Abstract–The most active area in brain PET ligand development and imaging continues to involve receptor/transporter studies involving reversible binding. The focus of this work has been to develop direct 4D parametric image reconstruction techniques for reversible binding imaging. Based on a recent graphical analysis formulation [1], we developed a closed-form 4D EM algorithm to directly reconstruct distribution volume (DV) parametric images using a plasma input model. Furthermore, while previous work in the area of 4D imaging has been primarily limited to plasma input models, we sought to also develop reference tissue model schemes whereby distribution volume ratio (DVR) parametric images were reconstructed by the reference tissue model within the 4D image reconstruction task (using the cerebellum as reference). The means of parameters estimated from 55 human 11C-raclopride dynamic PET studies were used for simulation (22 realizations) using a mathematical brain phantom. Images were reconstructed using standard FBP or EM methods followed by modeling, as well as the proposed direct methods. Noise vs. bias quantitative measurements were performed in various regions of the brain. Direct 4D EM reconstruction resulted in substantial visual and quantitative accuracy improvements (over 100% noise reduction, with matched bias, in both plasma and reference-tissue input models). Notable improvements were also observed in the coefficient of variation (COV) of the estimated binding potential (BP) values, including even for the relatively low BP regions of grey and thalamus, suggesting the ability for robust parameter estimation even in such regions.

I. INTRODUCTION ynamic PET imaging provides the ability to extract important physiological and biochemical parameters of

interest [2]. There have been a number of different formulations to directly estimate such kinetic parameters from sinogram data (e.g. see reviews by Tsoumpas et al. [3] and Rahmim et al. [4]). The direct approach seeks to move beyond the conventional independent frame image reconstruction approach, and is also able to accurately model the Poisson noise distribution in PET data (whereas indirect methods face the difficult task of estimating noise correlations in the reconstructed images - as a result, space-dependent noise variance and inter-voxel correlations are commonly neglected or oversimplified in practice).

Manuscript received Nov 5, 2009. A. Rahmim, Y. Zhou, J. Tang, and D. F. Wong are with the Department

of Radiology, Johns Hopkins University, Baltimore, MD 21205 (first author information: telephone: 410-502-8579, e-mail: arahmim1@jhmi.edu).

In the context of graphical modeling methods, the Patlak linear model was investigated by a number of groups [5, 6] including our own [7] wherein direct statistical 4D parametric reconstruction algorithms were successfully developed. These approaches were however only applicable to tracers modeled as effectively irreversible (e.g. 18F-FDG, 18F-FDOPA).

Nonetheless, the most active area in brain PET ligand development and imaging continues to involve receptor/transporter studies involving reversible binding. The focus of the present work is therefore to develop a direct 4D parametric image estimation scheme in reversible binding. Furthermore, while previous works in the area of 4D imaging have been primarily limited to plasma input models [3, 4], we seek to also develop a reference tissue model scheme within the 4D image reconstruction task.

II. REVERSIBLE-BINDING 4D PARAMETRIC IMAGING

The standard Logan graphical formulation [8] for reversible binding can be written as:

( ) ( )

( ) ( )

t t pj

j jj j

C d C dDV B

C t C t

τ τ τ τ′= +� �0 0

(1)

where jDV and jB ′ are the slope and intercept parameters

at a voxel j, and ( )pC τ and ( )jC τ are the plasma and target

tissue tracer concentrations, respectively, at time τ. The slope paramter DV is the distribution volume, which is the parameter of interest. At the same time, the Logan method is known to result in a noise-induced negative bias in the estimates of DV [9-14]. This is related to the fact that this formulation involves a non-trivial dependence on the concentration term ( )jC t , which for our purposes, would also much complicate the 4D reconstruction formulation. Instead, we have based our work on a recent graphical formulation by Zhou et al. [1] that does not exhibit the aforementioned noise-induced bias due to very favorable linear properties. The authors have shown that once the system reaches equilibrium relative to plasma input at t� t*, the following expression is valid:

0 0

( ) ( )

( ) ( )

t t pj

j jp p

C d C dDV B

C t C t

τ τ τ τ= +� �

(2)

Direct 4D parametric image reconstruction with plasma input and reference tissue models in

reversible binding imaging Arman Rahmim, Yun Zhou, Jing Tang, and Dean F. Wong

D

2009 IEEE Nuclear Science Symposium Conference Record M03-8

9781-4244-3962-1/09/$25.00 ©2009 IEEE 2516

Having the advantage of being linear in 0( )

t

jC dτ τ� , this new

formulation renders itself to a closed-form 4D EM algorithm, as we show next.

