Fundamental x-ray interaction limits in diagnostic imaging detectors:
Frequency-dependent Swank noise
G. Hajdoka兲
Imaging Research Laboratories, Robarts Research Institute, P.O. Box 5015, London, Ontario N6A 5K8,
Canada, London Regional Cancer Program, London Health Sciences Centre, London, Ontario,
N6A 4L6, Canada, and Department of Medical Biophysics, University of Western Ontario,
London, Ontario, N6A 3K7, Canada
J. J. Battista
Departments of Medical Biophysics and Oncology, University of Western Ontario, London, Ontario,
N6A 3K7, Canada and London Regional Cancer Program, London Health Sciences Centre, London,
Ontario, N6A 4L6, Canada
I. A. Cunningham
Imaging Research Laboratories, Robarts Research Institute, P.O. Box 5015, London, Ontario N6A 5K8,
Canada, Departments of Diagnostic Radiology and Nuclear Medicine, London Health Sciences
Centre, London, Ontario, N6A 5W9, Canada, and Department of Medical Biophysics, University
of Western Ontario, London, Ontario, N6A 3K7, Canada
共Received 6 November 2007; revised 30 March 2008; accepted for publication 24 April 2008;
published 19 June 2008兲
A frequency-dependent x-ray Swank factor based on the “x-ray interaction” modulation transfer
function and normalized noise power spectrum is determined from a Monte Carlo analysis. This
factor was calculated in four converter materials: amorphous silicon 共a-Si兲, amorphous selenium
共a-Se兲, cesium iodide 共CsI兲, and lead iodide 共PbI2兲 for incident photon energies between 10 and 150
keV and various converter thicknesses. When scaled by the quantum efficiency, the x-ray Swank
factor describes the best possible detective quantum efficiency 共DQE兲 a detector can have. As such,
this x-ray interaction DQE provides a target performance benchmark. It is expressed as a function
of 共Fourier-based兲 spatial frequency and takes into consideration signal and noise correlations
introduced by reabsorption of Compton scatter and photoelectric characteristic emissions. It is
shown that the x-ray Swank factor is largely insensitive to converter thickness for quantum efficiency values greater than 0.5. Thus, while most of the tabulated values correspond to thick converters with a quantum efficiency of 0.99, they are appropriate to use for many detectors in current
use. A simple expression for the x-ray interaction DQE of digital detectors 共including noise aliasing兲 is derived in terms of the quantum efficiency, x-ray Swank factor, detector element size, and fill
factor. Good agreement is shown with DQE curves published by other investigators for each
converter material, and the conditions required to achieve this ideal performance are discussed. For
high-resolution imaging applications, the x-ray Swank factor indicates: 共i兲 a-Si should only be used
at low-energy 共e.g., mammography兲; 共ii兲 a-Se has the most promise for any application below 100
keV; and 共iii兲 while quantum efficiency may be increased at energies just above the K edge in CsI
and PbI2, this benefit is offset by a substantial drop in the x-ray Swank factor, particularly at high
spatial frequencies. © 2008 American Association of Physicists in Medicine.
关DOI: 10.1118/1.2936412兴
Key words: Swank factor, noise power spectrum, diagnostic x-ray detectors, diagnostic x-ray imaging
I. INTRODUCTION
The early work1–5 of Rose provided an important contribution to imaging science by establishing that image noise is
fundamentally limited by the statistical nature of image
quanta. Image noise was subsequently described in terms of
the noise equivalent number of quanta,6,7 ideally equal to the
number of interacting photons; and detector performance in
terms of the detective quantum efficiency 共DQE兲, the fraction of incident photons detected, ideally equal to the quantum efficiency. However, Rose assumed image quanta to be
uncorrelated and follow Poisson statistics. In diagnostic im3194
Med. Phys. 35 „7…, July 2008
aging, x rays have a spectrum of energies, and of greater
importance, a random fraction of the incident energy may
escape the detector in the form of Compton scatter or fluorescent x-ray photons. Swank8 showed that variable x-ray
energy absorption results in increased image noise and defined the noise-equivalent absorption as the product of the
quantum efficiency ␩ and what we now call the x-ray Swank
statistical factor 共Ix兲,8
Ix =
M 21
,
M 0M 2
0094-2405/2008/35„7…/3194/11/$23.00
共1兲
© 2008 Am. Assoc. Phys. Med.
