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Sensitivity of Photon-Counting Based ${ m K}$-Edge Imaging in X-ray Computed Tomography

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1678 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 9, SEPTEMBER 2011
Sensitivity of Photon-Counting Based
-EdgeImaging in X-ray Computed Tomography
Ewald Roessl*, Bernhard Brendel, Klaus-Jürgen Engel, Jens-Peter Schlomka, Axel Thran, and Roland Proksa
Abstract—
The feasibility of
-edge imaging using energy-re-solved, photon-counting transmission measurements in X-raycomputed tomography (CT) has been demonstrated by simu-lations and experiments. The method is based on probing thediscontinuities of the attenuation coefﬁcient of heavy elementsabove and below the
-edge energy by using energy-sensitive,photon counting X-ray detectors. In this paper, we investigate thedependence of the sensitivity of
-edge imaging on the atomicnumber of the contrast material, on the object diameter ,on the spectral response of the X-ray detector and on the X-raytube voltage. We assume a photon-counting detector equippedwith six adjustable energy thresholds. Physical effects leading to adegradation of the energy resolution of the detector are taken intoaccountusingtheconceptofaspectralresponsefunction
for which we assume four different models. As a validation of ouranalytical considerations and in order to investigate the inﬂuenceof elliptically shaped phantoms, we provide CT simulations of an anthropomorphic Forbild–Abdomen phantom containing agold-contrast agent. The dependence on the values of the energythresholds is taken into account by optimizing the achievablesignal-to-noise ratios (SNR) with respect to the threshold values.We ﬁnd that for a given X-ray spectrum and object size the SNRin the heavy element’s basis material image peaks for a certainatomic number . The dependence of the SNR in the high-basis-material image on the object diameter is the natural, ex-ponential decrease with particularly deteriorating effects in thecase where the attenuation from the object itself causes a totalsignal loss below the
-edge. The inﬂuence of the energy-responseof the detector is very important. We observed that the optimalSNR values obtained with an ideal detector and with a CdTe pixeldetector whose response, showing signiﬁcant tailing, has beendetermined at a synchrotron differ by factors of about two tothree. The potentially very important impact of scattered X-rayradiation and pulse pile-up occurring at high photon rates on thesensitivity of the technique is qualitatively discussed.
Index Terms—
Computed tomography (CT), photon-counting,spectral
-edge imaging.
I. I
NTRODUCTION
D
IRECT-CONVERSION semiconductor detectors op-erated in photon-counting mode allow the selectiveimaging and quantiﬁcation of contrast materials containing a
Manuscript received December 27, 2010; revised March 28, 2011; acceptedMarch 30, 2011. Date of publication April 15, 2011; date of current versionAugust 31, 2011.
Asterisk indicates corresponding author.
*E. Roessl is with Philips Research Europe–Hamburg, D-22335 Hamburg,Germany (e-mail: ewald.roessl@philips.com).B.Brendel,J.-P.Schlomka,A.Thran,andR.ProksaarewithPhilipsResearchEurope–Hamburg, D-22335 Hamburg, Germany.K.-J. Engel is with Philips Research Europe–Aachen, D-52066 Aachen,Germany.Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/TMI.2011.2142188
heavy element from energy-sensitive transmission measure-ments. From a basis material decomposition in the projectiondomain followed by a conventional ﬁltered-back-projectionreconstruction, images of the distribution of the contrast mate-rial separate from the anatomical background can be obtained.The feasibility of this imaging technique, called “ -edgeimaging” in the following, has been demonstrated recentlyboth by simulations [1] and experiments [2]. In this context itis important to investigate the method’s sensitivity in the ﬁeldof X-ray computed-tomography (CT), both for applicationsin micro-CT and human CT. It is clear that the sensitivity of the method will strongly depend on the object size and thetube voltage setting. While the object size will determine thedegree of beam-hardening in the transmitted beam, the tubevoltage will determine the highest possible atomic number forwhich the method can be used at all. Therefore, for a givencombination of object size and tube-spectrum, there