Photon-counting spectral phase-contrast mammography
E. Fredenberg, E. Roessl, T. Koehler, U. van Stevendaal, I. Schulze-Wenck, N. Wieberneit, M. Stampanoni, Z. Wang, R. A. Kubik-Huch, N. Hauser, M. Lundqvist, M. Danielsson, M. Aslund
This is the submitted manuscript of:
Fredenberg E, Roessl E, Koehler T, van Stevendaal U, Schulze-Wenck I, Wieberneit N, Stampanoni M, Wang Z, Kubik-Huch RA, Hauser N, Lundqvist M., Danielsson M., Åslund M. , “
Photon-counting spectral phase-contrast mammography ,” Proc. SPIE
The published version of the manuscript is available at: https://doi.org/10.1117/12.910615 See also:
Fredenberg E, Danielsson M, Stayma n JW, Siewerdsen JH, Åslund M. Ideal‐observer detectability in photon‐counting differential phase‐contrast imaging using a linear‐systems approach. Medical physics. 2012 Sep;39(9):5317-35. https://doi.org/10.1118/1.4739195 All publications by Erik Fredenberg: https://scholar.google.com/citations?hl=en&user=5tUe2P0AAAAJ hoton-Counting Spectral Phase-Contrast Mammography
E. Fredenberg, a,b
E. Roessl, c T. Koehler, c U. van Stevendaal, c I. Schulze-Wenck, d N. Wieberneit, d M. Stampanoni, e,f
Z. Wang, e R. A. Kubik-Huch, g N. Hauser, h M. Lundqvist, b M. Danielsson, a,b
M. ˚Aslund ba Department of Physics, Royal Institute of Technology (KTH), AlbaNova University Center,106 91 Stockholm, Sweden; b Philips Women’s Healthcare, Smidesv¨agen 5, 171 41 Solna, Sweden; c Philips Technologie GmbH Innovative Technologies, Research Laboratories, R¨ontgenstrasse24, 22335 Hamburg, Germany; d Philips Healthcare, R¨ontgenstrasse 24, 22335 Hamburg, Germany; e Swiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland; f Institute for Biomedical Engineering, University and ETH Z¨urich, 8092 Z¨urich, Switzerland; g Department of Radiology, Kantonsspital Baden, 5404 Baden, Switzerland; h Department of Gynecology and Obstetrics, Interdisciplinary Breast Center Baden,Kantonsspital Baden, 5404 Baden, Switzerland
ABSTRACT
Phase-contrast imaging is an emerging technology that may increase the signal-difference-to-noise ratio in med-ical imaging. One of the most promising phase-contrast techniques is Talbot interferometry, which, combinedwith energy-sensitive photon-counting detectors, enables spectral phase-contrast imaging. We have evaluateda realistic system for spectral phase-contrast mammography by cascaded-systems analysis. Phase contrast im-proved detectability compared to absorption contrast, in particular for fine tumor structures. This result wassupported by images of human mastectomy samples that were acquired with a conventional detector. The opti-mal incident energy was slightly higher in phase contrast than in absorption contrast. In Talbot interferometry,the optimum was sharp and located at the setup design energy. Detectability may be further improved by energyweighting, which for phase contrast was found to have a weaker energy dependence than absorption contrast.Optimal weighting in Talbot interferometry had a steeper energy dependence, with a maximum at the setupdesign energy. Spectral material decomposition was not facilitated by phase contrast, but phase informationmay be used instead of spectral information.
Keywords: mammography; phase contrast; spectral imaging; detectability index; Talbot interferometry; photoncounting
1. INTRODUCTION
Phase-contrast imaging is an emerging technology in medical x-ray imaging that may increase the signal-difference-to-noise ratio compared to conventional absorption contrast.
One of the most promising phase-contrast techniques for medical imaging is Talbot interferometry.
Benefits of Talbot interferometry in amedical imaging context include low coherence requirements, a compact setup, phase and absorption contrastare readily separated, and good photon economy. A challenge of the technique is that Talbot interferometers areoptimized only at a single energy – the design energy.
Electronic mail: [email protected] n absorption contrast, there exists an optimal incident energy because of the tradeoff between contrast anddose, which both increase towards lower energies. The same kind of tradeoff exists in phase-contrast imagingwith a resulting optimal energy. Photon-counting detectors are fast enough to measure the energy of individual photons and can be employedfor single-shot spectral imaging. There are two well-investigated applications of spectral imaging: (1) Energyweighting is optimization of the signal-to-noise ratio by weighting photons according to information content. (2) Material decomposition extracts information about the object constituents by the material-specific absorptionenergy dependence.
16, 17
A special case is energy subtraction, which aims at reducing the impact of anatomical-structure overlap, so-called anatomical noise.
