Energy resolution of a photon-counting silicon strip detector
Erik Fredenberg, Mats Lundqvist, Bjorn Cederstrom, Magnus Aslund, Mats Danielsson
This is the accepted manuscript of:
Fredenberg, E., Lundqvist, M., Cederström, B., Åslund, M. and Danielsson, M., 2010. Energy resolution of a photon-counting silicon strip detector. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 613(1), pp.156-162.
The published version of the manuscript is available at: https://doi.org/10.1016/j.nima.2009.10.152 An accepted manuscript is the manuscript of an article that has been accepted for publication and which typically includes author-incorporated changes suggested during submission, peer review, and editor-author communications. They do not include other publisher value-added contributions such as copy-editing, formatting, technical enhancements and (if relevant) pagination. All publications by Erik Fredenberg: https://scholar.google.com/citations?hl=en&user=5tUe2P0AAAAJ © 2009. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ nergy resolution of a photon-counting silicon strip detector
Erik Fredenberg ∗ ,a , Mats Lundqvist b , Bj(cid:246)rn Cederstr(cid:246)m a , Magnus ¯slund b , Mats Danielsson a a Department of Physics, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden b Sectra Mamea AB, Smidesv(cid:228)gen 5, SE-171 41 Solna, Sweden
AbstractA photon-counting silicon strip detector with two energy thresholds was investigated for spectral x-ray imaging in amammography system. Preliminary studies already indicate clinical bene(cid:28)t of the detector, and the purpose of thepresent study is optimization with respect to energy resolution. Factors relevant for the energy response were measured,simulated, or gathered from previous studies, and used as input parameters to a cascaded detector model. Thresholdscans over several x-ray spectra were used to calibrate threshold levels to energy, and to validate the model. The energyresolution of the detector assembly was assessed to range over ∆ E/E = 0 . to 0.26 in the mammography region.Electronic noise dominated the peak broadening, followed by charge sharing between adjacent detector strips, and achannel-to-channel threshold spread. The energy resolution may be improved substantially if these e(cid:27)ects are reducedto a minimum. Anti-coincidence logic mitigated double counting from charge sharing, but erased the energy resolutionof all detected events, and optimization of the logic is desirable. Pile-up was found to be of minor importance at typicalmammography rates.Key words: Spectral x-ray imaging, Mammography, Silicon strip detector, Photon counting, Energy resolution,Cascaded detector model1. IntroductionX-ray mammography is an e(cid:27)ective and wide-spreadmethod to diagnose breast cancer, but it is also techni-cally demanding [1]. Two major challenges that face themodality are the small signal di(cid:27)erences between lesionsand breast tissue, and the lumpy backgrounds that arecaused by superposition of glandular structures.Spectral imaging is a method to extract informationabout the object constituents by the material speci(cid:28)c en-ergy dependence of x-ray attenuation [2, 3]. In mammog-raphy, there are at least two potential bene(cid:28)ts of this ap-proach compared to non-energy resolved imaging. (1) Thesignal-to-quantum-noise ratio may be optimized with re-spect to its energy dependence; photons at energies withlarger agent-to-background contrast can be assigned a greaterweight [4, 5]. (2) The signal-to-background-noise ratio canbe optimized by minimization of the background cluttercontrast. A weighted subtraction of two images acquiredat di(cid:27)erent mean energies cancels the contrast betweenany two materials (adipose and glandular tissue) whereasall other materials (lesions) to some degree remain visible.The contrast in the subtracted image is greatly improvedif the lesion is enhanced by a contrast agent with an ab-sorption edge in the energy interval, which provides a largedi(cid:27)erence in attenuation [6, 7, 8, 9, 10]. ∗ One way of obtaining spectral information is to usetwo or more input spectra. For imaging with clinical x-ray sources, this most often translates into several expo-sures with di(cid:27)erent beam qualities (di(cid:27)erent accelerationvoltages, (cid:28)ltering, and anode materials) [6, 7, 8]. Resultsof the dual-spectra approach are promising, but the ex-amination may be lengthy with increased risk of motionblur and discomfort for the patient. This may be solvedby instead using an energy sensitive detector, which hasbeen pursued with sandwich detectors [11, 12]. For bothof the above approaches, however, the e(cid:27)ectiveness maybe impaired due to overlap of the spectra, and a limited(cid:29)exibility in choice of spectra and energy levels. In recentyears, photon-counting silicon detectors with high intrinsicenergy resolution and, in principle, an unlimited numberof energy levels (electronic spectrum-splitting) have beenintroduced as another option [9, 10, 13, 14].An objective of the EU-funded HighReX project is toinvestigate the bene(cid:28)ts of spectral imaging in mammogra-phy [15]. The systems used in the HighReX project arebased on the Sectra MicroDose Mammography (MDM)system, which is a scanning multi-slit