Observer model optimization of a spectral mammography system
Erik Fredenberg, Magnus Aslund, Bjorn Cederstrom, Mats Lundqvist, Mats Danielsson
This is the submitted manuscript of:
Fredenberg, E., Åslund, M., Ceder ström, B., Lundqvist, M. and Danielsson, M., “Observer model optimization of a spectral mammography system,” Proc. SPIE 7622, Medical Imaging 2010: Physics of Medical Imaging, 762210 (2010).
The published version of the manuscript is available at: https://doi.org/10.1117/12.845480 See also:
Fredenberg, E., Hemmendorff, M., Cederström,
B., Åslund, M. and Danielsson, M., 2010. Contrast‐enhanced spectral mammography with a photon‐counting detector. Medical physics, 37(5), pp.2017-2029. https://doi.org/10.1118/1.3371689 All publications by Erik Fredenberg: https://scholar.google.com/citations?hl=en&user=5tUe2P0AAAAJ bserver model optimization of aspectral mammography system
Erik Fredenberg, a Magnus ˚Aslund, b Bj¨orn Cederstr¨om, a Mats Lundqvist b andMats Danielsson aa Department of Physics, Royal Institute of Technology, AlbaNova, 106 91, Stockholm, Sweden b Sectra Mamea AB, Smidesv¨agen 2, 171 41 Solna, Sweden
ABSTRACT
Spectral imaging is a method in medical x-ray imaging to extract information about the object constituentsby the material-specific energy dependence of x-ray attenuation. Contrast-enhanced spectral imaging has beenthoroughly investigated, but unenhanced imaging may be more useful because it comes as a bonus to theconventional non-energy-resolved absorption image at screening; there is no additional radiation dose and no needfor contrast medium. We have used a previously developed theoretical framework and system model that includequantum and anatomical noise to characterize the performance of a photon-counting spectral mammographysystem with two energy bins for unenhanced imaging. The theoretical framework was validated with synthesizedimages. Optimal combination of the energy-resolved images for detecting large unenhanced tumors correspondedclosely, but not exactly, to minimization of the anatomical noise, which is commonly referred to as energysubtraction. In that case, an ideal-observer detectability index could be improved close to 50% compared toabsorption imaging. Optimization with respect to the signal-to-quantum-noise ratio, commonly referred to asenergy weighting, deteriorated detectability. For small microcalcifications or tumors on uniform backgrounds,however, energy subtraction was suboptimal whereas energy weighting provided a minute improvement. Theperformance was largely independent of beam quality, detector energy resolution, and bin count fraction. It isclear that inclusion of anatomical noise and imaging task in spectral optimization may yield completely differentresults than an analysis based solely on quantum noise.
Keywords: model observer, spectral imaging, mammography, photon counting, energy subtraction, energyweighting, anatomical noise, detectability index
1. INTRODUCTION
The energy dependence of x-ray attenuation is material specific because of (1) different dependence on the atomicnumber for the photo-electric and Compton cross sections ( σ ∝ Z and Z respectively), and (2) discontinuitiesin the photo-electric cross section at absorption edges. Spectral imaging is a method in medical x-ray imagingthat takes advantage of the energy dependence to extract information about the object constituents.
1, 2
Phantom studies ? , ? , 26, 38 and clinical trials have proven contrast-enhanced spectral imaging to be a promis-ing approach for enhanced tumor detectability. Injection of contrast agent is, however, probably not motivatedfor regular screening, and contrast-enhanced spectral imaging is expected to be an alternative mainly for diag-nostic mammography at recalls. Enhancement of lesions without iodine uptake would be useful, since, in thecase of electronic spectrum splitting, it comes as a bonus on top of the conventional absorption image with noadditional dose to the patient. Spectral imaging could potentially increase detectability of obscured lesions, anddiscriminate between solid and cystic lesions already in the screening image, for instance.Previous studies in this field have focused mainly on calcifications, predominantly with encouraging re-sults, although some are more moderate. The main difficulty appears to be that amplified quantum noisein the subtracted image may reduce detectability of small details. For tumor imaging, a few clinical investigationshave been presented.
