Empirical dual energy calibration EDEC for cone-beam computed tomography

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1 Empirical dual energy calibration EDEC for cone-beam computed tomography Philip Stenner, Timo Berkus, and Marc Kachelriess Institute of Medical Physics, University of Erlangen-Nürnberg, Henkestrasse 91, Erlangen, 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 material decomposition functions. These require the precise knowledge of the detected x-ray spectra, and even if they are exactly known the reliability of DECT will suffer from scattered radiation. We propose an empirical method to determine the proper decomposition function. In contrast to other decomposition algorithms our empirical dual energy calibration EDEC technique requires neither knowledge of the spectra nor of the attenuation coefficients. The desired material-selective raw data p 1 and p 2 are obtained as functions of the measured attenuation data q 1 and q 2 one DECT scan =two raw data sets by passing them through a polynomial function. The polynomial s coefficients are determined using a general least squares fit based on thresholded images of a calibration phantom. The calibration phantom s dimension should be of the same order of magnitude as the test object, but other than that no assumptions on its exact size or positioning are made. Once the decomposition coefficients are determined DECT raw data can be decomposed by simply passing them through the polynomial. To demonstrate EDEC simulations of an oval CTDI phantom, a lung phantom, a thorax phantom and a mouse phantom were carried out. The method was further verified by measuring a physical mouse phantom, a half-and-half-cylinder phantom and a Yin-Yang phantom with a dedicated in vivo dual source micro-ct scanner. The raw data were decomposed into their components, reconstructed, and the pixel values obtained were compared to the theoretical values. The determination of the calibration coefficients with EDEC is very robust and depends only slightly on the type of calibration phantom used. The images of the test phantoms simulations and measurements show a nearly perfect agreement with the theoretical values and density values. Since EDEC is an empirical technique it inherently compensates for scatter components. The empirical dual energy calibration technique is a pragmatic, simple, and reliable calibration approach that produces highly quantitative DECT images American Association of Physicists in Medicine. DOI: / 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 is scanned with two different x-ray spectra. Typically, one would acquire an object with two different tube voltages U 1 and U 2 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- and material-selective reconstruction and, as a side effect, it can be used to remove beam hardening. 1 6 While radiographic dual energy techniques are popular in baggage screening or in bone densitometry there has been only one clinical DECT product implementation. 4 In this case the dual spectrum was achieved by rapidly switching between U 1 and U 2 from projection to projection. Recently, a clinical dual source spiral cone-beam CT scanner Somatom Definition, Siemens Medical Solutions, Forchheim, Germany became available Fig. 1. It naturally allows us to perform dual energy scans without tube voltage switching, and it is likely that there will be a renaissance of DECT in the near future. In addition to clinical CT our aim is to investigate the potential of preclinical DECT with respect to a high-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 linear attenuation coefficient r,e, that is a function of location r and photon energy E, can be decomposed as r,e = f 1 r 1 E + f 2 r 2 E, with two a priori known independent energy dependencies i E and the two material images f 1 r and f 2 r. One example of these energy dependencies is the decomposition of the linear attenuation coefficient into contributions of photoelectric 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. Another 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 Med. Phys. 34 9, September /2007/34 9 /3630/12/$ Am. Assoc. Phys. Med. 3630

