Analytical calculation of the lower bound on timing resolution for PET scintillation detectors comprising high-aspect-ratio crystal elements

Two recent publications (Vinke et al 2014, Gundacker et al 2014) and an associated thesis (Gundacker 2014) have demonstrated how the CRLB on timing resolution can be calculated for scintillators with non-negligible photon transport using Monte-Carlo generated photon transit time distributions. In this work, we formulate the CRLB for HAR scintillators with a purely mathematical expression by using a closed-form solution for optical transit time spread. Such an approach has great utility, as it allows methods for achieving ultra-precise timing resolution with HAR scintillators with a simple mathematical expression, rather than relying on exhaustive Monte-Carlo studies. Researchers can easily adapt this methodology to their own applications. The final expression also represents an elegant form of the statistical limit on timing resolution for scintillation detectors with non-negligible photon transport.

The probability of a forward traveling scintillation photon having a path length between $x$ and $x+\text{d}x$ in a crystal of rectangular parallelepiped geometry with width $w$ and length $L$, at a distance $h$ from the bottom of the geometry can be expressed with equation (2).

$$\begin{eqnarray}f{(x)_{{\rm{for}}}} = \left\{ {\begin{array}{*{20}{l}} {0,}&{\quad x < h}\\ {h\frac{{\left( {\frac{x}{{x - {\rm{d}}x}} - 1} \right)}}{{2x}}(1 - {R_x}),}&{\quad h \leqslant x \leqslant \frac{h}{{{\rm{sin}}\left( {\frac{\pi }{2} - {{\rm{\Theta }}_{{c_x}}}} \right)}}} \end{array}} \right.\end{eqnarray} \tag{ 2 }$$

${{R}_{x}}$ represents Fresnel's coefficient of reflectivity at the exit boundary where a crystal-grease interface exists. The lower bound of this expression is length $h$, which is the distance traveled directly from the position of interaction to the exit surface. The upper bound represents the path length of a photon that reflects by total-internal-reflection (TIR) off the side of the scintillator due to the crystal-air interface and impinges the exit boundary at the critical angle of the crystal-grease interface, ${{\Theta}_{{{c}_{x}}}}$. The probability for a path length between $x$ and $x+\text{d}x$ is given by projecting the conic area these path lengths form at the exit boundary onto a unit-sphere and comparing the surface area of this projection to the $4\pi$ emissions of scintillation light, as illustrated in figure 1. It is also noted that equation (2) is shown in a reduced form.

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The probability of a scintillation photon to travel away from the photosensor with a total path length between $x$ and $x+\text{d}x$ can be described with equation (3).

$$\begin{eqnarray}f{(x)_b} = \left\{ {\begin{array}{*{20}{l}} {0,}&{x < 2L - h}\\ {(2L - h)\frac{{\frac{x}{{x - {\rm{d}}x}} - 1}}{{2x}}{R_r},}&{2L - h \leqslant x \leqslant \frac{{2L - h}}{{{\rm{cos}}({{\rm{\Theta }}_{{c_g}}})}}}\\ {(2L - h)\frac{{\frac{x}{{x - {\rm{d}}x}} - 1}}{{2x}}(1 - {R_x}),}&{\frac{{2L - h}}{{{\rm{cos}}({{\rm{\Theta }}_{{c_g}}})}} < x \leqslant \frac{{2L - h}}{{{\rm{sin}}\left( {\frac{\pi }{2} - {{\rm{\Theta }}_{{c_x}}}} \right)}}} \end{array}} \right.\end{eqnarray} \tag{ 3 }$$

Equation (3) equals zeros below a distance of $2L-h$, which is the shortest distance a back-reflected photon can have, traveling straight upward and then straight back downward to the exit boundary. Path lengths of back-reflected photons longer than this value represent those that impinge the top of the crystal at an angle less than the critical angle (${{\Theta}_{{{c}_{g}}}}$) for the scintillator-air interface until ${{\Theta}_{{{c}_{g}}}}$ is reached. In this region, the reflection coefficient, ${{R}_{r}}$, is determined by the wavelength and angular dependent properties of the reflector material. Once ${{\Theta}_{{{c}_{g}}}}$ is reached, transport to the exit boundary is again guided by TIR until the critical angle for the scintillator-grease interface at the exit is reached. Good illustrations of photon probability per path length in the direction opposite of the crystal's exit surface can be found in Figures 2.19 and 2.22 of Yang (2012). The expressions for forward and back-reflected photons overlap in regard to distance traveled for certain positions of interaction and crystal dimensions. Therefore, a complete expression for optical path length is given by equation (4).