12, 13
There are also phantom studies that indicate feasibility for soft tissue imaging,
14, 15 but
Send correspondence to M. ˚Aslund, e-mail: [email protected] he minimal attenuation difference between glandular and tumorous tissue, which is likely the main challenge, does not seem to be addressed.In this work, we have investigated spectral imaging of tumors and microcalcifications without contrast agent,henceforth referred to as unenhanced imaging. There are at least three potential benefits of this approach com-pared to conventional non-energy resolved imaging, henceforth referred to as absorption imaging. (1) Energyweighting refers to optimization of the signal-to-quantum-noise ratio with respect to its energy dependence;photons at energies with larger agent-to-background contrast can be assigned a greater weight.
17, 18 (2) Energysubtraction (or dual-energy subtraction) refers to optimization of the signal-to-background-noise ratio by min-imization of the background clutter contrast. The contrast between any two materials (adipose and glandulartissue) in a weighted subtraction of images acquired at different mean energies can be reduced to zero, whereasall other materials to some degree remain visible.
3, 4, 6, 15, 19 (3) A third possible benefit of spectral imaging isquantification of the target, e.g. evaluation of microcalcification thickness. One way of obtaining spectral information is to use two or more input spectra. For imaging with clinicalx-ray sources, this most often translates into several exposures with different beam qualities (different accelera-tion voltages, filtering, and anode materials).
3, 4, 6
Results of the dual-spectra approach are promising, but theexamination may be lengthy with increased risk of motion blur and discomfort for the patient. This problem canbe solved by a simultaneous exposure with different beam qualities, or by using an energy sensitive sandwichdetector.
8, 9
For all of the above approaches, however, the effectiveness may be impaired due to overlap of thespectra, and a limited flexibility in choice of spectra and energy levels. In recent years, photon-counting silicondetectors with high intrinsic energy resolution, and, in principle, an unlimited number of energy levels (electronicspectrum-splitting) have been introduced as another option.
15, 19
An objective of the EU-funded HighReX project is to investigate the benefits of spectral imaging in mammog-raphy. The systems used in the HighReX project are based on the Sectra MicroDose Mammography system(Sectra Mamea AB, Solna, Sweden), which is a scanning multi-slit full-field digital mammography system witha photon-counting silicon strip detector.
22, 23
One advantage of this geometry in a spectral imaging context isefficient intrinsic scatter rejection. We have investigated a prototype detector and system developed within the HighReX project for unen-hanced spectral imaging. A semi-empirical cascaded system model and a framework for system characterizationhave been presented previously and is used also in the present study. We have used an ideal-observer detectabil-ity index, which includes quantum and anatomical noise, as a figure of merit to investigate feasibility and foroptimization.
2. MATERIAL AND METHODS2.1. Background
Figure 1 (Left) shows a photograph and schematic of the multi-slit system. The x-ray beam is collimated to a fanbeam matching the pre-collimator. The pre-collimator transforms the beam to several equidistant line beams.Beneath the breast support there is a detector box containing a post-breast collimator and the x-ray detector.The detector is comprised of several lines of Si-strip detectors matching the line beams exiting the breast. Thefan beam, pre-collimator, post-breast collimator and detector are scanned together laterally across the breast toobtain a full field image.The detector that was used for measurements and simulations of spectral imaging is a prototype photon-counting detector, developed within the HighReX project and mounted on a Sectra MicroDose Mammographyunit. A previous publication provides an investigation of the detector energy response, which is a prerequisiteto accurately model spectral imaging. Figure 1 (Right) shows a schematic of the detector. A bias voltage isapplied over the detector material, so that when a photon interacts, charge is released and drifts as electron-holepairs towards the anode and cathode respectively. Each strip is wire bonded to a preamplifier and shaper, whichare fast enough to allow for single photon-counting. The preamplifier and shaper collect the charge and convertit to a pulse with a height that is proportional to the charge and thus to the energy of the impinging photon. reastx-ray beam Si-strip detector linespre-collimatorcompression platebreast support _ +
HV rejectionhighlowASICpre-collimatorbreastx-ray tube detector yzx scanscan
Figure 1. Left:
Photograph and schematic of the Sectra MicroDose Mammography system.
Right:
The image receptorand electronics.
Pulses below a few keV are regarded as noise and are rejected by a low-energy threshold in a discriminator. Allremaining pulses are sorted into two energy bins by an additional high-energy threshold, and registered by twocounters. A preamplifier, shaper, and discriminator with counters are referred to as a channel, and all channelsare implemented in an application specific integrated circuit (ASIC). Anti-coincidence (AC) logic in the ASICdetects double counting from charge sharing by a simultaneous detection in two adjacent channels, and the eventis registered only once in the high-energy bin of the channel with the largest signal. Spatial resolution and imagenoise is thus improved, but all energy information of charge-shared photons is lost.