full-(cid:28)eld digitalmammography system with a photon-counting silicon stripdetector [16, 17, 18]. An advantage of this geometry in aspectral imaging context is e(cid:30)cient intrinsic scatter rejec-tion [19, 20, 21].We have investigated the energy response of a proto- Sectra Mamea AB, Solna, SwedenPreprint submitted to Nuclear Instruments and Methods A September 22, 2009 ype detector for the HighReX project on a system level.The major factors that a(cid:27)ect the energy response havebeen identi(cid:28)ed, and used as input to a cascaded detectormodel. The purpose of the model is detector optimizationwith respect to energy resolution. Optimal energy reso-lution will improve performance when the detector is em-ployed for spectral imaging within the HighReX project,in particular when using a K-edge contrast agent such asiodine. Knowledge of the energy resolution will also be es-sential for simulating contrast- and noncontrast-enhancedspectral imaging with the detector.2. Materials and Methods2.1. Description of the system and detectorBecause the systems used in the HighReX project aremodi(cid:28)cations of the MDM system, and because an MDMsystem was used for testing the detector, we abridge ourdiscussion to consider only the MDM system. It comprisesa tungsten target x-ray tube with aluminum (cid:28)ltration, apre-collimator, and an image receptor, all mounted on acommon arm (Fig. 1). The image receptor consists of sev-eral modules of silicon strip detectors with correspondingcollimator slits in the pre-breast collimator. To acquire animage, the arm is rotated around the center of the source sothat the detector modules and pre-collimator are scannedacross the object. In Fig. 1 and henceforth, x refers to thedetector strip direction and y to the scan direction.The detector modules were fabricated on 500 µ m thickn-type silicon wafers with p-doped strips at a pitch of50 µ m. Each strip thus forms a separate PIN-diode, whichis depleted by a 150 V bias voltage. Aluminum strands areDC-coupled to the strips, and wire bonded to the read-out electronics. To obtain high quantum e(cid:30)ciency despitethe relatively low atomic number of silicon, the modulesare arranged edge-on to the x-ray beam [13]. Interactionsin the guard ring are avoided by irradiating the detectormodules at a slight angle [17], which yields an e(cid:27)ectivethickness of approximately 4 mm. Scatter shields betweenthe modules block detector-to-detector scatter. The sili-con detector modules are in many ways similar to the onesthat are used in the MDM system [18, 17], whereas theread-out electronics to a larger degree are di(cid:27)erent fromprevious versions [17, 22].Each detector strip is connected to a preampli(cid:28)er andshaper, which are fast enough to allow single photon count-ing. The pulse height depends on the released charge inthe silicon, and thus on the energy of the impinging pho-ton. An average of 273 electron-hole pairs are createdfor each keV photon energy, whereas the equivalent noisecharge is in the order of a few hundred electrons, and alow-energy threshold at a few keV in a discriminator fol-lowing the shaper ensures that the electronic noise doesnot a(cid:27)ect the number of detected counts. All remainingpulses are sorted into two energy bins by an additionalhigh-energy threshold, and registered by two counters. A preampli(cid:28)er with discriminator and counters are referredto as a channel, and all channels are implemented in anapplication speci(cid:28)c integrated circuit (ASIC). The gain ofthe preampli(cid:28)er varies slightly between the channels, andto compensate for this, the threshold levels of individualchannels were trimmed in 3 bits towards either the elec-tronic noise (cid:29)oor or some discontinuity in the input spec-trum. On-chip current-based 8-bit digital-to-analog con-verters de(cid:28)ne the global high- and low-energy thresholdlevels.Charge sharing between adjacent detector strips mayincrease image noise at low spatial frequencies and degradethe spatial resolution if the charge is large enough to beregistered by both channels (double counting). The energyresolution is also a(cid:27)ected because all charge is not collectedinto a single pulse. The present ASIC implements anti-coincidence (AC) logic, which distinguishes charge-sharedevents by a simultaneous detection of pulses that reachover the low-energy threshold in adjacent channels. The(cid:28)rst detected pulse, which is generally the largest one, in-crements the high-energy bin, whereas the slower pulse isrejected. Double counting is thus avoided, which improvesspatial resolution and noise, but all energy information islost. The AC logic cannot be turned o(cid:27) in the presentASIC, but is disabled by masking every other channel.This procedure reduces the e(cid:30)ciency and is an option forphysical evaluation only, not for clinical imaging.2.2. Modeling the detectorThe energy response function of the detector was mod-eled using the MATLAB software package as a semi-empirical cascade of several detector e(cid:27)ects. These weregrouped into 8 categories, which are outlined in the bot-tom part of Fig. 1 and described in detail below. Some ofthe steps in the cascade require measured input parame-ters, and the procedures to (cid:28)nd these are described in thenext section.