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Accurate Image Reconstruction with Computed System Response Matrix for a High-Sensitivity Dual-Head PET Scanner Chien-Min Kao, Yun Dong, Qingguo Xie and Chin-Tu Chen (submitted to IEEE Transaction on Medical Imaging, May 2008)
Abstract—We have recently proposed a compact, dual-head PET scanner configuration for providing high-sensitivity imaging of small animals. Although the scanner is able to reach a sensitivity of about 30% at the center of imaging field-of-view, its compact configuration produces substantial depth-of-interaction (DOI) blurring and results in significantly degraded spatial resolution. It is known that DOI blurring can be reduced if the system response matrix (SRM), which describes the individual sensitivity of the detection channels of the scanner to image voxels defined in its field of view, can be incorporated in reconstruction. In practice, however, the huge dimension of the SRM found in modern PET scanners make its computation and storage extremely challenging. In this paper, we show that for the dualhead scanner configuration one can employ cubic image voxels having a proper size to create substantial symmetries in the SRM, as a result producing drastic reductions in computation and storage. We have applied this strategy to the proposed highsensitivity small-animal PET scanner (µPET) to enable accurate computations of its SRM by using Monte-Carlo simulation. Our results with simulated data indicate that the proposed µPET scanner can achieve an isotropic and uniform spatial resolution of ~1.2 mm after incorporating the SRM in reconstruction. In contrast, the image resolution deteriorates significantly and becomes non-isotropic when not employing the SRM: at the center of the scanner, its resolution is ~1.8 mm in directions parallel to the detectors and becomes ~3.2 mm in the direction normal to the detectors. Furthermore, the resolution in the latter direction deteriorates considerably when moving away from the scanner’s center. Images generated by with employing the SRM in reconstruction also show substantially better noise properties than those generated without. When applied to a real dataset, considerable enhancement to the image resolution and contrast is obtained when using the SRM in reconstruction. Index Terms—positron emission tomography, small-animal PET, positron emission mammography, iterative image reconstruction, DOI blurring
I. I NTRODUCTION CHIEVING high spatial resolution and high detection sensitivitiy for PET systems has been a major research interest in medical-imaging research. One physical factor that
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C.-M. Kao is with the Department of Radiology, The Univesity of Chicago, Chicago, IL 60637, USA. (phone: 773-702-6273; fax: 773-702-3766; e-mail:
[email protected]). Y. Dong is with the Department of Biomedical Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA. Q. Xie is with the Department of Radiology, The Univesity of Chicago, Chicago, IL 60637, USA, and currently with the Department of Biomedical Engineering, Huazhong University of Science and Technology, Hubei, China. C.-T. Chen is with the Department of Radiology, The Univesity of Chicago, Chicago, IL 60637, USA.
has limited PET imaging performance is the so-called depthof-interaction (DOI) blurring, or the parallax errors. DOI blurring refers to the increase in the apparent aperture of a pair of scintillation crystals (and hence degradation in the spatial resolution) as the incidence angle of the gamma ray increases. The concern for DOI blurring is particularly relevant in developing PET systems that desire both high resolution and high sensitivity, such as with the case in developing dedicated small-animal PET (µPET) systems [1]–[12]. This is because, for such systems one prefers the use of narrow and long crystals for yielding high intrinsic detector resolution and efficiency, as well as the use of compact configurations for yielding large detection solid-angles to annihilation photons originating within the scanner’s field-of-view (FOV). The former choice produces large variations in the apparent aperture with the gamma-ray incident angle and the latter design more oblique gamma rays entering the systems. The appearance of DOI blurring in PET image depends on the scanner geometry. For the conventional ring-based systems, it appears as a progressive degradation in the radial resolution of the image with the radial distance [13,14]. One widely-used practice for limiting DOI blurring in ring-based PET systems is therefore to constrain, often substantially, the transaxial FOVs (TFOVs) of the scanners. This practice increases the number of detection channels needed for imaging a given volume (hence increasing the system cost) and reduces the detection solid-angle for a given scanner length (hence compromising the system sensitivity). Moreover, it does not resolve the fundamental issues with DOI blurring and one is still often forced to use relatively short scintillators (hence limiting the system sensitivity). To overcome design limitations caused by DOI blurring, detectors that are capable of producing DOI measurements have been developed and substantial efforts are being invested in this active research topic [15]–[26]. Despite their significance, DOI detectors have been used for building only a few PET systems so far [7]–[12]. In theory, resolution degradations in a PET system, including DOI blurring, can be eliminated in image reconstruction if the spatial response of the scanner, i.e., the system response matrix (SRM), can be accurately known [27]–[39]. The task of obtaining accurate SRMs is however a significant challenge for modern PET systems due to the huge numbers of detection channels found with these systems. Qi et al. reduces the complexity of the task by factoring the SRM into a product of several sub-matrices that are substantially easier to compute