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Hajdok, Battista, and Cunningham: Fundamental noise limits in diagnostic imaging detectors
where M j is the jth moment of the absorbed x-ray energy
distribution 共AED兲. The AED describes the probability per
unit energy that an incident x ray will deposit a certain energy within the detector, and Ix describes the corresponding
reduction in DQE. If no other noise sources are significant,9
DQE=␩Ix .
共2兲
Swank8 analytically calculated Ix using x-ray energyabsorption coefficients. Chan and Doi9 used Monte Carlo
simulations of photon transport to quantify x-ray energy absorption noise in common phosphor materials 共CaWO4 and
Gd2O2S兲 used in film-screen systems. Others10–17 have experimentally measured pulse-height distributions to quantify
the total conversion-gain noise for both phosphor 共e.g., CsI兲
and photoconductor 共e.g., a-Se兲 based imaging detectors.
More recently, Badano et al.18,19 used Monte Carlo simulations of x-ray and optical photon transport to determine the
combined effect of x-ray energy absorption and optical noise
in indirect detectors for breast imaging. These investigations
showed that Swank noise can reduce the DQE by 5%
− 50%, depending on the incident x-ray energy and converter
thickness. This reduction is primarily from variations in
K-fluorescent x-ray escape following a photoelectric interaction.
A significant limitation of these investigations is that
while they described variations in the total energy deposited,
they did not consider statistical correlations in image noise
introduced by the random spatial location of reabsorption,
and therefore described degradations in the zero-spatialfrequency DQE value only. For example, K-fluorescence reabsorption following a photoelectric interaction further degrades the DQE with increasing spatial frequency.20,21 More
recently, it has been shown22 that the Swank factor has a
strong spatial-frequency dependence, and
DQE共k兲 = ␩Ix共k兲,
共3兲
in the absence of other sources of noise, where k is a radial
spatial frequency and Ix = Ix共0兲.
In this article, we use Monte Carlo simulations of x-ray
photon and electron transport to examine the fundamental
limitations imposed by the various x-ray interaction processes in terms of the frequency-dependent Swank factor for
direct 共amorphous silicon, amorphous selenium, lead iodide兲
and indirect 共cesium iodide兲 conversion detector materials.
For each converter material, the importance of each x-ray
interaction process and their corresponding secondary radiation 共secondary x-ray or electron兲 is identified and quantified
as a function of x-ray energy and converter thickness. In
addition, selected Monte Carlo results are compared with
recently published experimental data to determine how close
existing detectors are to these fundamental limits, which can
serve as target benchmarks for the design and development
of future digital x-ray detectors.
The scope of this study is limited to the spatial distribution of x-ray energy absorption noise. The subsequent effect
of secondary image-forming quanta production and transport
within the converter material, is not included, but discussed.
Medical Physics, Vol. 35, No. 7, July 2008
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II. METHODS
In general, the performance of an x-ray detector can be
characterized using the frequency-dependent DQE, as given
by4
DQE共k兲 =
q̄Ḡ2MTF2共k兲 MTF2共k兲
=
,
NPS共k兲
NNPS共k兲
共4兲
where q̄ is the mean number of incident x-ray quanta per unit
area, Ḡ is the mean detector gain relating q̄ to the average
zero-mean pixel value d̄, MTF共k兲 is the modulation transfer
function 共MTF兲, and NPS共k兲 is the image NPS. The normalized NPS, defined here as NNPS共k兲 = q̄NPS共k兲 / d̄2, is convenient to use as it is dimensionless and often has a value close
to unity. Combining Eqs. 共3兲 and 共4兲 gives
Ix共k兲 =
MTF2x 共k兲
,
␩NNPSx共k兲
共5兲
where MTFx共k兲 and NNPSx共k兲 are the MTF and NNPS associated with x-ray energy deposition in the detector, and can
be determined from Monte Carlo methods. The conditions
required to achieve this performance limit, where DQE共k兲
⬇ ␩Ix共k兲, are discussed in Sec. IV.
II.A. Monte Carlo code
The latest version of the Electron Gamma Shower
共EGSnrc兲 Monte Carlo code23,24 was used to simulate the
coupled photon electron transport within typical x-ray converter materials. The user code DOSXYZnrc 共Ref. 25兲 was
used to determine the spatial distribution of absorbed dose
共energy per unit mass兲 within a rectilinear slab geometry.
The particle transport parameters, PCUT and ECUT, which
represent the minimum total energy 共kinetic plus rest mass兲
below which no radiation transport takes place, were set to 1
and 512 keV for photons and electrons, respectively. These
values were chosen to ensure that photon-electron transport
was modeled as accurately as possible within the detector
volume. In some simulations, the effect of electron transport
was suppressed by setting ECUT equal to the incident photon energy 共i.e., “on-the-spot” energy deposition兲.