$$\begin{eqnarray}&&f(x)=\frac{{{f}_{\text{for}}}(x)+{{f}_{b}}(x)}{{\int}_{0}^{\infty}\left({{f}_{\text{for}}}(x)+{{f}_{b}}(x)\right)\text{d}x}\end{eqnarray} \tag{ 4 }$$

Equation (4) expresses the optical transit of scintillation light in terms of distance traveled, but this PDF can be transformed to represent the optical transit time of scintillation photons by considering that the transit time is simply the distance traveled divided by $(c/n)$, where c is the speed of light in a vacuum and n is the index of refraction of the crystal. The optical transport PDF will be represented as a function of time, $f(t)$, for the rest of this manuscript. Equation (4) represents the optical transport of scintillation light for a particular position of interaction, but a measured CTR between two detectors is characterized by interactions at multiple positions along a scintillation crystal. The probability distribution of optical transit time over all positions of interaction can be expressed by

$$\begin{eqnarray}&&f(t)=\underset{i=0}{\overset{L}{\sum}}\,f{{\left({{t}_{g}}+t\right)}_{i}}\times {{w}_{i}}\end{eqnarray} \tag{ 5 }$$

where ${{w}_{i}}$ represents the weight for a particular position of interaction given from the attenuation of annihilation photons in the scintillation crystal, calculated according to ${{\text{e}}^{\mu (L-h)}}$, and ${{t}_{g}}$ is the time-of-flight of an annihilation photon from the top of the crystal to a position of interaction.

With an analytical PDF for optical transport in place, the contribution to timing uncertainty made by the optical transit time spread was determined. The standard deviation of the propagation time of scintillation photons can be calculated by taking the square root of the difference in the first and second moments of the expectation value of the attenuation-weighted photon transit time PDF, equation (5), as shown in equation (6).

$$\begin{eqnarray}&&\sigma _{\text{trans}}^{2}=E\left[f{{(t)}^{2}}\right]-E{{\left[f(t)\right]}^{2}} {{\sigma}_{\text{trans}}}=\sqrt{{\int}_{-\infty}^{\infty}{{t}^{2}}\times f(t)\text{d}t-{{\left({\int}_{-\infty}^{\infty}t\times f(t)\text{d}t\right)}^{2}}}\end{eqnarray} \tag{ 6 }$$

The temporal distribution of a scintillator's emissions, ${{p}_{{{t}_{e}}}}(t)$, can be expressed as a linear combination of bi-exponential models for any number of luminescence processes $i$ that occur with probability ${{P}_{ec,i}}$. Here, ${{P}_{ec,i}}$ represents a number between 0 and 1 that properly weights each bi-exponential component.

$$\begin{eqnarray}&&{{p}_{{{t}_{e}}}}(t\mid \Theta)=\underset{i}{\sum}\,{{P}_{ec,i}}\frac{1}{\left({{\tau}_{d,i}}-{{\tau}_{r,i}}\right)}\left[{{\text{e}}^{-\frac{t-\Theta}{{{\tau}_{d,i}}}}}-{{\text{e}}^{-\frac{t-\Theta}{{{\tau}_{r,i}}}}}\right]\end{eqnarray} \tag{ 7 }$$

In equation (7), ${{\tau}_{r}}$ and ${{\tau}_{d}}$ are the rise and decay constants that characterize the emissions distribution, and $\Theta$ represents the interaction time of incident radiation in the scintillator. Performing a convolution of equation (4), expressed as a function of time, with equation (7) will produce the distribution of arrival statistics at the exit boundary of the crystal, ${{p}_{\text{trans}}}$.

$$\begin{eqnarray}&&{{p}_{{{t}_{\text{trans}}}}}(t\mid \Theta)={\int}_{0}^{\infty}{{p}_{{{t}_{e}}}}\left(t-\tilde{t}\mid \Theta\right)\times f\left(\tilde{t}\right)\text{d}\tilde{t}\end{eqnarray} \tag{ 8 }$$

In order to calculate the lower bound on timing resolution, the variance in optical transport spread must be considered for all positions of interaction, as in equation (5). A convolution of equation (8) with a normal distribution having a width that corresponds to the charge transit spread in a photosensor will produce a distribution of primary triggers.

$$\begin{eqnarray}&&{{p}_{{{t}_{\text{pt}}}}}(t\mid D)={\int}_{0}^{\infty}g\left(t-\tilde{t}\right)\times {{p}_{{{t}_{\text{trans}}}}}\left({{\tilde{t}}_{\text{trans}}}\right)\text{d}\tilde{t}\end{eqnarray} \tag{ 9 }$$

Equation (9) represents the distribution of primary triggers from the photosensor, in the absence of noise, under the condition $D$ that the photons are detected.

