Designing a compact high performance brain PET scanner—simulation study

We performed a simulation study to examine the effect of a compact geometry on image quality. We simulated four single-ring scanners with different diameters. The main parameters are listed in table 1 and the scanner geometries are illustrated in figure 2. Each detector is made of an array of $12\left(\text{transaxial}\right)\times 1\left(\text{axial}\right)$ LSO crystals. The size of each crystal is $2.5\times 2.5\times 15$ mm³. When modeling detector with depth-of-interaction (DOI) capability, the crystals were divided into multiple layers in the radial direction with equal length. The projection matrix $\boldsymbol{P}$ for each scanner was calculated using numerical integration of the solid angle effect with an accurate modeling of the photon penetration (Huesman et al 2000). To compare the performance of these four scanners, we computed the Cramér–Rao (CR) bound, which is the lower bound of the variance for an unbiased estimator (DeGroot and Schervish 2002). We used 2.5 mm square image pixels and an image matrix size of $101\times 101$. The activity distribution was a uniform cylinder with diameter of 225 mm.

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The Fisher information matrix was computed by

$$\begin{eqnarray}&&\boldsymbol{F}=\boldsymbol{P}^{{\prime}}\text{diag}\left[\frac{1}{{\bar{y}}}\right]\boldsymbol{P}\end{eqnarray} \tag{ 1 }$$

where $\overline{\boldsymbol{y}}=\boldsymbol{P}\boldsymbol{x}$ is the expectation of the PET data and $\boldsymbol{x}$ is the image of the uniform cylinder. The inverse of the Fisher information matrix, $\boldsymbol{F}^{{-1}}$, is the CR variance bound of an unbiased estimate of the image. Using the above configurations, the resulting Fisher information matrix for every scanner is invertible and thus provides a finite CR lower bound.

The variance of the total uptake in an ROI can be calculated by

$$\begin{eqnarray}&&{{V}_{\text{CR}}}=\boldsymbol{f}^{{\prime}}\boldsymbol{F}^{{-1}}\boldsymbol{f}\end{eqnarray} \tag{ 2 }$$

where $\boldsymbol{f}$ is the ROI indicator function, whose ith element f_i is equal to one if voxel i is inside the ROI, and zero otherwise (Qi and Huesman 2006). We computed the variance bound for a 2.5 mm ROI and 10 mm ROI at increasing radial positions.

Guided by the CR bound study, we proposed a helmet scanner design with DOI detectors as shown in figure 3, which is similar to the design in (Tashima et al 2013). The helmet scanner has three components: a top panel, six side rings with varying diameters, and a bottom panel. The detector block parameters and the diameter of the bottom ring of the helmet scanner are the same as those in the Siemens brain MR-PET brain scanner (Kolb et al 2012). Parameters of the helmet scanner are listed in table 2 in comparison with those of the brain PET and mCT scanner (Jakoby et al 2011). The number of blocks and the diameter in each side ring of the helmet scanner were chosen to approximate a hemisphere. The bottom panel contains $4\times 4$ detector blocks and the top panel contains 52 detector blocks. Distance between the bottom panel and the bottom ring is about 160 mm and the axial gap between the top panel and the top ring is 2.5 mm. GATE V6.2 (Jan et al 2004) was used for the simulations because of its flexibility in modeling irregular scanner geometries. The 'generic repeater' function in GATE was used to construct the helmet scanner. To cross validate our GATE simulation results, we also simulated the cylindrical brain PET and the helmet scanner with only side rings using SimSET (Harrison et al 2002).

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Table 2. Geometric parameters of the simulated helmet PET scanner.