II.B. Detector geometry
The modeled detector geometry, shown schematically in
Fig. 1, consisted of a broad, parallel beam of x-ray photons
normally incident on a planar uniform slab of x-ray converter
material. Each slab was subdivided into either 2048⫻ 2048
共or 4096⫻ 4096兲 voxels, whereby each voxel had a planar
area of 10⫻ 10 ␮m2 共or 50⫻ 50 ␮m2兲. Four types of x-ray
converter materials were used, spanning a wide range of
atomic numbers: 共i兲 amorphous silicon 共a-Si兲, 共ii兲 amorphous
selenium 共a-Se兲, 共iii兲 cesium iodide 共CsI兲, and 共iv兲 lead iodide 共PbI2兲. The mass density of each material was chosen to
agree with that achieved in practice.
The thickness of each converter material, t␩, was based on
a specific quantum efficiency value, ␩, as given by
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Hajdok, Battista, and Cunningham: Fundamental noise limits in diagnostic imaging detectors
y
edge effects. The 2D x-ray interaction NPS, NPSx共u , v兲,
where u and v are spatial-frequency conjugates of the x and
y directions, was estimated using27
Broad Uniform
Monoenergetic
X-ray Beam
x
3196
z
Detector
Material
2.048 or 20.48
cm
NPSx共u, v兲 =
Primary
Interaction Site
t
2.048 or 20.48 cm
Deposition of Kinetic
Energy Along
Electron Path
Re-absorption of
Secondary X-ray
FIG. 1. Detector geometry 关in Cartesian coordinates 共x , y , z兲兴 modeled in the
Monte Carlo calculations. The thickness t for each converter material was
calculated 关see Eq. 共6兲兴 for several quantum efficiency values at each incident photon energy.
t␩共h␯,Z兲 = −
ln共1 − ␩兲
,
␮共h␯,Z兲
共6兲
where ␮ is the linear attenuation coefficient at photon energy
h␯ and atomic number Z. Various quantum efficiency values
ranging from 0.10 to 0.99 were examined for each incident
photon energy 关Eq. 共6兲 is accurate for thick detectors, but for
thin detectors the thickness is understated by a few percent兴.
In general, it is reasonable to expect a dose-efficient x-ray
detector to achieve ␩ ⱖ 0.5 at most x-ray energies of importance. Further increases of thickness could be expected to
increase costs with modest return on image quality and may
reduce resolution due to obliquely incident x rays.
II.C. X-ray interaction modulation transfer function
The x-ray interaction MTF was determined from Monte
Carlo simulations of the point spread function using an infinitesimal pencil beam of x rays incident on a converter
material. Details of these simulations have been described
previously.26
II.D. X-ray interaction normalized noise power
spectrum
II.D.1. Spatial fluctuations in absorbed energy
Monte Carlo simulations were used to determine the twodimensional 共2D兲 dose 共absorbed energy per unit mass兲 distribution d共x , y兲 in each converter material. A set of ten independent dose distributions were generated, each using 108
incident x-ray histories, which was sufficient to ensure that
statistical errors in the absorbed dose per scoring voxel was
less than 10%. Each d共x , y兲 was then mean-subtracted to give
the zero-mean fluctuation in absorbed dose, ⌬d̃共x , y兲
= d̃共x , y兲 − d̄. Simulations were conducted for monoenergetic
x-ray energies ranging from 10 to 150 keV in 5 keV intervals.
II.D.2. Noise power spectrum
Each set of ⌬d̃共x , y兲 was subdivided into many nonoverlapping realizations 共ranging from approximately 100 to
1000 for each set兲 from a region of interest chosen to avoid
Medical Physics, Vol. 35, No. 7, July 2008
xoy o
具兩DFT兵⌬d关nx,ny兴其兩2典,
N xN y
共7兲
where xo and y o represent the scoring voxel center-to-center
spacings in the x and y directions, Nx and Ny are the number
of scoring voxels in each realization, 具¯典 represents an ensemble average, DFT 兵¯其 represents the discrete 2D Fourier
transform operator, and 关nx , ny兴 are voxel locations. The units
of NPSx共u , v兲 are Gy2 mm2. The standard error in our calculations of NPSx共u , v兲 ranged between 3% and 10%, depending on the number of realizations used in the ensemble
average.28 In order to further reduce the NPS uncertainty,29
NPSx共u , v兲 was radially averaged 共due to circular symmetry兲
to yield a one-dimensional radial x-ray interaction NPS,
NPSx共k兲.
