Channel model of pinhole gamma camera

Transcription

1 Artcle Nuclear Scence & Technology September 0 Vol.56 No.5: do: 0.007/s Channel model of pnhole gamma camera DAI QuSheng Suzhou Insttute of Bomedcal Engneerng and Technology, Chnese Academy of Scences, Suzhou 563, Chna Receved February 8, 0; accepted May 3, 0 The pnhole gamma camera s the bass of the newly developed method of hgh-resoluton pnhole sngle photon emsson computed tomography (SPECT). In ths paper, a pnhole gamma camera system was examned usng the channel model of nformaton theory. The model pnhole gamma camera was smplfed from a nosy, lossy channel to a noseless, lossless channel by assumng a plane source. An approxmate formula of average mutual nformaton for the pnhole gamma camera was then derved from the model. Imagng experments valdated the applcablty of the analytc formula. The fndngs demonstrated that the pnhole gamma camera can be adequately descrbed usng the channel model of nformaton theory, and that average mutual nformaton can be consdered a fgure of mert for the optmzng desgn of the pnhole gamma camera. pnhole gamma camera, channel model, average mutual nformaton Ctaton: Da Q S. Channel model of pnhole gamma camera. Chnese Sc Bull, 0, 56: , do: 0.007/s Hgh-resoluton pnhole sngle photon emsson computed tomography (SPECT) magng plays an mportant role n the rapdly growng feld of molecular magng [,]. A varety of hgh-performance pnhole SPECT magng systems have been constructed [3 6], but the desgn theory of these magng systems s developng relatvely slowly. In the desgn of pnhole SPECT systems, the performance of magng systems can be predcted wth smulaton methods [7 ]. These methods can be used to quantfy the spatal resoluton, senstvty and feld of vew (FOV) of the magng systems [0 ]. These characterstcs are, however, lmted by each other. The mprovement of any two parameters always occurs at the cost of a thrd parameter, so t s dffcult to dentfy the optmal tradeoff. Therefore, t s valuable to use only one fgure of mert n evaluatng the overall performance of the pnhole SPECT system. Accordng to general nformaton theory, the magng process s a process of nformaton transmsson, snce ts man purpose s to obtan nformaton. Accordng to a communcaton model (Fgure ), an magng object can be regarded as an nformaton source, whle the observer can emal: qushengda@gmal.com be regarded as a recever. As such, the magng system can be regarded as a channel, contanng the functon of an encoder. An mage reconstructon algorthm s thus equvalent to a decoder. Many magng studes smply mprove the decodng capablty of already exstng algorthms, whch does not ncrease the amount of nformaton transferred by the magng system [3 6]. If we wsh to obtan more nformaton from an nformaton source, however, t s necessary to optmze the desgn of the magng system from the perspectve of the channel. Fgure shows a smplfed verson of the channel model shown n Fgure. Ths verson was used for convenence the current study, whch used the channel model and assocated theory to examne a gamma photon magng system. In ths smplfed model, the magng object s stll an nformaton source, the magng system s a channel, the computer recevng projecton data can be regarded as the recever, and the projecton data consttute the elements n recever space. No mage reconstructon algorthm was ncluded n the smplfed communcaton model, to avod any potental effects on the performance evaluaton of the magng system. In the channel model, we drectly used the average mutual nformaton to represent the nformaton transmsson The Author(s) 0. Ths artcle s publshed wth open access at Sprngerlnk.com csb.scchna.com

2 Da Q S Chnese Sc Bull September (0) Vol.56 No Fgure Schematc dagram of a communcaton model for photon magng. ( ) ( ) pn eff n R = d + M + R M, () ( ) g = d cos θ 6 z, () 3 pn eff Fgure Schematc dagram of a communcaton model for a pnhole magng system. performance of a gamma photon magng system. The channel capacty s related to the statstcal characterstcs of the channel, but the channel can acheve the greatest amount of nformaton transmsson only when t s used for a partcular nformaton source. Snce the nformaton source of a gamma photon magng system cannot be changed freely, the channel capacty cannot be used to accurately characterze the nformaton transmsson capacty of the magng system. The pnhole gamma camera s the bass of the newly developed pnhole SPECT system. As the SPECT system tself s more complcated than the gamma camera, ths ntal nvestgaton examnes the pnhole gamma camera to establsh the theoretcal foundaton for future studes of the pnhole SPECT system. The schematc geometry of pnhole magng s shown n Fgure 3. The spatal resoluton, R pn, and geometrc effcency, g pn, can be approxmated by eqs. () and () [7]: Fgure 3 Schematc geometry of pnhole magng. μh e α where deff = d + tan s the effectve dameter of μ pnhole collmator [8], d s the geometrc dameter, h s the channel heght, a s the openng angle, μ s the attenuaton coeffcent, M = f / z s the magnfcaton, z s the source-tocollmator dstance, f s the focal length, R n s the ntrnsc resoluton of detector, and L s the sze of detector. The formula apples to the pnhole axs regon. In ths paper, the approxmate formula of average mutual nformaton for the pnhole gamma camera was frst derved, followed by dscusson of ts mplcatons. Fnally, the results of magng experments are reported. Channel model of pnhole gamma camera In ths secton, the channel type of pnhole gamma camera s frst analyzed from the perspectve of mappng, then descrbed wth a mathematcal model. In ths magng model, the magng object s dvded nto a 3D voxel matrx, and the projecton mage nto a D pxel matrx. In 3D magng space, the same radaton dstrbuton can be mapped onto dfferent projecton mages at dfferent tmes due to statstcal fluctuaton, Compton scatterng and background radaton. Thus, the pnhole SPECT system consttutes a nosy channel. For the same reasons, a varety of dstrbutons may be mapped onto the same projecton mage. Thus, the pnhole SPECT system also consttutes a lossy channel. To summarze, the gamma camera can be consdered a nosy and lossy channel. Accordng to nformaton theory, the followng mathematcal language can be used to descrbe the channel model of the magng system. Let X X, X X P = p( X), p( X) p( X) be the probablty space of nput sgnal set of channel, (3)

