IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 10, OCTOBER 2013
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Analytic Model of Energy-Absorption Response
Functions in Compound X-ray Detector Materials
Seungman Yun, Ho Kyung Kim*, Hanbean Youn, Jesse Tanguay, and Ian A. Cunningham
Abstract—The absorbed energy distribution (AED) in X-ray
imaging detectors is an important factor that affects both energy
resolution and image quality through the Swank factor and
detective quantum efficiency. In the diagnostic energy range
(20–140 keV), escape of characteristic photons following photoelectric absorption and Compton scatter photons are primary
sources of absorbed-energy dispersion in X-ray detectors. In
this paper, we describe the development of an analytic model of
the AED in compound X-ray detector materials, based on the
cascaded-systems approach, that includes the effects of escape
and reabsorption of characteristic and Compton-scatter photons.
We derive analytic expressions for both semi-infinite slab and
pixel geometries and validate our approach by Monte Carlo
simulations. The analytic model provides the energy-dependent
X-ray response function of arbitrary compound materials without
time-consuming Monte Carlo simulations. We believe this model
will be useful for correcting spectral distortion artifacts commonly
observed in photon-counting applications and optimal design and
development of novel X-ray detectors.
or electron-hole pairs in a photoconductor, and liberation of
these secondary quanta is determined by the random processes
of X-ray energy deposition [1]. For example, at diagnostic
energies (20–140 keV), variations in absorbed energy due to
random escape of characteristic photons following a photoelectric interaction (PE) or scatter photons following a Compton
(incoherent) interaction are primary causes of absorbed energy
dispersion [2], [3] that increase variability in deposited energy
in both energy-integrating [4] and photon-counting X-ray systems [3], [5]. Accurate and precise measurements of incident
photon energy are particularly important for photon-counting
techniques such as K-edge imaging [6].
Over the past several decades, many researchers [4], [7], [3],
[8], [9] have characterized the effects of energy dispersion on
X-ray image quality in terms of the absorbed energy distribution
(AED) [keV ] given by
Index Terms—Absorbed energy distribution (AED), compound
semiconductor, Compton scattering, detector response function, digital radiography, fluorescence, photoelectric absorption,
photon-counting imaging, X-ray convertor.
(1)
I. INTRODUCTION
T
HE SIGNAL generated by X-ray detectors is normally
proportional to the charge collected by each element
in an array of detector elements. This charge is proportional
to the number of secondary image quanta liberated in the
X-ray convertor, which may be optical photons in a phosphor
Manuscript received April 04, 2013; revised April 30, 2013; accepted May
24, 2013. Date of publication June 03, 2013; date of current version October 02,
2013. This work was supported by the Basic Science Research Program through
the National Research Foundation (NRF) of Korea funded by the Ministry of
Education, Science and Technology (2011-0009769), the Canadian Institutes of
Health Research (CIHR) operating grant and Natural Sciences and Engineering
Research Council (NSERC) discovery grant (386330-2010). Asterisk indicates
corresponding author.
S. Yun is with the Mechanical Engineering Department, Pusan National
University, Busan 609-735, Korea, and also with Imaging Research Laboratories, Robarts Research Institute, London, ON, N6A 5K8 Canada (e-mail:
[email protected]).
*H. K. Kim is with the Mechanical Engineering Department and the Center
for Advanced Medical Engineering Research, Pusan National University, Busan
609-735, Korea (e-mail: [email protected]).
H. Youn is with the Mechanical Engineering Department, Pusan National
University, Busan 609-735, Korea (e-mail: [email protected]).
