Cascaded systems analyses of photon-counting x-ray detectorsbml.pusan.ac.kr/PublishFrame/Publication/PDF/IntCon/089... · 2013-05-28 · Single-photon counting (SPC) x-ray imaging

Cascaded systems analyses of photon-counting x-ray detectors

Jesse Tanguay,a,b Seungman Yun,c,d Ho Kyung Kimd and Ian A. Cunninghama,b

aRobarts Research Inst., Western University, London, Ontario, N6A 5C1 Canada;bDept. of Medical Biophysics, Western University, London, Ontario, N6A 5C1 Canada;

cDept. of Biomedical Engineering, Western University, London, Ontario, N6A 5C1 Canada;dSchool of Mechanical Engineering, Pusan National University, Jangjeon-dong, Geumjeong-gu, Busan 609-735,

Republic of Korea

ABSTRACT

Single-photon counting (SPC) x-ray imaging has the potential to improve image quality and enable new advancedenergy-dependent methods. Recently, cascaded systems analysis (CSA) has been extended to the description ofthe detective quantum e�ciency (DQE) of SPC detectors. In this article we apply the new CSA approach to thedescription of the DQE of hypothetical direct-conversion selenium (Se) and cadmium zinc telluride (CdZnTe)detectors including the e�ects of poly-energetic x-ray spectra, stochastic conversion of x-ray energy to electron-hole (e-h) pairs, depth-dependent collection of e-h pairs using the Hecht relation, additive electronic noise, andthresholding. Comparisons are made to an energy-integrating model. For this simple model, with the exceptionof thick (1 - 10 mm) Se-based convertors, we found that the SPC DQE was 5-20 % greater than that of the energy-integrating model. This trend was true even when additive noise was included in the SPC model and excludedfrom the energy-integrating model. However, the DQE of SPC detectors with poor collection e�ciency (such asthick (>1 mm) Se detectors) and high levels of additive noise can be degraded by 40-90 % for all energies andx-ray spectra considered. While photon-counting approaches are not yet ready for routine diagnostic imaging,the available DQE is equal to or higher than that of conventional energy-integrating detectors under a wide rangeof x-ray energies and convertor thickness. However, like energy-integrating detectors, the DQE of SPC detectorswill be degraded by the combination of poor collection e�ciency and high levels of additive noise.

1. INTRODUCTION

The prospect of single-photon counting (SPC) x-ray detectors for medical imaging has identi�ed potential bene�tsover conventional designs where detector signals are proportional to energy deposited during image-acquisitionintervals.1�4 These include reduced image noise from the stochastic processes of image formation includingescape or reabsorption of Compton scatter and photoelectric emssions, conversion to secondary quanta, sec-ondary spreading and collection, additive noise, broad-spectrum imaging, and others.5 Scatter reabsorptionand secondary-quanta sharing between detector elements is a form of cross-talk that can reduce the detectivequantum e�iciency (DQE)6 and energy discrimination7 of SPC detectors if not addressed. Adaptive-binningtechniques may mitigate this e�ect in measurements of total energy deposited by summing signals from a num-ber of elements surrounding each interaction.2 However, the e�ects of stochastic image-forming processes willcontinue to degrade the DQE of SPC systems.

Cascaded systems analyses (CSA) are widely used to describe the DQE of conventional energy-integratingsystems and recently have been applied to SPC detectors for the description of signal and noise.8,9 Withthis approach, the DQE of SPC systems can be described under certain conditions with the additional newrequirement of transfering the probability density function (PDF) describing the number of image formingquanta through each stage of a cascade of elementary image-forming processes (plus conventional signal and noisetransfer). Assuming linear and wide-sense-stationary (WSS) or wide-sense-cyclostationary (WSCS) processes,the SPC DQE is given by

DQE (k) =c2T2 (k)

qoWc (k)(1)

where c is the mean SPC detector signal (proportional to the number of interacting photons), T (k) is themodulation transfer function (MTF), qo [mm−2] is the mean number of incident x-ray quanta per unit area, andWc (k) is the SPC NPS. In a previous work we presented a CSA model of an x-ray detector that included the

Medical Imaging 2013: Physics of Medical Imaging, edited by Robert M. Nishikawa, Bruce R. Whiting, Christoph Hoeschen, Proc. of SPIE Vol. 8668, 86680S · © 2013 SPIE · CCC code: 1605-7422/13/$18 · doi: 10.1117/12.2008132

Proc. of SPIE Vol. 8668 86680S-1

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/22/2013 Terms of Use: http://spiedl.org/terms

e�ects of stochastic conversion gain, incomplete collection e�ciency, additive electric noise, and thresholding,assuming low count rates, negligible spreading of quanta between neighboring detector elements, and mono-energetic x-ray spectra. In this article we extend this model to include depth dependent collection of chargesliberated in a photoconductor and poly-energetic x-ray spectra. To accomplish this goal we develop new transferrelationships for image-forming processes where gain and/or selection variables are functions of a random variable,such as the depth of interaction.