14, 15, 18–21
Spectral absorption-contrast imaging is well investigated, whereas spectral phase contrast is a relativelyunexplored area. We evaluate two aspects of the x-ray energy spectrum on phase contrast in general, and ona mammography system based on photon-counting Talbot interferometry in particular: (1) optimization of theincident spectrum with respect to energy, and (2) utilization of the transmitted spectrum (spectral imaging).The framework that we will use for evaluation has been presented previously, and is introduced for non-spectralimaging.
2. MATERIAL AND METHODS2.1. A photon-counting phase-contrast mammography setup
We have investigated a photon-counting Talbot-interferometry setup for phase-contrast mammography withsilicon strip detectors and geometry similar, but not identical, to the Philips MicroDose Mammography system(Philips Digital Mammography AB, Solna, Sweden).
23, 24
The setup will be further described in an upcomingpublication, but the main components are outlined in Fig. 1 and described in the following under the assumptionof a parallel beam.A π -shifting beam splitter (phase grating) introduces interference fringes at D n = np / λ d , where n =1 , , . . . is the Talbot order, p is the beam-splitter pitch, and λ d is the wave length for which the setup isdesigned (the design wave length). A phase gradient in the object causes a phase shift (displacement) of thefringes, which can be measured in the direction perpendicular to the grating slits ( x ) to obtain the differentialphase shift that is caused by the object. Absorption contrast can be measured in parallel to phase contrast byaveraging over the fringes.The fringe period is generally very short (in the order of p / We have developed a cascaded-systems framework to evaluate the performance of phase-contrast imaging thatwill be presented in an upcoming publication. The framework is based on the noise-equivalent number ofquanta (NEQ) and an ideal-observer detectability index ( d (cid:48) ): NEQ( f ) = (cid:104) I (cid:105) T ( f ) S ( f ) and d (cid:48) = (cid:90) Ny NEQ( f ) × W ( f ) d f , (1)where f is the spatial frequency vector, (cid:104) I (cid:105) is the expected image signal, T is the modulation transfer function(MTF) of the system, and S is the noise-power spectrum (NPS). W = C × F is the task function with contrast C and signal template F .To compare phase and absorption contrast side-by-side, the differential phase-contrast signal was integratedin the x direction, and the absorption-contrast signal was taken as the logarithm of the detected number of bject 2 object 2sourcesource grating source grating D n L source Λ D n xzy detectorpixels scanscannedanalyzergratingbeam splitter strip detectorsscan yzx detected signalPhase-contrast direction Absorption-contrast directionobject 1 object 1 Figure 1.
Schematic of the photon-counting Talbot-interferometry setup for phase-contrast mammography.
Left:
Abeam splitter (phase grating) illuminated by an x-ray source that is covered by a source grating induces interferencefringes. The fringes are displaced by the phase gradient in an object. A fine-pitch analyzer grating can be used todemodulate the high-frequency fringes into lower frequencies so that the fringe displacement and hence the phase shiftcan be measured by the photon-counting silicon strip detector elements.
Right:
The silicon strip detectors are scannedin the other direction to cover the full field-of-view. photons. Hence, the signal difference between target material c and background material g in phase contrast(subscript Φ) and absorption contrast (subscript µ ) is ∆ s Φ = |(cid:104) I Φ c (cid:105) − (cid:104) I Φ g (cid:105)| = k | δ c − δ g | d c ≡ k ∆ δ cg d c and ∆ s µ = |(cid:104) I µc (cid:105) − (cid:104) I µg (cid:105)| = | µ c − µ g | d c ≡ ∆ µ cg d c , (2)where k = 2 π/λ is the wave number; δ c and δ g are the real parts of the complex refractive index for the respectivematerials; µ is the linear attenuation coefficient; ∆ δ cg and ∆ µ cg are the differences in µ and δ ; d c is the targetthickness.The MTF in phase and absorption contrast was derived as T Φ ( f ) = T ( f ) and T µ ( f ) = T ( f ) , (3)where subscript 0 indicates the system MTF. The MTF is hence equal for phase and absorption contrast. TheNPS on the other hand differs, and because of the one-dimensional detection of the phase derivative, the phase-contrast NPS differs in the x - and y -directions: S Q Φ ( f ) = π × n p × (cid:20) λ d λ (cid:21) × A × f x × N × S Q ( f x )2 N × S Q ( f y ) and S Qµ ( f ) = 1 N × S Q ( f ) , (4)where N is the number of photons incident on the analyzer grating. Λ ∈ (0 ,
