1Stefan SeifertDelft University of Technology
Two Approaches to Modeling the Time Resolution of Scintillation
DetectorsS. Seifert, H.T. van Dam, D.R. Schaart
2Stefan SeifertDelft University of Technology
Outline
A common starting pointModeling (analog) SiPM timing response
Extended Hyman modelThe ideal photon counter
Fisher information and Cramér–Rao Lower Boundfull time stamp informationSingle time stamp information1-to-n time stamp information
Important disclaimersDiscussionSome (hopefully) interesting experimental data
Conclusions
3Stefan SeifertDelft University of Technology
A Common Starting Point
4Stefan SeifertDelft University of Technology
Com
mon
Sta
rtin
gPo
int
(γ-)Source
Emitted Particle(γ-Photon)
ScintillationCrystal
Sensor
Emission
Absorption
Emission of optical photonsDetection of optical photons
SignalElectronics
Timestamp
The Scintillation Detection Chain
5Stefan SeifertDelft University of Technology
Com
mon
Sta
rtin
gPo
int
Emission
Absorption
Emission
Detection
γ-Source
γ-Photon
ScintillationCrystal
Sensor
SignalElectronics
Timestamp
Necessary Assumptions:Scintillation photons are statistically independent and identically distributed in timePhoton transport delay, photon detection, and signal delay are statistically independentElectronic representations are independent and identically distributed
Assumptions
6Stefan SeifertDelft University of Technology
Com
mon
Sta
rting
Poin
t
Emission at t = Θ Absorption
Emission of optical photons
Registration of optical photons
random delay (optical + electronic)
pdf p(tr|Θ) describing the distribution of registration times of independent scintillation photon signals
Estimate on Θ
Registration Time Distribution p(tr|Θ)
7Stefan SeifertDelft University of Technology
Com
mon
Sta
rtin
gPo
int Assumptions
Assumptions that make life easier:Instantaneous γ-absorptionDistribution of scintillation photon delays is independent on location of the absorption OR,simplest case distribution of scintillation photon delays is negligible
Emission at t = Θ
Absorption
Emission of optical photons
Distribution of registration times
random delay (optical + electronic)
Electronics
Timestamp
8Stefan SeifertDelft University of Technology
Registration Time DistributionEmission at t = Θ
Absorption
Emission of optical photons
Distribution of registration times
random delay (optical + electronic)
Delay
~200 ps
Prob
abilit
y De
nsity
Electronics
Timestamp
Com
mon
Sta
rtin
gPo
int
9Stefan SeifertDelft University of Technology
Registration Time Distribution
Delay
~200 ps
Prob
abilit
y De
nsity
d, r,e
ec,d, r,
0 : ( )
( | ) 1 : ( )i i
t tt
ii i i
t t
p tP e e t t
d, r,e ec,
d, r, d, r,d, r,
0 : ( )
|: ( )i i
t tit
i i i ii ii
t t
PP te e t t
n trans e
n trans e0
| |
| |
t t t
t
t t t
p t p t p t t dt
P t p t P t t dt
2trans
2trans
trans
21 e2
t t
tp t
Com
mon
Sta
rtin
gPo
int
10Stefan SeifertDelft University of Technology
Θ = γ-interaction
time (here 0 ps)ptn(t|Θ) =
time stamp pdf Ptn(t|Θ) = time stamp
cdf
ptn(t|Θ) x40
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
Exemplary ptn(tts|Θ) and Ptn (tts|Θ) for LYSO:Ce
Com
mon
Sta
rtin
gPo
int
11Stefan SeifertDelft University of Technology
An analytical model for time resolution of a scintillation detectors with analog SiPMs
12Stefan SeifertDelft University of Technology
Analog SiPM response to single individual scintillation photons
Anal
og S
iPM
s
13Stefan SeifertDelft University of Technology
Analog SiPM response to single individual scintillation photons
Anal
og S
iPM
s
Some more assumptions– SPS are additive– SPS given by (constant)
shape function andfluctuating gain:
pdf to measure a signal v at a given time t given:
( , ) ( )v t a a f t
sps pt ptstp tr
0
| , D | ,v t v ptp v t p t p v t t dt
14Stefan SeifertDelft University of Technology
Analog SiPM response to single individual scintillation photons
Anal
og S
iPM
s
sps pt ptstr tr tr
0
| , D | ,v t vp v t p t p v t t dt
spsE | ,Dv t
spsvar | ,Dv t
Calculate expectation value and variance for SPS:
15Stefan SeifertDelft University of Technology
Response to Scintillation PulsesAn
alog
SiP
Ms
pt spsE | E | ,Dv t N v t
SPS are independent and additive
pt N
22pt sps pt spsvar | var | ,D E | ,Dv t N v t v t
with average number of detected scintillation photons(‘primary triggers’)standard deviation of Npt (taking into account the intrinsic energy resolution o the scintillator)
pt
tot th
t
thth
var |
E |
v t
v tt
Linear approximation of the timing uncertainty
16Stefan SeifertDelft University of Technology
Response to Scintillation PulsesAn
alog
SiP
Ms
pt spsE | E | ,Dv t N v t
SPS are independent and additive
pt N
22pt sps pt spsvar | var | ,D E | ,Dv t N v t v t
with average number of detected scintillation photons(‘primary triggers’)standard deviation of Npt (taking into account the intrinsic energy resolution o the scintillator)
pt
tot th
t
thth
var |
E |
v t
v tt
Linear approximation of the timing uncertaintyHere, we can add electronic noise in a simple manner