(1) Quantum e(cid:30)ciency was calculated with publishedlinear absorption coe(cid:30)cients [23]. Charge collection on thealuminum strands was assumed ideal so that the full en-ergy deposition of photo-electric events was detected. Thelow-energy threshold was assumed to reject the detectionof Compton scattered events so that scattering only con-tributed to (cid:28)ltering of the beam. Secondary photo-electricinteractions of scattered photons in adjacent detector mod-ules was eliminated by scatter shields, and secondary inter-actions in the detector module of the (cid:28)rst interaction wasignored because of the large angular spread of Comptonscattering. Rayleigh scattering was excluded altogetherbecause of a relatively small cross section at hard x-rayenergies. Fluorescence is generally a minor problem insilicon detectors at hard x-ray energies [24], and it wasignored in the model. Elaborate motivations for ignoring The MathWorks Inc., Natick, Massachusetts zx breastSi-strip detector linespre-collimatorcompression platebreast support _ + HV rejectionhighlowASIC
CS high thresholdACPUPU low thresholdlow threshold high thresholdscanpre-collimatorbreastx-ray tube detectorENEN i n t r i n s i c E r es . Q E i n t r i n s i c E r es . Q E a bc x-ray beamscan Figure 1: a: In the MDM system, the arm is rotated around thecenter of the source to acquire an image. b: Closeup of the de-tector assembly and the electronics. c: Block diagram of the cas-caded detector model for two adjacent channels. The model includes:(1) Quantum e(cid:30)ciency (QE). (2) Intrinsic energy resolution of sili-con. (3) Charge sharing (CS). (4) Electronic noise (EN). (5) Pile-upin the shaper (PU). (6) Nonlinearity of the shaper and thresholds,and bit resolution. (7) Channel-to-channel spread of the thresholds.(8) Anti-coincidence logic (AC) with leakage and chance coincidence(CC). scattering and (cid:29)uorescence can be found at the end of thissection.(2) With a relatively large number of released chargepairs at each photon conversion, silicon has good intrin-sic energy resolution. The peak was modeled as normaldistributed with standard deviation (in units of eV) σ i = √ Eη(cid:178) , where (cid:178) = 3 . eV is the mean energy needed tocreate an electron-hole pair in silicon, and η = 0 . isthe Fano factor for silicon [24]. The full-width-at-half-maximum (FWHM) of the peak is 0.2(cid:21)0.3 keV in the in-terval 20(cid:21)40 keV.(3) Charge sharing results in loss of detected chargeand a corresponding spread towards lower pulse heights inthe channel of interaction, and a reversed probability fordetection of charge from interactions in adjacent strips.We used the probability distribution from a previously de-veloped computer model to predict the e(cid:27)ects of chargesharing [18]. With a 50 µ m strip pitch, charge sharinghas a relatively large impact on energy resolution withpeak widths ranging from 1.8 to 1.4 keV FWHM in the20(cid:21)40 keV interval and with heavy tails towards lower en- ergies.(4) Electronic noise generally has a negligible e(cid:27)ect onthe number of detected events in a photon counting detec-tor, but the energy resolution is a(cid:27)ected. The equivalentnoise charge in a similar ASIC without silicon detector at-tached has been found to be σ add = 200 electrons r.m.s. at[17], but we can expect a higher level in this study becauseof the added detector capacitance and leakage current.(5) Pile-up occurs mainly in the shaper. For typicalmammography rates of R < kHz, and shaper deadtimes τ s < ns, the product Rτ s (cid:191) , and pile-up isa relatively small e(cid:27)ect. In that case, there is no needto distinguish between paralyzable and non-paralyzableshaper behavior [24]. Ignoring multiple pile-up, the de-tected count-rate is then r pu ≈ R − R τ s , (1)where R is the true rate without pile-up. The distribu-tion of two piled-up pulses with partial overlap was sim-pli(cid:28)ed into a rect function extending from min( E , E ) to sum( E , E ) , where E and E are the energies of the im-pinging photons.(6) The combined energy response of shaper and dis-criminator is approximately linear at low energies and thensaturates. The nonlinearity at higher energies was foundempirically to be well described by an inverse power func-tion so that the threshold level ( T ) as a function of energy( E ) is T ( E ) = (cid:40) C E + C for E < C C E − C + C for E ≥ C , (2)where the coe(cid:30)cients C (cid:21) C are free parameters. A re-duction to only four parameters is achieved by requiring T and d T / d E to be continuous.(7) Small deviations in the threshold levels of individ-ual channels remained after trimming because of a limitedbit depth and slightly di(cid:27)erent energy dependence of thechannels. This resulted in an energy dependent channel-to-channel spread, which was modeled as normal distrib-uted and increasing away from the trimming point. Thespread in a single module of a similar detector has beenmeasured to approximately 0.9 keV FWHM [17].