1) ranges from 0 for an object atthe detector to 1 for an object at or upstream of the beam splitter. Hence, the object may be located after theeam splitter (Λ <
1) to keep the setup compact and avoid beam-splitter absorption, but the cost is increasednoise (equivalent to reduced phase-contrast sensitivity). Moreover, problems arise with thick objects that areplaced after the beam splitter since the displacement of the interference fringes will depend on distance fromthe detector. A ∈ (0 ,
1) is the amplitude of the interference fringes (often referred to as “visibility”), which isaffected by coherence. The inverse frequency dependence of S Q Φ in the x -direction is caused by the integrationfrom differential phase contrast to phase contrast. The inverse wavelength dependence is induced by the smallerangular deviation at shorter wavelengths.Moreover, a phase-propagation simulation framework was implemented in order to validate the analyticalcalculations. In a recently published study native, human breast mastectomy samples were scanned with a prototype Talbotinterferometer designed at the Paul-Scherrer Institute (PSI) in Villigen, Switzerland. After radical mastectomyat the Interdisciplinary Breast Center, Baden, Switzerland, the samples were transported in a dedicated cooledsample holder to PSI. Imaging in the Talbot interferometer was performed within a time frame of 2 hours fromresection and were returned immediately to the clinic to proceed with the histopathological examination.The x-ray setup consisted of an x-ray generator (Seifert ID 3000) and an unfiltered tungsten line-focustube operated at 40 kVp with mean energy of 28 keV and a current of 25 mA. Further, a flat panel CMOSdetector (Hamamatsu C9732DK) with a 12 × field-of-view and 50 × µ m pixel size was used. A Talbotinterferometer similar to the one depicted in Fig. 1 was used to acquire phase information by the phase-steppingapproach. Attenuation, differential phase, and dark-field information was obtained by means of phase-retrieval. In order to simplify the evaluation task for the radiologist and to illustrate the improvement brought byhigh-frequency information in the phase contrast image, a dedicated fusion algorithm was developed by PhilipsResearch and applied to the attenuation data and differential phase data after phase retrieval. The algorithmis designed to achieve a noise optimal weighting of each independent Fourier component of the attenuation dataand differential phase data. Its details will be published elsewhere. Absorption and refraction are described by the complex index of refraction; n = 1 − δ + iβ , where δ describesrefraction and β is related to absorption. With efficient scatter rejection and away from absorption edges, thereal part of the refractive index and linear absorption follow approximately δ ∝ E − ρ and µ ∝ (cid:26) E − Z . ρ at low Eρ at high E , (5)where ρ is the mass density and Z represents atomic number. Linear absorption is divided into two regionsdominated by the photo-electric effect and Compton scattering at low and high energies, respectively. Thecrossing between the two interaction processes depends on atomic number.When discussing energy dependence, we consider two cases. Firstly, a (ideal) setup with a monochromaticbeam and design energy that follows the incident energy, i.e. E d = E , by adaption of D n . Hence, the influenceof the setup is minimized and A = 1. This case is related to the intrinsic properties of phase contrast ratherthan specific for Talbot interferometry, and general conclusions to other phase-contrast techniques can be drawn.Secondly, we consider a setup with design energy locked at the optimal energy for phase contrast, i.e. E d = E ∗ ,which implies A ≤
1. This case is specific for Talbot interferometry and reflects the particular tradeoffs that areassociated with a more realistic system based on the technique.