	Helmet scanner	Cylindrical brain scanner	mCT
Crystal size (mm)	$2.5\times 2.5\times 20$	$2.5\times 2.5\times 20$	$4\times 4\times 20$
Number of crystals per block	$12\times 12$	$12\times 12$	$13\times 13$
Number of transaxial blocks	[38, 38, 36, 33, 29, 26]	32	48
Ring diameter (mm)	[376.5, 366.4, 346.4, 316.8, 282.7, 248]	376.5	849
Number of axial blocks	6	6	4
Axial gap (mm)	2.5	2.5	4
Axial FOV (mm)	190	190	218

A uniform water cylinder phantom with radius 12.25 cm and 19 cm long was first simulated to compare the photon detection efficiency among different scanners. The attenuation coefficient of the cylinder material was 0.0095 mm⁻¹. True and scattered events were simulated using both the GATE and SimSET. The overall sensitivity was calculated as the ratio between the number of detected true events and the total number of annihilations in the phantom. The spatial variation in sensitivity was also calculated for every voxel based on the source location from the root file in the GATE simulation.

In addition, we used a 3D Hoffman phantom (Hoffman et al 1990) to mimic an activity distribution in brain imaging and examined the effective sensitivity as a function of axial position. The activity ratio of the grey matter:white matter:cerebrospinal fluid (CSF) was set to $5:1:0$. The attenuation image in each slice was a uniform ellipse fitted to cover the emission activity with an attenuation coefficient of 0.0095 mm⁻¹. True events in each slice were collected based on the source location. The sensitivity in each slice was computed by dividing the number of detected true events originated in the slice by the total activity in the slice and the results were compared across different scanners. Three helmet configurations were evaluated to understand the contribution from each component of the helmet PET: helmet with only side detectors, helmet with side and top detectors, and helmet with side, top and bottom detectors.

Since sensitivity measures the overall coincident-photon detection efficiency, we also compared directly the variance in reconstructed images at matched spatial resolution. The variance and spatial resolution were estimated using the theoretical formula derived by Qi and Leahy (2000) as well as Monte Carlo reconstruction of multiple datasets. The theoretical formula allows us to study a wide range of reconstruction parameters, while Monte Carlo reconstructions provide validation for the theoretical predictions at selected points.

The reconstruction method was in the penalized likelihood framework with a quadratic penalty function:

$$\begin{eqnarray}&&\widehat{\boldsymbol{x}}\left(\,\boldsymbol{y}\right)=\text{arg}\underset{\boldsymbol{x}\geqslant 0}{{\max}}\,L\left(\,\boldsymbol{y}|\boldsymbol{x}\right)-\frac{1}{2}\beta \boldsymbol{x}^{{\prime}}\boldsymbol{R}\boldsymbol{x}.\end{eqnarray} \tag{ 3 }$$

where $L\left(\,\boldsymbol{y}|\boldsymbol{x}\right)$ is the Poisson log-likelihood function, β is the regularization parameter that controls the strength of the regularization, and $\boldsymbol{R}$ is a semi-positive definite matrix. Following the previous work (Fessler and Rogers 1996), the local impulse response at pixel j can be expressed by

$$\begin{eqnarray}&&{{l}^{j}}\left(\widehat{\boldsymbol{x}}\right)={{\left[\boldsymbol{F}+\beta \boldsymbol{R}\right]}^{-1}}\boldsymbol{F}\boldsymbol{e}_{{j}}\end{eqnarray} \tag{ 4 }$$

and the covariance matrix by

$$\begin{eqnarray}&&\text{Cov}\left(\widehat{\boldsymbol{x}}\right)={{\left[\boldsymbol{F}+\beta \boldsymbol{R}\right]}^{-1}}\boldsymbol{F}{{\left[\boldsymbol{F}+\beta \boldsymbol{R}\right]}^{-1}}.\end{eqnarray} \tag{ 5 }$$

We used the peak intensity of the local impulse response, which is referred to as the contrast recovery coefficient (CRC) of a point source, to measure the spatial resolution. Using locally shift-invariant approximation and fast Fourier transform, the CRC at pixel j and the pixel variance can be quickly evaluated by Qi and Leahy (2000)

