Spectral analysis of fundamental signal and noise performancesin photoconductors for mammography

Ho Kyung Kima)

School of Mechanical Engineering, Pusan National University, Jangjeon-dong, Geumjeong-gu,Busan 609-735, Republic of Korea and Center for Advanced Medical Engineering Research,Pusan National University, Jangjeon-dong, Geumjeong-gu, Busan 609-735, Republic of Korea

Chang Hwy LimSchool of Mechanical Engineering, Pusan National University, Jangjeon-dong, Geumjeong-gu,Busan 609-735, Republic of Korea

Jesse TanguayImaging Research Laboratories, Robarts Research Institute, University of Western Ontario, 100 Perth Drive,London, Ontario N6A 5K8, Canada

Seungman YunSchool of Mechanical Engineering, Pusan National University, Jangjeon-dong, Geumjeong-gu,Busan 609-735, Republic of Korea and Imaging Research Laboratories, Robarts Research Institute,100 Perth Drive, London, Ontario N6A 5K8, Canada

Ian A. CunninghamImaging Research Laboratories, Robarts Research Institute, University of Western Ontario, 100 Perth Drive,London, Ontario N6A 5K8, Canada

(Received 27 September 2011; revised 24 March 2012; accepted for publication 26 March 2012;

published 13 April 2012)

Purpose: This study investigates the fundamental signal and noise performance limitations

imposed by the stochastic nature of x-ray interactions in selected photoconductor materials, such as

Si, a-Se, CdZnTe, HgI2, PbI2, PbO, and TlBr, for x-ray spectra typically used in mammography.

Methods: It is shown how Monte Carlo simulations can be combined with a cascaded model to deter-

mine the absorbed energy distribution for each combination of photoconductor and x-ray spectrum.

The model is used to determine the quantum efficiency, mean energy absorption per interaction, Swank

noise factor, secondary quantum noise, and zero-frequency detective quantum efficiency (DQE).

Results: The quantum efficiency of materials with higher atomic number and density demonstrates

a larger dependence on convertor thickness than those with lower atomic number and density with

the exception of a-Se. The mean deposited energy increases with increasing average energy of the

incident x-ray spectrum. HgI2, PbI2, and CdZnTe demonstrate the largest increase in deposited

energy with increasing mass loading and a-Se and Si the smallest. The best DQE performances are

achieved with PbO and TlBr. For mass loading greater than 100 mg cm�2, a-Se, HgI2, and PbI2

provide similar DQE values to PbO and TlBr.

Conclusions: The quantum absorption efficiency, average deposited energy per interacting x-ray,

Swank noise factor, and detective quantum efficiency are tabulated by means of graphs which may

help with the design and selection of materials for photoconductor-based mammography detectors.

Neglecting the electrical characteristics of photoconductor materials and taking into account only

x-ray interactions, it is concluded that PbO shows the strongest signal-to-noise ratio performance of

the materials investigated in this study. VC 2012 American Association of Physicists in Medicine.

[http://dx.doi.org/10.1118/1.3702455]

Key words: cascaded-systems analysis, detective quantum efficiency, mammography, Monte Carlo,

photoconductor, quantum efficiency, Swank noise

I. INTRODUCTION

Breast cancer is the most common cancer in women,1 affect-

ing one in nine women in North America. Early detection by

mammography is challenging due to the need to visualize low-

contrast lesions and microcalcifications. Analog mammog-

raphy systems use film-screen technology to detect transmitted

x-rays as well as record and display image information. Digital

mammography, often using large-area semiconductor-based

detectors, is able to address limitations of film-screen systems

for imaging dense breasts,2 as demonstrated in a study by

Pisano et al.,3 which is particularly important when screening

younger populations where the benefits of screening mammog-

raphy remain controversial.4,5 Despite much progress in the

development of alternative systems,6 such as dedicated breast

computed tomography, digital tomosynthesis and magnetic

resonance imaging, two-dimensional (2D) mammography

remains the clinical standard for breast cancer screening.7

2478 Med. Phys. 39 (5), May 2012 0094-2405/2012/39(5)/2478/13/$30.00 VC 2012 Am. Assoc. Phys. Med. 2478

In any x-ray imaging task, the detector must provide

images of high diagnostic quality while maintaining a safe

level of patient x-ray exposure. This balance is best

described by the detective quantum efficiency (DQE) of the

detector.8–10 In mammography, high spatial resolution and

low detector electronic noise are also critical.11 However,

even if detector noise is negligible, the stochastic nature of

quantum-based imaging systems remains a source of image

noise and reduced DQE.12 This is particularly true for high-

resolution imaging where a secondary quantum sink may

exist at high spatial frequencies if the number of secondary

quanta contributing to final image signal per interacting

x-ray photon is much less than approximately 100.10,12 In

general, both 2D and three-dimensional x-ray imaging tech-

niques are optimized only with appropriate selection of

high-gain x-ray convertor materials.

