INSTITUTE OF PHYSICS PUBLISHING Phys. Med. Biol. 5 (7) 3 7 PHYSICS IN MEDICINE AND BIOLOGY doi:.88/3-955/5/4/6 A new timing model for calculating the intrinsic timing resolution of a scintillator detector Yiping Shao Department of Nuclear Medicine, State University of New York at Buffalo, 5 Parker Hall, 3435 Main St., Buffalo, NY 44, USA E-mail: Shao@buffalo.edu Received August 6, in final form 5 December 6 Published 5 January 7 Online at stacks.iop.org/pmb/5/3 Abstract The coincidence timing resolution is a critical parameter which to a large extent determines the system performance of positron emission tomography (PET). This is particularly true for time-of-flight (TOF) PET that requires an excellent coincidence timing resolution ( ns) in order to significantly improve the image quality. The intrinsic timing resolution is conventionally calculated with a single-exponential timing model that includes two parameters of a scintillator detector: scintillation decay time and total photoelectron yield from the photon electron conversion. However, this calculation has led to significant errors when the coincidence timing resolution reaches ns or less. In this paper, a bi-exponential timing model is derived and evaluated. The new timing model includes an additional parameter of a scintillator detector: scintillation rise time. The effect of rise time on the timing resolution has been investigated analytically, and the results reveal that the rise time can significantly change the timing resolution of fast scintillators that have short decay time constants. Compared with measured data, the calculations have shown that the new timing model significantly improves the accuracy in the calculation of timing resolutions.. Introduction It is well known that the improvement of coincidence timing resolution can significantly improve the performance of positron emission tomography (PET), since it will allow a narrower timing window to minimize the random events and to significantly improve the image quality with enhanced overall signal-to-noise ratio. In particular, an excellent coincidence timing resolution ( ns) is essential to develop practical time-of-flight (TOF) PET that can significantly improve the image quality when compared with conventional non-tof PET. A scintillator detector normally consists of a scintillator crystal and a photon sensor that are optically coupled together. The intrinsic timing resolution, defined as the timing resolution 3-955/7/43+5$3. 7 IOP Publishing Ltd Printed in the UK 3
4 YShao associated with primary photoelectrons generated from the photon electron conversion and without being further amplified inside the photon sensor, is the theoretical limit of timing resolution that a practical scintillator detector can achieve. Therefore, it is valuable to calculate the intrinsic timing resolution of a scintillator detector as a first-order approximation to estimate its timing performance without simulating more sophisticated detector timing processes that involve photoelectron amplification, electrical signal processing and timing pickoff, etc (Lynch 966, Binkly 994,Heroet al 99, Clinthorne et al 99, Petrick et al 99). Conventionally, the intrinsic timing resolution is calculated with two parameters of a scintillator detector: scintillator decay time and total photoelectron yield (Lynch 966, Binkly 994, Post and Schiff 95). For slow scintillators with coincidence timing resolution in the range of 4 ns or larger, this calculation is pretty accurate (Lynch 966, Binkly 994). However, for recently developed fast scintillators with coincidence timing resolution reaching to ns or less, the calculation has led to significant errors as demonstrated by the latest studies (Glodo et al 5, Moszynski et al 998). It has been suggested by several groups that the effect of scintillation rise time can no longer be ignored for these fast scintillators, and should be included in the calculation in order to improve the accuracy (Glodo et al 5, Moszynski et al 996, 997, 998, Ludziejewski et al 995). Intuitively, there is always a non-zero rise time due to a complicated luminescence process that leads to the scintillation, and it can be contributed from one of several luminescence processes, such as exciton luminescence, dopant luminescence, charge-transfer luminescence and core valence luminescence, etc. For instance, the luminescence process of LaBr 3 :Ce involves the energy transfer time from the host to activator, and decreased charge collection efficiency by cerium ions due to the diffusion of charge carriers since the radiation trapping effects, either self-trapped hole or excitons (Glodo et al 5). Due to this kind of excitation and de-excitation, most scintillation emissions are observed with a non-zero rise time constant, as illustrated in figure. Therefore, rise time can affect the timing property of a scintillator detector and should be included in the calculation of timing resolution. In this study, we have developed a new timing model that includes the rise time, and investigated its effects on the timing resolution at various conditions. The calculations are also compared with the measurements of several newly developed fast scintillators to study the improvement of calculation accuracy by the new timing model.. Method.. Conventional single-exponential model It is known that the probability of N photoelectrons generated from the photon electron conversion between the time and t can be well described as a Poisson distribution (Post and Schiff 95), P(t) N = f(t) N e f(t) /N!, () where f(t)is the mean or expected number of photoelectrons between the time and t, and f() =. If τ and τ are scintillator rise and decay time constants, f(t)is conventionally defined as t f(t)= a e t/τ dt, () where a is a constant. After the integration, we have f(t)= R[ e t/τ ], (3) where R = f( ) = aτ is the total photoelectron yield.
