Empirical dual energy calibration „EDEC… for cone-beamcomputed tomography

Philip Stenner, Timo Berkus, and Marc KachelriessInstitute of Medical Physics, University of Erlangen-Nürnberg, Henkestrasse 91, Erlangen,91052 Germany

�Received 1 February 2007; revised 25 June 2007; accepted for publication 17 July 2007;published 24 August 2007�

Material-selective imaging using dual energy CT �DECT� relies heavily on well-calibrated materialdecomposition functions. These require the precise knowledge of the detected x-ray spectra, andeven if they are exactly known the reliability of DECT will suffer from scattered radiation. Wepropose an empirical method to determine the proper decomposition function. In contrast to otherdecomposition algorithms our empirical dual energy calibration �EDEC� technique requires neitherknowledge of the spectra nor of the attenuation coefficients. The desired material-selective raw datap1 and p2 are obtained as functions of the measured attenuation data q1 and q2 �one DECT scan=two raw data sets� by passing them through a polynomial function. The polynomial’s coefficientsare determined using a general least squares fit based on thresholded images of a calibrationphantom. The calibration phantom’s dimension should be of the same order of magnitude as the testobject, but other than that no assumptions on its exact size or positioning are made. Once thedecomposition coefficients are determined DECT raw data can be decomposed by simply passingthem through the polynomial. To demonstrate EDEC simulations of an oval CTDI phantom, a lungphantom, a thorax phantom and a mouse phantom were carried out. The method was furtherverified by measuring a physical mouse phantom, a half-and-half-cylinder phantom and a Yin-Yangphantom with a dedicated in vivo dual source micro-CT scanner. The raw data were decomposedinto their components, reconstructed, and the pixel values obtained were compared to the theoret-ical values. The determination of the calibration coefficients with EDEC is very robust and dependsonly slightly on the type of calibration phantom used. The images of the test phantoms �simulationsand measurements� show a nearly perfect agreement with the theoretical � values and densityvalues. Since EDEC is an empirical technique it inherently compensates for scatter components.The empirical dual energy calibration technique is a pragmatic, simple, and reliable calibrationapproach that produces highly quantitative DECT images. © 2007 American Association of Physi-cists in Medicine. �DOI: 10.1118/1.2769104�

Key words: dual energy CT, material decomposition, flat-panel detector CT, C-arm CT, micro-CT,artifacts, image quality

I. INTRODUCTION

Dual energy CT �DECT� is a modality where one object isscanned with two different x-ray spectra. Typically, onewould acquire an object with two different tube voltages U1

and U2 but other possibilities such as different prefiltration,postfiltration, or stacked or sandwich detectors are in use,too. Basically, DECT can be used to perform energy- andmaterial-selective reconstruction and, as a side effect, it canbe used to remove beam hardening.1–6

While radiographic dual energy techniques are popular inbaggage screening or in bone densitometry there has beenonly one clinical DECT product implementation.4 In thiscase the dual spectrum was achieved by rapidly switchingbetween U1 and U2 from projection to projection. Recently, aclinical dual source spiral cone-beam CT scanner �SomatomDefinition, Siemens Medical Solutions, Forchheim, Ger-many� became available �Fig. 1�. It naturally allows us toperform dual energy scans without tube voltage switching,

and it is likely that there will be a renaissance of DECT in

3630 Med. Phys. 34 „9…, September 2007 0094-2405/2007/34

the near future. In addition to clinical CT our aim is to in-vestigate the potential of preclinical DECT with respect to ahigh-speed in vivo dual source cone-beam micro-CT scanner�TomoScope 30s Duo, VAMP GmbH, Erlangen, Germany,Fig. 1�.

Dual energy CT relies on the assumption that the linearattenuation coefficient ��r ,E�, that is a function of location rand photon energy E, can be decomposed as

��r,E� = f1�r��1�E� + f2�r��2�E� ,

with two a priori known independent energy dependencies�i�E� and the two material images f1�r� and f2�r�. One ex-ample of these energy dependencies is the decomposition ofthe linear attenuation coefficient into contributions of photo-electric and Compton scattering interactions ��E� and ��E�.The corresponding spatial dependencies f��r� and f��r�would then represent the cross sections of these effects. An-other possibility is to choose two basis materials, say water�i=1� and bone �i=2�. Let �i�E�= �� /��i�E� be the mass

attenuation coefficient for material i. Then, one can recon-

3630„9…/3630/12/$23.00 © 2007 Am. Assoc. Phys. Med.