Direct-conversion x-ray imaging methods, which use a

photoconductive material to convert incident x-ray quanta

directly into image forming quanta, have better spatial resolu-

tion characteristics than indirect-conversion methods that use

optical scintillators.13–17 However, important drawbacks of

large-area photoconductors include their instability over

time, sensitivity to environmental temperatures, and difficult

reproducibility of uniform electronic properties over large

areas during production.17–21 Thus far, amorphous selenium

(a-Se) is the primary photoconductor used in commercialized

direct-conversion digital mammography detectors.14–17,21

However, a-Se suffers from low x-ray sensitivity due to a

high effective ionization energy or w-value (�45 eV) and

much research effort is being invested in the development of

alternatives. Several candidate compound photoconductive

materials have been identified,17,22–27 although limitations

imposed by electrical properties, such as mean drift length,

leakage current, material stability, reproducibility, and large-

area availability hamper their development.27–29

Optimal image quality can be achieved only when the

DQE is as close to unity as possible. For example, x-ray and

optical scatter in a scintillator impact on both spatial resolution

and image noise,30–32 resulting in a degraded DQE. Sakellaris

et al. used Monte Carlo methods to investigate x-ray interac-

tion physics in a-Se and their effect on spatial resolution,33

and extended their study to a wide range of photoconductor

materials.34 Hajdok et al.35 investigated the DQE and x-ray

Swank noise term in Si, a-Se, CsI, and PbI2 for a wide range

of monoenergetic incident photons. The energy width of x-ray

spectra also has a negative impact on the DQE.36

Cascaded-systems analysis (CSA) is an effective tool for

describing how the physical processes that contribute to the

final image affect the DQE and image quality. Since CSA is

Fourier based, it applies to systems that are linear and shift-

invariant and involve only wide-sense stationary (WSS) or

wide-sense cyclostationary noise processes.10 Monte Carlo

(MC) calculations are sometimes required to account for

complex physical process or geometries. For example, at

megavoltage (MV) energies, multiple scattering events can

be difficult to represent using CSA.

Some investigators have used MC methods to calculate

absorbed energy distributions (AED) and then used CSA

methods to examine the effect of blur and other factors on

image quality and the DQE.37–39 While this approach has

been used widely, it remains an approximation that does

not correctly take into consideration the statistical nature of

secondary quanta (electron-hole pairs or optical quanta) gen-

eration and collection, and therefore ignores a possible degra-

dation in the DQE due to a secondary quantum sink. At MV

energies, this is generally a reasonable approximation due to

the very large number of secondary quanta generated for

each interacting x-ray photon. At the lower mammographic

energies, however, statistics of secondary quanta generation

and collection efficiencies must be carefully considered as

part of the design process. There are several commercial

phosphor-based detectors in current use that suffer from a

secondary quantum sink due to this problem. The require-

ment for high conversion gain will likely become more strin-

gent with the development of photon-counting technologies.

In this paper, we introduce a method of combining the

best of each approach that incorporates MC methods to

determine the AED with a CSA model to calculate the DQE,

and discuss optimal photoconductors for digital mammog-

raphy based on the simulation results. Other considerations,

including grain boundary problems inherent in polycrystal-

line materials such as PbI2 and HgI2 which may affect reso-

lution and charge collection,22 are not specifically addressed.

II. THEORY

When an x-ray photon is incident on a detector, both the

probability of interaction and the energy deposited (and

therefore, the average number of secondary quanta generated)

are functions of the energy of the incident photon. This viola-

tes a requirement of the CSA approach that each process in

the cascade of processes be statistically independent. In other

words, since the photon energy determines both the probabil-

ity of interaction and the conversion gain, we cannot repre-

sent these as two different cascaded processes. Our solution,

inspired in part by Rabbani and Van Metter’s description of

an input-labeled random amplification process,40 is to repre-

sent both the selection of interacting quanta and subsequent

conversion to secondary quanta as a single process with the

additional feature that both the probability of interaction and

the mean conversion gain are functions of a random variable

representing the energy of each incident photon. We call this

new process a “photon-interaction” process. The input is a

spatial point distribution of incident quanta and the output is

a spatial point distribution of secondary quanta at the loca-

tions where they are liberated within the photoconductor.