A new timing model for calculating intrinsic timing resolution 5..8 probability.6.4 excitation de-excitation decay emission. 3 4 5 6 time (ns) Figure. Conceptual curves of two scintillation timing processes: () an excitation and deexcitation process characterized by a non-zero rise time constant; () a decay process characterized by a decay time constant. These two processes lead to a combined light emission that increases rapidly but not instantaneously to the maximum value, then decreases afterwards. This timing model has been used appropriately to estimate the timing resolution of a scintillator detector that has a slow scintillator, such as NaI, whose rise time constant is much smaller than its long decay time constant and can be neglected in the calculation (Lynch 966, Binkly 994, Post and Schiff 95). However, for newly developed fast scintillators, such as LSO and LaBr 3 :Ce, their rise time constants are no longer negligible compared with their short decay time constants. Recent experiments have shown that the smaller the rise time constant, the better the timing resolution (Glodo et al 5, Moszynski et al 996, 997, 998, Ludziejewski et al 995, van Loef et al ). Therefore, the effects of rise time can be very significant to the timing resolution of fast scintillators... Bi-exponential timing model We propose a new bi-exponential timing model that includes scintillation rise time with the following definition of the mean number of photoelectrons: f(t)= a t e t/τ ( e t/τ ) dt, (4) where a is a constant. After the integration of equation (4), we have [ f(t)= R τ + τ e t/τ + τ ] e t/ τ τ τ +τ τ τ where R = f( ) = a τ τ +τ is the total photoelectron yield. Several conclusions can be immediately obtained from equation (5): () If τ =, equation (5) becomes equation (3). Therefore, the conventional singleexponential timing model is included as a special case of the new timing model when the rise time τ =. (5)
6 YShao () If τ butτ /τ is very small, equation (5) will be approximate to equation (3), which explains that the single-exponential model is appropriate for a slow scintillator that has a large τ value. (3) For a fast scintillator, τ is significantly smaller compared with the value of a slow scintillator, and τ /τ is not small enough to be neglected in equation (5), which explains that the single-exponential model is not appropriate for a fast scintillator with a relatively small τ value. To further quantitatively understand the timing effects of rise time, the following important timing distributions and resolutions are calculated from both single- and bi-exponential timing models. Two functions of the mean number of photoelectrons f(t)are shown in figure. The rise and decay time constants are.5 and 4 ns respectively, which are similar to the corresponding values of the LSO scintillator. It is clear that the main characteristic difference between the two f(t)functions is at the beginning of the time profiles, within the region that is approximately equal to the rise time constant. The corresponding Poisson timing distribution P N (t) associated with each individual photoelectron is calculated with the two timing models and shown in figure 3. For illustration purposes, only the distributions that correspond to the first to the fifth photoelectron are shown. These timing distributions indicate that a small difference in two f(t) functions calculated from two timing models can lead to significant differences in the timing distribution of a photoelectron. This further exemplifies that rise time can have a profound impact on the timing property of a scintillator detector. The above timing distribution is associated with a single detector and is defined as the single-detector timing distribution. However, for PET, the coincidence timing resolution between the two detectors, t c, is a more important performance parameter, and is generally measured as the FWHM (full-width at half-maximum) of the coincidence timing distribution between the two detectors. This coincidence timing distribution can be calculated as the convolution of two single-detector timing distributions that usually correspond to the pair of same order of photoelectrons, assuming that two detectors are identical (Binkly 994). The calculated coincidence timing distributions that correspond to the pair of the