Report

Two Approaches to Modeling the Time Resolution of Scintillation Detectors S. Seifert, H.T. van Dam, D.R. Schaart Stefan Seifert Delft University of Technology 1 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 Stefan Seifert Delft University of Technology 2 A Common Starting Point Stefan Seifert Delft University of Technology 3 Common Starting Point The Scintillation Detection Chain (γ-)Source Emitted Particle (γ-Photon) Scintillation Crystal Emission Absorption Emission of optical photons Detection of optical photons Sensor Electronics Signal Stefan Seifert Delft University of Technology Timestamp 4 Common Starting Point Assumptions Necessary Assumptions: γ-Source γ-Photon Emission Absorption Scintillation Crystal Sensor Scintillation photons are statistically independent and identically distributed in time Photon transport delay, photon Emission Detection detection, and signal delay are statistically independent Electronic representations are independent and identically Signal distributed Electronics Timestamp Stefan Seifert Delft University of Technology 5 Common Starting Point Registration Time Distribution p(tr|Θ) Emission at t = Θ Absorption Emission of optical photons pdf p(tr|Θ) describing the distribution of registration times of independent scintillation photon signals random delay (optical + electronic) Registration of optical photons Estimate on Θ Stefan Seifert Delft University of Technology 6 Common Starting Point Assumptions Emission at t = Θ Absorption Assumptions that make life easier: Instantaneous γ-absorption Emission of optical photons random delay (optical + electronic) Distribution of registration times Distribution of scintillation photon delays is independent on location of the absorption OR, simplest case distribution of scintillation photon delays is negligible Electronics Timestamp Stefan Seifert Delft University of Technology 7 Emission at t = Θ Absorption Emission of optical photons random delay (optical + electronic) Distribution of registration times Electronics Probability Density Common Starting Point Registration Time Distribution ~200 ps Delay Timestamp Stefan Seifert Delft University of Technology 8 Probability Density Common Starting Point Registration Time Distribution ~200 ps Delay pt (t | ) e t : (t ) 0 1 P ec, i i d,i r,i t t d, i e e r,i pt trans t t : (t ) 1 2 e 2 t t trans 2 2 trans Pt t | e t : (t ) 0 i Pec, i d, i r, i t t e d,i e r,i d, i r, i d, i r, i pt n t | pt trans t t t | dt t Pt t t | dt pt e t : (t ) t Pt n t | pt trans e 0 Stefan Seifert Delft University of Technology 9 Common Starting Point Exemplary ptn(tts|Θ) and Ptn (tts|Θ) for LYSO:Ce Θ = γ-interaction time (here 0 ps) ptn(t|Θ) = time stamp pdf Ptn(t|Θ) = time stamp cdf Ptn(t|Θ) ptn(t|Θ) Ptn(t|Θ) Parameters: rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian): σ = 125 ps ptn(t|Θ) x40 Stefan Seifert Delft University of Technology 10 An analytical model for time resolution of a scintillation detectors with analog SiPMs Stefan Seifert Delft University of Technology 11 Analog SiPMs Analog SiPM response to single individual scintillation photons Stefan Seifert Delft University of Technology 12 Analog SiPMs Analog SiPM response to single individual scintillation photons Some more assumptions – SPS are additive – SPS given by (constant) shape function and fluctuating gain: v (t , a ) a f (t ) pdf to measure a signal v at a given time t given: p vsps v | t , D p t p v | t , t dt t pt tp v pts tr 0 Stefan Seifert Delft University of Technology 13 pt Analog SiPMs Analog SiPM response to single individual scintillation photons p vsps v | t , D p t p v | t , t dt t pt tr v pts tr tr 0 Calculate expectation value and variance for SPS: E v sps | t , D var v sps | t , D Stefan Seifert Delft University of Technology 14 Analog SiPMs Response to Scintillation Pulses SPS are independent and additive E v | t N pt E v sps | t , D var v | t N pt var v sps | t , D with N pt pt 2 pt E v sps | t , D 2 average number of detected scintillation photons (‘primary triggers’) standard deviation of Npt (taking into account the intrinsic energy resolution o the scintillator) Linear approximation of the timing uncertainty t vartot v | tth tth Stefan Seifert Delft University of Technology E v | tth 15 Analog SiPMs Response to Scintillation Pulses SPS are independent and additive E v | t N pt E v