A. The proposed 4D framework We can manipulate the above term to arrive at an expression for the accumulated image n

jx in voxel j of frame n ( n

startt to nendt ). This can be written as:

( )d ( ) ( )nend

nstart

tn p pj j j jt

x C DV C n B nτ τ= = + Δ� (3)

where

( ) ( )dnend

nstart

tp p

tC n C τ τ�� (4)

( ) ( )( )p p n p nend startn C t C tΔ −� (5)

Then, based on the 3D system matrix relation n n=g Px relating the accumulated activity in each frame n to the measured data using the system matrix P, we propose to build the 4D relation G = PD where we have defined:

C (1) (1), ,

C (N) (1)

p p

p p

� � � �Δ� �� � � � � �� � Δ � �

1

n

g P PDV

G P DB

g P P� � � � � �

(6)

Upon application of the expectation maximization (EM) scheme, utilizing a new hidden complete-data EM formulation (as we have elaborated in [7]), this can be shown to result in the following iterative update algorithm:

oldnew

old old1

1

oldnew

old old1

1

( )( , )( )

( )( , )( )

nNp T

N npn

nnN

p TN np

nn

C nC n

nn

==

==

=

= ΔΔ

��

��

DV gDV Pg DV Bs

B gB Pg DV Bs

(7)

where ( )old old old old( , ) ( ) ( )n p pC n n= + Δg DV B P DV B

(8)

is the mean estimated data for each frame n based on previous estimates of the slope DV and intercept B images, and s is the standard sensitivity image. Note that the sensitivity term can incorporate both the normalization and attenuation terms. For modeling of scatter and randoms, see Sec. II-B.

As a result, the above iterative scheme allows direct estimation of the parametric DV image from the data. For a standard two-tissue compartment model (Fig. 1), the DV is

given by ( )31 1

2 4 2

1 1kK K BPk k k

� �+ = +�

�, where 3

4

kBPk

= is the

binding potential. Now, it is known that assuming the existence of a reference region with no specific tracer binding (i.e. 3 ~ 0k ), and also assuming that it has a similar

1 2/ref ref refDV K k= ratio as the 1 2/K k value for the target region, it follows that the distribution volume ratio (DVR) is given by: target ref/ 1DVR DV DV BP= +� , eliminating the

blood-brain barrier (BBB) components K1 and k2, and allowing estimation of BP.

Fig. 1: Depiction of the commonly utilized two tissue compartment

model.

This technique, however, still requires the somewhat

invasive approach of plasma input sampling. An alternative derivation using the reference tissue model is possible. It can be shown [1] that if such reference tissue exists and is in relative equilibrium with the plasma input, we have:

0 0( ) ( )

( ) ( )

t t refj

j jref ref

C d C dDVR

C t C t

τ τ τ τθ= +� �

(9)

where we note that the slope parameter is now the DVR. We are then able to derive an iterative 4D solution very similar to expression (5), by replacing ( )pC n and ( )p nΔ with:

( ) ( )d nend

nstart

tref ref

tC n C τ τ��

(10)

( ) ( )( )ref ref n ref nend startn C t C tΔ −�

(11)

An approach we investigated in this work is to estimate ( )refC n in (10) (the accumulated reference tissue activity in

each frame) from initial, fast reconstructions of the dynamic images (e.g. using FBP), and to also approximate

( )ref nendC t

a

at the end of each frame as the average between

the mean accumulated integral values in the next and existing frames. Following this, the 4D iterative technique can be applied, from which the binding potential at each voxel can be calculated by 1j jBP DVR= − .

B. Scatter and random correction It is also desired to incorporate corrections for random and

scatter terms within the proposed 4D-EM framework. A straightforward approach is to do so, very similar to the conventional ordinary Poisson (OP) EM algorithm [15], by modifying (8) to:

( )old old

old old

( , )

( ) ( )

n

p p n nC n n

=

+ Δ + +

g DV B

P DV B R S (12)

where Rn and Sn represent the randoms and scatter contributions in each frame, which can be estimated as is done in conventional reconstruction by analysis of each dynamic frame.