(8) Chance coincidence in the AC logic occurs at a rate r ch = R [1 − exp( − Rτ ac )] ≈ R τ ac , where τ ac is the ACtime window and the approximation is for Rτ ac (cid:191) [24].The count-rates in the two bins are then r lo = R lo − r ch ≈ R lo − R τ ac , and r hi = R hi + r ch (1 + ξ cc ) ≈ (3) ≈ R hi + 2 R τ ac (1 + ξ cc ) , where ξ cc is the leakage of the logic. Combining Eqs. (1)and (3), the total count-rate is r sum ≈ R − R [ τ s + 2 τ ac (1 − ξ cc )] , and we note that the impact of chance coincidence istwice that of pile-up if there is no leakage. A preliminary3lectronics test revealed, however, that if two simultaneouspulses are similar in size, the AC logic cannot make a cor-rect decision and both pulses are directed to the respectivehigh-energy bins. This is a relatively unlikely situation forcharge-shared events because it requires interaction closeto the border between two strips. It is, however, morelikely in the case of chance coincidence because the ener-gies are higher, which results in similar-sized pulses dueto the nonlinear shaper output. We can thus expect twoleakage coe(cid:30)cients of the AC logic, ξ cc > ξ cs , for chancecoincidence and charge sharing, which have to be deter-mined separately.Published mammography spectra [25] were used as in-put to the model for comparison to measurements. Theenergy resolution of the high-energy threshold was evalu-ated as ∆ E/E , where ∆ E is the FWHM of the predictedresponse to a delta peak.To verify the assumption that Compton scattered pho-tons pose a minor problem, a simple geometrical modelwas set up that traced a photon through the center of adetector module. The Klein-Nishina cross section was usedto calculate a probability function for scattering angle anddeposited energy [26]. Accordingly, energy deposition in-creases with incident photon energy, and for the hardestspectrum considered in this study (40 kV and 3 mm alu-minum (cid:28)ltration), the mean deposited energy was found tobe 1.5 keV with a maximum (Compton edge) of 5.4 keV.It is hence safe to assume that scattered events are re-jected in the detector strip of the primary interaction fortypical low-energy threshold levels. The detected scatter-to-primary ratio for (cid:28)rst order secondary interactions ofscattered photons was 2.1%, with a maximum of 2.8% for40 keV photons. We ignored this amount, which is similarto what may detected from scattering in an object; 2.1%was measured for a 50 mm breast at 38 kV and 0.5 mmaluminum (cid:28)ltration in a similar geometry [21].It cannot be excluded that (cid:29)uorescent photons escapethe relatively narrow strips of the detector. Therefore,the size of the escape peak was calculated according toa previous, experimentally veri(cid:28)ed, study [27]. In sum-mary, 92% of the absorbed photons eject a K-electron, theK-(cid:29)uorescent yield of silicon is 4.3%, and the energy ofthe (cid:29)uorescent photon is 1.74 keV. A similar geometry asfor calculating scattering was used. We found that therelative intensity of the escape peak was 0.3% at 15 keVand decreasing with energy because of deeper interactionsin the silicon. 15 keV is in the lowermost region of typi-cal mammography spectra, and (cid:29)uorescence can hence becon(cid:28)dently ignored.2.3. Measurements on the detectorA complete detector assembly with a total of 89856channels, was mounted on a standard MDM system. Thelow-energy threshold levels were trimmed towards the elec-tronic noise to minimize the variance in count-rates be-tween channels. The high-energy thresholds were trimmedagainst the steep derivative at the K absorption edge (33.2 keV)of an iodine (cid:28)ltered 40 kVp spectrum. The air kerma wasmonitored with an ion chamber, and, knowing the expo-sure time, converted to (cid:29)ux using published spectra [25],attenuation and energy absorption coe(cid:30)cients [23].Integral pulse height spectra were acquired by scanningthe high- and low-energy thresholds of 144 channels overseveral incident energy spectra. The tungsten spectrumwas (cid:28)ltered with a total of 3 mm aluminum to make itrelatively distinct. In the following, the words (cid:16)thresholdscan(cid:17) and (cid:16)integral pulse height spectrum(cid:17) are used in-terchangeably when relating to this procedure. The meanpulse height spectra between all channels were used to cal-ibrate the global threshold level to energy and estimatingthe electronic noise by (cid:28)tting the coe(cid:30)cients of Eq. (2)and σ add for each energy bin separately, keeping all othermodel parameters (cid:28)xed except the amplitude. A secondpurpose of the (cid:28)t was to visually validate the model cor-respondence to data.To quantify the spread in threshold levels, the pulseheight spectrum of each individual channel was (cid:28)tted tothe mean using amplitude and a translation in thresholdlevel as free parameters. The translation represents theresidual from trimming, and was assumed to increase lin-early as a function of mean threshold level with a min-imum at the trimming point. From the residuals of theindividual channels, the channel standard deviation couldbe calculated, which hence also increases linearly from thetrimming point. Standard deviations calculated from sev-eral spectra acquired with di(cid:27)erent kVp were combinedwith weights provided by statistical errors. When measur-ing on the low-energy threshold, the high-energy thresholdwas set to its maximum value so that it would not in(cid:29)uencethe measurement, and the sum of the high- and low-energybins was recorded. When measuring on