# A Tunable Energy Filter for Medical X-Ray Imaging

Authors: Erik Fredenberg; Björn Cederström; Magnus Åslund; Carolina Ribbing; Mats Danielsson
Journal: X-Ray Optics and Instrumentation
 (2008)

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## Abstract

A multiprism lens (MPL) is a refractive X-ray lens, and its chromatic properties can be employed in an energy filtering setup to obtain a narrow tunable X-ray spectrum. We present the first evaluation of such a filter for medical X-ray imaging. The experimental setup yields a 6.6 gain of flux at 20 keV, and we demonstrate tunability by altering the energy spectrum to center also around 17 and 23 keV. All measurements are found to agree well with ray-tracing and a proposed geometrical model. Compared to a model mammography system with absorption filtering, the experimental MPL filter reduces dose 13–25% for 3–7 cm breasts if the spectrum is centered around the optimal energy. Additionally, the resolution is improved 2.5 times for a 5 cm breast. The scan time is increased 3 times but can be reduced with a slightly decreased energy filtering and resolution.


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## Body

## 1. Introduction

The absorbed dose necessary to obtain a medical X-ray image with sufficient signal-to-noise
ratio varies with object composition, thickness, and X-ray energy spectrum
[1–3]. In fact, for a certain object composition and
thickness there exists an optimal energy; low-energy photons are to a great
extent absorbed and contribute to dose, whereas photons of higher energies pass
the object unaffected or scattered and add to noise without giving much useful
information. A spectrum that is narrow and centered around the optimal energy
is hence the most dose-efficient one.The positive effects of energy filtering to optimize
the spectrum were observed already at the advent of medical X-ray imaging. It
was noted that putting a thin layer of material, such as a piece of leather,
between the patient and the X-ray tube resulted in less irritation of the
patient's skin [4].
Since then, energy filtering has not changed drastically. Absorption filtering
is still the dominant method to filter out low-energy photons, often in
combination with a limited X-ray tube acceleration voltage to cut off photons
of higher energy. In some cases the material of the absorption filter is chosen
so as to have an absorption edge above the optimal energy to further reduce the
high-energy part of the spectrum. In principle, the spectrum can be made
arbitrarily narrow using heavy filtration, but only at the cost of a severe
reduction in flux.To optimize the spectrum beyond what is practically
achievable with absorption filtering, several options have been proposed for
mammography and computed tomography. Some X-ray sources are capable of
producing a high enough flux to allow for very efficient monochromators, for
example, synchrotrons [5–7],
laser plasma sources [8], channeling radiation sources [9], parametric X-ray sources
[10], and sources
producing X-rays by inverse Compton scattering on free electron lasers
[11]. Authors have
reported promising results as to image quality and dose efficiency.
Nevertheless, the high complexity and cost of such sources limit the
feasibility for routine clinical X-ray imaging.It is also possible to apply less stringent energy
filtering to a regular X-ray tube, and filtering based on mosaic crystals has,
for instance, been proposed for mammography [12, 13]. Such crystals have small imperfections in the
structure, resulting in a broader energy spectrum and higher flux compared to
perfect crystals. The reduced flux compared to conventional mammography is,
however, still identified as a problem, and a decreased resolution in the plane
of diffraction of the crystal has been noted. To somewhat mitigate the reduction
in flux, mosaic crystals can be used in combination with polycapillary optics
to gather the radiation into a quasiparallel beam [14].The multiprism lens (MPL) is a refractive X-ray lens.
It distinguishes itself from other X-ray lenses, such as parabolic lenses,
compound refractive lenses, and fresnel lenses, by a simplified manufacturing
process, and a tunable focal length [15]. Experimental measurements have been performed at
synchrotron facilities for various lens materials such as epoxy and silicon
[16], and lithium
[17]. Studies have
also been performed with X-ray tubes using MPLs made of, for example, vinyl
[15] and plexiglas
[18].Chromatic properties of the MPL
make it possible to employ it as an X-ray energy
filter, and in the following, we present a study of such an MPL filter. It is
intended as an alternative to the mentioned filtering methods for small-scale
applications, such as medical imaging, and we use an X-ray tube for the
evaluation. We construct theoretical models of the MPL filter and compare the
results to measurements. We show tunability in a fixed geometry for the first
time experimentally, and we estimate the feasibility and potential advantages of
introducing MPL filters in medical imaging.


## 2. Methods

### 2.1. Background

MPLs consist of two rows of prisms put on an angle in relation to each other and symmetrically
arranged around the optical axis. The rows touch at the entrance side of the
lens and are separated from the optical axis by a distancedg at
the exit side (Figure 1). Peripheral rays
entering the lens encounter a larger number of prisms than central ones and are
therefore refracted more. It can be shown that the amount of lens material in
the direction of the optical axis has a parabolic projection [16]. Since MPLs consist of only
flat surfaces, manufacturing is relatively easy, and the lenses are therefore
suitable for small-scale applications.Figure 1
Left: energy filtering with the multiprism
lens. The chromatic lens focuses radiation on a slit, and the energy
corresponding to a sharp image is preferentially transmitted. We use a setup
withso=755 mm and si=314 mm, yielding an image size of di=ds=10.0 μm with a 24 μm source. Right: close-up of the lens. It has a
tooth height dt=100 μm a length L=61 mm, and a focal length that is tuned by
changing dg.The focal length of an MPL with lengthL and tooth height dt is F=dgdt/δL, where δ is the decrement of the real part of the
refractive index from unity [16]. Imaging is one-dimensional and follows the Gaussian
lens formula for a thin lens, F−1=so−1+si−1, where so and si are the object and image distances,
respectively. F can be made short (in the order of 0.1 m), but
note that the parabolic projection holds over the whole aperture only as long
as dg is larger than dt.Sinceδ in the considered energy interval varies as E−2, the lens is chromatic. Thus, if a slit is
placed in the image plane of a particular X-ray energy (Epeak), radiation of that energy is transmitted
through the slit, whereas other energies are out of focus and preferentially
blocked (Figure 1). The focal length, and so Epeak, can be tuned by changing dg in an otherwise fixed geometry. Henceforth,
the MPL and slit together are referred to as the MPL filter.A primary figure of merit of the MPL filter is the
energy dependent gain of flux, which is the transmitted flux from radiation
focused by the MPL (Φfoc) normalized to the flux from divergent
radiation in an identical setup but without lens (Φdiv);(1)G(E)=Φfoc(E)Φdiv(E).An important MPL parameter is the effective aperture (De), which is the width of a slit with the same
transmission as the lens. Dividing the effective aperture by the physical
aperture of the lens yields the average transmission (Nt). Both these parameters quantify absorption
in the MPL.In this study, we chose to investigate an epoxy MPL (Z≈6). Beryllium or lithium with lower atomic
numbers might yield better results, but because of manufacturing and handling
problems, these lens materials are ruled out for small-scale applications.


