Prospects for quantitative computed tomography imaging in the

Prospects for quantitative computed tomography imaging in the presenceof foreign metal bodies using statistical image reconstruction

Jeffrey F. Williamsona)

Department of Radiation Oncology, Washington University, St. Louis, Missouri 63110

Bruce R. WhitingDepartment of Radiology, Washington University, St. Louis, Missouri 63110

Jasenka Benac and Ryan J. MurphyDepartment of Electrical Engineering, Washington University, St. Louis, Missouri 63130

G. James BlaineDepartment of Radiology, Washington University, St. Louis, Missouri 63110

Joseph A. O’SullivanDepartment of Electrical Engineering, Washington University, St. Louis, Missouri 63130

David G. PolitteDepartment of Radiology, Washington University, St. Louis, Missouri 63110

Donald L. SnyderDepartment of Electrical Engineering, Washington University, St. Louis, Missouri 63130

~Received 19 March 2002; accepted for publication 5 August 2002; published 30 September 2002!

X-ray computed tomography~CT! images of patients bearing metal intracavitary applicators orother metal foreign objects exhibit severe artifacts including streaks and aliasing. We have system-atically evaluated via computer simulations the impact of scattered radiation, the polyenergeticspectrum, and measurement noise on the performance of three reconstruction algorithms: conven-tional filtered backprojection~FBP!, deterministic iterative deblurring, and a new iterative algo-rithm, alternating minimization~AM !, based on a CT detector model that includes noise, scatter,and polyenergetic spectra. Contrary to the dominant view of the literature, FBP streaking artifactsare due mostly to mismatches between FBP’s simplified model of CT detector response and thephysical process of signal acquisition. Artifacts on AM images are significantly mitigated as thisalgorithm substantially reduces detector-model mismatches. However, metal artifacts are reduced toacceptable levels only when prior knowledge of the metal object in the patient, including its pose,shape, and attenuation map, are used to constrain AM’s iterations. AM image reconstruction, incombination with object-constrained CT to estimate the pose of metal objects in the patient, is apromising approach for effectively mitigating metal artifacts and making quantitative estimation oftissue attenuation coefficients a clinical possibility. ©2002 American Association of Physicists inMedicine. @DOI: 10.1118/1.1509443#

Key words: x-ray transmission computed tomography, metal artifact reduction, brachytherapy,statistical image reconstruction, filtered backprojection

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I. INTRODUCTION

Metallic and other dense objects are often placed in theman body, including implants and applicators usedbrachytherapy, cochlear simulators and receivers,1 implant-able orthopedic appliances, surgical clips and staples,dental restorations. These high-atomic number, high-denobjects attenuate x rays in the diagnostic energy range mmore strongly than soft tissue or bone so that far fewer ptons traversing these objects reach the detectors. This crstrong artifacts~see Fig. 1! in the form of dark and brightstreaks spread across the entire image reconstructedconventional filtered back projection~FBP! in x-ray com-puted tomography~CT! images. These artifacts can masoft tissue structures not just in the immediate vicinity of tdense object, but also throughout the entire image, rendeit of limited use.

2404 Med. Phys. 29 „10…, October 2002 0094-2405 Õ2002Õ29„

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In brachytherapy, the severe streaking artifacts arisfrom commercial intracavitary applicators make accursegmentation of bladder and rectal walls, uterus, cervix,small bowel difficult if not impossible.2,3 These applianceshave transverse-plane dimensions as large as 3 cm,stainless steel walls of 0.5 to 1.0 mm thickness, and ofcontain thicker stainless steel or brass components. Mvaginal colpostats contain 3 to 5 mm thick tungsten-alshields intended to reduce bladder and rectal doses.dose sparing4 is considered essential by many practitionersmaintain normal tissue injury at acceptable levels. Attemto circumvent metal artifacts on CT images include the uof plastic unshielded applicators5 and use of bone-imagingwindow and level settings,2 which diminish visualization ofsoft-tissue organ boundaries. Also, custom-made FletchSuit applicators with afterloading tungsten shields canfabricated.6 However, limiting local shielding to afterload

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uter Lucite

2405 Williamson et al. : Prospects for quantitative CT imaging 2405

FIG. 1. Image resulting from application of our FBP algorithm to measured sinogram data extracted from a Siemens Somatom Plus 4 scanner. The oannulus was omitted from our simulations. The simulation phantom~Ref. 42! contains steel, brass, Teflon, and Al rods.

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able configurations, solely to obtain good image quality,verely compromises the utility of internal shielding asoptimization strategy. New low-energy7–9 brachytherapysources, such as Am-241, Yb-169, and Sm-145 have binvestigated, in part, because of the ease with which senstissues can be shielded by means of flexible high-atonumber foils. Thus, therapeutic outcome may be improby customizing shield design for each patient so asachieve an optimal dose distribution.10 Metal artifacts also

Medical Physics, Vol. 29, No. 10, October 2002

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hinder CT imaging for other types of brachytherapy procdures, including template-guided interstitial implants usimetal needles,11 permanent prostate implants using metalI-125 or Pd-103 seeds,12 and head and neck implants in thpresence of dental prostheses. Clearly, successful mitigaof severe streaking artifacts through improved image recstruction or processing techniques would be of significvalue, not only for image-based brachytherapy planning,for many other applications of CT imaging as well.

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Most published analyses of FBP metal artifacts assuthat such artifacts are due to missing data, i.e., gaps intransmission sinogram arising from near-complete attention of the x-ray beam by foreign metal bodies present inpatient. According to this view, metal artifact mitigation involves identifying regions of the sinogram exhibiting higattenuation followed by processing to approximately replthe lost information. Information replacement strategiesclude ‘‘rubout,’’ 13 linear interpolation,14 waveletinterpolation,15 and adaptive filtering.16 Such techniques careduce streaking, but are less effective for complex objectanatomy and do not ensure quantitative values from replasinogram regions. As pointed out by Medoff,17 successfulreconstruction of incomplete data requiresa priori knowl-edge to deal with ‘‘null spaces’’ in the sinogram.