the high-energythreshold, the low-energy threshold was set to approxi-mately half the acceleration voltage to reject virtually allcharge shared events but still detect most of the spectrum.Leakage of the AC logic associated with charge sharinga(cid:27)ects image noise, and can be measured with the noisepower spectrum (NPS). If a fraction χ photons are doublecounted in each channel, three uncorrelated processes canbe identi(cid:28)ed, namely, single counting of the photon witha probability (1 − χ ) , or double counting in the right orleft adjacent channel with probabilities χ/ each. In ourcase, the latter two are equivalent, and for a large numberof photons, the autocovariance in the detector directionof the image is, K ( x ) = (1 − χ ) K s ( x ) + χK d ( x ) , where K s and K d are the autocovariance functions for single anddouble counting. For single counting, the image functionis a Dirac function ( δ ), and so is the autocovariance [28],i.e. K s ( x ) = σ δ ( x ) , where σ is the variance. If thequanta are poisson distributed, σ = G N , where N is theexpectation value of the true number of counts without type 23344 and electrometer Unidose E, PTW, Freiburg, Ger-many G is the large area gain of the sys-tem. In the case of double counting, the image functionis instead represented by two Dirac functions, separatedby the strip pitch ( p ), and the autocovariance is hence K d = G N [2 δ ( x ) + δ ( x − p ) + δ ( x + p )] . The expectationvalue of number of detected counts in the channel, includ-ing double counting, is n = GN (1 + χ ) . Combining theabove, and since the NPS of a stationary system is theFourier transform of the autocovariance [28], S ( u ) n = (1 − χ ) (cid:99) K s ( x ) + χ (cid:99) K d ( x ) GN (1 + χ ) == G χ [1 + 2 cos(2 πu/p )]1 + χ , (4)where S is the NPS, u is the spatial frequency in the x -direction, and Fourier transforms are denoted by the cir-cum(cid:29)ex. In particular, S (0) = G (1 + 3 χ ) / (1 + χ ) whennormalized with the mean channel signal, which has beenderived previously for G = 1 [17].The NPS was measured and calculated in a way simi-lar to standardized methodology as applied to the MDMgeometry [16]. 1000 100 ×
100 pixel regions of interest (ROI’s)were acquired from a (cid:29)at-(cid:28)eld image of 0.5 mm aluminumand 40 mm polymethyl methacrylate at 28 kVp. The NPSwas then calculated as the mean of the squared fast Fouriertransform of the di(cid:27)erence in image signal from the meanin each ROI. χ was determined from Eq. 4 with the meanROI signal as n . In case the NPS is measured in the high-energy bin and chance coincidence is negligible, χ = ξ cs .The (cid:29)ux in the MDM setup was limited due to tech-nical constraints, and to measure the detector linearity, asimilar setup but with a single 128-channel detector mod-ule was used. A tungsten target x-ray tube at 33 kVp was(cid:28)ltered with 0.5 mm aluminum, and the (cid:29)ux was controlledwith the anode current and an adjustable slit in front ofthe detector. Levels of the low-energy threshold in indi-vidual channels were again trimmed towards the electronicnoise, and the global threshold level was set relatively highto reject all noise and most charge-shared events, whereasthe global high-energy threshold was set to the maximumvalue to detect AC events exclusively. The mean of allchannels as a function of (cid:29)ux was recorded for both en-ergy bins, with and without AC. In the former case, non-linearity is introduced by pile-up and chance coincidence,but without AC, pile-up only contributes. τ pu , τ ac , and ξ cc were found from Eqs. (1) and (3).Error estimates of the measurements described abovewere calculated from the scatter of several data pointsaround the (cid:28)tted curve assuming a normal distribution, aspropagated statistical errors, or as the maximum of thesetwo in case both were available [29]. The estimates are inall cases presented as ± standard deviation. Fitting tomeasured data was done in a least-squares sense, weightedwith propagated statistical errors where applicable. Philips PW2274/20 with high tension generator PW1830 R [kHz] r [ k H z ] r pu r sum r lo r hi Figure 2: Linearity of the detector as a function of true count-rate( R ). r pu is the count-rate without anti-coincidence logic. r lo , r hi ,and r sum are count-rates with anti-coincidence logic in the high- andlow-energy bins, and the sum of the two. Measurement points areindicated by crosses, and (cid:28)ts to these by Eqs. (1) and (3) are shownwith lines.