The normalized NPS, NNPSx共k兲, is therefore given by
NNPSx共k兲 =
q̄NPSx共k兲
d̄2
,
共8兲
where q̄ represents the number of incident x-ray histories per
unit area in our simulations.
II.E. X-ray interaction Swank factor
II.E.1. Frequency-dependent Swank factor Ix„k…
Once the x-ray interaction MTF and NNPS were determined from the Monte Carlo simulations, the frequencydependent x-ray Swank factor was calculated for each converter material using Eq. 共5兲.
II.E.2. Zero-frequency Swank factor Ix
The zero-frequency Swank factor significantly understates
the actual effect of Swank noise. The extent of the limitation
of Ix was determined for each converter material by independent Monte Carlo simulations using Eq. 共1兲. The DOSRZnrc
user code30 was used to calculate the AED by: 共i兲 scoring the
histogram of total energy deposited in the converter material
by each incident photon, and 共ii兲 scoring the frequency of
each energy pulse amplitude over many histories. The energy
bin width of the AED was set to 0.5 keV. The AED was
determined with 107 photon histories, which was sufficient to
reduce the statistical uncertainty in each bin to less than 1%.
II.F. Comparison with published experimental
data
The x-ray Swank factor provides a target benchmark of
system performance, but is not directly measurable. Rather,
performance can be measured in terms of the DQE and compared with a prediction based on Eq. 共3兲. The quantum efficiency, as determined by the area under the AED, and
frequency-dependent x-ray Swank factor using Eq. 共5兲, were
determined for four prototype detectors31–34 using the experimental conditions listed in Table I. The incident x-ray spectra
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Hajdok, Battista, and Cunningham: Fundamental noise limits in diagnostic imaging detectors
3197
TABLE I. X-ray beam and detector parameters used in the prototype systems.
Detector material
Beam
quality
共kV兲
Beam
filtration
共mm Al兲
Detector
exposure
共mR兲
Converter
thickness
共␮m兲
Detector
element pitch
共␮m兲
Fill factor
a-Si a
a-Se b
CsIc
PbI2 d
26
70
80
90
0.4
2.0
20.0
2.0
50
3.8
1
23
1000
300
350
86
50
160
200
100
⬃1
0.7
⬃1
0.67
a
See Ref. 31.
See Ref. 32.
See Ref. 33.
d
See Ref. 34.
b
c
were modeled using an in-house MATLAB 共Mathworks, Matick, MA兲 code based on the semiempirical Tucker–Barnes
model.35
The measured DQE of a digital detector includes effects
such as detector-element aperture width, fill factor ␥, and
DQEdig共k兲 =
noise aliasing. Assuming a detector element center-to-center
spacing xo, and a uniform sensitivity aperture of width ␥xo,
an estimate of the “digital” DQE, DQEdig共k兲, based on the
x-ray Swank factor is given by36
MTF2x 共k兲sinc2共␥xok兲
⬁
冉 冊 冉 冋 册冊
,
n
n
NNPSx共k兲sinc 共␥xok兲 + 兺 NNPSx k ⫾
sinc2 ␥xo k ⫾
xo
xo
n=1
2
where spatial-frequency k is defined up to the sampling cutoff frequency, and sinc共x兲 = sin共␲x兲 / ␲x.
It will be shown that NNPSx共k兲 is largely independent of
spatial frequency under most conditions of practical importance, giving
DQEdig共k兲 ⬇ ␩␥Ix共k兲sinc2共␥xok兲,
共10兲
since the sum of sinc2共␥xok兲 and its aliases is equal to 1 / ␥.37
III. RESULTS
III.A. X-ray interaction NNPS
Figure 2 shows x-ray interaction NNPS for each converter
material 共0.99 quantum efficiency兲 tested at selected incident
monoenergetic photon energies between 10 and 100 keV. In
a-Si, the NNPS is flat and close to unity at both 10 and 30
keV, which indicates that x-ray interaction noise is uncorrelated and the zero-frequency Swank factor value is near unity
共i.e., full reabsorption of incident x-ray energy due to dominance of photoelectric interactions and low Si K edge of 1.4
keV兲. As the incident energy increases, Compton interactions
become more probable, which increases NNPS共0兲 above
unity as a result of increased backscatter escape of Compton
scatter photons. At high energies 共e.g., 70 and 100 keV兲,
reabsorption of Compton scatter x rays is responsible for the
drop in NNPS with increasing spatial frequency.