Since β decreases monotonically with energy (except at absorption edges), the optimal incident energy in absorp-tion contrast is affected by the tradeoff between high contrast at low energies and low noise (high transmission)at high energies. δ also decreases with energy according to Eq. (5), but the dependence is weaker and we canexpect the optimal incident energy to be slightly higher than for absorption contrast.n view of Eq. (5), Eq. (2) with explicit energy dependence becomes∆ s Φ ( E ) = k ( E )∆ δ cg ( E ) d c ∝ E − ρd c and ∆ s µ ( E ) = ∆ µ cg ( E ) d c ∝ E − ρd c , (6)where we have used λ ∝ E − and assumed dominance by the photo-electric effect. In an ideal photon-countingsystem, the detected number of photons are N ( E ) = N ( E ) exp[ − d b µ g ( E )], where d b is the breast thickness, µ g is the linear attenuation of breast tissue, and N is the incident number of photons. Further, the noise isuncorrelated in an ideal system so that S Q ( E ) = N ( E ). Assuming case 1 above, i.e. E d = E , λ d = λ , and A = 1, Eq. (4) yields S Q Φ ( E ) ∝ N ( E ) exp[ − CE − ρd b ] and S Qµ ( E ) ∝ N ( E ) exp[ − CE − ρd b ] , (7)where C is a constant. Equation (7) shows that the phase- and absorption-contrast NPS have identical energydependencies for this case, and there is no directional energy dependence.If we require the dose to be equal at each energy, N ( E ) ∝ /D ( E ), and further assume the dose to beproportional to the reciprocal of the energy, N ( E ) ∝ /D ( E ) ∝ E and the detectability index (Eq. (1))becomes d (cid:48) ( E ) ∝ exp( − CE − d b ) × E − d c and d (cid:48) µ ( E ) ∝ exp( − CE − d b ) × E − d c . (8)A maximum of d (cid:48) with respect to energy (the optimal energy; E ∗ ) can be found for instance with differentiation,i.e. by setting ∂d (cid:48) /∂E = 0, which evaluates to E ∗ Φ ∝ (3 × Cd b ) / and E ∗ µ ∝ (3 / × Cd b ) / . (9)As expected, the optimal incident energy in phase contrast is slightly higher than for absorption contrast (afactor of 5 / ∼ . E ∗ is independent of target thickness ( d c ) and material ( µ ), which isin agreement with previous results for absorption contrast. If we instead consider case 2 according to above, i.e. E d = E ∗ , which is closer to a realistic system based onTalbot interferometry, the situation becomes more complicated than the general result obtained in Eq. (9). Theoptimal energy is additionally affected by reduced amplitude of the interference fringes away from the designenergy ( E ∗ ) of the setup according to A ∝ (cid:20) (cid:94) (cid:18) π E d E n (cid:19)(cid:21) , (10)where ∧ is the continuous triangle function, defined here as ∧ ( θ ) ≡ /π × (cid:82) θ sgn[sin( φ )] dφ − Reducedamplitude leads to reduced fringe visibility and increased noise, which favors imaging at the design energy.There is, however, an additional energy dependence on S Q Φ because λ d /λ in Eq. (4) does not cancel. Thismeans that the increased noise towards lower energies is mitigated, whereas the increase towards higher energiesis amplified. In Talbot interferometry, the phase-contrast NPS is asymmetric according to Eq. (4) so that theseeffects appear in only one direction, and the influence is reduced by the square root. For a given incident spectrum, detected photons can be weighted according to their information content, i.e.low-energy photons are assigned a higher weight.
For an ideal photon-counting system with several energybins indexed by Ω, the detected number of photons are N = (cid:80) N Ω = q (cid:80) φ Ω , where q is the total number ofcounts and φ Ω accounts for the spectrum ( (cid:80) φ Ω = 1). Equation (2) for this system becomes∆ s µ = |(cid:104) I µc (cid:105) − (cid:104) I µg (cid:105)| = (cid:12)(cid:12)(cid:12) ln (cid:104) q (cid:88) φ Ω w Ω exp( − ∆ µ cg Ω × d c ) (cid:105) − ln (cid:104) q (cid:88) φ Ω w Ω (cid:105)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ln (cid:20) (cid:80) φ Ω w Ω exp( − ∆ µ cg Ω × d c ) (cid:80) φ Ω w Ω (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) (cid:39) (cid:80) φ Ω w Ω × ∆ µ cg Ω d c (cid:80) φ Ω w Ω , and∆ s Φ = |(cid:104) I Φ c (cid:105) − (cid:104) I Φ g (cid:105)| (cid:39) (cid:80) φ Ω w Ω × k Ω ∆ δ cg Ω d c (cid:80) φ Ω w Ω , (11)here ∆ µ cg , Ω ≡ | µ c , Ω − µ g , Ω | and ∆ δ cg , Ω ≡ | δ c , Ω − δ g , Ω | are the differences in effective µ and δ over the energybin. The weight factor w Ω is applied before the logarithm or phase calculation. For small signal differences itwould, however, be approximately equivalent to propagate the energy bins through logarithm / phase calculationbefore weighting. With uncorrelated noise, S Q = N ω = q × φ Ω . For case 1, i.e. E d = E ⇒ λ d = λ and A = 1, Eq. (4)becomes S Qµ ( f ) = (cid:88) ∂I∂N Ω (cid:12)(cid:12)(cid:12)(cid:12) N Ω × S Q Ω ( f ) = 1 q × (cid:80) φ Ω w Ω ) × (cid:88) φ Ω w and S Q Φ ( f ) = π × n p × f x × q × (cid:80) φ Ω w Ω ) × (cid:88) φ Ω w q × (cid:80) φ Ω w Ω ) × (cid:88) φ Ω w . (12)Note that the energy dependencies of S Qµ and S Q Φ are equal and independent