3. Results and DiscussionFigure 2 shows the linearity measurement, with (cid:28)ts toEqs. (1) and (3) for r pu , r lo , r hi , and r sum = r lo + r hi . R was extrapolated from the approximately linear curvethrough points at low count-rates. The shaper dead timewas found to be τ s = 189 ± ns, and the AC time windowand chance coincidence leakage were τ ac = 138 ± . nsand ξ cc = 0 . ± . , respectively. In all cases, the errorestimates from the scatter of the data correspond closelyto what is expected from the counting statistics, indicatingthat the errors are primarily random. r pu and r sum almostcoincide, which illustrates that the high leakage results inonly a small loss of counts to chance coincidence.The NPS divided by the mean ROI signal is shown inFig. 3 for both energy bins. The (cid:29)ux was 33 kHz, whichis low enough for pile-up and chance coincidence to benegligible (Fig. 2). Double counting in the high-energybin results in a bent NPS in the detector direction, andby (cid:28)tting to Eq. 4, the leakage of the AC logic was foundto be ξ cs = 0 . ± . . The error estimates from thescatter of the data points are small and correspond closelyto expectations from statistics, which suggests that Eq. 4describes the data well. A (cid:29)at NPS indicates uncorrelatedpixels, which, as expected, is the case for the low-energybin in the detector direction and for both bins in the scandirection.Figure 4 shows an example of a threshold scan of thehigh-energy threshold over a 25-kVp spectrum. The scanis shown as a function of global threshold level, which isrelated to photon energy through Eq. (2), and the crosssection at T hi = 50 is shown as a histogram to the right.5 u [mm −1 ] N PS / n detector hi scan hi detector lo scan lo Figure 3: NPS as a function of spatial frequency ( u ) divided by themean ROI signal ( S/n ) of the high- and low-energy bins in the slitand scan directions. Fitting to Eq. (4) is shown with a solid line. r [ a . u . ] T hi [a.u.] 0.5 203040 c h a nn e l s r at T hi = 500.3 0.7 Figure 4: Example of a threshold scan for the high-energy thresholdand a 25-kVp spectrum. The thin lines are individual channels, andthe mean is indicated in the center. The cross section at T hi = 50 isshown to the right. There is a vertical spread in amplitudes, and a horizontalspread in threshold levels. The former can be compensatedfor in an image by (cid:29)at-(cid:28)eld calibration, but the thresholdspread inevitable reduces energy resolution. Also shownin Fig. 4 is the mean of all channels, which was used asexpectation value when estimating the threshold spreadand for (cid:28)tting the model.Scans of the low-energy threshold are shown in Fig. 5for the high-energy bin ( r hi ) and for both bins summed( r sum ). Measurement points are approximately twice asdense as indicated. The high-energy threshold was at itsmaximum value so that r hi contains AC events exclusively,and the increase towards lower threshold levels is due toincreased detection of charge-shared events and increasedchance coincidence. Fitting of the model to scans of fourspectra in the range 20(cid:21)40 kVp yielded estimates of theelectronic noise and the coe(cid:30)cients ( C ) of Eq. (2). Ahigh (cid:29)ux of ≤ kHz (decreasing with kVp) was used r [ a . u . ] E [keV] T l o [ a . u . ] r hi r sum scan and fit:20, 30, 40 kVpthreshold level Figure 5: Scans of the low-energy threshold over 20(cid:21)40 kVp spectrawith the high-energy threshold at its maximum value. Two groupsof curves are shown corresponding to counts in the low- and high-energy bins summed ( r sum ), and in the high-energy bin only ( r hi ).The low-energy threshold level as a function of energy is shown as adashed line. to cause some amount of pile-up and chance coincidenceto challenge the model. The (cid:28)t is shown in Fig. 5 forthree of the spectra, and the global threshold level as afunction of energy is superimposed on the (cid:28)gure. Theelectronic noise was found to be σ add = 4 . keV FWHM(505 electrons r.m.s.). Using the relationship of Eq. (2),the threshold spread was translated from threshold levelsinto σ lo = 2 . ± . to . ± . keV FWHM in the in-terval 1(cid:21)20 keV, which is where the low-energy thresholdis supposed to operate. Error estimates were propagatedfrom the statistical uncertainty of the threshold spread. Inunits of threshold levels, the spread was found to be fairlyconstant with global threshold level, and the increase to-wards higher energies is mainly due to the nonlinearity ofEq. (2).Scans of the high-energy threshold are shown in Fig. 6for r hi . r sum can in this case be assumed constant andis therefore not shown. All measurement points are indi-cated. The low-energy thresholds were set to 10.7, 12.3,18.9, 20.6, and 22.6 keV for the (cid:28)ve 20(cid:21)40 kVp spectra,with levels and spread determined by the low-energy thresh-old scan. A constant background is evident for scans above30 kVp, which is due mainly to chance coincidence andnot charge sharing because the low-energy thresholds wererelatively high. The electronic noise and the coe(cid:30)cientsof Eq. 2 were (cid:28)tted, with the resulting model predictionand relationship between threshold and energy shown inFig. 6. σ add was 2.9 keV FWHM (339 electrons r.m.s.).The spread of the thresholds was assumed to be σ hi = 0 at 33.2 keV. Below the trimming point, the spread found amaximum of σ hi = 1 . ± . keV FWHM at 20 keV, and it6 r [ a . u . ] E [keV]
