Two Approaches to Modeling the Time Resolution of Scintillation Detectors

1Stefan SeifertDelft University of Technology

Two Approaches to Modeling the Time Resolution of Scintillation

DetectorsS. Seifert, H.T. van Dam, D.R. Schaart


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


A Common Starting Point


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


Com

mon

Sta

rtin

gPo

int

Emission

Absorption

Emission

Detection

γ-Source

γ-Photon


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


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)


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


Distribution of registration times


Electronics

Timestamp


Registration Time DistributionEmission at t = Θ

Absorption


Distribution of registration times


Delay

~200 ps

Prob

abilit

y De

nsity

Electronics

Timestamp

Com

mon

Sta

rtin

gPo

int


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


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


Analog SiPM response to single individual scintillation photons

Anal

og S

iPM

s



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



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:


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


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


Comparison to MeasurementsAn

alog

SiP

Ms


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


Lower Bound on the time resolution of ideal scintillation photon counters


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


(γ-)Source

Emitted Particle(γ-Photon)


Sensor

Emission

Absorption

Emission of optical photonsDetection of optical photons

SignalElectronics

Timestamp

The Scintillation Detection Chainid

eal p

hoto

n co

unte

rs


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


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?


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


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


idea

l pho

ton

coun

ters


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)



idea

l pho

ton

coun

ters


n n

2ln | |

nt t tI p t p t dt

def

Information in independent samples is additive:

Θ = γ-interaction

timetn =

(random) time


stamp pdf

n n

2ln | |

NT t tI N p t p t dt



idea

l pho

ton

coun

ters


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


stamp pdf



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


idea

l pho

ton

coun

ters


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


T(N) = ordered set of N time stamps


idea

l pho

ton

coun

ters



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





t(n) = nth element of T(N)

f(n)|N(t|Θ)= pdf for t(n)


idea

l pho

ton

coun

ters


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








idea

l pho

ton

coun

ters



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






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


idea

l pho

ton

coun

ters

Single Time Stamp vs. Full Information



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?


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




T(n) = subset containing the first n elements of T(N)



idea

l pho

ton

coun

ters


…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


tn = (random) time stamp




t(n) = nth element of T(N) f(n)|N(t|Θ)= pdf for t(n)



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








idea

l pho

ton

coun

ters


idea

l pho

ton

coun

ters


…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







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


idea

l pho

ton

coun

ters


…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


tn = (random) time stamp




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


idea

l pho

ton

coun

ters




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


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)


idea

l pho

ton

coun

ters

Calculated Lower Bound vs. Literature Data


The

lowe

r lim

it on

the

timin

g re

solu

tion CRT limit vs. detector parameters


Some (hopefully) interesting experimental data


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


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


Mon

olith

ic c

ryst

al

dete

ctor

s Timing performance of monolithic scintillator detectors


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


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


Some backup


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


Monolithic crystal detectorsM

onol

ithic

cry

stal

de

tect

ors


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


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


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


Paired CollimatorM

onol

ithic

cry

stal

de

tect

ors


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


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

Two Approaches to Modeling the Time Resolution of Scintillation Detectors

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Transcript of Two Approaches to Modeling the Time Resolution of Scintillation Detectors