C. Sign-constraint considerations The EM algorithm imposes a non-negativity constraint.

This can appear as problematic because, unlike the intercept estimate in the irreversible Patlak model which is positive, the intercept estimate in the Logan model, or in this case in the Zhou model (i.e. B in (2)), is often negative. However,

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with further inspection, one observes that the ( )p nΔ term (5) in the abovementioned 4D model is itself negative, because typically at this late stage of the acquisition, for a bolus injection, the plasma input curve would be decreasing, resulting in a negative value for ( )p nΔ . As a result, what we have done in this work has been to change the sign of

( )p nΔ to positive, in turn allowing positive estimates for the intercept term B within the EM framework.

A limitation imposed, however, is that in practice it is common to perform the integration in (3) from 0 to the end of each of those late frames n

endt (and not from the beginning of

each frame nstartt ); i.e. to consider overall accumulated

activity from the very beginning of the scan until the end of each frame that is considered in the analysis. This has the potential to obtain less noisy parametric images. However, within the 4D framework, this would imply that

( )( )p p nendn C tΔ = which is positive, thus not allowing

aforementioned work-around with the sign constraint. As a result, the present work does not consider accumulation of activity from the beginning of the scan, which will only be explored in the future with more flexible reconstruction algorithms.

For the reference tissue model (9), as shown by Zhou et al. [1], the intercept term θ is proportional to

( )refB DVR B− × . In practice, the first term B commonly

has a larger magnitude than the second term refDVR B× , and therefore, θ has the same sign as B, and as a result, above approach can be employed in this model as well. A notable consideration, however, is in the reference tissue itself, in which case θ = 0. In other words, the reference tissue in 4D-EM estimation behaves similar to a cold region in standard EM methods, which is known to result in a positive bias due to the non-negativity constraint inherent in the EM algorithm.

In future work, we shall explore reconstruction algorithms that do not impose the aforementioned sign constraint, aiming for reduced bias performance in the cerebellum, and allowing generalized incorporation of parameters of varying signs across the image. In any case, as we shall see next, the proposed 4D EM method is able to achieve very favorable quantitative performance.

III. EXPERIMENTAL DESIGN

A. Simulations Mean regional slope and intercept parametric estimates

obtained from 55 11C-raclopride dynamic PET human scans were input into a mathematical brain phantom [16]. The two resulting parametric brain phantoms, and the plasma tracer concentration ( )pC τ as measured for a particular patient over time, were then employed by (2) to generate dynamic frame images, followed by analytical simulation of each to generate dynamic datasets, using 22 noise realizations.

The data were simulated for the geometry of the Discovery RX PET/CT scanner [17], except with the transaxial dimensions (crystal dimensions and field of view) scaled by 0.5 to simulate a dedicated brain scanner.

Five 5-min frames, covering 40-65 min of the simulated study, were considered, with the dynamic data reconstructed up to 40 iterations (21 subsets) for both the plasma input and reference tissue models. For the latter, time activity curves for the cerebellum, used as reference, were estimated using initial FBP reconstructions. Conventional FBP and EM reconstructions followed by modeling were then compared with the proposed direct 4D methods for both the plasma input and reference tissue models.

1) Figures of Merit For each given region of interest (ROI) of known uniform intensity, the normalized standard deviation (NSD) was defined across multiple noise realizations:

( )2

1 1

1 11

NSD = 100%

M Ntj j

j tX X

M NX

= =−

−×

� � (13)

where t

jX is the intensity in the specified ROI at measurement/realization t (t=1...N) and in a voxel j (j=1...M),

and 1

1 Nt

j jt

X XN =

�� is the mean intensity in voxel j as

averaged over the multiple realizations, and 1

1 M

jj

X XM =

�� is

the overall mean ROI intensity. Similarly, the regional bias (RB) for a given ROI of

known uniform intensity Xtrue was defined as:

RB = true

true

X XX−

(14)