will be arange of optimal atomic numbers, i.e., elements, for which themethod is most sensitive.In this paper we provide estimates for the achievablesignal-to-noise ratio (SNR) in the -edge basis-material im-ages in a CT environment as a function of the object size,high-voltage setting and the atomic number of the con-trast-generating element. Moreover, we study the inﬂuence of the energy resolution, or more generally the energy response,of the detection system on the -edge imaging sensitivity. Wewill consider only circular objects in our analytical estimationof the achievable SNR values. As a veriﬁcation that our resultsare, with some modiﬁcations, equally applicable to medicalCT, i.e., for elliptically-shaped objects, we also simulated ananthropomorphic phantom with a gold contrast insert. Forthis case we also verify the sensitivity as a function of tubevoltage as well as the fact that by combining the entire spectralacquisition data it is always possible to recover a conventionalCT image of diagnostic quality.The performance of photon-counting systems in the contextof third-generation CT imaging is strongly inﬂuenced by twophysical effects that were not taken into account in our consid-eration: the degrading impact of pulse pile-up and of scatteredradiation on the measured X-ray spectra. Both effects are im-portant in a clinical environment, as usually very high X-rayﬂux rates and large coverage detectors are used in order to keepthe total scanning time low. We thus emphasize that the pre-sented results could potentially be too optimistic when consid-ering third-generation CT.The structure of this paper is the following: in Section IIwe brieﬂy summarize the method of -edge imaging in com-puted tomography using energy-resolved photon-counting data.
0278-0062/$26.00 © 2011 IEEE
ROESSL
et al.
: SENSITIVITY OF PHOTON-COUNTING BASED -EDGE IMAGING IN X-RAY COMPUTED TOMOGRAPHY 1679
This includes the description of the model for X-ray attenuationand detection and the way the basis-material decomposition isperformed. In Section III we outline our analytical model forthe computation of the SNR in the -edge basis images, basedon the Cramér-Rao lower bound (CRLB). We also deﬁne theimaging parameters and the phantom geometry which form thebase of our analytical model considerations. Moreover, we de-scribe the simulation of the anthropomorphic Forbild abdomenphantom together with the framework used for the investiga-tion of the applicability of the CRLB to our problem. In Sec-tion IV we present our results and we ﬁnally close with a dis-cussionof theirconsequencesinSectionVtogetherwithaqual-itative evaluation of the potential impact of scattered radiationand pulse-pile-up.II. B
ACKGROUND
The aim of this paper is to provide realistic estimates for thesensitivity of the -edge imaging technique based on energy-sensitive, photon counting detectors in an X-ray CT system.This technique has been recently described in [1] and its ex-perimental feasibility has been demonstrated [2]. The methodis a simple generalization of the Alvarez-Macovski [3] idea of dual-energy imaging to the case of a three-dimensional basismaterial decomposition [4](1)The ﬁrst two terms in the above equation describe the en-ergy-dependent contributions of the photo-electric effect andtheCompton-effecttothetotalattenuationoftheobjectresulting from everything but the contrast element. This is anexcellent approximation for all types of biological materials inthe clinically relevant energy regime of computed tomographywhichisabout20–150keV.Thethird termtakestheattenuationcaused by a heavy element with mass attenuation coefﬁcientinto account. Note that contains not only thecontribution of the -electrons to the total attenuation but alsothe photo-electric effect and the Compton effect contributionsof all other atomic electrons. Obviously, the minimal numberof spectrally distinct transmission measurements required toestimate the line-integrals of the basis-material densitiesalong the geometrical line(2)increases from two for the dual-energy case to three.Althoughthetechniquecaninprinciplebeappliedtoavarietyof data acquisition schemes using different types of detectors,we will focus here on photon-counting detectors.The expectation values of the number of recordedcountsinacertain(two-sided)energybin(orenergywindow)of an energy-sensitive, photon-counting detector can be describedby [1], [2](3)where , , , and