Time-of-Flight Requirements to Mitigate Blurring Induced by Annihilation Photon Acollinearity

Abstract

One of the limiting factors of spatial resolution in positron emission tomography (PET) imaging is annihilation photon acollinearity (APA). For whole-body PET scanners, APA induces a blur ranging from 1.7 to 2.2 mm FWHM. For long axial field-of-view (FOV) scanners, this range increases even more, depending on the maximum ring difference. It was previously shown that perfect time-of-flight (TOF) resolution sharpens the APA-induced blur by altering its expected Gaussian shape into a profile resembling a 1/r function, thereby reducing its contribution to spatial resolution loss. This suggests that the conventional theoretical limit of PET spatial resolution could be overcome if sufficient TOF resolution can be achieved. However, the requirements to achieve an observable gain in spatial resolution have yet to be explored. We propose an investigation of these requirements for whole-body and long axial FOV scanners, in terms of TOF resolution and count statistics. Using a fictive 81-cm diameter scanner with 2-mm wide detectors, we show that ultrafast TOF resolution—13 ps FWHM—enables an observable gain in spatial resolution for a range of count statistics. In addition, we show that lower TOF resolutions (i.e., higher TOF values of 27 or 67 ps) could mitigate APA for the oblique tubes of response of long axial FOV systems subjected to larger APA blurring. This last observation is of particular interest as it suggests that the nonstationary nature of spatial resolution in PET imaging can be further mitigated when such TOF precision is achieved.

I. Introduction

Annihilation photon acollinearity (APA) is one of the limiting factors of spatial resolution in positron emission tomography (PET) imaging [1]. It is predominantly relevant for whole-body and long axial field-of-view (FOV) systems since APA blurring is proportional to the length of the tubes of response (TORs) [2]. For example, APA results in a blurring of approximately 2 mm FWHM for an 80-cm diameter scanner. This effect becomes even more significant for oblique TORs in long axial FOV systems, which are defined as having an axial length between 1 and 2 m [3]. In recent years, interest in these systems has increased, mainly due to their enhanced sensitivity [4].

Recently, it was shown that time of flight (TOF) can mitigate the effect of detector size on spatial resolution [5]. In particular, TOF precision at least equivalent to the detector size enabled an observable gain in spatial resolution. However, the achievable gain also depends on the count statistics. It was also observed that ultrafast TOF—13 ps FWHM—changes the shape of the blurring induced by APA, enabling a gain in spatial resolution even without modeling APA in the reconstruction process [6]. However, part of this study used a PET simulation software that has recently been shown to incorrectly model APA [7]. Nevertheless, the study in [6] also demonstrated this behavior for perfect TOF resolution via analytical modeling, thus it is very likely that ultrafast TOF does indeed mitigate the blur induced by APA, but it remains to be seen whether the achievable gain is perceptible for PET imaging.

In this work, TOF resolution in the range of 13–67 ps, corresponding to spatial precision between 2 and 10 mm, will be considered. This is significantly better than the current TOF resolution of 214 ps (32 mm) achieved by clinical PET/CT scanners [8] or around 100 ps (15 mm) in the laboratory setting [9], [10]. New detector concepts are also being explored to push the timing performance, for instance by exploiting the prompt Cherenkov radiation [11] or the Purcell effect [12]. While efforts to reach ultrafast TOF resolution—i.e., about 10 ps FWHM [13]—are underway, there is still debate as to whether these efforts are justified for the best impact on PET imaging [14]. Thus, it is of interest to investigate the TOF resolution required to mitigate the blur induced by APA.

This study aims to provide insights in the requirements on TOF resolution and count statistics needed to mitigate the APA-induced blurring in the context of whole-body and long axial FOV systems. It relies on simulations of two types of phantom to achieve this aim. The first is a pair of custom Hot Spot phantoms that enables a quantitative analysis of spatial resolvability. The second is a Brain phantom which will provide a more accurate perception of the gain that could be achieved. Lastly, the difference in APA blurring between whole-body and long axial FOV systems mainly results from the addition of longer oblique TORs in the latter. Thus, the phantoms were studied in two regimes of APA blurring: the one expected in whole-body systems and another that mimics what is observed only in long axial FOV systems.