first to the pair of the fifth photoelectrons are shown in figure 4. These distributions show that the rise time can significantly affect the coincidence timing resolutions between two fast scintillator detectors..3. The calculation of timing resolutions As shown in figure 4, it is expected to have the best coincidence timing resolution (or smallest FWHM value) with the pair of first photoelectrons, and degraded values with the pair of the second and subsequent photoelectrons. Therefore, in theory, the detector timing trigger should be set as low as possible at the first photoelectron level in order to have the best timing resolution. However, in reality, this may not always be possible because of practical constraints, such as the high noise level. Experimentally, it can be difficult to accurately estimate the exact photoelectron level of the detector trigger, or to obtain such information from the published data. Nevertheless, the coincidence timing resolution that corresponds to the pair of first photoelectrons will provide an ultimate theoretical limit that any pair of practical detectors can achieve, and is used in this paper to compare with the experimental data. In order to compare with the measured data of single-detector timing resolution that is deduced from coincidence timing measurements of two detectors, a theoretical single-detector
A new timing model for calculating intrinsic timing resolution 7 mean number of photoelectrons f(t) mean number of photoelectrons f(t) 8 6 4 5 f(t) with rise and decay time f(t) with decay time only τ =.5 ns τ = 4 ns R = 5 5 time (ns) (a) 5 f(t) with rise and decay time f(t) with decay time only τ =.5 ns τ =4ns R = 3 4 5 6 time (ns) (b) Figure. The mean number of photoelectrons f(t)as a function of time, calculated with singleand bi-exponential timing models, plotted with two different time scales in (a) and (b). The main difference between the two functions is at the beginning of the time, within the region that is approximately equal to the rise time constant. timing resolution, t s, can be deduced from the calculated coincidence timing resolution t c ; by assuming that the two detectors are identical, we have t s = t c /. In addition, since the timing resolution is experimentally measured with PMT output signals, the measured detector timing resolution has included the timing dispersion contributed from PMT itself. This PMT timing dispersion is generally measured as a transit-timedispersion (TTS). Since PMT timing dispersion is independent from the scintillation and photon electron conversion, a detector timing resolution, t s,det, is calculated with TTS and t s as t s,det = (TTS) + ( t s ). (6)
8 YShao.5 probability P N (t).4.3. P (t) - P (t) with rise and decay time 5 P (t) - P (t) with decay time only 5 τ =.5 ns τ = 4 ns R =...4.6.8 time (ns) Figure 3. The single-detector timing distributions for different photoelectrons (ranging from the first to the fifth), calculated with single- and bi-exponential timing models and the parameters of τ =.5 ns, τ = 4 ns and R =. The peak position of the distributions increases from the first to the fifth photoelectron. 4 Convolution of P -P 5 with rise and decay time Convolution of P N 8 6 4 Convolution of P -P 5 with decay time only τ =.5 ns τ = 4 ns R =.5.5 time (ns) Figure 4. The coincidence timing distributions between two detectors, calculated with single- and bi-exponential timing models. Each coincidence timing distribution is the convolution of a pair of single-detector timing distributions with the same number of photoelectrons. The peak position of the distributions increases from the pair of the first to the pair of the fifth photoelectrons. In practice, an emission time profile of a scintillator can contain multi-time components with several different rise and decay time constants. For instance, combinations of two rise and three decay time constants were used to fit the emission profile measured from LaBr 3 :Ce at different Ce concentrations (Glodo et al 5). However, since t c is mainly determined by the first photoelectrons, only the beginning part of the time profile will significantly affect t c, which is mainly characterized by () the effective rise time from the combination of multitime components and () the decay time with the component that has the dominant intensity.