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and store [27]–[29]. To obtain an accurate SRM for a µPET system, Rafecas et al. uses Monte-Carlo (MC) methods for modeling the physical processes in PET imaging [34,35]. These work concerns with ring-based systems; for them, one can exploit their rotational symmertry to reduce computation. Recently, polar voxels are introduced to create additional symmetries in the SRMs of ring-based systems for further reducing the computation [38,39]. Efforts to derive SRMs from measured point-source responses and employ them in reconstruction have also been attempted [36,37]. Kao et al. and Boning et al. have shown that it is possible to eliminate, in reconstruction, the substantial DOI blurring found with scanners that adopt TFOVs as larger as their detector rings [31]–[33]. The results of these work have demonstrated that, by employing accurate SRMs, one can effectively correct for the effects of finite detectors, including DOI blurring, to produce PET images of improved resolution. Recently, we have investigated the development of a highly sensitive µPET scanner by using two large-area detectors and adopting a stationary, compact scanner configuration [40]– [44]. While this µPET scanner can reach a central sensitivity (the sensitivity to a point source placed at the center of the scanner’s FOV) of ~30% and an extended imaging volume as large as 27x15x5 cm3 [42], its compact configuration leads to significant DOI blurring and substantially degrades the image resolution. The ability to effectively correct for the DOI blurring in reconstruction is thus an important requirement for this high-sensitivity scanner. The scanner, however, has more than 224 million lines-of-response (LORs) and the image volume can contain more than 11 million voxels. To compute the resulting SRM by brute force will take about a month if the computation for each element of the SRM would require only 1 nanosecond. The storage for the SRM, and its effective access for reconstruction, is also a great concern. Independent of the introduction of polar voxels for ring-based PET systems, to overcome the practical challenges discussed above we have proposed the use of cubic voxels having a proper size to create substantial symmetries in the SRMs of dual-head PET scanners, making it practically possible to employ MC techniques to compute the SRMs accurately [42,44]. In this paper, we will present and discuss this concept for computing the SRM, and investigate the degree of improvement to the image resolution that can be achieved for the dual-head µPET scanner when incorporating the SRM in reconstruction. The rest of this paper is organized as follows. In Section II, we discuss the symmetry properties available with a dual-head PET scanner when using cubic image voxels that are properly sized. In Section III, we discuss the configuration of our dualhead µPET scanner, the computation of its SRM by using MC techniques and employing symmetries, the reconstruction algorithms, and the simulated- and real-data considered in our studies. In Section IV, we present our simulation and real-data results. Concluding remarks are given in Section V. II. S YMMETRY P ROPERTIES FOR D UAL -H EAD S CANNERS Without loss of generality, we consider two square detection planes containing 1 mm2 square detection elements (referred
y o
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x +R
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(a) 3D view
(b) 2D projection views
Figure 1. The geometric configuration of a dual-head PET scanner and the definition of the coordinate system. The detectors are parallel to the y − z plane and their normals are along the x axis. The entire space enclosed by the detectors is the scanner’s active imaging volume. For reconstruction, the volume is divided into non-overlapping cubic voxels that are assumed to contain uniform radioactivity concentrations within. To anticipate the directional dependence of the effects of DOI blurring, we will designate an image plane parallel to the y − z plane (parallel to the detectors) as a p-plane and a plane parallel to the x − z plane (vertical to the detectors) a v-plane.
to as crystals below). The detection planes are parallel to the y − z plane, located at positions x = ±R (see Fig. 1). Denote by ~cu = (cu,y , cu,z ) and ~cl = (cl,y , cl,z ) the indices that identify a crystal in the upper and lower detection planes, respectively. We align the coordinate system with the detectors such that the centers of the crystals given by ~cu =~cl = (0, 0) are located at y = z = 0. A detection channel of the system, commonly referred to as an LOR in the literature, is formed by a pair of crystals, one from each detection plane. Hence, a detection channel, or an LOR, is identified by a paired indices given by (~cu ,~cl ). In this work, the entire space enclosed by the detector planes is the scanner’s active imaging volume. To anticipate the directional dependence of the effects of DOI blurring, we will designate an image plane parallel to the y − z plane (parallel to the detectors) as a p-plane and a plane parallel to the x − z plane (vertical to the detectors) a v-plane. A. Symmetry properties Consider a point ~r = (rx , ry , rz ) inside the scanner’s active imaging volume and denote by h(~cu ,~cl ;~r) the sensitivity of the LOR (~cu ,~cl ) to ~r. Evidently, the system is invariant with respect to moving a distance equal to an integral multiple of the crystal size along the y- or z-axis; hence, we have h(~cu ,~cl ;~r) = h(~cu + ~m,~cl + ~m;~r + ~m0 ),
(1)
where ~m = (my , mz ), my , mz ∈ Z, and ~m0 = (0, my , mz ). Similarly, the system is invariant under the exchange of the y and z axes. Hence, denoting by Syz the operator that swaps the y and z components of its operand (i.e., Syz~c = (cz , cy ) and Syz~r = (rx , rz , ry )), we have h(~cu ,~cl ;~r) = h(Syz~cu , Syz~cl ; Syz~r).
(2)
Let Rx , Ry , and Rz denote the operators that negate the x, y, and z components of its operand, respectively (e.g., Rx~r = (−rx , ry , rz ) and so on). The invariance of the system under reflection with respect to the coordinate axes then implies h(~cu ,~cl ;~r) = h(~cl ,~cu ; Rx~r)
(3)
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and h(~cu ,~cl ;~r) = h(Ra~cu , Ra~cl ; Ra~r), a = y, z.
(4)
The above point symmetries will allow us to derive equivalent symmetries for cubic image voxels when they are appropriately sized and positioned. We propose to partition the scanner’s imaging volume into non-overlapping voxels having the size dx × d × d mm3 , where d = N −1 and N is an even integer.1 Let ~i = (ix , iy , iz ), ix , iy , iz ∈ Z, denote the index vector identifying an image voxel and V (~i) denote its volume. We will position the voxelized image volume such that the center of the voxel ~i = (0, 0, 0) is at the origin of the coordinate system. Denote by h(~cu ,~cl ;~i) the sensitivity of the detection channel (~cu ,~cl ) to voxel ~i. By definition, we have h(~cu ,~cl ;~i) =
Z V (~i)
d 3~r h(~cu ,~cl ;~r).