15 20 25 30 35 4075150 T h i [ a . u . ] r hi scan and fit:20, 25, 30, 35, 40 kVpthreshold level Figure 6: Scans of the high-energy threshold measured in the high-energy bin ( r hi ) over 20(cid:21)40 kVp spectra with low-energy thresholds atapproximately half the acceleration voltage. The high-energy thresh-old level as a function of energy is shown as a dashed line. increased rapidly and monotonically above the trimmingpoint, reaching σ hi = 1 . ± . keV at 40 keV. Again, thespread towards higher energies is strongly enhanced by thenonlinearity of T ( E ) .As a general observation it can be said that the modelagrees reasonably well with measured data. Statisticalerrors of the scans in Figs. 5 and 6 are small, and it isclear that systematic errors, caused by assumptions in themodel and errors in all input parameters, dominate for the(cid:28)t. Valid error estimates on C and σ add are thus hard toobtain.Figure 7 illustrates the energy response of the high-energy threshold to delta peaks at low count-rates (nopile-up or chance coincidence). The experimental detec-tor was evaluated with the low-energy threshold at 7 keV.Response functions at 20, 30, and 40 keV are plotted with ∆ E = 5 . , 4.6, and 4.9 keV. As expected, peak widths in-crease away from the trimming point because of increasedthreshold spread, whereas charge sharing spreads all peakstowards lower energies. Simulation points are plotted insteps of the maximum bit depth of the threshold, and theincreasing spread at higher energies re(cid:29)ects the contribu-tion to peak width caused by the nonlinear shaper output.In fact, one step in threshold level corresponds to 1.4 keVat 40 keV, but only 0.13 keV at 20 keV. The peak heightscorrespond to the relative amount of energy resolved in-formation, and the decline towards higher energies is dueto decreased quantum e(cid:30)ciency and increased detection ofcharge-shared events that results in a constant backgroundin the high-energy bin. ∆ E/E is plotted in Fig. 7 as a function of energy.For the 20, 30, and 40 keV peaks, ∆ E/E = 0 . ,
15 20 25 30 35 40 45 500.51 r [ a . u . ] E [keV]
15 20 25 30 35 40 45 500.10.20.4 ∆ E / E keV keV keV keV keV keV keV keV keV energy resol u tion:experiment a l optimized Figure 7: Energy response on monochromatic delta peaks. Theplotted peaks are for the experimental detector. Energy resolution( ∆ E/E ) is shown for the experimental detector, and for an improveddetector with high AC e(cid:30)ciency and low threshold spread and elec-tronic noise. random errors was the threshold spread, propagated rela-tive errors on the energy resolution were less than 4.5%,and it is hence likely that systematic errors dominate. Insummary, the largest contributors to the peak broaden-ing are the electronic noise (2.9 keV FWHM), followed bythreshold spread (1.2(cid:21)1.9 keV FWHM), and charge sharing(1.8(cid:21)1.4 keV FWHM). Note that our particular choice ofFWHM as ∆ E measure slightly underestimates the contri-bution by charge sharing because peak tails are neglected.The energy resolution of the present detector is lowerthan some previously reported results on similar siliconstrip detectors [14], which is, however, due mainly to thefact that we have considered a full system in this study. Forinstance, the small strip pitch needed for high-resolutionmammography causes relatively large amounts of electronicnoise and charge sharing, the double-threshold con(cid:28)gura-tion adds complexity and electronic noise, and channel-to-channel threshold spread reduces the energy resolutionwhen more than one channel is considered. The predictedenergy resolution of an improved detector with half thethreshold spread and electronic noise, and with a 3.5-keVlow-energy threshold and no leakage of the AC logic is alsoshown in Fig. 7 for comparison. Improvements of 1.9(cid:21)1.7times are seen at 20(cid:21)40 keV. Note that this is still substan-tially worse than the intrinsic energy resolution of silicon,which is in the order of 0.01 in the interval.One aspect of energy resolution that is not capturedby the ∆ E/E measure is the constant background of ACevents that are put in the high-energy bin. For the ex-perimental detector in Fig. 7, charge sharing results in abackground intensity of 24(cid:21)63% for the three peaks. Low-ering the low-energy threshold, as for the near-ideal case7n Fig. 7, results in more e(cid:30)cient AC and a narrower peak,but also more background without energy information. Athigh intensities, pile-up and chance coincidence would alsocontribute to a more or less constant background.4. ConclusionsMeasurements, simulations, and published data wereused as input parameters to a cascaded detector model,which was validated by comparison to threshold scans overseveral input spectra. Using the model, the energy re-sponse of the detector assembly could be assessed on asystem level without monochromatic radiation, and theimpact of various parameters could be estimated.The energy resolution was found to be ∆ E/E = 0 . (cid:21)0.26 in the relevant energy range. The major factors con-tributing to the width of the response function were foundto be electronic noise, followed by charge sharing, and achannel-to-channel threshold spread that was boosted bya nonlinear shaper output. Additionally, a relatively largeconstant background of charge-shared photons detected bythe AC logic was added to the high-energy bin. The shaperdead time and AC time window were both less than 200 ns,and pile-up and chance coincidence were found to be ofminor importance at mammography count-rates. Fluores-cence and scattering e(cid:27)ects in the silicon were estimatedto be negligible.Relatively large improvements of the energy resolutionare within reach. Minimization of the electronic noise ishighly important to reduce the peak broadening. Thetrimming point should be chosen close to the point of op-eration of the threshold, and variations between