A new timing model for calculating intrinsic timing resolution 9.8 Coincidence timing resolution t c (FWHM, ns).7.6.5.4.3.. R = 5 R = R =..4.6.8 Scintillator rise time τ (ns) Figure 5. The coincidence timing resolution as a function of scintillator rise time, calculated with a pair of first photoelectrons, with τ = 4 ns, R = 5, and. Therefore, a method to include multi-time components in equation (5) is to use the measured or calculated effective rise time constant, and the decay time constant with dominant intensity. 3. Results 3.. The impact of rise time on the coincidence timing resolution To quantitatively understand how the new timing model reflects the important effect of rise time on the coincidence timing resolution, equation (5) is used for the following exemplified calculations of t c, with the scintillation parameters being similar to those measured from LSO crystals. As shown in figure 5, t c increases monotonically and rapidly with the rise time, with a minimum value at τ =. At R =, a nominal increase of τ from. to. ns will increase t c from.66 to. ns, a significant 33% change. In figure 6(a), t c is shown as a function of decay time with different rise time constants. The results clearly show that t c increases with τ at any given τ. To more accurately measure the effects of rise time, the per cent change of t c is defined as t c(τ ) t c (τ =) t c (τ %, =) which reflects the relative change of t c compared to its values at τ =.. Figure 6(b) shows that t c increases with τ /τ. For a given τ, this per cent change of t c increases almost exponentially with the decrease of τ, indicating a significant impact of rise time on fast scintillators (τ < ns). For example, for a scintillator with τ =.5 ns and R =, the per cent change of t c increases from % when τ = 3 ns to 3% when τ = 4 ns. In figure 7, t c is shown as a function of R with different rise times. The results show a well-known trend that the coincidence timing resolution improves quickly with the increase of total photoelectron yield. These changes are particularly rapid at the low R values, but flatten at high R values. Calculations also show that these t c functions with different rise times are almost parallel to each other at different R values, indicating that t c can be significantly improved if the rise time can be reduced (e.g. with different Ce concentration)
YShao Coincidence timing resolution t c (FWHM,ns) Per cent changes of coinc. timing resolution (%) τ =. ns τ =. ns.5 τ =.5 ns τ =.8 ns.5 5 5 5 3 Scintillator decay time τ (ns) (a) 6 τ =. ns 4 τ =.5 ns τ =.8 ns 8 6 4 5 5 5 3 Scintillator decay time τ (ns) (b) Figure 6. (a) The coincidence timing resolution as a function of scintillator decay time, calculated with a pair of first photoelectrons, with τ =.,.,.5,.8 ns, and R =. (b) Corresponding per cent changes between the coincidence timing resolutions calculated from two timing models, defined as tc(τ ) t c(τ =) t c(τ =) %. It measures the impact of rise time on the coincidence timing resolution. while total photoelectron yield can be maintained. This conclusion matches well with the recent quantitative measurements of LaBr 3 :Ce, as shown in table (Glodo et al 5). 3.. Comparison between the measurement and calculation for LaBr 3 :Ce Recent studies have measured the timing resolution of LaBr 3 :Ce at different Ce concentrations, as well as detailed rise and decay time constants, total photoelectron yield and intensities of
A new timing model for calculating intrinsic timing resolution Coincidence timing resolution t c (FWHM, ns).5.5 τ =. ns τ =. ns τ =.5 ns τ =.8 ns 5 5 Total photoelectron yield R Figure 7. The coincidence timing resolution as a function of total photoelectron yield, calculated with a pair of first photoelectrons, with τ =.,.,.5,.8 ns and τ = 4 ns. multi-time components (Glodo et al 5). In the measurement, the timing resolution t s of a single LaBr 3 :Ce detector was estimated from the measurement of a coincidence timing resolution between a LaBr 3 :Ce detector and a fast BaF detector. These measured data are listed in table. It is clear that the main difference among scintillators with different Ce concentrations is the rise time, which leads to different timing resolutions among them. In the calculation, the measured scintillation parameters are