2. THEORY

2.1 SPC image signal

An SPC image is generated by identifying x-ray interaction events in a series of fast detector readout operationshaving intervals at [s]. We use c†(r) to represent a digital image consisting of a uniform two-dimensional grid ofδ functions at positions rnm scaled by the SPC signals from each element, cnm, giving

c† (r) =

∞∑n=−∞

∞∑m=−∞

cnm δ (r− rnm) . (2)

We use the overhead ˜ to indicate a random variable (RV) and † to indicate a scaled δ function. For WSCSconditions, the mean sampled SPC image signal is given by8,9

c† (r) =

∞∑n=−∞

∞∑m=−∞

c δ (r− rnm) (3)

where c represents the mean number of photon counts detected in an element. We let d (r) represent thepresampling and prethresholding signal for one fast readout of the x-ray detector and let pd(d; r) represent thePDF for d(r). The quantity d (r) is proportional to the total energy deposited in a detector element centered atposition r during one readout interval. We assume d (r) is a WSS RV and therefore has the PDF pd(d; r) = pd(d).Thus while d is a function of r, its PDF is not, and the mean signal from an SPC detector element is given by8,9

c = M

ˆ ∞t

pd (d) dd (4)

whereM is the number of readouts during an x-ray exposure. The above expression demonstrates that the meansignal from an SPC detector is shift invariant and can be determined simply from the PDF of d.

2.2 SPC autocovariance and Wiener noise power spectrum

For WSS or WSCS conditions, the NPS of an SPC image is given by the Fourier transform of the autocovarianceof c†:

Wc† (k) = F {Kc† (τ )} (5)

We have previously demonstrated that c† is a WSCS RV and therefore has autocovariance function given by

Kc† (τ ) =

∞∑n=−∞

∞∑m=−∞

Kc (τ ) δ (τ − rnm) (6)

where Kc (τ ) represents the presampling autocovariance function, given by8,9

Kc (τ ) = M

ˆ ∞t

ˆ ∞t

pd (d, d′; τ ) dddd′ −M[ˆ ∞

t

pd (d) dd

]2(7)

where pd (d, d′; τ ) is the joint PDF for two elements separated by τ . Equations (3) to (7) completely describethe �rst and second order statistics of an SPC image. The above expression shows that the SPC autocovarianceis determined by the joint PDF of d (r) and d (r + τ ).



2.3 Special case: Low count rates and no charge sharing

As a special case, we consider a low-count-rate limit where the probability that two photons deposit energy inthe same element during a single readout is negligible (no pulse pile-up) and no charge sharing between elements.This may be a good assumption for photoconductor-based detectors such as Se, but possibly not for CsI-basedsystems where optical scatter will cause sharing of energy from a single x-ray photon between more than oneelement. However, if adaptive binning is implemented to sum signals from elements surrounding a primaryinteraction, this assumption may also be valid. In a previous work we showed that under these conditions thepresampling mean SPC image signal and NPS can be expressed as

c = qoaξ1 (t)

[1 +

1− λλ

ξ0 (t)

ξ1 (t)

](8)

and

Wc (k) = σ2ca sinc2 (axu) sinc2 (ayv) (9)

where u and v are spatial frequencies in the x and y directions, respectively, and σ2c is the variance in the number

of counts detected in a single detector element, given by

σ2c = qoaξ1 (t)

[1 +

(1− λ)2

λ

(1

1− λ− ξ0 (t)

)ξ0 (t)

ξ1 (t)− 2 (1− λ) ξ0 (t)

](10)

where λ = ¯qoaat is the average number of photons incident on a detector element of area a during a readoutinterval of length at, and ξ0 (t) and ξ1 (t) represent the probability that the signal generated in a detector elementis greater than the threshold t for the case of zero or one photons incident on an element, respectively. Thesecond term in parenthesis in Eq. (8) corresponds to false-count events from additive electronic noise.

As demonstrated by Eqs. (8) and (9), calculation of the mean image signal and NPS requires the PDFdescribing the signal generated in a detector element for the case of zero and one incident photons. Previously,we introduced PDF transfer relationships that can be used to propagate of the PDF describing the total numberof quanta through elementary image forming processes. In the following sections we provide a simple extensionto the situation where gain and selection parameters are functions of a random variable.

2.4 PDF transfer through input-labeled elementary processes

In many detectors the number of liberated secondary image quanta (eg. electron-hole (e-h) pairs libeartedin a photoconductor) and collection e�ciency is a random function of depth of x-ray interaction within theconvertor material. Here we describe how to transfer the PDF through these input-labeled10 quantum selectionand quantum gain processes.

For each case, we let pin(Nin) and pout(Nout) represent the PDFs for the total number of input quanta Nin

and output quanta Nout, respectively. In general, the PDF for Nout is given by

pout (Nout) =

ˆ ∞0

pout (Nout|Nin) pin (Nin) dNin (11)

where pout(Nout|Nin) is the PDF for Nout given Nin. Since Nin and Nout only assume integer values, pout(Nout)can be expressed as

pout (Nout) =

∞∑n′=0

prout (Nout = n′) δ (Nout − n′) (12)

where

prout (Nout) =

∞∑n′′=0

prout (Nout|Nin = n′′) prin (Nin = n′′) (13)

is the probability mass function (PMF) for Nout.



2.4.1 Input-labeled quantum selection

An input-labeled quantum selection stage describes the process of randomly selecting quanta from an input dis-tribution when the selection probability is a function of an independent RV associated with each input quantum.For example, this will be used to allow the collection of electron-hole pairs to be a function of x-ray interactiondepth which is itself a RV.

