Prism-array lenses for energy filtering in medical x-ray imaging
Erik Fredenberg, Bjorn Cederstrom, Carolina Ribbing, Mats Danielsson
This is the submitted manuscript of:
Fredenberg, E., Cederström, B., Ribbing, C. and Danielsson, M. , “
Prism-array lenses for energy filtering in medical x-ray imaging ,” Proc. SPIE
The published version of the manuscript is available at: https://doi.org/10.1117/12.713767 See also:
Fredenberg, E., Cederström, B., Nillius, P., Ribbing, C., Karlsson, S. and Danielsson, M., 2009. A low-absorption x-ray energy filter for small-scale applications. Optics Express, 17(14), pp.11388-11398. https://doi.org/10.1364/OE.17.011388 All publications by Erik Fredenberg: https://scholar.google.com/citations?hl=en&user=5tUe2P0AAAAJ rism-array lenses for energy filtering in medical x-rayimaging
Erik Fredenberg, a Bj¨orn Cederstr¨om, a Carolina Ribbing, b and Mats Danielsson aa Department of Physics, Royal Institute of Technology, AlbaNova, 106 91 Stockholm, Sweden; b The ˚Angstr¨om Laboratory, Uppsala University, 751 21 Uppsala, Sweden;
ABSTRACT
Conventional energy filters for x-ray imaging are based on absorbing materials which attenuate low energyphotons, sometimes combined with an absorption edge, thus also discriminating towards photons of higherenergies. These filters are fairly inefficient, in particular for photons of higher energies, and other methods forachieving a narrower bandwidth have been proposed. Such methods include various types of monochromators,based on for instance mosaic crystals or refractive multi-prism x-ray lenses (MPL’s). Prism-array lenses (PAL’s)are similar to MPL’s, but are shorter, have larger apertures, and higher transmission. A PAL consists of anumber of small prisms arranged in columns perpendicular to the optical axis. The column height decreasesalong the optical axis so that the projection of lens material is approximately linear with a Fresnel phase-platepattern superimposed on it. The focusing effect is one dimensional, and the lens is chromatic. Hence, unwantedenergies can be blocked by placing a slit in the image plane of a desired energy. We present the first experimentaland theoretical results on an energy filter based on a silicon PAL. The study includes an evaluation of the spectralshaping properties of the filter as well as a quantification of the achievable increase in dose efficiency comparedto standard methods. Previously, PAL’s have been investigated with synchrotron radiation, but in this study amedical imaging setup, based on a regular x-ray tube, is considered.
Keywords: x-ray imaging; energy filtering; x-ray optics; prism-array lens; mammography
1. INTRODUCTION
It is well known that the absorbed dose necessary to obtain an x-ray image with sufficient signal-to-noise ratiovaries with object thickness and x-ray energy spectrum.
In fact, for a certain object thickness there existsan optimal energy; lower energy photons will to a large extent be absorbed and contribute to dose, whereasphotons of higher energies will pass the object merely unaffected and add to noise without giving much usefulinformation. A spectrum that is narrow and centered around the optimal energy is hence the most dose efficientone.Today, absorption filtering is the dominant method to narrow the x-ray tube spectrum. A thin film of materialis used to filter out low energy photons, often in combination with a limited x-ray tube acceleration voltage inorder to cut off higher energy photons. In principle, the spectrum can be made very narrow by squeezing themaximum energy, set by the acceleration voltage, towards a wall of heavy filtration, however, only at the cost ofa severe reduction in flux. The material of the absorption filter can also be chosen so as to have an absorptionedge above the optimal energy to further reduce the high energy part of the spectrum. In general, however, themethod of absorption filtering has not changed much since Pfahler, at the advent of x-ray imaging, discoveredthat putting a piece of leather between the patient and the x-ray tube resulted in less irritation on the skin ofthe patient. Nevertheless, several methods have in recent years been proposed to optimize the x-ray spectrum beyondwhat is practically achievable with absorption filtering. Very efficient monochromators can be applied to highflux x-ray sources, for instance synchrotrons, laser plasma sources, channeling radiation sources, parametricx-ray sources, and sources producing x-rays by inverse Compton scattering on free electron lasers. Although
Electronic mail: [email protected] mage quality and dose efficiency are improved, the high complexity and cost of such sources limit the feasibilityfor routine clinical x-ray imaging.Therefore it has also been proposed to apply less stringent energy filtering to a regular x-ray tube, resultingin a broader spectrum but also a better photon economy. Mosaic crystals, for instance, have small imperfectionsin the crystal structure and can be employed to achieve a spectrum optimized for mammography.