3631 Stenner, Berkus, and Kachelriess: EDEC 3631

struct the density distributions of water and bone f i�r�=�i�r�.

To reconstruct the material images f i�r� one needs toknow the corresponding line integrals along line L:

pi�L� = Rf i = �L

dlf i�r� .

For convenience, we avoid stating the explicit dependencieson L in the following. R denotes the Radon transform opera-tor in two dimensions, and the x-ray transform operator inthree dimensions, respectively. The measurements, however,rather yield

qj = − ln � dE wj�E�e−p1�1�E�−p2�2�E� + Sj ,

where w1�E� and w2�E� are the two detected spectra usedduring the scan, normalized to unit area. Obviously qj

=qj�p1 , p2�. Note that the index i counts the basis functions�basis materials� while the index j counts the detected spec-tra involved.

The main issue in material decomposition is to find theinverse pi= pi�q1 ,q2� and relies on exactly knowing wj�E�and �i�E�, which are difficult to measure. Even if wj�E� and�i�E� were known, scattered radiation Sj would impair thesuccess of a straightforward inversion of the measurementequation.

To avoid these difficulties, empirical calibration tech-niques are in use that correlate measurements of known ab-sorbers with known positioning to the analytically derivedintersection length of the rays and the absorbers.3 For ex-ample two wedges of different basis materials, of known sizeand orientation, may be used to carry out such an empiricalcalibration. We did not follow this approach because of tworeasons: On the one hand it is an error prone and lengthyprocess since it requires scanning of all possible ray paths

7

through both of the two materials. On the other hand it is

Medical Physics, Vol. 34, No. 9, September 2007

very difficult to manufacture these wedge phantoms on asmall scale, and it is nearly impossible to correctly positionthem in a micro-CT scanner.

We propose a novel empirical calibration algorithm. Incontrast to other methods, EDEC does not require us to knowthe spectra, to know the attenuation coefficients, nor does itneed to have exact knowledge of the calibration phantomgeometry, size, and position. EDEC is based on similar ideasas the empirical cupping correction �ECC� and the interestedreader may refer to Ref. 8.

In the following section we will explain the theory ofEDEC. In order to verify the method we carried out simula-

FIG. 2. Basis images obtained from a DECT scan of a calibration phantom.Note that the gray scale window is chosen individually for each image todisplay the range mean ±1.5 sigma from black to white, where mean is themean value, and sigma is the standard deviation of all pixels within an

FIG. 1. Dual source CT scanners can be used to acquireDECT data �left: clinical CT, right: micro-CT�. Typi-cally, both source-detector systems are mounted underan angle of 90°.

image.

˜

3632 Stenner, Berkus, and Kachelriess: EDEC 3632

tions in the scale of clinical and micro-CT and obtained mea-surements of several micro-CT phantoms. The simulated andphysical phantoms are introduced in Sec. III. We present anddiscuss the results in Sec. IV. Section V contains the sum-mary and conclusion.

II. METHOD

Let qj be the polychromatic CT raw data and pi the de-sired monochromatic, material-specific raw data value. Theindex j=1,2 counts the detected spectra, and i=1,2 countsthe materials to decompose for. We define

pi = Di�q1,q2� ,

where Di is some yet unknown decomposition function. Let

Di�q1,q2� = �n=0

N−1

cinbn�q1,q2� = ci · b�q1,q2�

be a linear combination of basis functions bn�q1 ,q2�. In ourcase we use the polynomials bn�q1 ,q2�=q1

kq2l as basis func-

tions with k=0, . . . ,K and l=0, . . . ,L and n=k�L+1�+ l. Thetotal number of basis functions is N= �K+1��L+1�. K=L=4 turns out to be an adequate choice for our data and hence25 basis functions are used here. The total number of coef-

FIG. 3. Images of the calibration phantom consisting of water �i=1� and alucalibration scans. The rows of the 2�5 image matrix illustrate the calibrationdepict the weight wi�r�, the template ti�r�, the weighted template wi�r�ti�r− ti�r��. The standard reconstructions are windowed to �C=0 HU/W=200=20% �, where 100% corresponds to the theoretical value of water, and theand weighted template images are binary and shown as black and white.

ficients in each coefficient vector ci is 25 for each material,

Medical Physics, Vol. 34, No. 9, September 2007

which makes 50 unknown coefficients in total. The purposeof EDEC is to determine the coefficients ci. Note that EDECis not restricted to polynomials. If desired other basis func-tions can be used.

