>

*t** and therefore, the DVR estimates obtained are less than but approaching the ‘true’ values in the limit (

Logan et al., 1990,

1996;

Slifstein and Laruelle, 2000). The use of ordinary least squares (OLS) may also contribute to bias estimation of DVR or DV because OLS estimation assumes that the independent variables are noise-free and the dependent variable has uncorrelated random errors with mean zero (

Draper and Smith, 1981). Our data show that the bias associated with the Logan DVR estimates was further exaggerated in the presence of noise in the independent variables, and the effect was particularly pronounced at the voxel level ( and ). To this end, several strategies have been proposed recently to reduce the bias introduced into Logan graphical analysis by errors-in-variables in the regression model (

Draper and Smith, 1981;

Seber and Wild, 1989) through the use of different statistical estimation methods (

Logan et al., 2011;

Ogden, 2003;

Varga and Szabo, 2002), iterative data smoothing (

Logan et al., 2001), or mathematical rearrangement of the independent and dependent variables (

Ichise et al., 2002,

2003). However, there was only a modest to moderate bias reduction at the expense of increased parameter variability and computational burden as well as less rapid in the generation of parametric images (

Ichise et al., 2002;

Logan, 2003). Given the limited improvement in bias reduction using the aforementioned strategies, we simply used a bilinear form in conjunction with WLS estimation for both graphical analyses because 1) it avoids an error magnification process of numerical division of the independent and dependent variables by a noisy random variable (e.g.,

*C*_{T}(

*t*)), which may possess very high noise levels particularly on a voxel basis and 2) it allows data weights based on noise variance estimates or assumed error variance models to be incorporated in the regression process (

Carson, 1993). Nevertheless, further work on reducing bias in graphical methods is still ongoing and remains an active area of research.

Total scan duration is one of the most important considerations when evaluating a molecular imaging probe for clinical uses. The scan duration needs to be short enough to allow for reliable estimation of physiological or pharmacological parameters but it has to be long enough to minimize excessive measurement noise due to low counting statistics, especially toward the end of the scan. To minimize patient discomfort, a shorter scan is more desirable if the results are not compromised. To evaluate the effect of using short time windows on the DVR estimation, the brain datasets were truncated by progressively removing the late frames from 125-min to 65-min post-injection, and as such the number of data points used in linear regression was also reduced. The DVR

_{PAR} estimates () and parametric DVR images () obtained by Logan graphical analysis were found to be significantly deteriorated due to the high noise levels of voxel TACs, particularly when short time windows were used, resulting in a large number of unreliable linear fittings. Conversely, the use of a short time window had less impact on DVR

_{ROI} estimates as compared to DVR

_{PAR} estimates derived by either graphical method. This is mainly attributed to averaging the voxel TACs within the target and the reference regions, where the noise levels are much lower than a single voxel TAC. Moreover, it was also noted that DVR

_{ROI} estimates were not necessarily consistent with DVR

_{PAR} estimates for Logan analysis (see

Appendix A), and their discrepancies were much larger when short time windows were used ( and ).

The validity of the binding parameters derived with the RE plot approach depends on the relative equilibrium condition that needs to be verified a priori in practice. In general, a longer time than the Logan analysis is required for the RE plot to achieve linearity (

Logan, 2003;

Zhou et al., 2009). However, it may sometimes be problematic to assure that the ‘true’ steady-state condition is reached and the RE plot be used, particularly for radiotracers having short half-lives (such as [

^{11}C]-labeled compounds) and slow kinetics, and for regions slowly equilibrating with the plasma or the reference tissue. In the first case, the noise level could be very high at late times and a long data acquisition time may not be possible, whereas in the latter case it may take hours for a specific region to reach equilibrium with the plasma or the reference tissue that is impractical in clinical situation. Irrespective of statistical noise, choosing a

*t** value that is earlier than the time after which asymptotic linearity is attained in graphical methods will lead to underestimation of DVR. To address the non-equilibrium issue when applying the RE plot approach to radiotracers having slow dynamics,

Zhou et al. (2010) proposed a bi-graphical approach that made use of the ratio between the influx constant (K

_{i}) and the DV of the exchangeable space (DV

_{E}) calculated with the Patlak graphical analysis (

Patlak and Blasberg, 1985;

Patlak et al., 1983) to correct for the underestimation of the total DV. However, this technique involves numerical division between K

_{i} and DV

_{E} estimates and thereby inducing noise to the total DV estimates especially on a voxel level. Spatial smoothing is thus required (

Zhou et al., 2010), but it needs to be optimized with respect to the noise level and the radiotracer used. Moreover, because two graphical methods (RE and Patlak plots) and plasma input are used, the computation time is unavoidably longer and the technique is limited only to the calculation of total DV estimates.

In this work, partial volume correction was not performed on the PET images. It can be seen from the equations of both graphical methods (

Eqs. (1) and

(8)) that the resultant DVR estimates would be affected in the same way by partial volume effect (PVE). For the Logan graphical analysis, the ‘time’ abscissa range would be stretched but the ordinate values would not be changed by PVE in the target tissue measurements, resulting in lower slope (i.e., lower DVR estimate). For the RE plot, the ‘time’ abscissa values would not be changed but the ordinate range would be compressed by PVE in the target tissue TAC and thus the DVR estimate (equal to the slope of the plot) would also be lowered by the same scale as in the Logan plot.

Some differences in reaching equilibrium condition with the cerebellum among the brain tissue, blood vessels, lateral ventricles, and scalp deserve mention. The time to attain an ‘effective’ equilibrium with respect to the reference tissue (cerebellum) was similar for all gray matter regions (). This is probably because the rates for [^{18}F]FDDNP and any possible nonpolar metabolite to cross the BBB may be similar in these regions, although slower in subcortical white matter, where longer time is needed to reach equilibrium with the cerebellum. In contrast, there exists no BBB for the blood vessels, scalp, and lateral ventricles. The rate for which the activity concentration accumulated in the scalp and lateral ventricles was also shown to be slow ( and ). Further studies are clearly required to elucidate the differences in transport mechanism for these regions.

As seen from the absolute difference images shown in , there was a better differentiation between the intracranial, scalp, and lateral ventricle regions due to the suppression of the DVR values by the RE plot approach in the two latter regions. Our data show that this effect was more noticeable with the time window of 45–65 min, most likely because of the slow-equilibrium issue associated with the scalp and the lateral ventricle regions, where a longer time is required to reach steady-state equilibrium as compared to the brain tissues (). Therefore, the RE plot approach would produce systematically lower DVR estimates in those regions when

*t** is not long enough (e.g.,

*t** ≤ 45 min), as the linearity of the curve is not yet achieved (). This is different from the Logan graphical analysis, in which the linearity of the plot is normally attained before the true steady-state equilibrium is reached (

Logan et al., 1990,

1996;

Wong et al., 2010). The suppression of DVR values in the scalp and the blood vessel regions, however, was no longer apparent when a longer

*t** (>65 min) was used in the RE plot approach (). Nonetheless, the need of a longer

*t** does not necessarily impose a limitation on the applicability of the RE plot approach because the scalp and the blood vessels are not regions that are of particular interests in brain aggregate imaging with [

^{18}F]FDDNP PET. Indeed, the use of a shorter

*t** (<65 min) in the RE plot to suppress the scalp and the lateral ventricles on the parametric DVR images is potentially useful for providing a better visualization of the pattern of β-amyloid plaques and neurofibrillary tangles in the brain tissues of progressive AD. For example, the construction of cortical surface maps (

Protas et al., 2010) could be facilitated without the unwanted signals on the scalp and the lateral ventricles.