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \text{cr}{{\text{c}}_{j}}\approx \frac{1}{N}\underset{i=0}{\overset{N-1}{\sum}}\,\frac{{{\lambda}_{i}}}{{{\lambda}_{i}}+\beta {{\mu}_{i}}} \end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\operatorname{var}}_{j}}\approx \frac{1}{N}\underset{i=0}{\overset{N-1}{\sum}}\,\frac{{{\lambda}_{i}}}{{{\left({{\lambda}_{i}}+\beta {{\mu}_{i}}\right)}^{2}}} \end{array}\end{eqnarray} \tag{ 7 }$$

where vectors $\boldsymbol{\lambda }$ and $\boldsymbol{\mu }$ are the Fourier transform of the image formed by the jth column of $\boldsymbol{F}$ and $\boldsymbol{R}$, respectively. Readers are referred to (Qi and Leahy 2000) for more details.

In the Monte Carlo validation process, a 3D phantom was created by replicating one slice of the Hoffman phantom (figure 4) in the axial direction. We used a cylindrical object to reduce the variability caused by the activity distribution when evaluating the performance in different axial slices. In order to perform the Monte Carlo reconstructions within a reasonable amount of time⁷, the voxel size was set to $5\times 5\times 10$ mm³ and the image matrix size was $49\times 49\times 19$. The projection matrix $\boldsymbol{P}$ was computed using a single-ray tracing method. While we would use smaller voxels and more sophisticated projection model in practice (see next section), we expect that the relative performance between different geometries remains similar. We evaluated the system performance with and without TOF. The simulated TOF resolution was 200 ps and the time bin width was 40 ps. The resulting time window function was a convolution between a Gaussian function (FWHM  =  200 ps) and a rectangular function (width  =  40 ps), which is given by

$$\begin{eqnarray}&&T(x)=\frac{1}{2}\left[\text{erf}\left(\frac{x+s/2}{\sqrt{2}\sigma}\right)-\text{erf}\left(\frac{x-s/2}{\sqrt{2}\sigma}\right)\right],\end{eqnarray} \tag{ 8 }$$

where x is the distance in mm from the voxel to the spatial center of the time bin, $s=40\times 0.15$ mm is the spatial width of the time bin, $\sigma =0.15\times 200/\left(2\sqrt{2\ln 2}\right)$ mm is the standard deviation of the Gaussian function, and $\text{erf}(\centerdot )$ denotes the error function.

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For data generation, a histogram was first created by forward projecting the phantom using the projection matrix, and then 200 independent and identically distributed (i.i.d) realizations were generated by introducing Poisson noise to the histogram. For CRC calculation, a single-voxel source with a contrast of 2 was added to the phantom at a radial offset of 5 cm from the center (see the hot spot in figure 4). Each data set was reconstructed independently using a maximum a posteriori-expectation maximization (MAP-EM) algorithm (De Pierro 1994) with 200 iterations and three different β values.

For image quality evaluation, GATE simulations were used to generate list mode data of the 3D Hoffman brain phantom and a mini Derenzo phantom. The tracer was Fluorine-18. Since the Hoffman phantom has only 19 slices, the image matrix size was set to $196\times 196\times 19$ with a voxel size of $1.25\times 1.25\times 10$ mm³ in both cases. List-mode reconstruction was used for the helmet reconstruction because of the large number of lines of response and lack of symmetry for storing a pre-computed projection matrix. Single-ray tracing method (Siddon 1985) is often preferred for on-the-fly calculations due to its fast speed and simple implementation, but it does not account for the parallax error caused by photon penetration. To model the detector response, image-domain point spread function (PSF) can be used in conjunction with the single-ray tracing method (Zhou and Qi 2011). However due to the lack of rotational symmetry of the helmet scanner, it requires measuring the PSF for a huge number of voxels, which is very difficult. Hence we adopted an on-the-fly multi-ray tracing method (Huesman et al 2000) with a crystal division of $2~\left(\text{transaxial}\right)\times 1~\left(\text{axial}\right)\times 6$ (radial) to model the solid angle and crystal penetration effect. The axial division was set to 1 because the image voxel size in the axial direction is larger than the detector crystal size. For DOI detectors, three crystal layers were simulated. The division for each crystal layer was $2\times 1\times 2$ to match that for the non-DOI detector. All list-mode data were reconstructed using an ordered subset expectation maximization (OSEM) algorithm with 10 subsets and 2 iterations.