It should be mentioned that the decomposition must becarried out in the raw data domain unless one chooses basisfunctions that decompose as

bn�q1,q2� = b̂n�q1� + b̃n�q2� .

The latter can be achieved by choosing b̂n�q1�=q1k and

bn�q2�=q2l with k=0, . . . ,K and l=1, . . . ,L, for example �K

+L+1 coefficients per material in total�. Such a situationwould allow us to sum the j=1 data and the j=2 data in theimage domain and therefore belongs to the class of the so-called energy subtraction methods. Due to the lower preci-sion and image quality of those energy subtractionmethods—for example energy subtraction cannot removebeam hardening artifacts—we do not follow this strategy. Werather allow for the full flexibility of raw data-based�projection-based� DECT reconstruction and do not make

m �i=2�. The pair of images on top are standard reconstructions of the twos and results for water �material 1� and aluminum �material 2�. The columnsfinal decomposition results f i�r�, and the weighted difference wi�r��f i�r�

. The window setting for the material-selective images is �C=100% /Whted difference is displayed as �C=0% /W=5% �. The weight and template

minustep

�, theHU�

weig

any further assumptions on bn�q1 ,q2�.

3633 Stenner, Berkus, and Kachelriess: EDEC 3633

To simplify notation let us drop the material index i andlet us determine the decomposition coefficients c=c1 for ma-terial 1. Decomposing for material 2 can be done in completeanalogy.

Making use of the linearity of the x-ray transform R a setof �K+1��L+1� basis images is defined as fn�r�=R−1bn�q1 ,q2�, i.e., fn is the reconstruction of the raw dataq1 and q2 after they have been passed through the basis func-tions bn. The reconstruction of the material-selective rawdata can now be written as a linear combination of thesebasis images

f�r� = R−1D�q1,q2� = R−1�n

cnbn�q1,q2�

= �n

cnR−1bn�q1,q2� = �

n

cnfn�r� = c · f�r� .

We want to find the set of coefficients c that minimizes theleast square deviation

E2 =� d2r w�r��f�r� − t�r��2,

between the linearly combined basis images f�r�=c · f�r� anda given template image t�r�. The weight image w�r� is usedto avoid unwanted structures of the calibration object, whichwould impair the optimization procedure.

Medical Physics, Vol. 34, No. 9, September 2007

To obtain the basis images, the template image and theweight image a DECT scan �two raw data sets with differentdetected spectra� of a calibration phantom is performed. Thecalibration phantom contains homogeneous areas with suffi-cient amounts of material 1 and material 2 and ensures thatall reasonable path length combinations of material 1 andmaterial 2 are acquired. To obtain the basis images these rawdata, q1 and q2, are passed through the �K+1��L+1� basisfunctions and are reconstructed. With our choice of basisfunctions we generate 25 sinograms whose entries are q1

kq2l

and reconstruct those �Fig. 2�. A standard reconstruction ofthe calibration phantom is used to determine t�r� and w�r� bysimple thresholding as follows.

Basically, t�r� shall represent our a priori knowledge ofthose regions that correspond to material 1 and those regionsthat definitely do not contain material 1 �and thus containmaterial 2 or air�:

t�r� = �1 for r � material 1,

0 for r � air and material 2.�

Note that the template need not be defined in regions thatmay contain a mixture of both basis materials 1 and 2 sincethese regions are suppressed by the weight image. Theweight image is set to one whenever we are sure about thecontents of voxel r. This is the case in material 1 regions, in

FIG. 4. The photograph shows the mouse phantom�left� and the two-cylinders phantom that was used forcalibration �right�.

FIG. 5. The photographs show the half-and-half phan-tom �left� and the Yin-Yang phantom �right�.

3634 Stenner, Berkus, and Kachelriess: EDEC 3634

material 2 regions, and in air regions. We set w�r� to zerowherever we are outside the field of measurement, wheneverwe expect point spread function effects �those regions are atthe edges of the homogeneous areas and are found by ero-sion� or whenever we encounter a material that is neitherbasis material nor air:

w�r� = �1 for r � material 1 or 2 or air after erosion,

0 for r � eroded boundaries, unknown material.�

Now we are ready to minimize E2. Differentiating E2 withrespect to cn yields the linear system a=B ·c with

an =� d2r w�r�fn�r�t�r� ,

Bnm =� d2r w�r�fn�r�fm�r� .