For task-specific system performance, we can define an ideal-observer detectability index d (cid:48) = 2 π (cid:90) Ny GNEQ( ω ) × C × F ( ω ) × ω d ω, (1)where the integral is taken over the Nyqvist region and ω is the spatial frequency in the radial direction. Polarcoordinates have been used for notational convenience, but all calculations were done in cartesian coordinateswhere appropriate, i.e. rotational symmetry was not assumed in general. C = ∆ s/ (cid:104) I (cid:105) is the target-to-backgroundcontrast in the image for signal difference ∆ s = (cid:104)| I background − I target |(cid:105) , where the angle brackets denote theexpectation value and I is the image signal per unit area. F is the signal template, which integrates to thearea of the target for unit contrast. GNEQ is the generalized noise-equivalent quanta that includes detectorand anatomical noise according to Richard and Siewerdsen. It is a reasonable extension of the standardnoise equivalent quanta (NEQ) that measures the detector noise performance, because the dominant source ofdistraction for many imaging tasks in mammography is the variability of the anatomical background, which isreferred to as anatomical or structured noise.
31, 32
For a quantum-limited system,GNEQ( ω ) = (cid:104) I (cid:105) T ( ω ) S Q ( ω ) + S A ( ω ) , (2)where T is the modulation transfer function (MTF) of the system, and S Q and S A are the power spectra (NPS)of quantum and anatomical noise respectively. .1.3. A theoretical framework for spectral imaging A framework to characterize the performance of the multi-slit system for spectral mammography has beenpresented previously, and is summarized below.As was mentioned above, two spectral optimization schemes that appear in the literature are energy weightingand energy subtraction. Somewhat simplified, energy weighting ignores S A and maximizes C /S Q , whereasenergy subtraction instead minimizes S A . Although S A and S Q can be expected to have completely differentfrequency distributions, and depending on the particular F , these two extremes are often good approximations,it is clearly a simplification. In this work a general optimization of Eq. (1) is instead considered.If the low- and high-energy images are normalized with the expected number of counts from mean breasttissue, a combined image with zero mean can be formed according to I ( x, y ) = w n lo ( x, y ) (cid:104) n lo (cid:105) + n hi ( x, y ) (cid:104) n hi (cid:105) − ( w + 1) (cid:39) w ln (cid:183) n lo ( x, y ) (cid:104) n lo (cid:105) (cid:184) + ln (cid:183) n hi ( x, y ) (cid:104) n hi (cid:105) (cid:184) , (3)where w is a weight factor. The approximation is valid for | n Ω − (cid:104) n Ω (cid:105)| (cid:191)
1, i.e. small signal differences. Writtenthis way, it is evident that a linear combination, which is the common form for energy weighting, is approximatelyequal to combination in the logarithmic domain, which is often used for energy subtraction. Energy weighting andenergy subtraction can therefore be regarded special cases of a general image combination. Normalization withthe expected number of counts in Eq. (3) is, however, a detour to make this point and to convey the derivationsbelow. A consequence is that Eq. (2) does not make sense because (cid:104) I (cid:105) = 0, but the product GNEQ × C is stillvalid. In the practical case, a combination of the non-normalized images is more handy, i.e. I (cid:48) ( x, y ) = w (cid:48) n lo ( x, y ) + n hi ( x, y ) or I (cid:48)(cid:48) ( x, y ) = w (cid:48)(cid:48) ln n lo ( x, y ) + ln n hi ( x, y ) . (4)The image mean of I , I (cid:48) , and I (cid:48)(cid:48) differ, but assuming small signal differences, the detectability indices calculatedwith all three image combinations are the same for w (cid:48) = ζ hi /ζ lo × w and w (cid:48)(cid:48) = w. (5)Note that I (cid:48) ( w (cid:48) = 1) is a conventional non-energy-resolved absorption image.If we assume no correlation between the energy bins, the quantum noise in the combined image is S Q ( ω ) = (cid:88) Ω ∂I∂n Ω (cid:175)(cid:175)(cid:175)(cid:175) n Ω × S QΩ ( ω ) (cid:39) q (cid:183) w ζ lo + 1 ζ hi (cid:184) . (6)where Ω ∈ { lo, hi } denotes the detector energy bin, q is the incident number of quanta, and ζ is the expectedfraction of incident counts to be detected. The approximation in Eq. (6) is for spatially uncorrelated noise.The anatomical noise in an x-ray image of breast tissue is caused by the variation in glandularity, which istransferred to the