### 2.2. Geometrical and Ray-Tracing Models

We derive a theoretical model ofG(E),De, and Nt, based on geometrical optics and a thin lens
approximation. In an energy filtering setup according to Figure 1 with a lens
of tooth height dt, the transmitted flux can be described as
(2)Φfoc(E)=Φ0(E)12π∫−dtdtηt(E)ηi(E)sody1,
where ηt and ηi account for transmission through the lens and
the slit, respectively, Φ0 is the X-ray tube flux, y1 refers to the lateral position in the plane of
the lens, and the integral is taken over the lens aperture.An MPL with linear attenuation coefficientμ has a Gaussian transmission
profile(3)ηt(E)=exp⁡(−y122σ2(E))with rms σ=Fδ/μ [16].If the image of the source from each pointy1 on the lens is described by the function fi(E,y1,y2), the part of the radiation that is not blocked
by the slit is fi integrated over the slit opening (ds),
(4)ηi(E)=∫dsfi(y1,y2,E)dy2,
where y2 refers to the lateral position in the plane of
the slit. In this plane, which is the image plane of Epeak,fi(Epeak,y1,y2) from all points on the lens coincide to form a
sharp image, whereas fi contributions from any other energy are spread
out and sum to a blurred total image.The flux from divergent radiation through the slit can
be calculated as(5)Φdiv(E)=Φ0(E)12πdsso+si
by assuming a point source.Equations (2) and (5) yield the gain according to (1).
The effective aperture is obtained by integrating (3) over the physical lens
aperture, which in turn yields the average transmission;(6)Nt(E)=De(E)2dt,whereDe(E)=∫−dtdtηt(E)dy1.The described geometrical model assumes the lens to be
thin and free of any aberrations except its chromaticity. To estimate the
impact of those approximations, we implement the setup in a custom-made
ray-tracing model using the MATLAB software package (The MathWorks Inc.,
Natick, Mass, USA). A large number of rays (∼106) emerging from the source at random positions
and angles in energy steps of 0.5 keV are traced through each tooth of the
lens. Published linear absorption coefficients [19] and semiempirical data on
atomic scattering factors [20] serve as input to calculate absorption and
refraction. Compton scattering is treated as absorption, and the lens is
assumed flawless.


### 2.3. Measurements

The experimental setup is depicted in Figure2. We use a tungsten anode X-ray tube
(Philips PW2274/20), which is water cooled and can be run continuously. The
acceleration voltage for all measurements is 33 kVp, the anode current is 10 mA, and the anode angle is 3.5°. At this angle, the X-ray tube focal spot size is 24 μm, as specified by the manufacturer. An edge scan
of the source confirmed the size to be in the interval 22–27 μm FWHM. The shape is approximately trapezoidal
with a base-to-FWHM ratio of 1.6.Figure 2
Schematic of the experimental setup used
for the measurements. To simplify alignment, we measure on only one lens-half,
and instead of using a slit, a tantalum edge is scanned across the image plane.The epoxy MPL was molded from a silicon master, which
in turn was produced by wet anisotropic etching [16]. It has a tooth height of dt=100 μm and a length of L=61 mm. To simplify alignment, we use only one
lens-half, which does not change anything in principle but reduces the gain by
a factor of two. The lens-half is mounted on precision stages for lateral
translation and tilting, and it is collimated by a 100 μm collimator slit.We use a CZT compound solid-state detector (Amptek
XR-100T-CZT) to record the energy spectrum transmitted through the filter. It has
a near 100% detection efficiency, negligible hole-tailing, and an energy
resolution better than 0.5 keV around 20 keV.The distancesso=755 mm and si=314 mm are chosen in accordance with the Gaussian
lens formula so as to produce a di=10.0 μm image of the source. We obtain the image of
the source by scanning a tantalum edge in steps of 1 μm over the image plane and differentiating the
recorded profile.To findΦfoc, we fit a straight line to the edge scan
profile over the interval with the fastest rate of change. The slope of the
line is the derivative over an interval where the scan profile is essentially
linear, and the length of the line corresponds to the size of the slit. To
determine Φdiv, the same procedure is repeated without an MPL.
The gain is calculated from Φfoc and Φdiv, and Epeak is at the maximum gain. We do not use a slit
in the image plane of our setup to avoid the additional uncertainty introduced
by a narrow slit. Nt is measured without the edge as the fraction
of flux with and without lens. De is calculated from Nt.We demonstrate tunability of the MPL filter by
repeating the measurements for several angles, corresponding to a range of peak
energies. The absolute gap at the rear end of the lens (dg) cannot be determined by the experiment, but
the relative gap between the measurements (Δdg) is monitored.