The total number of output quanta is expressed as

Nout =

Nin∑i=1

αi(bi) (14)

where αi is a Bernoulli RV that assumes values of either 1 (probability α) or zero (probability 1 − α) and is afunction of bi which is a RV characterized by the PDF pb (b). The PMF for αi given bi is given by the Bernoullidistribution:11

pr(αi = k| bi

)=

(1k

)α(bi)

k[1− α(bi)

]1−kk ∈ {0, 1}

where α(bi) is the expected value of αi(bi) given bi. The PMF for Nout given Nin and {bi, i = 1..Nin} is obtainedby convolving11 pr(αi = k|bi) with pr(αj = k|bj) for all j resulting in n′′ − 1 convolutions:

prout (Nout = n′|Nin = n′′, b1, ..., bNin) = P

n′′∑i=1

αi

(bi

)= n′

(15)

= pr(α1 = k| b1

)∗ pr

(α2 = k| b2

)∗ ... ∗ pr

(αn′′ = k| bn′′

).(16)

Averaging over {bi, i = 1..Nin} yields

prout (Nout = n′|Nin = n′′) =

ˆ...

ˆ [pr(α1 = k| b1

)∗ pr

(α2 = k| b2

)∗

... ∗ pr(αn′′ = k| bn′′

)]p (b1, b2..., bn′′) db1...dbn′′ (17)

where p(b1, b2..., bn′′) is the joint PDF for {bi, i = 1..Nin}. The above equation is a general expression forprout(Nout = n′|Nin = n′′) for input-labeled quantum selection.

Special Case I: bi = bj = b. We consider the simplifying case where bi = bj = b for all i and j correspond-ing, for example, to the situation of incomplete collection of secondary quanta liberated at the same depth ofinteraction by one interacting photon. In this case Eq. (17) reduces to


ˆ [pr(α1 = k| b

)∗ pr

(α2 = k| b

)∗ ... ∗ pr

(αn′′ = k| b

)]p (b) db (18)

where pb (b) is the PDF describing b. The quantity in square brackets is simply the binomial distribution:11


(n′′

n′

)ˆα (b)

n′(1− α (b))

n′′−n′pb (b) db n′ ∈ {0, 1, .., n′′} . (19)

Special Case II: αi(bi) = αi. This case corresponds to the conventional quantum selection process where theselection variable is not a function of a random variable, and, therefore


(n′′

n′

)αn′(1− α)

n′′−n′n′ ∈ {0, 1, .., n′′} (20)

which is the same expression derived in previous work.9



2.4.2 Input-labeled quantum gain

A quantum gain stage represents the process of producing or liberating gi quanta, such as electron-hole pairsin a semiconductor, for the ith input quantum. Similar to the input-labeled quantum selection stage, the totalnumber of output quanta is given by

Nout =

Nin∑i=1

gi(bi). (21)

The PMF of Nout given Nin and {bi, i = 1..Nin} is then given by11

prout (Nout = n′|Nin = n′′, b1, ..., bNin) = prg

(g1 = k| b1

)∗ prg

(g2 = k| b2

)∗ ... ∗ prg

(gn′′ = k| bn′′

)where prg(gi|bi) is the PMF for g given bi. Averaging over {bi, i = 1..Nin} yields


ˆ...

ˆ [pr(g1 = k| b1

)∗ pr

(g2 = k| b2

)∗

... ∗ pr(gn′′ = k| bn′′

)]p (b1, b2..., bn′′) db1...dbn′′ . (22)

Combining Eqs. (12), (13) and (22) yields the PDF transfer relationship for input-labeled quantum gain.

Special Case Ia: bi = bj = b. Similar to quantum selection, in the case that bi = bj = b we have


ˆ (prg (g|b) ∗(n

′′−1) prg (g|b))pb (b) db

∣∣∣∣g=n′

(23)

where prg (g|b) ∗(n′′−1) prg (g|b) represents the discrete convolution of prg(g|b) with itself n′′ − 1 times.

Special Case Ib: bi = bj = b and Poisson gain. When gi is Poisson-distributed for all i, the expression inparenthesis in Eq. (22) is also Poisson-distributed,11 therefore


ˆ (∑n′′

n′=1 gi (b))n′

n′!exp

− n′′∑n′=1

gi (b)

p (b) db. (24)

Special Case IIa: gi(bi) = gi. When gi is not a function of a RV we have

prout (Nout = n′|Nin = n′′) =(

prg (g) ∗(n′′−1) prg (g)

)∣∣∣g=n′

(25)

which is the same expression derived in previous work.9

Special Case IIb: gi(bi) = gi and Poisson gain. Similar to Eq. (24), the PMF of Nout given Nin is Poisson

with mean value∑n′′

n′=1 gi:


(∑n′′

n′=1 gi

)n′n′!

exp

− n′′∑n′=1

gi

. (26)



2.5 Application to simple detector model

We apply the theoretical formalism developed above to the description of a simple model of a direct-conversionsemiconductor-based x-ray detector. We consider both mono and poly-energetic x-ray �uence and the specialcase of fast readouts and negligible secondary quanta relocation, as discussed above.