2 3631 Stenner, Berkus, and Kachelriess: EDEC 3631 FIG. 1. Dual source CT scanners can be used to acquire DECT data left: clinical CT, right: micro-ct. Typically, both source-detector systems are mounted under an angle of 90. struct the density distributions of water and bone f i r = i r. To reconstruct the material images f i r one needs to know the corresponding line integrals along line L: p i L = Rf i = L dlf i r. For convenience, we avoid stating the explicit dependencies on L in the following. R denotes the Radon transform operator in two dimensions, and the x-ray transform operator in three dimensions, respectively. The measurements, however, rather yield very difficult to manufacture these wedge phantoms on a small scale, and it is nearly impossible to correctly position them in a micro-ct scanner. We propose a novel empirical calibration algorithm. In contrast to other methods, EDEC does not require us to know the spectra, to know the attenuation coefficients, nor does it need to have exact knowledge of the calibration phantom geometry, size, and position. EDEC is based on similar ideas as the empirical cupping correction ECC and the interested reader may refer to Ref. 8. In the following section we will explain the theory of EDEC. In order to verify the method we carried out simula- q j = ln de w j E e p 1 1 E p 2 2 E + S j, where w 1 E and w 2 E are the two detected spectra used during the scan, normalized to unit area. Obviously q j =q j p 1, p 2. Note that the index i counts the basis functions basis materials while the index j counts the detected spectra involved. The main issue in material decomposition is to find the inverse p i = p i q 1,q 2 and relies on exactly knowing w j E and i E, which are difficult to measure. Even if w j E and i E were known, scattered radiation S j would impair the success of a straightforward inversion of the measurement equation. To avoid these difficulties, empirical calibration techniques are in use that correlate measurements of known absorbers with known positioning to the analytically derived intersection length of the rays and the absorbers. 3 For example two wedges of different basis materials, of known size and orientation, may be used to carry out such an empirical calibration. We did not follow this approach because of two reasons: On the one hand it is an error prone and lengthy process since it requires scanning of all possible ray paths through both of the two materials. 7 On the other hand it is 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 to display the range mean ±1.5 sigma from black to white, where mean is the mean value, and sigma is the standard deviation of all pixels within an image.

3 3632 Stenner, Berkus, and Kachelriess: EDEC 3632 FIG. 3. Images of the calibration phantom consisting of water i=1 and aluminum i=2. The pair of images on top are standard reconstructions of the two calibration scans. The rows of the 2 5 image matrix illustrate the calibration steps and results for water material 1 and aluminum material 2. The columns depict the weight w i r, the template t i r, the weighted template w i r t i r, the final decomposition results f i r, and the weighted difference w i r f i r t i r. The standard reconstructions are windowed to C=0 HU/W=200 HU. The window setting for the material-selective images is C=100%/W =20%, where 100% corresponds to the theoretical value of water, and the weighted difference is displayed as C=0%/W=5%. The weight and template and weighted template images are binary and shown as black and white. tions in the scale of clinical and micro-ct and obtained measurements of several micro-ct phantoms. The simulated and physical phantoms are introduced in Sec. III. We present and discuss the results in Sec. IV. Section V contains the summary and conclusion. II. METHOD Let q j be the polychromatic CT raw data and p i the desired monochromatic, material-specific raw data value. The index j =1,2 counts the detected spectra, and i =1,2 counts the materials to decompose for. We define p i = D i q 1,q 2, where D i is some yet unknown decomposition function. Let N 1 D i q 1,q 2 = c in b n q 1,q 2 = c i b q 1,q 2 n=0 be a linear combination of basis functions b n q 1,q 2.Inour case we use the polynomials b n q 1,q 2 =q k l 1 q 2 as basis functions with k=0,...,k and l=0,...,l and n=k L+1 +l. The total number of basis functions is N= K+1 L+1. K=L =4 turns out to be an adequate choice for our data and hence 25 basis functions are used here. The total number of coefficients in each coefficient vector c i is 25 for each material, which makes 50 unknown coefficients in total. The purpose of EDEC is to determine the coefficients c i. Note that EDEC is not restricted to polynomials. If desired other basis functions can be used. It should be mentioned that the decomposition must be carried out in the raw data domain unless one chooses basis functions that decompose as b n q 1,q 2 = bˆ n q 1 + b n q 2. k The latter can be achieved by choosing bˆ n q 1 =q 1 and l b n q 2 =q 2 with k=0,...,k and l=1,...,l, for example K +L+1 coefficients per material in total. Such a situation would allow us to sum the j=1 data and the j=2 data in the image domain and therefore belongs to the class of the socalled energy subtraction methods. Due to the lower precision and image quality of those energy subtraction methods for example energy subtraction cannot remove beam hardening artifacts we do not follow this strategy. We rather allow for the full flexibility of raw data-based projection-based DECT reconstruction and do not make any further assumptions on b n q 1,q 2.