### 2.4. Medical Imaging

To evaluate the
MPL filter for medical imaging, we use a theoretical model of a scanning
digital mammography system and compare an array of filters (the MPL system) to
the same number of precollimators in an absorption-filtered multislit geometry
(the reference system). The comparison is made in terms of dose efficiency,
scan time, and resolution. A scanning system is ideal for implementation of the
MPL filter since the line focus of each lens can be matched to a linear
detector. One example of a clinical mammography system based on a scanning
multislit geometry is the Sectra MicroDose Mammography (MDM) system (Sectra
Mamea AB, Kista, Sweden) [21–23].The reference system, similar but not identical to the
MDM system, is absorption filtered and has a450×600 μm2 X-ray tube spot size. Each detector has a
corresponding 60 μm precollimator slit, which is placed before the
object, 650 mm from the source, and 100 mm upstream of the detector. The point
spread function (PSF) of the system is calculated as a convolution of the slit
and the source, using the measured source shape.In the MPL system, each lens-detector pair is similar
to the setup in Figure1, with a distance sd=100 mm for the object between the slit and the
detector. The PSF is the image of the source when assuming perfect lens
imaging.The modulation transfer functions (MTFs) of the
systems are calculated as the Fourier transforms of the respective PSFs, and we
define the resolution in the scanning direction (ν) as the spatial frequency in the detector
plane when the MTF has dropped to 0.1. In the direction perpendicular to the
scanning (parallel to the lenses or slits), the width of the source in the MPL
system is chosen so that the resolution of the two systems is equal. A
published method to find the maximum load to rotary tungsten anodes as a
function of source size [24]
is used to calculate the relative scan time (T^=Tmpl/Tref).In both systems, the object used for the calculations
consists of breast tissue of 50% glandularity with a 5 mm skin layer and an
embedded 300μm microcalcification. The image receptor is a
3.6 mm thick silicon-strip detector, which affects the quantum efficiency but
is assumed ideal in terms of resolution.To estimate dose efficiency, we use the spectral
quantum efficiency (SQE) as a figure of merit. It is defined as [22]
(7)SQE=SDNR2AGD⋅AGDmonoSDNRmono2,
calculated with the
signal-difference-to-noise ratio (SDNR) between the microcalcification and
surrounding tissue, and the average glandular dose (AGD). The SQE is normalized
to the ideal monochromatic case. For a quantum limited system and a given
spectrum, it is an exposure-independent quantity, inversely proportional to the
dose needed to obtain a certain SDNR.Normalized glandular dose coefficients [25] and X-ray attenuation
coefficients [19] are
used to calculate dose and absorption. For anode materials other than tungsten,
published spectra are used as input [26]. The influence of scattering is neglected, which is a
fair assumption for a multislit geometry [23].


## 2.1. Background

MPLs consist of two rows of prisms put on an angle in relation to each other and symmetrically
arranged around the optical axis. The rows touch at the entrance side of the
lens and are separated from the optical axis by a distancedg at
the exit side (Figure 1). Peripheral rays
entering the lens encounter a larger number of prisms than central ones and are
therefore refracted more. It can be shown that the amount of lens material in
the direction of the optical axis has a parabolic projection [16]. Since MPLs consist of only
flat surfaces, manufacturing is relatively easy, and the lenses are therefore
suitable for small-scale applications.Figure 1
Left: energy filtering with the multiprism
lens. The chromatic lens focuses radiation on a slit, and the energy
corresponding to a sharp image is preferentially transmitted. We use a setup
withso=755 mm and si=314 mm, yielding an image size of di=ds=10.0 μm with a 24 μm source. Right: close-up of the lens. It has a
tooth height dt=100 μm a length L=61 mm, and a focal length that is tuned by
changing dg.The focal length of an MPL with lengthL and tooth height dt is F=dgdt/δL, where δ is the decrement of the real part of the
refractive index from unity [16]. Imaging is one-dimensional and follows the Gaussian
lens formula for a thin lens, F−1=so−1+si−1, where so and si are the object and image distances,
respectively. F can be made short (in the order of 0.1 m), but
note that the parabolic projection holds over the whole aperture only as long
as dg is larger than dt.Sinceδ in the considered energy interval varies as E−2, the lens is chromatic. Thus, if a slit is
placed in the image plane of a particular X-ray energy (Epeak), radiation of that energy is transmitted
through the slit, whereas other energies are out of focus and preferentially
blocked (Figure 1). The focal length, and so Epeak, can be tuned by changing dg in an otherwise fixed geometry. Henceforth,
the MPL and slit together are referred to as the MPL filter.A primary figure of merit of the MPL filter is the
energy dependent gain of flux, which is the transmitted flux from radiation
focused by the MPL (Φfoc) normalized to the flux from divergent
radiation in an identical setup but without lens (Φdiv);(1)G(E)=Φfoc(E)Φdiv(E).An important MPL parameter is the effective aperture (De), which is the width of a slit with the same
transmission as the lens. Dividing the effective aperture by the physical
aperture of the lens yields the average transmission (Nt). Both these parameters quantify absorption
in the MPL.In this study, we chose to investigate an epoxy MPL (Z≈6). Beryllium or lithium with lower atomic
numbers might yield better results, but because of manufacturing and handling
problems, these lens materials are ruled out for small-scale applications.


## 2.2. Geometrical and Ray-Tracing Models

We derive a theoretical model ofG(E),De, and Nt, based on geometrical optics and a thin lens
approximation. In an energy filtering setup according to Figure 1 with a lens
of tooth height dt, the transmitted flux can be described as
(2)Φfoc(E)=Φ0(E)12π∫−dtdtηt(E)ηi(E)sody1,
where ηt and ηi account for transmission through the lens and
the slit, respectively, Φ0 is the X-ray tube flux, y1 refers to the lateral position in the plane of
the lens, and the integral is taken over the lens aperture.An MPL with linear attenuation coefficientμ has a Gaussian transmission
profile(3)ηt(E)=exp⁡(−y122σ2(E))with rms σ=Fδ/μ [16].If the image of the source from each pointy1 on the lens is described by the function fi(E,y1,y2), the part of the radiation that is not blocked
by the slit is fi integrated over the slit opening (ds),
(4)ηi(E)=∫dsfi(y1,y2,E)dy2,
where y2 refers to the lateral position in the plane of
the slit. In this plane, which is the image plane of Epeak,fi(Epeak,y1,y2) from all points on the lens coincide to form a
sharp image, whereas fi contributions from any other energy are spread
out and sum to a blurred total image.The flux from divergent radiation through the slit can
be calculated as(5)Φdiv(E)=Φ0(E)12πdsso+si
by assuming a point source.Equations (2) and (5) yield the gain according to (1).
The effective aperture is obtained by integrating (3) over the physical lens
aperture, which in turn yields the average transmission;(6)Nt(E)=De(E)2dt,whereDe(E)=∫−dtdtηt(E)dy1.The described geometrical model assumes the lens to be
thin and free of any aberrations except its chromaticity. To estimate the
impact of those approximations, we implement the setup in a custom-made
ray-tracing model using the MATLAB software package (The MathWorks Inc.,
Natick, Mass, USA). A large number of rays (∼106) emerging from the source at random positions
and angles in energy steps of 0.5 keV are traced through each tooth of the
lens. Published linear absorption coefficients [19] and semiempirical data on
atomic scattering factors [20] serve as input to calculate absorption and
refraction. Compton scattering is treated as absorption, and the lens is
assumed flawless.