Figure 5 compares the CR bound for the four scanners with different ring diameters. As expected, a more compact geometry can improve photon detection sensitivity and reduce variance of ROI quantification at the center of the field of view. However, it also results in higher variance near the edge of the scanner without DOI information and the effect is more pronounced for the smaller ROI (figure 5(a)). Note that when there is no DOI, the ROI variance at large radial offsets in the most compact geometry is almost the same for 2.5 mm ROI and 10 mm ROI because the neighboring pixels are highly correlated. Using crystals longer than 15 mm in a compact geometry increases the variance in the peripheral regions even further. This indicates that without DOI measurement, a compact geometry is not suitable for quantifying uptake in small peripheral regions (e.g. cortical surface). When the scanners are equipped with DOI detectors, a compact scanner can outperform a large scanner (figures 5(c) and (d)) and thus reducing the material cost and weight. Comparing the plots in figures 5(c) and (d), we can see that higher DOI resolution has a minimal effect on the largest scanner (scanner 4, R  =  288 mm), but is more critical for the most compact scanner (scanner 1, R  =  135 mm).

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For the cylinder phantom simulation, the sensitivity comparisons between the GATE and SimSET for true events are shown in table 3. The difference between the GATE and SimSET results are within 2–3%. The results show that the sensitivity of the full helmet scanner is about 4.2 times that of the cylindrical brain PET, and 5.4 times that of the mCT scanner. The helmet PET with only side rings has a sensitivity improvement of $73\%$ over the cylindrical brain PET. Comparing the brain PET with the mCT scanner, the brain PET has 30% higher sensitivity even though the diameter of the mCT scanner is about twice that of the brain PET. The improvement factor is less than two because the mCT has a longer axial FOV and smaller gaps in each ring.

Table 3. True coincidence sensitivity comparison.

Model	mCT scanner (%)	Cylinder brain PET (%)	Helmet (side only) (%)	Helmet (%)
SimSET	$(0.6692\pm 0.0002)$	$(0.8705\pm 0.0002)$	$(1.5072\pm 0.0002)$
GATE	$(0.6610\pm 0.0092)$	$(0.8554\pm 0.0105)$	$(1.4813\pm 0.0138)$	$(3.5830\pm 0.0214)$

Note: GATE results have greater uncertainty because fewer number of events were simulated.

Table 4 compares the scatter fractions for all simulated scanners. The values for the mCT and the cylindrical brain PET are very close to those reported in the literature, $33.2\pm 0.7$% for the mCT (Jakoby et al 2011) and 38% for the Siemens brain PET (Kolb et al 2012). The cylindrical brain PET and helmet with only side rings have higher scatter fractions than the mCT scanner because of their larger solid angle coverage in the axial direction. Adding the top and bottom panel reduces the scatter fraction in the helmet scanner because a large number of lines of response formed by the top panel with side detectors are short in length and thus have lower-than-average scatter fractions.

To visualize the spatial variant sensitivity, figure 6 shows a sagittal view of the sensitivity image for the cylinder brain PET and the helmet scanner. We can see that the helmet scanner substantially increases the sensitivity in the upper FOV; however, the largest improvements occur near the peripheral regions that may be less valuable in practice. To further examine the sensitivity improvement for brain imaging, we computed the sensitivity for each axial slice (19 slices in total) using the Hoffman phantom simulation data. The results are plotted in figure 7(a) for four different scanners: the cylindrical brain PET, helmet with only side detectors, helmet with side and top detectors, and the full helmet. The ratios between these scanners with respect to the cylindrical brain PET are plotted in figure 7(b). The results show that the maximum improvement is near the top FOV with the maximum sensitivity ratio between the full helmet and the cylindrical PET being 35, while the minimum improvement is near the lower middle FOV (planes 7–8) with a sensitivity ratio of 1.6. The bottom panel mainly improves the sensitivity in the lower FOV (planes 1–5), while the top panel mainly increases the sensitivity in the upper FOV (planes 13–19). Therefore, if we are interested only in the top part of the brain, the bottom panel is not needed; if we care about the lower part of the brain, such as the cerebellum and the hippocampus, then bottom panel can help to improve the sensitivity.