The amount of information carried by a subset of $N$ samples from a given PDF about a parameter $\Theta$ is described by the Fisher information. For this application, the expression for the Fisher statistic represents the amount of information on $\Theta$ carried by all available primary triggers and is written as

$$\begin{eqnarray}&&{{I}_{N}}(\Theta)=N\times {\int}_{-\infty}^{\infty}{{\left[\frac{\partial}{\partial \Theta}\ln {{p}_{{{t}_{\text{pt}}}}}(t\mid \Theta)\right]}^{2}}\frac{1}{{{p}_{{{t}_{\text{pt}}}}}(t\mid \Theta)}\text{d}t.\end{eqnarray} \tag{ 10 }$$

The Cramér–Rao relationship states that the variance of an unbiased estimator, ${{\Xi}_{N}}$, on $\Theta$, the interaction time, is at least as large as the inverse of the Fisher information.

$$\begin{eqnarray}&&\text{var}\left({{\Xi}_{N}}\mid \Theta\right)\geqslant \frac{1}{{{I}_{N}}(\Theta)}\end{eqnarray} \tag{ 11 }$$

Equation (10) inherently assumes an ideal case where a timestamp is available from each detected photon and therefore forms the lower bound on the variance of the unbiased estimator on $\Theta$. For comparison to achievable timing resolution in ToF-PET the CRLB is reported as the FWHM of a CTR between two identical detectors, according to equation (12), for the remainder of the manuscript.

$$\begin{eqnarray}&&\text{CRLB}=\frac{1}{\sqrt{{{I}_{N}}(\Theta)}}\times 2.35\times \sqrt{2}\end{eqnarray} \tag{ 12 }$$

To validate the optical transport PDF, the time of arrival of optical photons was simulated with the Monte Carlo component of optical transport code (ZEMAX 2015). For these simulations, polished LYSO:Ce and ESR specular reflector were modeled with the optical properties outlined in Lörincz et al (2010). One hundred thousand photons with a wavelength of 420 nm were generated isotropically at various depths along the center of crystal geometries. Two different scintillator geometries were simulated, and the propagation time of scintillation light from its origin to the exit of the crystal was compared with the time of arrival calculated by the optical transport PDF. Specifically, scintillation crystals of pixel geometries were simulated with $3\times 3\times 20$ and $3\times 3\times 30$ mm$^{3}$ dimensions. The two pixel geometries were chosen to assess the accuracy of $f(t)$ for modeling HAR crystal geometries.

Simulations were also performed with the reflector material having a Lambertian scatter profile for the $3\times 3\times 20$ mm$^{3}$ geometry to assess the accuracy of the analytical model in predicting the photon transport time distribution for scintillators wrapped in diffuse reflectors, such as Teflon tape. Note that as the goal of this work is to optimize timing resolution, we have focused our studies on scintillators with polished surfaces; scintillators with their surfaces roughened are unanimously reported to have lower light collection (Heinrichs et al 2002, Lörincz et al 2010, Pauwels et al 2012).

Coincidence timing resolution was measured between pairs of $3\times 3\times 5$ mm$^{3}$ and $3\times 3\times 20$ mm$^{3}$ LYSO:Ce crystals optically coupled to Hamamatsu S12572-50C silicon photomultipliers (SiPMs). The surfaces of the crystals were either wrapped in ESR reflector or Teflon tape in separate experiments, with an air gap left between the reflector and crystal surface. Teflon tape, a diffuse reflector, was used for the measurement with the 5 mm long crystals and one of two measurements with the 20 mm long crystals. An illustration of the experimental setup is shown in figure 2. The SiPM read-out electronics and experimental setup used were the same as described in Yeom et al (2013a). The SiPMs were ac coupled to Minicircuits MAR-3+ preamplifiers, and their signals were split into timing and energy channels, then connected to an Agilent DSO940 2 GHz oscilloscope. Detector signals were sampled at 20 GSa sec$^{-1}$, and time-pickoff was performed off-line by leading edge discrimination on a full cubic spline of the timing signals. A baseline offset correction algorithm was used to compensate for shifts due to dark pulses (Vinke et al 2009). Coincidence events were selected within the full width at tenth maximum of a normal distribution fit to the 511 keV photopeak of each detector.

