Dark-field Tomography: Modeling and Reconstruction

Transcription

1 Dark-field Tomography: Modeling and Recontruction W. Cong, F. Pfeiffer, M. Bech, O. Bunk, C. David, and G. Wang Abtract: Dark-field image are formed by mall-angle cattering of x-ray photon. The mall-angle cattering ignal i particularly enitive to tructural variation and denity fluctuation on a length cale of everal ten to hundred nanometer, offering a new contrat mechanim to reveal ubtle tructural variation of object. In thi paper, we derive a novel phyical model to decribe x-ray aborption and mall-angle cattering, and ue the propoed model to recontruct the volumetric mall-angle cattering image. The numerical experiment and tet experiment demontrate that the recontructed cattering image reveal unique feature with a high contrat reolution. The propoed approach ha great potential in biomedical imaging, nondetructive detection, and other application. Key Word: X-ray imaging, mall-angle cattering, dark-field imaging, dark-field tomography. 1

2 1. Introduction The conventional x-ray computed tomography (CT) i baed on x-ray attenuation, and yield ufficient contrat for ubtance with high denity difference. However, thi technique cannot achieve atifactory enitivity and pecificity for low attenuation matter, uch a oft biological tiue [1, 2]. In contrat to attenuation mechanim, mall-angle cattering i particularly enitive to tructural variation and denity fluctuation on a length cale of everal ten to hundred nanometer. It provide a new contrat to reveal ubtle tructural variation of matter. Pfeiffer and coworker propoed a grating interferometer technique to produce high quality dark-field image uing a hopitalgrade x-ray tube. The dark-field image of biological pecimen preent ignificantly higher contrat reolution than conventional attenuation-baed image [3]. Recently, a ingle grating wa ued to modulate the x-ray beam. The modulated image contain a primary image and a grid harmonic image. The ratio between the harmonic and primary image reveal a pure cattering image [4]. Moreover, a fan-beam wa propoed to illuminate an object lice for acquiition of coherent cattering data with multiline detector [5]. The central detector row receive the tranmitted radiation while the out-ofcenter row record only cattered radiation. The technique i able to perform a rapid canning for the object and provide a ignificant increment in image contrat for quantitative analye. Strobl et al propoed a method to imulate the broadening of the angular ditribution of mall angle cattering for dark field tomographic imaging. Thi broadening i related to both microcopic tructure and multiple cattering along the path length through a matter [6, 7]. Harding et al directly applied the filtered backprojection algorithm to recontruction the cattering contrat image from dark field data [5]. However, the propagation of x-ray photon through matter i a complex proce, which experience both aborption and cattering imultaneouly. The cattering i caued by the change in the refractive index, and the aborption depend on the denity of matter. A photon propagation model 2

3 decribe photon interaction with matter, and i eential for tomographic imaging. In thi paper, we derive a novel phyical model to decribe x-ray beam aborption and mall-angle cattering. Then, we apply the propoed model to recontruct the volumetric mall-angle cattering image. 1. X-ray mall-angle cattering In quantum mechanic, light i conidered with both wave and particle behavior. Hence, the x-ray photon tranmiion can be treated a a beam of particle propagating through an object. A x-ray photon interact with the object, ome photon would be deflected from the original direction due to a difference in the refractive index to generate a cattering ignal. Thu, x-ray photon can be divided into tranmitted photon traveling along a traight line in a direction and cattered photon deflected from the original direction. The propagation of tranmitted photon along the direction can be well decribed by the Beer-Lambert law, where t R ar Rar texp tr d (1) t a r i the light intenity along the direction, and t the attenuation coefficient defined a a um of aborption coefficient a and cattering coefficient, that i t a. Eq. (1) can be reduced to a Radon tranform, a ln a r r R R t r d (2) where r and R a a r expre the intenity value from the x-ray ource and upon the detector repectively after the x-ray propagation along the direction through the object. Baed on the attenuation-baed CT technology, the unknown x-ray attenuation coefficient t can be 3

4 recontructed uing the conventional recontruction algorithm, uch a the filtered backprojection (FBP) algorithm [8, 9]. While an x-ray beam propagate in matter along a traight line, ome photon would experience a mall-angle forward cattering. The o-called dark-field image i formed through the mall-angle cattering of x-ray. The cattered photon intenity r depend on both aborption and cattering coefficient of object. According to the energy conervation principle, the difference d r of the cattered photon intenity between the oppoite ide of an elementary volume with a cro ectional area da and length dh along the direction i equal to the difference between intenity of cattered photon from primary beam ubtracting the intenity of photon aborbed by matter, which can be expreed a follow, r r r. (3) d da da dh da dh a a Since d rda r dhda, a differential equation in term of the cattered photon intenity can be obtained from Eq. (3): r r r, (4) a a where r repreent the lo of the cattering intenity due to aborption, and r i the a a quantity of cattered photon from the primary beam a r. In other word, Eq. (4) decribe the balance of the photon between the input and output of an elementary volume at the given direction. Becaue Eq. (4) i a linear firt-order differential equation, it olution can be obtained in the cloed form: R R r R r ta r texpa r ddt. (5) t 4