denote the bin-sensi-tivity function (of the th energy bin), the incident X-ray spec-trum,the thattenuationbasisfunctionsandthedetectorabsorp-tion efﬁciency, respectively. The constants multiplying the inte-gral, , and denote the solid angle subtended by the detectoras seen from the focal spot, the measurement time during whichthe detector acquires data and the anode tube current during themeasurement, respectively.The argument of the exponential function in (3) is given bythe negative line-integral of the attenuation given in (1). Theindex runsfrom with beingthetotalnumberofen-ergy bins (energy thresholds) implemented in the detector elec-tronics.Instead of describing a given detector directly by the bin-sensitivity functions , we use the concept of the spectraldetector response to characterize its spectral perfor-mance, see, e.g., [5]. The quantity equals the prob-ability for generating a pulse of height between andin the detector electronics caused by an interaction of a singleincident X-ray photon of energy in the sensor material of thedetector.Theabovebin-sensitivitiesarethenrelatedtothespec-tral detector response via(4)The quantities , with denotethe pulse-height thresholds, deﬁning the two-sided energy win-dows. The response function is normalized to unity for everyvalue of the incident energy(5)The physical meaning of this normalization is that the interac-tionofanyparticleinthesensormaterialwillgiverisetoexactlyone pulse of deﬁnite height. The advantage of working with thespectral detector response as compared to the bin-sensitivitiesis that the latter depend on the threshold values while theformer does not. The threshold values will also strongly inﬂu-ence the sensitivity of the technique for the imaging of a givenelement, see, e.g., [6], [7], but this inﬂuence was eliminated bya threshold optimization on a case-by-case basis.Assuming that the measurements from (3) can be consid-ered as independent Poisson random variables one can writedown the negative log-likelihood for the measurement resultgiventhebasismaterialdecomposition ,see,e.g.,[1](6)
1680 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 9, SEPTEMBER 2011
Minimizing (6) for a given measurement with respect to thegives the maximum likelihood estimates from which the im-ages can be reconstruction using, e.g., conventional ﬁl-tered-back-projection.Whereasthephoto-electricandComptonimages can be recombined into a conventional CT image, ac-cording to (1), the image provides quantitative informa-tion about the density distribution of the contrast generatingelement.In this paper we compute estimates for the achievable SNRin thecontrast density image . For simplicity,we considerthe case of ﬁrst generation CT scanners. The data acquisition isrealized by the simultaneous translation of the X-ray tube anda single X-ray detector element perpendicular to the beam foreach of the angular views over 180 . Moreover, we assumethat our objects have cylindrical symmetry and that we are onlyinterested in the SNR in the iso-center of our scanner whichcoincides with the object’s symmetry axis.Two physical effects were neglected in our study althoughtheycertainlydohaveanimportantinﬂuenceonthe -edgesen-sitivity in particular if applied to third-generation CT systemswhere very high ﬂux rates are commonly employed: the lossof energy-information due to pulse pile-up and the impact of scattered radiation on the measured X-ray spectra. A thoroughdiscussion of the potential impact of both effects is provided inSection V where we also emphasize the need for further inves-tigations. In the following we describe in detail our analyticalapproach to the computation of the SNR in the basis materialimageIII. M
ATERIALS AND
M
ETHODS
A. Estimation of SNR in Basis Images