Iterative reconstruction algorithms constitute anothstrategy for mitigating metal artifacts. Such algorithms aable to incorporate prior knowledge, including the geomecomposition, and location of the metal objects in the patieas constraints on the reconstructed images. Using a deteristic iterative deblurring technique developed by our group18

Wang19 demonstrated substantial reduction of streaking afacts in the presence of opaque, convex objects of knolocation. Our more general approach,20 called ‘‘object-constrained CT~OCCT!,’’ accommodates highly attenuatinobjects of arbitrary geometry and unknown location by cobining estimation of metal object pose with iterative recostruction of the surrounding tissue attenuation map in a ufied framework.

Model-based image reconstruction techniques rejecthypothesis that metal streaking artifacts are solely or eprimarily due to missing information. Rather, these methoassume that metal artifacts, indeed all artifacts, arise fdiscrepancies between the actual CT signal formation pcess and the model assumed by the reconstruction algoriDespite the dominance of the ‘‘missing information’’ view ometal artifacts, a significant literature exists demonstratthat measurement artifacts such as noise,16 scatter,21,22 beamhardening,23,24 and high contrast edge effects25 cause streaksto appear in images reconstructed by FBP. An importearly effort to reduce detector model mismatches for tramission CT was the convex algorithm proposed by Lanand Carson.26 Their iterative expectation-maximization~EM!algorithm was based on the assumption that CT transmismeasurement is an inherently stochastic process that isscribed by the Poisson distribution. More recently, iteratreconstruction algorithms have been proposed that incthe scanner x-ray spectrum in their data models27,28 andtherefore accommodate beam hardening. Our group29 has de-veloped a novel iterative alternating minimization algorithhereafter denoted as AM, which includes background evee.g., scattered radiation, as well as beam hardening andsurement noise, in its underlying signal formation model.

From the model-based image formation perspective, Fis a discrete implementation of an analytic model, whassumes that CT transmission measurements are nois


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and are linear functions of the attenuation line integrals alothe corresponding primary photon trajectories throughpatient.30 When scanning subjects comprised of only antomically native materials under normal conditions, retively simple corrections31 to the raw sinogram are sufficiento assure that these assumptions are at least approximtrue, resulting in FBP images free of visually obvious arfacts. However, in regions shadowed by highly attenuatobjects, noise and nonlinear detector responses due to scand spectral hardening dramatically increase, giving risepronounced streaking artifacts. The hypothesis of this pais that image reconstruction algorithms derived from sigacquisition models accounting for these nonlinear effewill yield higher quality images in the presence of meobjects.

In this paper, the performance of three algorithmsconventional FBP, Snyder’s deterministic iterative deblurri~IDB!, and O’Sullivan’s model-based iterative algorith~AM !—is compared when metal objects are present inscanning subject. Numerical experiments are performeding simulated sinogram data in which the level of noisspectral polychromaticity and scatter are controlled. Oursults clearly demonstrate that metal streaking artifactsFBP images are dominated by photon scatter, beam haring, and propagation of noise, arising from low-count singram regions, rather than by missing information. In adtion, we show that the model-based AM algorithsubstantially reduces streaking artifacts, demonstratingimportance of basing image reconstruction upon a realimodel of the CT signal formation process. However, useprior knowledge of metal geometry to constrain iterative iage reconstruction is also shown to be necessary toequately mitigate streaking artifacts. Finally, we show thAM has the potential to support quantitatively accurateinvivo measurements of tissue attenuation coefficients witpatients. This has important implications for radiotheratreatment planning of low-energy photon brachytherapy32–34

and other treatment modalities, which produce dose distrtions sensitive to tissue composition heterogeneities.

II. METHODS AND MATERIALS

A. Problem definition

Figure 2 illustrates a typical third-generation CT gantThe thin fan-shaped x-ray beam traverses the patient anopposed by a detector array, which measures the photonence emerging from the patient. LetX denote the set of discrete spatial positions~pixels! x in the patient supporting theattenuation coefficient-valued image,m(x,E). The data,d(y), acquired by the detector array are indexed by a vecof discrete values,y[(b,g), where b denotes the sourceposition andg denotes the detector location in the gantThe set$d(y):yPY% is called the sinogram. The mean fluence sinogram,g(y) can be modeled by

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m~y,E!5(x

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whereI 0(y,E) is the source intensity~the number of photonsof energyE detected by the detector aty in the absence of anattenuating subject!; h(yux) is the scanner’s point-spreafunction; m(x,E) is the linear attenuation coefficient~inunits of mm21! of the patient at positionx and energyE; andthe functions(y) denotes the mean number of backgrouevents due to scattered photon fluence and extrafocal ration associated with the radiation field.g(y) denotes themean of all possible independent measurements,d(y),which are assumed to be Poisson random variables. Ifdata are noiseless, thend(y)5g(y) for knownm ands. Thepoint-spread function describes the attenuating effectvoxel x on the photons incident upon the detector-sourcey and includes such effects as finite detector size and vdiscretization. The dimensionless quantitym(y,E) denotesthe mean photon attenuation~in units of mean-free path!along the straight-line photon trajectory,y.