To quantify overall noise vs. bias performance for the entire image, NSD values for all the ROIs (r=1...R) were averaged as a measure of average noise. As a measure of overall bias, the normalized mean squared error (NMSE) was defined as:

2

1

1NMSE = ( )R

overallr

RB rR =

� (15)

Furthermore, for BP values as averaged across a given ROI for multiple noise realizations, we quantified regional BP variance and bias as follows:

( )2

1

1Var =1

Nt

BPt

BP BPN =

−− � (16)

Bias = trueBP

true

BP BP

BP

− (17)

where tBP is the BP estimate at a noise realization t, BP is the mean BP estimate in the ROI across all realizations, and

trueBP is the true, simulated value. The difference between (13) and (16) is that the former first quantifies variations of a

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given voxel intensity across different noise realizations, followed by averaging in each ROI, while the latter first averages the parameter of interest, in this case the BP, across each ROI, followed by measurement of variation.

B. Application to patient study We considered application of the proposed method on the

second generation high resolution research tomograph (HRRT) [18]. We modified the existing HRRT reconstruction code to enable both: (i) standard kinetic modeling after each iteration through independent reconstruction of the dynamic frames (conventional method) as well as (ii) the proposed 4D direct parametric image reconstruction. Corrections for random and scatter events were also performed (see Sec. II-B), as estimated using the singles rate method and the standard single scatter simulation, respectively, for the HRRT scanner [15].

A 90-minute 11C- raclopride study on a 27 year-old make subject was considered. The reference tissue model was applied for the 40-60 min duration of the study (four 5-min frames), using the cerebellum as reference. Up to twenty iterations (16 subsets each) of both aforementioned conventional and proposed approaches were studied.

Since the true DVR values are not known, bias measurement as such was not performed. Instead, we plotted, for a variety of ROIs, the DVR vs. NSD values, where NSD was defined as in (13). This would allow one to compare how the noise performances compare given similarly obtained values of DVR.

IV. RESULTS

A. Simulations

1) Plasma input model Fig. 1 shows typical reconstructed images for the three

techniques, with increasing noise from left to right, as obtained in FBP reconstruction using different filters (Hann, Hamming, Cosine, Shepp-Logan, and Ram-Lak) and in the EM methods with increasing iterations. It is seen that while the FBP and standard EM approaches result in noisy images, this trend is more controlled and qualitatively improved for the proposed 4D method.

For a quantitative analysis, Fig. 3 depicts overall NSD vs. NMSE (see Sec. III) plots for the various parametric DV images shown in Fig. 2. Fig. 4 plots regional NSD vs. regional bias (RB) curves for eight individual regions of the brain (grey, caudate, putamen, thalamus, corpus callosum, white, nucleus accumbens and cerebellum). It is clearly seen that the proposed direct 4D EM reconstruction results in substantial quantitative accuracy improvements. The observation that the FBP approach performs relatively poorly in terms of bias can be attributed to its poorer resolution performance, thus higher partial volume effect, translating to higher bias.

Fig. 2: Estimated parametric DV images of increasing noise (from left to

right): generated for FBP using different filters, and for EM using increasing iterations of 3, 5, 10, 20 and 40 (21 subsets were used; shown with no post-filtering).

Fig. 3: Overall noise (NSD) vs. bias (NMSE) results comparing standard

FBP and EM methods with proposed 4D techniques (plasma input model). Points on each curve correspond to the images in each row of Fig. 2.

Fig. 4: Regional noise (NSD) vs. bias (RB) curves for FBP, standard EM

and proposed direct 4D EM techniques (plasma input model). Points on each curve correspond to the images in each row of Fig. 2.

Fig. 5 depicts plots of variance (VarBP) vs. bias (BiasBP) in

regional BP estimates, as defined in (16) and (17), for five regions of grey, caudate, putamen, thalamus, and nucleus accumbens. It is worth noting that in conventional imaging, coefficients of variation (COV) of BP estimates are too high

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for relatively low uptake BP regions of grey and thalamus, and are typically not reliable. We have seen improvement by factors of ~3 in the COV for the various regions, thus suggesting the possibility of imaging low BP images more feasibly.

Fig. 5: Variance vs. bias in regional BP estimates for FBP, standard EM

and proposed direct 4D EM techniques (plasma input model). Points on each curve correspond to the images in each row of Fig. 2.