4 3633 Stenner, Berkus, and Kachelriess: EDEC 3633 FIG. 4. The photograph shows the mouse phantom left and the two-cylinders phantom that was used for calibration right. To simplify notation let us drop the material index i and let us determine the decomposition coefficients c=c 1 for material 1. Decomposing for material 2 can be done in complete analogy. Making use of the linearity of the x-ray transform R a set of K+1 L+1 basis images is defined as f n r =R 1 b n q 1,q 2, i.e., f n is the reconstruction of the raw data q 1 and q 2 after they have been passed through the basis functions b n. The reconstruction of the material-selective raw data can now be written as a linear combination of these basis images f r = R 1 D q 1,q 2 = R 1 c n b n q 1,q 2 n = n c n R 1 b n q 1,q 2 = c n f n r = c f r. n We want to find the set of coefficients c that minimizes the least square deviation E 2 = d 2 rw r f r t r 2, between the linearly combined basis images f r =c f r and a given template image t r. The weight image w r is used to avoid unwanted structures of the calibration object, which would impair the optimization procedure. To obtain the basis images, the template image and the weight image a DECT scan two raw data sets with different detected spectra of a calibration phantom is performed. The calibration phantom contains homogeneous areas with sufficient amounts of material 1 and material 2 and ensures that all reasonable path length combinations of material 1 and material 2 are acquired. To obtain the basis images these raw data, q 1 and q 2, are passed through the K+1 L+1 basis functions and are reconstructed. With our choice of basis functions we generate 25 sinograms whose entries are q k l 1 q 2 and reconstruct those Fig. 2. A standard reconstruction of the calibration phantom is used to determine t r and w r by simple thresholding as follows. Basically, t r shall represent our a priori knowledge of those regions that correspond to material 1 and those regions that definitely do not contain material 1 and thus contain material 2 or air : = 1 for r material 1, t r 0 for r air and material 2. Note that the template need not be defined in regions that may contain a mixture of both basis materials 1 and 2 since these regions are suppressed by the weight image. The weight image is set to one whenever we are sure about the contents of voxel r. This is the case in material 1 regions, in FIG. 5. The photographs show the half-and-half phantom left and the Yin-Yang phantom right.

5 3634 Stenner, Berkus, and Kachelriess: EDEC 3634 FIG. 6. A simulated 32 cm CTDI test phantom consisting of water and aluminum. The material decomposition uses the coefficients determined from the calibration data of Fig. 3. Tube voltages are 80 and 140 kv. The standard reconstructions and the monochromatic image are windowed to C=0 HU/W=200 HU. The window setting for the material-selective images is C =100%/W=20%. material 2 regions, and in air regions. We set w r to zero wherever we are outside the field of measurement, whenever we expect point spread function effects those regions are at the edges of the homogeneous areas and are found by erosion or whenever we encounter a material that is neither basis 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 E 2. Differentiating E 2 with respect to c n yields the linear system a=b c with a n = d 2 rw r f n r t r, B nm = d 2 rw r f n r f m r. The solution to the optimization problem is simply given as c=b 1 a. An actual implementation must avoid explicitly computing B 1 and should rather use Gauss-Jordan elimination to invert the linear system. Figure 3 shows the intermediate steps necessary to acquire the calibration coefficients. The difference of the water image to the water template and of the aluminum image to the aluminum template show that our method works very well. The phantom shown as well as all other simulated and physical phantoms are invariant under translation along the z axis, hence we show only single slices and apply EDEC to this two-dimensional 2D case. EDEC also works in three dimensions as long as the segmentation of the phantoms is done in three dimensions. Under these circumstances all three appearances of d 2 r have to be replaced by d 3 r. Our method is an empirical correction, and thus it corrects beam-hardening-based cupping as well as cupping due to scattered radiation, but EDEC cannot provide a channeldependent correction. In the presented method the decomposition function D q 1,q 2 does not depend on the line of integration L. In order to account for the effects of an x-ray spectrum that varies across the detector the decomposition function also needs to be a function of L. The reasons for this are manifold, such as the heel effect anode angle effect, the use of a bow-tie filter, x-ray scatter, or the varying detector efficiency as a function of the intersection length of the x ray and the scintillator material. 