image through I ( g ( x, y ) ) , with g ( x, y ) being the glandular volume fraction as a function ofspatial image coordinates x and y . We therefore adopt the power spectrum of g ( x, y ) as a glandularity NPS( S A g ( ω )), which is transferred to the image NPS ( S A ( ω )) according to S A ( ω ) (cid:39) (cid:42) d I d g (cid:175)(cid:175)(cid:175)(cid:175) (cid:43) × S A g ( ω ) T ( ω ) (cid:39) d [ w ∆ µ ag ,lo + ∆ µ ag ,hi ] × S A g ( ω ) T ( ω ) , (7)where d b is the breast thickness, ∆ µ ag , Ω ≡ µ a , Ω − µ g , Ω is the difference in effective linear attenuation betweenadipose and glandular tissue, and the angle brackets represent the expectation value over the glandularity range.The first approximation of Eq. (7) is for piecewise linearity of I ( g ). The second approximation assumes linearityacross the range of glandularities, image combination according to Eq. (3), and small signal differences.Maximization of ∆ s /S Q and 1 /S A yields the optima for energy weighting and energy subtraction, respec-tively: w ∗ s /S Q = ζ lo ∆ µ bc ,lo /ζ hi ∆ µ bc ,hi , and w ∗ /S A = − ∆ µ ag ,hi / ∆ µ ag ,lo . (8) A can in practice not be completely eliminated according to Eq. (8) because the latter is based on the linearapproximation of I ( g ) in Eq. (7), and a better estimate is to instead use the piecewise linear approximation inEq. (7). Calculation of the expectation value, however, requires the probability density function ( λ g ) accordingto (cid:42) d I d g (cid:175)(cid:175)(cid:175)(cid:175) (cid:43) = (cid:90) d I d g (cid:175)(cid:175)(cid:175)(cid:175) × λ g d g (cid:39) (cid:90) (cid:90) d I d g (cid:175)(cid:175)(cid:175)(cid:175) × g ( x, y )d x d y. (9)The approximation in Eq. (9) assumes a glandularity map ( g ( x, y )) to be a representative estimate of the densityfunction. The previously developed model of the spectral imaging system
25, 26 was extended to include imaging of unen-hanced tumors, microcalcifications and cysts in an anatomical background. Equation (1) was used as a figureof merit for optimization and for comparison to conventional absorption imaging. We assumed that the spectralimage must come as a bonus on top of an optimal absorption image, which limited the choice of incident spec-tra and dose range. Compared to contrast-enhanced spectral imaging, the split energy does not have to bematched to an absorption edge of the contrast agent, but can be chosen to minimize quantum noise. In addition,the sensitivity to a limited energy resolution can be expected to be lower in unenhanced imaging because of nodiscontinuities in the attenuation spectrum.Energy resolved images were synthesized since clinical or phantom data was not available. The purpose ofthe images was twofold; to measure the noise in combined images for verification of Eqs. 6 and 7, and to visualizethe result of image combination. A tungsten target x-ray tube with 0.5 mm aluminum filtration and 30 kVpacceleration voltage was assumed if not otherwise stated. The object was a 50 mm breast with 5 mm skin thicknessand embedded lesions. Published x-ray spectra, x-ray attenuation coefficients, and dose coefficients wereused as input. Glandular structure was generated using the clustered lumpy background technique. Thestructure was chosen to range over all glandularities with a 50% glandularity mean. Tumor x-ray attenuationwas gathered from Johns and Yaffe, calcium phosphate (Ca P O ) microcalcifications and cysts consisting of100% glandular tissue were assumed. The tumors and cysts were 20 mm thick and had diameters of 20 mm,30 mm and 40 mm. Equal thickness means equal contrast, but the different diameters are affected differentlyby anatomical noise. Relatively large tumors were assumed to compensate for low detectability, but it has beenshown that tumors larger than 20 mm constitute approximately 30% of all missed breast cancers. The diameterof the calcifications was 100 µ m. The objects were imaged at a dose of 1 mGy, and quantum noise was addedwith a fraction double-counted photons calculated by the detector model. We assumed that there were no deadchannels in the detector. × × ∼ ×