## 2.3. Measurements

The experimental setup is depicted in Figure2. We use a tungsten anode X-ray tube
(Philips PW2274/20), which is water cooled and can be run continuously. The
acceleration voltage for all measurements is 33 kVp, the anode current is 10 mA, and the anode angle is 3.5°. At this angle, the X-ray tube focal spot size is 24 μm, as specified by the manufacturer. An edge scan
of the source confirmed the size to be in the interval 22–27 μm FWHM. The shape is approximately trapezoidal
with a base-to-FWHM ratio of 1.6.Figure 2
Schematic of the experimental setup used
for the measurements. To simplify alignment, we measure on only one lens-half,
and instead of using a slit, a tantalum edge is scanned across the image plane.The epoxy MPL was molded from a silicon master, which
in turn was produced by wet anisotropic etching [16]. It has a tooth height of dt=100 μm and a length of L=61 mm. To simplify alignment, we use only one
lens-half, which does not change anything in principle but reduces the gain by
a factor of two. The lens-half is mounted on precision stages for lateral
translation and tilting, and it is collimated by a 100 μm collimator slit.We use a CZT compound solid-state detector (Amptek
XR-100T-CZT) to record the energy spectrum transmitted through the filter. It has
a near 100% detection efficiency, negligible hole-tailing, and an energy
resolution better than 0.5 keV around 20 keV.The distancesso=755 mm and si=314 mm are chosen in accordance with the Gaussian
lens formula so as to produce a di=10.0 μm image of the source. We obtain the image of
the source by scanning a tantalum edge in steps of 1 μm over the image plane and differentiating the
recorded profile.To findΦfoc, we fit a straight line to the edge scan
profile over the interval with the fastest rate of change. The slope of the
line is the derivative over an interval where the scan profile is essentially
linear, and the length of the line corresponds to the size of the slit. To
determine Φdiv, the same procedure is repeated without an MPL.
The gain is calculated from Φfoc and Φdiv, and Epeak is at the maximum gain. We do not use a slit
in the image plane of our setup to avoid the additional uncertainty introduced
by a narrow slit. Nt is measured without the edge as the fraction
of flux with and without lens. De is calculated from Nt.We demonstrate tunability of the MPL filter by
repeating the measurements for several angles, corresponding to a range of peak
energies. The absolute gap at the rear end of the lens (dg) cannot be determined by the experiment, but
the relative gap between the measurements (Δdg) is monitored.


## 2.4. Medical Imaging

To evaluate the
MPL filter for medical imaging, we use a theoretical model of a scanning
digital mammography system and compare an array of filters (the MPL system) to
the same number of precollimators in an absorption-filtered multislit geometry
(the reference system). The comparison is made in terms of dose efficiency,
scan time, and resolution. A scanning system is ideal for implementation of the
MPL filter since the line focus of each lens can be matched to a linear
detector. One example of a clinical mammography system based on a scanning
multislit geometry is the Sectra MicroDose Mammography (MDM) system (Sectra
Mamea AB, Kista, Sweden) [21–23].The reference system, similar but not identical to the
MDM system, is absorption filtered and has a450×600 μm2 X-ray tube spot size. Each detector has a
corresponding 60 μm precollimator slit, which is placed before the
object, 650 mm from the source, and 100 mm upstream of the detector. The point
spread function (PSF) of the system is calculated as a convolution of the slit
and the source, using the measured source shape.In the MPL system, each lens-detector pair is similar
to the setup in Figure1, with a distance sd=100 mm for the object between the slit and the
detector. The PSF is the image of the source when assuming perfect lens
imaging.The modulation transfer functions (MTFs) of the
systems are calculated as the Fourier transforms of the respective PSFs, and we
define the resolution in the scanning direction (ν) as the spatial frequency in the detector
plane when the MTF has dropped to 0.1. In the direction perpendicular to the
scanning (parallel to the lenses or slits), the width of the source in the MPL
system is chosen so that the resolution of the two systems is equal. A
published method to find the maximum load to rotary tungsten anodes as a
function of source size [24]
is used to calculate the relative scan time (T^=Tmpl/Tref).In both systems, the object used for the calculations
consists of breast tissue of 50% glandularity with a 5 mm skin layer and an
embedded 300μm microcalcification. The image receptor is a
3.6 mm thick silicon-strip detector, which affects the quantum efficiency but
is assumed ideal in terms of resolution.To estimate dose efficiency, we use the spectral
quantum efficiency (SQE) as a figure of merit. It is defined as [22]
(7)SQE=SDNR2AGD⋅AGDmonoSDNRmono2,
calculated with the
signal-difference-to-noise ratio (SDNR) between the microcalcification and
surrounding tissue, and the average glandular dose (AGD). The SQE is normalized
to the ideal monochromatic case. For a quantum limited system and a given
spectrum, it is an exposure-independent quantity, inversely proportional to the
dose needed to obtain a certain SDNR.Normalized glandular dose coefficients [25] and X-ray attenuation
coefficients [19] are
used to calculate dose and absorption. For anode materials other than tungsten,
published spectra are used as input [26]. The influence of scattering is neglected, which is a
fair assumption for a multislit geometry [23].


