A ProjectorBackprojector with Slice-to-Slice Blurring for Efficient 3D Scatter Modeling

Transcription

1 A ProjecorBackprojecor wih Slice-o-Slice Blurring for Efficien 3D Scaer Modeling Gengsheng L. Zeng, Chuanyong Bai, and Gran T. Gullberg Deparmen of Radiology, Universiy of Uah 729 Arapeen Drive, Sal Lake Ciy, Uah 84108, US,4 Absrac Scaer correcion is an imporan facor in single phoon emission compued omography (SPECT). Many scaer correcion echniques, such as muliple-window subracion and inrinsic modeling wih ieraive algorihms, have been under sudy for many years. Previously, we developed an efficien slice-o-slice blurring echnique o model aenuaion and sysem geomeric response in a projecorhackprojecor pair, which was used in an ML-EM algorihm o reconsruc SPECT daa. This paper proposes a projecor/backprojecor ha models he 3D firs-order scaer in SPECT, also using an efficien slice-o-slice blurring echnique. The scaer response is esimaed from a known, non-uniform, aenuaion disribuion map. is assumed ha he probabiliy of a scaer even a a pixel is proporional o he aenuaion coefficien value a ha pixel. Mone Carlo simulaions of poin-sources and a MCAT orso phanom were used o verify he accuracy of he proposed projecorhackprojecor model. For a 64 x 64 x 64 image volume, i ook 8.7 seconds o perform each ieraion per slice, when modeling 3D scaer, aenuaion, and sysem poin response funcions. The main advanage of he proposed mehod is is ease o implemen and efficiency in compuer calculaion.. NTRODUCTON There are several facors ha affec SPECT (single phoon emission compued omography) images, such as low couning saisics, phoon aenuaion, sysem disance-dependen blurring, and phoon scaer. This paper will presen a mehod o provide a more accurae image by modeling he firs-order Compon scaer wihin he paien s body. Aenuaion and sysem poin response funcion are also included in he proposed model. This mehod significanly reduces he compuaion ime in 3D scaer correcion. n fac, many mehods have already been developed for scaer correcion in SPECT. Mehods (see, for example, [ 1-61) ha use muliple acquisiion energy windows o esimae he scaered phoons and subrac he esimaed phoons from he projecion daa have found applicaions in research and clinical sudies. These pre-processing mehods are efficien and effecive, bu he pre-subracing mehods may increase noise and may inroduce negaive or zero values a locaions where projecion values are posiive. An alernaive mehod o avoid subracion is o add he esimaed scaer evens o forward projecion of he curren reconsruced image in an ieraive algorihm [7]. Subracing or adding daa will increase he noise level in he daa. eraive reconsrucion mehods [8-141 can model scaer physics in he projecorhackprojecor and have been shown o provide more accurae reconsrucions han subracindadding mehods [ 141. The ieraive mehods require he spaially varian scaer response funcions wihin he projecorhackprojecion. Pre-soring a complee se of scaer poin response funcions for each paien is no feasible in pracice. Research has been done o approximae he response funcions in an objec by he waer equivalen deph mehod [lo, 131. However, hese approximaion mehods are sill no efficien, and do no work well for non-uniform objecs. f a paien s aenuaion map is available via a ransmission scan [ 151, he Compon scaer poin response funcion can be esimaed, for example, by he Klein-Nishina formula [ 161. Recenly Welch e al. [12] developed a simple mehod o esimae he projecions of firs-order Compon scaer. Their mehod gives a fairly accurae esimae of he scaered projecions fro an inhomogeneous media, and has been implemened in 2Dl [12]. This paper proposes implemenaion of Welch s mehod hrough he use of a fas slice-o-slice blumng echnique and exension of Welch s Compon scaer model ino 3D. n addiion, he sysem geomeric poin response correcion is also included. The new iimplemenaion is efficien. correcs for 3D firs-order Cornpon scaer, 3D deph dependen sysem poin response, and non-uniform aenuaion. 