J. Tanguay, and I. A. Cunningham are with Imaging Research Laboratories,
Robarts Research Institute, and Department of Medical Biophysics, University
of Western Ontario, London, ON, N6A 5K8 Canada (e-mail: jessetan@robarts.
ca; [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMI.2013.2265806
where
represents the incident X-ray photon energy,
[keV ] describes the spectral distribution of incident photons,
[keV ]
represents the absorbed X-ray energy, and
is the energy response function describing the average spectral
given an incident photon with
density of absorbed energy
.
energy
Experimental determinations of the AED, such as the work of
Blevis et al. [10], provide performance information of both conventional and photon-counting detectors. Alternatively, Monte
Carlo (MC) methods can be used to accommodate complex
geometries but generally require substantial computational effort. Analytic models are attractive as they can provide more
insight into the performance limitations of these systems. For
example, LeClair et al. [11] described the AED of a CdZnTe
spectroscopy detector but did not consider the effects of X-ray
scatter. Scatter is particularly important in pixelated detectors
used for imaging as scatter escaping from an element reduces
the energy deposited and absorbed scatter from a nearby element
gives the appearance of additional low-energy incident photons.
In this study, we describe a new approach for determining
analytic models of absorbed energy distributions based on the
established methods of linear cascaded-systems analysis (CSA)
[12]–[20]. The CSA method accounts for spatial correlations
and stochastic properties of energy-absorbing processes (photoelectric and Compton interactions) and has been successful
for describing signal and noise performance and the detective
quantum efficiency (DQE) of many X-ray detectors [20]. We
apply it to a description of the AED of the five common convertor materials Si, Se, CsI, CdTe, and HgI , and validate the
0278-0062 © 2013 IEEE
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 10, OCTOBER 2013
The probability that an incident X-ray photon with energy
interacts in the X-ray convertor and deposits energy is given by
the quantum absorption efficiency
(2)
Fig. 1. Schematic representation of the geometry used to calculate the reabsorption probabilities of characteristic and Compton-scatter photons for a semiinfinite slab with thickness .
results with MC calculations. The method may be useful in an
evaluation of other promising candidate compound convertor
materials such as PbI , PbO, or TlBr.
II. THEORY
Fig. 1 illustrates an X-ray interaction in which a scatter
photon is emitted. Energy is deposited at the primary interaction site and potentially at a remote reabsorption site. Scatter
escape (from either front or rear detector surfaces) will result
in a partial loss of incident energy and thereby dispersion of
absorbed energy. In this model, all incident photons are treated
identically and photoelectric and Compton interactions are
each viewed as special cases of this simple interaction model.
Rayleigh (coherent) scatter interactions do not deposit energy
and are not included [20].
Similar to Chan and Doi [21], the X-ray convertor is subdivided into layers and
solid-angle increments to derive
analytic expressions for reabsorption probabilities. In all cases,
incident photons are assumed normal to the X-ray convertor
plane. Characteristic emissions and Compton scatter photons
that are reabsorbed are assumed to deposit all of their energy
with the exception of reabsorption of high-Z characteristic photons in low-Z atoms where another characteristic photon may
be emitted. To keep the model simple, we consider only K-shell
transitions and use the K photon energy averaged by the fraction of K-L and K-M transition similar to the approach used by
Hajdok et al. [2].
A. X-ray Interaction Model
Using the parallel cascades method of Yao and Cunningham
[16], Fig. 2 illustrates the possible combinations of energy-depositing events in compound X-ray detector materials for a slab
geometry. Each connecting arrow represents a spatial distribution of points that connect the output of one process to the input
of the next and describe where in the image plane each process
takes place. Each item connected by arrows represents a point
process that acts on each point in its input distribution to produce a new distribution of points at its output. The initial input
distribution represents the spatial distribution of X-ray photons
incident on the detector. The final output distributions represent
spatial distributions of energy-depositing events.
is the sum of photoelectric and inwhere
coherent linear interaction coefficients, respectively, and represents convertor-material thickness. The coherent (Rayleigh)
scatter interaction coefficient is not included in
since those
interactions do not deposit energy or change the photon energy
and do not affect the probability of photoelectric or incoherent
scatter interactions. The probabilities of photoelectric
and
Compton scatter
interactions are, therefore, given by
(3)
(4)
1) Photoelectric Interaction: In compound X-ray detector
materials, PE absorption may occur in either the high or low-Z
atom with probability
given by
(5)
where Z denotes high or low-Z atoms when replaced with H
and
represent the fractional weight,
or L, and
density, and photoelectric attenuation coefficient, respectively.