The solution to the optimization problem is simply givenas c=B−1 ·a. An actual implementation must avoid explicitlycomputing B−1 and should rather use Gauss-Jordan elimina-tion to invert the linear system. Figure 3 shows the interme-diate steps necessary to acquire the calibration coefficients.The difference of the water image to the water template andof the aluminum image to the aluminum template show thatour method works very well. The phantom shown as well asall other simulated and physical phantoms are invariant un-der translation along the z axis, hence we show only singleslices and apply EDEC to this two-dimensional �2D� case.EDEC also works in three dimensions as long as the segmen-tation of the phantoms is done in three dimensions. Underthese circumstances all three appearances of d2r have to bereplaced by d3r.

Our method is an empirical correction, and thus it correctsbeam-hardening-based cupping as well as cupping due toscattered radiation, but EDEC cannot provide a channel-dependent correction. In the presented method the decompo-sition function D�q1 ,q2� does not depend on the line of inte-

gration L. In order to account for the effects of an x-ray

Medical Physics, Vol. 34, No. 9, September 2007

spectrum that varies across the detector the decompositionfunction also needs to be a function of L. The reasons for thisare manifold, such as the heel effect �anode angle effect�, theuse of a bow-tie filter, x-ray scatter, or the varying detectorefficiency as a function of the intersection length of the x rayand the scintillator material.8 A family of precorrection func-tions D�q1 ,q2 ,L� that depend on each ray L may be derivedanalytically if one makes assumptions on several parameters,e.g., the anode material and angle, the prefiltration, and thedetector absorption. One may combine EDEC with such ananalytical precorrection to obtain a hybrid approach similarto the technique presented in Ref. 9. For the measurementsand experiments at hand EDEC alone proved to be sufficient.

III. SIMULATIONS AND MEASUREMENTS

We performed simulations in the scale of clinical CT andmicro-CT and carried out measurements with our dual sourcemicro-CT. Measurements with our clinical DSCT were omit-ted due to the costly production of large-scale calibrationphantoms.

III.A. Simulations

Semiempirical spectra were used for all simulations.10

III.A.1. Test phantoms

To validate our calibration technique in the scale of clini-cal CT we simulated three test phantoms. �For more infor-mation please refer to our website www.imp.uni-erlangen.de/phantoms.� The first one is a generalized 32 cm�16 cm ovalCTDI phantom with water equivalent background and fivealuminum inserts. The other test phantoms were a lung phan-tom consisting of fat, soft tissue, cortical, and spongiousbone and a thorax phantom consisting of soft tissue, contrastagent, cortical, and spongious bone. All simulations werecarried out at 80 and 140 kV.

A mouse phantom was simulated whose size

FIG. 6. A simulated 32 cm CTDI test phantom consist-ing of water and aluminum. The material decomposi-tion uses the coefficients determined from the calibra-tion data of Fig. 3. Tube voltages are 80 and 140 kV.The standard reconstructions and the monochromaticimage are windowed to �C=0 HU/W=200 HU�. Thewindow setting for the material-selective images is �C=100% /W=20% �.

�width=32 mm, height=24 mm� meets the requirements of a

3635 Stenner, Berkus, and Kachelriess: EDEC 3635

micro-CT scan. The phantom body consisted of waterequivalent plastic, and the two high contrast inserts weremade up of different concentrations of iodine. The two smallbones consisted of hydroxiapatite �200 mg/mL� and thethree large bones of hydroxiapatite �400 mg/mL�. Themouse phantom was simulated at tube voltages of 30 and65 kV.

III.A.2. Calibration phantoms

We simulated a 45 cm Yin–Yang phantom consisting ofwater and aluminum to illustrate the various steps of ourmethod. This simulation was performed at 80 and 140 kVand is shown in Fig. 3. The Yin–Yang phantom also served

FIG. 7. The bottom row shows standard reconstructions at 120 kV that havimages in the 3�3 image matrix. The monochromatic images are density wartifacts in the standard reconstruction images are marked with arrows. Oncombinations through both materials exist perform well. The decompositionposition results whereas the decomposition using a phantom where materiaartifacts connecting the aluminum inserts. All images are windowed to �C=

as a calibration phantom for the CTDI phantom. In order to

Medical Physics, Vol. 34, No. 9, September 2007

examine the effects of different geometries of the calibrationphantom on the decomposition of the test phantom we fur-ther simulated three additional types of calibration phantoms:the cylinder-in-cylinder phantom, the three-quarter-and-quarter-cylinder phantom and the two-cylinders phantom.The first two have a diameter of 32 cm. The diameters of thelatter are 16 and 24 cm, respectively. The material combina-tions were water with aluminum and water with bone, de-pending on the type of test phantom the calibration was per-formed for. All simulations were carried out at 80 and140 kV.