II. Materials and Methods

A. Simulation Configurations

A 2-D and a 3-D fictive whole-body scanner geometries were studied. The 2-D, i.e., single slice, scanner had an inner diameter of 81 cm and it was shaped as a regular 40-sided polygon. Its sides were subdivided uniformly into 32 detectors, each 2 mm wide. In order to avoid parallax effect and DOI timing bias [15], [16], the depth of the detectors was fixed at 0.1 mm. A fictitious high-density material was defined for the detectors to reduce simulation time by ensuring that incident annihilation photons were almost certainly detected. The detectors were 4-mm wide along the axial direction for the 2-D scanner to shorten the time required for the simulations. The 3-D scanner consisted of stacking axially 128 replicas of the 2-D scanner, except that the detectors were 2-mm wide along the axial direction, resulting in a 256-mm axial FOV. No decoding error was simulated in both geometries.

The data acquisition was simulated with the Geant4 Application for Tomographic Emission (GATE) software (v9.2) [17]. GATE implementation of APA in this version results in a significant underestimation of its effect on spatial resolution, because it was incorrectly represented as a Gaussian divided by the norm of its argument. A GATE simulation incorporating this implementation of APA produces a spatial blur of 0.4 mm for a PET scanner having an inner radius of 81 cm, whereas theory predicts a Gaussian blur with a 2.1-mm FWHM, assuming that APA has an FWHM of 0.58° [18]. Thus, we modified that part of the code, as described in [7], so that the blurring induced by APA had a Gaussian shape, as observed in experimental studies [19], [20], [21]. Back-to-back sources were used for the GATE simulations. This source type directly generates pairs of annihilation photons traveling in opposite directions, thus neglecting the effects of the positron emission and range of the PET emission process. It also allows the user to choose the APA FWHM. Only the photoelectric process was enabled for the annihilation photons, meaning that attenuation and scatter corrections can be neglected for the reconstructions.

Two values of APA FWHM were considered. The first value was 0.55°, which is a Gaussian approximation of APA in PET imaging [21], herein referred to as basicTOR. The second value was chosen as an artifice to simulate the oblique TORs that are present only in long axial FOV systems while using the unmodified 2-D and 3-D fictive scanners described previously. After all, simulating a long axial FOV system would mix the contributions of TORs available in a whole-body scanner with those unique to a long axial FOV scanner. Another artifice that could have been used would be to increase the scanner radius accordingly. However, this would increase the geometric sampling of the resulting scanner. For these reasons, we opted to increase the APA FWHM without modifying the scanner geometry. The APA FWHM was chosen such that it would be equivalent, in terms of blur induced in the image space of an 81 cm diameter scanner, to having an APA FWHM of 0.55° for TORs that are 102 cm long. The TORs length of 102 cm was chosen to correspond to our estimation of the median length of oblique TORs in the EXPLORER scanner [22]. The corresponding APA FWHM was 0.68° (details provided in Appendix A), herein referred to as obliqueTOR. The term TOR type will be used from now on when we are referring to the concept of basicTOR versus obliqueTOR.

For all simulations, only true events were kept for the reconstruction. Thus, the only sources of spatial blurring in these simulations were the size of the detectors and APA. The theoretical spatial resolution of a PET scanner can be approximated as

$$\alpha \sqrt{{\left(\frac{d}{2}\right)}^2+b^2+{\left(\mathit{\text{sin}}\left(\frac{\varphi }{2}\right)\times R\right)}^2+s^2}$$ (1)

with the reconstruction factor (α), the detector width (d), the detector decoding error (b), the scanner radius (R), the APA FWHM (ϕ), and the positron range (s) [2]. Hereafter, the value of (1) for a given PET scanner will be referred to as its theoretical spatial resolution. Note that s = 0 and b = 0 in this study. In [2], the contribution of APA is expressed as 0.0044 × R since it is assumed that ϕ = 0.5°. For basicTOR and obliqueTOR, the constant becomes 0.0048 and 0.0058, respectively. As a result, the theoretical spatial resolution is 2.2-mm FWHM for basicTOR and 2.6-mm FWHM for obliqueTOR, as summarized in Table I.

TABLE I.

Expected FWHM of the Sources of Spatial Blur Affecting the Simulations of the Two TOR Types Under Study

Source of blur	Detector size	APA	Theoretical
basicTOR	1 mm	1.9 mm	2.2 mm
obliqueTOR	1 mm	2.4 mm	2.6 mm

Four TOF resolutions were studied, which corresponded to having a precision of 2, 4, 10, and 20 mm FWHM. Their values in coincidence timing resolution (CTR) are provided in Table II.