11. METHODS A. Slice-o-Slice Bluirring Model n order o obain quaniaive SPECT images, i is necessary o correc fior aenuaion, scaer, and sysem geomeric poin response. has been previously demonsraed ha he slice-o-slice blurring projecorhackprojecor is efficien and effecive when used in an ieraive ML-EM algorihm [ 17-19]. The basic procedure in he slice-o-slice projecor is as follows: A each projecion view, he voxelized image volume roaes wih he deecor. We define a layer o be a slice of he voxelized image volume parallel o he deecor. Saring from he layer farhes from he deecor, each layer is convolved The research work presened in his manuscrip was parially suppored by NH Gran RO1 HL39792 and Picker nemaional /98/$ EEE 173 1

2 ~ Array wih a convoluion kernel. The resulan 2D image is added o he nex layer. The nex layer now conains blurred versions of previous layers. This procedure is repeaed unil he deecor plane is reached. The very las layer is he projecion image and he previous layers are discarded. The convoluion kernels are disance-dependen and can be deermined from he sysem poin response funcion [ 181. f aenuaion correcion is needed, each pixel in he curren layer is scaled by is own aenuaion facor before he convoluion is performed as illusraed in Fig. 1. Here he aenuaion facor is he negaive exponenial of he aenuaion coefficien a ha pixel. Slice-by-slice blurring can be expressed as: proj. = (((Ll*hl +L2)*h2+L3)*h3+L4)*h4+... (1) where L, is he nh layer, h, is he blurring kernel for he nh layer, * is he 2D convoluion operaor. Slice-by-slice blurring wih aenuaion modeling can be expressed as: proj. = (((((((L, xal)*hl +L2)XAZ)*h2 +L3)XA3)*h3 + L4) x A,)*h, +... (2) where A, is he array of aenuaion facors in he nh layer, x is he poin-by-poin muliplicaion. The convoluion kernel beween wo layers is very small. Thus, he projecorhackprojecor wih incremenal slice-oslice blurring model is very efficien in an ieraive algorihm. The small kernels are usually 3x3 or 5x5 squares and can be implemened as wo 1D convoluions of size 3 or 5 o reduce compuaion ime. Those wo convoluions are in horizonal and verical direcions, respecively [20]. The kernels can also have he shape of a cross. Secion 1.B will invesigae a mehod in which he slice-o-slice blurring mehod is used o model he 3D scaer efficienly. B. 30 Scaer Model has been repored ha in a Tc-99m emission window 80% - 90% of he scaered evens are of he firs order [21]. has also been repored [21 ha he probabiliy of he firs order scaer is proporional o (i) aciviy in he source-voxel, (ii) he aenuaion facor from he source-voxel o he scaervoxel, (iii) he aenuaion coefficien of he scaer-voxel, (iv) a funcion of Compon scaer angle, and (v) he aenuaion facor from he scaer-voxel o he deecor (see Fig. 2). The following symbolic expression is given in [12]: where S.. is he number of scaered evens deeced in J projecion bin, i, originaing from a source voxel, j, and pk is he aenuaion coefficien a scaering sie, k. The Gaussian scaering probabiliy G(Qk) is a funcion of he scaering angle, Qk, and is deermined by a leas-squares fi using Mone Carlo sudies. A ray-racing echnique was uilized o implemen (3) in [12]. The curren paper will use a fas slice-o-slice blumng echnique o implemen (3). Fig. 3 and Fig. 4 show he flowchars of proposed projecor and backprojecor, respecively. The implemenaion procedure of he projecor is described as follows: Sep 1: The image is slice-o-slice diffused owards he deecor, as described in Secion.A, wih an excepion being ha afer he diffusion reaches he deecor, ALL he layers are o be used o generae he scaered phoons. The diffusion kernel used in his sep is 5x5 and implemened as wo orhogonal 1D kernels. The Gaussian funcion G(+k) is used k o evaluae he kernels. The aenuaion facor exp(-j. pl,dx) J is included during diffusion. Therefore, Sep 1 implemens he Blurring beween layers source-voxel (ii) of aenuaion facors Figure 1. llusraion of slice-o-slice blurring projecor ha models he sysem geomeric poin response and aenuaion. (The space beween layers is for presenaion purposes only.) Deecoi z 1 Figure 2. llusraion of firs-order Compon scaer. 1732