(5)
Applying Eqs. (1)-(4) to Eq. (5), we obtain h(~cu ,~cl ;~i) = h(~cu + ~m,~cl + ~m;~i + N · ~m0 ) = h(Syz~cu , Syz~cl ; Syz~i)
(a)
(6) (7)
= h(~cu ,~cl ; Rx~i)
(8)
= h(Ry~cu , Ry~cl ; Ry~i) = h(Rz~cu , Rz~cl ; Rz~i).
(9)
z
(10)
Below, we will show that these voxel symmetries will result in tremendous reduction in the computation and storage of the SRM.
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o B. Reduction in computation and storage for the SRM The SRM is the collection of all the voxel sensitivity functions h(~cu ,~cl ;~i). Consider an image p-plane defined given by a specific ix . The translation symmetry given by Eq. (6) indicates that one only needs to consider N × N voxels in this plane, regardless of the actual number of voxels in the plane. The reflection symmetries given by Eqs. (9) and (10) reduce this number to (N/2) × (N/2). Application of Eq. (7) further decreases it to N N +1 . (11) Nt = 4 2 Next, because of Eq. (8) one only needs to compute elements associated with non-negative ix . Figure 2(a) illustrates, for the 2D case, how the voxel symmetries obtained with N = 2 can be applied to relate elements of the SRM. The proposed µPET scanner has a crystal pitch of 2.4 mm and in the present work we will consider a detector spacing of 5 cm. We therefore conveniently choose a voxel size of 0.6×0.6×0.5 mm3 to be able to support an image resolution of ~1.2 mm. With this choice, we have N = 4 and 100 p-planes for reconstruction. As a result, we need to consider Nt = 3 voxels per p-plane and 51 p-planes in computing the SRM, yielding a total of 151 image voxels to consider. There are multiple ways to select the three basic voxels in a p-planes; one possibility is shown in Fig. 2(b). In comparison, the selected voxel size
(b) Figure 2. Symmetry of a dual-head PET scanner. (a) Consider the 2D case with N = 2. The black rectangles in (i)-(iv) identify a detection channel (~cu ,~cl ) and an image voxel ~i, hence an element of the SRM, h(~cu ,~cl ;~i). The element shown in (i) is equal to those shown in (ii)-(iv) after sequentially applying Eq. (6), Eq. (8) with Ry , and Eq. (6). Therefore, it can be shown that one only needs to consider the single image voxel shown in (iv) for a given image plane. (b) Consider the case 3D case with N = 4 and an arbitrary p-plane. The solid-line grid indicates the projection of the crystal grid on the p-plane and the dashed-line grid the image voxels. By applying voxel symmetries, only the three voxels indicated by the solid black lines need to be considered for a given p-plane for computing the SRM, regardless of the actual number of voxels in the plane.
generates about 12 million image voxels for the active imaging volume of the µPET scanner. Therefore, by exploiting voxel symmetry we are able to reduce the complexity in computing the SRM, and its storage, by a factor of ~8×104 ! It is also noted that that the SRM is sparse, i.e., for a given image voxel ~i, we have h(~cu ,~cl ;~i) = 0 for most detection elements (~cu ,~cl ). There is no need to store these zero matrix elements. III. M ETHODS A. The UofC dual-head µPET scanner
1 Note
that the crystal cross-section is 1 × 1 mm2 by assumption. Therefore, in the y and z directions the number of crystal is an even integral multiple of the number of the image voxel.
We have proposed, and are investigating, the use of two large-area detectors with a stationary, compact configuration
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B. Computation of the SRM by Monte-Carlo simulations
Figure 3. A µPET scanner consists of two HRRT detector heads (having a detection-sensitive surface of about 25×17 cm2 ) in a stationary, compact configuration (the setup shown has an ~5 cm spacing between the detector heads) to provide a large detection solid-angle and an extended imaging volume. Simulation results and actual measurements indicate that the scanner can yield a central sensitivity of ~30%.
for developing high-sensitivity µPET systems [40]–[42]. Our current prototype employs two HRRT (High Resolution Research Tomograph [12]) detector heads (see Fig. 3). These heads have a detection-sensitive surface of ~25×17 cm2 . Each head contains an 104×72 array of ~2.1×2.1×20 mm3 doublelayered LSO/LYSO phoswich crystals (the LSO layer is closer to the subject), with a crystal pitch of ~2.4 mm. Each crystal layer is 10 mm thick; therefore, the detector can generate binary DOI measurements having a depth resolution of ~10 mm. The two HRRT detectors are placed at a short distance apart (5 cm in the present work) to provide a large detection solid-angle for imaging small animals. The scanner remains stationary to allow the use of the entire space between the heads for imaging. We have shown by computer simulations and actual measurements that this µPET scanner can achieve a central sensitivity of ~30% [40,42]. In comparison, the central sensitivity of existing µPET scanners is typically below 5% [2,3]. The much higher sensitivity of the proposed scanner with respect to other systems can allow the use of lower imaging dose and shorter imaging time, and can lead to higher temporal resolution and quantitatively more accurate results. As already mentioned above, while our compact µPET scanner yields a high sensitivity, its compact geometry creates great concern for DOI blurring. It is important to note that, although the HRRT detector heads are capable of DOI measurements, the 10 mm thickness of the individual crystal layer is comparable with, or larger than, the thickness of the crystals found in most other µPET scanners. Therefore, the main function of using the DOI detectors in this scanner is not to eliminate DOI blurring, but to increase its detection efficiency. Specifically, we expect this scanner to yield an intrinsic spatial resolution no better than ~2.3 mm, which is the resolution reported for the non-compact HRRT system designed for imaging human brains [12]. By incorporating an accurate SRM in reconstruction, however, we expect the scanner to reach a resolution of ~1.2 mm, which is estimated by applying an empirical rule of thumb to the 2.4 mm crystal pitch of the HRRT detector [13]. Consequently, this compact µPET scanner will be able to provide a central sensitivity that is almost an order of magnitude higher than most existing µPET systems while providing a comparable image resolution.
