1. 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

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

5. 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

6. 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

7. 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

8. 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

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

10. 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

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 and fluctuating gain:
pdf to measure a signal v at a given time t given:

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
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

16. 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

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. 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

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 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?

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 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

26. 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)

27. 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:

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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 –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 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)

36. 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!

37. 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

38. 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)

39. 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)

40. 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

41. 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)

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

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

44. Some (hopefully) interesting experimental data

45. 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

46. 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.

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 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 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

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

51. 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

52. Monolithic crystal detectors

53. 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).

54. 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

55. Detector test & calibration stage
Monolithic crystal detectors
Reference
detector
PDPC dSiPM (DPC )
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

56. Monolithic crystal detectors
Paired Collimator
Monolithic crystal detectors

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

58. 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