TABLE II.

TOF Resolution Considered in This Study and the Size of TOF Bin Used for Each of Them

TOF resolution (FWHM)	2 mm	4 mm	10 mm	20 mm
CTR (FWHM)	13.3 ps	26.6 ps	66.6 ps	133.3 ps
TOF bin size	0.9 mm	2.0 mm	4.6 mm	11.3 mm

B. Image Reconstruction

All simulations with the 2-D scanner were reconstructed with an in-house implementation of the TOF-MLEM algorithm. The image domain was an 80 × 80 mm² square divided in 0.3125 mm wide pixels. Highly accurate representation of the detector response function in the system matrix was possible since the reconstructions were in 2-D [5]. The TOF system matrices were obtained by weighting its TOF-less counterpart with a discretized representation of the TOF information, which corresponds to [23, eq. (10b)]. The TOF response function was modeled as a 1-D Gaussian function varying only along the TORs, and the TOF domain was discretized into uniform bins. Thus, the coincidences of each simulation were stored as a histogram where each element corresponds to a unique combination of TOR and TOF discretization. The TOF bin size, provided in Table II, was chosen so that the discretization was fine enough to have a negligible impact on the accuracy of the TOF weights.

Highly accurate representation of the detector response function is far more complex to achieve for a 3-D scanner and it is not supported by our in-house software. Thus, all simulations with the 3-D scanner were reconstructed with version 3.1.1 of the CASToR software [24]. The simulations were saved as list mode and reconstructed using the MLEM algorithm, i.e., without ordered subsets, and without discretization of the TOF information. The multi-Siddon projector, with 25 samples, was chosen to model the geometric aspect of the system matrix while TOF was configured as continuous with its default options [23]. The image space was a volume of 23.36 × 23.36 × 18.96 cm³ discretized uniformly in 584 × 584 × 474 voxels.

For all the reconstructions, correction factors for attenuation, scatter, and randoms were not applied since they were neglected in the simulations. Note that due to the choice of detectors, the TORs are defined almost purely by their geometry and thus normalization factors, albeit naturally included in the MLEM algorithm, were not computed for either the 2-D and 3-D reconstructions. Lastly, while the simulations of this study were affected by both the detector geometry and APA, only the former was modeled in the system matrix for both the 2-D and 3-D reconstructions.

C. Evaluation of Resolving Power With the Hot Spot Phantoms

Two custom Hot Spot phantoms were created for this study. Their body was 60 mm diameter and 4 mm high cylinder. The diameters of the spots for the two phantoms covered the range of 1.0–3.0 mm with steps of 0.2 mm. The activity in the phantoms was distributed so that the spot-to-background ratio would be 4:1, thus making the resolvability task more realistic.

Since TOF can also mitigate the effect of detector size on spatial resolution and since the achievable gain depends on the count statistics [5], three levels of count statistics were investigated. The largest was our approximation of the expected count statistics for a “typical” acquisition on a hypothetical whole-body scanner that can achieve very fast TOF resolution, see Appendix B for details. This resulted in 7M coincidences, herein referred to as FullStat simulation, after taking into account the volume of the Hot Spot phantom. Acquisition with half and a 10^th of that count statistics, respectively, referred to as HalfStat and 1/10thStat, were also studied. Thus, the analysis of each custom Hot Spot phantom consisted of 24 simulations: basicTOR versus obliqueTOR, four TOF resolutions, and three count statistics. All these acquisitions were simulated with the 2-D scanner.

The spatial resolution achieved in an acquisition was defined as the width of the smallest spots for which its sector was deemed resolved for the best reconstruction. The best reconstruction was selected as the iteration which had the smallest mean-squared difference with the true activity distribution, which is readily available with GATE simulations. A sector was considered resolved if at least 90% of its valley-to-peak ratio (VPR) satisfied the Rayleigh resolution criterion (i.e., VPR ≤ 0.735), as proposed in [25]. VPRs were calculated from all adjacent pairs of spots to ensure that this characterization of resolvability took into account anisotropies in spatial resolution. The analysis was done with the script provided in [26] and described in [27], with the values of the peaks and valleys being defined as the mean of the central 30% of their size. Visual examples of the previously described methodology are provided in [26].