## 3. Results

### 3.1. Filter Parameters

Measurements are presented atEpeak=23,20, and 17 keV. Table 1 shows the
corresponding values of dg for the models, and the measured Δdg to obtain the 20 and 17 keV peaks relative to
the 23 keV peak. Δdg agrees within 1 μm with the geometrical model, and within 2 μm with ray-tracing.Table 1
Summary of the measurements and model predictions. The
average transmission (Nt), image size (di), gain (G), and the gap at the rear end of the lens (dg) are presented at Epeak=17, 20, and 23 keV.
Geometrical modelRay tracingMeasurementEpeakdiNtdgGGMdiNtdgGRTdi(a)NtΔdgGM(a)ΔGGGMΔGGRT[keV][μm][μm][μm][μm][μm][μm][%][%]2310.00.59637.49.30.57536.011.1
(7.8–13.8)0.5006.2±0.1−16+32010.00.60837.69.20.57746.811.0
(7.9–13.8)0.52206.6±0.1−13−21710.00.591157.59.80.571037.112.9
(9.6–16.1)0.54525.9±0.1−21−17(a)Error
estimates correspond to ±1 standard deviation.Figure3 shows the sharp image of the source at 20 keV
as determined by the differentiated edge-scan, fitted to a trapezoid with a
similar shape as the source. The FWHMs of the
fitted profiles (di) for all three measurements are tabulated in
Table 1 and compared to model predictions. A somewhat smaller di is expected from the models, but the
discrepancy is not large and falls within one standard deviation of the fit.
Small-angle scattering, which is treated as absorption in the models, and air
bubbles in the epoxy might account for parts of the increased image size.Figure 3
An image of the source atEpeak=20 keV. The differentiated edge-scan profile is
fitted to a 11.0 μm FWHM trapezoid. The dotted profiles
correspond to ±1 standard deviation of the fitted width.As seen in Table1, Nt in the considered interval is almost
independent of lens angle with a measured mean of 0.52. De is thus slightly more than half the physical
aperture.The transmitted flux through a 10μm slit as a function of energy is plotted in
Figure 4, with an MPL (Φfoc) and without any filter (Φdiv). Also shown is Φdiv normalized to the peak of Φfoc to illustrate the difference in shape. Φfoc and Φdiv yield the gain by division, with the result in
Figure 5 for Epeak=23,20, and 17 keV along with the predicted
results from the models. Also shown in the figure is the calculated gain for 22
and 27 μm sources, which
corresponds to one standard deviation of the source
size measurements. It is evident that the gain decreases with the source size,
and the uncertainty in the source size measurement therefore adds an additional
overall uncertainty to the models.Figure 4
The measured MPL-filtered spectrum atEpeak=20 keV (Φfoc) compared to the unfiltered 33 kVp tungsten
spectrum (Φdiv). Also shown is Φdiv normalized to the peak value of Φfoc.Figure 5
Gain of flux as a function of energy forEpeak=17, 20, and 23 keV. The error bars correspond to ±1 standard deviation of the measured values. Also shown are results from the geometrical (GM) and ray-tracing (RT)
models for a 24 μm source, and the geometrical model for source
sizes 22–27 μm.Values of the peak gain and deviations of the measured
values from the models are presented in Table1. Since the ray-tracing results
agree well with the measurements, deviations of the geometrical model can be
explained to a large extent by model approximations.


### 3.2. Medical Imaging

Figure6 shows
the SQE as a function of breast thickness for the measured MPL-filtered spectra
(Φfoc) at Epeak=17,20, and 23 keV, and for a spectrum at Epeak=25 keV calculated with the geometrical model. The
20, 23, and 25 keV spectra are optimized for breast thicknesses 3, 5, and 7 cm
with SQE values of 0.83, 0.85, and 0.82, respectively, whereas the 17 keV
spectrum falls outside the mammographic region. Figure 6 thus illustrates the
importance of a tunable filter, in particular for thinner breasts.Figure 6
Filter performance in mammography using the
SQE as a figure of merit. The measured  and calculated MPL-filtered spectra
are denoted by anode material—Epeak. Absorption filtered spectra are denoted by
anode—filter materials. The aluminum and rhodium filters are 0.5 mm and 25 μm thick.Also plotted in Figure6 is the SQE of the reference
system with 0.5 mm aluminum and 25 μm rhodium absorption filters, calculated with
the measured tungsten spectrum (Φdiv) as input. The rhodium-filtered spectrum is
optimal for a 3 cm breast, whereas the aluminum-filtered spectrum is relatively
constant over the range of breast thicknesses. SQE values of 0.72, 0.66, and
0.61 are obtained with the best absorption filter for breast thicknesses of 3,
5, and 7 cm. The MPL filter thus improves the SQE 15, 29, and 34%,
corresponding to dose reductions of 13, 22, and 26%.The SQE of the reference system can be raised by
switching to a rhodium anode. That is, however, the case also for the MPL
system and the improvement is still 15% for a 3 cm breast, hence independent of
the anode material. A molybdenum-molybdenum anode-filter combination yields an
SQE that is inferior or equal to the rhodium-rhodium spectrum across the range
of breast thicknesses, and it is therefore not included in the comparison.Table2 shows ν,
SQE, and T^ for a 5 cm breast and a range of MPL-system
parameters at Epeak=23 keV. We used the measured MPL-filtered
spectrum and the geometrical model to calculate T^ for the experimental and the hypothetical
setups, respectively. The reference system employs a 0.5 mm-aluminum-filtered
tungsten spectrum. Since ν is in the scanning direction, it can be
improved by oversampling.Table 2
Comparison of a scanning MPL-filtered mammography
set-up atEpeak=23 keV to a clinical reference system with
absorption filtration for a 5 cm breast. The spectral quantum efficiency (SQE)
and the relative scan time (T^) are presented for the experimentally
obtained spectrum (first row), along with a number of calculated MPL-filtered
spectra (subsequent rows) with different focal lengths (F), source-to-lens and
lens-to-slit distances (so and si), and resolution in the scanning direction (ν). The parameters of the reference system are
shown in the bottom row.
FSource sizesosiνSQET^[mm][μm2][mm][mm][mm−1]21924×932755308200.853.021965×932755308140.771.021924×812360560150.791.015324×600245405120.771.015345×60024540580.720.63450×600650(a)80.661.0(a)The
source-to-collimator distance.Referring to Table2, an MPL system with parameters
corresponding to the current experimental setup yields a longer scan time than
the reference system (row 1), but the resolution in the scanning direction is
2.5 times higher. By either increasing the size of the source (row 2) or by
moving the lens closer to it (row 3), a scan time that matches the reference
system is achieved. Compared to the reference system, the resolution is still
1.75–1.88 times higher. The SQE is 17–20% higher with corresponding dose
reductions of 14–16%. Standard mammography systems are shorter than the setup
we used for the experiment, but with a shorter focal length, a more compact MPL
system is obtained (row 4). If the resolution of the two systems are matched
(row 5), the MPL filter is able to reduce scan time compared to the reference
system, still at an improved SQE.


