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Two Approaches to Modeling the Time Resolution of Scintillation Detectors

Two Approaches to Modeling the Time Resolution of Scintillation Detectors. S. Seifert , H.T. van Dam, D.R. Schaart . Outline. A common starting point Modeling (analog) SiPM timing response Extended Hyman model The ideal photon counter Fisher information and Cramér – Rao Lower Bound

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Two Approaches to Modeling the Time Resolution of Scintillation Detectors

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  1. Two Approaches to Modeling the Time Resolution of Scintillation Detectors S. Seifert, H.T. van Dam, D.R. Schaart

  2. Outline • A common starting point • Modeling (analog) SiPM timing response • Extended Hyman model • The ideal photon counter • Fisher information and Cramér–Rao Lower Bound • full time stamp information • Single time stamp information • 1-to-n time stamp information • Important disclaimers • Discussion • Some (hopefully) interesting experimental data • Conclusions

  3. A Common Starting Point

  4. The Scintillation Detection Chain Common StartingPoint (γ-)Source Emission Emitted Particle (γ-Photon) Absorption Scintillation Crystal Emission of optical photons Detection of optical photons Sensor Electronics Signal Timestamp

  5. Assumptions Common StartingPoint Necessary Assumptions: • Scintillation photons are statistically independent and identically distributed in time • Photon transport delay, photon detection, and signal delay are statistically independent • Electronic representations are independent and identically distributed γ-Source Emission γ-Photon Absorption Scintillation Crystal Emission Detection Sensor Signal Electronics Timestamp

  6. Registration Time Distribution p(tr|Θ) Common StartingPoint Emission at t = Θ pdfp(tr|Θ) describing the distribution of registration times of independent scintillation photon signals Absorption Emission of optical photons random delay (optical + electronic) Registration of optical photons Estimate on Θ Delay distribution

  7. Assumptions Common StartingPoint Emission at t = Θ Absorption Assumptions that make life easier: • Instantaneous γ-absorption • Distribution of scintillation photon delays is independent on location of the absorption OR, • simplest case distribution of scintillation photon delays is negligible Emission of optical photons random delay (optical + electronic) Distribution of registration times Electronics Timestamp

  8. Registration Time Distribution Common StartingPoint Emission at t = Θ Absorption Emission of optical photons random delay (optical + electronic) Distribution of registration times ~200ps Probability Density Electronics Timestamp Delay Delay distribution

  9. Registration Time Distribution Common StartingPoint ~200ps Probability Density Delay

  10. Ptn(t|Θ) ptn(t|Θ) Ptn(t|Θ) ptn(t|Θ) x40 Exemplary ptn(tts|Θ) and Ptn(tts|Θ) for LYSO:Ce Common StartingPoint Θ= γ-interaction time (here 0 ps) ptn(t|Θ) = time stamp pdf Ptn(t|Θ) = time stamp cdf Parameters: rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian):σ= 125 ps

  11. An analytical model for time resolution of a scintillation detectors with analog SiPMs

  12. Analog SiPM response to single individual scintillation photons Analog SiPMs

  13. Analog SiPM response to single individual scintillation photons Analog SiPMs • Some more assumptions • SPS are additive • SPS given by (constant) shape function andfluctuating gain: • pdfto measure a signal v at a given time tgiven:

  14. Analog SiPM response to single individual scintillation photons Analog SiPMs Calculate expectation value and variance for SPS:

  15. Response to Scintillation Pulses Analog SiPMs • SPS are independent and additive • average number of detected scintillation photons • (‘primary triggers’) • with • standard deviation of Npt (taking into account the • intrinsic energy resolution o the scintillator) • Linear approximation of the timing uncertainty

  16. Response to Scintillation Pulses Analog SiPMs • SPS are independent and additive • average number of detected scintillation photons • (‘primary triggers’) • with • standard deviation of Npt (taking into account the • intrinsic energy resolution o the scintillator) • Linear approximation of the timing uncertainty • Here, we can add electronic noise in a simple manner

  17. Comparison to Measurements Analog SiPMs

  18. Some properties of the model: Analog SiPMs • compares reasonably well to measurements • reduces to Hyman model for Poisson distributed Npt, negligible cross-talk, and negligible electronic noise • absolute values for time resolution BUT • many input parameters are more difficult to measure than CRT • predictive power strongly depends on the accuracy of the input parameters

  19. Lower Bound on the time resolution of ideal scintillation photon counters

  20. one timestamp for the nth detected scintillation photon The Ideal Photon Counter and Derivatives ideal photon counters • Detected scintillation photons are independent and identically distributed (i.i.d.) • Capable of producing timestamps for individualdetected photons • ‘Ideal’ does not mean that the timestamps are noiseless n timestamps for the first n detected scintillation photons timestamps for all detected scintillation photons