Medical Physics, Vol. 35, No. 7, July 2008
共9兲
The other materials show similar increases in the zerofrequency NNPS value, due to backscatter escape of
K-fluorescent photons, primarily at energies just above the K
edge. However, the effect is generally less in high-Z materials since there is less variability in the energy of backscatter
escape photons from Compton events than with characteristic photons from photoelectric interactions. Reabsorption of
these photons is also responsible for the subsequent drop in
NNPS with increasing frequency.
The NNPS results changed only slightly when electron
transport was excluded from the Monte Carlo simulations.
Hence, electron transport plays a minor role in the NNPS of
these materials at energies below 100 keV.
III.B. X-ray interaction Swank factor
III.B.1. Dependence on spatial-frequency
The x-ray interaction Swank factor is plotted in Fig. 3. as
a function of spatial frequency for each converter material
共0.99 quantum efficiency兲 at selected incident monoenergetic
photon energies below 100 keV. At 10 keV in a-Si, Ix共k兲 is
close to unity and has little frequency dependence. However,
it drops to approximately 0.7 and 0.2, at 30 and 70 keV,
respectively, for spatial frequencies greater than 1 cycle/ mm
because of the substantial low-frequency drop in the MTF
共e.g., 55% at 70 keV, due to reabsorption of Compton scatter
photons兲.26 Since Ix共k兲 is proportional to the squared MTF,
such a low-frequency drop has important implications on
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3198
FIG. 2. Monte Carlo x-ray interaction NNPS for each converter material 共␩ = 0.99兲 at selected incident monoenergetic photon energies 共note that ordinate axis
is plotted up to 10 cycles/ mm for a-Si and 25 cycles/ mm for the others兲. The zero-frequency value is greater than unity when 共random兲 backscatter escape
of Compton scatter or K-fluorescent photons is significant. Reabsorption of Compton 共a-Si兲 and K-fluorescent 共others兲 x rays causes the observed drop with
frequency.
potential image quality from detectors using a silicon-based
converter material for all but low x-ray energies. In the other
materials, reabsorption of K-fluorescent x-ray photons causes
a substantial decrease in the x-ray Swank factor, particularly
at energies just above the K edge.
III.B.2. Dependence on incident x-ray energy
Figure 4 shows x-ray interaction Swank factors as a function of incident photon energy for each material 共0.99 quantum efficiency兲 tested at selected spatial-frequencies below
10 cycles/ mm. In the case of a-Si, the x-ray Swank factor at
nonzero frequencies decreases significantly from approximately 0.95 to 0.15 between 10 and 80 keV. This is due to
Compton events becoming more probable, and in the diagnostic energy range, Compton scatter x rays retain, on average, ⬃80% of the incident photon energy.38 As a result, these
secondary x rays may be reabsorbed far from the primary
interaction site, causing significant degradation of the x-ray
interaction MTF, and hence, Swank factor, at low spatial
frequencies.26 In the case of the other materials, a substantial
drop in the Swank factor occurs at the K edge. At energies
greater than approximately 20 keV above the K edge, this
drop is largely recovered.
Medical Physics, Vol. 35, No. 7, July 2008
III.B.3. Dependence on x-ray converter thickness
The x-ray interaction Swank factor is shown in Fig. 5 as a
function of converter thickness 共expressed in terms of quantum efficiency兲 for each material tested at selected incident
photon energies between 20 and 100 keV. Column I represents the conventional 共zero-frequency兲 x-ray Swank factor,
as calculated from the moments of the AED 关see Eq. 共1兲兴;
while column II represents the frequency-dependent x-ray
Swank factor evaluated at 10 cycles/ mm, as determined
from the MTF and NNPS 关see Eq. 共5兲兴. Comparison of these
two versions is meant to show how significant the Swank
factor is at nonzero spatial frequencies.
In the case of the zero-frequency Swank factor, a quantum
efficiency of 0.99 represents a best-case scenario. In general,
Ix共0兲 increases with thickness as reabsorption of secondary x
rays increases, thereby reducing the variability in total energy deposited. However, this dependence is modest for all
materials and energies investigated. a-Si is the possible exception, but only for energies greater than 40 keV, where it is
not likely to be used due to the need for a very thick converter. Therefore, these zero-frequency Swank values are applicable to any detector of practical importance.