## 3.1. Filter Parameters

Measurements are presented atEpeak=23,20, and 17 keV. Table 1 shows the
corresponding values of dg for the models, and the measured Δdg to obtain the 20 and 17 keV peaks relative to
the 23 keV peak. Δdg agrees within 1 μm with the geometrical model, and within 2 μm with ray-tracing.Table 1
Summary of the measurements and model predictions. The
average transmission (Nt), image size (di), gain (G), and the gap at the rear end of the lens (dg) are presented at Epeak=17, 20, and 23 keV.
Geometrical modelRay tracingMeasurementEpeakdiNtdgGGMdiNtdgGRTdi(a)NtΔdgGM(a)ΔGGGMΔGGRT[keV][μm][μm][μm][μm][μm][μm][%][%]2310.00.59637.49.30.57536.011.1
(7.8–13.8)0.5006.2±0.1−16+32010.00.60837.69.20.57746.811.0
(7.9–13.8)0.52206.6±0.1−13−21710.00.591157.59.80.571037.112.9
(9.6–16.1)0.54525.9±0.1−21−17(a)Error
estimates correspond to ±1 standard deviation.Figure3 shows the sharp image of the source at 20 keV
as determined by the differentiated edge-scan, fitted to a trapezoid with a
similar shape as the source. The FWHMs of the
fitted profiles (di) for all three measurements are tabulated in
Table 1 and compared to model predictions. A somewhat smaller di is expected from the models, but the
discrepancy is not large and falls within one standard deviation of the fit.
Small-angle scattering, which is treated as absorption in the models, and air
bubbles in the epoxy might account for parts of the increased image size.Figure 3
An image of the source atEpeak=20 keV. The differentiated edge-scan profile is
fitted to a 11.0 μm FWHM trapezoid. The dotted profiles
correspond to ±1 standard deviation of the fitted width.As seen in Table1, Nt in the considered interval is almost
independent of lens angle with a measured mean of 0.52. De is thus slightly more than half the physical
aperture.The transmitted flux through a 10μm slit as a function of energy is plotted in
Figure 4, with an MPL (Φfoc) and without any filter (Φdiv). Also shown is Φdiv normalized to the peak of Φfoc to illustrate the difference in shape. Φfoc and Φdiv yield the gain by division, with the result in
Figure 5 for Epeak=23,20, and 17 keV along with the predicted
results from the models. Also shown in the figure is the calculated gain for 22
and 27 μm sources, which
corresponds to one standard deviation of the source
size measurements. It is evident that the gain decreases with the source size,
and the uncertainty in the source size measurement therefore adds an additional
overall uncertainty to the models.Figure 4
The measured MPL-filtered spectrum atEpeak=20 keV (Φfoc) compared to the unfiltered 33 kVp tungsten
spectrum (Φdiv). Also shown is Φdiv normalized to the peak value of Φfoc.Figure 5
Gain of flux as a function of energy forEpeak=17, 20, and 23 keV. The error bars correspond to ±1 standard deviation of the measured values. Also shown are results from the geometrical (GM) and ray-tracing (RT)
models for a 24 μm source, and the geometrical model for source
sizes 22–27 μm.Values of the peak gain and deviations of the measured
values from the models are presented in Table1. Since the ray-tracing results
agree well with the measurements, deviations of the geometrical model can be
explained to a large extent by model approximations.