  21. The Scintillation Detection Chain ideal photon counters (γ-)Source Emission Emitted Particle (γ-Photon) Absorption Scintillation Crystal Emission of optical photons Detection of optical photons Sensor Electronics Signal Timestamp

  22. The Scintillation with the (full) IPC ideal photon counters Emission again, considered to be instantaneous at t = Θ Absorption Emission of NSC optical photons Te,N= {te,1, te,2 ,…,te,N} Detection of N optical photons TN = {t1, t2,…,tN} Ξ(Estimate of Θ)

  23. What is the best possible Timing resolution obtainable for a given γ-Detector? ideal photon counters What is minimum variance of Ξ for a given set TN?

  24. Fisher Information and the Cramér–Rao Lower Bound ideal photon counters Our question can be answered if we can find the (average) Fisher Information in TN(or a chosen subset)

  25. The Fisher Information for the IPCa ) full time stamp information ideal photon counters Average information in a (randomly chosen) single timestamp: Θ= γ-interaction time tn= (random) time stamp

  26. The Fisher Information for the IPCa ) full time stamp information ideal photon counters Average information in a (randomly chosen) single timestamp: Θ= γ-interaction time tn= (random) time stamp ptn(t|Θ) = time stamp pdf pdf describing the distribution of time stamps after a γ-interaction at Θ(as defined earlier)

  27. The Fisher Information for the IPCa ) full time stamp information ideal photon counters Average information in a (randomly chosen) single timestamp: Θ= γ-interaction time tn= (random) time stamp ptn(t|Θ) = time stamp pdf Information in independent samples is additive:

  28. The Fisher Information for the IPCa ) full time stamp information ideal photon counters Average information in a (randomly chosen) single timestamp: Θ= γ-interaction time tn= (random) time stamp ptn(t|Θ) = time stamp pdf Information in independent samples is additive: Regardless of the shape of ptn(t|Θ)or the estimator

  29. The Fisher Information for the IPCb ) single time stamp information ideal photon counters Introducing order in TN Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps

  30. The Fisher Information for the IPCb ) single time stamp information ideal photon counters creating an ordered setT(N)= {t(1), t(2),…, t(n)}t(1) < t(2) … t(N-1) < t(N) Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps

  31. The Fisher Information for the IPCb ) single time stamp information ideal photon counters creating an ordered setT(N)= {t(1), t(2),…, t(n)}t(1) < t(2) … t(N-1) < t(N) Find the pdff(n)|N(t|Θ) describing the distribution of the ‘nth order statistic’ (which fortunately is textbook stuff) Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps t(n) = nth element of T(N) f(n)|N(t|Θ)= pdffor t(n) H. A. David 1989, “Order Statistics” John Wiley & Son, Inc, ISBN 00-471-02723-5

  32. n = 1 n = 5 n = 10 n = 15 n = 20 The Fisher Information for the IPCb ) single time stamp information ideal photon counters Exemplary f(n)|N(t |Θ) for LYSO Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps t(n) = nth element of T(N) f(n)|N(t|Θ)= pdffor t(n) Parameters: rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian):σ= 120 ps

  33. The Fisher Information for the IPCb ) single time stamp information ideal photon counters creating an ordered setT(N)= {t(1), t(2),…, t(n)}t(1) < t(2) … t(N-1) < t(N) Find the f(n)|N(t|Θ) The rest is formality: Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps t(n) = nth element of T(N) f(n)|N(t|Θ)= pdffor t(n) I(n)|N(Θ)= FI regarding Θ carried by the nth time stamp Essentially corresponds to the single photon variance as calculated by Matt FishburnM W and Charbon E 2010 “System Tradeoffs in Gamma-Ray Detection Utilizing SPAD Arrays and Scintillators” IEEE Trans. Nucl. Sci. 57 2549–2557

  34. Single Time Stamp vs. Full Information ideal photon counters Best possible single photon timing rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian):σ= 125 ps Primary triggers: N = 4700 This limit holds for all scintillation detectors that share the properties used as input parameters We probably, the intrinsic limit can be approached reasonably close, using a few, early time stamps, only – but how many do we need?

  35. The Fisher Information for the IPCc ) 1-to-nth time stamp information ideal photon counters …where things turn nasty …. Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps T(n) = subset containing the first n elements of T(N) t(n) = nth element of T(N)

  36. n = 1 n = 5 n = 10 n = 15 n = 20 The Fisher Information for the IPCc ) 1-to-nth time stamp information ideal photon counters …where things turn nasty …. Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps T(n) = subset containing the first n elements of T(N) t(n) = nth element of T(N) f(n)|N(t|Θ)= pdf for t(n) Exemplary f(n)|N(t|Θ) for LYSO:Ce t(n) are neither independent nor identically distributed!

  37. The Fisher Information for the IPCc ) 1-to-nth time stamp information ideal photon counters …where things turn nasty …. Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps T(n) = subset containing the first n elements of T(N) t(n) = nth element of T(N) f(n)|N(t|Θ)= pdffor t(n) t(n) are neither independent nor identically distributed FI needs to be calculated from the joint distribution function of the t(n), which is an n-fold integral. Not at all practical

  38. The Fisher Information for the IPCc ) 1-to-nth time stamp information ideal photon counters …where things turn nasty, ... or not, if someone solves the problem for you and shows that Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps T(n) = subset containing the first n elements of T(N) t(n) = nth element of T(N) f(n)|N(t|Θ)= pdffor t(n) F(n)|N(t|Θ)=cdffor t(n) I(1…n)|N(Θ)= FI regarding Θ carried by the first ntime stamps S. Park, ‘On the asymptotic Fisher information in order statistics’ Metrika, Vol. 57, pp. 71–80 (2003)

  39. The Fisher Information for the IPCc ) 1-to-nth time stamp information ideal photon counters …where things turn nasty, ... or not, if someone solves the problem for you and shows that Θ= γ-interaction time tn= (random) time stamp TN = set of Ntime stamps T(N) = ordered set of N time stamps T(n) = subset containing the first n elements of T(N) t(n) = nth element of T(N) f(n)|N(t|Θ)= pdffor t(n) F(n)|N(t|Θ)=cdffor t(n) I(1…n)|N(Θ)= FI regarding Θ carried by the first ntime stamps S. Park, ‘On the asymptotic Fisher information in order statistics’ Metrika, Vol. 57, pp. 71–80 (2003)

  40. The Fisher Information for the IPCc ) 1-to-nth time stamp information ideal photon counters LYSO:Ce LaBr3:5%Ce rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian):σ= 125 ps Primary triggers: N = 4700 rise times:τr1 = 280ps (71%); τr1 = 280ps (27%) decay times: τd1 = 15.4 ns (98%) τd1 = 130 ns (2%) TTS (Gaussian):σ= 125 ps Primary triggers: N = 6200

  41. Three Important Disclaimers ideal photon counters • pdf’s must be differentiablein between -0 and ∞ (e.g. h(t|Θ)=0 for a single-exponential-pulse) • Analog light sensors never trigger on single photon signals (even at very low thresholds)only the calculated “intrinsic limit” can directly be compared • In digital sensors nthtrigger may not correspond to t(n) (do to conditions imposed by the trigger network)

  42. Calculated Lower Bound vs. Literature Data ideal photon counters

  43. CRT limit vs. detector parameters The lower limit on the timing resolution

  44. Some (hopefully) interesting experimental data

  45. Fully digital SiPMs digital SiPMs dSiPM array Philips Digital Photon Counting • As analog SiPMs but with actively quenched SPADs • negligible noise at the single photon level • comparable PDE • excellent time jitter (~100ps) • 16 dies (4 x 4) • 16 timestamps • 64 photon count values

  46. Timing performance of monolithic scintillator detectors Monolithic crystal detectors • Reconstruction of the 1stphoton arrival time probability distribution function for each (x,y,z) position

  47. Timing performance of monolithic scintillator detectors Monolithic crystal detectors

  48. Timing performance of monolithic scintillator detectors Monolithic crystal detectors Use of MLITE method to determine the true interaction time Timing spectrum of the 16x16x10mm3monolithic crystal (with a 3x3x5mm3 reference) Using only the earliest timestamp: CRT ~ 200 ps – 230 ps FWHM H.T. van Dam, et al. “Sub-200 ps CRT in monolithic scintillator PET detectors using digital SiPM arrays and maximum likelihood interaction time estimation (MLITE)”, PMB at press

  49. Conclusions • The time resolution of scintillation detectors can be predicted accurately with analytical models • …as long as we do not have to include the photon transport • which can be included but that requires accurate estimates of the corresponding distributions • FI-CR formalism is a very powerful tool in determining intrinsic performance limits and the limiting factors • ..where the simplest form (full TN information) is often the most interesting • The calculation of IN is as simple as calculating an average • ML methods make efficient use of the available information (but require calibration)

  50. Some backup

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