At 10 cycles/ mm, the frequency-dependent Swank factor
decreases with increasing converter thickness. However, this
dependence is again modest with the exception of a-Si at
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3199
FIG. 3. Monte Carlo x-ray interaction Swank factor as a function of incident photon energy for each converter material 共␩ = 0.99兲 at selected incident
monoenergetic photon energies below 100 keV. The zero-frequency value is decreased from unity by backscatter photon escape. The drop with increasing
frequency is due to reabsorption of Comphon 共a-Si兲 and K-fluorescent 共other materials兲 photons. Note that the zero-frequency value of the NNPS was
extrapolated for each Swank factor shown.
high energies, justifying our use of a 99% quantum efficiency thickness as a reasonable approximation for most detectors of importance.
stead that a secondary optical quantum sink exists, as discussed below, that would also degrade the DQE with increasing frequency.
III.C. Comparison with published experimental data
IV. DISCUSSION
Other physical factors than x-ray interaction processes
may degrade the DQE of a detector in practice, such as electronic noise, detector element fill factor, noise aliasing, and
secondary quanta spread and collection. In Fig. 6, the experimental DQE of various prototype detectors based on each
converter material are compared with the x-ray interaction
DQE. In each material, the x-ray interaction Swank factor
has only a minor frequency dependence, and when scaled by
the calculated quantum efficiency and detector fill factor,
gives a good estimate of the low-frequency DQE in a-Si and
a-Se. The agreement is slightly less in PbI2. At higher
spatial-frequencies, noise aliasing has a substantial effect on
the three photoconductors. In these materials, the digitaldetector DQE in terms of the x-ray Swank factor in Eq. 共9兲
describes the measured DQE very well and is indistinguishable from the simple form in Eq. 共10兲. However, in CsI, there
is less agreement with the simple model, and it is surprising
that noise aliasing could have the large effect observed, as
optical blur tends to reduce noise aliasing. It is possible in-
While the conventional 共zero-frequency兲 Swank factor
has been used for many years to describe detector performance, it is surprising to see how strong its frequency dependence is under many conditions of practical importance.
In general, Ix共k兲 drops dramatically at very low frequencies
共⬃0.3 cycles/ mm兲 when Compton interactions are important. This occurs in low-Z materials, such as a-Si, and energies above 40 keV. Although it is generally accepted that
a-Si may be impractical at these higher energies due to the
required detector thickness, this result quantifies how badly
any low-Z detector would perform, even if converter density
could be significantly increased. The x-ray Swank factor also
drops significantly at energies just above the K edge for frequencies greater than 1 or 2 cycles/ mm for high-Z materials. This will offset the potential improvements in DQE that
may result from a corresponding increase in quantum efficiency at these energies. As there is only a modest frequency
dependence 共⬍20%兲 in the NPS for most of the conditions
investigated, x-ray interaction noise remains relatively un-
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Hajdok, Battista, and Cunningham: Fundamental noise limits in diagnostic imaging detectors
3200
FIG. 4. Monte Carlo x-ray interaction Swank factor as a function of incident photon energy for each conveter material 共␩ = 0.99兲 at selected spatial frequencies.
The 0 cycles/ mm profile represents the conventional x-ray Swank factor as calculated from Eq. 共1兲.
correlated and the shape of the x-ray Swank factor is determined largely by the squared MTF through Eq. 共5兲.
The modest dependence of the x-ray Swank factor on
converter thickness 共Fig. 5兲, for quantum efficiency values
greater than 0.5, shows that it is representative of most detectors. An exception may be a-Si at high energy, where this
detector would have a very low DQE, and hence, not be
practical. Thus, the values reported in Figs. 3 and 4 can be
interpreted as reasonable estimates for any detector using
these materials, regardless of converter density or thickness,
as long as quantum efficiency is close to 0.5 or greater. In
practice, converter thickness can be an important consideration if the transport of secondary quanta 共e.g., optical light兲
becomes depth dependent. For example, in the case of a
phosphor, this depth dependence can lead to an increase in
the optical component of Swank noise and result in a decrease in DQE at nonzero spatial frequencies due to the Lubberts effect.39–43
The frequency-dependent x-ray Swank factor describes an
upper limit of the DQE for any detector. Thus, it forms the
basis of predicting how successful a new 共or existing兲 detector might be for special imaging applications. Table II shows
spatial-frequency values at which the x-ray Swank factor
drops to 0.75 at energies representative of mammography
共20 keV兲, ␮-CT 共40 keV兲, general radiography 共60 keV兲,
and chest radiography 共80 keV兲. These values indicate the
maximum spatial frequency at which each converter could
共potentially兲 have a high DQE at each energy. It is shown
Medical Physics, Vol. 35, No. 7, July 2008
that a-Si is not appropriate for any application other than
mammography, a-Se could be used for all applications, and
that both CsI and PbI2 have limited potential for highresolution imaging at energies just above their respective K
edges.
Although the x-ray Swank factor provides tremendous insight into how a detector could perform in principle, reaching this level of performance can be a challenge. For example, a large number of secondary quanta 共e.g., optical
light兲 must be generated and collected with each interacting
x-ray to avoid secondary quantum sinks.44 A simple
cascaded-systems analysis shows that a secondary quantum
sink can be avoided if the following condition is satisfied
␤ MTFs2共k兲 Ⰷ
W
ĒabNNPSx共k兲
⬇
W
Ēab
,
共11兲
where ␤ is the collection and detection efficiency of secondary quanta, MTFs共k兲 describes the spread of secondary
quanta before being detected, W is the converter work energy
in keV 共average energy to liberate each secondary quantum兲,
Ēab is the average energy absorbed in keV per interacting x
ray, and NNPSx共k兲 is the normalized x-ray interaction NPS
reported here, which is always of order unity. In the case of
a CsI-based mammography detector, W ⬃ 0.02 keV and the
system MTF may be dominated by optical spread. If we
choose the frequency at which the MTF has a value of 0.3 to
be our maximum frequency of interest, we obtain the condi-
3201
Hajdok, Battista, and Cunningham: Fundamental noise limits in diagnostic imaging detectors
Silicon
Silicon
1
0.9
0.8
0.7
0.6
0.5
0.4
20 keV
40 keV
60 keV
80 keV
100 keV
0.3
0.2
0.1
10
(a)
20
30
40
50
60
70
80
Swank Factor (10 cy/mm)
Swank Factor (0 cy/mm)
1
0
0
0.6
0.5
0.4
0.3
0.2
10
20
30
70
80
Selenium
0.7
0.6
0.5
0.4
0.3
0.2
90 100
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
10
20
30
40
50
60
70
80
0
0
90 100
10
20
30
40
50
60
70
80
Quantum Efficiency [%]
Quantum Efficiency [%]
Cesium Iodide
Cesium Iodide
90 100
1
Swank Factor (10 cy/mm)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
10
(c)
20
30
40
50
60
70
80
0
0
90 100
10
Quantum Efficiency [%]
20
30
40
50
60
70
80
90 100
Quantum Efficiency [%]
Lead Iodide
Lead Iodide
1
Swank Factor (10 cy/mm)
1
Swank Factor (0 cy/mm)
60
1
(b)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(d)
50
Selenium
0.8
0
0
40
Quantum Efficiency [%]
Swank Factor (10 cy/mm)
Swank Factor (0 cy/mm)
0.7
0
0
90 100
0.1
Swank Factor (0 cy/mm)
0.8
0.1
1
0
0
0.9
Quantum Efficiency [%]
0.9
0
0
3201
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
10
20
30
40
50
60
70
80
90 100
Quantum Efficiency [%]
0
0
10
20
30
40
50
60
70
80
90 100
Quantum Efficiency [%]
FIG. 5. Monte Carlo x-ray interaction Swank factor as a function of converter thickness 共expressed as a quantum efficiency兲 for each converter material at
selected incident photon energies and 0 共column I兲 and 10 共column II兲 cycles/ mm.
tion ␤ Ⰷ 0.01 to avoid a secondary quantum sink. This can be
extremely difficult to achieve in practice for phosphor-based
detectors.
Medical Physics, Vol. 35, No. 7, July 2008
Even when electronic noise is negligible, digital detectors
present additional challenges to achieve the performance predicted by the x-ray Swank factor due to detector fill factor
3202
Hajdok, Battista, and Cunningham: Fundamental noise limits in diagnostic imaging detectors
3202
FIG. 6. Comparison of “ideal” DQE based on quantum efficiency 共QE兲 and frequency-dependent x-ray Swank factor with the digital DQE based on Eq. 共9兲,
simplified digital DQE based on Eq. 共10兲 and experimental 共EXP兲 data for: 共a兲 a-Si,31 共b兲 a-Se,32 共c兲 CsI,33 and 共d兲 PbI2 共Ref. 34兲 prototype detectors.
and noise-aliasing issues. A nonunity fill factor effectively
reduces quantum efficiency. This is shown in Figs. 6共b兲 and
6共d兲 where the zero-frequency DQE value is reduced from a
calculated quantum efficiency of 0.60 to 0.4 in a-Se and
from 0.56 to 0.3 in PbI2, consistent with measured DQE
values. In both cases, the zero-frequency x-ray Swank factor
was close to unity. Noise aliasing increases the normalized
NPS at frequencies near the sampling cutoff frequency36 and
is responsible for most of the DQE drop with frequency from
the zero-frequency value for a-Si, a-Se, and PbI2.