5 Subtituting Eq. (1) into Eq. (5), we obtain R R r R r a R r texp r ddt. (6) t Uing a variable tranformation, Eq. (6) can be implified to a Radon tranform with repect to the cattering coefficient ditribution: ln 1 r R R a R r r d (7) where r R and R a r are the intenity of tranmiion photon and the intenity of mallangle cattering photon on the detector, repectively. Eq. (7) decribe the relationhip between the cattering characteritic of matter and meaured photon tranmiion and mall-angle cattering data. Eq. (7) i alo a tandard Radon tranform for the cattering coefficient, o claical recontruction algorithm, uch a filtered backprojection (FBP) algorithm, algebraic recontruction technique (ART), can be applied to recontruct the cattering coefficient ditribution tomographically [8, 9]. 2. Numerical imulation 2.1. Shepp-Logan' phantom 5

6 We firt employed Shepp-Logan' phantom to evaluate the propoed dark-field tomographic imaging method [9]. The phantom parameter were lited in Table I. Ten ellipe wa included in the phantom to mimic the ubtle tructure of the head tiue. In Table I, a and b are the emi-axe of an ellipe, x and y pecify the center of an ellipe, t denote an x-ray attenuation coefficient, and the x-ray cattering coefficient, and the angle (in degree) between the horizontal emi-axi of the ellipe and the x-axi of the recontruction ytem. The phantom had a le denity variation and low attenuation contrat. The projection data of the dark-field and bright-field at different poition and angle were computed uing our in-houe Monte Carlo imulator. The number of projection wa 18. The recontruction matrix wa et to 512 by 512. Then, the FBP algorithm baed on the propoed cattering model wa implemented to recontruct the cattering and attenuation coefficient ditribution, repectively. A a reult, the recontructed cattering coefficient image howed a high contrat reolution for different tiue tructure, a hown in Fig.1 (a) with interface highlighted due to dark-field imaging. In contrat, the conventional attenuated-baed recontruction hown in Fig.1 (b) No. t Table I. Parameter of the Shepp-Logan Phantom a b x y

7 could not dicern the different type of tiue becaue of an inufficient attenuation contrat Breat phantom The propoed dark-field tomographic imaging technique wa alo numerically evaluated uing the digital breat phantom [1]. The breat phantom repreent an uncompreed breat of a half ellipoidal hape, containing anatomical and pathological feature of different ize and contrat. The three ellipoidal emiaxe were et to 5, 5, and 1 mm. The kin thickne wa et to 2.5 mm. The fibroe were modeled a cylinder, the calcification and ma a ball, a hown in Fig.2. The phantom wa poitioned in the nonnegative pace, attached to the chet wall defined on z =. Mae, Fibroe and calcification were centered on the plane of z =3 mm and z=5 mm, repectively. Table II-IV lit the geometrical parameter, attenuation and cattering propertie at 38 kev of the mae, fibroe, and calcification, repectively [1]. The breat phantom wa of low attenuation contrat between different breat tiue. We adopted the parallel-beam mode to can the phantom, and imulated the dark-field and bright-field data at different poition and angle uing our x-ray Monte Carlo imulator. Then, the recontruction algorithm wa again ued to recontruct the cattering and attenuation image. Similarly, the recontructed cattering image had an excellent contrat to differentiate the tiue tructure. Fig.3 (a) and Fig.4 (a) howed the recontructed image at the cro ection of z=3mm and z=5mm, repectively. In contrat, the conventional attenuated-baed recontruction hown in Fig.3 (b) and Fig.4 (b) could not dicriminate the tumor from the normal breat tiue. 7

8 Table II. Dimenion, attenuation and cattering propertie of mae. Ma No Radiu (mm) t Table III. Dimenion, attenuation and cattering propertie of fibroe. Fibrou No Radiu (mm) Height (mm) t Table IV. Dimenion, attenuation and cattering propertie of fibroe. Calcification No Radiu (mm) t Tet experiment To tet our theoretical model on experimental dark-field data obtained with a grating interferometer, we carried out ome tet experiment at the beamline ID19 of the European Synchrotron Radiation Facility (ESRF, Grenoble). A mall beatle (ex-vivo) wa ued a a on a trongly cattering tet ample. A monochromatic x-ray beam of 24.9 kev ( =.498 nm) wa ued for the meaurement. The interferometer wa placed at a ditance of 14 m from the wiggler ource (ee [11] for more detail on the grating parameter). The field-of-view wa matched to the ize of the pecimen and wa mm2. To achieve a very high angular, and thu phae enitivity, the ditance between G1 and G2 wa choen to be a 8