In the scale of micro-CT we simulated a second two-cylinders phantom that served as a calibration phantom forthe mouse phantom. The large cylinder consisted of water

en acquired with the same patient dose as the nine monochromatic DECTted and have been simulated at 80 and 140 kV. Significant beam hardeningibration phantoms where ray paths through either material 1 or 2 and path

on the three-quarter-and-quarter-cylinder phantom yields excellent decom-fully embedded in material 1 like the cylinder-in-cylinder phantom shows/W=200 HU�.

e beeigh

ly calbasedl 2 is0 HU

equivalent plastic and the small cylinder of hydroxiapatite

ation

3636 Stenner, Berkus, and Kachelriess: EDEC 3636

�400 mg/mL�. The cylinders’ diameters were 20 and 10 mm.The phantom was simulated at tube voltages of 30 and65 kV.

III.B. Measurements

All physical phantoms had a length of 40 mm and weremanufactured by QRM �Möhrendorf, Germany�. The rawdata were acquired with a dual source cone-beam micro-CTscanner and a Feldkamp-type algorithm was used to recon-struct the images. Our method is applicable to various kindsof CT scanners, although cone-beam CT is the method ofchoice in the present article.

III.B.1. Test phantoms

We performed micro-CT measurements of three 32 mmtest phantoms. The physical mouse phantom was built ac-cording to the previously described simulated version, exceptthat the concentrations of hydroxiapatite for the bones werereduced by 50% to guarantee moderate attenuation �Fig. 4�.The phantom was scanned at 40 and 50 kV. The half-and-half phantom consisted of two half cylinders, one of waterequivalent plastic, the other of a mixture of calcium hydrox-

FIG. 8. Quantitative results achieved at various tube voltage combinations �rand for density images �columns�. The tube current was chosen to keep thedirectly comparable. The values given in the images are mean � standard devin the water background close to the upper right edge of the phantom. The vvoltage combinations that are far apart �here for the 80 and 140 kV combin

iapatite �200 mg/mL� and water equivalent plastic �Fig. 5�.

Medical Physics, Vol. 34, No. 9, September 2007

The half-and-half phantom was scanned at 40 and 140 kV.The third physical test phantom was a small Yin–Yang phan-tom that was also made up of two materials, one being waterequivalent plastic and the other being a mixture of calciumhydroxiapatite �400 mg/mL� and water equivalent plastic�Fig. 5�. The employed voltages were 80 and 140 kV.

III.B.2. Calibration phantoms

The mouse phantom was calibrated with a physical two-cylinders phantom that also matches its simulated version insize and geometry. The large cylinder was made up of waterequivalent plastic and the small cylinder consisted of hy-droxiapatite �200 mg/mL�. The two-cylinders phantom wasalso measured at 40 and 50 kV and is shown in Fig. 4. Thedecomposition coefficients for the half-and-half-cylinderphantom and the Yin–Yang phantom were determined fromthe respective phantom itself, thus the test and calibrationphantom were identical in these two cases.

The effects of cross-scatter constitute a major drawbacksince scattered radiation from one tube-detector system willaffect the measurements of the other tube-detector system. Inorder to eliminate cross-scatter we performed two single-energy scans instead of one dual-energy scan. The phantoms

for monochromatic images of the attenuation coefficient at 70 and 511 keVpatient dose for all three voltage pairs constant, hence the noise levels are

n measured in an ROI centered in the central aluminum insert and in an ROIare further normalized to water. As expected, image noise is lowest for tube�. The calibration was performed with the two-cylinders phantom.

ows�totaliatio

alues

were scanned twice at two different voltages with one of the

3637 Stenner, Berkus, and Kachelriess: EDEC 3637

tubes turned off during both scans. When using both tubes atthe same time a proprietary scanner-internal scatter precor-rection compensates for the cross-scatter effects.