If the incident photon energy
is greater than the binding
energy of a K-shell electron, a PE interaction may result in the
emission of a characteristic photon with probability
where
and represent the K-shell participation factor and fluorescence yield, respectively [22]. While there may be a characteristic photon emitted from other electron transitions, Hajdok et
al. [2] showed that L-shell transitions are negligible.
2) Reabsorption of Characteristic Emission: Based on the
technique described by Chan and Doi [21], we first consider the
characteristic reabsorption probability for the semi-infinite slab
geometry. The probability of generating a K photon in th layer
is given by
(6)
where
and
represent the sublayer
thickness and index, respectively. The probability of generating
a K photon in an X-ray convertor with thickness is then given
by
(7)
The characteristic photon will be emitted isotropically in the
azimuthal angle . This symmetry allows us to use a quarter
sphere (corresponding to
and
) to calculate reabsorption properties. The set of solid angle elements
corresponding to the first quadrant in
YUN et al.: ANALYTIC MODEL OF ENERGY-ABSORPTION RESPONSE FUNCTIONS IN COMPOUND X-RAY DETECTOR MATERIALS
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Fig. 2. Schematic diagram describing the cascades of possible energy-depositing processes considered in this work including photoelectric emission escape or
reabsorption, and Compton scatter escape or reabsorption. Each arrow represents a spatial distribution of points that connect the output of one process to the input
of the next, describing where in the image plane each process takes place. Each item connected by arrows represents a process that acts on each point of the input
distribution. For example, the first (left-most) triangle represents a quantum-selection process where each input point represents a photon that is either passed or
not passed to the output according to the binomial probability . Each diamond represents a branch where each input point is passed to only one of the outputs
according to the specified probabilities. AED is given by the combined probability of entries in each column (divided into photo peak, escape peak, and Compton
continuum contributions for clarity). The shaded area describes reabsorption of a high-Z characteristic emission in a low-Z atom which results in an additional
low-Z characteristic emission.
azimuthal angle plane and
is given by
change in polar angle
The probability that a characteristic photon generated in the th
layer is reabsorbed in
is given by
This result gives the probability that an incident photon undergoes a photoelectric interaction and generates a characteristic
X-ray that is reabsorbed in the convertor material. It is dependent on
to reflect the energy dependence of interaction depth
in the convertor. The probability of reabsorption per generated
characteristic photon
is then given by
(9)
(13)
represents the
Equation (13) gives the average probability that a characteristic
photon is reabsorbed in the convertor material used to describe
reabsorption in Fig. 2.
3) Compton Interaction: In a Compton-scatter event, energy
is deposited at the primary interaction site and potentially at a remote reabsorption site and has recently been described in terms
of cascaded-system analyses [20]. Unlike photoelectric emissions, the energy of Compton-scatter photons
is a random
variable determined by
and scatter angle [22]
(8)
where
represents the K-photon energy, and
traveling path of the K photon (Fig. 1)
(10)
Summing (9) over all solid angles gives the reabsorption probability for a characteristic photon emitted from the th layer
(11)
(14)
which is independent of the incident photon energy
. The
average K-photon reabsorption probability for a convertor with
thickness is, therefore, given by
(12)
keV is the electron rest enwhere
ergy and is a random variable with probability density function (PDF) given by [22]–[24]
(15)
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where
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 10, OCTOBER 2013
represents the classical radius of the electron
m and the Compton cross section per electron is given
by [25]
(16)
The Klein–Nishina factor
, is given by [22]
(17)
This expression for Klein–Nishina factor does not include the
effects of bound electrons that might be important for high-Z
atoms and low energies. The spectral density of Compton-scattered photons is then given by
Fig. 3. Schematic representation of the geometry used to calculate the reabsorption probabilities of characteristic and Compton-scatter photons in a pixel
with thickness and pixel pitch . The main difference between the pixel and
slab geometries is the traveling path ( for the pixel and for the slab) of the
characteristic/Compton-scatter photon.