The computation of the SNR in the high- element basisimage is performed in several steps. In a ﬁrst step, wecompute the expected mean values of the number of counts tobeobservedineachenergybinwithinoneangularviewofatyp-ical CT acquisition scan. Ina second step,we computethenoiseinduced by the basis-material decomposition on the projectiondata using the Cramér-Rao lower bound (CRLB) [6], [8]–[10](7)(8)with being the so-called Fischer information matrix. Wemake use of the diagonal terms on the right-hand side of for-mula (8) to compute the variance of the line-integral of the density of the contrast-generating element. The derivativesin (7) are evaluated using the line-integrals correspondingto the known composition of the object along the ray. We willverify later, that the CRLB provides a reasonable estimate inanalytical form for the actual variance of the maximum likeli-hood estimate introduced above. Nevertheless, we must bear inmindthatourresultswillconstituteupperboundsforthe -edgeimaging sensitivity, i.e., the achievable SNR in the basis-mate-rial image . Since we assume cylinder symmetry of theobject and perfect alignment of the scanner’s and the object’ssymmetry axis throughout this work, the noise estimates arevalid for all angular views.In a third and ﬁnal step, we use the standard ﬁltered-back-projection formula for parallel geometry to compute the noisepropagationfromtheprojectiondomaintotheimagedomainforthe high- element basis image. This yields the noise varianceof the contrast material density in the iso-center of the scannerwhich together with the known density allows us to provide anestimate for the achievable SNR in the high- -element basis-material image
1
(9)The estimate for the image noise in the iso-center of a CTscanner is presented in many textbooks, see, e.g., [11]. Adaptedto our case we have(10)where and denote the number of angular projec-tions and the number of discrete sampling points of the ﬁlterkernel in the spatial domain, respectively. is the sampling in-terval,thedistancebetweentwoconsecutivepositionsofthede-tector element along the direction perpendicular to the X-rays.Note that in the case of ﬁxed total scanning time (the case weconsider), the above sum is independent of since the vari-ance on the projections is proportional to (via the in-verse proportionality between and acquisition time per an-gularview).Weneglecteffectsofinterpolationduringback-pro- jection on the variance estimate of (10). The discretized versionof the ﬁlter kernel for the back-projection is denoted bywhere we use the standard “ramp ﬁlter” by Ramachandran andLakshminarayanan, see, e.g., [12]evenodd.(11)The SNR values computed from the CRLB strongly dependonthevalueschosenfortheenergythresholdswiththestrongestdependencecomingfromthethresholdclosesttothe -edgeen-ergyofthecontrastmaterial,see[6]fordetails.Inordertoobtaina reliablemeasure forthe sensitivity ofthetechnique, thevaluesof the energy thresholds were optimized for each point in pa-rameter space, i.e., each distinct combination of values chosenfor the atomic number, object size, tube voltage, and detectormodel. Our optimization strategy was the following: for eachpoint in parameter space we started seven instances of a func-tionminimization,onemorethanthetotalnumberofthresholds.The function to be minimized was the CRLB variance of thegold line-integral expressed in (8). In the ﬁrst minimization run,the starting values of all thresholds were at energies above the-edge energy of the particular element studied. In the second
1
We assume implicitly that our imaging system is limited by statistical errorsand that our estimate for the high-
element density in the iso-center
is unbiased. Therefore, we use the known true values for the densities in ourSNR estimate. In a real system imperfect modeling and/or system calibrationwill always induce systematic errors or bias (image artifacts) in addition to thestatistical errors.
ROESSL
et al.
: SENSITIVITY OF PHOTON-COUNTING BASED -EDGE IMAGING IN X-RAY COMPUTED TOMOGRAPHY 1681
Fig. 1. Schematic view of the cylinder-symmetric phantom with diameter
and the a contrast material insert of diameter
used for the simulation.
minimization run, the starting value of all but one of the thresh-olds were at energies above the -edge energy, and so on. Thelocal minima in the CRLB variance were recorded for all of theseven runs and the global minimum was found among the sevenlocal ones. We used the program MINUIT from the CERN Pro-gramLibrary
2
tonumericallyminimizetheCRLBvariancewithrespect to the energy thresholds. The MINUIT processor MI-GRAD was used throughout which is an implementation of avariation of the Davidon–Fletcher–Powell variable metric algo-rithm. For details on the algorithm see, e.g., [13].Theresultingoptimalthresholdvalueswerethenusedtocom-pute the variances for all positions of the detector ele-ment in a given angular view of the scan. This procedure guar-antees a correct comparison of sensitivity for different objectdiameters ,atomicnumbers ,X-rayspectra,andspectral de-tector response behavior.