IV. RESULTS

The results of the simulations were analyzed both quali-tatively and quantitatively. For the measurements we per-formed only qualitative analysis, since the exact compositionof the employed materials is unknown.

IV.A. Simulations

The CTDI test phantom was correctly decomposed intowater and aluminum �Fig. 6�, and the severe beam hardeningartifacts present in the images of the polychromatic raw-datawere almost completely removed. The CTDI phantom wasdecomposed using the decomposition coefficients c1 and c2

obtained from the Yin-Yang calibration phantom from Fig. 3.Figure 7 shows that the decomposition of the CTDI phan-

tom also works with coefficients c1 and c2 that were acquiredwith calibration phantoms other than the Yin–Yang phantom.In addition to the CTDI phantom we applied EDEC to thelung and thorax test phantoms. The following three calibra-tion phantoms were used: the cylinder-in-cylinder phantom,the three-quarter-and-quarter-cylinder phantom and the two-cylinders phantom. Our experiments showed that all phan-

TABLE I. The first column shows the density values that were obtained fromthree ROIs in the combined monochromatic image of the simulated mousephantom. The second column shows the density values that were used forthe simulation.

Material

Density valuesfrom

ROI/mg mm−3

Density valuesused for

simulation/mg mm−3

Water 1.00±0.02 1.00Hydroxiapatite �400 mg/mL� 1.28±0.02 1.27Hydroxiapatite �200 mg/mL� 1.15±0.02 1.14

Medical Physics, Vol. 34, No. 9, September 2007

toms were suited for calibration although two conditionsmust be met to achieve excellent decomposition: The cali-bration phantom must allow for line integrals that cover �a�both materials exclusively and �b� all path length combina-tions through material 1 and material 2 simultaneously.

For the DECT images of the oval CTDI phantom theEDEC calibration was carried out for water and aluminum.For the other two test phantoms the calibration phantomswere filled with water and bone. The DECT scan was takenat 80 and 140 kV. The bottom row of Fig. 7 further showsthe images that are obtained from a standard 120 kV scanand reconstruction. The tube currents were chosen: �a� toensure the same patient dose for the 120 kV scan and for theDECT scan �two spectra� and �b� to optimally distribute theavailable dose among the 80 and 140 kV exposures. We bal-ance the tube current between the low and the high energyscan in a way to minimize the variance in the final destina-tion image �automatic exposure control �AEC� for DECT�.11

The effects can be best seen regarding the oval CTDIphantom images in Fig. 7. The standard reconstruction at120 kV shows significant beam hardening artifacts, whichare marked with arrows: dark bands of high amplitude thatconnect the aluminum inserts. These artifacts should be com-pletely removed after projection-based DECT reconstruction�they would not be removable if only image-based energysubtraction techniques were used�.

Obviously, the cylinder-in-cylinder calibration phantomproduces the least accurate coefficients: The dark bands aregone but white bands of low amplitude appear at the sameplace. The other two calibration phantoms, the three-quarter-and-quarter-cylinder and the two-cylinders phantom,achieved superior results. Here, rays through either one ofthe basis materials as well as rays that run through bothmaterials exist, and the monochromatic images of the testphantom are close to perfect. A very close look still revealssome artifacts for the three-quarter-and-quarter-cylinderphantom. This is probably due to the fact that the number of

FIG. 9. A simulated 32 mm�24 mm mouse test phan-tom. Tube voltages are 30 and 65 kV. The standardreconstructions and the monochromatic image are win-dowed to �C=0 HU/W=200 HU�. The window settingfor the material-selective images is �C=100% /W=20% �.

ray paths that intersect only the aluminum region is much

es is

3638 Stenner, Berkus, and Kachelriess: EDEC 3638

smaller in the three-quarter-and-quarter-cylinder phantomcompared to the two-cylinders phantom. From these obser-vations one can conclude that a suitable test phantom mustallow for rays that run in either material as well as in bothmaterials, and the amount of basis material in the calibrationphantom should approximate the expected imaging situation.

Given that the two-cylinders phantom provides the bestimage quality quantitative values were analyzed, too. Figure8 shows reconstructions of the oval CTDI phantom that werecalibrated with the two-cylinders phantom. The simulationswere carried out at 120 and 140 kV, at 80 and 120 kV, and at80 and 140 kV. The decomposed data were combined toyield � images at 70 keV, which is the effective energy typi-cal for clinical CT, at 511 keV, the energy that is importantfor PET attenuation correction, and they were combined toyield density images �� images�.