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The calculated statistical limit on timing resolution was also compared with the measured CTR for the two LYSO:Ce detectors. The margin between the measured CTR for the $3\times 3\times 5$ mm$^{3}$ crystals and the calculated CRLB for the measurement should be consistent with that observed between the CTR for $3\times 3\times 20$ mm$^{3}$ crystals and the CRLB for that case, if optical photon transport is accurately incorporated into the calculation. Additionally, the CTR measurement with $3\times 3\times 20$ mm$^{3}$ crystals wrapped in Teflon tape should match the value measured with ESR reflector for the assertion that this optical model can be applied to crystals wrapped with diffuse reflector to be validated.

The average light output of the LYSO:Ce crystals was previously measured using a single photon calibrated Hamamatsu R9779 PMT to be $9808\pm 469$ and $7684\pm 722$ photons/511 keV for the 5 mm and 20 mm long crystal, respectively. The average number of primary triggers induced in the SiPMs from the interaction of annihilation photons in the crystals was found by integrating the PDE information on the SiPM's data sheet (Hamamatsu 2015), at the appropriate ${{V}_{ob}}$, over the emissions spectrum of LYSO:Ce. This effective PDE does not include the influence of dark counts and after pulsing. The single photon timing resolution (SPTR), 250 ps (FWHM), was also taken from the sensor's data sheet. The intrinsic rise and decay constants for the LYSO:Ce crystals were assumed to be $72\pm 3$ ps and $43\pm 0.6$ ns, respectively, as measured in Seifert et al (2012b).

Figure 3(a) shows the calculated optical transport PDFs (equation (4)) for a $3\times 3\times 20$ mm$^{3}$ crystal with the optical properties of LYSO:Ce and reflector properties of Vikuiti 3M ESR (2015) for depths of interaction (DOI) at 2, 10, and 17 mm from the top of the crystal. The PDFs show a larger variance in the time of arrival of scintillation light at interaction locations close to the photosensor, with decreasing variance as the position of interaction occurs closer to the top of the crystal. This behavior is in agreement with the assertions and measurements presented in Moses et al (1999) and Yeom et al (2013a), and it is noted that the optical transport PDF matches simulation results for the same geometry and positions of interaction in Yeom et al (2013a). Figure 3(b) shows the optical transport PDF when all positions of interaction are taken into account (equation (5)). This distribution is in agreement with a simulation of the arrival time of scintillation light from a simulation of a similar geometry in Gundacker et al (2013b) ($2\times 2\times 20$ mm$^{3}$ LSO:Ce,Ca(0.4%) was simulated in that work).

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The calculated transit time spread (equation (6)) of scintillation photons in crystals with a $3\times 3$ mm$^{2}$ cross-section and index of refraction of LYSO:Ce is shown for various lengths in figure 4. The transit time spread versus crystal length was found to trend according to a second-order polynomial.

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In figure 5(a), two examples are given for ${{p}_{\text{trans}}}$, where $f(t)$ has been chosen for DOI at 1 mm and 19 mm from the top of a $3\times 3\times 20$ mm$^{3}$ LYSO:Ce crystal. The inset of this figure shows the entire distributions, and the major portion of the figure shows the distributions at an early time window containing the statistics used for an estimate on time of interaction. An example of how the convolution of equations (6)–(8) affects the time of arrival of photon statistics is displayed in figure 6(b). The PDF for LYSO:Ce's temporal emissions profile, ${{p}_{e}}(t)$, is shown in blue, and the optical transport distribution, $f(t)$, is shown for a $3\times 3\times 20$ mm$^{3}$ crystal in green. The convolution of these two distributions, ${{p}_{\text{trans}}}$, is shown in red. When ${{p}_{\text{trans}}}$ is convolved with a distribution representing charge transit spread, $g(t)$ in aqua, the resulting distribution for the temporal profile of the creation of primary triggers, ${{p}_{\text{pt}}}$, is shown in purple.

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Plots of the analytically calculated (equation (8)) and simulated time of arrival of scintillation photons for the $3\times 3\times 20$ mm$^{3}$ crystal at DOI of 10 mm and 1 mm from the top of the crystal are displayed in figures 6(a) and (b), respectively. The same profiles are shown for 15 and 3 mm DOI for the $3\times 3\times 30$ mm$^{3}$ geometry in figures 6(c) and (d). The optical transport PDF showed excellent agreement with the light transport simulations, validating $f(t)$ as an accurate analytical expression for the time of arrival of scintillation photons for crystals of these geometries.

Figure 7 shows examples of the comparison between the simulated and analytical photon arrival times for a $3\times 3\times 20$ mm$^{3}$ LYSO:Ce pixel wrapped in a diffuse reflector. It can be seen that while the scatter from the diffuse reflector induces some blurring in photons having long path lengths in the crystal, the overall agreement is still quite good. For this crystal/reflector configuration, photons initially propagating towards the photosensor reach the critical angle of the exit interface before falling below the critical angle for the crystal/air boundary at the sides of the crystal. Therefore, few photons initially traveling in the forward 2$\pi$ that reach the photosensor have interacted with the surrounding reflector. In fact, when considering the annihilation photon attenuation corrected optical photon arrival time for all positions of interaction (figure 7(d)), there is very little difference in the analytical and simulated distributions. This suggests that, at least for polished crystals, the difference in the optical transport time variance for crystals with specular or diffuse reflectors might only have a very small effect on CTR. This assertion is tested in section 3.3.

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Figure 8 shows the measured CTR data between pairs of $3\times 3\times 5$ mm$^{3}$ and $3\times 3\times 20$ mm$^{3}$ LYSO:Ce crystals coupled to analogue SiPMs at the optimum LED threshold and bias ${{V}_{br}}+4.6$ V. The FWHM of the resulting distributions were $135\pm 1$ ps (5 mm length + Teflon), $162\pm 1$ ps (20 mm length + ESR), and $165\pm 1$ ps (20 mm length + Teflon). The error on these values was taken from the 95% confidence interval on the fit of the normal distribution to the data. The factor of $\sim$6 reduction in dark counts with the S12572-50C SiPM, relative to the previous commercial version of this device (S10362-33-050C) in combination with the baseline correction algorithms used in determining the LED threshold allowed the devices to be biased much higher than typically reported. This resulted in an integral PDE for LYSO:Ce of 41% (Hamamatsu 2015).

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The measured CTR was compared with the calculated lower bound on timing resolution for these detectors as an experimental validation of the CRLB expression for HAR scintillation detectors. Figure 8 also shows the CRLB and the error associated with the calculation. The calculated lower bound on CTR for the $3\times 3\times 5$ mm$^{3}$ crystals was $125\pm 8$ ps, and $152\pm 8$ ps was calculated for the $3\times 3\times 20$ mm$^{3}$ crystals. The error on these values was calculated in the manner outlined in Seifert et al (2012a). The measured CTRs for the $3\times 3\times 20$ mm$^{3}$ crystals approached the calculated statistical limit within 10 ps (within 6.5%) and 13 ps (within 8.6%) for the ESR and Teflon wrapped measurements, respectively. This margin is consistent with the 10 ps (8%) margin between the calculated CRLB and measured CTR with the Teflon wrapped $3\times 3\times 5$ mm$^{3}$ crystals.

The measured CTR and the statistical limit on timing performance are not expected to overlap, as the experimental setup contains unavoidable processes such as noise. As mentioned in section 2.2, the CRLB assumes the availability of timing information from every detected photon. This is not the case with the analogue SiPMs used in this experimental setup. Previous studies with short ($\leqslant$5 mm in length) scintillation crystals have shown that the calculated CRLB without optical transport can be closely approach with analogue photosensors using digital pulse processing of signals and an optimized LED threshold (Post et al 1950, Seifert et al 2012a, Cates et al 2013). This same tendency was observed in figure 8 with HAR crystals.

The parameters in the CRLB calculation were varied to investigate the influence of various physical process on CTR with this experimental setup. Figure 9(a) shows the calculated statistical limit on CTR as a function of the normalized input parameters. For LYSO:Ce, improving the scintillation rise time has little influence on achievable timing resolution, as has also been reported in Gundacker et al (2013a) and Seifert et al (2012a). With an experimentally measured rise time of $72\pm 3$ ps (Seifert et al 2012b), this material has extremely fast early luminescence processes that do not dominate the contribution to timing uncertainty. The statistical limit on timing performance showed inverse relationships between the CRLB and the $\sqrt{N}$ and $\sqrt{{{\tau}_{d}}}$ dependence for photon statistics and decay time, respectively. This relationship is well known and has been previously reported in Hyman et al (1964), Fishburn et al (2010) Kronberger et al (2010) and Gundacker et al (2013a). Calculated timing performance was found to improve linearly with SPTR. The calculation of the CRLB in figures 8 and 9(a) at normalized parameter value 1 uses the manufacturer's reported SPTR of 250 ps FWHM. Values reported for the SPTR of Hamamatsu MPPCs have varied from $\sim$190–280 ps FWHM (Seifert et al 2012a, Gundacker et al 2013c). However, these are reported for an older version of the photosensor used in this work, as noted in section 3.3. Therefore, we have taken the manufactures value to be the most reasonable estimate SPTR for the SiPMs here. It is noted that large variations in SPTR of 20–30% lead to $\sim$10–15 ps variations in the calculated CRLB. This can also be seen in figure 9(a).

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Figure 9(b) shows the calculated CRLB on the CTR between two $3\times 3$ mm$^{2}$ cross-sectional area LYSO:Ce crystals as the length was varied, holding all other parameters constant. This shows the influence of photon transport variance on achievable timing resolution. The CRLB showed a linear decrease below 20 mm until reaching 5 mm, where the influence of photosensor SPTR becomes dominant. It is noted that the value calculated at a crystal length of 5 mm precisely matches that calculated from Seifert's formalism of the CRLB (Seifert et al 2012a), where no photon transport is incorporated, within $<$1%. The increase above 5 mm then represents the margin of error when using a formulation of the CRLB that does not account for optical photon transit and applying it to cases with non-negligible photon transit time variance. This figure also illustrates the utility of calculating the CRLB for HAR crystals with this approach. To compare with other efforts using a Monte-Carlo generated photon transit time distributions, a separate simulation would have to be run for each of these crystal lengths, totaling 38 separate simulations to perform this same investigation that was generated from simple calculations.

Figure 10 shows a colormap of the lower bound on CRT for $3\times 3\times 20$ mm$^{3}$ LYSO:Ce scintillators, where available photon statistics and photosensor temporal response were parametrically varied. A rise time of 72 ps and decay time of 43 ns, respectively were assumed for the calculation (Seifert et al 2012b), and light output was considered to be 27 000 photons/MeV (Haas and Dorenbos 2008). Note that the $x$-axis of this figure represents total detection efficiency, and it is comprised of both light collection efficiency at the exit interface of the crystal and the PDE of the photosensor used to convert scintillation light into charge. The light collection at the exit of crystal elements with polished surfaces and lengths of 20 mm is typically reported between 40–60% (Levin 2002, Bauer et al 2009, Pauwels et al 2009, Lörincz et al 2010, Auffray et al 2013). The PDE of a photosensor tends to be between 20–30% for photomultipliers with bialkili photocathodes, and analogue SiPMs typically have an effective PDE of 30–35% when operated at a sufficient margin above the breakdown voltage, which is the case when the bias is optimized for timing measurements. Taking these factors into account, one arrives at a total optical photon detection efficiency in the range of 8–18%. The calculated CRLB in this region is consistent with the trend in literature of CTRs just above or below 200 ps when 20 mm length crystals are coupled to fast photosensors (SPTR $\leqslant$300 ps FWHM).

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The lines in figure 10 labeled (1)–(3) represent the CRLB for scenarios where advancements in crystal read-out or photosensor PDE were assumed. Scenario 1 represents the CRLB for the photon detection efficiency reported in section 2.4. At a SPTR of 250 ps FWHM we have the calculated lower bound reported in figure 8(b). To approach a coincidence timing resolution of 100 ps FWHM, the timing performance of the photosensor must be significantly improved. For example, if the SPTR could be improved to values achievable with micro-channel plate photomultipliers (25–40 ps FWHM), the lower bound improves to $\sim$110 ps. Scenario (2) represents the CRLB improvement for the case where light collection efficiency at the exit boundary is increased by 50%. This light collection increase represents a reported simulation value for increased light collection via photonic crystal techniques, for an LSO:Ce/grease exit interface (Lecoq et al 2013). At this level of light collection, the lower bound approaches 110 ps for SPTR values $<$200 ps. This improvement of the statistical limit on timing resolution for this scenario underscores the importance of ongoing research to improve the light extraction from inorganic scintillators. Scenario 3 shows the CRLB improvement when light collection is increased by 50% and photosensor PDE is improved to 50%. This PDE value is almost achievable with super-bialkali photocaothoes in PMTs (Nakamura et al 2010), and the Philips DPC-3200 has a reported average effective PDE of nearly 40% at 420 nm (Philips Digital Photon Counting 2014). Therefore, an assertion that photosensor technology could improve to have a PDE of 50% is not unrealistic. With these improvements, the calculated CRLB approaches 100 ps for SPTR values $<$200 ps.