B. Phantom Geometry, Scan Parameters, and Spectral Detector Characteristics
A schematicof the phantom geometryis shown in Fig.1. Thephantom is made of a ﬁrst, outer cylinder of diameter com-posed of average (female) soft tissue as deﬁned in [14]. Withintheinner,concentric,cylindricalregionofdiameter weassumeahomogenousmixtureofthesamesoft-tissuematerialasbeforeand one contrast generating element with atomic number in aconcentration of 0.25 mol/l. As mentioned in Section III-A, ourphantom is placed in perfect alignment with the symmetry axisof the scanner.We included a wide range of object diameters and atomicnumbers in our study. The diameters varied between 20 to250 mm for a tube voltage of 60 kVp and from 50 to 400 mmfor tube voltages of 90 and 130 kVp, always in steps of 10 mm.The atomic number was varied between 45 and 85 (rhodiumto astatine). The corresponding -edge energies range from23.2 keV for rhodium to 96.1 keV for astatine.Our study was intended to cover geometries for both, clin-ical and pre-clinical CT. For each of the two scenarios we used
2
http://wwwasdoc.web.cern.ch/wwwasdoc/minuit/minmain.htmlFig. 2. X-ray tube spectra used throughout the simulations for high-voltagesettings of 60 kVp (solid line), 90 kVp (dashed line), and 130 kVp (dash–dottedline). The model described in [15] was used for their computation.
different values for the acquisition geometry. For clinical CTwe used typical parameters of commercial systems, for the pre-clinical case we followed as much as possible the geometry of a recently built spectral CT scanner, see, [2]. The source-de-tector distances were set to 1040 and 600 mm for the clinicaland pre-clinical geometries, respectively. The physical detectorpixel sizes in-plane and through-plane were 1.4 mm 1.1 mmand 0.4 mm 1.2 mm, respectively. The (in-plane) samplinginterval was 0.7 mm and 0.2 mm, as demanded by the sam-pling theorem. In the clinical case we used a contrast agent in-sert with a diameter of mm, in the pre-clinical casemm. The tube-loading (product of anode-tube currentand total scan time per transverse detector position) was 168mAs in both cases.
3
Our model for the detected number of X-ray photons is givenby (3). The value of can be computed from the detector sizeand the distance between the detector and the X-ray source. Theproductof and isdeterminedbythetubeloadinggivenabovewith themeasurementtimeduring whichthedetectoracquireddata for one transverse sampling position. To investigate the de-pendence on the source spectrum , we implemented thetheoretical model for the generation of X-ray tube spectra, asdescribed in [15], to generate spectra for three different valuesof the kVp setting: 60 kVp for pre-clinical applications or theimaging of small objects, 90 kVp, e.g., for a medical head-scanand 130 kVp, e.g., for a medical thorax scan. The three spectraare depicted in Fig. 2. In all cases an inherent ﬁltration of 1.5mm beryllium and 2.7 mm aluminum was assumed.We worked with four different models for the spectral de-tector response , three of which are shown in Fig. 3in the case of an incident X-ray energy of 100 keV.•
Ideal detector response
To obtain an idea of the sensitivity of the imaging tech-nique for the case of the most ideal imaging detector, weconsidered ﬁrst a delta-function response representing thecase of inﬁnite energy resolution. In this case the detector
3
Note that due to our parallel acquisition geometry the actual tube loadingwould be larger than the above cited value in an experimental realization by thenumber of sampling points per measured angular projection.