The tube currents were selected using the same method asbefore. Therefore we can ensure the same patient dose for all

FIG. 10. The physical 32 mm mouse test phantom that was decomposed withscan was performed with 40 and 50 kV spectra. The original polychromaticlarge bones deteriorate image quality. The borders of the phantom body ycombined monochromatic image. The standard reconstructions, the monoch=0 HU/W=600 HU�, and the window setting for the material-selective imag

three tube current combinations. Note that this implies that

Medical Physics, Vol. 34, No. 9, September 2007

different tube currents were simulated for the 70 keV, for the511 keV and for the � images, altogether the figure showsnine different DECT scans.

Two ROIs were place into each image. One ROI that mea-sures the mean and standard deviation within the central alu-minum insert and one ROI to determine the statistics for thephantom background �water�. As demonstrated in Fig. 8 im-age noise decreases for tube voltage combinations that arefar apart, i.e. it decreases from top to bottom. Since the totalpatient dose for all scans remained constant the values aredirectly comparable. The theoretical attenuation and densityvalues are as follows:

��Al, 70 keV���H2O, 70 keV�

= 3.224,

��Al, 511 keV�= 2.355,

ficients obtained from the two-cylinders calibration phantom. The micro-CTes show strong beam-hardening artifacts: Dark bands connecting the threeigher CT values due to cupping. Both effects are compensated for in thetic image, and the image of the calibration phantom are windowed to �C�C=100% /W=60% �. The monochromatic image is � weighted at 25 keV.

coefimag

ield hroma

��H2O, 511 keV�

3639 Stenner, Berkus, and Kachelriess: EDEC 3639

��Al���H2O�

= 2.699.

Regarding the mean values of the ROIs we immediatelyfind that a very high accuracy is achieved. Water should be at1000 since the images are normalized to the theoretical watervalue. The deviation of the reconstructed water value is lessthan 0.2% in all cases. Given the theoretical values we findthat aluminum should be at 3224 for the 70 keV, at 2355 forthe 511 keV, and at 2699 for the density image. The valuesactually measured are very close to the theoretical predic-tions. Obviously, the 25 calibration coefficients determinedby EDEC from the two-cylinders calibration phantom scansare highly accurate, and our choice K=L=4 appears to besufficient.

EDEC also delivers excellent results for the micro-CTscale. Figure 9 shows that the mouse phantom was accu-rately decomposed into water �basis material 1� and hydrox-iapatite �400 mg/mL, basis material 2�. The combined imageis density weighted. Three circular ROIs were placed in thephantom: one in the phantom body, one in the lower leftbone, and one in the upper left bone. The correspondingmaterials are water, hydroxiapatite �400 mg/mL� and hy-droxiapatite �200 mg/mL�, respectively. The density valuesthat were acquired in the combined monochromatic imagefor those ROIs are shown in Table I. The comparison to thedensity values that we used for the simulation confirms thatour method works very well.

IV.B. Measurements

Similarly, EDEC achieves an accurate decomposition incase of real micro-CT data. The physical mouse phantomwas decomposed with the calibration coefficients obtainedfrom the two-cylinders phantom. Figure 10 shows that de-

compositions based on these coefficients yield excellent re-

Medical Physics, Vol. 34, No. 9, September 2007

sults: The material-selective image for basis material 1 �wa-ter� solely shows the phantom body with traces of the highcontrast inserts. The material-selective image for basis mate-rial 2 �hydroxiapatite, 200 mg/mL� shows the three largebones, as expected. The two small bones are not visible dueto the window setting used. The original polychromatic im-ages are subject to strong beam-hardening artifacts: Darkbands connecting the three large bones deteriorate imagequality. The borders of the phantom body yield higher CTvalues due to cupping. Both effects are compensated for inthe combined monochromatic image. At the lower right sideof the phantom body the decomposition produces an over-shoot, which is caused by scatter. Other than that EDEC isapparently not prone to effects of scatter that could arisefrom the different phantom sizes.

Both the half-and-half-cylinder phantom and the Yin-Yang phantom have been correctly decomposed into theirtwo materials �Figs. 11 and 12�. In addition to the correctdecomposition EDEC removes cupping artifacts that are wellpronounced in the original polychromatic raw data. Figure13 illustrates this for the case of the Yin-Yang phantom. Itshows the CT values of two row profiles from the polychro-matic and the monochromatic images taken from Fig. 12.The profiles of the polychromatic image show significantcupping whereas the profiles of the monochromatic imageare perfectly flat.