Anotherapproach is to use chromatic refractive x-ray optics, e.g. a multi-prism lens (MPL), in a filtering set-up.
An MPL consists of two arrays of prisms put on an angle in relation to each other. Peripheral rays enteringthe lens will encounter a larger number of prisms than will central ones, hence experiencing a greater refraction.Since the refractive index of the lens material varies with x-ray energy the lens is chromatic and unwantedenergies can be blocked by placing a slit in the image plane of the desired energy. The problem with MPL filtersis the high absorption, in particular at the periphery of the lens, which limits the usable aperture.The prism-array lens (PAL) is a further elaboration of the MPL, with material corresponding to a phase shiftof integer steps of 2 π removed, resulting in a greater aperture, higher transmission, and shorter lens. PAL’s arethus similar to Fresnel phase-plates, but with smaller aspect ratios, which facilitates manufacturing. In the past,PAL’s have been investigated at synchrotron facilities,
19, 20 but not with regular x-ray tubes in the context ofmedical x-ray imaging.In the following, an investigation of the PAL for medical x-ray imaging will be presented. Experimentalmeasurements of the focusing and filtering capabilities of the PAL will be compared to theoretical estimatesobtained by diffraction and ray-tracing models. Furthermore, the feasibility of introducing the obtained spectrain a medical imaging set-up for dose reduction or increased signal-to-noise ratio will be evaluated, theoreticallyas well as experimentally.
2. THEORETICAL BACKGROUND2.1. Prism-array lenses
Prism arrays can be set up in several ways, and for this study a design according to Figs. 1b) and c) was chosen.It consists of a number of small prisms arranged in columns in the y -direction (the height direction of the lens,perpendicular to the optical axis), which are displaced in the y -direction with an increasing displacement alongthe x -direction (the direction of the optical axis). The design is altered from the one presented in Ref. 19by converting the regular prisms into right-angled half-prisms, and the sparser arrangement was found to befavorable for manufacturing due to fewer narrow corners and acute angles. Support structures were added atthe entrance of the lens and for each prism column.In analogy with Ref. 19, the projected amount of lens material in the x -direction ( X ) as a function of thedistance from the optical axis ( y ) for a lens with N columns can be described as X ( y ) = 2 T + N (cid:88) j =0 x j ( y ) x j ( y ) = (cid:40) mod (cid:179) | y |− d (0 . j )tan θ , b (cid:180) + t y ≥ d (0 . j ) t y < d (0 . j ) , (1)where x j ( y ) is the projection in the j :th column. T and t are the support structure thicknesses at the entranceof the lens and at each column respectively. The columnar displacement is denoted d, and b and θ are the prismbase and angle. The modulus function represents the remainder after division. To minimize absorption, b ischosen so as to correspond to a phase shift of a low multiple, ideally 1, of 2 π. The projection is approximatelylinear with a Fresnel phase-plate pattern superimposed on it, which results in a focusing effect in one dimension.In other words, peripheral rays encounter a larger number of prisms on their way through the lens than do centralones, thus also experience a greater refraction. The projected phase-plate pattern is an approximation built upof straight line segments, which is better the smaller d is, given a certain prism size.Imaging with a PAL follows for a thin lens the gaussian lens formula, F − = s − + s − , where s o and s i arethe object and image distances respectively. The focal length is F = d tan θ/δ, where θ is the prism angle, and δ is the decrement of the real part of the refractive index from unity. At energies and for lens materials of interestin this study, δ varies approximately as E − , and therefore the lens is chromatic, thus having different focal tT d d a Lb hd sourceslit polychromatic radiation y y a d i s o s i object detector partially blockednon-peak energy imagetransmittedpeak energy image object partially blockednon-peak energy imagepartially blockedtransmittedpeak energy image detectorlens slit or edge objectcollimator slitsource a s i s o b quasi-monochromatic radiation y x x y xy dc d o Figure 1. a)