of direction, similar to Eq. (7).If we combine Eqs. (11) and (12), Eq. (1) evaluates to d (cid:48) µ ∝ ( (cid:80) φ Ω w Ω × ∆ µ cg Ω d c ) (cid:80) φ Ω w and d (cid:48) ∝ ( (cid:80) φ Ω w Ω × k Ω ∆ δ cg Ω d c ) (cid:80) φ Ω w , (13)where we have assumed that the MTF is independent of energy; a fairly good approximation in our case. A global maximum of d (cid:48) with respect to weight can be found for instance by differentiation, i.e. by setting ∂d (cid:48) /∂w n = 0. For two energy bins (i.e. Ω ∈ lo, hi ), w ∗ Φ ,lo w ∗ Φ ,hi = ∆ δ lo λ hi ∆ δ hi λ lo ⇒ w ∗ Φ ∝ E − and w ∗ µ,lo w ∗ µ,hi = ∆ µ lo ∆ µ hi ⇒ w ∗ µ ∝ E − , (14)where we have used Eq. (5) to evaluate energy dependence. Hence, photons used for absorption contrast shouldbe weighted according to E − , which is in accordance with previous results. For phase-contrast imaging ingeneral, photons should be weighted according to E − . In most practical cases, however, phase-contrast efficiencyis reduced away from the setup design energy, which needs to be taken into account; photons close to the designenergy should be weighted higher. For Talbot interferometry, optimal weighting is affected according to thefringe amplitude in Eq. (10). X-ray attenuation in the medical imaging domain follows approximately µ = µ P E + µ C , where µ P E and µ C arethe linearly independent contributions from photo absorption and Compton scattering as given by Eq. (5). Because of the number of interaction effects, the proportions of not more than two materials in a mixture maybe determined with measurements at different x-ray energies, i.e. measurements at more than two energies areredundant. Talbot interferometry simultaneously detects absorption and phase contrast, which together withspectral imaging has the potential to add independent interaction processes for separation of more materials. Infact, a Talbot interferometer with optimal operation at several energies for inter alia this purpose has alreadybeen suggested. However, according to Eq. (5), x-ray refraction follows δ ∝ E − × ρ so that (1) all materials have the sameenergy dependence and (2) the material dependence of δ is limited to density and is hence correlated to Comptonscattering. Therefore, (1) material decomposition is in principle not possible in phase contrast without absorptioncontrast and (2) phase contrast does not add information to absorption-contrast spectral imaging. The sameis true for contrast-enhanced imaging since the overall energy dependencies of Compton scattering and δ areunaffected by absorption edges.Nevertheless, phase together with absorption contrast, as obtained in e.g. Talbot interferometry, may beused as a substitute for spectral imaging. This possibility is not further pursued in the present study, but beeninvestigated by other authors. . RESULTS AND DISCUSSION3.1. Non-spectral imaging Previous results on non-spectral Talbot interferometry are summarized in this section to illustrate the frameworkthat was introduced in Sec. 2.2.1. For more details we refer to Ref. 22.Phase-contrast imaging did not exhibit a general signal-difference-to-noise improvement relative to absorptioncontrast, but the performance was found to be highly task dependent. Two of the observed effects are illustratedin Fig. 2. Firstly, the intrinsic detection of the phase differential caused correlation of the noise when integratingto phase contrast, and the NPS decreased rapidly with spatial frequency according to Eq. (4). This brown noisethat shows up as streaks in Fig. 2 is less disturbing at higher spatial frequencies, and phase contrast was beneficialfor small and sharp targets, e.g. tumor spicula rather than solid tumors, and for discrimination tasks rather thandetection tasks. This is illustrated in Fig. 2 (a) by phase-propagation simulations of two target sizes (300 µ mand 5 mm); the small target is easier to distinguish in phase contrast whereas the large target is better visualizedin absorption contrast. Note that the printed pixel size varies in these two images in order to be able to displaythem side-by-side. Figure 2 (c) further illustrates the effect by means of the detectability benefit ratio ( d (cid:48) Φ /d (cid:48) )of phase over absorption contrast for a range of target sizes; it is evident that the benefit of phase contrast goesup for smaller radii. ph a s ec on t r a s t a b s o r p ti on c on t r a s t ph a s ec on t r a s t a b s o r p ti on c on t r a s t d e t ec t a b ilit y r a ti o ( d Φ ’ / d ’) tumorMCglandularair Figure 2.