Calculation of the mean image signal and NPS from Eqs. (8) and (9) requires the PDF for the total numberof collected secondaries given one and zero incident photons. In the case of zero incident photons the PDF issimply given by the PDF describing the additive electronic noise e. We obtain the PDF given one incidentphoton using the transfer relationships described in the previous sections with p0(N0) = δ(N0 − 1). We describethe model assuming monoenergetic x-ray �uence and then extend to poly-energetic �uence by averaging over thespectrum of incident x-ray photons.

Stage 1: Interaction of incident x-ray quanta in convertor. Each incident x-ray photon interacts in theconvertor material with probability α equal to the detector quantum e�ciency. The PDF describing the totalnumber of interacting x-ray photons N1 can be obtained by application of Eq. (13) with prN1

(N1|N0 = 1) givenby Eq. (20):

pr1 (N1 = n′|N0 = 1) = αn′(1− α)

n′′−n′. (27)

The PDF describing the number of interacting x-ray photons given one incident is then given by

p1 (N1|N0 = 1) = (1− α) δ (N1) + αδ (N1 − 1) . (28)

Stage 2: Liberation of secondary quanta in convertor. We assume that the ith interacting x-ray photonliberates on average g = E/w e-h pairs where w is the e�ective ionization energy of the convertor material. Inaddition, we assume that liberation of e-h pairs obeys Poisson statistics. The PDF for gi is then given by thePoisson distribution and we use the transfer relationship given by Eq. (26), yielding

pr2 (N2 = n′|N1 = n′′) =

{1 n′′ = 0

prg (n′) n′′ = 1. (29)

The PDF describing the number of liberated secondaries given one incident photon is then given by

p2 (N2|N0 = 1) = (1− α) δ (N2) + α

∞∑n′′=0

prg (N2) δ (N2 − n′′) . (30)

Stage 3: Depth-dependent coupling of secondary quanta to sensor elements. If an electron or hole istrapped or recombines, the charge detected will be only a fraction of the total charge liberated.12�14 We assumethat for a photon having energy E interacting at depth z, the average fraction of charge collected β (z) is givenby the Hecht relation:12�14

β (z) =µeτeε

L

(1− e−

L−zµeτeε

)+µhτhε

L

(1− e−

zµhτhε

)(31)

where L is the convertor thickness, µeτe and µhτh are the mobility-lifetime products [cm2V−1] for electrons andholes, respectively, ε [V cm−1] is the applied electric �eld, and we have assumed that the electrons travel towardsthe entrance surface and holes travel towards the exit surface. For a photon having energy E, the PDF for aphotoelectric interaction at depth z is given by

pz (z) =µpe (E) e−µtot(E)z

α (E)(32)



where α (E) is the quantum e�ciency, and µtot and µpe represent the total and photoelectric linear attenuation

coe�cients, respectively. For a given photon interaction, Eq. (19) describes the PMF for N3 given N2:

pr3 (N3 = n′|N2 = n′′) =

1 n′′ = 0(

n′′

n′

)ˆ L

0

β (z)n′′

(1− β (z))n′′−n′

pz (z) dz n′′ ≥ 1(33)

and

p3 (N3|N0 = 1) = (1− α) δ (N3)

+α

∞∑n′′=0

n′′∑n′=1

prg (g = n′′)

(n′′

n′

)ˆ L

0

β (z)n′′

(1− β (z))n′′−n′

pz (z) dzδ (N3 − n′) . (34)

Stage 4: Collection of secondary quanta in detector elements and additive noise. Readout signal dis proportional to the number of secondaries collected in each detector element plus an additive component dueto detector electronics:11

pd (d|1) =1

kp3

(d

k

)∗ pe (d) (35)

where pd (d|1) represents the PDF of readout signal d given one incident x-ray photon and pe(d) representsthe PDF of additive electronic noise evaluated at d. In all cases we assume that pe(e) is a zero-mean normaldistribution with variance σ2

e .

Stage 5: Sampling of detector elements. Generating discrete output signals from each detector elementis represented as a sampling process and does not change the PDF for d.

Stage 6: Thresholding of sampled detector signals. Calculation of both the mean signal and NPS fromin Eqs. (8) and (9) requires the probability that d is above the threshold given n incident quanta, ξn (t), giving:

ξ1 (t) = (1− α) ξF (t) + αξT (t) (36)

ξ0 (t) = ξF (t) (37)

where ξT (t) and ξF (t) represent contributions from true-count events and false-count events, respectively. Ex-pressions for ξT (t) and ξF (t) are developed in Appendix A.

2.6 DQE for mono-energetic x-ray �uence

In a previous article we demonstrated that for normally-distributed additive noise, the threshold t must be > 3σeto avoid false noise counts and increased image noise. When this condition is satis�ed, the mean SPC signalmaintains a linear relationship with the mean number of incident x-ray quanta. In this section we calculate theDQE under this assumption and compare with an energy-integrating detector. Calculation of the mean detectorsignal and NPS of an energy-integrating detector are given in Appendix B.

2.6.1 Energy-integrating DQE

The energy-integrating DQE assuming a total readout time T = Mat equal to that of the SPC detector describedabove. The corresponding additive readout noise is then given by M2σ2

e . Combining Eqs. (1), (54), and (55),the mono-energetic zero-frequency DQE is given by

DQEEI (0) =α

1 +σ2g

g2+

1− 〈β〉zg 〈β〉z

+Mσ2

e

g2 〈β〉2z λ

(38)



where we have used M = T/at and qo = M ¯qoaat = Mλ and 〈〉z represents a weighted average over interactiondepths z. The above expression demonstrates that low conversion gain, poor collection e�ciency and additiveelectronic noise degrade the mono-energetic DQE of conventional energy-integrating x-ray detectors. In addition,the above expression demonstrates that for a �xed readout time, the DQE decreases with decreasing number ofquanta.

2.6.2 Single-photon-counting DQE

The SPC DQE using Eq. (1). For t > 3σe, the zero frequency DQE becomes

DQESPC (0) =c

qoWc† (0)= αξT (t) . (39)

The above equation illustrates that the mono-energetic zero-frequency DQE of an SPC detector is equal to theprobability that a photon interacts in the x-ray detector multiplied by the probability an element collects moresecondaries than the threshold t.

2.7 DQE for poly-energetic x-ray �uence

2.7.1 Energy-integrating DQE

The poly-energetic DQE can be calculated by integrating Eqs. (54) and (56) over all x-ray energies, giving:

DQEEI (0) =〈αg 〈β〉z〉

2E⟨

〈β〉2z g2α⟩E

+⟨gσ2

βα⟩E

+Mσ2

e

λ

(40)

where 〈〉E represents a weighted average over x-ray energy. This result is similar to that recently reported by Kimet al.15 and demonstrates that, in addition to low conversion gain and poor collection e�ciency, use of a broad x-ray spectrum degrades the DQE of conventional energy-integrating detectors. A notable di�erence is this analysisincorporates the depth-dependence of the collection e�ciency, similar to the analysis of Mainprize et al.,16 andadditive electronic noise.

In the limit that all generated secondaries are collected (ie. β (z)→ 1) and qoa� T 2σ2e/a

2t , Eq. (40) results

in the form �rst described by Swank:17

DQEEI (0) ≈ 〈α〉E I (41)

where

I =〈αg〉2E⟨

αg2⟩E〈α〉E

(42)

is commonly referred to as the Swank x-ray noise term and is always less than unity.

2.7.2 Single-photon-counting DQE

Averaging Eqs. (8) and (9) over all energies, the poly-energetic zero-frequency DQE of an SPC x-ray detector isexpressed as

DQESPC (0) =c2

qoWc† (0)= 〈αξT (t)〉E . (43)

This result shows the zero-frequency DQE of an SPC detector is equal to the spectral average of the probabilitythat a photon interacts in the x-ray detector multiplied by the probability d is greater than the threshold. Inthe limit β (z)→ 1 and σadd → 0 we have

ξT (t) ≈ 1 (44)

andDQESPC (0) ≈ 〈α〉E (45)

giving the expected result that the DQE of SPC detectors with high collection e�ciency and low additive noiseis determined only by the quantum e�ciency.



Table 1. Physical and electrical properties including mass density, electron and hole mobility-lifetime products, appliedelectric �eld, and e�ective ionization energy w for CdZnTe and a-Se. Mobility-lifetime products for CdZnTe and Seare taken from Ruzin and Nemirovsky18 and Kasap,14 respectively. E�ective ionization energies are taken from Rowland.19

Se CdZnTe

density (g cm−3) 4.3 5.8w (eV) 45 5

µeτe [cm2V−1] 3×10−7 2×10−3

µhτh [cm2V−1] 1.2×10−5 1×10−5ε [V cm−1] 1×105 9×102

Table 2. Spectral information for IEC-compliant x-ray spectra used for theoretical and simulation studies.

Spectrum kV HVL (mm Al)

RQA-M1 25 0.56RQA-5 70 7.1RQA-7 90 9.1RQA-9 120 11.5

2.8 Monte Carlo validation

Monte Carlo simulations were performed to validate the detection model described above. We simulated x-ray detection with CdZnTe and Se-based semiconductor detectors with the physical and electrical propertiessummarized in Table 1. For each detector material we considered mono-energetic �uence at 20, 40, 60, and80 keV and poly-energetic x-ray �uence using standardized IEC-compliant x-ray spectra (RQA-M1, RQA-5,RQA-7, and RQA-9, summarized in Table 2) and detector thicknesses ranging from 0.1 to 10 mm. All spectrawere generated using an in-house MATLAB routine that implements algorithms published by Tucker et al.20

Simulation of x-ray interaction, conversion to secondary quanta, collection of secondary quanta, additive noise,and thresholding was performed using the seven-step algorithm described below:

1. Generate the number of Poisson-distributed photons incident on pixel i, j for each energy interval (1 keV).

2. Select photons that interact in the detector with probability of interaction α that depends on both photonenergy and detector thickness.

3. For each interaction, determine the depth of interaction by randomly sampling Eq. (32).

4. For each interaction, generate the number of Poisson-distributed secondary quanta liberated per-interactionby randomly sampling the Poisson distribution with mean gain g = E/w.

5. For each interacting photon, select the subset of liberated secondaries that are collected by sampling thebinomial distribution with number of trials equal to the number of liberated secondaries and probabilityof success equal to β(z) given by Eq. (31).

6. Simulate additive noise by adding or subtracting normally-distributed secondary quanta with zero meanand variance σ2

e .

7. For the photon counting model, if the number of secondaries collected per interaction is greater than t,increment the element signal by one. For the energy-integrating model, sum the total number of secondariescollected by all interacting photons.

We performed the above calculations for each element in a 32×32-element image and then calculated DQE (0)using the using the expression DQE (0) = SNR2

det/SNR2ideal where SNRdet is the detected signal-to-noise ratio

calculated from the mean and variance of the �nal image and SNRideal is the ideal signal-to-noise ratio calculatedfrom the mean and variance of the number of incident photons per detector element.



Single energy Broad spectrum

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

Se

20 keV

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

40 keV

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

60 keV

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

Thickness (mm)

80 keV

0.1 1 100

0.25

0.5

0.75

1

CdZnTe

20 keV

SPC, σadd2 =0

SPC, σadd2 =10

SPC, σadd2 =102

SPC, σadd2 =103

EI, σadd2 =0

0.1 1 100

0.25

0.5

0.75

1

40 keV

0.1 1 100

0.25

0.5

0.75

1

60 keV

0.1 1 100

0.25

0.5

0.75

1

Thickness (mm)

80 keV

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

Se

RQA−M1

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

RQA−5

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

RQA−7

0.1 1 100

0.25

0.5

0.75

1

DQ

E(0

)

Thickness (mm)

RQA−9

0.1 1 100

0.25

0.5

0.75

1

CdZnTe

RQA−M1

SPC, σadd2 =0

SPC, σadd2 =10

SPC, σadd2 =102

SPC, σadd2 =103

EI, σadd2 =0

0.1 1 100

0.25

0.5

0.75

1

RQA−5

0.1 1 100

0.25

0.5

0.75

1

RQA−7

0.1 1 100

0.25

0.5

0.75

1

Thickness (mm)

RQA−9

Figure 1. A graphical illustration of the dependence of DQE(0) on convertor thickness for Se and CdZnTe calculatedfor both single-energy (left) and broad spectra (right) for a range of additive noise levels. Lines and symbols representtheoretical and Monte Carlo calculations, respectively.

3. RESULTS

3.1 DQESPC(0) dependence on collection e�ciency and additive noise

The dependence of DQE(0) on convertor thickness is illustrated in Fig. 1 for selected photon energies andbroad spectra for a range of additive noise levels. In the absence of additive noise, DQESPC(0) is equal to thequantum e�ciency for all detector thicknesses and x-ray spectra considered. However, for non-zero additive noise,DQESPC(0) for Se reaches a maximum between 1.0-10 mm and then decreases by 40-90 % depending on x-rayenergy. This is due to the combined e�ect of high additive noise and poor charge-collection e�ciency (Fig. 2)since poor collection e�ciency reduces the number of collected secondaries to a value close to the threshold andhigh additive noise broadens the resulting distribution, resulting in the signal from low-energy photons fallingbelow the threshold. In the case of CdZnTe, which has higher electron and hole mobility-lifetime products thanSe, DQESPC(0) increases monotonically with convertor thickness and is within 5 % of the quantum e�ciency forall imaging conditions considered.

3.2 Comparison of SPC and energy-integrating x-ray detectors

Figure 1 also illustrates the dependence of DQEEI(0) on detector thickness in the absence of additive noise. Forall energies and detector thicknesses considered, DQEEI(0) is less than DQESPC(0). This is due to the fact that



Single energy Broad spectrum

0.1 1 10

0.25

0.5

0.75

1C

olle

ctio

n ef

ficie

ncy

Thickness (mm)

Se

20 keV80 keV

0.1 1 10

0.25

0.5

0.75

1

Thickness (mm)

CdZnTe

0.1 1 10

0.25

0.5

0.75

1

Col

lect

ion

effic

ienc

y

Thickness (mm)

Se

RQA−9RQA−M1

0.1 1 10

0.25

0.5

0.75

1

Thickness (mm)

CdZnTe

Figure 2. A plot of the collection e�ciency β (z) averaged over the depth of interaction z for both single-energy (left) andbroad-spectra (right).

stochastic conversion gain results in Swank noise, and therefore DQE degradation, for energy-integrating x-raydetectors,17,21�23 but not for SPC detectors. In SPC detectors, as long as the area under the curve describing thedistribution of detected secondaries is close to unity for all values above the threshold, the signal is independentof the width of the distribution.

Figure 1 illustrates the dependence of DQE(0) on thickness for RQA-M1, RQA-5, RQA-7, and RQA-9 spectrafor both SPC and energy-integrating detectors. Use of a broad spectrum appears to have a negligible a�ect onthe DQE for SPC x-ray detectors. As expected, use of a broad x-ray spectrum has further reduced the DQEof energy-integrating detectors for a similar reason as that discussed above in regard to the conversion gain. Itis interesting to note that even when additive noise is considered for SPC detectors and neglected for energy-integrating detectors, DQESPC(0) can still be 5-10% higher than DQEEI(0) for detector thicknesses ranging from0.1-1 mm corresponding to collection e�ciencies ranging from approximately 0.95-0.3 for Se and 0.95-1.00 forCdZnTe.