2) Reference tissue model The aforementioned study was also performed using the

proposed reference tissue based direct 4D approach, taking the cerebellum as reference. Fig. 6 shows noise (NSD) vs. bias (RB) curves, as generated with increasing iterations, for parametric DVR images obtained by standard EM+modeling vs. proposed 4D EM methods. Fig. 7 shows VarBP vs. BiasBP plots for the estimated regional BP values for grey, caudate, putamen, thalamus, and nucleus accumbens. Again, notable quantitative improvements for the proposed 4D method are readily observed.

Fig. 6: Regional noise (NSD) vs. bias (RB) curves for standard EM and

proposed direct 4D EM techniques (reference tissue model).

Fig. 7: Variance vs. bias in regional BP estimates for standard EM and

proposed direct 4D EM techniques (reference tissue model).

Fig. 8: Parametric images for a raclopride HRRT study with increasing

iterations of 1, 2, 4, 8, 13, 20 from left to right. (Top) Standard 3D reconstruction followed by modeling; (bottom) Proposed 4D reconstruction.

Fig. 9: NSD vs. DVR curves (generated by increasing iterations as seen in Fig. 7), comparing performance of conventional 3D+modeling with 4D reconstruction.

B. Application to HRRT patient study Applying the proposed reference tissue model to a

raclopride patient study on the HRRT scanner (as described in Sec. III.B), as a visual/qualitative comparison, Fig. 8 plots images of increasing iterations for the conventional 3D+modeling approach, as well as the proposed 4D direct parametric imaging technique.

To compare these results quantitatively, Fig. 9 plots regional noise (NSD) vs. DVR curves, as generated by increasing iterations, for nine regions of anterior caudate nucleus (right and left), posterior caudate nucleus (right and left), anterior putamen (right and left), posterior putamen

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(right and left) and the cerebellum. With the exception of the cerebellum, the proposed 4D methods is seen to at least match, and in a number of cases outperform 3D method, in the sense that for a given DVR value, improved noise is observed.

V. DISCUSSION We found that when initializing the 4D-EM algorithm, use

of appropriate ratios between the slope and intercept images played an important role for faster convergence. In other words, simply setting both images to uniform images of equal values can result in very slow convergence. Instead, one may obtain an initial ratio estimate using very fast (e.g. FBP) reconstruction, and set the ratios that way.

In the cerebellum, unlike other regions as seen in Fig. 9, the 4D method exhibited poorer quantitative performance. This can be attributed to the fact that for the employed reference tissue model (9), θ is zero for the reference tissue (cerebellum), and as a result, as elaborated in Sec. II-C, the non-negativity constraint inherent in the EM algorithm can result in a positive bias in the estimation of the intercept term θ. In turn, this can result in increased bias in the estimation of the slope term DVR for the reference tissue, as we have observed. At the same time, the proposed 4D-EM method was able to quantitatively outperform conventional reconstruction in a number of regions, and was not outperformed in any other analyzed region.

Our future work will consist of exploring other reconstruction algorithms that do not impose the aforementioned non-negativity constraint. This is motivated by the following: (i) to more effectively address the fact that θ equals zero for the reference tissue, as just discussed; (ii) in practice, while DV and DVR estimates in the plasma input and reference tissue models, respectively, are positive, each of the intercept terms B and θ do not necessarily have the same signs across the image; (iii) one would like to consider (3) with the integration performed from 0 (and not n

startt ) to nendt ; i.e. to consider accumulated activity from the beginning

of the scan, allowing for improved noise performance, as discussed in II-C.

VI. CONCLUSION This work proposed a closed-form iterative 4D-EM

algorithm applicable to both plasma input and reference tissue models, moving beyond the conventional approach of reconstruction followed by modeling. It was demonstrated that the proposed method exhibits improvements in quantitative performance, as demonstrated for simulations and a raclopride HRRT study. Extension of the present 4D-EM reconstruction algorithm to less restricted constraints is expected to further improve quantitative accuracy of the 4D reconstructed parametric images.

ACKNOWLEDGMENT This work was in part supported by the NIH grants

1S10RR023623, DA00412, MH078175 and AA12839. The

authors wish to thank Andrew Crabb for computational support.

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