Results on phase-contrast size and material dependence. (a) Size dependence: Simulated images of tumorstructures with diameters 300 µ m and 5 mm. Phase contrast (top row) and absorption contrast (bottom row). Thephase-contrast NPS decreases rapidly with frequency because of integration of quantum noise, which shows up as streaksin the vertical direction. (b) Material dependence: A 300- µ m-diameter air cavity (left column) and a microcalcification(right column). (c) Detectability benefit ratio of phase over absorption contrast ( d (cid:48) Φ /d (cid:48) ). Printed with permission fromMedical Physics. Secondly, phase contrast favored detection of materials that differ in density compared to the backgroundtissue, rather than materials with differences in atomic number that are efficiently probed by absorption contrast.For instance, the improvement of phase contrast in microcalcification detection was less than for tumor andglandular structures of the same size, which can be seen in Fig. 2 (c). The extreme case is a gaseous target,which is used for comparison to a 300- µ m-diameter microcalcification in Fig. 2 (b). Figure 3 shows images from the Talbot interferometer at PSI. It is a small section of a breast mastectomy samplefrom a female patient, aged 88 (case 3 in Ref. 30), with an invasive ductal breast carcinoma. The image fusionalgorithm operated on the absorption image shown in the left panel of Fig. 3 and the differential phase-contrastimage shown in the center panel, with the result shown in the right-hand panel. Note that the image fusionlgorithm accounts for the differential nature of the data by a division in Fourier space of the Fourier amplitudesby the frequency. The improvement in the visualization of fine details and interfaces can be appreciated from acomparison between the absorption image and the fused image. We attribute this improvement to the reducedphase-contrast NPS at high frequencies as described by Eq. (4).
Figure 3.
Images acquired at PSI of an invasive ductal breast carcinoma in a mastectomy sample.
Left:
Absorption-contrast image.
Center:
Differential phase-contrast image.
Right:
Image fusion of the absorption- and phase-contrastimages.
Figure 4 shows detectability for 200- µ m tumor structures and microcalcifications at 1 mGy as a function ofincident photon energy. Detectability was evaluated for the two cases described in Sec. 2.3: A setup with thedesign energy adapted to the incident energy, i.e. E d = E ; Talbot interferometry with design energy locked atthe optimal energy, i.e. E d = E ∗ . Detectability for absorption contrast is plotted for comparison.
20 30 40 50 60051015202530 photon energy [keV] E d = EE d = E * absorption contrast:phase contrast:20 30 40 50 6000.10.20.30.40.50.60.7 d e t ec t a b ilit y ( d ’) photon energy [keV] Figure 4.
Spectral optimization: Detectability at 1 mGy as a function of energy for phase contrast without effects of thesetup design energy ( E d = E ), Talbot interferometry including the design energy ( E d = E ∗ ), and absorption contrast. Left:
Detectability of a tumor structure.
Right:
Detectability of a microcalcification.
The two typical mammography targets in Fig. 4 have similar detectability energy dependence, which isexpected from Eq. (9) and is a well-known effect for absorption contrast. Detectability of the microcalcificationis substantially higher than for the tumor structure because of higher contrast, but there is no benefit of phasever absorption contrast. The latter is an effect of the higher atomic number and in accordance with Fig. 2 (c).The optimal energy of phase contrast is generally higher than for absorption contrast; 38 keV and 22 keVrespectively according to Fig. 4. The ratio of 38 / ∼ . E d = E ), imaging at the optimal energy improveddetectability by about 40% compared to a setup optimized for absorption contrast. If the design energy waslocked, however ( E d = E ∗ ), reduced amplitude modulation away from the design energy resulted in a sharpermaximum, and optimization for phase contrast yielded an improvement by a factor of 3 compared to imaging atthe optimum for absorption contrast. Figure 5 plots the weight that should be assigned to each photon as a function of energy to maximize detectability.The target was a 200- µ m tumor structure, but almost identical behavior was found for other targets. The samecases that were considered in Fig. 4 are represented also in Fig. 5: Phase contrast with the design energyadapted to the incident energy ( E d = E ); Talbot interferometry with design energy locked at the optimal energy( E d = E ∗ ). Optimal weighting for absorption-contrast is plotted for comparison.
20 30 40 50 6000.51.01.52.0 photon energy [keV] w e i gh t ( w ) E d = EE d = E * absorption contrast:phase contrast: Figure 5.