Energy filtering and imaging with the PAL. Vertical and horizontal axes are not to scale. b) A close-up ofthe PAL with lens parameters indicated. The lens used for this study had a total of 960 prisms, each with a base width b = 59 µ m , a height h = 6 µ m , and a prism angle θ = 5 . ◦ . c) SEM picture of the PAL used for the experiment. Theoptical axis is indicated by the dashed line. d) Schematic of the experimental set-up. The two 50 µ m slits which collimatethe beam in the z -direction are not shown. lengths for different x-ray energies. By placing a slit at the image plane ( s i ) of a particular x-ray energy ( E peak )that very energy is transmitted, whereas other energies are out of focus and will be preferentially blocked, seeFig. 1a). In a thin lens approximation, a PAL can be regarded as a one-dimensionally focusing Fresnel phase-plate, andits refractive effect can be described with Kirchhoff’s scalar diffraction theory (see for instance Ref. 21 for atreatment of the subject). In one dimension, the time independent optical disturbance ψ of a wave of length λ and number k = 2 π/λ at y in the image plane can be found, ψ ( y , y ) = − i (cid:114) ε λ (cid:90) d a exp [ ik ( ρ + r + nX )] √ ρr K d y , (2)where the integral is taken over the lens aperture. ε is the source strength per unit length, ρ is the distance from y at the source to y at the lens, and r is the distance from y to y . Phase shift and absorption are describedby the real ( δ ) and complex ( β ) parts of the index of refraction, n = 1 − δ + iβ. K, finally, is the obliquity factor,which depends on the angle between vectors ¯ ρ and ¯ r at y and is unitary in the forward direction. In the PALset-up presented here, distances in the y -direction are very small compared to distances in the x -direction and K is approximately unity. Furthermore, ρ and r in the denominator of Eq. (2) can be approximated with s o and s i . The same approximation is not fair for the exponential due to a large k, but binomial expansions give r = (cid:113) s + ( y − y ) ≈ s i + ( y − y ) / s i and ρ = (cid:112) s + ( y − y ) ≈ s o + ( y − y ) / s o , (3)hich leads to ψ ( y , y ) = − √ ε i exp [ ik (1 + s o + s i )] √ λs o s i (cid:90) d a exp (cid:183) ik (cid:181) ( y − y ) s o + ( y − y ) s i + ( iβ − δ ) X (cid:182)(cid:184) d y . (4)The intensity in the image plane (Φ foc ) over an interval corresponding to the image size ( d i ) when focusingradiation from a source of size d o is thusΦ foc = (cid:90) d o (cid:90) d i | ψ ( y , y ) | d y d y . (5)In a similar set-up without a lens, the intensity from divergent radiation (Φ div ) over the same interval isΦ div = d i s o + s i (cid:90) d o ε d y . (6)The fraction of Eqs. (5) and (6) yields the gain of flux behind a slit of size d i in the image plane compared to aset-up without lens, G ( E ) = Φ foc Φ div . (7)The average transmission of the PAL ( N t ) can be used to estimate the efficiency of the PAL filter, and isobtained by integrating and averaging the squared absorption part of Eq. (4) over the lens aperture, N t = (cid:90) d a exp( − kβX ) d a d y , (8)where − kβ corresponds to the linear attenuation coefficient of the lens material.The described phase-plate model assumes the lens to be thin compared to other distances in the x -direction,but the length of the lens is at least L = N b, which is typically in the order of a centimeter. To study the effectof such approximations a full ray-tracing model was set up, and the law of refraction was used to calculate thedeflection angle in each tooth of the lens. Roughly 10 rays were used, emerging from the source at randompositions and angles, and with a flat energy spectrum. The ray-tracing model still suffers from the approximationsthat no scattering occurs, and that the lens is flawless.