80 mm ) was chosen. The NPS was measured in absorption and combined images,and the radial NPS was found by converting to polar coordinates and averaging over 2 π . For comparison, theNPS in the synthesized absorption images was fitted to analytical expressions as described below, and the NPSin combined images was calculated using these fits and the expressions in Section 2.1.3.The NPS of anatomical backgrounds can be well described by an inverse power function, i.e. S A ( ω ) (cid:39) αω − β , (10)and this function was fitted to S A in synthesized absorption images in a region that was virtually unaffected bywindow artifacts. The anatomical noise in combined images was calculated using the fit and Eq. (7), and couldthen be compared to measurements in combined synthesized images. To calculate the detectability index, a flatdistribution of glandularities was assumed in Eq. (9), which can be expected to overestimate S A since a Gaussianistribution is more probable, and predictions are therefore moderate. For comparison, S A was also calculatedusing the known glandularity map from the synthesized images, which is likely to give a better prediction.Following the approximations in Eq. (7), the magnitude ( α ) of S A g ( ω ) is affected by x-ray imaging and imagecombination, but the frequency dependence ( β ) is intact. It has been found previously that β in images with largeattenuation differences is in fact affected by the particular image combination, which is likely to be at leastpartly caused by breakdown of the piecewise-linearity approximation in Eq. (7). For breast tissue, attenuationdifferences are relatively small and the approximation can be expected to hold better. The MTF of the imagingsystem, however, filters the image and therefore affects β , which is accounted for in Eq. (7).Quantum noise with a fraction χ double-counted events is not completely flat in the frequency domain butfollows S QΩ ( u ) = n Ω χ Ω [1 + 2 cos(2 πu/p )]1 + χ Ω , (11)where p is the pixel size and u is the spatial frequency in the detector direction. In the scan direction (spatialfrequency v ), S Q ( v ) was assumed flat because readouts are uncorrelated. S Q in combined images was calculatedfrom Eq. (11) via Eq. (6). The latter is minimized for a bin count fraction ξ lo = 1 − ξ hi = | w | | w | , (12)which could determine a suitable split energy. If not otherwise stated, a count fraction close to 0.5 was, however,chosen in order to reduce complexity of the optimization. The MTF was measured and fitted to an analytical function as previously described.
26, 38
An assumption ofequal MTF in both energy bins was adopted, although double counting degraded the resolution somewhat inthe high-energy bin, which generally affects the GNEQ. This simplifications was regarded justified because thedifference in MTF between the bins was small, and the major differences between optimally combined imagesand conventional absorption images are in the region where anatomical noise dominates, i.e. at low spatialfrequencies where T ( ω ) (cid:39) T lo ( ω ) (cid:39) T hi ( ω ) (cid:39) T (0) = 1.We used the designer nodule function, which was introduced by Burgess et al., to model the targets. Fortarget radius R and radial coordinate r , s ( r ) ∝ rect( ρ/ × (1 − ρ ) ν , where ρ = r/R . ν determines the shape ofthe function; s is a projected sphere for ν = 0 .
5, which we used to model microcalcifications, and approximatesa tumor for v = 1 .
5. The Fourier transform of s was used as task function in the model. For visualization of the optimal image combination, images with both quantum and anatomical noise weregenerated. Tumors and cysts, i.e. false positives, with profiles according to the designer nodule function wereinserted. All images were filtered with the MTF.Instead of the linear image combination that was used for system characterization, a polynomial-weightedlogarithmic subtraction was introduced to gain better background subtraction than with a constant weight factor,i.e. w = w ( n lo , n hi ) in Eq. (3). The polynomial was trimmed to minimize the variance of a phantom with sevenlevels of glandularity. A second degree polynomial was found to provide a good-enough minimization. It canbe noted that similar nonlinear techniques are employed for material-basis decomposition.
2, 5
More efficientoptimization schemes could be conceived, e.g. maximization of C /S A .Simulated images were low-pass filtered to reduce quantum noise. Equal filters were applied to both binsprior to forming the combined image. More advanced filtering methods have been shown to improve detectabilityconsiderably,
5, 29 and also to influence the optimization because unequal filtering of the bins does not cancel inthe GNEQ. igure 2.
Synthetic images of three 20 mm thick tumors with diameters 20, 30, and 40 mm at an AGD of 1.0 mGy.
Left:
Tumor locations.