used in the calculation of single timing resolutions with both timing models. The total photoelectron yield is chosen at the level that corresponds to the 5 KeV energy threshold which was applied in the measurement. The calculation uses the same 6 ps TTS as the PMT (Hamamatsu H53) used in the experiment (Glodo et al 5). Both calculated intrinsicand detector timing resolutions from single- and bi-exponential timing models are also shown in table. Comparisons between the measured and calculated timing resolutions are also shown in figure 8. For Ce concentrations at the.5% and 5% levels, the bi-exponential model significantly improves the accuracy in the calculation of timing resolutions, with calculation errors being 7% and 3%, respectively. By comparison, the corresponding calculation errors with the single-exponential model are 55% and 5%. This further demonstrates the importance of including the rise time in the calculation. In theory, the measured timing resolutions should be equal to or greater than 6 ps since the minimal timing resolution that can be experimentally measured is the contribution directly from the TTS itself. However, for LaBr3:Ce with Ce concentrations at the %, % and 3% levels, the measured timing resolutions are much less than 6 ps (table ). On the other hand, the calculated detector timing resolutions are greater than 6 ps since they do include both TTS and the calculated intrinsic timing resolution, and this has caused significant discrepancies among the measurements and calculations. Without knowing the details of the experiment setup and data processing in those studies (Glodo et al 5), it is difficult to understand the exact reason why the measured timing resolutions are less than 6 ps. Clearly, more experimental studies are needed to investigate this issue.
Table. Scintillation parameters and timing resolutions of LaBr 3 :Ce (Glodo et al 5). Timing resolution (ps) Ce 3+ Total photoelectron Effect of Measured Bi-exponential model Single-exponential model concentration yield Decay/rise times (intensity) rise time (%) (photoelectron/mev) (ns/ns (%)) (ns) Detector Intrinsic Detector Intrinsic Detector.5 5 3 9/5 (56%), 5./ (8%), 55 (6%) 9.4 36 9 33 4 6 5. 5 6 5/.38 (7%), 5/. (7%), 55 (3%).93 4 96 87 6. 4 664 6.5/.5 (89%), 4.5/.5 (5%), 55 (6%).5 6 79 79 3 6. 4 35 7.5/.6 (89%), 4.5/.5 (5%), 55 (6%).6 97 49 67 4 6 3. 4 58 8/. (9%),.5/. (4%), 55 (6%). 73 55 69 4 6 YShao
A new timing model for calculating intrinsic timing resolution 3 5 4 measurement bi-exp model single-exp model.5% Timing resolution (ps) 3 % 3% % 5%. Effective rise time τ (ns) (a) 5 4 measurement bi-exp model single-exp model.5% Timing resolution (ps) 3 % 3% % 5%. Effective rise time τ (ns) (b) Figure 8. (a) Comparison between the measured detector timing resolution and calculated intrinsic timing resolution t s for LaBr 3 :Ce with different Ce concentrations (Glodo et al 5). (b) Comparison between the measured detector timing resolution and calculated detector timing resolution t s,det = ( t s ) + (TTS), with TTS = 6 ps. 3.3. Comparison between the measurement and calculation for other fast scintillators The timing resolutions are also calculated with both timing models and compared with the data measured from other newly developed fast scintillators. The measured scintillation parameters for LSO, GSO, LuAP and YAP are shown in table, along with their measured and calculated timing resolutions. In the calculations, the energy thresholds are set at 46 KeV to include only the events within 5 kev photoelectric peaks. The PMT (Philips Photonics XPQ)
Table. Scintillation parameters and timing resolutions of some newly developed fast scintillators (Moszynski et al 996, 997, 998). Timing resolution (ps) Total photoelectron Measured Bi-exponential model Single-exponential model Scintillator yield Decay/rise times material photoelectron/mev (intensity) (ns/ns (%)) Detector Intrinsic Detector Intrinsic Detector LSO (Moszynski et al 996) 4 46./.48 (%) 98 34 7 75 GSO (Moszynski et al 996) 8 4./4. (69%), 97 (3%) 96 887 895 5 4 LuAP (Moszynski et al 997) 85 7.4/.6 (9%), 88 (%) 33 6 4 38 3 YAP (Moszynski et al 998) 43 6.7/.38 (89%), 4 (%) 3 9 79 4 3 4 YShao