4. DISCUSSION AND CONCLUSIONS

The DQE is an important indicator of performance for SPC systems. Recent developments in understandingnoise transfer in SPC systems has resulted in a PDF transfer-theory approach that can be used to describethe DQE and other metrics of SPC system performance. We described how serial cascades of input-labeledprocesses can be incorporated in the PDF transfer theory approach. Input-labeled processes are required whenselection, gain, or scatter parameters are functions of a random variable, for example the depth of interaction,and we demonstrated they can be incorporated into the CSA approach. New transfer relationships were appliedto describe the DQE of a simple model of Se and CdZnTe SPC detectors including the e�ects of poly-energeticx-ray spectra, stochastic conversion of x-ray energy to e-h pairs, depth-dependent collection of e-h pairs, additiveelectronic noise, and thresholding.

While SPC detectors are not yet ready for routine use in diagnostic imaging, we show that, with the exceptionof thick (1-10 mm) Se convertors with high additive noise levels, SPC imaging using Se and CdZnTe has thepotential to provide equal or better zero-frequency DQE values than energy-integrating systems. This is mostlydue to the fact that the DQE of SPC detectors is less sensitive to the degrading e�ects of stochastic conversionto secondary quanta (e-h pairs), incomplete collection of secondary quanta, and noise sources associated with theuse of a broad x-ray spectrum, commonly referred to as Swank noise. However, in situations where the collectione�ciency is very low (such as in thick Se detectors) and additive noise levels are high, the zero-frequency DQEof SPC detectors can be much less than that of energy-integrating detectors because the signal from low-energyphotons may fall below the threshold

The formalism developed in this work enables a theoretical characterization of the frequency-dependent NPSof SPC x-ray detectors. However, we did not develop expressions for the frequency-dependent signal (describedin terms of the MTF) of SPC x-ray detectors. In addition, the detector model used in this work does not includethe e�ects of charge sharing or reabsorption of characteristic and Compton-scatter x rays in neighboring pixels.Therefore, the results presented here likely overestimate the DQE of both SPC and energy integrating x-ray



detectors. In the future, techniques for describing the MTF of SPC x-ray detectors will be useful for determiningthe e�ects of charge-sharing on the frequency-dependent DQE of SPC x-ray detectors.

Appendix A: Calculation of ξT (t) and ξF (t)

From Eqs. (34) and (35), the probability of a true count given a photon interaction ξT (t) is given by

ξT (t) =

ˆ ∞t

∞∑i=0

i∑j=1

prg (i)

(ij

)ˆ L

0

β (z)j

(1− β (z))i−j

p (z) dzδ

(d

k− j) ∗ pe (d) dd (46)

=

∞∑i=0

i∑j=1

prg (i)

(ij

)ˆ L

0

β (z)j

(1− β (z))i−j

p (z) dz

[ˆ ∞t

δ

(d

k− j)∗ pe (d) dd

](47)

=

∞∑l=t

∞∑j=1

prg (i)

(ij

)ˆ L

0

β (z)j

(1− β (z))i−j

p (z) dzpe (kj) (48)

and the probability of a false count given no photon interaction ξF (t) is simply given by

ξF (t) =

ˆ ∞t

pe (d) dd. (49)

Appendix B: Cascaded analysis of an energy-integrating detector

We brie�y summarize transfer of the mean signal and NPS for an energy-integrating detector through thetheoretical detection model considered in this work.

A1. Stage 1: Interaction of incident x-ray quanta in the convertor

The NPS after stage 1 is given byW1 (k) = q1 = αqo

where α is the quantum e�ciency.

A2. Stage 2: Generation of secondary quanta in convertor

The mean and NPS after stage 2 are given by

q2 = gq1 = gαqo (50)

and

W2 (k) = g2W1 (k) + σ2g q1

= g2αqo

(1 +

g2

σ2g

)(51)

respectively.

A3. Stage 3: Coupling of charge carriers to detector elements

The mean and NPS after stage 3 can be described in terms of 〈β〉z using the input-labeled transfer relationshipfor a quantum gain stage, as described by Van Metter and Rabanni:24

W3 (k) = [W2 (k)− q2] 〈β〉2z +⟨β2 + σ2

β − β⟩zq2 + 〈β〉z q2 (52)

= 〈β〉2z g2αqo

[1 +

σ2g

g2+

1− 〈β〉zg 〈β〉z

](53)

where we have used the relationship σ2β = β (1− β).



A4. Stage 4: Collection of charge carriers in detector elements

Collection of charge carriers in detector elements is expressed as a multiplication with a rectangle function. Thee�ect on the NPS is modulation with the square of a two dimensional sinc function.

d = kg 〈β〉z qoa (54)

W4 (k) = 〈β〉2z g2αqo

[1 +

σ2g

g2+

1− 〈β〉zg 〈β〉z

]a2sinc2 (axu) sinc2 (ayv) (55)

A5. Stage 5: Additive electronic noise

Additive electronic noise does not alter the mean signal from an energy-integrating x-ray detector but doesincrease the presampling NPS giving:

W5 (k) = 〈β〉2z g2αqo

[1 +

σ2g

g2+

1− 〈β〉zg 〈β〉z

]a2sinc2 (axu) sinc2 (ayv)

+T 2σ2e/a

2t asinc2 (axu) sinc2 (ayv) (56)

where T = Mat is the readout time and σ2e is the additive noise in an element having area a for a readout interval

of length T .