B. Iterative deblurring algorithm

Deterministic iterative deblurring, or IDB, assumes ththe measured data are noiseless, scatter-free, and arise fmonoenergetic source; i.e.,I 0(y,E)5I 0(y)dE,E0

. The CT re-construction problem is formulated as follows: find thetenuation image,m(x), that yields the predicted sinogramb(y:m)5(xPXh(yux)m(x), that most closely approximateits measured counterpart,a(y)[2 ln(d(y)/I0(y)). Iterativedeblurring uses Csisza´r’s I divergence to quantify the discrepancy betweena(y) andb(y:m),

FIG. 2. Geometry of CT sinogram measurement. The x-ray source idistanceD from the isocenter. The transmission pathy is defined by thegantry angle indexb and the detector fang.


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yPY@a~y!2b~y:m!#. ~2!

Unlike the more common least squares discrepancy measEq. ~2! is the only discrepancy measure which satisfiCsiszar’s axioms for rationally choosing the ‘‘best’’ solutionto an inverse linear problem when botha and b arenonnegative.35 For any positive initial estimate,$m (0)(x),xPX%, of the image, the sequence of progressively updaimages$m (k)(x),xPX,k51,2,...%,

m~k11!~x!

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converges to an image$m(x),xPX% that minimizesI (aib)18

whereH0(x)5(yPYh(yux). Equation~3! has the same formas the expectation-maximization algorithm used in emisstomography, which finds the solution that maximizes likehood of the measured emission rates, assuming Poissonton emission sources.36,37 However, the form of~3! is im-posed by the discrepancy measure~2!, not by the randomprocesses associated with the CT measurement process

Object-constrained CT~OCCT! refers to the general approach, developed by our group,20 for using known attenua-tion maps of metal applicators and other metal bodiesknown geometry to constrain the iterative tissue attenuamap solution. We consider the image to be composed ofcomponentsm(x)5mb(x)1ma(x:u), where mb(x) is theunknown attenuation map of the patient’s body andma(x:u)is the known map of the applicator at the pose,u ~locationand orientation in the patient!, which is assumed to be unknown. LetXa(u) denote the subset ofX that supports theapplicator at poseu. Then, mb(x)50 for xPXa(u), andma(x:u)50 for xPX2Xa(u). Our initial OCCT implemen-tation sought to identify the body attenuation map,mb(x),and applicator pose,u, that together minimize theI diver-genceI @ai(xh(•ux)m(x)#. Briefly, the OCCT method20 isinitialized by a first guess of the body attenuation,mb

(0)(x),and an estimated applicator pose,u (0). Iterations commencefrom these initial estimates, with the (k11)th estimateobtained from the kth estimate, m (k)(x)5mb

(k)(x)1ma(x: u (k)), by performing an iterative update, defined bEq. ~3!, yielding an intermediate functionmEM

(k11)(x), xPX.This intermediate function is then modified by replacingvalues forxPXa(u) by ma(x:u) for a candidate next poseu,yielding a m (k11)(x:u)5mEM

(k11)(x), xPX2Xa(u), andm (k11)(x:u)5ma(x:u), for xPXa(u). A local search is thenperformed in the neighborhood of the current poseu (k) tofind a u (k11) that reduces theI divergence I @ai(xh(•ux)m (k11)(x:u)#. These steps are then repeated until covergence is achieved. A more complete account, includhandling of partially occupied pixels near the applicaboundary, can be found elsewhere.20

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C. The alternating minimization algorithm

Alternating minimization, a family of iterative algorithmdeveloped by our group,29 is able to utilize prior informationconcerning the geometry of foreign metal bodies, but unlIDB, is not limited by the simple monoenergetic, noiseleand scatter-free measurement model of CT x-ray transmsion measurement. We assume38,39 that the attenuation coefficient of each voxel in the patient can be represented blinear combination of a small number,N, of constituent sub-stances, e.g., bone and fat equivalent basis vectors,

m~x,E!5(i 51

N

m i~E!ci~x!5m~E!•c~x!, ~4!

whereci(x)5r i(x)/r i8 is the partial specific gravity of thei th constituent of voxelx; and m i(E) and r i8 are the linearattenuation coefficient at photon energyE and mass densityrespectively, of that constituent in pure form. SubstitutiEq. ~4! into ~1!, the mean of all possible measured sinograbecomes

g~y:c!5(E

I 0~y,E!e2SxS i h~yux!m i ~E!ci ~x!. ~5!

In contrast to~1!, the indexE in Eq. ~5! ranges over thedummy energyE50, as well as the nonzero discrete enervalues of the spectrum, where by definitionI 0(y,0)[s(y)andm i(0)[0. The quantityI 0(y,E) is assumed to be knownWe further assume that the measuredd(y) are randomly dis-tributed about the average value,g(y), according to thePoisson distribution. Thus, the probability of observingmeasured sinogram,d(y), given an image of partial specifigravities,c(x), is

P~duc!5)y

e2g~y:c!g~y:c!d~y!

d~y!!. ~6!

The problem to be solved by AM is to find the vector-valuimage, c(x), which maximizes the log-likelihood of~6!,ln@P(duc)#. The AM algorithm is only briefly described herea more complete description and derivation is presenelsewhere.29 It is easy to show thatc(x) also minimizes theIdivergenceI @d(y)ig(y:c)#. In contrast to the IDB deriva-tion, hereI @•i•# operates on the photon transmission ratthan attenuation data space. We now define linear and enential families of functions,l(d) and «(I 0 ,H•m), respec-tively,

l~d!5H p:p~y,E!>0,(E>0

p~y,E!5d~y!J ,

~7!«~ I 0 ,H•m!5$q:q~y,E!5I 0~y,E!e2SxS i h~yux!m i ~E!ci ~x!%,

whereH is the matrix with elementsh(yux). The linear fam-ily l(d) consists of the set of all possible monoenergesinograms, parametrized as a function of energy, whoseover energy gives the measured sinogram, while«(I 0 ,H•m) is the set of data means, as a function of energy, geated by the totality of unknown functions,c(x). Given thedefinitions~7!, the minimization can be recast as follows:


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Equation ~8! leads to an algorithm in which the iterationalternate between estimatingp andq.