9 large a 361 mm (ninth fractional Talbot ditance). The image were recorded uing a 15 µm thick polycrytalline gadolinium oxyulfide cintillation creen with a magnifying optical len ytem and a cooled charge coupled device (CCD).We ued the FReLoN 2 (a fat-readout, low-noie CCD developed at the ESRF) with pixel and a µm 2 pixel ize (in the 2 2 binning mode). Due to the magnifying len ytem, the effective pixel ize in the recorded image wa µm 2. In total 721 projection angle over 18 degree were recorded. The Fringe-canning method wa ued to acquire eight image, I xy ( k 1, 2,, 8 ) at every projection angle by hifting moiré fringe. In the k Talbot interferometer, the hift wa attained by diplacing one of the grating in the direction parallel to it diffraction vector. The tranmiion intenity and dark-field ignal can be extracted from the meaured intenity image repectively a follow, and M a x, y Ik x, y k b x y I x y kx I x y kx M M, k, in g k, co g k1 k1 where kx g i the diplacement of the grating. The number of projection angle i 721. The dark-field tomography method wa then applied to recontruct the beetle tiue tructure from the acquired dark field ignal. Fig. 5(a) and Fig. 6(a) how dark-field tomographic image. It can be oberved that the new contrat mechanim helped identify tructural feature of the invetigated ample. The mallangle-cattering baed recontruction a derived from the dark-field data are particularly ueful for identifying tructure in an object on the cale of about a hundred nanometer to a few micrometer. 9

10 For comparion, the correponding attenuation-baed tomographic lice are hown in Fig.5 (b) and Fig.6 (b), which exhibit a low attenuation contrat of the biological ample. 4. Dicuion and concluion In ummary, we have developed a novel phyical model to decribe both x-ray attenuation and mallangle cattering. The numerical and biological experiment have hown that dark-field tomographic imaging can reveal detailed tructural variation of matter, producing a higher contrat reolution for low attenuation contrat feature than conventional attenuation-baed computerized tomography. Darkfield image are particularly enitive and pecific to boundarie and interface in the matter, producing trong dark-field ignal to reveal detailed tructural information of matter. Additionally, the radiation doe trongly relie on x-ray aborption propertie of matter. The probability of x-ray photoelectric aborption drop off rapidly a a function of the incident X-ray photon energy. Higher x- ray photon energy produce lower radiation aborption, reulting in a poor contrat for low aborption matter. Hence, attenuation-baed CT often ue lower x-ray photon energy to enhance contrat reolution for low attenuation media, inducing a coniderable radiation doe. In contrat, the principal advantage of our dark-field imaging method i that the contrat of mall-angle cattering imaging doe not depend olely on photon aborption, o x-ray energie can be choen to minimize radiation aborption in matter. The propoed approach ha a great potential for a wide range of application, including clinical and pre-clinical imaging, food inpection, ecurity creening, and indutrial nondetructive teting. Acknowledgement: G.W. and W. Cong acknowledge the upport from the National Intitute of Health (Grant CA , EB636, EB8476, and CA127189) and Tohiba Medical Sytem (Advanced Imaging Method Grant). We gratefully acknowledge the aitance of C. Kottler and P. Cloeten in the experiment. 1

11 Reference: 1. Momoe, A., et al., Phae-contrat X-ray computed tomography for oberving biological oft tiue (vol 2, pg 473, 1996). Nature Medicine, (5): p Chapman, D., et al., Diffraction enhanced x-ray imaging. Phyic in Medicine and Biology, (11): p Pfeiffer, F., et al., Hard-X-ray dark-field imaging uing a grating interferometer. Nature Material, 28. 7(2): p Wen, H., et al., Spatial harmonic imaging of X-ray cattering initial reult. IEEE Tranaction on Medical Imaging, (8): p Harding, G., X-ray catter tomography for exploive detection. Radiation Phyic and Chemitry, (3-4): p Strobl, M., W. Treimer, and A. Hilger, Small angle cattering ignal for (neutron) computerized tomography. Applied Phyic Letter, (3): p Strobl, M., et al., Neutron dark-field tomography. Phyical Review Letter, (12): p Natterer, F., The mathematic of computerized tomography. 1986, Stuttgart Chicheter: B.G. Teubner ; Wiley. x, Kak, A.C. and M. Slane, Principle of Computerized Tomographic Imaging. 1988, New York: IEEE Pre. 1. Zeng, K., et al., Cone-beam mammo-computed tomography from data along two tilting arc. Medical Phyic, (1): p Pfeiffer, F. et al., High-reolution brain tumor viualization uing three-dimenional x-ray phae contrat tomography, Phyic in Medicine and Biology, (23), p

12 Fig.1. Comparion between dark-field tomography and conventional CT. Image recontructed from (a) the dark-field data and (b) the tranmiion data. Fig.2. 3D breat phantom. (a) A volumetric rendering, (b) a cro ection at z=3mm, and (c) a cro ection at z=5mm. 12

13 Fig.3. Comparion between dark-field tomography and conventional CT for the breat phantom, image at z=3mm lice recontructed from (a) the dark-field data and (b) the tranmiion data. Fig. 4. Comparion between dark-field tomography and conventional CT for the breat phantom, image at z=5mm lice recontructed from (a) dark-field data and (b) tranmiion data. 13

14 Fig. 5. Comparion between dark-field tomography and conventional CT for Beatle. Image at 27 th lice recontructed from (a) dark-field data and (b) tranmiion data. Fig. 6. Comparion between dark-field tomography and conventional CT for Beatle. Image at 33 th lice recontructed from (a) dark-field data and (b) tranmiion data. 14