channelsshould be kept at a minimum in order to minimize thethreshold spread. This is particularly important for thehigh-energy threshold, which is meant to operate in high-intensity parts of the spectrum. An improvement in shaperand discriminator linearity at higher energies is also desir-able to reduce the e(cid:27)ects of threshold spread and limitedbit depth. Finally, the AC scheme can be improved bykeeping the energy resolution of detected events, or byrecording them in a separate bin.Preliminary studies already indicate clinical bene(cid:28)t forspectral imaging with the described detector [30]. Theinformation and model provided here will be crucial forthe ongoing system optimization.5. AcknowledgmentsThe authors wish to thank Magnus Hemmendor(cid:27) andAlexander Chuntonov at Sectra Mamea AB for discussionsand practical help with measurements, and Bj(cid:246)rn Svens-son, also at Sectra, for discussions on charge sharing. Thiswork was funded in part by the European Union throughthe HighReX project. References ectors for medical imaging. Nucl. Instr. and Meth. A, 549:199(cid:21)204, 2005.[20] M. Lundqvist, B. Cederstr(cid:246)m, V. Chmill, M. Danielsson, andB. Hasegawa. Evaluation of a photon-counting X-ray imagingsystem. IEEE Trans. Nucl. Science, 48(4):1530(cid:21)1536, 2001.[21] M. ¯slund, B. Cederstr(cid:246)m, M. Lundqvist, and M. Daniels-son. Scatter rejection in multi-slit digital mammography. Med.Phys., 33:933(cid:21)940, 2006.[22] E. Beuville, B. Cederstr(cid:246)m, M. Danielsson, L. Luo, D. Nygren,E. Oltman, and J. Vestlund. An application speci(cid:28)c integratedcircuit and data acquisition system for digital x-ray imaging.Nucl. Instr. and Meth. A, 406:337(cid:21)342, 1998.[23] M.J. Berger, J.H. Hubbell, S.M. Seltzer, J.S., Coursey, andD.S. Zucker. XCOM: Photon Cross Section Database. online:http://physics.nist.gov/xcom. National Institute of Standardsand Technology, Gaithersburg, MD, 2005.[24] G. F. Knoll. Radiation Detection and Measurement. John Wiley& Sons, 3rd edition, 2000.[25] J.M. Boone, T.R. Fewell, and R.J. Jennings. Molybdenum,rhodium, and tungsten anode spectral models using interpolat-ing polynomials with application to mammography. Med Phys,24(12):1863(cid:21)74, 1997.[26] B. Grosswendt. Basic aspects of photon transport through mat-ter with respect to track structure formation. Radiat. Environ.Biophys., 38:147(cid:21)161, 1999.[27] S. X. Kang, X. Sun, X. Ju, Y. Y. Huang, K. Yao, Z. Q. Wu,and D. C. Xian. Measurement and calculation of escape peakintensities in synchrotron radiation x-ray (cid:29)uorescence analysis.Nucl. Instr. and Meth. B, 192:365(cid:21)369, 2002.[28] I. A. Cunningham. Handbook of Medical Imaging, volume 1.Physics and Psychophysics, chapter 2. Applied Linear-SystemsTheory. SPIE Press, Bellingham, USA, 2000.[29] L. Lyons. Statistics for nuclear and particle physicists. Cam-bridge University Press, 1986.[30] E. Fredenberg, M. Lundqvist, M. ¯slund, M. Hemmendor(cid:27),B. Cederstr(cid:246)m, and M. Danielsson. A photon-counting detectorfor dual-energy breast tomosynthesis. In J. Hsieh and E. Samei,editors, Proc. SPIE, Physics of Medical Imaging, volume 7258,2009.ectors for medical imaging. Nucl. Instr. and Meth. A, 549:199(cid:21)204, 2005.[20] M. Lundqvist, B. Cederstr(cid:246)m, V. Chmill, M. Danielsson, andB. Hasegawa. Evaluation of a photon-counting X-ray imagingsystem. IEEE Trans. Nucl. Science, 48(4):1530(cid:21)1536, 2001.[21] M. ¯slund, B. Cederstr(cid:246)m, M. Lundqvist, and M. Daniels-son. Scatter rejection in multi-slit digital mammography. Med.Phys., 33:933(cid:21)940, 2006.[22] E. Beuville, B. Cederstr(cid:246)m, M. Danielsson, L. Luo, D. Nygren,E. Oltman, and J. Vestlund. An application speci(cid:28)c integratedcircuit and data acquisition system for digital x-ray imaging.Nucl. Instr. and Meth. A, 406:337(cid:21)342, 1998.[23] M.J. Berger, J.H. Hubbell, S.M. Seltzer, J.S., Coursey, andD.S. Zucker. XCOM: Photon Cross Section Database. online:http://physics.nist.gov/xcom. National Institute of Standardsand Technology, Gaithersburg, MD, 2005.[24] G. F. Knoll. Radiation Detection and Measurement. John Wiley& Sons, 3rd edition, 2000.[25] J.M. Boone, T.R. Fewell, and R.J. Jennings. Molybdenum,rhodium, and tungsten anode spectral models using interpolat-ing polynomials with application to mammography. Med Phys,24(12):1863(cid:21)74, 1997.[26] B. Grosswendt. Basic aspects of photon transport through mat-ter with respect to track structure formation. Radiat. Environ.Biophys., 38:147(cid:21)161, 1999.[27] S. X. Kang, X. Sun, X. Ju, Y. Y. Huang, K. Yao, Z. Q. Wu,and D. C. Xian. Measurement and calculation of escape peakintensities in synchrotron radiation x-ray (cid:29)uorescence analysis.Nucl. Instr. and Meth. B, 192:365(cid:21)369, 2002.[28] I. A. Cunningham. Handbook of Medical Imaging, volume 1.Physics and Psychophysics, chapter 2. Applied Linear-SystemsTheory. SPIE Press, Bellingham, USA, 2000.[29] L. Lyons. Statistics for nuclear and particle physicists. Cam-bridge University Press, 1986.[30] E. Fredenberg, M. Lundqvist, M. ¯slund, M. Hemmendor(cid:27),B. Cederstr(cid:246)m, and M. Danielsson. A photon-counting detectorfor dual-energy breast tomosynthesis. In J. Hsieh and E. Samei,editors, Proc. SPIE, Physics of Medical Imaging, volume 7258,2009.