The zeroth (k50) iteration to the solution of~8! beginswith assigning an initial value or guess to eachci

(0)(x).Given thekth estimate of partial specific gravity,ci

(k)(x), the(k11)th estimate is calculated as follows. First, the currestimates of the functionsq andp from the exponential andlinear families are computed:

q~k!~y,E!5I 0~y,E!•expF2(x

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Next the back projectionsbi(k) and bi

(k) of the current esti-mates ofp and q, are calculated

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tively,

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~11!if ci

~k11!~x!,0, then setci~k11!~x!50.

The functionZi(x) is a precomputed normalization functionwhich can be freely chosen subject to the constraintsviewed by O’Sullivanet al.29 in this work,

Zi~x!5Z05max~y,E!

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By virtue of the definition of the dummy energyE50, theestimate forq(y,0) is alwayss(y). Equation~9! becomes

p~k!~y,E!5q~k!~y,E!d~y!

(E8Þ0q~k!~y,E8!1s~y!, ~13!

demonstrating thatp(k)(y,E) is the expected number ocounts contributed tod(y) by photons of energyE that arenot part of the background distribution,s(y).

At least two other iterative reconstruction algorithms40,41

have been proposed for maximizing the likelihooln@P(duc)#. Both have been generalized to accommodatex-ray spectrum in their data models27,28 and can be extendeto accommodate background events as well. However, AMunique in that it is based upon an analytic maximizatirather than an approximation to the objective function. Thit can be proven that the AM iterative sequence convermonotonically. The performance advantages, if any, cferred by this unique mathematical property have not binvestigated.

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D. Scanning conditions and phantom

The phantom utilized in this study is shown in Fig. 1. Adescribed in more detail elsewhere,42 it consists of water andLucite cylinders~total diameter 30.5 cm! giving maximumattenuation and scatter equivalent to abdominal scans. R~1.27 cm diameter! of various materials can be inserted inthe phantom at accurately defined locations to provide martifacts or low contrast objects. For the studies presenhere, the four rods were composed of brass, iron, aluminand Teflon. From construction drawings, matching synthdata sets were generated43 and found to closely match measured data.44

To test the reconstruction algorithms, 2D synthetic singram data were generated typical of abdominal scanninging a commercial third-generation helical scanner~SiemensSomatom Plus 4, Siemens AG, Erlangen, Germany!. Thisscanner has a fan-beam geometry with a source-to-isocedistance of 570 mm and a;52° fan-beam arc of detectors oa radius of 1005 mm. The gantry was assumed to consis768 detector samples per gantry position and 1408 gapositions per revolution. A detector width of 1.2 mm apixel dimensions of 131 mm2 were assumed. By reviewina small number of patients of various sizes scanned wi120 kVp tube potential, we found that the maximum beattenuation in the abdomen ranged from 6–8. Modeling rresentative scan protocols revealed unattenuated flux leequivalent to several million noise equivalent quanta~NEQ!with an effective energy of;75 keV. Flux levels dropped toseveral thousand NEQ in the thickest areas of the abdomPrimary beam transmission through metal objects was calated to be less than exp(215). Since scattered radiationtypically several percent~2–4%! in abdomen areas, depening on longitudinal collimation, it dominates the measursignal in the 10–100 NEQ range.

For the polyenergetic spectrum data model, an acceleing potential of 120 kVp and an additional filtration of 2mm Al were assumed. The Birch–Marshall model45,46 wasused to generate a filtered Bremsstrahlung spectrum~includ-ing characteristic x rays! for a tungsten target tube withtarget angle of 7°. Based upon an experimental sinoganalysis,47 a 240 mAs exposure, 3 mm slice thicknessisocenter, and 1 s/gantry rotation results in an incident flux1.63106 photons on each detector in the absence of atteators. For a typical abdominal scan~30 cm water thickness!with an attenuation of 6 to 8, the scatter contribution repsents about 3% of the signal corresponding to 80 couThis was approximated by settings(y)580. For the mo-noenergetic data model, an energy of 75 keV, anI 0(y) of1.63106 photons, and a constants(y)580 were assumed.

E. Data model and algorithm implementation

Both the data models and the iterative algorithmsbased upon forward- and back-projection operators descrelsewhere.43 The phantom was assumed to consist of discrpixels, with circular cross sections of the rods approximaby a stair-step boundary. The array elements,h(•u•), of thepoint-spread function were precalculated using parame


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ray tracing. The fractional contribution of a pixel with itcenter at (x1 ,x2) and a detector centered atg for a gantryangle, is given by

h~b,gux1 ,x2!5~1/Dg!Eg2Dg/2

g1Dg/2

L~b,g8,x1 ,x2!dg8, ~14!

whereDg is the angle subtended by the detector with respto the source location andL(b,g,x1 ,x2) is the distance be-tween proximal and distal intersections of the ray path~b,g!with the pixel (x1 ,x2). In addition to ignoring edge gradieneffects, our simulation neglected finite focal spot size agantry motion blur. Becauseh(•u•) is a sparse matrix, theprecalculated matrix was saved as an indexed data struccontaining only nonzero elements. By restricting the summtion indicesx andy of the forward and back projection operators to nonzeroh(yux), the number of arithmetic operations needed to evaluate sums of the form(xPXh(•ux)m(x) can be reduced by 40-fold.43 Some of the calcu-lations were further accelerated by using the method ofdered subsets29,43 to evaluate the iterations of Eq.~11!.

