S. Seifert, H.T. van Dam, D.R. Schaart 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 full time stamp information Single time stamp information 1-to-n time stamp information Important disclaimers Discussion Some (hopefully) interesting experimental data Conclusions

A Common Starting Point

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

Assumptions Necessary Assumptions: Common Starting Point Emission Absorption Detection γ-Source γ-Photon Scintillation Crystal Sensor Signal Electronics Timestamp 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

Registration Time Distribution p(tr|Θ) Common Starting Point Emission at t = Θ pdf p(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

Assumptions Assumptions that make life easier: Common Starting Point Emission at t = Θ Absorption Emission of optical photons Distribution of registration times random delay (optical + electronic) Electronics Timestamp 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

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

Registration Time Distribution Common Starting Point Delay ~200 ps Probability Density

Exemplary ptn(tts|Θ) and Ptn (tts|Θ) for LYSO:Ce Common Starting Point ptn(t|Θ) x40 Ptn(t|Θ) ptn(t|Θ) 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

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

Analog SiPM response to single individual scintillation photons Analog SiPMs

Analog SiPM response to single individual scintillation photons Analog SiPMs Some more assumptions SPS are additive SPS given by (constant) shape function and fluctuating gain: pdf to measure a signal v at a given time t given:

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

Response to Scintillation Pulses Analog SiPMs SPS are independent and additive with 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

Response to Scintillation Pulses Analog SiPMs SPS are independent and additive with 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 Here, we can add electronic noise in a simple manner

Comparison to Measurements Analog SiPMs

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

Lower Bound on the time resolution of ideal scintillation photon counters

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 individual detected photons ‘Ideal’ does not mean that the timestamps are noiseless one timestamp for the nth detected scintillation photon timestamps for all detected scintillation photons n timestamps for the first n detected scintillation photons

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

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)

What is minimum variance of Ξ for a given set TN? 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?

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)

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

The Fisher Information for the IPC a ) 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)

The Fisher Information for the IPC a ) 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:

The Fisher Information for the IPC a ) 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

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

The Fisher Information for the IPC b ) single time stamp information ideal photon counters creating an ordered set T(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 N time stamps T(N) = ordered set of N time stamps

The Fisher Information for the IPC b ) single time stamp information ideal photon counters creating an ordered set T(N) = {t(1), t(2),…, t(n)} t(1) < t(2) … t(N-1) < t(N) Find the pdf f(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 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

The Fisher Information for the IPC b ) 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 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) n = 1 n = 5 n = 10 n = 15 n = 20 Parameters: rise time: τr = 75 ps decay time: τd = 44 ns TTS (Gaussian): σ = 120 ps

The Fisher Information for the IPC b ) single time stamp information ideal photon counters creating an ordered set T(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 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 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

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?

The Fisher Information for the IPC c ) 1-to-nth time stamp information ideal photon counters …where things turn nasty …. Θ = γ-interaction time tn = (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)

The Fisher Information for the IPC c ) 1-to-nth time stamp information ideal photon counters …where things turn nasty …. Θ = γ-interaction time tn = (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) n = 1 n = 5 n = 10 n = 15 n = 20 Exemplary f(n)|N(t|Θ) for LYSO:Ce t(n) are neither independent nor identically distributed!

t(n) are neither independent nor identically distributed The Fisher Information for the IPC c ) 1-to-nth time stamp information ideal photon counters …where things turn nasty …. Θ = γ-interaction time tn = (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) 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

The Fisher Information for the IPC c ) 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 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)

The Fisher Information for the IPC c ) 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 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)

The Fisher Information for the IPC c ) 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

Three Important Disclaimers ideal photon counters 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)

Calculated Lower Bound vs. Literature Data ideal photon counters

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

Some (hopefully) interesting experimental data

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

Timing performance of monolithic scintillator detectors Monolithic crystal detectors Reconstruction of the 1st photon arrival time probability distribution function for each (x,y,z) position Put a geometry picture and say that these plots are for calibration data to reconstruct the first photon arrival time pdf.

Timing performance of monolithic scintillator detectors Monolithic crystal detectors

Timing performance of monolithic scintillator detectors Monolithic crystal detectors Use of MLITE method to determine the true interaction time Timing spectrum of the 16x16x10 mm3 monolithic crystal (with a 3x3x5 mm3 reference) 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 Point out that in Valencia the time resolution was 350 ps because there was no electronic skew correction and ML method If only electronic skew correction is used, time resolutions in the order of 200-230 ns can be obtained using the earliest timestamp Using ML gives the best results Point out that all these values are already estimated values for equal detector in coincidence 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

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)

Some backup

Timing performance with small scintillator pixels (reference) digital SiPMs 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 Detector size (mm3) CRT FWHM (ps) Photopeak position (# fired cells) (# primary triggers) 3 × 3 × 5 121 2141 3835 120 2147 3862 131 2133 3799 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

Monolithic crystal detectors

Interaction position encoding Monolithic crystal detectors x z crystal light sensor Light distribution depends on the position of interaction … crystal including the depth of interaction (DOI).

Interaction position encoding Monolithic crystal detectors x z crystal light sensor In reality there is: photon statistics detector noise reflections in crystal crystal Light intensity distribution high low

Detector test & calibration stage Monolithic crystal detectors Reference detector PDPC dSiPM (DPC-3200-44-22) LSO:Ce (LSO:Ce,Ca) crystals (Agile) Source: 22Na in a tungsten collimator beam  ~0.5 mm Wrapped with Teflon Temperature chamber: -25°C Sensor temperature stabilization system Detector under test Irradiation with  0.5mm 511keV beam

Monolithic crystal detectors Paired Collimator Monolithic crystal detectors

x-y-Position Estimation in monolithic scintillator detectors: Improved k-NN method

24 × 24 ×20 mm3 LSO on dSiPM array irradiated with  0.5mm 511keV beam Monolithic crystal detectors FHTM = 1.61mm FWTM = 5.4 mm FHTM = 1.64 mm FWTM = 5.5 mm Irradiation with  0.5mm 511keV beam