8 A family of precorrection functions D q 1,q 2,L that depend on each ray L may be derived analytically if one makes assumptions on several parameters, e.g., the anode material and angle, the prefiltration, and the detector absorption. One may combine EDEC with such an analytical precorrection to obtain a hybrid approach similar to the technique presented in Ref. 9. For the measurements and experiments at hand EDEC alone proved to be sufficient. III. SIMULATIONS AND MEASUREMENTS We performed simulations in the scale of clinical CT and micro-ct and carried out measurements with our dual source micro-ct. Measurements with our clinical DSCT were omitted due to the costly production of large-scale calibration phantoms. 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 clinical CT we simulated three test phantoms. For more information please refer to our website phantoms. The first one is a generalized 32 cm 16 cm oval CTDI phantom with water equivalent background and five aluminum inserts. The other test phantoms were a lung phantom consisting of fat, soft tissue, cortical, and spongious bone and a thorax phantom consisting of soft tissue, contrast agent, cortical, and spongious bone. All simulations were carried out at 80 and 140 kv. A mouse phantom was simulated whose size width=32 mm, height=24 mm meets the requirements of a

6 3635 Stenner, Berkus, and Kachelriess: EDEC 3635 FIG. 7. The bottom row shows standard reconstructions at 120 kv that have been acquired with the same patient dose as the nine monochromatic DECT images in the 3 3 image matrix. The monochromatic images are density weighted and have been simulated at 80 and 140 kv. Significant beam hardening artifacts in the standard reconstruction images are marked with arrows. Only calibration phantoms where ray paths through either material 1 or 2 and path combinations through both materials exist perform well. The decomposition based on the three-quarter-and-quarter-cylinder phantom yields excellent decomposition results whereas the decomposition using a phantom where material 2 is fully embedded in material 1 like the cylinder-in-cylinder phantom shows artifacts connecting the aluminum inserts. All images are windowed to C=0 HU/W=200 HU. micro-ct scan. The phantom body consisted of water equivalent plastic, and the two high contrast inserts were made up of different concentrations of iodine. The two small bones consisted of hydroxiapatite 200 mg/ml and the three large bones of hydroxiapatite 400 mg/ml. The mouse phantom was simulated at tube voltages of 30 and 65 kv. III.A.2. Calibration phantoms We simulated a 45 cm Yin Yang phantom consisting of water and aluminum to illustrate the various steps of our method. This simulation was performed at 80 and 140 kv and is shown in Fig. 3. The Yin Yang phantom also served as a calibration phantom for the CTDI phantom. In order to examine the effects of different geometries of the calibration phantom on the decomposition of the test phantom we further simulated three additional types of calibration phantoms: the cylinder-in-cylinder phantom, the three-quarter-andquarter-cylinder phantom and the two-cylinders phantom. The first two have a diameter of 32 cm. The diameters of the latter are 16 and 24 cm, respectively. The material combinations were water with aluminum and water with bone, depending on the type of test phantom the calibration was performed for. All simulations were carried out at 80 and 140 kv. In the scale of micro-ct we simulated a second twocylinders phantom that served as a calibration phantom for the mouse phantom. The large cylinder consisted of water equivalent plastic and the small cylinder of hydroxiapatite