The monoenergetic data model was evaluated using~1! byrestricting the sum to a single energy of 75 keV. Each vam(x,E) was evaluated atE575 keV directly for the materiallocated atx, rather than by the finite constituent model, E~4!. To add noise to this model, the data means, includingconstant background contributions, were passed throughrandom number generator, which sampled a signal vafrom the appropriate Poisson distribution. The process cated some zero counts in the simulated data and, while thacceptable to the alternating minimization algorithm, it psents a problem for FBP and IDB algorithms since a negalogarithm results. To avoid ln~0! evaluations, zero transmission values were replaced with ones prior to taking the lorithm. The choice of 1, rather than a smaller value, was min order to keep the noise variance at a reasonable level,to avoid atypical conditions for FBP and IDB. In a clinicascanner, photon scattering adds a positive offset to eachtector response, on average equivalent to about 100 detequanta, making the likelihood of encountering zero-counttector readings extremely low.

For the polyenergetic photon spectrum model, the numof constituents in Eq.~4! was set to one, i.e.,N51. Theunitless partial specific gravity function,c(x), was computedas the ratio of the attenuation coefficient of the materialthe pixelx to that of water,mwat(E575 keV), both evaluatedat 75 keV. This results in an object attenuation functiom(x,E)5mwat(E)•c(x). The high-density water-equivalenphantom thus created will have attenuation values simulathose of the Lucite and metal inserts at the appropriate cdinates~as shown in Fig. 1!. To add noise to this model, thdata means@from Eq. ~5!# were passed through the Poisssampling procedure.

Snyder’s OCCT approach20 can be used to iteratively estimate the pose of metal objects of known geometryAM48 as well as IDB. However, for the purposes of thstudy, the influence of the prior information~attenuation mapof the correctly localized object! provided by OCCT on al-

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gorithm performance was simulated by settingm (k11)(x:u)5ma(x:u), for the known rod pose and location,u, when-everxPXa(u) and using Eqs.~3! or ~11! only for x¹Xa(u).Thus, an error-free OCCT localization of the metal rod powas assumed by our simulations.

The filtered backprojection algorithm~FBP! was imple-mented for the Siemens fan-beam geometry.49 The ramp fre-quency function was modified by a Gaussian filter corsponding to a spatial convolution of the image with a twdimensional circularly symmetric spatial Gaussian functwith a FWHM of 1 mm.

F. Quantification of algorithm performance

Algorithm performance was assessed by viewing imaof the quantitymwat(E575 keV)• c(x) with a viewing win-dow of ~0.016, 0.024! mm21 ~620% of the water attenuation!. This is equivalent to Hounsfield window and level setings of 400 and 0, respectively. Horizontal profiles, placemm below the edge of the aluminum rod, were also plottFinally, several quantitative measures were evaluated,cluding mean percent bias, mean percent absolute bias, mI divergence, and mean squared error:

Mean percent bias5MPB51

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Mean square error5MSE51

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where i indexes individual pixels,ctrue( i ) denotes the trueimage ~from which the data model was derived!, and c( i )denotes the reconstructed image. Each error measurecomputed separately over the different regions~see Fig. 1!,including the outer Lucite cylindrical shell, the central Lucicylinder~excluding the area occupied by the metal rods!, andthe intervening water annulus. Average and percent relanoise were evaluated for the three regions as follows:

Mean noise5A( i 51N ~ c~ i !2 cnoiseless~ i !!2

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Mean percent noise5MPN5100%

•Mean noise/Mean~ cnoiseless! ,

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where c( i ) is the reconstructed image intensity~includingthe effects of synthetic noise in the modeled data! andcnoiseless( i ) denotes the corresponding image reconstrucusing the same algorithm and data model, but with nosuppressed.

III. RESULTS

Figure 3 shows the influence of noise, spectral polychmaticity, and scatter on images reconstructed by FBP. Fig


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4 compares the performance of AM and IDB for variomonoenergetic sinogram data models while Fig. 5 compaAM and FBP for data models based upon the polyenergphoton spectrum. Figure 6 shows representative profithrough selected images. Table I presents quantitative imquality metrics, defined by Eqs.~15! and ~16!, for the innerLucite cylinder and surrounding water annulus for variocombinations of algorithms and sinogram data models.

Under near-ideal conditions~monoenergetic spectrum, nnoise, no scatter!, Fig. 3~a! demonstrates that FBP performreasonably well in the presence of metal rods, exhibitinfaint Moire-type pattern of intensity oscillations, giving risto MPAEs of 1%–2%~Table I!. The MPB is very low~,0.1%! due to cancellation of negative and positive erroThis artifact, which diminishes as the detector density anumber of gantry views increases~data not shown!, is thewell-characterized aliasing artifact.50 Figure 3~b! demon-strates that measurement noise dramatically exacerbmetal streaking artifacts, producing 8%–31% mean absoerrors, and 15%–64% relative noise. Photon [email protected]~c!# produces streaks of similar geometry, but without chacteristic shot noise and even larger absolute errors. Wstreaks associated with beam hardening appear more s@polyenergetic spectrum without noise or scatter; Fig. 3~d!#,the mean percent bias is higher, indicating that this artifaresults in systematic overestimation, as well as oscillationthe attenuation function. Finally, Fig. 5~a! illustrates FBPperformance given the most realistic data model investiga~noise, polyenergetic spectrum and scatter!. It closely re-sembles the actual image~Fig. 1! obtained from the Siemenscanner for the cylindrical phantom. Note that regardlessthe noise level, the polyenergetic sinogram model with simlated scatter yields mean absolute errors of 16% for waand 43% for the Lucite cylinder. Finally, note that the preence of scatter reduces propagation of noise across theage. When no scatter is present, noise from the photon-pregions shadowed by the metal rods propagates acrossimage. Adding scatter increases photon flux in these regiwhich reduces noise propagation errors but at the expensreduced accuracy due to more prominent deterministreaks.

Our numerical experiments support our view that oserved metal artifacts are not due solely or even mostly‘‘missing’’ sinogram data, but do indeed arise from systenonlinearities, not included in the simple data acquisitimodel assumed by FBP. Of these, scatter and noise appebe the dominant sources of streaking, while beam hardencontributes most to deviation of the mean of the recostructedm(x) from the expected value. It is somewhat suprising that signal noise in the high attenuation regions ofsinogram can dominate metal streaking artifacts, particulain the absence of scatter.

Figure 4 compares the performance of the IDB and Aiterative reconstruction algorithms for the monoenergeticnogram model. Comparing columns 1 and 2 of Fig. 4 deonstrates that supplyinga priori information regarding thelocation and attenuation map of the high-density objects

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FIG. 3. Images reconstructed with FBP, based on different simulated sinogram models:~a! monoenergetic, noiseless, scatterless;~b! monoenergetic, noisy(1.63106 photons), scatterless;~c! monoenergetic, noiseless, with scatter;~d! polyenergetic, noiseless, scatterless;~e! and ~f! polyenergetic, scatterless datwith noise for~e! 105 photons and~f! 1.63106 photons/detector.

FIG. 4. Comparison of IDB and AM algorithms for monoenergetic, scatter-free data model. ‘‘Unknown rod attenuation map’’ means that no prior inforegarding composition and location of the high-density rods was supplied. ‘‘Known map’’ means that each updated image is modified to reflect the kodattenuation values and locations.



FIG. 5. Images reconstructed from simulated data including polyenergetic spectrum, noise (1.63106 photons unless otherwise specified! and scatter.~a! FBP;~b! and ~c! AM with beam-hardening and scatter corrections with 13105 photons for~b! rods unknown and~c! rods known;~d! AM with scatter but nobeam-hardening correction, known rods;~e! AM with scatter and beam-hardening corrections, known rods.

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the phantom substantially improves the performance of bIDB and AM. Surprisingly, whena priori information is notused with otherwise idealized sinogram data, IDB and Fperform reasonably well, producing relatively subtle arfacts. In contrast, the corresponding AM image@Fig. 4~e!#,exhibits significant streaking, resulting in 1% and 5% meabsolute errors in the water annulus and central Luciteinder, respectively. Inclusion of noise in the data model dmatically reduces IDB performance,~6%–20% MPAE and9%–38% MPN! regardless of the use of prior informatioThe deterministic signal acquisition model assumed by Icauses noise from the low photon-count regions to propathroughout the image, producing artifacts approachingseverity of FBP streaking. In contrast, AM is able to formuch smoother, streak-free images without the aid of relarization. The streaking artifacts that result when prinformation is ignored are unaffected by the presencenoise.

Our numerical experiments demonstrate that a realitreatment of detector counting statistics is essential toreconstruction algorithm that hopes to overcome the martifact-streaking problem. Noise degrades the performaof deterministic iterative algorithms as well as FBP whdense metal objects are present. Although beam harde


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and scatter also reduce the accuracy of both IDB and Fimages, the presence of scatter does mitigate noise proption errors.

Figures 5 and 6 illustrate the performance of AM and Fwith more realistic data models. The AM algorithm incorprating beam hardening and scatter corrections yields hquality images~MPNs of 2%–3%! even when the photonfluence is unrealistically low@Fig. 5~c!#. With clinically re-alistic levels of emitted photons/detector, MPN falls belo1%. Incorporating knowledge of the metal rods and thattenuating properties reduces MPAE’s to less than 1%eliminates nearly all evidence of streaking@Figs. 5~e! and6~b!#. However, when this information is not [email protected]~b!#, significant streaking artifacts arise, resulting in mepercent absolute errors of 6% and 2% in the Lucite cylinand water annulus, respectively. The local errors instreaks range from210% to123% @Fig. 6~c!#. Finally, ap-plying the monoenergetic AM algorithm to polyenergedata yields average absolute errors of 9% to 11%, whiccomparable to the MPAE resulting from application of FBto noiseless polyenergetic data. Examination of image pfiles @Figs. 6~d! and 6~e!# reveals nearly identical ‘‘cupping’’artifacts.

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FIG. 6. Horizontal profiles through images derived from simulated data including polyenergetic spectrum, noise (1.63106 photons unless specified! andscatter.~a! FBP;~b! and~c! AM with scatter and beam hardening correction,~b! rods known and~c! rods unknown;~d! FBP operating on noiseless, scatter-frdata;~e! AM without beam-hardening correction and known rod geometries operating on noiseless, scatter-free data.

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IV. DISCUSSION

Our study demonstrates several important findings. Wregard to FBP images, commonly seen metal streakingfacts are not due solely or even mostly to ‘‘missing’’ singram data. Indeed, Fig. 3~a! demonstrates that in the preence of nearly opaque structures, FBP images are quite g


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when operating on otherwise idealized noiseless, scatter-monoenergetic sinograms that match the simplified sigformation model assumed by FBP. Rather, streaking artifaarise from system nonlinearities~scatter and beam hardening!. In addition, noise, in the form of large statistical flutuations in photon-starved sinogram regions shadowed


TABLE I. Quantitative performance of image reconstruction models for various data models~i.e., simulated sinogram datasets!. FBP, filtered backprojection;IDB, deterministic iterative deblurring; AM1, alternating minimization assuming scatter-free monoenergetic data model; AM2, alternating minimizationassuming scatter-free polyenergetic data model; AM3, alternating minimization assuming polyenergetic data model and scatter correction.