sps | t , D var v | t N pt var v sps | t , D with N pt pt 2 pt E v sps | t , D 2 average number of detected scintillation photons (‘primary triggers’) standard deviation of Npt (taking into account the intrinsic energy resolution o the scintillator) Linear approximation of the timing uncertainty t vartot v | tth tth Stefan Seifert Delft University of Technology E v | tth Here, we can add electronic noise in a simple manner 16 Analog SiPMs Comparison to Measurements Stefan Seifert Delft University of Technology 17 Analog SiPMs Some properties of the model: 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 Stefan Seifert Delft University of Technology 18 Lower Bound on the time resolution of ideal scintillation photon counters Stefan Seifert Delft University of Technology 19 ideal photon counters The Ideal Photon Counter and Derivatives Detected scintillation photons are independent and identically distributed (i.i.d.) Capable of producing timestamps for individual detected photons ‘Ideal’ does not mean that the timestamps are noiseless timestamps for all detected scintillation photons Stefan Seifert Delft University of Technology one timestamp for the nth detected scintillation photon n timestamps for the first n detected scintillation photons 20 ideal photon counters The Scintillation Detection Chain (γ-)Source Emitted Particle (γ-Photon) Scintillation Crystal Emission Absorption Emission of optical photons Detection of optical photons Sensor Electronics Signal Stefan Seifert Delft University of Technology Timestamp 21 ideal photon counters The Scintillation with the (full) IPC Emission Absorption again, considered to be instantaneous at t = Θ 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) Stefan Seifert Delft University of Technology 22 ideal photon counters What is the best possible Timing resolution obtainable for a given γ-Detector? What is minimum variance of Ξ for a given set TN? Stefan Seifert Delft University of Technology 23 ideal photon counters Fisher Information and the Cramér–Rao Lower Bound var 1 I TN Our question can be answered if we can find the (average) Fisher Information in TN (or a chosen subset) Stefan Seifert Delft University of Technology 24 ideal photon counters The Fisher Information for the IPC a ) full time stamp information Average information in a (randomly chosen) single timestamp: I tn def ln p t n t | Stefan Seifert Delft University of Technology 2 p t n t | dt Θ tn = γ-interaction time = (random) time stamp 25 ideal photon counters The Fisher Information for the IPC a ) full time stamp information Average information in a (randomly chosen) single timestamp: I tn def ln p t n t | 2 p t n t | dt pdf describing the distribution of time stamps after a γ-interaction at Θ (as defined earlier) Stefan Seifert Delft University of Technology Θ = γ-interaction time tn = (random) time stamp ptn(t|Θ) = time stamp pdf 26 ideal photon counters The Fisher Information for the IPC a ) full time stamp information Average information in a (randomly chosen) single timestamp: I tn def ln p t n t | 2 p t n t | dt Information in independent samples is additive: I TN N ln p t n t | 2 Θ = γ-interaction time tn = (random) time stamp ptn(t|Θ) = time stamp pdf p t n t | dt Stefan Seifert Delft University of Technology 27 ideal photon counters The Fisher Information for the IPC a ) full time stamp information Average information in a (randomly chosen) single timestamp: I tn def ln p t n t | 2 Θ = γ-interaction time tn = (random) time stamp ptn(t|Θ) = time stamp pdf p t n t | dt Information in independent samples is additive: I TN N ln p t n t | 2 p t n t | dt varL B ( ) 1 N t LB Stefan Seifert Delft University of Technology 1 N Regardless of the shape of ptn(t|Θ) or the estimator 28 ideal photon counters The Fisher Information for the IPC b ) single time stamp information 1. Introducing order in TN Θ tn TN Stefan Seifert Delft University of Technology = γ-interaction time = (random) time stamp = set of N time stamps 29 ideal photon counters The Fisher Information for the IPC b ) single time stamp information 1. creating an ordered set T(N) = {t(1), t(2),…, t(n)} t(1) < t(2) … t(N-1) < t(N) Θ tn TN T(N) Stefan Seifert Delft University of Technology = γ-interaction time = (random) time stamp = set of N time stamps = ordered set of N time stamps 30 ideal photon counters The Fisher Information for the IPC b ) single time stamp information 1. creating an ordered set T(N) = {t(1), t(2),…, t(n)} t(1) < t(2) … t(N-1) < t(N) 2. Find the pdf f(n)|N(t |Θ) describing the distribution of the ‘nth order statistic’ (which fortunately is textbook stuff) Θ tn = γ-interaction time = (random) time stamp TN = set of N time stamps T(N) = ordered set of N time stamps t(n) = nth element of T(N) f(n)|N(t|Θ)= pdf for t(n) H. A. David 1989, “Order Statistics” John Wiley & Son, Inc, ISBN 00-471-02723-5 Stefan Seifert Delft University of Technology 31 ideal photon counters The Fisher Information for the IPC b ) single time stamp information Exemplary f(n)|N(t |Θ) for LYSO Θ tn n n n n n = = = = = 1 5 10 15 20 = γ-interaction time = (random) time stamp TN = set of N time stamps T(N) = ordered set of N time stamps t(n) = nth element of T(N) f(n)|N(t|Θ)= pdf for t(n) Parameters: rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian): σ = 120 ps Stefan Seifert Delft University of Technology 32 ideal photon counters The Fisher Information for the IPC b ) single time stamp information 1. creating an ordered set T(N) = {t(1), t(2),…, t(n)} t(1) < t(2) … t(N-1) < t(N) 2. Find the f(n)|N(t |Θ) 3. The rest is formality: I ( n )| N def Θ tn ln f ( n )| N t | f ( n )| N t | dt 2 = γ-interaction time = (random) time stamp TN = set of N time stamps T(N) = ordered set of N time stamps t(n) = nth element of T(N) f(n)|N(t|Θ)= pdf for t(n) I(n)|N(Θ) = FI regarding Θ carried by the nth time stamp varL B ,( n )| N ( ) 1 I ( n )| N Essentially corresponds to the single photon variance as calculated by Matt Fishburn M W and Charbon E 2010 “System Tradeoffs in Gamma-Ray Detection Utilizing SPAD Arrays and Scintillators” IEEE Trans. Nucl. Sci. 57 2549–2557 Stefan Seifert Delft University of Technology 33 ideal photon counters Single Time Stamp vs. Full Information Best possible single photon timing This limit holds for all scintillation detectors that share the properties used as input parameters rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian): σ = 125 ps Primary triggers: N = 4700 We probably, the intrinsic limit can be approached reasonably close, using a few, early time stamps, only – but how many do we need? Stefan Seifert Delft University of Technology 34 ideal photon counters The Fisher Information for the IPC c ) 1-to-nth time stamp information …where things turn nasty …. Θ tn TN T(N) T(n) t(n) Stefan Seifert Delft University of Technology = γ-interaction time = (random) time stamp = set of N time stamps = ordered set of N time stamps = subset containing the first n elements of T(N) = nth element of T(N) 35 ideal photon counters The Fisher Information for the IPC c ) 1-to-nth time stamp information …where things turn nasty …. Exemplary f(n)|N(t|Θ) for LYSO:Ce n=1 n=5 n = 10 n = 15 n = 20 Θ tn TN T(N) = = = = γ-interaction time (random) time stamp set of N time stamps 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) t(n) are neither independent nor identically distributed! Stefan Seifert Delft University of Technology 36 ideal photon counters The Fisher Information for the IPC c ) 1-to-nth time stamp information …where things turn nasty …. 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. Θ tn = γ-interaction time = (random) time stamp TN = set of N time 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) Not at all practical Stefan Seifert Delft University of Technology 37 ideal photon counters The Fisher Information for the IPC c ) 1-to-nth time stamp information …where things turn nasty, ... or not, if someone solves the problem for you and shows that I (1...n)|N ln h t | P r t n 1 t | p t n t | dt 2 Θ tn = γ-interaction time = (random) time stamp TN = set of N time 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) F(n)|N(t|Θ)=cdf for t(n) I(1…n)|N(Θ)= FI regarding Θ carried by the first n time stamps S. Park, ‘On the asymptotic Fisher information in order statistics’ Metrika, Vol. 57, pp. 71–80 (2003) Stefan Seifert Delft University of Technology 38 ideal photon counters The Fisher Information for the IPC c ) 1-to-nth time stamp information …where things turn nasty, ... or not, if someone solves the problem for you and shows that I (1...n)|N ln h t | P r t n 1 t | p t n t | dt 2 t | h t | 1- Pt t | n pt n 1 - F n 1 | N t | Θ tn TN T(N) = = = = γ-interaction time (random) time stamp set of N time stamps 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) F(n)|N(t|Θ)=cdf for t(n) I(1…n)|N(Θ) = FI regarding Θ carried by the first n time stamps S. Park, ‘On the asymptotic Fisher information in order statistics’ Metrika, Vol. 57, pp. 71–80 (2003) Stefan Seifert Delft University of Technology 39 ideal photon counters The Fisher Information for the IPC c ) 1-to-nth time stamp information LYSO:Ce rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian): σ = 125 ps Primary triggers: N = 4700 Stefan Seifert Delft University of Technology LaBr3:5%Ce 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 40 ideal photon counters Three Important Disclaimers pdf’s must be differentiable in 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 nth trigger may not correspond to t(n) (do to conditions imposed by the trigger network) Stefan Seifert Delft University of Technology 41 ideal photon counters Calculated Lower Bound vs. Literature Data Stefan Seifert Delft University of Technology 42 The lower limit on the timing resolution CRT limit vs. detector parameters Stefan Seifert Delft University of Technology 43 Some (hopefully) interesting experimental data Stefan Seifert Delft University of Technology 44 digital SiPMs Fully 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 Stefan Seifert Delft University of Technology 45 Monolithic crystal detectors Timing performance of monolithic scintillator detectors Reconstruction of the 1st photon arrival time probability distribution function for each (x,y,z) position Stefan Seifert Delft University of Technology 46 Monolithic crystal detectors Timing performance of monolithic scintillator detectors Stefan Seifert Delft University of Technology 47 Monolithic crystal detectors Timing performance of monolithic scintillator detectors Use of MLITE method to determine the true interaction time Crystal size (mm3) CRT FWHM (ps) 16 x 16 x 10 157 16 x 16 x 20 185 24 x 24 x 10 161 24 x 24 x 20 184 Timing spectrum of the 16x16x10 mm3 monolithic crystal (with a 3x3x5 mm3 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 Stefan Seifert Delft University of Technology 48 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) Stefan Seifert Delft University of Technology 49 Some backup Stefan Seifert Delft University of Technology 50 digital SiPMs Timing performance with small scintillator pixels (reference) • three LSO:Ce:Ca crystals 3×3×5 mm3 on different dSiPM arrays • all combinations measured to determine CRT for two identical detectors • best result: 120 ps FWHM H.T. van Dam, G. Borghi, “Sub-200 ps CRT in monolithic scintillator PET detectors using digital SiPM arrays and maximum likelihood interaction time estimation (MLITE)”, in submitted to PMB Stefan Seifert Delft University of Technology Detector size (mm3) 3×3×5 3×3×5 3×3×5 CRT FWHM (ps) 121 120 131 Photopeak position (# fired cells) 2141 2147 2133 Photopeak position (# primary triggers) 3835 3862 3799 51 Monolithic crystal detectors Monolithic crystal detectors Stefan Seifert Delft University of Technology 52 Monolithic crystal detectors Interaction position encoding x z crystal crystal light sensor Light distribution depends on the position of interaction … including the depth of interaction (DOI). Stefan Seifert Delft University of Technology 53 Monolithic crystal detectors Interaction position encoding x In reality there is: photon statistics z detector noise reflections in crystal crystal crystal light sensor Light intensity distribution high low Stefan Seifert Delft University of Technology 54 Monolithic crystal detectors Detector test & calibration stage • PDPC dSiPM (DPC-3200-44-22) • LSO:Ce (LSO:Ce,Ca) crystals (Agile) • Source: 22Na in a tungsten collimator Reference detector Detector under test beam ~0.5 mm • Wrapped with Teflon • Temperature chamber: -25°C • Sensor temperature stabilization system Stefan Seifert Delft University of Technology 55 Monolithic crystal detectors Paired Collimator Stefan Seifert Delft University of Technology 56 x-y-Position Estimation in monolithic scintillator detectors: Improved k-NN method Stefan Seifert Delft University of Technology 57 Monolithic crystal detectors 24 × 24 ×20 mm3 LSO on dSiPM array irradiated with 0.5mm 511keV beam FHTM = 1.61mm FWTM = 5.4 mm FHTM = 1.64 mm FWTM = 5.5 mm Stefan Seifert Delft University of Technology 58