7 3636 Stenner, Berkus, and Kachelriess: EDEC 3636 FIG. 8. Quantitative results achieved at various tube voltage combinations rows for monochromatic images of the attenuation coefficient at 70 and 511 kev and for density images columns. The tube current was chosen to keep the total patient dose for all three voltage pairs constant, hence the noise levels are directly comparable. The values given in the images are mean standard deviation measured in an ROI centered in the central aluminum insert and in an ROI in the water background close to the upper right edge of the phantom. The values are further normalized to water. As expected, image noise is lowest for tube voltage combinations that are far apart here for the 80 and 140 kv combination. The calibration was performed with the two-cylinders phantom. 400 mg/ml. The cylinders diameters were 20 and 10 mm. The phantom was simulated at tube voltages of 30 and 65 kv. III.B. Measurements All physical phantoms had a length of 40 mm and were manufactured by QRM Möhrendorf, Germany. The raw data were acquired with a dual source cone-beam micro-ct scanner and a Feldkamp-type algorithm was used to reconstruct the images. Our method is applicable to various kinds of CT scanners, although cone-beam CT is the method of choice in the present article. III.B.1. Test phantoms We performed micro-ct measurements of three 32 mm test phantoms. The physical mouse phantom was built according to the previously described simulated version, except that the concentrations of hydroxiapatite for the bones were reduced by 50% to guarantee moderate attenuation Fig. 4. The phantom was scanned at 40 and 50 kv. The half-andhalf phantom consisted of two half cylinders, one of water equivalent plastic, the other of a mixture of calcium hydroxiapatite 200 mg/ml and water equivalent plastic Fig. 5. The half-and-half phantom was scanned at 40 and 140 kv. The third physical test phantom was a small Yin Yang phantom that was also made up of two materials, one being water equivalent plastic and the other being a mixture of calcium hydroxiapatite 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 twocylinders phantom that also matches its simulated version in size and geometry. The large cylinder was made up of water equivalent plastic and the small cylinder consisted of hydroxiapatite 200 mg/ml. The two-cylinders phantom was also measured at 40 and 50 kv and is shown in Fig. 4. The decomposition coefficients for the half-and-half-cylinder phantom and the Yin Yang phantom were determined from the respective phantom itself, thus the test and calibration phantom were identical in these two cases. The effects of cross-scatter constitute a major drawback since scattered radiation from one tube-detector system will affect the measurements of the other tube-detector system. In order to eliminate cross-scatter we performed two singleenergy scans instead of one dual-energy scan. The phantoms were scanned twice at two different voltages with one of the

8 3637 Stenner, Berkus, and Kachelriess: EDEC 3637 FIG. 9. A simulated 32 mm 24 mm mouse test phantom. Tube voltages are 30 and 65 kv. The standard reconstructions and the monochromatic image are windowed to C=0 HU/W=200 HU. The window setting for the material-selective images is C=100%/W =20%. tubes turned off during both scans. When using both tubes at the same time a proprietary scanner-internal scatter precorrection compensates for the cross-scatter effects. IV. RESULTS The results of the simulations were analyzed both qualitatively and quantitatively. For the measurements we performed only qualitative analysis, since the exact composition of the employed materials is unknown. IV.A. Simulations TABLE I. The first column shows the density values that were obtained from three ROIs in the combined monochromatic image of the simulated mouse phantom. The second column shows the density values that were used for the simulation. Material Density values from ROI/mg mm 3 Density values used for simulation/mg mm 3 Water 1.00± Hydroxiapatite 400 mg/ml 1.28± Hydroxiapatite 200 mg/ml 1.15± The CTDI test phantom was correctly decomposed into water and aluminum Fig. 6, and the severe beam hardening artifacts present in the images of the polychromatic raw-data were almost completely removed. The CTDI phantom was decomposed using the decomposition coefficients c 1 and c 2 obtained from the Yin-Yang calibration phantom from Fig. 3. Figure 7 shows that the decomposition of the CTDI phantom also works with coefficients c 1 and c 2 that were acquired with calibration phantoms other than the Yin Yang phantom. In addition to the CTDI phantom we applied EDEC to the lung and thorax test phantoms. The following three calibration phantoms were used: the cylinder-in-cylinder phantom, the three-quarter-and-quarter-cylinder phantom and the twocylinders phantom. Our experiments showed that all phantoms were suited for calibration although two conditions must be met to achieve