Data model AlgorithmRod

m(x,u)

Lucite core Water interior

% relbias

% abserror

% relnoise

% relbias

% abserror

% relnoise

MonoenergeticNo NoiseNo Scatter

FBP N/A 0.04 2.24 N/A 0.09 0.76 N/AIDB N 20.18 1.01 N/A 20.07 0.23 N/AIDB Y 20.04 0.19 N/A 20.07 0.17 N/AAM1 N 0.38 4.97 N/A 0.01 0.91 N/AAM1 Y 20.06 0.23 N/A 0.03 0.22 N/A

MonoenergeticNoise (I 051.63106)

No scatter

FBP N/A 2.02 31.28 63.60 0.1 8.4 15.4IDB N 1.19 19.77 37.60 0.07 5.80 9.90IDB Y 0.33 18.76 35.40 0.10 5.57 9.30AM1 N 0.37 5.16 2.20 0.01 0.96 0.60AM1 Y 0.04 0.53 0.60 0.02 0.42 0.40

MonoenergeticNo noiseScatter

FBP N/A 3.00 37.75 N/A 0.09 7.15 N/A

PolyenergeticNo noiseNo scatter

FBP N/A 7.90 9.50 N/A 10.10 11.22 N/AIDB Y 7.84 8.98 N/A 9.95 11.04 N/AAM1 Y 7.86 8.73 N/A 10.00 11.13 N/AAM2 Y 20.03 0.25 N/A 0.01 0.29 N/A

PolyenergeticNoise (I 0513105)

No scatter

FBP N/A 10.60 48.57 71.00 10.10 17.28 16.50AM1 Y 7.87 8.81 2.50 10.00 11.12 1.70AM2 Y 20.02 2.40 3.00 0.01 1.57 2.00

PolyenergeticNoise (I 051.63106)

No scatter

FBP N/A 9.63 38.77 57.70 10.10 15.79 13.50AM1 Y 7.85 8.73 0.60 10.00 11.12 0.40AM2 Y 20.02 0.79 0.90 0.01 0.56 0.60

PolyenergeticNo noiseScatter

FBP N/A 10.30 42.68 N/A 10.00 15.74 N/AIDB Y 23.47 24.75 N/A 9.46 12.11 N/AAM2 Y 20.30 0.99 N/A 20.04 0.35 N/AAM3 Y 20.03 0.25 N/A 0.01 0.29 N/AAM3 N 0.34 5.31 N/A 0.02 0.83 N/A

PolyenergeticNoise (I 0513105)

Scatter

FBP N/A 10.30 43.52 8.90 10.00 16.20 6.90IDB Y 22.77 29.36 17.70 9.49 14.20 11.60AM2 Y 20.31 2.71 3.00 20.04 1.57 2.00AM3 Y 20.03 2.42 3.00 0.01 1.56 2.00AM3 N 0.34 6.00 3.00 0.02 1.78 2.00

PolyenergeticNoise (I 051.63106)

Scatter

FBP N/A 10.30 42.74 2.00 10.00 15.74 1.50IDB Y 23.45 24.83 2.40 9.46 12.13 1.60AM2 Y 20.33 1.49 1.00 20.05 0.61 0.60AM3 Y 20.03 0.74 0.80 0.01 0.56 0.60

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dense metal objects, is an important source of streakingfacts. We use the phrase ‘‘detector-model mismatch’’ tonote discrepancies between the nonlinear response of CTtectors to attenuation line integrals along the source-deteray path and the idealized linear detector response assuby the reconstruction algorithm. As noted in the Introductioeach of these detector-model mismatch mechanisms haspreviously identified by multiple authors as causes of Fimage artifacts in general. Besides this paper, only DeMarecent simulation study30 claims that detector-model mismatch is the dominant cause of FBP metal streaking artifspecifically.

Our experience demonstrates that incorporating


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known position and attenuation map of metal objects to cstrain the iterative image formation process does imprimage quality for both the deterministic iterative deblurrinand stochastic AM algorithms. However, in the presencebeam hardening, noise, and scatter, IDB images exhstreaking artifacts comparable to FBP images. As the Fand IDB algorithms suffer from identical detector modmismatches, this finding demonstrates that iterative as was FBP algorithms require accurate modeling of the CT snal acquisition process to mitigate metal artifacts. In cotrast, the AM algorithm, which accounts for scatter, the poenergetic photon spectrum, and Poisson counting statisyields images with markedly reduced metal artifacts wh

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FIG. 7. Comparison of sinograms calculated by discrete forward projection~FP! of the image, Fig. 4~e!, ~AM reconstruction without rod location knowledge!and the original discrete image used to form the simulated sinogram measurements for Fig. 4. The comparison is based upon 500 AM iterations.~a! Sinogramprofiles (s50) from forward projections of Fig. 4~e! ~broken line! and original image~solid line!; ~b! image of difference between noiseless Fig. 4 dasinogram and forward projection of Fig. 4~e!.


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presented with synthetic sinograms. Using a differmaximum-likelihood ~ML ! reconstruction algorithm withbeam hardening corrections, DeMan27 has also demonstratethat reducing detector-model mismatches mitigates mstreaking artifacts.