V. SUMMARY AND CONCLUSION

The empirical dual energy calibration �EDEC� is a simple,effective and accurate way to calibrate dual energy CT. Itrelies on a calibration phantom that must provide path lengthvariations through material 1, material 2, and combinationsof path lengths through both materials. In contrast to otherraw-data-based calibration techniques the exact shape, posi-

FIG. 11. A physical 32 mm half-and-half-cylinder testphantom consisting of water equivalent plastic and hy-droxiapatite. The micro-CT scan was performed with40 and 140 kV spectra, and the material decompositionuses the coefficients determined from the phantom it-self. The standard reconstructions and the monochro-matic image are windowed to �C=0 HU/W=600 HU�.The window setting for the material-selective images is�C=100% /W=60% �.

tion, and orientation of the phantom does not have to be

3640 Stenner, Berkus, and Kachelriess: EDEC 3640

known. EDEC determines the material decomposition coef-ficients in the image domain. The coefficients are then typi-cally used in the raw-data domain by passing pairs of attenu-ation values q1 and q2 �same ray at two tube voltages�through a simple polynomial:

pi = �k=0

K

�l=0

L

cinq1kq2

l with n = k�L + 1� + l .

The raw data p1 for basis material 1 and p2 for basis material2 then undergo the standard image reconstruction procedure�e.g., a Feldkamp-type cone-beam reconstruction�.

If desired, one may avoid products of q1 and q2 in thebasis functions and may then perform image-based DECTdecomposition �not shown in this article�. This kind of en-ergy subtraction calibration is likely to be more efficient andmore accurate than today’s procedure of simply having twowater-precorrected images �at two tube voltages� being lin-early combined.

We have demonstrated that the quality of EDEC decom-position slightly depends on the choice of the calibrationphantom and we have identified phantom designs that arewell suited. It was further shown that the sets of coefficientsobtained achieve precise quantitative results.

Above all EDEC was validated with measurements per-formed with a dual source micro-CT. The physical phantomswere correctly decomposed into the two basis materials. Inorder to eliminate the effects of the scanner-internal scatterprecorrection the phantoms were scanned twice at two dif-ferent voltages with one of the x-ray tubes turned off duringboth scans.

Whenever detector position-dependent corrections are re-quired where the spectrum w�L ,E� varies with each rayL—this is, for example, the case with bow-tie prefiltration—

FIG. 12. A physical 32 mm Yin–Yang test phantomconsisting of water equivalent plastic and hydroxiapa-tite. The micro-CT scan was performed with 80 and140 kV spectra, and the material decomposition usesthe coefficients determined from the phantom itself.The standard reconstructions and the monochromaticimage are windowed to �C=0 HU/W=600 HU�. Thewindow setting for the material-selective images is �C=100% /W=60% �.

one may combine EDEC with analytical precorrection or

Medical Physics, Vol. 34, No. 9, September 2007

FIG. 13. Row profiles at two different y positions �see phantom inserts� ofthe physical Yin–Yang phantom. The profiles of the polychromatic imageare taken from the measurement at voltage 1 of Fig. 12. The profiles of themonochromatic image are also taken from Fig. 12. M1 corresponds to waterand M2 to hydroxiapatite. Significant cupping is present in the regions ofM2 and moderate cupping in the areas of M1. EDEC reduces cupping inboth regions and profiles, and the resulting CT-value for M1 �water� yields

0 HU.

3641 Stenner, Berkus, and Kachelriess: EDEC 3641

material decomposition techniques and obtain a hybrid ap-proach similar to the technique presented in Ref. 9.

Altogether one may regard EDEC as being a simple andversatile method to obtain the desired material decomposi-tion functions.

ACKNOWLEDGMENTS

The study was supported in part by the DeutscheForschungsgemeinschaft �DFG� under grant FOR 661. Wethank Laura Konerth, Maria Loibl, and Peng Wang for per-forming the scans of the phantoms. We also thank ProfessorGisela Anton �Physikalisches Institut IV, University ofErlangen-Nürnberg� for her support.

a�Author to whom correspondence should be addressed. Telephone: 49�9131� 8522957; Fax: 49 �9131� 8522824. Electronic mail:marc.kachelriess@imp.uni-erlangen.de

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