## 3.2. Medical Imaging

Figure6 shows
the SQE as a function of breast thickness for the measured MPL-filtered spectra
(Φfoc) at Epeak=17,20, and 23 keV, and for a spectrum at Epeak=25 keV calculated with the geometrical model. The
20, 23, and 25 keV spectra are optimized for breast thicknesses 3, 5, and 7 cm
with SQE values of 0.83, 0.85, and 0.82, respectively, whereas the 17 keV
spectrum falls outside the mammographic region. Figure 6 thus illustrates the
importance of a tunable filter, in particular for thinner breasts.Figure 6
Filter performance in mammography using the
SQE as a figure of merit. The measured  and calculated MPL-filtered spectra
are denoted by anode material—Epeak. Absorption filtered spectra are denoted by
anode—filter materials. The aluminum and rhodium filters are 0.5 mm and 25 μm thick.Also plotted in Figure6 is the SQE of the reference
system with 0.5 mm aluminum and 25 μm rhodium absorption filters, calculated with
the measured tungsten spectrum (Φdiv) as input. The rhodium-filtered spectrum is
optimal for a 3 cm breast, whereas the aluminum-filtered spectrum is relatively
constant over the range of breast thicknesses. SQE values of 0.72, 0.66, and
0.61 are obtained with the best absorption filter for breast thicknesses of 3,
5, and 7 cm. The MPL filter thus improves the SQE 15, 29, and 34%,
corresponding to dose reductions of 13, 22, and 26%.The SQE of the reference system can be raised by
switching to a rhodium anode. That is, however, the case also for the MPL
system and the improvement is still 15% for a 3 cm breast, hence independent of
the anode material. A molybdenum-molybdenum anode-filter combination yields an
SQE that is inferior or equal to the rhodium-rhodium spectrum across the range
of breast thicknesses, and it is therefore not included in the comparison.Table2 shows ν,
SQE, and T^ for a 5 cm breast and a range of MPL-system
parameters at Epeak=23 keV. We used the measured MPL-filtered
spectrum and the geometrical model to calculate T^ for the experimental and the hypothetical
setups, respectively. The reference system employs a 0.5 mm-aluminum-filtered
tungsten spectrum. Since ν is in the scanning direction, it can be
improved by oversampling.Table 2
Comparison of a scanning MPL-filtered mammography
set-up atEpeak=23 keV to a clinical reference system with
absorption filtration for a 5 cm breast. The spectral quantum efficiency (SQE)
and the relative scan time (T^) are presented for the experimentally
obtained spectrum (first row), along with a number of calculated MPL-filtered
spectra (subsequent rows) with different focal lengths (F), source-to-lens and
lens-to-slit distances (so and si), and resolution in the scanning direction (ν). The parameters of the reference system are
shown in the bottom row.
FSource sizesosiνSQET^[mm][μm2][mm][mm][mm−1]21924×932755308200.853.021965×932755308140.771.021924×812360560150.791.015324×600245405120.771.015345×60024540580.720.63450×600650(a)80.661.0(a)The
source-to-collimator distance.Referring to Table2, an MPL system with parameters
corresponding to the current experimental setup yields a longer scan time than
the reference system (row 1), but the resolution in the scanning direction is
2.5 times higher. By either increasing the size of the source (row 2) or by
moving the lens closer to it (row 3), a scan time that matches the reference
system is achieved. Compared to the reference system, the resolution is still
1.75–1.88 times higher. The SQE is 17–20% higher with corresponding dose
reductions of 14–16%. Standard mammography systems are shorter than the setup
we used for the experiment, but with a shorter focal length, a more compact MPL
system is obtained (row 4). If the resolution of the two systems are matched
(row 5), the MPL filter is able to reduce scan time compared to the reference
system, still at an improved SQE.


## 4. Discussion

### 4.1. Filter Parameters

All aspects of the experimental results—image size, transmission, tunability,
and gain—agree well with model predictions.A possible reason for the relatively large deviation
of the 17 keV peak in terms of image size and gain is a sudden increase in size
or a movement of the X-ray tube focal spot. The focal spot is affected by the
cooling water temperature, which varied randomly during the several-hour
measurements.In an MPL, deflection occurs not only at two surfaces
as in a parabolic lens, but all along the way through the lens. Central rays
are refracted closer to the entrance side than peripheral rays, and the rear
principal plane therefore depends on the lateral position of the incident ray.
In addition, deflected rays miss teeth towards the exit side of the lens, and
the projection for parallel incident rays is hence not entirely parabolic. Both
these effects result in the same kind of aberration: an elongated focal length,
in particular for peripheral rays, and, as a consequence, a stretched image in
the direction of the optical axis.The∼10μm smaller dg predicted by ray-tracing compared to the
geometrical model for a certain focal length (Table 1) can be explained by the
fact that the former takes the elongated focal length into account, whereas the
latter assumes the lens to be thin. It is likely that the measurements follow
the ray-tracing model, although dg was not monitored directly. A smaller dg results in more absorption, which explains the
somewhat lower transmission and gain from ray tracing and measurements compared
to the geometrical model.


### 4.2. Medical Imaging

Table2 shows
that the MPL filter can reduce dose at the same scan time as the reference
system, and if a longer scan time is accepted, the energy filtering in terms of
SQE can be improved.Note that the optimal energy is system specific; we
might arrive at different results with detector materials other than silicon.
Previous studies on spectrum optimization indicate that the optimal energy also
depends on breast size and composition but is fairly independent of lesion type
[3]. Our results for
microcalcifications should therefore be similar for masses.The presented values on the SQE for
absorption-filtered spectra are slightly higher than previously reported
results from a similar system [22]. The reason for the better dose efficiency is
substantial self-filtration at the steep anode angle, and a correspondingly
harder input spectrum. In addition, since the acceleration voltage is not tuned
with respect to breast thickness in the present study, the SQE is less constant
across the range of thicknesses. For an MPL system, tuning of the acceleration
voltage is of less importance since the high-energy end of the spectrum is also
filtered. One possibility is therefore to use a higher voltage with the MPL
system to achieve higher flux at a constant or improved dose efficiency. This
option is not further investigated here.Other types of X-ray optics have been investigated for
energy filtering in medical imaging, in particular mosaic crystals. The
measured average transmission of the MPL is 0.54 at 17 keV, which for an MPL
with two lens-halves corresponds to an effective aperture of 108μm. Mosaic HOPG (002) crystals with a mosaic spread of 0.4° give a similar energy spectrum as the MPL filter but a lower peak
reflectivity at 17.5 keV: 0.38 according to theoretical estimates for an
infinitely thick crystal [27], and 0.22 as measured in a set-up relevant for
mammography [13]. At
higher energies, the reflectivity drops further. The crystal can be expected to
have a larger aperture than the MPL, but that effect is canceled if an array of
lenses is considered. MPL filters are therefore competitive for scanning
multi-slit systems.Since the MPL is one-dimensional, it requires a
line-shaped source as thin as 24μm and with an aspect ratio as high as 1:35 to
keep a reasonable scan time. In general, the thinner the source, the better the
energy filtering. Our calculations indicate that microfocal line sources with
enough power for a competitive scan time are in principle feasible. We use a
model that only takes heating of the anode into account, but more thorough
studies, which include also heating of the cathode and the space charge effect,
have concluded that microfocal sources are feasible for medical imaging
[28]. Still, however,
it is clear that less stringent demands on the focal spot size and/or the tube
power would be advantageous.In this study, we use mammography as an example.
Advantages of the energy filter are, however, likely to be found also within
other kinds of medical X-ray imaging, especially computed tomography where
doses are high, and the broad energy spectrum leads to image artifacts such as
beam hardening [6].
Small animal X-ray imaging is another field where dose reduction is needed
since the required high resolution demands almost lethal dose levels [29]. In this case, the high
resolution provided by an MPL filter is an additional advantage. Moreover, the
tunability of the filter opens the possibility to conduct dual-energy imaging
for contrast enhancement and detection of contrast agents at high sensitivity.