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Figures 8(a)–(c) show the theoretical predicted CRC-variance curves and the Monte Carlo results for three axial slices (top, center, and bottom). The curves are plotted on a log–log scale to show the variance reduction over a large range of CRC (a measure of spatial resolution). For all locations and all three β values, the Monte Carlo results match the theoretical predictions very well. The fact that the CRC-variance curves in each plot are almost parallel to each other in the log–log scale indicates that the variance reduction factor is nearly independent of CRC. Therefore, we used the theoretical results to calculate the variance reduction ratio (cylindrical brain PET/helmet PET) at a fixed CRC value of 0.2 and plotted the results in figure 8(d). We can see that the variance reduction is the greatest in the top planes and is the lowest near plane 8, which is consistent with the sensitivity ratio plot in figure 7. Adding 200 ps TOF information provides relatively uniform improvement across all axial slices. The very top planes (planes 16–19) have a slightly less TOF benefit because the LORs formed between the top panel and side detectors are shorter in length than those formed between the side detectors and TOF gain in each LOR is nearly proportional to the length of the LOR.

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For the hoffman phantom data, the total simulated activity was 1.6e9 annihilations, which resulted in 18 million detected true events for the cylindrical brain scanner and 46 million events for the helmet scanner. The count level is comparable to a 10 min scan of a patient with 5 mCi injection in terms of the noise equivalent counts. Figure 9 compares the reconstructed images using the single-ray tracing projector and multi-ray tracing projector for both the cylindrical brain PET and the helmet PET. The multi-ray tracing projector reduces image noise and reveals more structure details and the improvements are more pronounced for the cylindrical brain PET. Based on these results, the multi-ray tracing projector was used for all reconstructions hereinafter.

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Figures 10(a) and (b) compare three representative (top, middle and bottom) slices of the Hoffman phantom reconstructions for the cylindrical brain PET and the helmet PET using the multi-ray tracing projector. In general, the helmet PET produces images with higher contrast and lower noise than the cylindrical brain PET. The most obvious improvement is in the top planes which have far less noise. This is consistent with the sensitivity improvement discussed earlier. In addition, we investigated the effect of TOF on the image quality. Figures 10(c) and (d) show the reconstructed images of the helmet PET with 400 ps TOF resolution and 200 ps TOF resolution, respectively. We see that TOF information further improves the image resolution and reduces noise with the same number of OSEM iterations. Improving timing resolution from 400 ps to 200 ps also results in noticeable improvement in image quality, indicating better timing resolution is desirable for brain imaging.

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The number of detected true events for the cylindrical brain scanner was 3.3 million and for the helmet scanner was 10.1 million. The reconstructed Derenzo phantom images are shown in figure 11. We compared one center slice and one top slice between the cylindrical brain PET and helmet PET. For the center slice, the helmet PET has a similar spatial resolution to the cylindrical brain PET, but with lower noise. Neither scanner can resolve the smallest rods because of the parallax error at the radially off-center location. For the top slice, the helmet PET has much better resolution and one can resolve the smallest rods. The improved resolution in the top slice is partly because the LORs formed by the top panel and the side blocks do not suffer as much parallax error in the radial direction as the LORs formed by detectors in the side rings and as a result, the radial resolution in the top planes is improved. Figures 11(e)–(h) show the reconstructed images with 3-layer DOI. Vertical profiles through the smallest rods are plotted in figure 12 with the ground truth. The added DOI information improves the spatial resolution of both scanners. However, the images from cylindrical brain PET appear to be limited by noise and cannot take full advantage of the higher resolution offered by the DOI detectors. In comparison, the images from the helmet scanner are much clearer with far less noise. This also illustrates the importance of high sensitivity in achieving high-resolution PET imaging.

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