REFERENCES

1. B. Mikulec, �Development of segmented semiconductor arrays for quantum imaging,� Nucl. Instrum. Meth.A 510, pp. 1�23, 2003.

2. R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, and L. Tlustos, �The Medipix3 prototype, a pixel readoutchip working in single photon counting mode with improved spectrometric performance,� IEEE Trans. Nucl.Sci. 54, pp. 1824�1829, 2007.

3. J. Yorkston, �Recent developments in digital radiography detectors,� Nucl. Instrum. Meth. A 580, pp. 974�985, 2007.

4. P. M. Shikhaliev, �Projection x-ray imaging with photon energy weighting: Experimental evaluation with aprototype detector,� Phys. Med. Biol. 54, pp. 4971�4992, Aug 2009.

5. J. Tanguay, H. K. Kim, and I. A. Cunningham, �The role of x-ray swank factor in energy-resolving photon-counting imaging,� Med. Phys. 37, pp. 6205�6211, 2010.

6. R. J. Acciavatti and D. A. Maidment, �A comparative analysis of OTF, NPS, and DQE in energy integratingand photon counting digital x-ray detectors,� Med. Phys. 37, pp. 6480�6495, 2010.

7. P. M. Shikhaliev, S. G. Fritz, and J. W. Chapman, �Photon counting multienergy x-ray imaging: E�ect ofthe characteristic x rays on detector performance,� Med. Phys. 36, pp. 5107�5119, Nov 2009.

8. J. Tanguay, S. Yun, H. K. Kim, and I. A. Cunningham, �Extension of cascaded systems analysis to single-photon-counting x-ray detectors,� Proc. of SPIE 8313, pp. 8313101�83131013, 2012.

9. J. Tanguay, S. Yun, H. K. Kim, and I. A. Cunningham, �The detective quantum e�ciency of photon-countingx-ray detectors using cascaded-systems analyses,� Submitted to Med. Phys. (Manuscript number 12-1750) ,2013.

10. M. Rabbani and R. L. VanMetter, �Analysis of signal and noise propagation for several imaging mechanisms,�J Opt Soc Am A 6, pp. 1156�1164, 1989.

11. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, Inc., 3 ed., 1991.

12. K. Hecht, �Zum mechanismus des lichtelektrischen primarstromes in isolierenden kristallen,� Z. Phys. 77,pp. 235�245, 1932.

13. G. Knoll, Radiation Detection and Measurement, Wiley, New York, 2000.

14. S. O. Kasap, �X-ray sensitivity of photoconductors: Applications to stabilized a-Se,� J. Phys. D: Appl.Phys. 33, pp. 2853�2865, 2000.



15. H. K. Kim, C. H. Lim, J. Tanguay, S. Yun, and I. A. Cunningham, �Spectral analysis of fundamental signaland noise performances in photoconductors for mammography,� Med. Phys. 39, pp. 2478�2490, May 2012.

16. J. G. Mainprize, D. C. Hunt, and M. J. Ya�e, �Direct conversion detectors: The e�ect of incomplete chargecollection on detective quantum e�ciency,� Med. Phys. 29, pp. 976�990, Jun 2002.

17. R. K. Swank, �Absorption and noise in x-ray phosphors,� J. Appl. Phys. 44, pp. 4199�4203, 1973.

18. A. Ruzin and Y. Nemirovsky, �Methodology for evaluations of mobility-lifetime product by spectroscopymeasurements using CdZnTe spectrometers,� J. Appl. Phys. 82, pp. 4166�4171, 1997.

19. J. A. Rowlands and J. Yorkston, Handbook of Medical Imaging, ch. 4, pp. 223�329. SPIE Press, 2000.

20. D. M. Tucker, G. T. Barnes, and D. P. Chakraborty, �Semiempirical model for generating tungsten targetx-ray spectra,� Med Phys 18(2), pp. 211�218, 1991.

21. R. K. Swank, �Measurement of absorption and noise in an x-ray image intensi�er,� J. Appl. Phys. 45,pp. 3673�3678, 1974.

22. M. Drangova and J. A. Rowlands, �Optical factors a�ecting the detective quantum e�ciency of radiographicscreens,� Med. Phys. 13(2), pp. 150�157, 1986.

23. G. Hajdok, J. J. Battista, and I. A. Cunningham, �Fundamental x-ray interaction limits in diagnostic imagingdetectors: Frequency-dependent Swank noise,� Med. Phys. 35, pp. 3194�3204, Jul 2008.

24. R. V. Metter and M. Rabbani, �An application of multivariate moment-generating functions to the analysisof signal and noise propagation in radiographic screen-�lm systems,� Med. Phys. 17(1), pp. 65�71, 1990.



Documents

Cascaded systems analyses of photon-counting x-ray detectorsbml.pusan.ac.kr/PublishFrame/Publication/PDF/IntCon/089... · 2013-05-28 · Single-photon counting (SPC) x-ray imaging