II.A. Photon-interaction process

In this process, selection of interacting photons is

described as multiplication by the Bernoulli random variable

~anð ~E0nÞ which can have sample values 1 or 0 only, and where~E0n is a random variable describing the photon energy where

n identifies the nth incident photon. At each energy, this

represents a gain stage where the gain mean and variance

are given by aðE0Þ and aðE0Þ � a2ðE0Þ, respectively.10 Each

interacting x-ray photon subsequently produces ~gnð ~E0nÞ

2479 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2479

Medical Physics, Vol. 39, No. 5, May 2012

secondary quanta with mean and variance �gðE0Þ and r2gðE0Þ,

respectively. Propagation of the mean and Wiener noise

power spectrum (NPS) through this new photon-interaction

stage are developed in Appendix. For the special case of a

WSS input, the mean of ~qoutðrÞ takes the form

�qout ¼ a�gh is �qin; (1)

where �h is indicates an average over energy-dependent terms

weighted by the incident x-ray spectrum sðE0Þ. The NPS

takes the form

WoutðuÞ ¼ a�gh i2s WinðuÞ � �qin½ � þ ar2g þ a�g2

D Es�qin: (2)

Equations (1) and (2) are new contributions and describe the

transfer of mean signal and noise, respectively, through the

photon-interaction process. In terms of mathematical nota-

tion, the overhead tilde is used to indicate a random variable,

and the boldface r and u indicate vector-form variables in

spatial and spatial-frequency domains, respectively.

II.B. Cascaded model of photoconductor detector

Starting with the above photon-interaction process, we

describe each photoconductor detector using a CSA model

consisting of a cascade of five linear stages as shown in

Fig. 1. The input to the first stage is a distribution of x-ray

quanta with expected value �q0 (quanta per square millimeter)

and NPS W0ðuÞ ¼ �q0.

II.B.1. X-ray interaction stage

This stage describes x-ray interaction and conversion to

secondary quanta within the photoconductor. Using the

transfer relationships described above, we obtain

�q1 ¼ a�gh is �q0; (3)

in per square millimeter and

W1ðuÞ ¼ a�gh i2s W0ðuÞ � �q0½ � þ ar2g þ a�g2

D Es�q0

¼ ar2g þ a�g2

D Es�q0; (4)

in per square millimeter. Here, the term r2g accounts for vari-

ability in the number of charge carriers generated at a partic-

ular x-ray energy and a�g2 accounts for secondary quantum

noise due to the finite number of secondary quanta. For a

monoenergetic beam, they are equivalent to a conventional

cascade of selection and quantum gain processes. The above

expression can be rewritten as

W1ðuÞ ¼ ag2

D Es�q0; (5)

using the relation r2g ¼ g2 � �g2.

Equations (3) and (5) are particularly important because

they identify what information must be provided by the

Monte Carlo analysis for incorporation into a cascaded

model, and we show here that this comes from moments of

the AED. This is seen by letting AðE;E0Þ ¼ aðEÞRðE;E0Þrepresent the probability that an incident photon of energy E0

deposits energy E in the photoconductor where RðE;E0Þ is

the probability that an interacting photon of energy E0 depos-

its energy E. The AED for a spectrum of x-ray energies AðEÞis obtained by averaging AðE;E0Þ over sðE0Þ as schematically

illustrated in Fig. 2. The jth moment of AðE;E0Þ for incident

energy E0 is

MjðE0Þ ¼ð1

0

EjaðE0ÞRðE;E0ÞdE ¼ aðE0ÞEjðE0Þ; (6)

where �EðE0Þ is the mean deposited energy per interacting

photon. Averaging over sðE0Þ gives moments of the AED for

a spectrum of incident photons

Mj ¼ð1

0

sðE0ÞMjðE0ÞdE0 ¼ aEjD E

s: (7)

Assuming that �g ¼ �E=w and g2 ¼ E2=w2 where w is the

effective energy required to liberate one electron-hole pair in

the photoconductor gives

�q1 ¼M1 �q0

wand W1ðuÞ ¼

M2 �q0

w2: (8)

The above expressions are also new contributions. They pro-

vide a rigorous description of how MC calculations of the

AED can be incorporated into a cascaded model and identify

the requisite assumptions.

II.B.2. Collection of charge carriers by collectingelectrodes

It is assumed that on average a fraction b of all liberated

charge carriers will be collected at the electrodes. Factors

such as electron-hole recombination and charge trapping

will result in a decrease in b. While in general b may depend

on the depth of interaction,41 for simplicity, we ignore this

dependence. This process can be represented as a quantum

selection stage, giving

�q2 ¼ b �q1 ¼bM1 �q0

w; (9)

in per square millimeter and

W2ðuÞ ¼ b2W1ðuÞ þ r2b�q1 ¼ b2 �q0

M2

w2þ 1� b

bM1

w

� �; (10)

in per square millimeter. This result shows how secondary

quantum noise and potentially a secondary quantum sink

caused by poor collection efficiency12 affect the NPS.

II.B.3. Random relocation of secondary quanta

Electron-hole pairs liberated in the photoconductor will

undergo a number of random interactions and charge diffu-

sion under the applied electric field, resulting in random

relocation of each secondary quantum in the image plane

prior to being collected at the collecting electrodes. We

FIG. 1. A simple cascaded model describing signal and noise propagation in

a photoconductor detector. The overhead tilde designates a random variable.

The symbol “*s” is the quantum scatter operator.

2480 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2480

Medical Physics, Vol. 39, No. 5, May 2012

assume all secondary quanta are relocated independently and

with a probability described by the same point-spread func-

tion, giving:12,40

�q3 ¼ �q2 ¼bM1 �q0

w; (11)

and

W3ðuÞ ¼ W2ðuÞ � �q2½ � TðuÞj j2þ�q2;

¼ b2 �q0

M2

w2�M1

w

� �TðuÞj j2þ bM1 �q0

w; (12)

where TðuÞ is the Fourier transform of the scatter point-

spread function.

II.B.4. Collection of charges in detector elements

The detector generates a signal ~d proportional to the num-

ber of charge carriers that are collected in each detector ele-

ment, where

�d ¼ kaxay �q3 ¼ kaxaybM1 �q0

w; (13)

(unitless), and the presampling NPS (Ref. 10) is given by

WdðuÞ ¼ k2W3ðuÞa2xsinc2ðaxuxÞa2

ysinc2ðayuyÞ

¼ k2b2 �q0

M2

w2�M1

w

� �TðuÞj j2a2

xsinc2ðaxuxÞ

� a2ysinc2ðayuyÞ þ k2 bM1 �q0

wa2

xsinc2ðaxuxÞ

� a2ysinc2ðayuyÞ; (14)

in per square millimeter where u ¼ ðux; uyÞ.

II.B.5. Detective quantum efficiency

The DQE is therefore given by10

DQEðuÞ ¼�d2MTF2ðuÞ

�q0 WdðuÞ þPn

Pm

Wd u6ð nax; m

ayÞ

� �� � ; (15)

where the MTF is the product of TðuÞ and the sinc terms.

The zero-frequency DQE value is then

DQEð0Þ ¼ 1

M2

M21

þ w

M1

1� bb

� �

¼ 1

1þ r2

M21

þ w

M1

1� bb

� � ; (16)

where r2 ¼ M2 �M21. The above relationship expresses the

zero-frequency DQE value in terms of moments of the AED

obtained by Monte Carlo methods.

In Swank’s original work, he showed that

DQEð0Þ ¼ aIAED; (17)

where a ¼ M0 and where

IAED ¼M2

1

M0M2

; (18)

describes variations in the absorption process,42 sometimes

just called the Swank factor. While this result has been used

in a many subsequent investigations,43–46 our result shows

that secondary quantum noise can be incorporated into the

DQE calculation as

DQEð0Þ ¼ 1

1

ah isIAEDþ w

M1

1� bb

� � ; (19)

which includes a second term in the denominator and

accounts for secondary quantum noise. Equation (19) shows

that low conversion gain or poor collection efficiency will

degrade the DQE if the noise term exceeds a value of

approximately 0.1, giving the requirement that

b >1

1þ M1

10w

; (20)

to avoid a secondary quantum sink. This consideration will

be most important at low x-ray energies.

FIG. 2. Schematic illustration of the construction of

absorbed energy distribution in a photoconductor detec-

tor for polyenergetic incident x-ray quanta. Possible

energy-deposition events are depicted. Probable partial

energy deposition due to loss of energy through fluores-

cence escape makes a peak in the absorbed energy

distribution.

2481 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2481

Medical Physics, Vol. 39, No. 5, May 2012

III. METHODS

III.A. Photoconductor materials

The performance of seven candidate semiconductor mate-

rials for photoconductor-based mammography is evaluated

using MC methods to determine the quantum efficiency ah isand Swank noise term IAED. All material information was

taken from Kabir et al.47 and is summarized in Table I. All

materials, except Si, are assumed to be prepared as amor-

phous or polycrystalline forms and have reduced physical

densities compared to crystalline structures.

III.B. Monte Carlo simulations

We carried out virtual pulse-height spectroscopy measure-

ments using the latest version of Monte Carlo N-Particle

transport code (MCNP version 5, the Radiation Safety Informa-

tion Computational Center or RSICC, Oak Ridge, TN) to

simulate the coupled photon–electron transport within a de-

tector. X-ray photon energies were sampled from three repre-

sentative mammographic spectra (Mo=Mo, Rh=Rh, and

W=Al).48 Each spectrum was computer-generated by adjust-

ing tube voltages to achieve the required half-value layers.49

Detailed spectral information is summarized in Table II. In

all cases, a pencil-like photon beam was perpendicularly inci-

dent upon the center point of the top surface of the detector.

We modeled two different detector geometries, as illus-

trated in Fig. 3: (1) A cylindrical slab geometry having a ra-

dius of 200 mm and (2) a hexahedral geometry having an

aperture of 0.1� 0.1 mm. In the former, we expect no lateral

escape of characteristic or Compton scatter x-rays and

instead expect escape only from the top and bottom surfaces.

This geometry can be regarded as an “infinitely-sized” pixel

element and we refer to it as the “a1” model. The hexahe-

dral geometry mimics typical pixel apertures (�100 lm) in

mammography detectors and we refer to it as the “a100”

model. We mainly report on the results of the a1 model and

the signal and noise characteristics on the a100 model are

indirectly addressed by calculating the relative difference

with respect to the a1 model

DDQE ¼ DQEa1 � DQEa100

DQEa1

� 100%: (21)

A wide range of mass loading (q� L) values was consid-

ered for the Monte Carlo simulations. The maximum mass

loading was set to �230 mg cm�2 (corresponding to

L¼ 1 mm for Si). Each simulation included 2� 106 incident

photon histories with energy-absorbing events summed in 1-

keV energy bins.

III.C. Calculation of the DQE from the AED

Evaluation of the DQE with Eqs. (18) and (19) requires

the moments M1 and M2, evaluated as

Mj ¼ð1

0

EjAðEÞdE; (22)

for the jth moment, where

AðEÞ ¼ð1

0

sðE0ÞAðE;E0ÞdE0: (23)

We also calculate the mean energy absorbed per x-ray

interaction

TABLE I. Summary of material and atomic elemental information. Physical

densities of photoconductor materials, and atomic number and shell binding

energies of each compositional element are summarized.

Shell binding energy (keV)

Material

Density

(g cm�3)

Atomic number

(Z) K L1 L2 L3

Si 2.33 14 1.84 – – –

a-Se 4.3 34 12.66 1.65 1.48 1.43

CdZnTe 5.8 48 (Cd) 26.71 4.02 3.73 3.54

30 (Zn) 9.66 1.19 1.04 1.02

52 (Te) 31.81 4.94 4.61 4.34

HgI2 6.3 80 (Hg) 83.10 14.84 14.21 12.28

53 (I) 33.17 5.19 4.85 4.56

PbI2 4.0 82 (Pb) 88.00 15.86 15.20 13.04

PbO 4.8

TlBr 7.5 81 (Tl) 85.53 15.35 14.70 12.66

35 (Br) 13.47 1.78 1.60 1.55

TABLE II. Mammographic beam qualities suggested by IEC and the obtained

beam qualities from the computational spectrum simulator.

IEC Computer generation

Beam quality kVp HVL (mmAl) Adjusted kVp HVL (mmAl)

Mo=Mo 28 0.600 30 0.596

Rh=Rh 28 0.740 28 0.737

W=Al 28 0.830 30 0.833

FIG. 3. Illustration of the two geometries used for the Monte Carlo simulations. The cylindrical slab with radius 200 mm mimics an infinite pixel element and

the hexahedron with aperture of 0.1� 0.1 mm mimics a typical pixel element employed in mammography detectors.

2482 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2482

Medical Physics, Vol. 39, No. 5, May 2012

a �Eh isah is¼ M1

M0

: (24)

Values of the effective energy w required to liberate each

electron-hole pair for each material are obtained from

Refs. 25 and 47 and listed in Table III.

III.D. Validation

III.D.1. Comparison with theory

We compared the DQE obtained from the Monte Carlo

simulations with those calculated from a simple analytic

approach using the XCOM photon cross-sections library

(The National Institute of Standards and Technology, MD).

We assume that only photoelectric interactions result in

TABLE III. Minimum values of secondary quanta collection efficiency brequired to prevent a zero-frequency secondary quantum sink for each pho-

toconductor=spectral combination and selected detector thicknesses. As a

rule, the collection efficiency should be 10� these values to prevent high-

frequency quantum sinks (Ref. 10).

Mo=Mo Rh=Rh W=Al

q� L (mg cm�2) w-value (eV) 17 70 17 70 17 70

Si 5 0.003 0.003 0.003 0.003 0.003 0.003

a-Se 45 0.025 0.023 0.024 0.022 0.022 0.021

CdZnTe 5 0.003 0.003 0.003 0.003 0.002 0.002

HgI2 5 0.003 0.003 0.003 0.003 0.002 0.002

PbI2 5 0.003 0.003 0.003 0.003 0.002 0.002

PbO 8–20 0.008 0.008 0.008 0.008 0.007 0.007

TlBr 6.5 0.003 0.003 0.003 0.003 0.003 0.003

FIG. 4. Computer-generated Mo=Mo spectrum and AED curves of various photoconductors obtained from Monte Carlo simulations.

2483 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2483

Medical Physics, Vol. 39, No. 5, May 2012

energy absorption events and all the photon energy is depos-

ited. In this case the AED is given by

AðEÞ ¼ sðEÞaðEÞ ¼ sðEÞ 1� e�ltotðEÞL� �

; (25)

and jth moment of the AED is given by

Mj ¼ð1

0

sðEÞ 1� e�ltotðEÞL� � lpeðEÞ

ltotðEÞEjdE; (26)

where ltot, lpe, and L denote the total linear attenuation coef-

ficient, photoelectric attenuation coefficient, and thickness of

the x-ray convertor, respectively. Equations (25) and (26)

account for variability in image signal due to the spectral

width of the incident photons, but not the escape of scattered

photons, and therefore are expected to overestimate the DQE

determined from the Monte Carlo data.

III.D.2. Comparison with energy absorption tally

We verified the energy-moments method by comparing

the total energy absorption tally for each simulation with the

first moment of the pulse-height distribution [Eq. (22)]. The

largest discrepancy was 0.69%, corresponding to a 0.7 mm-

thick Si detector with a Mo=Mo spectrum, indicating the MC

simulations are reliable in terms of their statistical precision.

IV. RESULTS

IV.A. Absorbed energy distribution

Distributions of deposited energy obtained from the

Monte Carlo simulations are illustrated in Figs. 4–6 for

Mo, Rh and W target materials, respectively, for material

FIG. 5. Computer-generated Rh=Rh spectrum and AED curves of various photoconductors obtained from Monte Carlo simulations.

2484 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2484

Medical Physics, Vol. 39, No. 5, May 2012

thicknesses of 0.05 mm and 0.2 mm. The events in A(E) for

E< 12 keV are primarily due to partial energy deposition

from escape of Compton-scattered and characteristic pho-

tons. Energy absorption events in energy bins greater than

�12 keV increase with increasing thickness while events in

lower bins decrease due to scatter self-absorption. The AED

curves corresponding to Mo and Rh targets show more struc-

ture than those from W targets due to the sharp spectral

peaks resulting from characteristic emissions from Mo and

Rh. In all cases, these structures are primarily the result of

escape of characteristic emissions (K or L emissions,

depending on atomic number and spectral energy).

As a generalization, we expect DQE values to be largest

for AED curves that are distributed over the smallest energy

range, indicating a complex relationship between detector

material and spectral shape.

IV.B. Quantum absorption efficiency

Figure 7 illustrates the quantum efficiency ah is for each

photoconductor material and spectral combination as a func-

tion of mass loading calculated both theoretically and from

MC data. Excellent agreement is obtained. The quantum effi-

ciency of materials with higher atomic number Z and density

q shows a larger dependence on convertor thickness than

those with lower atomic number and density with the excep-

tion of a-Se which has lower Z and q than CdZnTe but dem-

onstrates a larger dependence on convertor thickness

because of its lower K-shell binding energy.

FIG. 6. Computer-generated W=Al spectrum and AED curves of various photoconductors obtained from Monte Carlo simulations.

2485 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2485

Medical Physics, Vol. 39, No. 5, May 2012

IV.C. Energy deposition per interaction

The average deposited energy per interacting x-ray quan-

tum for each photoconductor material is plotted as a function

of mass loading in Fig. 8. Mean deposited energy increases

with increasing average energy of the incident x-ray spectrum

indicated by dashed lines in Fig. 8 and, as expected, never

exceeds the average energy of the incident x-ray spectrum.

The dependence of deposited energy on the detector ma-

terial is different from that of quantum absorption efficiency

which is dependent upon both Z and q. HgI2, PbI2, and

CdZnTe demonstrate the largest increase in deposited energy

with increasing mass loading and a-Se and Si the smallest.

The deposited energy of each material increases as the frac-

tion of photoelectric absorption events per x-ray interaction

increases. Partial energy deposition due to escape of charac-

teristic and Compton-scattered photons reduces the mean de-

posited energy from the analytic model using Eq. (26).

IV.D. Swank noise term

Figure 9 illustrates the Swank factor IAED for the various

photoconductor materials. In general, the dependence of the

Swank factor on mass loading is similar to that of the mean

deposited energy per interaction with the exception of Si. As

illustrated in Figs. 4–6, additional peaks in lower energy

bins produced by photon escape distort A(E), and hence

increase the variability in deposited photon energy and

decrease the Swank noise term.

As shown in Fig. 9, IAED increases with mass loading and

then plateaus. It is interesting to note that IAED of high-Zmaterials decreases slightly with larger mass loading. This

observation is more pronounced in HgI2 and PbI2 for the Mo

spectrum, likely due to increased interaction probability of

high-energy photons resulting in an increase in absorption

events due to complete absorption of primary photons and

partial reabsorption of secondary photons. This has the effect

of increasing the variability in deposited energy and decreas-

ing the Swank noise term.

IV.E. Zero-frequency DQE

The minimum collection efficiency required to avoid a

zero-frequency secondary quantum sink12 as determined by

Eq. (20) are summarized in Table III. In summary, an effi-

ciency of 0.02 is required for a-Se and less than 0.01 for all

other materials. At high spatial frequencies, this value is

likely 10� this value, or 0.2 for a-Se and 0.1 for all other

materials.

FIG. 7. Quantum efficiency ah is of various photoconductor materials obtained from Monte Carlo simulations as a function of mass loading for different mam-

mography spectra. Dashed lines illustrate the analytic calculation results.

FIG. 8. Average energy depositions per interacting x-ray quantum (M1=M0) of various photoconductor materials as a function of mass loading for different

mammography spectra. The thick solid line describes the analytic result for a-Se. The dashed lines show the average energy of each spectrum.

2486 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2486

Medical Physics, Vol. 39, No. 5, May 2012

For conditions that avoid a secondary quantum sink, the

zero-frequency DQE value determined using Eq. (19) is

illustrated in Fig. 10. The best DQE performances are

achieved with PbO and TlBr. For mass loading greater than

100 mg cm�2, a-Se, HgI2, and PbI2 provide similar DQE

values to PbO and TlBr.

IV.F. Effect of pixel-aperture size on the signal andnoise properties

While the zero-frequency DQE is slightly degraded (by

0%–4%) in the a100 model, as illustrated in Fig. 11, there is

in general very little dependence on the aperture size. This

suggests that lateral escape of characteristic and Compton-

scatter photons does not severely reduce the zero-frequency

DQE of the photoconductors investigated in this study for

mammography. However, it should be noted that this calcu-

lation does not take into account the noise correlations intro-

duced by reabsorption of characteristic and Compton scatter

x-rays in neighboring pixels.

V. DISCUSSION

The MC-based model described here is the first that

makes a direct link to the cascaded-systems approach includ-

ing the effect of secondary quantum noise, and showed that

approximately 20% of charges liberated in a-Se and 10% of

charges liberated in other materials must be collected to

avoid a secondary quantum sink. Under these conditions, the

quantum efficiency, average energy absorption, Swank noise

term, and DQE were derived from AED curves obtained

from Monte Carlo simulations. We considered a geometry

having a large enough aperture to neglect lateral escape of

Compton-scattered and characteristic photons and a geome-

try having an aperture size of 0.1� 0.1 mm. Differences in

the signal and noise properties between the two geometries

were less than a few percent. These results are consistent

with the works of Sakellaris et al.33,34 The large-area DQE is

given by the product of the quantum efficiency and Swank

noise terms, and since the range of IAED values in this study

was relatively small, the effect on DQE was modest.

A common selection and design criteria for photocon-

ductors for mammographic applications are to find high-Z,

high-q materials to maximize the quantum efficiency.

Although lower-Z, lower-q materials, such as a-Se, are eas-

ily made to have sufficient thickness to attain high quantum

absorption efficiency and DQE, this can result in loss of

spatial resolution due to oblique angle effects. Another

design strategy might be to use high-q materials that have a

K-edge energy within the range of photon energies of the

incident spectrum. This would have the effect of increasing

FIG. 9. Swank noise factors (IAED) of various photoconductor materials obtained from Monte Carlo simulations as a function of mass loading for different

mammography spectra.

FIG. 10. Calculated zero-frequency DQE of various photoconductor materials as a function of mass loading for different mammography spectra.

2487 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2487

Medical Physics, Vol. 39, No. 5, May 2012

the quantum efficiency while at the same time reducing the

migration of characteristic photons. Of the materials consid-

ered in this study, PbO had the highest zero-frequency DQE

for each of the x-ray spectra considered.

If a detector can be made to operate in an energy-

discriminating photon-counting mode, rather than conven-

tional energy-integrating, the selection of materials having a

higher Swank factor is of great importance because the

Swank factor is directly related to variability in the measure-

ment of photon energy.50 For the design of mammography

detectors capable of energy discrimination, therefore, it

would be preferred to select materials having a large photo-

electric-to-Compton interaction ratio.

The photon-counting approach with Si microstrip linear-

array detectors has recently been implemented in a multi-

slit-scanned digital mammography system.51 This system

implements a 3.6-mm thick “edge-on geometry” to increase

the quantum absorption efficiency. While we did not con-

sider thicknesses greater than 1 mm for Si, we expect that

increasing the thickness to 3.6 mm could result in a

15%–20% increase in the zero-frequency DQE and Si could

therefore have similar DQE values to PbO. The DQE

reported here is greater than their reported value of approxi-

mately 0.7,51 possibly due to other factors affecting the

Swank factor but not included in our MC calculations such

as the charge-sharing effect that can significantly reduce the

Swank factor.52

Mainprize et al.41 described a theoretical DQE model

and added a noise term assuming independent shot noise.

They investigated the effect of incomplete charge collection

and showed the DQE could be reduced by as much as 50%

if the trapping density is high, analogous to the secondary

quantum sink reported here. Similar results have been

reported by others.53 Charge collection properties are deter-

mined by the material quality, polarity and the bias voltage,

known as the mean drift length (ljsjF where lj and sj denote

mobility and lifetime for the charge carrier of j, respec-

tively, and F the electric field intensity),54 and CdZnTe and

HgI2 show the best characteristics.25,47 Although an

increase in bias voltage can enhance the charge collection

properties, it can also elevate leakage current which plays a

role as shot noise. Considering these factors, HgI2 could be

a good choice of photoconductor for mammography.18,55

Alternatively, consideration of a thinner photoconductor

with a higher quantum absorption efficiency might be a

solution to mitigate incomplete charge collection under

operation in lower bias voltage. Again, PbO would be a

good candidate.

VI. CONCLUSIONS

The signal-to-noise characteristics of various photocon-

ductor materials for standard mammographic spectra were

determined using Monte Carlo methods incorporated into a

cascaded-systems model. The model provides a condition

showing that 10%–20% of all liberated quanta must be col-

lected to avoid a secondary quantum sink. The quantum effi-

ciency, average deposited energy per interacting x-ray,

Swank noise factor, and zero-frequency DQE are tabulated

by means of graphs that may help in the design and selection

of materials for photoconductor-based mammography detec-

tors. Neglecting the electrical characteristics of photocon-

ductor materials, and taking into consideration only the

physics of x-ray interactions, it is concluded that, of materi-

als considered in this study, PbO shows the strongest zero-

frequency DQE performance. When electrical characteris-

tics, such as secondary quantum gain, charge collection and

dark current are considered, HgI2 may also be an excellent

photoconductor for mammography.

ACKNOWLEDGMENTS

This research was supported by Basic Science Research

Program through the National Research Foundation of Korea

(NRF) funded by the Ministry of Education, Science and

Technology (2011-0009769), and the Canadian Natural Sci-

ences and Engineering Research Council Grant No. 386330.

There is no conflict of interest with any author and this manu-

script is not under consideration for publication elsewhere.

APPENDIX: SIGNAL AND NOISE TRANSFERTHROUGH A PHOTON-INTERACTION PROCESS

In this section, transfer functions describing propagation

of the mean �q and NPS WðuÞ through a process that combines

both selection and gain stages are developed theoretically.

Both the probability of interaction and mean conversion gain

FIG. 11. Relative differences in the zero-frequency DQE between the two different detector (a1 and a100) models.

2488 Kim et al.: Spectral analysis of signal=noise performances in photoconductors 2488

Medical Physics, Vol. 39, No. 5, May 2012

are functions of a random variable describing the incident

photon energy. The mean and NPS can be described using

the input-labeled transfer relationships developed by Rabbani

and Van Metter40

�qout ¼ �mh is �qin ¼ a �gh is �qin; (A1)

where �m ¼ a �g is the energy-dependent mean gain of the

combined process and

a�gh is¼ð1

0

sðE0ÞaðE0Þ�gðE0ÞdE0; (A2)

denotes the average of aðE0Þ�gðE0Þ weighted by the distribu-

tion of incident x-ray photon energies sðE0Þ.The NPS is given by

WoutðuÞ ¼ �mh i2s WinðuÞ � �qin½ � þ r2m þ �m2

� s�qin: (A3)

Using the product rule to show r2m ¼ a2r2

g þ �g2r2a þ r2

ar2g

results in Eq. (2).

a)Author to whom correspondence should be addressed. Electronic mail:

hokyung@pusan.ac.kr; Telephone: 82-51-510-3511; Fax: 82-51-518-

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