A new timing model for calculating intrinsic timing resolution 5 8 measurement bi-exp model single-exp model GSO Timing resolution (ps) Timing resolution (ps) 6 4 YAP LuAP LSO. 8 6 4 Effective rise time τ (ns) measurement bi-exp model single-exp YAP LSO LuAP (a) GSO. Effective rise time τ (ns) (b) Figure 9. (a) Comparison between the measured detector timing resolution and calculated intrinsic timing resolution t s, for LSO, GSO, LuAP and YAP (Moszynski et al 996, 997, 998). (b) Comparison between the measured detector timing resolution and calculated detector timing resolution t s,det = ( t s ) + (TTS), with TTS = 5 ps. used in the measurements has 5 ps measured TTS (Moszynski et al 998), which has been used in the calculation of detector timing resolutions. As shown in figure 9, there is a good agreement between the measurements and calculations with the bi-exponential timing model, with calculation errors being in the range of 7 %. By comparison, the corresponding calculation errors with the single-exponential model are in the range of 8 75%.
6 YShao 4. Discussion and summary There are several additional factors that may contribute to the difference between the measured and calculated timing resolutions: In the calculation, the timing resolution is calculated with a pair of first photoelectrons, which provides the best timing resolution that a scintillator detector can achieve. However, in the measurement, the timing trigger may not always be set at the first photoelectron level, as was discussed in section.3, which can lead to the difference between the measurement and calculation. This also explains that most calculated timing resolutions are better than measured ones. The PMT TTS can be a function of photoelectron yield and may not be easy to be quantified during the measurement. Therefore, in principle, the discrepancy between the measurement and calculation can always exist with uncertain TTS contribution at different signal levels. To solve this problem, more accurate and comprehensive timing simulations that include the effects of PMT and signal electronics will have to be applied in order to accurately compare with the measured detector timing resolution (Binkly 994, Hero et al 99, Clinthorne et al 99, Petrick et al 99) The model is based on pure exponential curves characterized by rise and decay time constants. It works well for LSO that emits a single time-component. However, for scintillation emission with multiple time-components and not strictly following the exponential curves, such as those with LuAP and YAP (Moszynski et al 997, 998), the model can deviate from the measured values significantly. In the measurements, the energy thresholds that selected the 5 KeV photoelectric events are not specified in the literature ( Moszynski et al 996, 997, 998), except those measured with LaBr 3 :Ce (Glodo et al 5). In the calculation, a threshold at 46 kev has been applied which might differ from the experimental values and could cause the deviation from the correct total photoelectron yield. In summary, a new bi-exponential timing model is proposed and evaluated. The new timing model calculates the intrinsic timing resolution with three parameters of a scintillator detector: scintillation rise and decay time constants, and total photoelectron yield. It includes the conventional timing model as its special case when the rise time constant is zero. Except those measured data with timing resolutions less than the value of PMT TTS (which are worth further investigation), the new timing model significantly improves the accuracy in the calculation of the intrinsic timing resolution of a scintillator detector, and can be served as a first-order approximation to estimate the detector timing resolution of a scintillator detector. Acknowledgment This work was supported by a grant from the Sterbutzel Research Foundation. References Binkly D M 994 Optimization of scintillation-detector timing systems using Monte Carlo analysis IEEE Trans. Nucl. Sci. 4 386 93 Clinthorne N H et al 99 A fundamental limit on timing performance with scintillation detectors IEEE Trans. Nucl. Sci. 37 658 63 Glodo Jetal5 Effects of Ce Concentration on scintillator properties of LaBr3:Ce IEEE Trans. Nucl. Sci. 5 85 8 Hero A O et al 99 Optimal and sub-optimal post-detection timing estimation for PET IEEE Trans. Nucl. Sci. 37 75 9
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