3 par Of (3). Sep 3: f no deph-dependen sysem poin response Sep 2: ne resulan volume from Sep 1 is funcion correcion is necessary, his sep simply evaluaes he by he volume of aenuaion coefficiens, pk, in voxel-by-voxel ray-sum of he resulan volume from Sep 2 wih modeling of manner. This sep implemens he par of muliplicaion by pk aenuaion effecs, ha is* wih exp(-f k.x dx). in (3). Noe ha no sysem geomeric response effecs are considered he (i-l)h layer of image volume x he (i-1)h layer of a. facor (exp(-p)) volume => (voxel-by-voxel muliplicaion) + 1D convolve TEMP, row-by-row => TEMP 1 D convolve TEMP, column-by-column => TEM TEMP + he ih layer of he image volume => i he ih layea of he image volume (voxel-by-voxel summaion) i+l => i = (oal # of layers)? 1 image volume x a. coeff. (p) volume => image volume (voxel-by-voxel muliplicaion) 2=>i U he (i-l)h layer of image volume x he (i-l)h layer of a. facor (exp(-p)) volume => (voxel-by-voxel muliplicaion) + 2D convolve TEMP wih 5-poin cross kemel => TEMP TEMP + he ih layer of he image volume => he ih layer of he image volume (voxel-by-voxel summaion) ' i+l=> i No volume as projecions of scaered phoons Figure 3. Flow-char of proposed projecor which models non-uniform aenuaion, sysem poin response funcion and 3D scaer using slice-o-slice Gaussian diffusion echnique. 1733

4 in (3). f i is desired o model he sysem geomeric response This proposed scaer projecion model uses he slice-ofuncion (as in his paper), his sep is he same as ha slice blurring mehod wo imes. The firs ime, he described in Secion 1.A wih he resulan volume from Sep convoluion kernels are deermined by he Compon scaer 2 as he image volume. The final resul is he image of firs- angle funcion, G(qk) ; he second ime, he convoluion order scaered phoons. kernels are deermined by he sysem geomeric poin response he ih layer of image volume x he ih layer of a. facor (exp(-p)) volume => 2D TEMP array (voxel-by-voxel muliplicaion) + 2D convolve TEMP wih 5-poin cross kernel => TEMP 1 ~~ TEMP + he (i-1)h layer of he image volume => he (i-1)h layer of he image volume (voxel-by-voxel summaion) i-1 => i No image volume x a. coeff. (p) volume => * image volume (voxel-by-voxel muliplicaion) (# of layers) => i Se he ih layer of volume OUT o zero he ih layer of image volume x he ih layer of a. facor (exp(-p)) volume => 2D TEMP array (voxel-by-voxel muliplicaion) 1 D convolve TEMP, column-by-column => TEMP TEMP -> he (i-1)h layer of volume OUT TEMP + he (i-1)h layer of he image volume => he (i-1)h layer of he image volume (voxel-by -voxel summaion) il1 i-1 => i i=? backprojeced image is volume OUT Figure 4. Flow-char of proposed backprojecor which models non-uniform aenuaion, sysem poin response funcion and 3D scaer using slice-o-slice Gaussian diffusion echnique. 1734

5 ._ funcion (read [ 181 for deails). The projecor diffuses forward: he backprojecor diffuses backward hrough he voxelized image volume COMPUTER SMULATONS A. Poin Response Verijkaion n order o verify he accuracy of he proposed scaer model, poin source Mone Carlo sudies were performed. As shown in Fig. 5, wo poin-sources were planed in he MCAT phanom. The Mone Carlo sofware, PHG [22], was used o generae he projecions of he firs-order scaered phoons: a perfec collimaor was assumed. The peak phoon energy was 140 kev wih a 10% energy window. High-coun Mone Carlo daa were acquired and used as he gold sandard. Resuls from wo deecor posiions are shown in Fig. 6. The projecion profiles (Fig. 6) were ploed in log scale in order o visualize he differences beween he gold sandard and he proposed 