Center:
Absorption image with synthesized anatomical noise.
Right:
Optimally combined image,where all tumors are visible and all false positives are excluded.
3. RESULTS3.1. Images for visualization
Synthesized images at 1 mGy of three 20 mm thick tumors with diameters 20, 30, and 40 mm embedded in a50 mm breast are shown in Fig. 2. The left-hand image shows the tumors, which are hidden by anatomical andquantum noise in the center image. In addition, two cysts with equal size and attenuation as the tumors wereadded in the center and in the lower-left corner of the image. The polynomially combined image is shown to theright with all tumors visible and all false positives excluded. Anatomical noise was reduced at the cost of a lowercontrast-to-quantum-noise ratio, and low-pass filtering with a 1.5 mm Gaussian kernel was therefore applied. Itshould be noted that filtering of the absorption image would reduce high-frequency noise, but not exclude thefalse positives. In addition, a narrow display window helped visualize the tumors in the combined image, butwould not improve the absorption image. Smaller and thinner tumors than the ones imaged here were foundhard to visualize because of quantum noise dominance and reduced contrast respectively.
A logarithmic plot of the quantum and anatomical NPS in absorption images and images combined for maximumbackground subtraction is shown in Fig. 3 (Left). All 36 synthesized images were used in the calculations. S A in the absorption image crossed S Q at ω = 1 . − with β = 3 .
0. Image combination with a constant weightfactor reduced the quantum noise slightly, and reduced the anatomical noise more than three orders of magnitude.A polynomial weight factor reduced the anatomical noise another four orders of magnitude at equal quantumnoise. Nevertheless, the following derivation of detectability index was based on the linear combination in Eq. (3)to ensure linearity, although the polynomial combination can be expected to perform better.Figure 3 (Left) shows that S Q calculated by Eq. (6) corresponded well to measurements in the synthesizedimages. S A calculated with a flat glandularity distribution overestimated the noise, as expected, but a betterprediction was offered by the distribution estimated from the glandularity map. In the calculation of detectabilityindex below, the flat distribution was assumed so that predictions can be regarded moderate. The validity ofthe theoretical framework in Section 2.1.3 is further verified by Fig. 3 (Right), which plots the S A - S Q crossingas a function of weight factor calculated with the glandularity-map distribution. The agreement between modeland measurement was good at all weight factors. Figure 3 (Right) also plots α and β as a function of weightfactor. The assumption that β is independent of image combination seems fair; there is only a slight dip at w ∗ /S A , whereas α varies over several orders of magnitude. −1 −9 −8 −7 −5 −4 −3 ω [mm −1 ] N PS [ mm ] measurements model S Q model model S A , measured λ g S A , flat λ g combinedpolynomial w combinedconstant w absorption10 −6 −1.5 −1 −0.5 0 0.5 1 1.5 2−12−9−30 w S A - S Q c r o ss i ng [ mm - ] l og ( α ) o r −β −6 −β log( α ) S A - S Q crossing Figure 3. Left:
Logarithmic plot of quantum and anatomical NPS ( S Q and S A ) in synthesized absorption and com-bined images. Markers indicate measurements, and lines (denoted “model”) show fits to measurements or predictions bythe theoretical framework in Section 2.1.3. The results for two types of image combinations are presented: (1) a linearcombination with a constant weight factor (constant w , as was used for calculating the detectability index), and (2) com-bination with a polynomial weight factor (polynomial w , as was used for the righthand image in Fig. 2). For each imagecombination, calculations of S A are presented for (1) a flat glandularity distribution (flat λ g ) and (2) the distributionmeasured from the synthesized glandularity map (measured λ g ). Right:
The logarithm of the magnitude (log( α )) andthe exponent ( − β ) of S A in a combined image are plotted against the left axis as a function of a constant weight factor( w ). The crossing between anatomical and quantum noise ( S A - S Q ) for the same image combination is plotted againstthe right axis. Markers indicate measurements in the synthesized images and lines show predictions by the theoreticalframework in Section 2.1.3. The detectability index was calculated for the case shown in Fig. 2, and the NPS used for the calculationswas therefore measured in this image alone. The measurement provided a higher NPS than for all 36 imagescombined (as considered in Fig. 3) with an S A - S Q crossing at ω = 2 . − and β = 3 . µ m microcalcification inanatomical and quantum noise, and (3) a tumor on a flat background with only quantum noise. Note that inFig. 4 (Left), all detectability indices are normalized to the absorption image so that the plot shows the benefitof spectral imaging. In addition, positive and negative weight factors are reported as w (cid:48) and w (cid:48)(cid:48) according toEq. (5). Put together, this means that the absorption images of all three cases are located at (1,1).For the tumor in anatomical noise, optimal combination was found to be close, but clearly not identical, toenergy subtraction according to Eq. (8). Energy weighting was suboptimal because higher weighting of the low-energy photons also increased the anatomical noise. For the small microcalcification, however, energy subtractionwas suboptimal whereas energy weighting provided a minute improvement. A similar result was found for thetumor on a uniform background, and the optimal weight factors for these targets almost coincided. The optimalweight factor for energy weighting, and also the optimal energy, thus seems fairly independent of lesion type,which is in accordance with previous studies.
18, 39
Figure 4 (Right) shows the different parts of Eq. (1) at optimal image combination for the 20-mm tumorand the 100- µ m microcalcification in anatomical noise. The tumor contrast-to-noise ratio (GNEQ( ω ) × C ) ofthe combined image was relatively low, and benefit over the absorption image was found only at frequenciesbelow ∼ . − , where, however, the tumor task function ( F ( ω ) × ω ) was located. The microcalcificationtask function, on the other hand, increased with spatial frequency due to the two-dimensional integration, and d e t ec t a b ilit y i nd e x w’’ w´ tumor + S A + S Q tumor + S Q MC + S A + S Q energy subtraction ( w ) *1/S A energy weighting ( w ) /S*C absorption image −2 −1 ω [mm -1 ] F x ω [ mm o r mm x - ] GN E Q x C [ mm - ] tumor:absorption tumor:combinedMC:absorption +combinedGNEQ x C F x ω tumor MC Figure 4. Left:
Detectability as a function of weight factor ( w ) normalized so that a conventional absorption image islocated at (1,1). Three cases are presented: (1) a 20-mm tumor in anatomical and quantum noise (tumor + S A + S Q ),(2) the tumor on a homogenous background with quantum noise only (tumor + S Q ), and (3) a 100- µ m microcalcificationin both types of noise (MC + S A + S Q ). Markers indicate weight factors for the absorption image and for optimal imagecombinations according to Eq. (8). Right:
The task functions ( F × ω ) for a 20-mm tumor and a 100- µ m microcalcification(MC) are plotted against the left axis. The microcalcification task function is multiplied with 10 to make it visible inthe figure. Logarithmic plots of the contrast-to-noise ratio squared (GNEQ × C ) for absorption and optimally combinedimages of these two targets are shown against the right axis. The microcalcification plots virtually coincide. A combinationof F × ω and GNEQ × C illustrates integration to detectability index in Eq. (1). was hence virtually unaffected by the anatomical noise. Energy subtraction was therefore suboptimal and asmall benefit was provided by energy weighting, which increased the contrast-to-noise ratio almost equally forall spatial frequencies.A split energy of 21 keV provided a bin count fraction of approximately ξ lo = ξ hi = 0 .
5, and was used for allof the above cases. A scan of split energies at the optimal weight factor for tumor imaging (-0.57) revealed thatthe optimum for this case was 18.0 keV, which, however, improved the detectability on the order of 1%, and thespectrum can hence be safely split at the center. It can be noted that 18.0 keV yields ξ lo = 0 .
29, which is closeto ξ lo = 0 .
36 as predicted by Eq. (12).Table 1 presents detectability indices for absorption and optimally combined images. The first row concernsthe cases considered above: a 20-mm tumor and a 100- µ m microcalcification in quantum and anatomical noiseimaged with 30 kV and the experimental detector. Detectability of the tumor can be improved 50% by op-timal combination, which seems reasonable when comparing to Fig. 2. Optimal combination for imaging themicrocalcification yielded only a slight improvement on the order of 1%.The subsequent rows show two cases of relatively straightforward system changes that might influence de-tectability: a change in beam quality (higher acceleration voltage), and an optimized detector with electronicnoise, channel-to-channel threshold spread, dead time, and AC leakage reduced by a factor of two. Neither beamquality nor improved detector performance was found to influence the result materially for any of the cases.For combined images, a harder spectrum reduced detectability of both tumors and microcalcifications slightly,whereas optimized detector performance provided a minute improvement. For absorption images, the harderspectrum reduced detectability of microcalcifications, but improved the result for tumors because the reductionin background contrast, i.e. anatomical noise, was larger than the reduction in tumor contrast. able 1. Detectability index ( d (cid:48) ) for optimally combined and conventional absorption images of tumors and microcal-cifications (MCs). Optimization was done with acceleration voltage and the experimental compared to an optimizeddetector. detector acc. voltage Al filter E split d (cid:48) combined / absorption [kV] [mm] [keV] 20 mm tumor 0.1 mm MC1. experimental 30 0.5 21
2. experimental 40 0.5 25
3. optimized 30 0.5 21
4. DISCUSSION
It can be questioned if the level of the anatomical noise in the synthesized images corresponds to real breasttissue. The exponent ( β ) seems to be in the reasonable range; values between three and four have been publishedfor breast tissue.