Energy weighting: Optimal weight as a function of energy for phase contrast without effects of the setup designenergy ( E d = E ), Talbot interferometry including the design energy ( E d = E ∗ ), and absorption contrast. As was discussed in Sec. 2.3.1, absorption contrast should be weighted close to E − . For an incident energyspectrum that was rect-distributed between 16 and 40 keV, which is a realistic energy interval for mammography,optimal weighting improved detectability by 19% compared to unweighted photon counting (intrinsic weighting w ∗ µ ( E ) ∝
1) and by 33% compared to energy integrating detectors (intrinsic weighting w ∗ µ ( E ) = ∝ E ). Theseresults are in line with previous studies. Phase contrast on the other hand should be weighted according to E − for a setup without influence of thedesign energy ( E d = E ). Because of the weaker energy dependence, optimal weighting in phase contrast had asmaller impact than in absorption contrast; detectability was improved by 3.6% compared to photon countingand by 13% compared to energy integrating detectors with the rect-distributed incident spectrum. In Talbotinterferometry with the setup design energy taken into account ( E d = E ∗ ), the energy dependence was stronger,however. Optimal weighting dropped quicker towards higher energies and there was a superimposed maximumclose to the design energy. Accordingly, the improvement in detectability was larger in this case; 38% comparedto photon counting and 61% compared to energy integrating detectors. . CONCLUSIONS Cascaded-systems analysis enables comprehensive evaluation of phase-contrast efficiency. The benefit comparedto absorption contrast is highly dependent on task, in particular target size and material, with larger improve-ments for small structures, and for soft tissue rather than microcalcifications. Measurements on mastectomysamples with a conventional detector illustrated the improved detectability of fine tumor structures.The optimal incident energy is a factor of 1.7 higher in phase contrast than in absorption contrast becausethe phase shift drops slower with energy than does absorption. The difference is smaller than could be expected,however, partly because Compton scattering dominates absorption contrast at higher energies. Reduced phase-contrast efficiency away from the design energy in Talbot interferometry further sharpens the optimum in incidentenergy, and detectability was improved by a factor of 3 compared to a setup optimized for absorption contrast.Optimal weighting in phase contrast follows E − , compared to E − in absorption contrast. In Talbot in-terferometry, the energy dependence is stronger and there is a maximum at the setup design energy. Optimalweighting improved phase-contrast detectability by 3.6–38% compared to non-spectral photon counting detectorsand by 13–61% compared to energy integrating detectors.Spectral material decomposition was not facilitated by phase contrast, but phase may be used instead ofspectral information. ACKNOWLEDGMENTS
This research was funded in part by the Swedish agency for innovation systems (VINNOVA).
REFERENCES
1. Lewis, R. A., “Medical phase contrast x-ray imaging: current status and future prospects,”
Physics inMedicine and Biology (16), 3573–3583 (2004).2. Zhou, S. A. and Brahme, A., “Development of phase-contrast x-ray imaging techniques and potential medicalapplications,” Phys Med (3), 129–48 (2008).3. Keyrilainen, J., Bravin, A., Fernandez, M., Tenhunen, M., Virkkunen, P., and Suortti, P., “Phase-contrastx-ray imaging of breast,” Acta Radiologica (8), 866–884 (2010).4. Clauser, J. F., “Ultrahigh resolution interferometric X-ray imaging.” U.S. Patent 5,812,629 (1998).5. David, C., Nohammer, B., Solak, H. H., and Ziegler, E., “Differential x-ray phase contrast imaging using ashearing interferometer,” Applied Physics Letters (17), 3287–3289 (2002).6. Momose, A., Kawamoto, S., Koyama, I., Hamaishi, Y., Takai, K., and Suzuki, Y., “Demonstration of x-rayTalbot interferometry,” Japanese Journal of Applied Physics (7B), L866–L868 (2003).7. Pfeiffer, F., Weitkamp, T., Bunk, O., and David, C., “Phase retrieval and differential phase-contrast imagingwith low-brilliance x-ray sources,” Nature Physics (4), 258–261 (2006).8. Qi, Z. H., Zambelli, J., Bevins, N., and Chen, G. H., “A novel quantitative imaging technique for materialdifferentiation based on differential phase contrast CT,” in [ Proc. SPIE, Physics of Medical Imaging ], Hsieh,J. and Samei, E., eds., (2010).9. Motz, J. and Danos, M., “Image information content and patient exposure,”