17Stefan SeifertDelft University of Technology
Comparison to MeasurementsAn
alog
SiP
Ms
18Stefan SeifertDelft University of Technology
Some properties of the model:An
alog
SiP
Ms
compares reasonably well to measurementsreduces to Hyman model for Poisson distributed Npt, negligible cross-talk, and negligible electronic noiseabsolute values for time resolution
BUTmany input parameters are more difficult to measure than CRTpredictive power strongly depends on the accuracy of the input parameters
19Stefan SeifertDelft University of Technology
Lower Bound on the time resolution of ideal scintillation photon counters
20Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
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
one timestamp for the nth
detected scintillation photon
timestamps for all detected scintillation photons
n timestamps for the first n detected scintillation photons
21Stefan SeifertDelft University of Technology
(γ-)Source
Emitted Particle(γ-Photon)
ScintillationCrystal
Sensor
Emission
Absorption
Emission of optical photonsDetection of optical photons
SignalElectronics
Timestamp
The Scintillation Detection Chainid
eal p
hoto
n co
unte
rs
22Stefan SeifertDelft University of Technology
The Scintillation with the (full) IPCid
eal p
hoto
n co
unte
rs
again, considered to be instantaneous
Te,N = {te,1, te,2 ,…,te,N}
TN = {t1, t2 ,…,tN}
at t = Θ
Ξ (Estimate of Θ)
Emission
Absorption
Emission of NSC optical photonsDetection of N optical photons
23Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
What is the best possible Timing resolution obtainable for a given γ-Detector?
What is minimum variance of Ξ for a given set TN?
24Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters Fisher Information and the
Cramér–Rao Lower Bound
Our question can be answered if we can find the (average) Fisher
Information in TN (or a chosen subset)
N
1varTI
25Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCa ) full time stamp information
Average information in a (randomly chosen) single timestamp:
Θ = γ-interaction
timetn =
(random) time
stamp
n n
2ln | |
nt t tI p t p t dt
def
26Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
Average information in a (randomly chosen) single timestamp:
Θ = γ-interaction
timetn =
(random) time
stampptn(t|Θ) = time
stamp pdf
n n
2ln | |
nt t tI p t p t dt
def
pdf describing the distribution of time stamps after a γ-interaction at Θ (as defined earlier)
The Fisher Information for the IPCa ) full time stamp information
27Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
Average information in a (randomly chosen) single timestamp:
n n
2ln | |
nt t tI p t p t dt
def
Information in independent samples is additive:
Θ = γ-interaction
timetn =
(random) time
stampptn(t|Θ) = time
stamp pdf
n n
2ln | |
NT t tI N p t p t dt
The Fisher Information for the IPCa ) full time stamp information
28Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
Average information in a (randomly chosen) single timestamp:
n n
2ln | |
nt t tI p t p t dt
def
Information in independent samples is additive:
n n
2ln | |
NT t tI N p t p t dt
1 1var ( )LB LBtN N
Regardless of the shape of ptn(t|Θ) or the estimator
Θ = γ-interaction
timetn =
(random) time
stampptn(t|Θ) = time
stamp pdf
The Fisher Information for the IPCa ) full time stamp information
29Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCb ) single time stamp information
1. Introducing order in TNΘ = γ-
interaction timetn =
(random) time stamp
TN = set of N time stamps
30Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCb ) single time stamp information
1. 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 N time stamps
T(N) = ordered set of N time stamps
31Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCb ) single time stamp information
1. creating an ordered setT(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)
H. A. David 1989, “Order Statistics” John Wiley & Son, Inc, ISBN 00-471-02723-5
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)
32Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCb ) single time stamp information
Exemplary f(n)|N(t |Θ) for LYSO
n = 1n = 5n = 10n = 15n = 20
Parameters:rise time:
τr = 75 psdecay time:
τd = 44 nsTTS (Gaussian): σ = 120 ps
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)
33Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCb ) single time stamp information
1. creating an ordered setT(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:
2
( )| ( )| ( )|ln | |n N n N n NI f t f t dt
def
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 ,( )|( )|
1var ( )LB n Nn NI
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
34Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
Single Time Stamp vs. Full Information
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
Best possible single photon
timing
We probably, the intrinsic limit can be approached reasonably close, using a few, early time stamps, only – but how many do
we need?
35Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCc ) 1-to-nth time stamp information
…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)
36Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCc ) 1-to-nth time stamp information
…where things turn nasty ….
t(n) are neither independent nor identically distributed!
n = 1n = 5n = 10n = 15n = 20
Exemplary f(n)|N(t|Θ) for LYSO:Ce
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)
37Stefan SeifertDelft University of Technology
The Fisher Information for the IPCc ) 1-to-nth time stamp information
t(n) are neither independent nor identically distributed
…where things turn nasty ….
FI needs to be calculated from the joint distribution function of the t(n),
which is an n-fold integral.
Not at all practical
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)
idea
l pho
ton
coun
ters
38Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCc ) 1-to-nth time stamp information
…where things turn nasty,... or not, if someone solves the problem for you and shows that
S. Park, ‘On the asymptotic Fisher information in order statistics’ Metrika, Vol. 57, pp. 71–80 (2003)
n
2
(1...n)|N t1ln | Pr | |nI h t t t p t dt
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
39Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCc ) 1-to-nth time stamp information
…where things turn nasty,... or not, if someone solves the problem for you and shows that
S. Park, ‘On the asymptotic Fisher information in order statistics’ Metrika, Vol. 57, pp. 71–80 (2003)
n
2
(1...n)|N t1ln | Pr | |nI h t t t p t dt
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
n
n
||
1- |t
t
p th t
P t
1 |1- |n NF t
40Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
The Fisher Information for the IPCc ) 1-to-nth time stamp information
rise time:τr = 75 ps
decay time:τd = 44 ns
TTS (Gaussian): σ = 125 psPrimary 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 psPrimary triggers: N = 6200
LYSO:Ce LaBr3:5%Ce
41Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
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)
42Stefan SeifertDelft University of Technology
idea
l pho
ton
coun
ters
Calculated Lower Bound vs. Literature Data
43Stefan SeifertDelft University of Technology
The
lowe
r lim
it on
the
timin
g re
solu
tion CRT limit vs. detector parameters
44Stefan SeifertDelft University of Technology
Some (hopefully) interesting experimental data
45Stefan SeifertDelft University of Technology
Fully digital SiPMs
As analog SiPMs but with actively quenched SPADs
negligible noise at the single photon level
comparable PDE
excellent time jitter (~100ps)
digi
tal S
iPM
s
dSiPM array Philips Digital Photon Counting
16 dies (4 x 4)
16 timestamps 64 photon count
values
46Stefan SeifertDelft University of Technology
Timing performance of monolithic scintillator detectorsReconstruction of the 1st photon arrival time probability distribution function for each (x,y,z) position
Mon
olith
ic c
ryst
al
dete
ctor
s
47Stefan SeifertDelft University of Technology
Mon
olith
ic c
ryst
al
dete
ctor
s Timing performance of monolithic scintillator detectors
48Stefan SeifertDelft University of Technology
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
Use of MLITE method to determine the true interaction time
Crystal size (mm3)
CRT FWHM (ps)
16 x 16 x 10 15716 x 16 x 20 185
24 x 24 x 10 16124 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
Mon
olith
ic c
ryst
al
dete
ctor
s Timing performance of monolithic scintillator detectors
49Stefan SeifertDelft University of Technology
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 interestingThe calculation of IN is as simple as calculating an average
ML methods make efficient use of the available information (but require calibration)
Conclusions
50Stefan SeifertDelft University of Technology
Some backup
51Stefan SeifertDelft University of Technology
digi
tal S
iPM
s
Timing performance with small scintillator pixels (reference)
Detector size
(mm3)
CRT FWHM
(ps)
Photopeak position
(# fired cells)
Photopeak position(# primary triggers)
3 × 3 × 5 121 2141 38353 × 3 × 5 120 2147 38623 × 3 × 5 131 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
• 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
52Stefan SeifertDelft University of Technology
Monolithic crystal detectorsM
onol
ithic
cry
stal
de
tect
ors
53Stefan SeifertDelft University of Technology
Light distribution
depends on the position of
interaction …
including the depth of
interaction (DOI).
Interaction position encodingM
onol
ithic
cry
stal
de
tect
ors
x
zcrystallight
sensor
crystal
54Stefan SeifertDelft University of Technology
x
zcrystallight
sensor
crystal
Light intensity distribution
high
low
In reality there is:photon statisticsdetector noisereflections in crystal
Interaction position encodingM
onol
ithic
cry
stal
de
tect
ors
55Stefan SeifertDelft University of Technology
Mon
olith
ic c
ryst
al
dete
ctor
s
Referencedetector• 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
Detector test & calibration stage
56Stefan SeifertDelft University of Technology
Paired CollimatorM
onol
ithic
cry
stal
de
tect
ors
57Stefan SeifertDelft University of Technology
x-y-Position Estimation in monolithic scintillator detectors: Improved k-NN method
58Stefan SeifertDelft University of Technology
24 × 24 ×20 mm3 LSO on dSiPM arrayirradiated with 0.5mm 511keV beam
FHTM = 1.64 mm
FHTM = 1.61mm
FWTM = 5.4 mm
FWTM = 5.5 mm
Mon
olith
ic c
ryst
al
dete
ctor
s