energy
by
is reabsorbed in the convertor material is then given
(25)
(18)
4) Reabsorption of Compton-Scatter Photons: The reabsorption probability of Compton-scatter photons is determined using
a technique similar to above. The probability of a Compton
event in the th layer of thickness
is given by
Summing over all layers gives the probability that the Comptonscatter photon is reabsorbed
(26)
(19)
and the total probability of a Compton event is given by
(20)
The scatter spectrum is divided numerically into bins of energy
and width
keV with corresponding solid-angle
increments of
(21)
where
(22)
(23)
Subscripts lb and ub represent scatter angle lower and upper
limits for a specified scatter-photon energy and the minimum
scatter energy occurs when
. The probability that a
Compton-scatter photon with energy
scattered in the th
layer is emitted into solid angle
is given by
This result gives the probability that an incident photon interacts
through the Compton effect and produces a Compton photon
with energy
that is reabsorbed in the convertor material. The
probability of reabsorption per scattered photon is then given by
(27)
5) Extension to Pixel Geometry: We extend the semi-infinite slab model to pixel geometries by defining square-shaped
pixel boundaries in the
plane, such as the regions of charge
collection defined by electric fields in photoconductor-based direct-conversion detectors as illustrated in Fig. 3. For simplicity,
we assume that incident photons arrive at the center of each element and ignore the effect of charge sharing (or optical scatter
in a phosphor) between elements. While this may appear to
be a limiting simplification, it is well recognized that photoncounting detectors must include some form of element “binning” to accurately determine the total deposited energy surrounding each interaction [26], [27], [5]. Our assumption therefore corresponds to the case of summing the signals from elements surrounding the primary interaction location.
The horizontal distance in Fig. 3 is given by
(24)
where
is given by (8), is the polar angle index which
is a function of
through
and
, and is given by (9).
The probability that a scattered photon from the th layer with
(28)
where is the pixel pitch,
in azimuthal angle and
represents the change
is the azimuthal angle
YUN et al.: ANALYTIC MODEL OF ENERGY-ABSORPTION RESPONSE FUNCTIONS IN COMPOUND X-RAY DETECTOR MATERIALS
index. The traveling path for the pixel geometry
therefore, given by
(Fig. 3) is,
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a step at
. After differentiating, the probability density
of depositing energy into the photopeak given a Compton event,
, is given by
(29)
and
represent the polar
where
angle index and change in polar angle, respectively [see (10)].
Note that
is independent of the depth index and therefore
only accounts for the element boundaries in the
plane.
By choosing the smaller of
and
our model accounts for
pixel boundaries in all three dimensions. For example, if the
incident photon interacts at the bottom of the X-ray convertor
and the scattering angle is very low, then the photon will escape
through the bottom surface and the traveling path is given by .
Alternatively, if a secondary photon is emitted at an scattering
angle close to
then it may escape the pixel in the lateral
direction and the traveling path is given by . The solid angle
modified for the pixel geometry is given by
(33)
with area equal to the avresulting in a -function at
erage reabsorption probability as a function of scatter energy
and weighted by the scatter spectrum.
The Compton continuum results when the scatter photon is
not reabsorbed and only energy
is deposited as represented by the entry in the Compton-continuum column. The
integral probability is given by
(34)
(30)
and differentiating with respect to energy gives the Compton
continuum as
The probability that a characteristic photon generated in the th
layer is reabsorbed in
is therefore given by
(35)
(31)
The probability that a Compton photon generated in the th layer
is reabsorbed in
is given by
1) Slab Geometry: Combining photopeak results for photoelectric and Compton interactions gives the photopeak component of
as
(32)
Summing over all , and gives the reabsorption probabilities
of
and Compton photons
for the pixel geometry
(similar to (13) and (27) for the slab geometry).
B. Analytic Response Function
is determined by
The analytic response function
considering all possible cascades in Fig. 2 resulting in energy
deposition for any one interacting photon. For example, photoelectric interactions result in six cascade combinations that contribute to the photopeak and three to escape peaks, with each
contribution corresponding to an entry in the columns labeled
photopeak and escape peak in Fig. 2. The probability of each
contribution is given by the product of all gains in the corresponding cascades and
is equal to the combined
probability density of depositing . To assist with these calculations, we define
as the integral probability of depositing any energy between 0 and and therefore
.
Compton scatter is more complicated since the scatter-photon
energy is represented by the random variable , resulting in
a photopeak and Compton continuum. The photopeak occurs
when the scatter photon is reabsorbed, as represented by the
lowest entry in the photopeak column. The corresponding integral probability of depositing energy is zero for
with
(36)
where
. Similarly, escape peaks are given by
(37)
and the Compton continuum component by
(38)
Combining these results gives the energy response function
(39)
2) Pixel Geometry: While the response function for a pixel
geometry has a form similar to (39), reabsorption probabilities
are lower due to shorter traveling paths which result in increased
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 10, OCTOBER 2013
Fig. 4. Schematic illustration showing the effect of multiple scatter on the response function for slab (a) and pixel (b) geometries. Numbers in converter
correspond to photon energy deposited with each interaction.
Fig. 5. Schematic illustration of MC parameters used to calculate the AED.
K-escape peak and Compton continuum, and absorption of characteristic and Compton-scatter photons escaping from neighboring pixels cause additional contributions. We represent these
additional contributions as
and let
represent the resulting distribution of absorbed energies in pixel geometries as
(40)
where
is given by (39) with the replacement of
and
with
and
. It is important to note that
while
describes the average distribution of energy
deposited per photon incident on that element,
includes absorption of characteristic emissions and Compton
scatter from neighboring elements and therefore describes the
average energy response from multiple photons incident on a
detector array.
By the principle of reciprocity, the average energy deposited
is equal to the average energy that escapes laterally and is deposited in other elements, giving the spectral density of reabsorbed peaks (rp) as
(41)
and reabsorbed Compton scatter (rs) as
(42)
The total contribution is therefore
.
Fig. 4 illustrates the effect of multiple scatter on response
functions. With the slab geometry, multiple scatter has little effect other than possibly altering the probability of scatter-photon
escape. However, with pixel geometries, multiple scatter increases energy dispersion by moving energy from the photopeak
to lower levels if a scatter photon leaves the element as well as
adding contributions from neighboring elements.
III. VALIDATION
The analytic model is validated by MC simulations of X-ray
absorption in Si, Se, CsI, CdTe, and HgI X-ray convertor
materials. While Si is not a compound material, it has a high
Compton and Rayleigh cross section with negligible characteristic photons over all diagnostic energy ranges and is therefore
an important test of the Compton model. Se is also selected for
validation as it has a single escape peak with negligible
and
energy differences. Both Se and CsI are widely used in
digital radiography detectors and have important escape peaks.
Material properties are taken from [22], [28], and [29].
A. Monte Carlo Simulation
X-ray interaction and energy absorption was simulated
using the particle tracking (pTrac) tally of the MC user code
MCNP5 (Radiation Safety Information Computational Center
(RSICC), Oak Ridge, TN, USA) for a uniformly distributed
histories) as
parallel-beam X-ray source (2 2 cm and
illustrated in Fig. 5 for both slab and pixel geometries. The
code was configured to generate data files describing all X-ray
interactions for each photon history excluding electron transport considerations. These data files were used to determine
the resulting distributions of energy-depositing events, scored
in 1-keV bins, for both slab and pixel geometries. While both
mono and poly-energetic (RQA5 [30]) X-ray spectra were used
for slab geometries, only mono-energetic spectra were used for
pixel geometries due to the need for very large storage requirements for each photon energy. The effects of multiple scatter
on the response functions were determined by first including
multiple scatter with each history and then repeating the analysis assuming all energy in the characteristic/scatter photons is
completely deposited when reabsorbed. Also, MC simulations
included Rayleigh scattering which was not included in the
analytic models.
Theoretical and MC calculations of the AED were compared
in terms of the Swank factor, given by [4]
(43)
where
where
, and
are the first three moments of the AED
. The X-ray Swank factor takes
YUN et al.: ANALYTIC MODEL OF ENERGY-ABSORPTION RESPONSE FUNCTIONS IN COMPOUND X-RAY DETECTOR MATERIALS
Fig. 6. Comparison of analytic and MC response functions, with and without
-thick
multiple scattering, for the slab (a) and pixel (b) geometries using 500X-ray convertors and 100-keV incident photons.
on values between zero and one with
corresponding to an
ideal detector, and is commonly used to describe the effects of
energy dispersion on X-ray image quality [10], [31], [32], [3].
IV. RESULTS
In the following results, sub-layer and angle indexes of
and
were used to minimize numerical errors which may be introduced by discretization of each
dimension.
A. Energy-Response Function
1) Semi-Infinite Slab Geometry: Fig. 6(a) shows a comparison of analytic and MC energy response functions, obtained
both with and without multiple scattering, using 100 keV incident photons with 500- m-thick convertor materials. Qualitatively, there is very good agreement between the shape of the
response functions in all cases with the differences being relatively small compared to other structures. All show the primary features of photo-peak, escape peaks and Compton continuum and differences are attributed to three considerations.
The first is that analytic results show a simpler pattern of escape peaks in high-Z materials, particularly in HgI , due to the
use of an average K-photon energy rather than separate
and
emission lines. The second is due to neglecting the effect
of orbital (bound) electrons in Compton scattering in the analytic model which misses some energy broadening around the
Compton edge [33]–[35]. The third is the effect of ignoring multiple scatter in the analytic model which is more difficult to see
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in Fig. 6 and is identified by comparing the MC results obtained
with and without multiple scatter enabled in the calculation.
Multiple scatter causes a small reduction in the energy of characteristic emissions and scatter photons that leave the detector,
resulting in a small decrease in the intensity of escape peaks
and the Compton continuum with a corresponding increase at
energies just above the escape peaks and continuum. The close
agreement with MC results gives confidence that Rayleigh scattering, ignored in the analytic model, has little influence on the
response functions.
2) Pixel Geometry: Fig. 6(b) illustrates energy-response
functions for the pixelated geometry for 100 keV incident
photon energy, 500- m-thick X-ray convertors and a pixel
pitch of 100 m. In general, the distribution of absorbed energies for the pixel geometry is more dispersed than for the
slab geometry due to new contributions from absorption of
characteristic and Compton photons from neighboring elements. In addition, escape peaks are larger due to the increased
probability of escape from small detector elements. In particular, absorption of Compton photons generated in neighboring
elements results in a continuous distribution of absorbed energies for all results at energies just below the photopeaks.
However, the analytic calculations do not predict reabsorption
of fluorescence and Compton photons from multiple scatter in
the 30–70 keV energy range, as seen in Fig. 6(b). In general,
scatter has a greater effect on the response function for pixel
geometries than slab geometries, as shown in Fig. 4.
3) Energy Dependency: Fig. 7 shows a comparison of the
response function for slab geometry expressed as a function of
incident photon energy from analytic and MC calculations. The
color scales show the display range of each paired comparison.
In all cases, there is good agreement between the analytic model
and MC results with minor discrepancies in escape peaks for the
same reasons discussed above. The Compton continuum is not
visible in all but Si results due to the very low response value
compared to the photopeak height.
B. Absorbed Energy Distribution
Fig. 8 shows AED curves obtained using analytic and MC calculations including multiple scatter for an RQA-5 X-ray spectrum. The solid and dotted lines correspond to analytic and MC
results respectively for the slab geometry. Overall, there is excellent agreement between these two methods. Escape peaks are
not obvious due to the broad X-ray spectrum. Similar to energy
response results above, MC calculations show a broadening of
the Compton edge although this is visible in only the Si and
Se results where Compton plays a substantial role. The effect
of multiple scatter depends on escape peak energies and is less
obvious for the broad spectrum, but generally causes a reduction in AED values in the 15–35 keV range, depending on escape-peak energies. Very little effect is seen for HgI since the
K-edge energy of Hg is above the RQA-5 spectrum maximum
energy (70 kV).