Due to the huge dimension of the SRM, it is extremely challenging, if not impossible, to accurately compute the SRM of the proposed dual-head scanner by brute force. However, by applying the above-discussed voxel symmetries, the amount of computation can be drastically reduced, therefore making it practically acceptable to employ MC techniques for accurately modeling the physical processes and geometric effects in PET event detection. We employed the GATE package [45] for accurately modeling the dual-head configuration of the scanner, and the detection characteristics of the HRRT detector heads. GATE is a public-domain MC simulation package specifically developed for nuclear-medicine research; it is been widely used for PET research and its accuracy has been validated for many existing PET scanners. In the present work, we assumed a spacing of 5 cm between the detectors, an energy resolution of 20% at 511keV, an energy window of 350-650 keV, and a coincidence window of 6 ns. We employed a voxel size of 0.6×0.6×0.5 mm3 and needed to consider only 151 image voxels for computing the SRM (see Section II.C). We filled these voxels with a uniform activity of back-to-back 511keV γ-photons. The HRRT detector head has 104×72 crystals. However, in the simulation we extended the detector head to have 208×208 crystals to enable the use of voxel symmetries to populate the calculated results to all elements of the scanner’s SRM. We did not include subject scattering, positron range, and photon alinearity in simulation. Also, random events were excluded and scanner dead-time was ignored. Gamma-ray penetration of the crystals and intercrystal scattering, on the other hand, were accurately modeled by GATE. The HRRT detector heads have two crystal layers. In this paper, we considered only the front LSO layers; the method developed can be readily extended to deal with two layers. When using MC techniques for calculating the SRM, a large number of events need to be produced in order to reduce the statistical fluctuations inherent in the simulation. By comploying four Athlon x64 2.0 GHz PCs, we are able to generate about 1.5 billion true events for the front layers in 90 days to obtain reasonably good statistics for the resulting SRM. As discussed above in Section II.C, one would need to spend a factor of 8×104 more computation time in order to generate the SRM with similar statistical accuracy by brute force. It is difficult to image that one can successfully carry out such computations in practice, even with specialized computation hardware. The results obtained for each of the 151 image voxels were separately stored as sparse matrices to facilitate access in reconstruction. C. Image reconstruction algorithm, data generation, and numerical phantoms We implemented an OSEM algorithm [46] that can employ the MC-calculated SRM or the SRM calculated based on the idealized line-integral model. For the latter approach, the intersection lengths of a given LOR with the image voxels were computed by using the Siddon’s ray-tracing technique [47]. The LORs of the scanner were defined by connecting the
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centers of the front faces of two crystals. Given a numerical phantom, we employed the MC-calculated SRM to produce its noise-free projection data. Subject attenuation, scatter and randoms were not included. Poisson noise was then introduced to the noise-free dataset for generating noisy datasets having desired numbers of event. The simulated datasets were reconstructed by use of the OSEM algorithm employing the MCcalculated or line integral-based SRM. The reconstructions were terminated at certain numbers of iteration when the images were empirically observed to show little changes with iteration. For simplicity, below we will refer to the method employing the MC-calculated SRM (and the results generated) as the MC method (and MC results). Similarly, the Raytracing method (and Raytracing results) refer to the method employing the line integral-based SRM (and the results generated). Unless mentioned otherwise, the images reported below were obtained by use of 36 subsets and 2 iterations in the OSEM algorithm. The subsets had equal size and contained randomly selected LORs. For studying image resolution, we considered two numerical phantoms. The first phantom consisted of three identical 5×3 point-source arrays (see Fig. 5). Each source in the phantom occupied 2×2×2 voxels, corresponding to 1.2×1.2×1.0 mm3 in size. These arrays, covering an area of ~12×8 cm2 , were positoned at p-planes given by x = 0, 1 and 2 cm. Therefore, the phantom spanned a volume of about 12×8×2 cm3 . We placed one corner of the phantom at the center of the scanner for investigating the resolution properties in one quadrant of its ~25×17×5 cm3 active imaging volume. Because of scanner symmetry, the resolution properties over the entire imaging volume of the scanner were in fact investigated. We also constructed a numerical version of the microDerenzo phantom [48]. The phantom consisted of six groups of rod sources having diameters of 4.8, 4.0, 3.2, 2.4, 1.6, and 1.2 mm (see Fig. 7). Inside each group, the distance between two neighboring rod sources was twice the rod diameter. Our numerical phantom had a thickness of 10.5 mm. No background activity was introduced in the phantoms. The center of the phantom was aligned with the center of the scanner, with the rod sources being placed along the normal direction of the detectors. Three datasets were generated and studied: a noise-free dataset, a low-noise dataset and a highnoise dataset. The low- and high-noise datasets contained approximately 9×106 and 3.8×104 events, respectively. D. Real data We also applied the MC and Raytracing methods to real data for a spiral phantom acquired with our prototype system shown in Fig. 3. The spiral phantom was constructed by winding a ~1 mm-diameter plastic tube around a plastic screw and filling the tube with FDG solution. The turn-to-turn distance of spiral is about 3 mm. Only the events associated with front crystal layers were selected and a total of 1.3×109 events was collected. When computing the SRM by MC and Siddon’s ray-tracing methods, we have assumed that all crystals of the scanner have identical sensitivity response to gamm-ray interactions taking place inside them. This is not true in reality
