Exploring the Potential of a Cherenkov TOF PET Scanner: A Simulation Study

The detection of annihilation photons in positron emission tomography (PET) is based on scintillation light detection, but an interesting alternative is detection based on Cherenkov photons. Dense Cherenkov radiators provide an opportunity for high gamma detection efficiency—due to their high stopping power and photofraction—and excellent coincidence time resolution (CTR). However, because only a few tens of Cherenkov photons follow a gamma interaction in the radiator, the detection efficiency, and the energy resolution of a pure Cherenkov detector are an issue. This work explores gamma detection efficiency and CTR of PbF2-based detectors with different surface treatments and photodetectors covering one, two, or all crystal faces. Following the detector simulation analysis, we investigate the potential performance of a full-size Cherenkov PET scanner and quantitatively compare image quality with a commercial clinical PET scanner. We demonstrate that even though pure Cherenkov scanners have basically no energy resolution, the scatter fraction of around 50% is not prohibitively large, and images comparable to the state-of-the-art clinical PET scanner can be achieved due to improved efficiency and CTR attainable with PbF2.

, [5], [6] in the development of PET detectors invests a large effort to improve the system signal-to-noise ratio (SNR) since low SNR is regarded as one of the biggest technical limitations of PET imaging today [2].

A. Cherenkov Light and PbF 2
One of the ways to improve SNR is to increase the detector's efficiency for converting an incident 511-keV gamma photon into a detected event. The fraction of the incident gamma rays that interact in the detector is determined by the linear attenuation coefficient of the detector material. In this work, we study PbF 2 as a potentially excellent material for stopping and detecting gammas due to its high density and high effective atomic number resulting in photofraction and attenuation coefficient higher than that of L(Y)SO-the most widely used scintillator in PET (Table I). PbF 2 is a pure Cherenkov radiator. Cherenkov radiation is produced when a charged particle, such as an electron in a dielectric medium, moves faster than the phase velocity of light in that same medium [3]. An important consequence of having a pure Cherenkov radiator as a gamma detector in PET is that the overall number of produced photons is small (a few tens), which necessitates a photodetector with very high detection efficiency. Furthermore, the Cherenkov photons are produced promptly-radiated at the timescale of several picoseconds. This makes Cherenkov radiation a very attractive mechanism to be exploited for fast timing application, as there is a negligible contribution of the emission process to the overall time resolution of the detector.

B. Time-of-Flight
Using time-of-flight (TOF) information is another way to improve SNR as precise measurement of the difference in the arrival times of the two annihilation photons helps localize the emission point along the line-of-response (LOR), This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ leading to reduced noise correlations in the reconstructed image [7], [8]. The best current whole-body PET coincidence time resolution (CTR) is about 200-ps FWHM, achieved by the Siemens Biograph Vision scanner [9]. A CTR of 58-ps FWHM was recently reported using a pair of small LSO:Ce:Ca crystals (2 × 2 × 3 mm 3 ) coupled to silicon photomultipliers (SiPMs) [10]. This, presently the best CTR among the scintillation crystals, was obtained using unique high-power readout electronics, which is not practical for clinical PET systems. However, very recent work done by Krake et al. [11] demonstrated that it is possible to find low-power frontend solutions for the highest timing performance in TOF-PET, with a minimum power consumption of 17-mW per channel.
The use of Cherenkov light to detect gamma rays in PET was first discussed by Ooba et al. [12], where an improvement in the time resolution was proposed by using Cherenkov light produced in a silica aerogel with a refractive index of 1.2. Unfortunately, such a radiator would have a very low gamma detection efficiency due to its low density and low refractive index resulting in low Cherenkov light yield. PbF 2 was first experimentally studied by Korpar et al. [13], where a 95-ps FWHM CTR was measured for 15-mm long crystals coupled to a microchannel plate photomultiplier (MCP-PMT) as the light sensor. Recently, a Cherenkov-radiator-integrated micro-channel plate photomultiplier tube (CRI-MCP-PMT), where there are no optical boundaries between the radiator and photocathode, achieved an outstanding CTR of around 30-ps FWHM [14] and direct (reconstruction-free) positron emission imaging was demonstrated using these fast detectors [15]. However, to achieve such resolution, strong cuts had to be made in the timing pick-off threshold and pulse area, meaning only a small fraction of events was used, and the authors acknowledge that these detectors do not satisfy the detection efficiency requirement of clinical PET detectors. Although MCP-PMTs have excellent timing properties, they have several drawbacks like high cost, bulky size, and low quantum efficiency of the photocathode. High detection efficiency of the photodetector is essential for a pure Cherenkov PET detector since if none of the few produced Cherenkov photons is detected, the event is lost, resulting in reduced detector sensitivity.
SiPMs are promising photodetectors to be used with pure Cherenkov radiators. They have a higher photon detection efficiency (PDE) than MCP-PMTs, are compact, cost-effective, and can sense optical photons with a single photon detection time precision below 100 ps [16]. The use of SiPMs with PbF 2 crystals was studied by Dolenec et al. [17], where the best result obtained for TOF resolution was 297 ps. This value was shown to improve to 197-ps FWHM for 15-mm long crystal when selecting only single-cell hits but at the expense of lower efficiency [18]. By using a different detector chain (SiPM + electronics + digitization) a time resolution of 215-ps FWHM (142-ps FWHM) was recently obtained for 2 × 2 × 20 mm 3 (2 × 2 × 3 mm 3 ) sized PbF 2 crystals [6]. The above-mentioned Cherenkov detector studies also showed that the surface treatment of the crystals has an important influence on the detector's performance.
Effective (TOF-modified) noise equivalent count rate (NECR) and spatial resolution of a whole-body PbF 2 Cherenkov TOF-PET scanner was investigated by Alokhina et al. [19] using GATE/Geant4 simulations. Among the studied designs, they obtained the best effective NECR with 10-mm thick crystals with diffuse white coatings, which achieved TOF resolution of 180 ps.

C. Other PET Detectors Exploiting Cherenkov Light
Cherenkov light is also being studied in semiconductors such as TlBr and TlCl [20], [21]. Such detectors aim to combine excellent energy resolution from charge readout with timing measurement obtained by detecting Cherenkov light. There is also a renewed interest in BGO as a hybrid scintillator/Cherenkov radiator to be used as a cost-effective solution for TOF PET [22], [23]. While BGO was the scintillator of choice for use in whole-body PET scanners starting from the 1980s through the mid-2000s thanks to its high attenuation coefficient and photofraction (higher than L(Y)SO but lower than PbF 2 ), its low light output and slow scintillation signal were not good enough to perform TOF imaging. However, recent work has shown evidence that TOF imaging may be possible with BGO by detecting the prompt Cherenkov photons [24]. BGO is also attractive due to its much lower cost-about 1/3 that of Lu-based scintillators [25].

D. Total-Body PET
All the major commercial PET vendors use Lu-based crystals, and the cost of this crystal is the dominant component of the PET scanner cost. Crystal cost becomes especially important when considering long-axial total-body PET scanners [26]. First academic and commercial total-body PET scanners have been recently built, and they have demonstrated improved sensitivity, which can be used to image faster, better, or with lower doses [27], [28]. Their cost-to-benefit ratio will determine how well this technology will spread to clinical centers and whether their use will be broad or limited [29]. BGO and PbF 2 -which is even cheaper thanks to its low material cost (1/3 of BGO [30]) and lower melting point-might enable cost-effective total-body imaging.

E. Purpose of This Article
In our previous work, we showed that by using only true coincidences and the most likely position (MLP) reconstruction algorithm, images obtained with a simulated whole-body PbF 2 Cherenkov TOF-PET scanner are competitive to the images obtained with a whole-body LSO TOF-PET scanner [31]. This study expands our analysis of pure Cherenkov TOF PET scanners by using a clinically widely adopted TOF-OSEM algorithm and including all the coincidence events (true, scattered, and random) to evaluate the impact of scattered events on the image quality. It is accepted that in order for the detector to discriminate scattered photons from primary photons efficiently, the detector's energy resolution should be as high as possible [32]. The main research question we are trying to answer is: can a scanner based on PbF 2 -a pure Cherenkov radiator that has basically no energy resolutionprovide competitive image quality compared to the current state-of-the-art PET scanners. Keeping in mind that the negative impact of collecting more scattered events on the image quality can be compensated by collecting more true events, we also studied different Cherenkov detector designs with a potential for higher detection efficiency.

II. MATERIALS AND METHODS
We first performed a simulation study of different detector designs with SiPMs as photosensors, then selected a few of these designs to model whole-body Cherenkov PET systems. The performance of Cherenkov PET scanners was evaluated and compared to the reference scanner-our model of Siemens Biograph Vision clinical PET scanner. We assessed and compared count rates and image quality of the PET scanners following the national electrical manufacturers association (NEMA) NU 2-2018 standard. 1 Monte Carlo simulations were performed on the Slovenian national super-computing network (SLING) using GATE [33] version 8.1, a Geant4 [34] application for tomographic emission: a simulation toolkit for PET and SPECT medical imaging.

A. Detector Study
We studied detectors with 1, 2, and 6 sided crystal readouts (Table II). Detectors were based on the 3.2 × 3.2 × 20 mm 3 PbF 2 crystals, corresponding to the size of crystals used by the Siemens Biograph Vision scanner. Two detectors were placed back-to-back, and a point 511-keV gamma source was placed in-between as shown in Fig. 1. One million back-to-back gamma pairs were simulated for each detector design.
We used emstandard_opt4 physics list from Geant4, which uses accurate standard and low-energy models of electromagnetic interactions, making it suitable for medical physics applications [35]. The cut (production threshold) in the radiatorfor the charged particles producing Cherenkov photons-was set to 10 μm, and a maximum of one Cherenkov photon was produced in a step. Cherenkov photons were simulated in the range 250-1000 nm, and the simulated spectrum is shown in Fig. 2.
We considered two different surface treatments of the crystal in our simulations: 1) absorbing (black) and 2) reflective coating (reflector). The Geant4 unified model [36] was used The spectrum agrees well with the 1/λ 2 dependence, theoretically predicted by the Frank-Tamm equation [3].
to simulate the physical processes at the optical boundary. The black surface was modeled as smooth with a refractive index of 1.5 [37], and the optical photon was stopped if it was refracted and would thus exit the crystal. The reflective surface was modeled as slightly rough and air-coupled (refractive index-1) with a Lambertian (diffuse) reflector with 95% reflectivity. The surface roughness was modeled with sigmaalpha parameter, which describes the angular distribution (Gaussian) of the microfacets that make up the macro-surface, set to 5 • . In addition to the already mentioned references, the choice of optical parameters was also informed by Janecek et al. [38], [39] and Roncali and Cherry [40], as well as our own previous experimental work [13], [17].
The photodetectors were simulated with a PDE based on the Hamamatsu MPPC S14520 SiPM with a peak PDE of about 50% at around 450 nm (Fig. 2). The used SiPM also has good sensitivity in the near-ultraviolet part of the spectrumimportant for Cherenkov photons as they are more likely to be produced at lower wavelengths. The Cherenkov emission weighted PDE can be determined by weighting the photodetector's PDE with the Cherenkov emission spectrum, which gives a value of 22%. The effect of the optical interface between the crystal and the photodetector sensitive surface was considered by adding a quartz block-with a refractive index of 1.5-in-between, representing the window of a realistic photodetector. The window and the photodetector were modeled to be of negligible size (0.01-mm thick). The trigger was set to one photon, meaning that detecting one or more Cherenkov photons counted as a detection event. The noise of the SiPM (dark count rate and correlated noise) was not simulated at this point, as the primary focus of this work was the detrimental effects of scattering of gamma rays.
Coincidence detection efficiency was determined for the pair of detectors, as well as CTR. We studied CTR in an ideal and realistic case by setting the intrinsic single photon time resolution (SPTR) of the photodetector (e.g., SiPM) to 0 ps and 70 ps [10], respectively. In the ideal case, the time spread due to the interaction depth and different path lengths of Cherenkov photons in the PbF 2 crystal, was the primary Fig. 3. Scanner geometry simulated in GATE. The geometry of the simulated scanners corresponds to the geometry of the Siemens Biograph Vision PET/CT scanner. In addition to the simulated detector ring, lead shielding, carbon-fiber bed, and cylindrical NEMA NECR phantom can also be seen in the figure. limiting factor of the timing performance of the detector, as Cherenkov photons are produced promptly. Finally, we used a figure of merit (FOM) to compare and choose detector designs for a whole-body scanner simulation. FOM was defined as the ratio between the coincidence detection efficiency and CTR, detector parameters that directly impact the sensitivity of the scanner [2].

B. Image Quality Study
We selected three detector designs for a whole-body scanner study.
2) 2-sided readout with the photodetectors at the top and the bottom of the reflector wrapped crystal. 3) 6-sided crystal readout (all surfaces covered with photodetectors). 1-sided readout represents the standard, a 6-sided readout is a theoretical ideal, and a 2-sided readout is something that we view as still being practically feasible. The reference scanner and the Cherenkov scanners were modeled following the design of the Siemens Biograph Vision PET/CT scanner [41]. Details about the geometry can be seen in Fig. 3. We compared the performances of Cherenkov PET scanners among themselves as well as to the simulated results of the reference scanner based on the selected performance measures from the NEMA standard. We compared Siemens Biograph Vision PET/CT scanner's measured values from van Sluis et al. [41] with the reference scanner's values.
1) GATE Simulation: We modeled our reference scanner after Siemens Biograph Vision and we used information available online about its characteristics [41], [42]. In the GATE Monte Carlo simulation, the reference scanner's CTR was set to 214-ps FWHM, and a 4.7-ns coincidence time window was used for accepting events. The energy resolution was set to 10%, and a 435-585 keV energy window was used. The readout was simulated at the module level (Fig. 3)-signals inside the module were summed, and if the total deposited energy inside the module was within the specified energy window, the (singles) event was accepted, and its position within the module was determined with energy centroid policy (GATE TakeEnergyCentroid). In the case of multiple coincidences, all good coincidence pairs were included (GATE takeAllGoods policy). For the Cherenkov scanners, we used the same coincidence time window and coincidence forming and sorting policy as for the reference scanner, except there was no energy window applied (in this case, detection was based on optical photons instead of gammas-GATE opticaladded instead of adder). The SPTR of the photodetector was set to 70-ps FWHM in the whole-body scanner study. We did not take into account the intrinsic detector nor the acquisition dead-time. This can be justified by the fact that the dead-time of modern PET scanners is very low compared to the traditional PMTbased system due to minimal or no multiplexing. Therefore, any differences in count rate performance, at least in the range of activities used for clinical imaging, are primarily determined by the system sensitivity [25].
2) NECR and Scatter Fraction: NECR is a frequent metric used to compare the performance of PET systems. It characterizes the global SNR and acts as a surrogate for system-level image quality. Following the NEMA NU 2-2018 standard, the phantom used for this study consisted of a simulated line source of uniform activity inside of the 70-cm long polyethylene cylinder with a diameter of 20 cm. Because true, scatter, and random count rates were accurately known from the simulation, we determined NECR as where T, R, and S represent the true, random, and scatter coincidence count rates, respectively. These values can be affected by user-controlled parameters in an actual PET system [43].
In this work, we defined them in the following way: 1) true coincidences were considered those having both their singles initiated from the same annihilation event; 2) scatter coincidences were considered the true coincidences for which one of the two single photons (or both) interacted with the material before reaching the detector; and 3) random coincidences were those for which the coincidence event was formed by two gamma rays from different annihilation events. For every scanner, we collected approximately ∼10 6 prompt coincidences (at least 5 · 10 5 prompt counts are suggested by the NEMA standard) at each activity level. Scatter fraction (SF) was determined from the true and scatter counts as 3) Image Quality: We used a thorax-shaped phantom with hot spheres from the NEMA standard to determine percent contrast and percent background variability (BV). The background activity was set to 5.3 kBq/cm 3 (0.14 μCi/cm 3 ), and the ratio between the hot spheres and the background was 4:1. We used true sphere masks as regions of interest (ROIs) for contrast recovery calculations, and we also accounted for partial pixels. We positioned the phantom in the center of the scanner, and we simulated 4 min scans. We estimated the variance of contrast recovery coefficients and percent BV from 5 independently simulated 4 min scans.

4) Image Reconstruction:
We used open-source CASToR-Customizable and Advanced Software for Tomographic Reconstruction [44] for image reconstruction. Fig. 4. Graphical illustration of the method we used for scatter and random correction. The used notation is T-true, S-scatter, and R-random coincidences. Because this method assumes that the reconstruction method is linear, we performed a linearity test by comparing the reconstructed NEMA images using all coincidences (T, S, and R) with the images T + S + R obtained by separately reconstructing T and S and R images and then adding them. The comparison showed that OSEM is satisfactorily linear-only minor (negligible) differences in percent contrast (CRC-contrast recovery coefficient) and BV are observed for all the hot spheres.
Following the reconstruction parameters used in a clinical Siemens Biograph Vision scanner [41], the data was reconstructed using ordered subset expectation maximization (OSEM) 3D-iterative algorithm with eight iterations and five subsets, onto a 225 × 225 × 225 matrix with a voxel size of 1.6 × 1.6 × 1.6 mm 3 . Correction factors for normalization and attenuation correction were precomputed by CASToR and embedded in the data file used for reconstruction. We used the true attenuation map for computing attenuation correction factors.
The process of scatter and random correction is illustrated in Fig. 4. Our scatter and random correction method is based on the assumption that our reconstruction method is linear and that, we can separately reconstruct different types of coincidences (trues, scatters, and randoms). With this assumption, we obtained scatter and random corrected image by subtracting, from the image where we used all coincidences in the reconstruction, an image reconstructed from only scatter and random coincidences obtained from a separate simulation. Since OSEM [45] is not a linear algorithm, we evaluated the introduced error due to the linearity assumption. We compared the reconstructed NEMA images using all coincidences with the images obtained by separately reconstructing true and scatter and random images and then adding them. The comparison was made both for the reference and the Cherenkov scanner. Although the relative difference on the voxel level was up to a few percent, the impact on percent contrast for all the hot spheres was below 1%, and the impact on BV was about 1%, meaning that the OSEM algorithm behaves quite linearly, at least in the studied regime-using eight iterations and five subsets.
We also applied TOF information in the image reconstruction. By imaging a point 511-keV gamma source placed in the center of the scanner, we obtained the timing resolution of Cherenkov scanners from a histogram of differences in the detection times (t 2 − t 1 ) of the coincidence events. The histogram of detection time differences could not be well fitted with a single Gaussian, as was also observed by other groups performing timing measurements on PbF 2 [6], [46] and BGO [22], [24]. To account for "long tails," the distribution of time differences was modeled with a double Gaussian model (3) Fig. 5 shows that the time histograms are well fitted with a double Gaussian function, and the mixing ratio a 1 /a 2 is also shown.
The uncertainty of TOF measurements is usually modeled with a normalized Gaussian function. We modified the CASToR code to implement a double Gaussian TOF kernel used in the image reconstruction. We compared the resulting images with the images obtained with the standard single Gaussian TOF kernel, demonstrating better suitability of the double Gaussian TOF kernel.

A. Detector Study
We evaluated and compared the coincidence detection efficiency, CTR, and FOM of different detector designs. The results are shown in Table II. Black painted crystals achieve better CTRs but at a significantly lower coincidence detection efficiency than the crystals wrapped in a reflector, resulting in lower FOMs. Among the detectors with a 2-sided readout, the configuration with photodetectors at the top and the bottom of the crystal (Table II: 2-sided-top-bottom Cherenkov detector) resulted in the best combination of CTR and detection efficiency. We also simulated LSO crystals in the same backto-back crystal configuration with 10% energy resolution and a 435-585 keV energy window and obtained a coincidence detection efficiency of 17.6%. However, this value has to be compared to Cherenkov detectors with caution. The described setup does not test scatter rejection and does not account for intracrystal scatterings that can result in accepted events in a full-sized detector (scanner). Fig. 6 left shows the distribution of the emitted Cherenkov photons in PbF 2 crystal following a 511-keV gamma interaction. The direct photoelectric effect is followed by Fig. 5. Histogram of time differences (t 2 − t 1 ) of events forming the coincidences, obtained by imaging a point 511-keV gamma source placed in the center of the Cherenkov scanner. Note that the histogram of time differences for a whole-body scanner is slightly different from a single back-to-back detector pair, as annihilation gammas can hit the detectors at different angles, and there is also an impact of intracrystal scattering. Due to the long tails in the distribution, a double Gaussian (red) fits the histogram data better than a single Gaussian function (black). The FWHMs and their mixing ratios (a 1 /a 2 ) are also shown for each Cherenkov detector design. The SPTR of the photodetector was set to 70-ps FWHM.    energy of the incident gamma-ray is shown in Fig. 7. Both the detection efficiency and the average number of detected Cherenkov photons per registered event increase when using detectors with multisided readout. Gamma rays with lower energies are less likely to be detected, with no detection probability below the threshold of Cherenkov photon production, which in PbF 2 is about 100 keV. This represents a built-in intrinsic mechanism suppressing an important part of events, scattered in the patient's body.

B. Whole-Body Scanner Study
1) NECR and Scatter Fraction: NECR curves and SFs of the studied scanner designs are shown in Fig. 8. 1-sided Cherenkov scanner and the reference scanner have very similar NECR curves, while 6-sided Cherenkov scanner achieves the best NECR values. SFs of just below 50% are observed for the Cherenkov scanners, while our simulations resulted in an SF of 32.5% for the reference scanner.
2) Image Quality: Fig. 9 shows transverse images of the reconstructed phantom for different scanner designs, where a double Gaussian TOF kernel was used in the image reconstruction for Cherenkov scanners. On the other hand, using a standard single Gaussian TOF kernel with Cherenkov scanners resulted in poorer percent contrast and residual counts in the lung insert of the NEMA phantom. This effect can be seen in a profile through the selected reconstructed images in Fig. 10.  The relations between percent contrast and BV for the studied scanners are shown in Fig. 11. We calculated the relations from a series of images filtered by a Gaussian post-filter of different widths.

IV. DISCUSSION
In the detector study, we investigated Cherenkov detectors with multisided crystal readout and with two different surface treatments: 1) black and 2) reflective coating. Crystals with black surface treatments achieve better CTRs, but this does not outweigh the decreased coincidence detection efficiency compared to the crystals wrapped with a reflector (Table II). The average number of emitted photons in PbF 2 following an event that can be detected was 14.5 and 19.4 in the case of a direct photoelectric effect (Fig. 6). These simulated results agree well with the predicted intrinsic Cherenkov photon yield of 16.5 ± 3.3 estimated for PbF 2 from measurements [6].
Among the studied 2-sided crystal readouts, the design with photodetectors at the top and bottom achieved the best Fig. 11. Percent contrast versus BV for a 13 and 22 mm diameter hot sphere. Gaussian post-filters with different widths were used to vary the BV. The measured value of the Siemens Biograph Vision scanner is added for [41]. combination of detection efficiency and CTR. The results for a 2-sided-top-bottom detector design are comparable to that of the 6-sided crystal readout, which served as an idealized reference that probably has limited practical application. We did not use any time corrections on the trigger levels, which could improve CTRs, and would especially benefit the 2-sided-front-back configuration. Following the design of the reference detector, we modeled the 2-sided readouts with 20-mm long crystals, but there is much room for improvement of the 2-sided designs, as for example, the crystal could be split in two, which could improve CTR and also give us some depth-of-interaction (DOI) information.
We selected 1-sided, 2-sided-top-bottom, and 6-sided detector designs for a whole-body scanner study. We first performed an NECR comparison between the Cherenkov scanner and the reference Siemens Biograph Vision scanner (Fig. 8). Focusing only on the count rates, we did not modify the NECR with TOF information (effective NECR), which would benefit the Cherenkov scanners. The NECR curve of the Cherenkov scanner with a 1-sided readout is very similar to that of the reference scanner, even though it has a notably higher SF-47.3% to reference scanner's 32.5% SF (Fig. 8). The increased detection of scatter coincidences in the Cherenkov scanners is compensated by a higher detection efficiency of true coincidences leading to the same NECR values. The SF for the Cherenkov scanners is high because no energy window is used but would be even higher if there was no intrinsic suppression-scattered gammas with lower energy have a lower probability of being detected (Fig. 7). There is a good agreement between the simulated NECR values for the reference scanner and the values measured on the Siemens Biograph Vision scanner at lower (clinically relevant FDG) activities, but a little less so at higher activities, which was expected and can be attributed to the omission of the detector's dead-time in the simulation. 6-sided readout achieved the best NECR values, but the values of the 2-sided design were not lower by much.
The order of performance of the studied scanners predicted by the NECR analysis was repeated in the NEMA image quality study (Fig. 11). Cherenkov scanner with 1-sided readout had similar TOF performance and achieved very similar image quality as the reference scanner. To the best of our knowledge, this is the first time that it has been shown that despite having no energy resolution to be used for scatter suppression, Cherenkov PET can achieve image quality competitive to the current state of the art. By using multisided detector designs, even better image quality was achieved. For best image quality, a double Gaussian TOF kernel-as opposed to the conventional single Gaussian-had to be used because the time difference of the detected events in the Cherenkov detectors has long tails (Fig. 10). A recent study has also successfully implemented a double Gaussian TOF kernel for BGO detectors while also exploring Gaussian mixture models [47].
In this study, the photodetectors were modeled to be of negligible size, meaning that the detector with readouts at the sides (2-sided-top-bottom and 6-sided) had basically the same size compared to the detector with the readout at the back of the crystal. Although this is an ideal case, practically the SiPM can be made very thin, and the detectors with side readouts could be implemented in sparse designs, where not all axial rings are filled with detectors. A feasibility study was done by Zhang et al. [48], showing that removing 50% of detectors in the transverse or the axial direction did not have a major impact on the standard uptake values (SUVs) for a Philips Vereos scanner. A recent Monte Carlo study of the Siemens Biograph Vision PET with extended axial field-of-view (AFOV) using sparse detector module rings configuration [49] reports that sparse design allows extending the current limited AFOV of conventional PET systems by more than 100% at no additional detector material costs and without significantly affecting NEMA contrast recovery, system sensitivity, and transaxial spatial resolution. The concept of sparse design is therefore also viewed as an option of creating total-body PET systems with reduced cost [29].
The main limitation of our simulation study is not including the noise in the simulation-especially the dark count events of the SiPMs-which could affect the image quality when the trigger is set to only a single detected optical photon, as it was in this simulation study. The noise of the photodetectors will increase the number of random events, which would decrease the NECR of the scanner, resulting in more noisy reconstructed images. Noise could also impact image quality by degrading the CTR or the spatial resolution, especially in detector designs that involve multiplexing of signals across many SiPMs. The dark count rate (primary noise) can be greatly reduced by cooling the SiPMs, although a remaining question would be the impact of correlated noise (afterpulsing and optical crosstalk [50]). Besides cooling, the noise could also be tackled by raising the trigger level to, e.g., two or three photons. However, the impact of higher trigger levels on the detection efficiency would have to be evaluated, and it would most notably affect the detector with lower number of detected Cherenkov photons on average (Fig. 6).
Potential exists to still improve the PDE, which would also reduce the issue of raising the trigger levels. One way to achieve this is by using existing photodetectors in multisided readout detectors, as demonstrated here (Fig. 6). Detection efficiency would also be improved if the optical coupling between the crystal and the photodetector is improved. Optical boundaries exist between the radiator and the photodetector, and mismatch in refractive index can lead to light lossreduced light transfer efficiency (LTE)-due to the internal trapping inside the crystal. Different methods to improve LTE were demonstrated. Ota et al. [14] improved Cherenkov photon transmission to the photocathode by removing the optical boundaries-using CRI-MCP-PMT. Pots et al. [51] improved LTE from an inorganic BaF 2 scintillator by coupling the crystal, with UV-transparent optical grease, to the photodetector. Photonic crystals are another possible solution that was proposed and is being investigated to improve the LTE through photonic nano-structuring of the different surfaces of the crystal [52].
The effects of noise, different trigger levels, and modified reconstruction approaches will be the subject of future research.

V. CONCLUSION AND PROSPECTS
In this work, we have investigated pure Cherenkov PET detectors and the performance of scanners based on them. Cherenkov detectors are commonly viewed as fast detectors that can achieve excellent CTR but are not feasible for clinical PET detectors due to their limited gamma detection efficiency and the fact that they provide very little energy information. Our Monte Carlo simulations show that Cherenkov scanners can achieve image quality comparable if not better to that of the current state-of-the-art PET scanners-even though they have a larger SF-due to improved efficiency and CTR attainable with PbF 2 . A more general message that can be drawn from our results is that a PET detector's reduced or absent energy resolution can be compensated by its gamma detection efficiency and/or CTR. The detection efficiency and CTR of Cherenkov detectors are improved even further by considering multisided crystal readout, as we demonstrate in this study. Our simulations show a similar performance of detectors with 2-sided readout (top-bottom) to theoretically ideal detectors with 6-sided readout. Detector with side readouts could practically be realized in sparse scanner designs.
The low-cost Cherenkov detectors could become especially interesting for total-body scanners as their current high cost is limiting their dissemination in hospitals and research clinics. Our future research will focus on the effects of SiPM noise and experimental verification of detector performance.