31, 36
The magnitude ( α ), on the other hand, was difficult to validate. Burgess reported thatthe anatomical noise in digitized mammograms dominated below ∼ − , which is slightly lower than the S A - S Q crossings of the synthesized images and indicates that we used a noise magnitude that was higher thanaverage but not totally arbitrary. A measurement of the NPS in clinical images with the multi-slit system wouldbe required to fully settle this issue.Another uncertainty concerns the agreement between the model and real observers. The ideal observer thatwas used in this study represents the upper limit of observer performance, and other models, such as the non-prewhitening observer has been shown to provide better agreement with human observers in some cases. Inaddition, it has been shown that the noise in mammograms is not completely random, as was assumed here, buthas a deterministic component that the radiologist after training may be able to see through.. Finally, factorsother than anatomical noise, such as variations in thickness or anatomy, might play a big role in the practicalcase. Observer studies in realistic anatomical backgrounds and clinical studies are needed to fully investigatethese effects.There are several potential improvements to the technology, which warrant further study. These includeoptimization of spatial filtering for noise reduction, optimization of incident spectrum,
15, 40 and nonlinearimage combination.
1, 2, 5
The results presented here for unenhanced spectral imaging also have implications to non-energy-resolvedbeam quality optimization. We saw in Table 1 that detectability for microcalcifications in the absorption imagewas reduced by a harder spectrum, which is in line with common optimization that only includes quantumnoise. The detectability for tumors, which are more affected by anatomical noise, however, rose with the harderspectrum. This is further illustrated in Fig. 4 (Left), where excluding the low-energy image altogether ( w = 0)resulted in higher detectability than was found for the absorption image. In addition, comparing Eqs. (6) and(7), we see that S A as opposed to S Q is independent of dose. Increasing the dose therefore does not improve theGNEQ if anatomical noise dominates, contrary to the standard NEQ.
5. CONCLUSIONS
Unenhanced spectral imaging has great potential because it comes as a bonus to the conventional non-energy-resolved absorption image at screening; there is no additional radiation dose to the patient and no need to injectcontrast medium. We have used a previously developed theoretical framework and system model to characterizethe performance of a photon-counting spectral imaging system with two energy bins for unenhanced spectralmammography. The model calculated a task-dependent ideal-observer detectability index via the generalizedNEQ (GNEQ), which includes quantum and anatomical noise. This figure of merit was used to find an optimalcombination of the energy-resolved images, to compare optimally combined images with absorption images, andto investigate the effect of system optimization. In addition, synthesized images with quantum and anatomicalnoise were generated with the system model and used to verify the theoretical framework and to illustrate thetechnique.ptimal combination for imaging of large unenhanced tumors in the presence of anatomical noise provideda 50% improvement in detectability compared to absorption imaging. The image combination correspondedclosely, but not exactly, to minimization of the anatomical noise, i.e. the energy subtraction scheme. Higherweighting of the more-information-dense photons, referred to as energy weighting, deteriorated detectability forthis task. For small microcalcifications or tumors on uniform backgrounds, however, the situation was reversed;energy subtraction was clearly suboptimal whereas energy weighting provided a small benefit on the order of 1%.The performance was largely independent of beam quality, detector energy resolution, and bin count fraction,which simplifies optimization in the practical case. Several potential improvements to the technique warrantfurther study, including spatial filtering and nonlinear image combination.Optimal image combination and the benefit of spectral imaging depended to a large extent on the anatomicalnoise and imaging task. This may have implications also on optimization of non-energy resolved imaging, whereit is common practice to consider quantum noise alone.
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