excellent decomposition: The calibration phantom must allow for line integrals that cover a both materials exclusively and b all path length combinations through material 1 and material 2 simultaneously. For the DECT images of the oval CTDI phantom the EDEC calibration was carried out for water and aluminum. For the other two test phantoms the calibration phantoms were filled with water and bone. The DECT scan was taken at 80 and 140 kv. The bottom row of Fig. 7 further shows the images that are obtained from a standard 120 kv scan and reconstruction. The tube currents were chosen: a to ensure the same patient dose for the 120 kv scan and for the DECT scan two spectra and b to optimally distribute the available dose among the 80 and 140 kv exposures. We balance the tube current between the low and the high energy scan in a way to minimize the variance in the final destination image automatic exposure control AEC for DECT. 11 The effects can be best seen regarding the oval CTDI phantom images in Fig. 7. The standard reconstruction at 120 kv shows significant beam hardening artifacts, which are marked with arrows: dark bands of high amplitude that connect the aluminum inserts. These artifacts should be completely removed after projection-based DECT reconstruction they would not be removable if only image-based energy subtraction techniques were used. Obviously, the cylinder-in-cylinder calibration phantom produces the least accurate coefficients: The dark bands are gone but white bands of low amplitude appear at the same place. The other two calibration phantoms, the three-quarterand-quarter-cylinder and the two-cylinders phantom, achieved superior results. Here, rays through either one of the basis materials as well as rays that run through both materials exist, and the monochromatic images of the test phantom are close to perfect. A very close look still reveals some artifacts for the three-quarter-and-quarter-cylinder phantom. This is probably due to the fact that the number of ray paths that intersect only the aluminum region is much

9 3638 Stenner, Berkus, and Kachelriess: EDEC 3638 FIG. 10. The physical 32 mm mouse test phantom that was decomposed with coefficients obtained from the two-cylinders calibration phantom. The micro-ct scan was performed with 40 and 50 kv spectra. The original polychromatic images show strong beam-hardening artifacts: Dark bands connecting the three large bones deteriorate image quality. The borders of the phantom body yield higher CT values due to cupping. Both effects are compensated for in the combined monochromatic image. The standard reconstructions, the monochromatic image, and the image of the calibration phantom are windowed to C =0 HU/W=600 HU, and the window setting for the material-selective images is C=100%/W=60%. The monochromatic image is weighted at 25 kev. smaller in the three-quarter-and-quarter-cylinder phantom compared to the two-cylinders phantom. From these observations one can conclude that a suitable test phantom must allow for rays that run in either material as well as in both materials, and the amount of basis material in the calibration phantom should approximate the expected imaging situation. Given that the two-cylinders phantom provides the best image quality quantitative values were analyzed, too. Figure 8 shows reconstructions of the oval CTDI phantom that were calibrated with the two-cylinders phantom. The simulations were carried out at 120 and 140 kv, at 80 and 120 kv, and at 80 and 140 kv. The decomposed data were combined to yield images at 70 kev, which is the effective energy typical for clinical CT, at 511 kev, the energy that is important for PET attenuation correction, and they were combined to yield density images images. The tube currents were selected using the same method as before. Therefore we can ensure the same patient dose for all three tube current combinations. Note that this implies that different tube currents were simulated for the 70 kev, for the 511 kev and for the images, altogether the figure shows nine different DECT scans. Two ROIs were place into each image. One ROI that measures the mean and standard deviation within the central aluminum insert and one ROI to determine the statistics for the phantom background water. As demonstrated in Fig. 8 image noise decreases for tube voltage combinations that are far apart, i.e. it decreases from top to bottom. Since the total patient dose for all scans remained constant the values are directly comparable. The theoretical attenuation and density values are as follows: Al, 70 kev H 2 O, 70 kev = 3.224, Al, 511 kev H 2 O, 511 kev = 2.355,