3D scaer model in he ail porions. is observed ha he proposed projecion model maches he Mone Carlo model fairly well. For a 64x64~64 image volume, generaing a 64x64 scaer projecion array required 5 seconds. The proposed 3D scaered projecion model has shown o be simple, efficien, and accurae for 3D modeling. B. MCAT Phanom Reconsrucion An MCAT orso phanom was used for he algorihm conparison sudies. The phanom had aciviies in he hear, liver, lungs, and background issues. s non-uniform aenuaion map was he same as he one shown in Fig. 5. Two ses of Mone Carlo projecion daa were generaed wih he PHG sofware [223. The peak phoon energy was 140 kev wih a 10% energy window. The firs se conained primary projecions only, and no aenuaion was assumed. The purpose of he firs se of projeciion daa was o provide a gold sandard for aenuaion and :scaer correcions. The second se combined primary and firs-order Compon scaered daa wih non-uniform aenuaion effecs. Each se had 120 views uniformly disribued over 360", and he daa a each view were sored in a 64x64 array. The pixel size was cm. A parallel collimaor was 22 cm away from he axis of roaion. The collimaor had a hiickness of 4 cm, hole radius of 0.2 cm and sepal hickness of 0.02 cm. Fig. 7 shows one slice of he 3D reconsrucion of he MCAT using 5 ieraions of OS-EM algorihm [23]; four views, 90" apar, formed a sub-se. Fig. 7 (a) is he gold sandard reconsruced wih he scaer-free and aenuaionfree daa. Figs. 7 (b, c, and d) are he reconsrucions using he daa wih scaer and aenuaion. Neiher aenuaion nor scaer was correced dluring he reconsrucion in Fig. 7 (b). Aenuaion effecs were correced in Fig. 7 (c), bu no scaer was correced. n Fig. '7 (d), boh aenuaion and scaer were modeled in he reconsrucion. The profiles illusrae he effeciveness of he proposed algorihm. All reconsrucions were performed on a SUN Ulra- Enerprise. The compuaion ime for 3D aenuaion and sysem poin response correcion was 5.2 seconds per slice per ieraion. The compuaion ime for 3D scaer, aenuaion and sysem poin response correcion was 8.7 seconds per slice per ieraion. Usually, abou 5 ieraions were used in rouine OS- EM reconsrucions. Figure 5. The poin sources and he aenuaion map used in he Mone Carlo simulaions and he proposed projecor. Posiion 1 Deecor posiion 1 Posiion (3> F. -. a> Phanom Ei 10-2 FP c; ;h *..- $ :. Deecor posiion 2 Mone Carlo Esimaed low4 (sandard) Deecor Bin Figure 6. Mone Carlo simulaed (blue) and esimaed (red) scaer response for wo posiions in he non-uniformly aenuaing phanom in Fig si U2

6 V. DSCUSSON The abiliy o implemen an exising 3D firs-order Compon scaer model by using a slice-o-slice Gaussian diffusion echnique is an imporan conribuion. This implemenaion is efficien; however, his scaer model is no wihou is drawbacks. For example, he higher-order Compon scaer is no considered. The Gaussian scaer blurring kernel is energy dependen and we do no have a closed-form formula o calculae i. Currenly, he kernel is esimaed by leassquares fi wih Mone Carlo sudies. n addiion o he advanage of efficien compuaion, he proposed scaer model can easily be exended ino fan-beam, cone-beam, and oher imaging geomeries by changing he very las sep (ha is, Sep 3) in he projecor/backprojecor according o he collimaor geomeries. ACKNOWLEDGMENTS The auhors hank he Biodynamics Research Uni, Mayo Foundaion, for use of heir Analyze sofware. We hank Sean Webb for echnical wriing assisance. REFERENCES B. Axelsson, P. Msaki, and A. sraslson, Subracion of Compon-scaered phoons in single phoon emission compued omography, J. Nucl. Med., vol. 25, pp , R. J. Jaszczak, K. L. Greer, C. E. Floyd, C. C. Harris, and R. E. Coleman, mproved SPECT quanificaion using compensaion for scaered phoons, J. Nucl. Med., vol. 25, pp , K. E Koral, X. Wang, E M. Swailem, S. Buchbinder, N. H. Clinhorne., W. L. Rogers, and B. M. W. Tsui, SPECT dual-energy-window Compon correcion: Scaer muliplier required for quanificaion, J. Nucl. Med., vol. 31, pp , [4] K. Ogawa, Y. Haraa, T. chihara, A. Kubo, and S. Hashimoo, A pracical mehod for posiion-dependen Compon-scaer correcion in SPECT, EEE Trans, Med. hag., vol. 10, pp , M. A. King, F. J. Hademenos, and S. J. Glick, A dual phoo-peak window mehod for scaer correcion, J. Nucl. Med., vol. 33, pp , [6] D. R. Gilland, R. J. Jaszczak, H. Wang, T. G. Turkingon, K. L. Greer, and R. E. Coleman, A 3D model of nonuniform aenuaion and deecor response compensaion for efficien reconsrucion in SPECT, Phys. Med, Biol., vol. 39, pp , [7] J. E. Bowsher, V. A. Johnson, T. G. Turkingon, R. J. Jaszczak, C. E. Floyd, and R. E. Coleman, Bayesian reconsrucion and use of anaomical a priori informaion for emission omography, EEE Trans. Med. mag., vol. 15, pp , [8] C. E. Floyd, R. J. Jaszczak, and R. E. Coleman, Maximum likelihood reconsrucion for SPECT wih Mone Carlo modeling: Asympoic behavior, EEE Trans. Nucl. Sci., vol. 34, pp , [9] J. E. Bowsher and C. E. Floyd, Treamen of Compon scaering in maximum-likelihood, expecaionmaximizaion reconsrucions of SPECT images, J. Nucl. Med., vol. 32, pp , [lo] E. C. Frey and B. M. W. Tsui, A pracical mehod for incorporaing scaer in a projecor-backprojecor for accurae scaer compensaion in SPECT, EEE Trans. Nucl. Sci., vol. 40, pp , (c) (4 Figure 7. 3D reconsrucion of he MCAT wih 5 ieraions of OS-EM algorihm. All images are correced by disance-dependen sysem geomeric response funcion. (a) The gold sandard using scaer-free and aenuaionfree daa is picured. Daa wih scaer and aenuaion arc used in b, c, and d. (b) No aenuaion and no scaer correcions are performed. (c) Aenuaion correced is performed, bu no scaer correcion is performed. (d) Boh aenuaion and scaer correcions are performed. Profiles are drawn from pixel (1 1, 1) o pixel (64,5) as shown. All images are scaled o heir own maximum and minimum values. 1736

7 [12] A. Welch, G. T. Gullberg, P. E. Chrisian, F. L. Daz, A ransmission-map-based scaer correcion echnique for SPECT in inhomogeneous media, Med. Phys., vol. 22, pp , F. J. Beekman, C. Kamphuis, and M. A. Viergever, mproved quaniaion in SPECT imaging using fully 3D ieraive spaially varian scaer compensaion, EEE Trans. Med. mag., vol. 15, pp , F. J. Beekman, C. Kamphuis, and E. C. Frey, Scaer compensaion mehods in 3D ieraive SPECT reconsrucion: A simulaion sudy, Phys. Med. Biol., vol. 42, pp , [15] C.-H. Tung, G. T. Gullberg, G. L. Zeng, P. E. Chrisian, E L. Daz, and H. T. Morgan, Non-uniform aenuaion correcion using simulaneous ransmission and emission converging omography, EEE Trans. Nucl. Sci., vol. 39, pp , [61 Z. Liang, R. J. Jaszczak, C. E. Floyd, K. L. Greer, and R. E. Coleman, Reprojecion and back projecion in SPECT image reconsrucion, Proc. EEE Energy nform. Technol. Souheas, vol. ETS-1, pp , [ 171 A. W. McCarhy and M.. Miller, Maximum likelihood SPECT in clinical compuaion imes using meshconneced parallel compuers, EEE Trans. Med. mag., vol. 10, pp , [18] G. L. Zeng, G. T. Gullberg, C. Bai, P. E. Chrisian, F. Trisjono, E. V. R. DiBella, J. W. Tanner, and H. T. Morgan, eraive reconsrucion of Fluorine- 18 SPECT Using geomeric poin response correcion, J. Nucl. Med., in press, [ 191 J. W. Wallis, T. R. Miller, M. M. Miller, and J. Hamill, Rapid 3-D projecion in ieraive reconsrucion using Gaussian diffusion, [absrac] J. Nucl. Med., vol. 37, p. 63P, [20] W. K. Pra, Digial mage Processing (2nd Ediion), John Wiley & Sons, nc., New York, [21] C. E. Floyd, R. J. Jaszczak, C. C. Harris, and R. E. Coleman, Energy and spaial disribuion of muliple order Compon scaer in SPECT: A Mone Carlo invesigaion, Phys. Med. Biol., vol. 29, pp , [22] S. Vannoy, The Phoon Hisory Generaor (User s Manual), The maging Research Lab, Universiy of Washingon Medical Cener, [23] H. M. Hudson and R. S. Larkin, Acceleraed EM reconsrucion using ordered subses of projecion daa, EEE Trans. Med. mag., vol. 13, pp ,