and real data need to be normalized with respect to sensitivity variations. Let β (~cu ,~cl ) denote the intrinsic efficiency of the detection channel (~cu ,~cl ) and hr (~cu ,~cl ;~i) the actual detectorpair sensitivity function. We can write hr (~cu ,~cl ;~i) = β (~cu ,~cl )h(~cu ,~cl ;~i),
(12)
where h(~cu ,~cl ;~i) is the calculated detector-pair sensitivity function. To estimate the normalization factor β (~cu ,~cl ), we employed a rectangular tank that has an area larger than the HRRT detector head. The tank was placed at the center of the scanner and filled with a uniform FDG concentration of c. Data were acquired for more than eight hours to improve the statistics of the measurements. We ignored scatter and randoms in the acquired data; the expected measurement obtained was therefore equal to m(~cu ,~cl ) = cτ × α(~cu ,~cl ) × β (~cu ,~cl ) × ∑ f (~i)h(~cu ,~cl ;~i), (13) ~i
where τ is the acquisition time, α(~cu ,~cl ) is the attenuation by the tank estimated from the intersection length of the LOR with the tank, and f (~i) is the fractional volume of the image voxel ~i convered by the tank. Both α(~cu ,~ci ) and f (~i) can be readily calculated from the geometry. Consequently, we estimated the normalization factor by β (~cu ,~cl ) =
m(~cu ,~cl ) . cτ × α(~cu ,~cl ) × ∑~i f (~i)h(~cu ,~cl ;~i)
(14)
We applied the above procedure to the MC-calculated and lineintegral based SRMs to obtain their respective normalization factors. The spiral-phantom dataset, without correcting for scatter, randoms and subject attenuation, was normalized by these factors before applying the MC or Raytracing methods for reconstruction. IV. R ESULTS A. The SRM and parallax errors We refer to the sensitivity function of a detection element to all image voxels, i.e., the collection of {h(~cu ,~cl ;~i); ∀i}, as a detector-pair response function (DRF). Figure 4(a) displays six sample DRFs, generated as described in Section III.B, for image voxels on the central v-plane given by y = 0. These DRFs show clear linear structures, of which the directions indicate their associated LORs projected on the y = 0 plane. As expected, these linear structures are wider for more oblique LORs. Figure 4(b) shows horizontal profiles of these DRFs at x = 0 cm (which is midway between the two HRRT detectors). These profiles show noticeable statistical fluctuations at the tails for DRFs associated with oblique LORs, reflecting the statistical nature of MC computations. We observe that these profiles are symmetric and their full-widths-at-halfmaximum (FWHMs), from the vertical to the most oblique LORs, are 1.4 mm, 3.0 mm, 5.0 mm, 6.0 mm, 8.1 mm and 11.4 mm, respectively. Figure 4(c), on the other hand, shows the profiles at x = 2.45 cm (i.e., on the p-plane nearest to the upper detector). The profiles of the oblique LORs now become considerably asymmetric. Also, all profiles grow wider with respect to those obtained at x = 0 cm: from the vertical
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HRRT detector x 5 cm
z
27 cm (a)
Figure 5. Contour plots of the MC (top row) and Raytracing (bottom row) images obtained for the point-source phantom at three p-planes given by x = 0 cm (left column, labeled as Array I), x = 1 cm (middle column, labeled as Array II), and x = 2 cm (right column, labeled as Array III). The lower-left source in Array I is closest to the center of the scanner and the upper-right source in Array III is farthest. The MC result shows isotropic and uniform spatial resolution of ~1.2 mm on the p-planes. The Raytracing result still shows an isotropic resolution on the p-planes. The resolution is ~1.8 cm for the source closest to the scanner center and becomes ~1.2 mm for the source farthest from the scanner center.
(b)
(c) Figure 4. DOI blurring in the proposed dual-head µPET scanner. (a) Images showing the DRFs on the y = 0 plane obtained for six crystal pairs. The HRRT detectors have 104 crystals along the z axis, indexed from −52 to 51, and 72 crystals along the y axis, indexed from −36 and 35. The DRFs shown are for the crystal pairs given by (i) ~cu =~cl = (0, 0); (ii) ~cu = (0, −10), ~cl = (0, 10); (iii) ~cu = (0, −20), ~cu = (0, −20); (iv) ~cu = (0, −30), ~cl = (0, 30); (v) ~cu = (0, −40), cl = ~(0, 40); (vi) ~cu = (0, −50), ~cl = (0, 50). Evidently, these DRFs show linear structures indicating the directions of their respective LORs. Also, more oblique LORs are wider due to DOI blurirng. (b) Horizontal profiles of these DRFs along the line at x = 0 cm. These profiles are symmetric; their FWHMs are: (i) 1.4 mm, (ii) 3.0 mm, (iii) 5.0 mm, (iv) 6.0 mm, (v) 8.1 mm, and (vi) 11.4 mm. (c) Horizontal profiles of these DRFs along the line at x = 2.45 cm. For oblique LORs, the profiles become asymmetric, indicating that the DRFs are depth dependent. The FWHMs of these profiles are also wider than those obtained at x = 0 cm, becoming: (i) 2.3 mm, (ii) 4.0 mm, (iii) 8.3 mm, (iv) 9.7 mm, (v) 11.2 mm, and (vi) 14.8 mm. A more than 8-fold increase in the FWHM is observed from the vertical LOR to the most oblique LOR been examined.
to the most oblique LORs, the FWHMs increase to 2.3 mm, 4.0 mm, 8.3 mm, 9.7 mm, 11.2 mm, and 14.8 mm. Therefore, the MC-calculated SRM shows that the DRFs of the propsoed µPET scanner are not depth independent (i.e., they are variant along the x direction) and their FWHMs exhibit an eight-fold increase from the vertical LOR to the most oblique LOR that we have examined, illustrating severe DOI blurring due to the scanner’s compact geometry. It is noted that the proposed µPET scanner contains even more oblique LORs than those shown in Fig. 4. B. Image resolution The noise-free dataset generated for the point-source phantom was reconstructed by using 3 iterations in MC reconstruction and 10 iterations in Raytracing reconstruction, at which points the images were observed to show little change with iteration. Figure 5 displays the contour plots of the images obtained at the p-planes given by x = 0, 1, 2 cm. Comparing the MC and Raytracing images, we observe that the MC method generates better image resolution on the p-planes than does the Raytracing method. In addition, the MC result shows isotropic resolution on the p-planes and the resolution appears constant over the volume covered by the phantom (which, as explained in Section III.C, implies an isotropic and uniform resolution on the p-planes over the entire imaging volume of the scanner). In comparison, the Raytracing result also shows an isotropic resolution on the p-planes. However, the resolution appears to vary with position. Quantitatively, the MC method yields an isotropic spatial resolution of ~1.2 mm FWHM on the pplanes for all the sources in the phantom. At the center of the scanner (the lower-left source in Array I in Fig 5), the Raytracing method yields an isotropic ~1.8 mm FWHM on the p-planes. For the source farthest away from the scanner center (the upper-right source in Array III in Fig. 5), the FWHM becomes an isotropic 1.2 mm.
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(a)
(b)
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III
(c) Figure 6. Reconstructed images and intensity profiles obtained for the point-source phantom. (a) Images obtained on the y = 0 v-plane by the MC method. (b) Images obtained on the y = 0 v-plane by the Raytracing method. The lower-left source in the array is closest to the scanner center. Evidently, the Raytracing image shows substantial resolution degradation along the x direction. (c) The intensity profiles along the x direction through the centers of the three leftmost sources shown in (a) and (b). The MC profile shows a good agreement with the truth. The Raytracing profile, on the other hand, shows much degraded resolutions. Also, the profile becomes asymmetric when moving away from the center.
Figures 6(a) and 6(b) show the resulting MC and Raytracing images obtained for the point-source phantom on the central vplane given by y = 0. While the MC result still shows good and uniform spatial resolution for all the point sources, substantial degradation along the x direction (which is vertical to the detectors) can be observed on the Raytracing result. Figure 6(c) shows the intensity profiles along x, through the centers of the three leftmost point sources shown. The MC result again shows a uniform resolution of ~1.2 mm FWHM, equal to the p-plane resolution observed above. The Raytracing result, on the other hand, shows much degraded and non-uniform resolution: the FWHMs along the x direction are ~3.2, 3.5 and 5.0 mm for the sources located at the planes given by x = 0, 1, 2 cm, respectively. The shape of the profile also becomes increasingly more asymmetric when moving away from the central plane. In summary, our results indicate that, when incorporating an accurate SRM in reconstruction, an isotropic and uniform spatial resolution of ~1.2 mm FWHM over an extended imaging volume of ~25×17×5 cm3 can be achieved for the propsoed µPET scanner. Failing to incorporate the SRM in reconstruction, distinctive resolution properties are obtained along the directions parallel and vertical to the detectors. In directions parallel to the detectors, isotropic resolution is obtained and the spatial resolution is ~1.8 mm FWHM at the center of the image volume (which is about 50% worse than the MC result) and becomes ~1.2 mm FWHM at the edge of the imaging volume. In the direction vertical to the detectors, the resolution is substantially worse: it is ~3.2 mm FWHM at the center of the imaging volume (which is 190%
Figure 7. Images obtained for the micro-Derenzo phantom from simulated noise-free data. The axis of the phantom is normal to the detectors of the scanner. From left to right, the three columns show, respectively, the MC images, the true images, and Raytracing images. From top to bottom, the images obtained at the p-planes given by x = 0, 2.5, and 5 mm are shown.
worse than the MC result) and degrades to ~5.0 mm at the edge of the imaging volume. It is interesting, and important, to note that, unlike ring-based systems, we observe that DOI blurring causes resolution degradation at the center of a dualhead scanner as well. Hence, it is not possible to minimize the effects of DOI blurring by restricting the TFOV of a dual-head scanner. C. Micro-Derenzo Phantom Figure 7 shows the results obtained from the noise-free dataset generated for the numerical micro-Derenzo phantom by using 50 (36 subsets) and 10 iterations in the MC and Raytracing methods, respectively. Both the MC and Raytracing images can resolve the smallest 1.2 mm-diameter rod sources (2.4 mm center-to-center spacing) in the phantom. Visually, however, the MC result agrees well with the truth, and it shows better resolution and contrast than does the Raytracing result. Figure 8 compares the intensity profiles of the MC and Raytracing images. Again, the MC result is quantitatively more accurate than the Raytracing result. Specifically, although Raytracing result shows distinct peaks corresponding to the rod sources, non-zero intensity is observed between rods, reflecting the effect of finite resolution. The MC result, in comparison, shows considerably lower intensity between rods, especially for smaller rod sources. Figures 8(c) and 8(d) show intensity profiles along the x direction. Again, the Raytracing result shows inferior resolution to the MC result along the x direction. Figure 9 and 10 shows results obtained from the two simulated noisy datasets of the numerical micro-Derenzo phantom. For the low-noise dataset, the smallest rods cannot be resolved in the Raytracing result but remains visible in the MC result. For the high-noise dataset, the Raytracing method cannot resolve 1.2 mm-, 1.6 mm-, and 2.4 mm-diameter rods. In comparison, MC method can clearly resolve the 1.6 mm-
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Figure 8. Sample intensity profiles obtained for the micro-Derenzo phantom from simulated noise-free data. Graphs (a) and (b) show the intensity profiles on the central p-plane images; the insert image shows the location where the profiles are obtained. Graphs (c) and (d) show the intensity profiles along the x-axis, through the center of the 1.2 mm-diameter rod source closest to the center of the phantom (c), and of the 4.8 mm-diameter rod source closest to the center of the phantom (d). The MC and Raytracing images are normalized to have the same total image intensity with the true image.
and 2.4 mm-diameter rods and somewhat resolve the 1.2 mm rods. Figures 10 compares intensity profiles obtained by the MC and Raytracing methods. Our results therefore suggest that, when working with noisy datasets, incorporation of an accurate SRM in reconstruction can actually improve the noise properties of the images, possibly by more accurately locating the events to their actual positions. D. Real data Figure 11 shows the maximum-intensity-projection (MIP) images of the images obtained for the spiral phantom with 8 and 10 iterations of MC and Raytracing reconstructions. Subjectively, the MC result shows considerably improved image resolution and contrast than does the Raytracing result. Since that the turn-to-turn spacing of the spiral is about 3 mm, the clear visualization of the spiral in the MC MIP image is consistent with the ~1.2 mm spatial resolution estimated from our simulation study discussed above in Section IV.B. It is noted that no corrections for attenuation, scatter and randoms have been applied to the real data. Also, at present the MCcalculated SRM has not been validated with measurements. Finally, no scanner and subject rotations are employed in imaging (the same stationary configuration is assumed in
above simulation studies) and hence the acquired (and simulated) data are theoretically incomplete. However, there are no noticeable image artifacts in both MC and Raytracing results, suggesting that the degree of data incompleteness is not significant with the proposed compact geometry. V. C ONCLUSIONS AND D ISCUSSION In developing high-resolution PET systems, a significant challenge is the presence of DOI blurring. This challenge becomes most critical if high sensitivity is also required, such as in the case of developing a high-sensitivity µPET scanner. We have proposed to employ two large-area detectors and adopt a stationary, compact geometry for building a µPET scanner that can provide a high sensitivity (our prototype can yield a central sensitivity of ~30%) and an extended imaging volume (up to 25×17×5 cm3 with the 5 cm detector spacing). Although the design successfully leads to high sensitivity, its compact geometry also produces substantial DOI blurring. Consequently, the ability to eliminate DOI blurring in reconstruction is an important requirement for the proposed scanner design to succeed. One key to correcting DOI blurring in reconstruction is to obtain an accurate SRM for the scanner, which is an overwhelming task due to the huge dimension
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Figure 9. Images obtained for the micro-Derenzo phatnom from simulated low-noise (left graph) and high-noise (right graph) datasets. The axis of the phantom is normal to the detectors of the scanner. From left to right, the two columns in each graph show the images obtained by using the MC (left) and Raytracing (right) methods, respectively. From top to bottom, the images shown are obtained at the p-planes given by x = 0, 2.5 and 5 mm.