3. MATERIAL AND METHODS3.1. Experimental set-up
PAL’s with various prism parameter settings were fabricated by a commercial company ∗ using deep reactive ionetching (DRIE) of silicon according to the so-called Bosch process. It is a plasma-based cyclic process whichcan be used to attain micro-structures with aspect ratios of up to 40:1, and the etching process can be tuned foroptimization of side-wall verticality and reduction of side-wall roughness. See Ref. 22 for a thorough descriptionof the process. For masking, the surface of the silicon wafer was oxidized, and the thin layer of silicon dioxideso created was patterned using standard lithographic methods. The mask was removed with hydrofluoric acidafter the etching.For the present study, a lens was chosen from the batch, having a total of 960 prisms, each with a prism base b = 59 µ m (which corresponds to a phase shift of 2 π in silicon), height h = 6 µ m , and prism angle θ = 5 . ◦ . SeeFig. 1b). There were 2 ×
16 prisms in the first column, which gives a physical aperture of 2 d a = 192 µ m , andthe displacement in the y -direction between two adjacent columns was d = 1 . µ m . These parameters optimizethe lens for 23 keV with a focal length of 172 mm at that energy. Support structures were added at the entranceand exit of the lens ( T = 100 µ m) and at each column ( t = 8 µ m). The N = 63 columns yielded a lens length of9 mm. In Fig. 2 the projection of lens material in the x -direction according to Eq. 1 is shown. The manufacturingprocess was not totally optimized and the lenses exhibited structure failure, in particular over-etch of the convexprism angles of up to 10 µ m , which is to be seen in Fig. 1c). This limited the depth of the lens to 165 µ m . ∗ Silex Microsystems (J¨arf¨alla, Sweden)
00 800 900 1000 1100 1200 1300 1400−30−20−100102030 X [ µ m] y [ µ m ] Figure 2.
Projected lens profile ( X ) as a function of the distance from the optical axis ( y ) for the PAL used in thestudy. Only a part of the aperture is shown. A tungsten anode x-ray tube, † acceleration voltage 33 kVp and anode current 50 mA, was employed in aset-up depicted in Fig. 1d). A CZT compound solid state detector ‡ with a near 100% detection efficiency andnegligible hole tailing around 23 keV (according to the manufacturer) was used. An energy resolution of 0.5 keVwas chosen for sufficient count rate without pile-up. To facilitate line-up, the lens was mounted on precisionstages for lateral translation and tilting. Upstream of the lens a 200 µ m collimator slit was placed to matchthe aperture of the lens and to reduce stray radiation. The beam was further collimated in the z -direction (thedepth direction of the lens, perpendicular to the optical axis) to 50 µ m by two slits; one in front of the lens andone in front of the detector.At the image plane ( s i ), a precision stage for translation in the y -direction was placed, on which could bemounted either a 12 µ m tungsten slit or a sharp tantalum edge depending on the measurement (see below).In order to determine the x-ray tube focal spot size in the y -direction, the anode angle was measured andthe source size could be determined from specifications by the manufacturer. This source size was used for themodels. For verification, an edge scan was performed with the tantalum edge in the image plane against theupper edge of the collimator slit prior to the experiments. The focal spot size in the z -direction was 12 mm, andcould be considered unlimited for the present measurements.The distance s o = 585 mm was chosen in accordance with the gaussian lens formula so as to produce a sharp23 keV image of the source at s i = 244 mm. These distances refer to the entrance of the lens, which is assumedto be the plane of the lens, but the lens is in practice not thin and a somewhat different image distance can beexpected from experiment and ray-tracing, even for a perfect lens. Therefore, the stage at s i could be moved inthe x -direction, and measurements were performed in steps along the optical axis in order to find the image planefor 23 keV radiation. Image sizes predicted by the models were recalculated for the measured image distance inorder to make comparisons meaningful. This was also done for the predicted gain, which varies with image sizeand distance. To