## 4.1. Filter Parameters

All aspects of the experimental results—image size, transmission, tunability,
and gain—agree well with model predictions.A possible reason for the relatively large deviation
of the 17 keV peak in terms of image size and gain is a sudden increase in size
or a movement of the X-ray tube focal spot. The focal spot is affected by the
cooling water temperature, which varied randomly during the several-hour
measurements.In an MPL, deflection occurs not only at two surfaces
as in a parabolic lens, but all along the way through the lens. Central rays
are refracted closer to the entrance side than peripheral rays, and the rear
principal plane therefore depends on the lateral position of the incident ray.
In addition, deflected rays miss teeth towards the exit side of the lens, and
the projection for parallel incident rays is hence not entirely parabolic. Both
these effects result in the same kind of aberration: an elongated focal length,
in particular for peripheral rays, and, as a consequence, a stretched image in
the direction of the optical axis.The∼10μm smaller dg predicted by ray-tracing compared to the
geometrical model for a certain focal length (Table 1) can be explained by the
fact that the former takes the elongated focal length into account, whereas the
latter assumes the lens to be thin. It is likely that the measurements follow
the ray-tracing model, although dg was not monitored directly. A smaller dg results in more absorption, which explains the
somewhat lower transmission and gain from ray tracing and measurements compared
to the geometrical model.


## 4.2. Medical Imaging

Table2 shows
that the MPL filter can reduce dose at the same scan time as the reference
system, and if a longer scan time is accepted, the energy filtering in terms of
SQE can be improved.Note that the optimal energy is system specific; we
might arrive at different results with detector materials other than silicon.
Previous studies on spectrum optimization indicate that the optimal energy also
depends on breast size and composition but is fairly independent of lesion type
[3]. Our results for
microcalcifications should therefore be similar for masses.The presented values on the SQE for
absorption-filtered spectra are slightly higher than previously reported
results from a similar system [22]. The reason for the better dose efficiency is
substantial self-filtration at the steep anode angle, and a correspondingly
harder input spectrum. In addition, since the acceleration voltage is not tuned
with respect to breast thickness in the present study, the SQE is less constant
across the range of thicknesses. For an MPL system, tuning of the acceleration
voltage is of less importance since the high-energy end of the spectrum is also
filtered. One possibility is therefore to use a higher voltage with the MPL
system to achieve higher flux at a constant or improved dose efficiency. This
option is not further investigated here.Other types of X-ray optics have been investigated for
energy filtering in medical imaging, in particular mosaic crystals. The
measured average transmission of the MPL is 0.54 at 17 keV, which for an MPL
with two lens-halves corresponds to an effective aperture of 108μm. Mosaic HOPG (002) crystals with a mosaic spread of 0.4° give a similar energy spectrum as the MPL filter but a lower peak
reflectivity at 17.5 keV: 0.38 according to theoretical estimates for an
infinitely thick crystal [27], and 0.22 as measured in a set-up relevant for
mammography [13]. At
higher energies, the reflectivity drops further. The crystal can be expected to
have a larger aperture than the MPL, but that effect is canceled if an array of
lenses is considered. MPL filters are therefore competitive for scanning
multi-slit systems.Since the MPL is one-dimensional, it requires a
line-shaped source as thin as 24μm and with an aspect ratio as high as 1:35 to
keep a reasonable scan time. In general, the thinner the source, the better the
energy filtering. Our calculations indicate that microfocal line sources with
enough power for a competitive scan time are in principle feasible. We use a
model that only takes heating of the anode into account, but more thorough
studies, which include also heating of the cathode and the space charge effect,
have concluded that microfocal sources are feasible for medical imaging
[28]. Still, however,
it is clear that less stringent demands on the focal spot size and/or the tube
power would be advantageous.In this study, we use mammography as an example.
Advantages of the energy filter are, however, likely to be found also within
other kinds of medical X-ray imaging, especially computed tomography where
doses are high, and the broad energy spectrum leads to image artifacts such as
beam hardening [6].
Small animal X-ray imaging is another field where dose reduction is needed
since the required high resolution demands almost lethal dose levels [29]. In this case, the high
resolution provided by an MPL filter is an additional advantage. Moreover, the
tunability of the filter opens the possibility to conduct dual-energy imaging
for contrast enhancement and detection of contrast agents at high sensitivity.


## 5. Conclusions

We conclude
that MPL filtering is a promising technology for medical imaging, in particular
for mammography, to reduce dose or to improve the signal-to-noise ratio. Our
proposed ray-tracing and geometrical models agree well with measurements and
can be used to predict the behavior of MPL filters.The investigated epoxy MPL generates an 11μm wide one-dimensional image of the source at
a peak energy of 20 keV, which is within one standard deviation of the
expectations from geometry. The gain of flux is measured to 6.6, which is also
in good agreement with the geometrical model. Ray tracing shows that deviations
in the gain can be explained almost entirely by approximations in the model.
Tunability is verified by altering the gap at the rear end of the lens to
achieve peak energies of 17, 20, and 23 keV. The change in gap agrees within 2 μm of model predictions.Compared to a reference mammography system, the MPL
filter reduces the dose by 13–25% for 3–7 cm breasts. This result depends on
the absorption filter of the reference system but is virtually independent of
the investigated anode materials, given the same anode in both systems. The
resolution in the scanning direction is improved 2.5 times for a 5 cm breast,
but the scan time is 3 times longer. If adapted to the same scan time as the
reference system, the MPL filter is able to reduce the dose 16% at a 1.88 times
better resolution. An MPL-based system does, however, require a thin line
source, and the feasibility of such sources has to be investigated further.



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*Source: 635024-2008-10-13.xml*