The Rayleigh resolution criterion does not take into account the existence of a background signal. This means that even with a perfect reconstruction, the VPRs would be low-bounded at 0.25 due to the 4:1 activity distribution in the phantom. To overcome this limitation, we propose to remove the theoretical baseline of the profile, which corresponds to the theoretical background of the phantom. Since this value is not always available, we approximated it with the mean value of the phantom background along the horizontal line passing through the origin.

D. Qualitative Evaluation With Brain Phantom

Acquisitions of a voxelized phantom of the human brain, the BigBrain [28], were simulated with the 3-D scanner to provide a qualitative overview of the impact of TOF resolution on the blur induced by APA. The analysis consisted of eight simulations: basicTOR versus obliqueTOR and four TOF resolutions. More specifically, this phantom was based on the “2015 3D Classified Volumes, MNI ICBM 152 Space” with 0.4 mm isotropic voxels,¹ herein referred to as BigBrain. The distribution of back-to-back sources in the phantom regions was configured such that it would correspond to the expected distribution of ¹⁸F-FDG in a human brain. Approximately 110 million coincidence events were acquired for the basicTOR and the obliqueTOR simulations. The motivation behind the number of coincidence events is provided in Appendix B.

Due to the fine image voxelization and the low number of coincidences, we expected the theoretical ground truth of an acquisition to differ significantly from the experimental ground truth. Thus, each TOR type was simulated only once and with perfect TOF information and the TOF variants were created by incorporating the appropriate TOF errors in the resulting list mode. Furthermore, their respective ground truth were post-smoothed with a 3-D isotropic Gaussian kernel of 1.0 mm sigma to mitigate local intensity fluctuations.

Better TOF resolution results in faster convergence speed of the reconstruction process, which means that comparison of images obtained at a fixed iteration number might provide an inadequate comparison. In this study, we chose to select the best image obtained for the best TOF cases by visual assessment, i.e., 2-mm TOF, while the candidates for the other TOF resolution were selected such that they had equivalent noise to the image chosen for 2-mm TOF. The noise in the reconstructed images was defined as the standard deviation of a region of interest (15 × 15 × 10 mm) placed inside the white matter and near one of the ventricles.

III. Results

A. Quantitative Evaluation of Resolvability

The resolvability of the 1.2–2.2 mm sectors for the basicTOR simulations are shown in Fig. 1. The 2.2 mm and larger spots were resolved in all cases, which was expected since it corresponds to the theoretical spatial resolution of basicTOR. From 1.4 to 2.0 mm spots, the dependence of resolvability on count statistics and TOF resolution can be appreciated. On the other hand, at most 30% of the 1.2 mm spots were resolved even with 2-mm TOF, indicating that a significant source of spatial blur remains even with ultrafast TOF. Overall, these results show that, even at a 10-mm TOF resolution, there is still a gain in resolvability over the APA-induced blur (cyan line). For the 2-mm TOF resolution case, a gain of 0.8 mm was achieved with all three count statistics, which is non-negligible since the theoretical spatial resolution of basicTOR is 2.2 mm.

Fig. 1.

Percentage of VPR satisfying the Rayleigh resolution criterion for the best images obtained from basicTOR simulations. Only the results for the spots ranging from 1.2 to 2.2 mm are shown. The values obtained from the three count statistics, FullStat, HalfStat, and 1/10thStat, are, respectively, shown as full blue, dashed green, and dotted red line. The black line shows the 90% threshold that is used to determine if a sector is resolved. The cyan line shows the expected blur (TOF-less) induced by APA for the basicTOR simulations, i.e., 1.9-mm FWHM (see Table I).

Some of the images used in the previous analysis are shown in Fig. 2. Overall, our visual assessments were in agreement with the observations made previously. However, we also noted some potential discrepancy, e.g., in Fig. 2(e), where some of the 1.8 mm spots could be easily mistaken for reconstruction artifacts. The images obtained from 20-mm TOF can be used as a baseline since their sectors resolvability (not shown) was in agreement with the expected values for a TOF-less reconstruction.

Fig. 2.

Best reconstructions from six simulations of the basicTOR case. FullStat (top row) and 1/10thStat (bottom row). From left to right, the CTR values were 13.3, 26.6, and 133.3 ps, respectively. Images in a row are displayed with the same gray scale.

The resolvability of the 1.2–2.2 mm sectors for the obliqueTOR simulations are shown in Fig. 3. The effect of the increased APA blurring can be observed in the loss of resolvability compared to basicTOR, e.g., with the 1.4 mm spots that were not resolved in any of the simulations. Furthermore, the effect of TOF resolution on resolvability can be observed over a greater range of spot size, most likely as a result of the relative size of APA blurring compared to TOF precision and detector size. Lastly, 1.8 mm spots of the 2-mm TOF reconstructions were resolved with the 1/10thStat count statistics even though its (TOF-less) theoretical spatial resolution was 2.6-mm FWHM, indicating a potential gain of at least 0.8 mm in spatial resolution even with low count statistics.

Fig. 3.

Percentage of VPR satisfying the Rayleigh resolution criterion for the best images obtained from obliqueTOR simulations. Only the results for the spots ranging from 1.2 to 2.2 mm are shown. The values obtained from the three count statistics, FullStat, HalfStat, and 1/10thStat, are, respectively, shown as full blue, dashed green, and dotted red line. The black line shows the 90% threshold that is used to determine if a sector is resolved.

B. Brain Phantom

Fig. 4 shows transverse slices of the reconstructions obtained for the BigBrain with the basicTOR simulations. Overall, an improvement in TOF resolution consistently leads to improved image sharpness. Indeed, while the structures observed in the ground truth were difficult to discern in the 20-mm TOF image, they were mostly well distinguished in the 2-mm TOF image. Nevertheless, the noisy nature of these images limited somewhat their interpretability, especially considering the already small value of the theoretical spatial resolution (green circle in Fig. 4(e)).

Fig. 4.

Study of the basicTOR case with the BigBrain phantom. The CTR values of the reconstructions shown in (a)–(d) were, respectively, 133.3, 66.6, 26.6, and 13.3 ps. Images (a)–(e) are expanded views (with trilinear interpolation) of the 4.6 × 4.6 cm² area centered at ~4.4 cm from the axial center, as shown in (f). The green circle in (e) is 2.2 mm in diameter, representing the theoretical spatial resolution of the basicTOR case. The maximum value for visualization of the reconstructed images was fixed at ~60% of the cumulative intensity histogram of 2-mm TOF reconstruction. The color map is shown in (f).

Fig. 5 shows transverse slices of the reconstructions obtained for the BigBrain with the obliqueTOR simulations. Again, an increase in sharpness was observed when TOF resolution improves. The gain in spatial resolution seemed greater here, which might be caused by the fact that the blur induced by APA was stronger for obliqueTOR simulations, thus making TOF information relatively more precise. The purple arrow shows a space between two gyri, i.e., sulcus, that seemed clearly perceptible only in the 2-mm TOF reconstruction.

Fig. 5.

Study of the obliqueTOR case with the BigBrain phantom. The CTR values of the reconstructions shown in (a)–(d) were, respectively, 133.3, 66.6, 26.6, and 13.3 ps. Images (a)–(e) are expanded views (with trilinear interpolation) of the 4.6 × 4.6 cm² area centered at ~5.6 cm from the FOV center, as shown in (f). The green circle in (e) is 2.6 mm in diameter, representing the theoretical spatial resolution of the obliqueTOR case. The maximum value for visualization of the reconstructed images was fixed at ~65% of the cumulative intensity histogram of 2-mm TOF reconstruction. The color map is shown in (f).

IV. Discussion

This study provides an estimation of the TOF resolution required to achieve a perceptible gain in spatial resolution by mitigating the blur induced by APA. Since the APA blur is about 2 mm for a whole-body PET scanner, it was expected that a TOF precision of around 13 ps would be necessary to observe any gain. This is corroborated by Fig. 1, where an enhancement of spatial resolution of around 0.8 mm is observed with the 2-mm TOF simulations. Nevertheless, gains in resolvability were also observed at worst TOF resolutions for obliqueTOR, which is more affected by APA. These observations are consistent with our visual evaluation of the images obtained with the BigBrain phantom, which is mostly apparent as an improvement in sharpness. Thus, achieving a TOF resolution of 67 ps or better would not only potentially improve the spatial resolution of PET imaging but might also contribute to a more stationary spatial resolution across the FOV by reducing the gap in precision between transaxial and axial TOR.

The study was based on simulations that neglected multiple sources of noise, e.g., scatter coincidences, to simplify the analysis. As such, the gain observed in this study should be interpreted as higher bounds of the gain that can be achieved in practice. Considering that APA is mostly affected by the size of the scanner, this study covers the largest blur that can be expected from APA, as long as PET imaging is not applied to animals larger than human. While a gain of at best 0.8 mm may appear limited, it is significant relative to the spatial resolution that can be achieved in a long axial FOV PET scanner. Furthermore, the gain achieved for obliqueTOR is greater than that for basicTOR, suggesting that TOF could help mitigate the detrimental effect of oblique TORs on the axial spatial resolution of a long axial FOV PET scanner. In our view, the key advantage of TOF highlighted in this study is its ability to improve the spatial resolution uniformity, i.e., to make it more stationary in the FOV.

Considering the focus of this study on whole-body and long axial FOV PET scanners, our choice of detector size, i.e., 2 mm, is on the low side and one could argue that it is nonviable due to the cost. This choice was made because the ability of very fast TOF resolution to mitigate the blur induced by detector size depends on count statistics [5]. This means that with larger detectors, e.g., if the blur induced by detectors is larger than that induced by APA, it is possible that the count statistics studied here would have been insufficient to achieve a spatial resolvability lower than the expected blur induced by APA. It would then be difficult to conclude whether the gain observed was due to the effect of very fast TOF resolution on the blur induced by APA or only on the blur induced by detector size. This study supports the hypothesis that very fast TOF resolution can mitigate the effect of APA on the spatial resolution limit of a PET scanner. The specific gain achievable for a given geometry can be easily investigated following the same procedure. Also, the developments of PET detectors based on monolithic crystals that can achieve a positioning resolution better than 2 mm are underway, and they are often less costly than their pixelated counterparts [29].

The analysis of the Hot Spot phantoms with the Rayleigh criterion provided a quantitative methodology to ascertain the variation in resolvability in the cases studied. However, one needs to keep in mind that this criterion was not designed for PET resolvability, which, as highlighted in Fig. 2, might result in discrepancy with the human perception. From our experience, it is generally in agreement with our visual assessments, except for reconstructions with large structured noise. It should also be noted that the gains in spatial resolution were observed using a 4:1 contrast, which gives credibility to these results, especially for lower count statistics.

While the enhancement in sharpness seen in Figs. 4 and 5 is evident, it is not always clear if this is due to the mitigation of APA blur or simply the result of TOF enhancement of signal-to-noise ratio (SNR). We believe that the methodology used for these qualitative assessments could still be improved, which might alleviate these concerns. First, there is the configuration of the BigBrain phantom. Considering that the best gain observed with the Hot Spots phantoms was 0.8 mm, the phantom voxelization being only 0.4-mm isotropic might have been insufficient. The BigBrain phantom is freely available up to an isotropic voxelization of 0.1 mm, but a simulation with such a fine voxelization would require considerably more computing resources. Second, there is the choice of the system matrix used for the reconstruction. The images obtained with the multi-Siddon projector were quite noisy, which hindered conclusive assessments of structure resolvability improvements. Thus, we expect that using a better representation of the system matrix or a reconstruction algorithm with better noise property, e.g., by including a regularization term, might be necessary to better appreciate the potential of TOF to mitigate the blur induced by APA.

One aspect we would like to emphasize is that this study does not show that TOF can remove the blur induced by APA. Our claim is that a sufficiently high TOF resolution can change the shape of the blur induced by APA, as shown in [6] when perfect TOF resolution is assumed, and that this change result in a gain in spatial resolution. Thus, this gain can be achieved without additional consideration in the reconstruction process (e.g., APA modeling in the system matrix), provided a PET scanner with a good enough TOF resolution is available.

One might then ask what the gain would be if APA was modeled in the system matrix. Any further gain would be in the realm of a discrete deconvolution and it would thus be correlated to the data available “per voxel,” which is limited in PET imaging. However, the fact that the response function of APA in image space seems sharper with better TOF resolution also means that one could expect better gain compared to when APA is modeled in the reconstruction process without TOF information. A potential element of error to consider when modeling APA in the system matrix is that it might have a non-negligible effect on the TOF response function, especially in oblique TORs since their length vary more in the axial dimension compared to the transaxial ones. Thus, one could limit the potential gain in spatial resolution or introduce artifacts if the effect of APA on TOF is not considered, as might be the cases for the 3-D reconstructions of this study.