obtain a precise measurement of the source size and of the gain, the tantalum edge was scanned in steps of1 µ m over the image plane. The data so obtained correspond to the integral of the intensity profile in the imageplane as a function of position, and so the actual profile was found by differentiation. A gauss function was fittedto the differentiated edge scan profile, and the image size ( d i ) defined here as the full-width-at-half-maximum(FWHM), was calculated. † Philips PW2274/20 with high tension generator PW1830 ‡ Amptek XR-100T-CZT o obtain Φ foc , a straight 12 µ m line was fitted to the edge scan profile over the interval with the fastestrate of change, which is where the edge passes the image of the source. The slope of the line was taken to bethe derivative, which is a good measure of the flux over an interval where the scan profile is essentially linear,and the length of the straight line corresponds to the size of the slit. A second edge scan without the lens wasperformed in order to determine Φ div . Since in this case the rate of change was much lower, the straight line wasfitted over 50 µ m to reduce statistical noise, and normalized to 12 µ m. An alternative way to calculate the gainwould have been to use the fitted gaussian profile, but such a calculation is more dependent on the actual sourceshape.The average transmission factor of the lens ( N t ) was derived from the edge scan as the fraction of the twonon-differentiated profiles at a point where there was no shadow of the edge. Although an edge scan avoids some of the problems compared to measurements with a slit, e.g. alignmentand change of slit size, an authentic filtering set-up does require a real slit in the image plane. Therefore suchmeasurements were also performed with a ∼ µ m tungsten slit.The spectral quantum efficiency for a certain mammographic spectrum and breast size can be defined as SQE = SDNR AGD s · AGD m SDNR . (9)Here, SDNR s and AGD s are the signal-difference-to-noise-ratio and average glandular dose for the investigatedspectrum. SDNR m and AGD m are the same quantities for the ideal monochromatic case. For a quantum limitedsystem, the SQE is an exposure independent quantity, inversely proportional to the dose needed to obtain acertain SDNR.Dose efficiency of the PAL filter was experimentally evaluated using the SQE metric with a simple mammog-raphy phantom. The phantom consisted of 6 cm tissue equivalent material (BR12), corresponding to a breastof approximately 50% glandularity, with an added 200 µ m aluminum foil, representing the contrast of a microcalcification. Values of the SDNR for the phantom were experimentally obtained for PAL and aluminum filteredspectra according to SDNR = | n − n |√ n + n , (10)where n and n are the signals from pixels with and without aluminum foil. To find the AGD, monte-carlocalculated normalized glandular dose coefficients were applied to the spectra. The ideal monochromatic SDNRwas obtained theoretically with material compositions, and x-ray attenuation coefficients. For comparison,values of the SQE were also calculated for the experimentally obtained spectra at a range of breast thicknesses,and the ray-tracing gain was applied directly to the tungsten spectrum to estimate the theoretically highestpossible SQE.The impact of the PAL filter on spatial resolution and image acquisition time in medical imaging will dependon the imaging geometry and is only briefly discussed in this paper.
4. RESULTS
At the 3 . ± . ◦ anode angle being used, the x-ray tube focal spot size was specified by the manufacturer to24 . ± . µ m . The edge scan yielded a 24 . . − . µ m FWHM fitted gaussian profile, and thus confirmedthe source size.The image plane of the set-up was found on a distance s i = 301 mm for 23 keV radiation, 57 mm fartherfrom the lens than expected.Fig. 3 shows the result of an edge scan at the image plane. The energy is 23 keV and the 12 µ m fittedline used to find the peak gain is indicated. In Table 1 the average transmission factors ( N t ) of models andexperiment are tabulated and can be compared to Eq. (8) which yields 0.35. Also shown are the FWHM’s of thefitted profiles ( i.e. d i ). Values of N t and d i derived from the models are recalculated for s i = 301 mm. c oun t r a t e [ s − ] Figure 3.