In contrast to previous studies, we have found that wout using a priori knowledge of metal rod locations anattenuation maps as constraints, significant streaking artifpersist in images formed by iterative reconstruction alrithms, even when the sinogram data is exactly matchethe algorithm’s assumptions as in the case of AM@Figs. 4~e!and 4~g! for idealized monoenergetic sinograms and F5~b! for more realistic sinograms#. These residual streaks introduce mean percent absolute errors of 5%–6% in recstructed Lucite core attenuation coefficients. Simulatiobased on the convex ML algorithm40 exhibit nearly identicalresidual streaks. DeMan’s51,27 simulations, based upon yeanother ML statistical algorithm,41 exhibit qualitatively simi-lar artifacts. In contrast, IDB images formed without priknowledge exhibit much less pronounced residual streakartifacts. Because our sinogram model is derived from a velized phantom and a discrete forward projection operathat exactly matches the projection operators used byalgorithms themselves, edge-gradient effects cannot expour findings. We conclude that eliminating detector-momismatch does not, in itself, sufficiently mitigate metal arfacts on images formed by iteratively maximizing the Poson transmission log-likelihood, ln@P(duc)#. In addition, priorknowledge of metal object attenuation maps must be useconstrain ML iterations. For images consisting only of nmal anatomic constituents, AM is able to reconstruct neaartifact-free images without the aid of prior knowledge.

Figure 7 demonstrates that the discrete forward projecof the streaky image@Fig. 4~e!# formed by unconstrained AMiterations is nearly identical to that of the ‘‘truth’’ imageused to synthesize the model sinogram, except for smallviations adjacent to the high density metal. Derivatives ofobjective function for the image in Fig. 4~e! are near zero,suggesting that AM is approaching a fixed point, represeing either a local or global minimum. As the number of Aiterations increases, the differences between the two sgrams continue to decrease, as do the streaks, althstreaks are still evident after 400 000 iterations. A possexplanation for the poor convergence of AM, or convergento an image other than the truth image, is that the iterasolution may contain some component of a null space imaThe null space is the set ofmn(x) such that*Xh(yux)3@m(x)1mn(x)#dx5*Xh(yux)m(x)dx. From prior workon the Radon transform,52–54we know that the null space oh(yux) contains many nonzero functions for discretey whichdepend on pixel size, detector discretization, and theproximation selected for evaluation of the forward operaAvoidance of null space solutions may be a mechanismwhich prior information improves statistical algorithm covergence. Unlike the nonlinear AM iterations, FBP and IDare essentially linear transformation algorithms, with sotion spaces orthogonal to the null space. We are continuinexplore the structure of the null space and its relation to


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AM iterations to deepen our understanding of this phenoenon and to identify more effective and flexible formsprior knowledge constraints.

Our study has a number of important limitations. Becauour sinogram data models are based upon discrete forwprojections of discrete phantoms, our analysis does notclude edge-gradient effects, finite focal spot effects, or mtion blur, some of which30 have been shown to be importacontributors to metal artifacts. Secondly, our AM simulatioare based upon idealized phantoms consisting of wequivalent materials@N51 in Eq. ~4!#. Our efforts55 to usedual energy CT imaging to measure photon cross sectionthe energy range of interest to brachytherapy~20–1000 keV!indicate that at least two constituents are needed to arately represent the photon cross sections of biologicalsues. Finally, this study omits treatment of regularizatioRegularization, either in the form of sieves56 or penalty-driven regularization schemes57 ~modified objective func-tions that penalize undesirable solution behavior, e.g., lavoxel-to-voxel fluctuations!, could significantly reduce metastreaking and noise-related artifacts usually at the expensother characteristics such as spatial resolution. Finally, Aalong with all other published ML transmission CT algrithms, ignores the fact that modern CT detectors are eneintegrating rather than photon-counting detectors. A mcomplex compound Poisson distribution is needed toscribe the stochastic behavior of energy-integratdetectors,44 although the simple Poisson distribution mayan acceptable approximation under many circumstances

An important goal of our future research is to test Aiterative reconstruction on sinograms experimentallyquired from clinical scanners with metal objects presentthe phantom scanned. Preliminary tests demonstrate thaages reconstructed from experimentally scanned rod phtoms exhibit streaks similar to Fig. 4~e!. Integration of AMreconstruction into the OCCT framework will permit iterative localization of high density rod locations, so that tvalue of constrained AM iterative image reconstruction cbe assessed. Validation of AM-based metal artifact reducrequires extending the algorithm to multiple-constituent mdia and may require addition of regularization and modelof edge-gradient effects. In parallel with our algorithm dvelopments, we are pursuing modeling of measured sgrams, to ensure that all important sources of detector mmatch are identified.

V. CONCLUSIONS

Contrary to the dominant view of the literature, the famiar metal streaking artifacts evident in filtered backprojectimages are almost completely due to discrepancies betwthe simplified detector model assumed by the algorithmthe actual CT signal formation process. Photon scatter, hening of the polychromatic spectrum, and especially noisephoton-starved areas of the sinogram, shadowed by dmetal objects, all contribute to this detector model mismaphenomenon. We have evaluated the performance of a nmaximum-likelihood algorithm, the alternating minimizatio

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~AM ! algorithm,29 which is based upon a statistical modelCT detector response and accounts for beam hardeningphoton scatter. Compared to filtered backprojection or deministic iterative models, AM dramatically reduces artifacby minimizing detector-model mismatch. However, evwhen model mismatch is completely eliminated, significastreaking artifacts remain. These artifactual images maypart of the null space of the discrete forward projection oerator. By using prior information regarding the locatioshape and attenuation map of dense metal objects tostrain AM’s iterative solutions, we are able to obtain neaartifact-free images. Utilization of such prior informationmade practical by our more general object-constrainedapproach,20 which combines an iterative search for the poof applicators and other rigid metal objects with AM itertions to estimate the tissue attenuation map.

ACKNOWLEDGMENTS

The study was supported, in part, by Washington Univsity and Grant No. R01 CA 75371~J. F. Williamson, Princi-pal Investigator! awarded by the U.S. National InstitutesHealth.

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