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