As mentioned in Section I, some of the results in [6], which showed that ultrafast TOF resolution could mitigate the blur induced by APA, were obtained with an incorrect representation of APA. Indeed, it was shown in [7] that in most, if not all, PET acquisition Monte Carlo simulation software, APA amplitude was incorrectly modeled as having a Gaussian shape, resulting in its deviation having a significantly sharper profile compared to the expected Gaussian shape. Here, APA was correctly modeled following the recommendation in [7] and the results obtained still indicate that having access to ultrafast TOF resolution does indeed provide a gain in spatial resolution.

We attempted to include an analysis of point sources in order to observe the effect of TOF resolution on the response function of APA in image space. Considering that TOF-less reconstruction results in a Gaussian blur and that direct image formation with perfect TOF resolution has been shown to result in a Gaussian divided by the norm of its argument [6], we expected to see a transition between these two following with an improvement in TOF resolution. However, the profiles obtained did not converge to a smooth response function even with the use of the methodology proposed in [30]. In particular, the profile first stabilized to the classic Gaussian blur of APA and then shifted to a highly noisy profile that was akin to a sinus cardinal function. This is probably due to an inconsistency between the data that are affected by APA and the system matrix that neglect the effect of APA. Indeed, it can be shown that, in theory, a point source affected by APA is equivalent in Radon space to a Gaussian-shaped source of pure collinear emission of its annihilation photons. Thus, the forward projection of its ground truth is consistent with its theoretical acquisition. This is not the case with ultrafast TOF since the forward projection tends to be a Gaussian divided by the norm of its argument, thus being inconsistent with its theoretical acquisition, which was brought to our attention [31]. To confirm this hypothesis, one could model APA in the system matrix and validate that, by following the methodology proposed in [30], it converges to a well-behaved response function.

Lastly, improved TOF resolution accelerates reconstruction convergence and enhances the SNR in reconstructed images [32], [33]. However, the interplay between these TOF benefits and the spatial resolution gains observed in this study remains to be elucidated. Notably, the number of iterations required to achieve spatial resolution improvements beyond instrumental limits may exceed those optimal for SNR. This potential tradeoff presents a methodological challenge that warrants further investigation to optimize both spatial resolution and noise in future studies.

V. Conclusion

This study strengthens the idea that TOF can mitigate the blurring induced by APA for whole-body and long axial FOV PET systems by providing some insights on the requirements, in terms of TOF resolution and count statistics, to achieve a perceptible gain in spatial resolution. This gain is obtained even though APA is not modeled in the system matrix, thus indicating that having a ≤67-ps TOF resolution sharpens the response function of APA in image space. TOF resolution approaching 10-ps FWHM brings an observable spatial resolution improvement for a range of count statistics in typical size whole-body PET systems. Simulations also suggest that TOF resolution up to 67-ps FWHM can mitigate APA blur for oblique TOR in long axial FOV PET systems, thus enhancing their contributions in the reconstruction process. These observations highlight additional benefits of achieving very fast to ultrafast TOF resolution in clinical PET scanners, especially those with long axial FOV.

Appendix A. APA FWHM for obliqueTOR

The expected blur induced by APA, assuming it has an FWHM of 0.55°, for a PET scanner 102 cm in diameter is

$$R\times sin\left(0.5*\tau \right)=0.5\times 1020\times sin\left(0.5\times 0.55\right)=2.4\text{mm}$$

with R being the radius of the scanner and τ the APA FWHM.

Thus, the APA FWHM required for a PET scanner 81 cm in diameter to be affected by an image space blur of 2.4-mm FWHM is

$$\text{arcsin}\left(2.4/\left(0.5\times 810\right)\right)/0.5=0.68°.$$

Appendix B. Count Statistics for Hypothetical Whole-Body PET Scanner With Very Fast TOF Resolution

We based our choice of count statistics on the ultrahigh-resolution (UHR) brain PET scanner that was developed at Sherbrooke Molecular Imaging Center [34]. It is based on the labPET II technology, which uses a 1:1:1 coupling between the crystals, photodetectors, and electronics. While this approach is advantageous in terms of spatial resolution, it also results in a suboptimal density in detector elements and thus a lower peak sensitivity. The UHR has a 39.8-cm ring diameter and a 23.5-cm axial length [34], the latter being similar to the fictive 3-D scanner axial length used in this study.

Acquisitions with the UHR using an ¹⁸F-FDG brain protocol typically yielded between 110M and 500M coincidences. The 110M case was chosen as a basis for the hypothetical system of interest, since its qualitative evaluation was based on the BigBrain phantom.