Analytic results for the pixel geometry are represented by the
dashed lines in Fig. 8. As expected, the higher escape probability
of emission/scatter photons results in more low-energy absorption events, and therefore an increase in energy dispersion. In
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 10, OCTOBER 2013
Fig. 7. Comparison of the 2-D response function calculated by analytic (a) and
MC (b) methods for incident energies ranging from 1 to 100 keV for a 500-thick semi-infinite slab.
Fig. 8. Comparison of analytic and MC AEDs for 500- m-thick X-ray convertors with an RQA5 X-ray spectrum. The solid, dotted, and dashed lines represent
calculations for the analytic semi-infinite slab, MC semi-infinite slab (including
multiple scatter), and analytic pixel geometry, respectively.
addition, reabsorption peaks from characteristic photons generated in neighboring elements appear in the AEDs because they
are independent of incident photon energy.
Table I summarizes the X-ray Swank factor values corresponding to AED curves in Fig. 8. In all cases, the analytic and
MC results for the slab geometry agree within 3% and show similar trends. Si has the lowest value due to the greatest Compton
continuum relative to the photopeak. For all other materials, MC
results are slightly greater than analytic because of less dispersion due to multiple scatter. It is interesting to note that for this
spectrum, the largest Swank factors for the slab geometry occur
with Se and HgI and are 6%–7% greater than CsI. However,
this difference is smaller with the pixel geometry.
C. Effects of Convertor Thickness and Pixel Size
Fig. 9(a) and (b) illustrates the energy-response functions for
CdTe with 100-keV incident photons and selected X-ray con-
TABLE I
X-RAY SWANK NOISE FACTORS DETERMINED FROM
AED CURVES IN FIG. 8 FOR RQA-5 SPECTRA
vertor thicknesses for slab and pixel geometries. Increased convertor thickness results in increased photo-peak, escape peaks,
and Compton continuum due to the greater quantum absorption
efficiency for both slab and pixel geometry.
Fig. 9(c) shows the effect of pixel size for CdTe with 100-keV
incident photons. Larger pixel sizes result in greater photopeaks
due to the longer path length of emission/scatter photons in the
YUN et al.: ANALYTIC MODEL OF ENERGY-ABSORPTION RESPONSE FUNCTIONS IN COMPOUND X-RAY DETECTOR MATERIALS
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Fig. 9. Comparisons of analytic and MC response functions for CdTe using slab (a) and pixel [(b) and (c)] geometries for selected convertor thicknesses and pixel
pitches for 100 keV incident photons.
primary pixel volume. The effect of photon escape from neighboring elements in the pixel geometry increases as the element
size is reduced, resulting in increased energy dispersion.
V. DISCUSSION AND CONCLUSION
We have presented an analytic technique for obtaining the
distribution of absorbed X-ray energies in compound X-ray detector materials for both semi-infinite slab and pixel geometries.
The pixel geometry will be particularly important in describing
the imaging performance of pixelated detectors used in conventional X-ray imaging and novel photon-counting applications.
The results from our analytic approach agreed well with results
from MC calculations. The strength of the analytic approach is
that it allows us to obtain the 2-D response function of arbitrary compound materials without time-consuming MC simulations. It does not include the effects of multiple scattering or
outer-shell (L or M) transitions although MC calculations show
these to be relatively small in the cases tested. Also, the analytic
model does not describe charge or optical photon transport phenomena (e.g., charge sharing in a photoconductor or spreading
of optical photons in a phosphor) and electrical properties (e.g.,
mobility and work energy) of X-ray detectors. While the analytic model provides some physical insight into detector performance, particularly in the design of photon-counting applications and detector optimization, it must also be considered an
approximation that may produce optimistic results.
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