10 3639 Stenner, Berkus, and Kachelriess: EDEC 3639 FIG. 11. A physical 32 mm half-and-half-cylinder test phantom consisting of water equivalent plastic and hydroxiapatite. The micro-ct scan was performed with 40 and 140 kv spectra, and the material decomposition uses the coefficients determined from the phantom itself. The standard reconstructions and the monochromatic image are windowed to C=0 HU/W=600 HU. The window setting for the material-selective images is C=100% /W=60%. Al H 2 O = Regarding the mean values of the ROIs we immediately find that a very high accuracy is achieved. Water should be at 1000 since the images are normalized to the theoretical water value. The deviation of the reconstructed water value is less than 0.2% in all cases. Given the theoretical values we find that aluminum should be at 3224 for the 70 kev, at 2355 for the 511 kev, and at 2699 for the density image. The values actually measured are very close to the theoretical predictions. Obviously, the 25 calibration coefficients determined by EDEC from the two-cylinders calibration phantom scans are highly accurate, and our choice K=L=4 appears to be sufficient. EDEC also delivers excellent results for the micro-ct scale. Figure 9 shows that the mouse phantom was accurately decomposed into water basis material 1 and hydroxiapatite 400 mg/ml, basis material 2. The combined image is density weighted. Three circular ROIs were placed in the phantom: one in the phantom body, one in the lower left bone, and one in the upper left bone. The corresponding materials are water, hydroxiapatite 400 mg/ml and hydroxiapatite 200 mg/ml, respectively. The density values that were acquired in the combined monochromatic image for those ROIs are shown in Table I. The comparison to the density values that we used for the simulation confirms that our method works very well. IV.B. Measurements Similarly, EDEC achieves an accurate decomposition in case of real micro-ct data. The physical mouse phantom was decomposed with the calibration coefficients obtained from the two-cylinders phantom. Figure 10 shows that decompositions based on these coefficients yield excellent results: The material-selective image for basis material 1 water solely shows the phantom body with traces of the high contrast inserts. The material-selective image for basis material 2 hydroxiapatite, 200 mg/ml shows the three large bones, as expected. The two small bones are not visible due to the window setting used. The original polychromatic images are subject to strong beam-hardening artifacts: Dark bands connecting the three large bones deteriorate image quality. The borders of the phantom body yield higher CT values due to cupping. Both effects are compensated for in the combined monochromatic image. At the lower right side of the phantom body the decomposition produces an overshoot, which is caused by scatter. Other than that EDEC is apparently not prone to effects of scatter that could arise from the different phantom sizes. Both the half-and-half-cylinder phantom and the Yin- Yang phantom have been correctly decomposed into their two materials Figs. 11 and 12. In addition to the correct decomposition EDEC removes cupping artifacts that are well pronounced in the original polychromatic raw data. Figure 13 illustrates this for the case of the Yin-Yang phantom. It shows the CT values of two row profiles from the polychromatic and the monochromatic images taken from Fig. 12. The profiles of the polychromatic image show significant cupping whereas the profiles of the monochromatic image are perfectly flat. V. SUMMARY AND CONCLUSION The empirical dual energy calibration EDEC is a simple, effective and accurate way to calibrate dual energy CT. It relies on a calibration phantom that must provide path length variations through material 1, material 2, and combinations of path lengths through both materials. In contrast to other raw-data-based calibration techniques the exact shape, position, and orientation of the phantom does not have to be