It should also be noted that although the Swank factor is
called a “factor,” the effect it has on the DQE, as used in Eq.
共3兲, is multiplicative only for Poisson noise sources.8
The spatial-frequency dependence of the x-ray Swank
factor has an additional important implication for energydiscriminating detectors that has not been addressed. New
detectors are currently being developed by a number of investigators using a-Si,45 CdZnTe,46 PbO,47 and other materials, which may result in new energy-weighted photoncounting imaging techniques. However, random relocation of
x-ray energy into different detector elements will seriously
compromise performance, and the frequency-dependent
x-ray Swank factor will almost certainly play an interesting
role.
V. CONCLUSIONS
Monte Carlo simulations were used to determine the spatial fluctuations in absorbed energy for various x-ray converter materials 共a-Si, a-Se, CsI, PbI2兲 as a function of incident photon energy in the diagnostic range and converter
thickness. From each simulation, the normalized “x-ray in-
TABLE II. Spatial-frequency values 共cycles/ mm兲 at which the x-ray Swank factor drops to 0.75 for each
converter material at selected energies, interpolated from data in Figs. 3 and 4. These values indicate the
maximum frequency at which these converters could have a high DQE at each energy.
Converter
material
Mammography
共20 keV兲
Micro-CT
共40 keV兲
General radiography
共60 keV兲
Chest radiography
共80 keV兲
a-Si
a-Se
CsI
PbI2
⬎10
⬃10
⬎25
⬎25
⬍1
⬎25
⬃1
⬃4
⬍0.3
⬃25
⬍2
⬃2.5
⬍0.3
⬃2
⬃5
⬃10
Medical Physics, Vol. 35, No. 7, July 2008
3203
Hajdok, Battista, and Cunningham: Fundamental noise limits in diagnostic imaging detectors
teraction” noise power spectrum was calculated and used to
determine the frequency-dependent x-ray Swank statistical
factor Ix共k兲 as a function of spatial frequency between 0 and
25 cycles/ mm 共0 and 10 for a-Si兲. This factor describes the
effect of random variations in absorbed energy, including
spatial correlations, associated with secondary x-ray and
electron reabsorption. It does not include the effect of engineering limitations or noise aliasing, but when scaled by the
quantum efficiency, it represents an upper limit on the detective quantum efficiency.
In low-Z materials 共a-Si兲, reabsorption of Compton scatter x rays severely degrades Ix共k兲 above 40 keV, while in
higher-Z materials 共a-Se, CsI, PbI2兲, reabsorption of
K-fluorescent x rays has a significant effect at their respective K edges. The effect of electron transport is generally
negligible except for a-Se and CsI above 100 keV. Converter
thickness has a modest effect 共ⱕ10%兲 on Ix共k兲 for each material having a quantum efficiency greater than 0.5, except in
a-Si at high energy.
To achieve performance approaching the x-ray Swankfactor prediction, secondary quantum sinks must be avoided.
A simple relationship is provided to ensure this condition is
met. Noise aliasing in digital detectors will also reduce the
DQE. A simple expression is derived giving the DQE in
terms of quantum efficiency, x-ray Swank factor, detector
element size, and fill factor. Published DQE results for digital detectors using each converter material show very good
agreement with the simple expressions, although the potential for a secondary optical quantum sink cannot be ruled out
in the CsI detector. Based on the calculated x-ray Swank
factors, it is concluded that: 共i兲 a-Si is well-suited for mammography only; 共ii兲 a-Se has a wide scope of imaging applicability; and 共iii兲 benefits from an increased quantum efficiency just above K-edge energies in CsI and PbI2 are offset
by a significant drop in the x-ray Swank factor, particularly
at high spatial frequencies.
ACKNOWLEDGMENTS
The authors are grateful for financial support from the
Canadian Institutes of Health Research 共CIHR兲 and Ontario
Research and Development Challenge Fund 共ORDCF/
OCEBCRI project兲.
a兲
Author to whom correspondence should be addressed. Present address:
Imaging Research Labs, Robarts Research Institute, P.O. Box 5015, 100
Perth Drive, London, Ontario N6A 5K8, Canada. Telephone: 共519兲 6858300 ext 34035; Fax: 共519兲 663-3900. Electronic mail:
[email protected]
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