Edge scan at 23 keV of the PAL filtered 33 kVp spectrum in the image plane of the set-up; count rate as afunction of position of the edge. The fitted 12 µ m line is indicated. Table 1.
Results from the filtering and focusing measurements at 23 keV, compared to the phase-plate and ray-tracingmodels for a 24.5 µ m FWHM gaussian source. The image size ( d i ) and peak gain ( G ) of the models are recalculated forthe image distance ( s i ) of the measurement. Also presented is the average transmission factor ( N t ). s i d i G N t [mm] [ µ m]Measurement 301 17.3 5.2 0.32Phase-plate model 236 13.5 6.7 0.34Ray-tracing model 246 12.4 6.5 0.31Relative flux after a 12 µ m slit as a function of energy is plotted in Fig. 4 for filtered and unfiltered spectra.The filtered spectrum has an FWHM of 4.7. These curves yield the gain by division, with the result shownin Fig. 5 along with the predicted results from the phase-plate model as well as from ray-tracing. Also shownin Fig. 5 is ray-tracing gain for 21.5 and 28.5 µ m sources, which correspond to ± s i = 301 mm.In Fig. 6 is plotted the experimentally obtained PAL and aluminum filtered spectra, and the unfilteredtungsten spectrum multiplied with the ray-tracing gain. The unfiltered spectrum is also shown. Values of theFWHM for the filtered spectra are presented in Table 2.The SQE values of PAL and 0.5 mm aluminum filtered spectra for the 6 cm breast phantom are indicated inFig. 7. Also presented is the calculated SQE as a function of breast thickness for the experimentally obtainedPAL and aluminum spectra, and for a PAL spectrum obtained from the ray-tracing model. The peak valuesof the SQE are tabulated in Table 2 along with the dose reduction of the PAL compared to aluminum filteredspectra.
5. DISCUSSION5.1. Focusing and filtering properties
The measured image of the source is larger than would be expected from the models, and corresponds to a sourcesize of 33.5 µ m . This is not covered within one standard deviation of the source size measurements, but it wasfound that the source size varied with the cooling water temperature, which could not be kept totally constant.The measurements were long compared to the variations, and a moving source will therefore show up as a larger r e l a t i v e f l u x with PAL filterwithout PAL filter Figure 4.
Relative flux after a 12 µ m slit as a functionof energy for the PAL filtered 33 kVp spectrum andfor the unfiltered spectrum ( i.e. Φ foc and Φ div ). TheFWHM of Φ foc is 4.7 keV.
15 20 25 3001234567 energy [keV] g a i n PP: 24.5 µ mRT: 24.5 µ mRT: 21.5 &28.5 µ mM Figure 5.
Measured gain (M) as a function of energycompared to the phase-plate (PP) and ray-tracing (RT)models for a 24.5 µ m source. Also indicated is ray-tracing gain for source sizes 21.5 and 28.5 µ m , whichcorrespond to ± ±
10 15 20 25 3000.10.20.30.40.50.60.70.80.91 energy [keV] r e l a t i v e f l u x PAL filter (E)PAL filter (RT)Al filter (E)unfiltered
Figure 6.
The experimentally obtained PAL and alu-minum filtered spectra (E), and the raw tungsten spec-trum multiplied with the ray-tracing gain (RT). Thelatter one corresponds to the best possible filtrationwith the current PAL. S Q E PAL filter (E)PAL filter (RT)Al filter (E)
Figure 7.
Experimentally obtained SQE of PAL and0.5 mm aluminum filtered spectra for the breast phan-tom (E). Also shown is the calculated SQE as a func-tion of breast thickness for the experimentally obtainedPAL and aluminum spectra, and for a PAL spectrumobtained from the ray-tracing model (RT). able 2.
Results from the measurements of dose efficiency. The SQE and dose reduction refer to 6 cm breasts. Theexperimentally obtained values (E) of the PAL and aluminum SQE are shown, along with the calculated SQE from theray-tracing model (RT). Also presented are the FWHM’s of the filtered spectra.
FWHM SQE Dose reduction[keV] [%]PAL (E) 8.6 0 . ± .
06 14 ± z -direction, which means that also thefocusing properties might vary with depth. A narrow beam of only 50 µ m , which is about one third of the lensdepth, was used to obtain uniform focusing, but still the effects of varying prism sizes cannot be excluded.Measured gain as a function of energy is in good agreement with the ray-tracing model at all energies. Thedeviation at the peak energy of 20% can again be explained by a larger than expected or moving source, whichwill result in a lower gain. The effect is illustrated in Fig. 5 by the ray-tracing gain for source sizes 21.5 and28.5 µ m . Imperfections in the lens structure might influence also the gain, by non-perfect focusing or absorptionat under etched narrow passages. It should also be noted that the slit size of 12 µ m was kept constant, althoughif adapted to the size of the image would yield a higher gain. The reason was to make comparisons with themodels, and with the results using the non-adjustable tungsten slit, more lucid.Scattered and reflected rays are treated as absorption in the ray-tracing model. Small angle Rayleigh scat-tering might, however, account for parts of the increased image size, whereas Compton scattering and reflectionshould add background radiation and reduce the gain. In fact, the edge scan in Fig. 3 does indicate a certainamount of background radiation since it has a superimposed slow rate of change and does not reach zero afterthe image. The scattering cross section increases fast with energy at the considered energy interval, whereastotal external reflection on the prism surfaces is more probable at low energies.As can be seen in Fig. 5, there is good agreement between the phase-plate and ray-tracing models for predictedgain at the peak energy 23 keV, but the phase-plate gain decreases slower when moving to other energies, inparticular to higher ones. The faster decrease towards lower energies is likely to be due to absorption in thesupport structures of the lens. This seems to indicate that the focal length predicted by the phase-plate modelvaries slower with energy than E , and since ray-tracing and measurement agree better, the reliability of thephase-plate model can be questioned. Since the lens length was almost 10 mm, the thin lens approximationmight have influenced the result; the lens profile shown in Fig. 2 assumes a thin lens, but deflection will occurall the way through the PAL and in particular peripheral rays with a large deflection might miss prisms towardsthe exit side.Moreover, the thin lens approximation can explain the difference in focal length between the two models, asseen in Table 1, since the phase-plate model assumes a thin lens but the ray-tracing does not. The measuredelongation of the focal length, on the other hand, is larger than predicted by ray-tracing and much larger thanthe length of the lens. Imperfections in the lens structure is again a probable reason for the deviation. According to Fig. 7 and Table 2, the PAL filter can reduce the dose in mammography compared to absorptionfiltering with 14% for a 6 cm breast. The measured dose reduction does, however, suffer from fairly largestatistical errors, and a more moderate approach would be to compare the SQE calculated from the measuredspectra. In that case a reduction in dose of 9% can be expected. An upper bound is set to 17% by the ray-tracingcalculation using only theoretical values for PAL and aluminum filtered spectra. As can also be seen in Fig. 7,the PAL filtered spectrum is better optimized for 7 cm breasts for which a larger dose reduction can be expected.ttenuation coefficients of real breast tissue and not BR12 was used to calculate the SQE, which might explainthe small deviation of the measured SQE from theoretical values for the aluminum filtered spectrum.As can be seen in Table 2 and Fig. 6, the PAL filtered spectrum is substantially broader than the spectrumpredicted by ray-tracing, which lowers the SQE. It is also broader than the spectrum obtained by the edge scanprocedure (Fig. 4). Therefore, except for the sources of error discussed above, it is fair to assume that the 12 µ mslit is broadening the spectrum. In fact, the thin walls of the slit were found to leak radiation, especially at higherenergies, and that would indeed contribute to the broadening. Additionally, the size of the slit was associatedwith some uncertainty.When interpreting the results it should also be kept in mind that the steep angle of the set-up leads to moreself-filtration and a somewhat harder spectrum than in diagnostic radiology, where an angle of about 17 ◦ iscommon. The support structures of the PAL design presented here were added to allow for through etch of the silicon wafer,but the etch depth was, as explained above, limited by structure failure. By removing the support structures, theaverage transmission at 23 keV could have been increased to 0.57 according to Eq. (8), which is an improvementin the transmission and the gain by a factor 1.6.Moreover, the atomic number, Z, of the lens material plays an important role for the transmission. The reasonfor choosing silicon in this study was that manufacturing methods are readily available and the transmissionis good enough for a proof of principle study. For improving the lens, however, a lower Z material, such as aplastic, would be preferable. If going from silicon ( Z = 14) to epoxy ( Z ≈
6) and increasing b so that θ = 2 . ◦ in order to keep the phase shift of 2 π, focal length, and aperture, the average transmission would become 0.93for a lens with no support structures. This is an improvement of an additional factor 1.6.Compared to its predecessor, the MPL, a PAL makes possible higher transmission since lens material cor-responding to a phase shift of 2 π, inactive for deflection but active for absorption, is removed. In a previousexperimental study, the average transmission of an epoxy MPL was found to be higher than the present PALtransmission (0.5 at 23 keV), but the MPL was made of epoxy and the result should be well within reach of animproved PAL. Additionally, the gaussian transmission of the MPL approaches zero faster towards the peripheryof the lens than does the approximately linear transmission profile of the PAL, which opens the possibility fora larger aperture and a higher gain. In fact, an aperture improvement factor (AIF) proportional to the im-provement in gain, was derived in Ref. 19 as AIF = 3 . · √ δF · ( √ µb tan θ ) − . For epoxy lenses at 23 keV theimprovement is a factor 23.To be feasible for medical imaging, the PAL filter must allow a photon flux and image acquisition timecomparable to existing imaging systems. Since the lens provides a line focus which is not limited in length, it maybe coupled to a row detector in a scanning system, such as computed tomography or scanned slit mammographysystems. Furthermore, several lenses in an array can be coupled to a row detector each in a multi slit geometry.The aperture of the PAL is approximately the same size as a collimator slit in a conventional such geometry, andthe transmission of the lens may therefore be compared to the transmission of an absorption filter. For instance,the transmission of a 0.5 mm aluminum filter at 23 keV is 0.73, which is roughly a factor 2 higher than thetransmission of the lens presented here, but within reach of an improved PAL with less support structures anda more optimized lens material. Additionally, since the PAL gathers radiation, the lens aperture can be madelarger than the collimating slit in a conventional system at the same resolution. The actual flux integrated overall energies does, however, depend on several specific parameters of the set-up including its length, the resolution,and the size of the source, and a more extensive study would be needed to address the issue of image acquisitiontime properly.
6. CONCLUSIONS
An energy filter for medical x-ray imaging based on a silicon prism-array lens with a peak energy at 23 keV hasbeen investigated, experimentally and theoretically.hen imaging a bremsstrahlung source, the lens produces a line shaped image with a size somewhat largerthan expected. Measured gain of flux is in good agreement with the ray-tracing and phase-plate models at thepeak energy, deviating -20% from ray-tracing. Deviations in both image size and gain can be accounted for byinstability of the x-ray tube focal spot size and position, imperfections in the lens structure, and approximationsin the models. For energies other than the peak energy the gain predicted by the ray-tracing model agreeswell with measurements, whereas the phase-plate model predicts less efficient filtering. This is likely due toapproximations in the model.An experimental model of a mammography system showed that the PAL filter can reduce dose by 14%compared to absorption filtering for 6 cm breasts. The dose reduction in this study was limited by imperfectionsin the set-up, and potentially a dose reduction of 17% is possible according to ray-tracing.Change in lens design can increase the transmission and gain by a factor 1 . × . , and the aperture of the PALcan be increased with more than an order of magnitude compared to its predecessor, the MPL. It is estimatedthat a filter based on such an improved lens will allow a photon flux comparable to conventional absorptionfilters. ACKNOWLEDGMENTS
We acknowledge the Swedish Research Council and and the foundation Lars Hiertas Minne for funding parts ofour work on prism-array lenses.
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