11 3640 Stenner, Berkus, and Kachelriess: EDEC 3640 FIG. 12. A physical 32 mm Yin Yang test phantom consisting of water equivalent plastic and hydroxiapatite. The micro-ct scan was performed with 80 and 140 kv spectra, and the material decomposition uses the coefficients determined from the phantom itself. The standard reconstructions and the monochromatic image are windowed to C=0 HU/W=600 HU. The window setting for the material-selective images is C =100%/W=60%. known. EDEC determines the material decomposition coefficients in the image domain. The coefficients are then typically used in the raw-data domain by passing pairs of attenuation values q 1 and q 2 same ray at two tube voltages through a simple polynomial: K L p i = k=0 c in q k l 1 q 2 l=0 with n = k L +1 + l. The raw data p 1 for basis material 1 and p 2 for basis material 2 then undergo the standard image reconstruction procedure e.g., a Feldkamp-type cone-beam reconstruction. If desired, one may avoid products of q 1 and q 2 in the basis functions and may then perform image-based DECT decomposition not shown in this article. This kind of energy subtraction calibration is likely to be more efficient and more accurate than today s procedure of simply having two water-precorrected images at two tube voltages being linearly combined. We have demonstrated that the quality of EDEC decomposition slightly depends on the choice of the calibration phantom and we have identified phantom designs that are well suited. It was further shown that the sets of coefficients obtained achieve precise quantitative results. Above all EDEC was validated with measurements performed with a dual source micro-ct. The physical phantoms were correctly decomposed into the two basis materials. In order to eliminate the effects of the scanner-internal scatter precorrection the phantoms were scanned twice at two different voltages with one of the x-ray tubes turned off during both scans. Whenever detector position-dependent corrections are required where the spectrum w L,E varies with each ray L this is, for example, the case with bow-tie prefiltration one may combine EDEC with analytical precorrection or FIG. 13. Row profiles at two different y positions see phantom inserts of the physical Yin Yang phantom. The profiles of the polychromatic image are taken from the measurement at voltage 1 of Fig. 12. The profiles of the monochromatic image are also taken from Fig. 12. M1 corresponds to water and M2 to hydroxiapatite. Significant cupping is present in the regions of M2 and moderate cupping in the areas of M1. EDEC reduces cupping in both regions and profiles, and the resulting CT-value for M1 water yields 0 HU.

12 3641 Stenner, Berkus, and Kachelriess: EDEC 3641 material decomposition techniques and obtain a hybrid approach similar to the technique presented in Ref. 9. Altogether one may regard EDEC as being a simple and versatile method to obtain the desired material decomposition functions. ACKNOWLEDGMENTS The study was supported in part by the Deutsche Forschungsgemeinschaft DFG under grant FOR 661. We thank Laura Konerth, Maria Loibl, and Peng Wang for performing the scans of the phantoms. We also thank Professor Gisela Anton Physikalisches Institut IV, University of Erlangen-Nürnberg for her support. a Author to whom correspondence should be addressed. Telephone: ; Fax: Electronic mail: marc.kachelriess@imp.uni-erlangen.de 1 R. Alvarez and A. Macovski, Energy-selective reconstructions in x-ray CT, Phys. Med. Biol. 21, R. Alvarez and E. Seppi, A comparison of noise and dose in conventional and energy selective computed tomography, IEEE Trans. Nucl. Sci. NS-26, A. Coleman and M. Sinclair, A beam-hardening correction using dualenergy computed tomography, Phys. Med. Biol. 30, W. A. Kalender, W. Perman, J. Vetter, and E. Klotz, Evaluation of a prototype dual-energy computed tomographic apparatus. I. Phantom studies, Med. Phys. 13, J. Vetter, W. Perman, W. A. Kalender, R. Mazess, and J. Holden, Evaluation of a prototype dual-energy computed tomographic apparatus. II. Determination of vertebral bone mineral content, Med. Phys. 13, W. A. Kalender, E. Klotz, and L. Kostaridou, An algorithm for noise suppression in dual energy CT material density images, IEEE Trans. Med. Imaging 7, C. K. Wong and H. Huang, Calibration procedure in dual-energy scanning using the basis function technique, Med. Phys. 10, M. Kachelriess, K. Sourbelle, and W. Kalender, Empirical cupping correction: A first order raw data precorrection for cone-beam computed tomography, Med. Phys. 33, K. Sourbelle, M. Kachelriess, M. Karolczak, and W. A. Kalender, Hybrid cupping correction HCC for quantitative cone-beam CT, Radiology 237(P), D. M. Tucker, G. T. Barnes, and D. P. Chakraborty, Semiempirical model for generating tungsten target x-ray spectra, Med. Phys. 18, M. Kachelriess, Automatic exposure control AEC for dual energy CT DECT, Eur. Radiol. 17, Suppl. 1: