spectral analysis of fundamental signal and noise...
TRANSCRIPT
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:
[email protected]; Telephone: 82-51-510-3511; Fax: 82-51-518-
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