Med. Phys. (1), 8–22 (1978).10. Engel, K. J., Geller, D., K¨ohler, T., Martens, G., Schusser, S., Vogtmeier, G., and R¨ossl, E., “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instr. and Meth. A (0),S202–S207 (2011).11. Fredenberg, E., Lundqvist, M., Cederstr¨om, B., ˚Aslund, M., and Danielsson, M., “Energy resolution of aphoton-counting silicon strip detector,”
Nucl. Instr. and Meth. A (1), 156–162 (2010).12. Tapiovaara, M. and Wagner, R., “SNR and DQE analysis of broad spectrum x-ray imaging,”
Phys. Med.Biol. , 519–529 (1985).13. Cahn, R., Cederstr¨om, B., Danielsson, M., Hall, A., Lundqvist, M., and Nygren, D., “Detective quantumefficiency dependence on x-ray energy weighting in mammography,” Med. Phys. (12), 2680–3 (1999).14. Fredenberg, E., Hemmendorff, M., Cederstr¨om, B., ˚Aslund, M., and Danielsson, M., “Contrast-enhancedspectral mammography with a photon-counting detector,” Med. Phys. (5), 2017–2029 (2010).5. Fredenberg, E., ˚Aslund, M., Cederstr¨om, B., Lundqvist, M., and Danielsson, M., “Observer model opti-mization of a spectral mammography system,” in [ Proc. SPIE, Physics of Medical Imaging ], Samei, E. andPelc, N. J., eds., (2010).16. Alvarez, R. and Macovski, A., “Energy-selective reconstructions in x-ray computerized tomography,”
Phys.Med. Biol. , 733–744 (1976).17. Lehmann, L. A., Alvarez, R. E., Macovski, A., Brody, W. R., Pelc, N. J., Riederer, S. J., and Hall, A. L.,“Generalized image combinations in dual KVP digital radiography,” Med. Phys. (5), 659–667 (1981).18. Johns, P., Drost, D., Yaffe, M., and Fenster, A., “Dual-energy mammography: initial experimental results,” Med. Phys. , 297–304 (1985).19. Lewin, J., Isaacs, P., Vance, V., and Larke, F., “Dual-energy contrast-enhanced digital subtraction mam-mography: Feasibility,” Radiology , 261–268 (2003).20. Baldelli, P., Bravin, A., Maggio, C. D., Gennaro, G., Sarnelli, A., Taibi, A., and Gambaccini, M., “Evalu-ation of the minimum iodine concentration for contrast-enhanced subtraction mammography,”
Phys. Med.Biol. (17), 4233–51 (2006).21. Bornefalk, H., Lewin, J. M., Danielsson, M., and Lundqvist, M., “Single-shot dual-energy subtractionmammography with electronic spectrum splitting: Feasibility,” Eur. J. Radiol. , 275–278 (2006).22. Fredenberg, E., Danielsson, M., Stayman, J. W., Siewerdsen, J. H., and ˚Aslund, M., “Cascaded-systemsanalysis of phase-contrast imaging,” Med. Phys. (2012). Submitted.23. Lundqvist, M.,
Silicon strip detectors for scanned multi-slit x-ray imaging , PhD thesis, Royal Institute ofTechnology (KTH), Stockholm (2003).24. ˚Aslund, M., Cederstr¨om, B., Lundqvist, M., and Danielsson, M., “Physical characterization of a scanningphoton counting digital mammography system based on Si-strip detectors,”
Med. Phys. (6), 1918–1925(2007).25. Sharp, P. F., Metz, C. E., Wagner, R. F., Myers, K. J., and Burgess, A. E., “ICRU Rep. 54 Medical imag-ing: the assessment of mage quality,” International Commission on Radiological Units and Measurements,Bethesda, MD (1996).26. Metz, C. E., Wagner, R. F., Doi, K., Brown, D. G., Nishikawa, R. M., and Myers, K. J., “Toward consensuson quantitative assessment of medical imaging-systems,”
Med. Phys. (7), 1057–1061 (1995).27. Siewerdsen, J. H., [ The handbook of medical image perception and techniques ], ch. 25. Optimization of 2Dand 3D radiographic imaging systems, Cambridge University Press, Cambridge (2010).28. Siewerdsen, J. H. and Jaffray, D. A., “Optimization of x-ray imaging geometry (with specific application toflat-panel cone-beam computed tomography),”
Med. Phys. (8), 1903–14 (2000).29. Cunningham, I. A., [ Handbook of Medical Imaging ], vol. 1. Physics and Psychophysics, ch. 2. AppliedLinear-Systems Theory, SPIE Press, Bellingham, USA (2000).30. Stampanoni, M., Wang, Z. T., Thuring, T., David, C., Roessl, E., Trippel, M., Kubik-Huch, R. A., Singer,G., Hohl, M. K., and Hauser, N., “The first analysis and clinical evaluation of native breast tissue usingdifferential phase-contrast mammography,”
Investigative Radiology (12), 801–806 (2011).31. Pfeiffer, F., Bech, M., Bunk, O., Kraft, P., Eikenberry, E. F., Bronnimann, C., Grunzweig, C., and David,C., “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials (2), 134–137 (2008).32. Roessl, E., Koehler, T., van Stevendaal, U., Martens, G., Hauser, N., Wang, Z., and Stampanoni, M., “Imagefusion algorithm for differential phase contrast imaging,” in [ Proc. SPIE, Physics of Medical Imaging ],(2012). Submitted.33. Roessl, E. et al., “Composition algorithm for differential phase contrast projection imaging,” In preparation.34. Cederstr¨om, B., Lundqvist, M., and Ribbing, C., “Multi-prism x-ray lens,”
Appl. Phys. Lett. (8), 1399–1401 (2002).35. Fahrig, R. and Yaffe, M. J., “Optimization of spectral shape in digital mammography: dependence on anodematerial, breast thickness, and lesion type,” Med Phys21