1682 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 9, SEPTEMBER 2011
Fig. 3. Various models for the detector response function used in the simu-lations: ideal detector response (not shown), shift-invariant, Gaussian response(FWHM10keV,solid line),shift-invariant,measuredresponsewith photo-peak (FWHM 7 keV), and escape peak (dashed line) and a measured response froma CdTe array detector (dash–dotted line). In all cases, the response to incidentphotons of energy 100 keV is shown.
allows to exactly determine the energy of each incomingphoton(12)•
Shift-invariant Gaussian response
Secondly,wesimulatedashift-invariantGaussianresponsefunction with a photo-peak of 10 keV FWHM (solid linein Fig. 3). By shift invariant we mean that the shape of thedetector response is independent of the incident energy. Inmathematical language(13)•
Shift-invariant response featuring photo-peak and -es-cape peak
Thirdly, we considered a shift-invariant detector responsefeaturing photo-, and -escape peaks of about 7 keVFWHM (dashed line in Fig. 3). Apart from the assumptionof shift invariance, this model is already quite realistic. Itwas obtained experimentally by acquiring a pulse-heightspectrum from the response of a CdTe detector togamma rays with an energy of about 59.5 keV.•
Measured response from Synchrotron
Eventually, we included a detector response measured atthe HASYLAB synchrotron facility (Hamburg, Germany)for a 1024 pixel array of 400 1200 CdTe detec-tors [16]. A detailed phenomenological parameterizationof this response model (dash–dotted line in Fig. 3) for dif-ferent incident energies can be found in [2].The number of total energy thresholds we considered was6 in all cases studied in this paper in order to make contact withmeasurement data [2]. Although this certainly restricts the gen-erality of our results, we feel that this choice reﬂects a reason-able compromise between aspects of practicability for CT andtheneedforasufﬁcientlyﬁnesamplingofthepulse-heightspec-trum by the comparator electronics. While is the abso-lute minimum for a successful differentiation between the three
Fig. 4. Anthropomorphic Forbild Thorax phantom with additional, centralgold contrast insert. The insert has a diameter of 10 mm and contains a mixtureof gold and water with a gold concentration of 0.2 mol/l. The smaller andlarger phantom diameters are 240 and 400 mm, respectively. In addition fourregions-of-interest are shown to compare noise and signal-to-noise ratios of a conventional CT image and an image obtained from a photon-countingdetector, see Fig. 13.
attenuation components, we typically observed that increasingto higher and higher values will only increase the detector’scomplexityandcost butwillnotlead toa considerableimprove-ment in the SNR in the -edge image, see [7] for a recent dis-cussion on the effects of binning on the spectral performance of energy-selective photon-counting detectors.The detector absorption efﬁciency was assumed to be 100%,i.e., , for all energies within our spectra. This as-sumption is very well satisﬁed in practice for conventional CTsystems due to reasons of dose efﬁciency. The same require-ments would certainly apply to future photon counting com-puted tomography as well. For very high X-ray energies this as-sumption will obviously be violated although for a 3-mm-thick cadmium-telluride (CdTe) detector crystal still about 67% of 150 keV photons are stopped in the sensor.
C. Forbild Abdomen Phantom Simulations
As a veriﬁcation that our analytical considerations can beapplied to more realistic phantoms, i.e., phantoms of ellipticalshape, we simulated a Forbild abdomen phantom [17] with agold contrast insert of 0.2 mol/l gold concentration as shown inFig. 4. The energy dependent tissue attenuation was modeledusing the values for the compositions and densities for the tis-sues
Liver
and
SkeletonSpongiosa
from[18]and
SkeletonVerte-bral Column D6 L3
and
Average Soft Tissue (female)
from [14].Astheconcentrationwassmallerthantheconcentrationusedforour analytical considerations (0.25 mol/l), we adjusted the tubeloading to 262.5 mAs. According to (9) this exactly compen-sates for the lower expectation in the SNR values as the CRLBvariance is inversely proportional to the tube loading. In orderto obtain a maximally reliable estimate for the SNR to be ex-pected in our CT scan, we ran 50 independent noise realizationsof the same simulation and estimated not only the variance butalso its uncertainty. Apart from this our simulations were per-formedwith theparametersdescribedin SectionIII-Baboveforthecaseof clinicalCT.The responsefunctionwas chosenasthemost realistic response measured at the synchrotron. In order tostudy the dependence on tube voltage and radiation dose, weperformed the above simulation series for tube voltages of 90,

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