of the SRM. In this work, we have accomplished this task by strategically choosing cubic image voxels having a proper size to create substantial symmetries for the SRM. As a result, we are able to reduce the complexity in computing the SRM, and its storage, by a factor of ~8×104 , making it practically acceptable to employ Monte-Carlo techniques to accurately model the response of the scanner. We have calculated the SRM for the our dual-head scanner and employed it to investigate its DOI blurring. Our results indicate that, if DOI blurring is not corrected for our prototypical scanner generates a spatial resolution of 1.2-1.8 mm FWHM in the directions parallel to the detectors, and a resolution of 3.2-5.0 mm FWHM in the direction vertical to the detectors. In contrast, after incorporating the SRM in reconstruction to correct for DOI blurring the resulting images show a uniform and isotropic spatial resolution of ~1.2 mm FWHM over the extended imaging volume of the scanner. Consequently, the proposed µPET scanner is able to provide both high resolution and high sensitivity. Our results with simulated noisy datasets also indicate that the image noise properties are improved when incorporating the SRM in reconstruction. We have also applied the calculated SRM for reconstructing a real dataset. Our initial result with the real data shows that the image resolution and contrast are improved after employing the calculated SRM in reconstruction. In addition, the apparent resolution of the resulting image is consistent with the ~1.2 mm FWHM resolution estimated from the simulation studies. We note that the symmetry properties created for the SRM may also be exploited for increasing the computation speed of reconstruction. Potentially, the symmetries can allow one to devise algorithms that minimize the number of slow I/O access to the SRM elements and image voxels when computing the forward and backward projections. The symmetries may also allow fast computations of the forward and backward projections by computing them in the Fourier space. In the present
work, we have not validated the accuracy of the calculated SRMs with measurements. Also, we need to conduct more comprehensive evaluations of the image noise properties when employing the SRM in reconstruction, and to investigate the effects of errors in the calculated SRM to images. Finally, the normalization method for real data discussed above in Section III.D is far from optimized. Specifically, the normalization data are contaminated with noise, scatter and randoms. These contaminations can introduce errors to the resulting normalization factors, and hence to the resulting images. In addition, the model given by Eq. (12) may not be accurate. The developments of normalization sources/phantoms and acquisition schemes to reduce noise, scatter and randoms in the normalization data, of more accurate normalization models, and of more accurate methods to estimate the normalization factors, are themselves important research topics. In-depth treatments of these issues mentioned above are beyond the scope of this paper; they will be addressed in future by a series of studies. R EFERENCES [1] Y.-C. Tai, A. F. Chatziioannou, Y. Yang, R. W. Silverman, K. Meadors, S. Siegel, D. F. Newport, J. R. Stickel, and S. R. Cherry, “MicroPET II: design, development and initial performance of an improved microPET scanner for small-animal imaging,” Physics in Medicine and Biology, vol. 48, no. 11, pp. 1519–1537, 2003. [2] R. Lecomte, “Technology challenges in small animal PET imaging,” Nuclear Instruments and Methods in Physics A, vol. 527, pp. 157–165, 2004. [3] Y. Tai and R. Laforest, “Instrumentation aspects of animal PET,” Annu Rev Biomed Eng, vol. 7, pp. 255–285, 2005. [4] Y.-C. Tai, A. Ruangma, D. Rowland, S. Siegel, D. F. Newport, P. L. Chow, and R. Laforest, “Performance evaluation of the microPET focus: A third-generation microPET scanner dedicated to animal imaging,” J. Nucl. Med., vol. 46, pp. 455–463, 2005. [5] R. Laforest, D. Longford, S. Siegel, D. F. Newport, and J. Yap, “Performance evaluation of the microPET FOCUS-F120,” IEEE Trans. Nucl. Sci., vol. 54, pp. 42–49, 2007. [6] M. C. Huisman, S. Reder, A. W. Weber, S. I. Ziegler, and M. Schwaiger, “Performance evaluation of the Philips MOSAIC small animal PET scanner,” Eur. J. Nucl. Med. Mol. Imaging., vol. 34, pp. 532–540, 2007. [7] J. Seidel, J. J. Vaquero, and M. Green, “Resolution uniformity and sensitivity of the NIH ATLAS small animal PET scanner: Comparison to simulated LSO scanners without depth-of-interaction capability,” IEEE Trans. Nucl. Sci., vol. 50, pp. 1347–1350, 2003. [8] N. Hagiwara, T. Obi, T. Yamaya, M. Yamaguchi, N. Ohyama, K. Kitamura, H. Haneishi, E. Yoshida, N. Inadama, and H. Murayama, “Threedimensional image reconstruction for JPET-D4 using suitable histogramming method to DOI detector,” in Conference Record of the 2004 IEEE Nuclear Science Symposium and Medical Imaging Conference, pp. 3822–3825, 2004. [9] J. Missimer, Z. Madi, M. Honer, C. Keller, A. Schubiger, and S.M. Ametamey, “Performance evaluation of the 16-module quadHIDAC small animal PET scanner,” PHys. Med. Biol., vol. 49, pp. 2069–2081, 2004. [10] K. P. Schafers, A. J. Reader, M. Kriens, C. Knoess, O. Schober, and M. Schafers, “Performance evaluation of the 32-module quadHIDAC small-animal PET scanner,” J. Nucl. Med., vol. 46, pp. 996–1004, 2005. [11] Y. Wang, J. Seidel, B. M. Tsui, J. J. Vaquero, and M. G. Pomper, “Performance evaluation of the GE healthcare eXplore VISTA dualring small-animal PET scanner,” J. Nucl. Med., vol. 47, pp. 1891–1900, 2006. [12] H. W. de Jong, F. H. van Velden, R. W. Kloet, F. L. Buijs, R. Boellaard, and A. A. Lammertsma, “Performance evaluation of the ECAT HRRT: an LSO-LYSO double layer high resolution, high sensitivity scanner,” Phys. Med. Biol., vol. 52, pp. 1505–1526, 2007. [13] S. R. Cherry and J. Sorenson, Physics in Nuclear Medicine. Elsevier Science, 2003. [14] M. Wernick and J. Aarsvold, eds., Emission Tomography: The Fundamentals of PET and SPECT. Elsevier Academic Press, 2004.
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Figure 10. Sample intensity profiles obtained for the micro-Derenzo phantom from simulated noisy datasets. Graphs (a) and (b) show the intensity profiles on the central p-plane images, obtained from the low-noise dataset. The insert image shows the location where the profiles are obtained. Similarly, graphs (c) and (d) show the intensity profiles on the central p-plane images, obtained from the high-noise dataset. The MC method improves the image noise properties, especially when applied to the high-noise dataset. These images are normalized to have the same total image intensity.
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(b) Figure 11. Results obtained from real data acquired for the spiral phantom. (a) The maximum-intensity projection (MIP) images obtained by the Raytracing method (left) and the MC method (right). The axis of the phantom is placed parallel to the detectors of the scanner. (b) Image intensity profiles through the center of the MIP images shown in (a). Subjectively, the MC result shows better image resolution and contrast than does the Raytracing result.