3 760 Da Q S Chnese Sc Bull September (0) Vol.56 No.5 where X denotes the th radaton dstrbuton, p(x ) s the correspondng probablty, and px ( ) =. Let Y Y, Y Y j P = p( Y), p( Y) p( Yj ) be the probablty space of the output sgnal set of the channel, where Y j denotes the jth projecton, p(y j ) s the correspondng probablty, and py ( j ) =. Hence, the probablty space of the channel s [X,P(Y X), Y], and PY ( X ) =. The average mutual nformaton Y can be calculated by I( X; Y) = H( X) + H( Y) H( XY). (5) In prncple, all of the radoactve dstrbutons and ther correspondng projecton mages could be enumerated to calculate the average mutual nformaton. However, under the condtons of current computng, there s not enough space to store all the elements of source space and recever space. Therefore, t s mpossble to calculate the average mutual nformaton of all knds of channels. To overcome ths dffculty, the problem was smplfed by assumng a plane source. Let R pn be the spatal resoluton of the magng system, D FOV be the FOV, and R n be the ntrnsc resoluton of crystal array detector. The plane source of D FOV D FOV can then be dvded nto a two-dmensonal pont source matrx model of n FOV n FOV accordng to the resoluton of the magng system, where n FOV = D FOV /R pn, the dstance between ponts, s R pn. Hence the number of pont sources nsde the FOV s πdfov =. (6) 4R Eq. () and D FOV = L/M are substtuted nto eq. (6) resultng n πl =. (7) 4 deff ( M + ) + Rn In actual magng, the concentraton of radoactve materal that organsms absorb has a maxmum value. Here, we let ρ max represent the maxmum concentraton of plane source, whch s at the th pont source. Thus, the maxmum actvty of the pont source matrx from plane source s a pn ρ R. max max pn (4) = (8) In the magng system, the count of correspondng detector unts through geometrc mappng s N = ξa gt, (9) max where ξ refers to the effcency of the detector, t represents measurement tme, and g represents the effcency of the correspondng detector pxel: where 3 Rn cos θ Rn, d 4π( z+ f) + M 3 Smd cos θ Rn Rn g =, d, (0) 4 π ( z+ f) + M + M d cos θ R 3 n, d < + 6z n M π R Smd = + M d 4 ( + M) d n arccos ( ) + R ( + M) d R. n The gamma photons passng through the pnhole aperture to the detector are manly dvded nto three components: drect, penetraton and scatterng. In general, the drect component s regarded as sgnal, and penetraton, scatterng and background radaton are regarded as nose. As such, geometrc aperture was used n eq. (0) rather than effectve aperture. The nformaton transmsson effcency of the collmator s gven by η = γ ( γ + γ + γ + γ ), () d d p s b where γ d s the drect component, γ p s the penetraton component, γ s s the scatterng component, and γ b s the background radaton component. Whle N >>, the standard devaton of N, σ, can be approxmated by N, here σ was used as a unt of gray scale. Consderng the effect of nose on the nformaton transmsson capacty of the magng system, η was used drectly to calbrate the effectve gray scale. Thus, the effectve gray scale of projecton mage s gven by N ξa max gt K = η + = η +. () N Therefore, one gray mage approxmately corresponds to only one plane source, and the pnhole gamma camera approxmates the noseless, lossless channel. Therefore, when a max s at the th pont source, the number of elements n recever space s K, and the mutual nformaton s n A I = ( n )log K. (3) A Snce [, n A ], the average mutual nformaton of the magng system s n I = I = log K n A A = = ξ amax gt = log η +. (4) n A =

4 Da Q S Chnese Sc Bull September (0) Vol.56 No.5 76 When n A >>, n A ξamax gt I log η +. (5) = Accordng to eq. (5), an ncrease of ξa max t, causes the average mutual nformaton to ncrease. Ths ndcates that the gamma photon magng system s a tme-dependent channel; thus, the recever space n the model above s not complete. If the value of eq. (5) s less than 0, the artfacts produced by nose would lead the mage to produce an error message. The average mutual nformaton s generally not less than 0 n nformaton theory, so n the followng calculaton, t s defned as I = 0, when I < 0. Comparng eq. () and (0) shows that the channel model ncreases the amount of nformaton about the geometrc relatonshp between the pnhole and the detector unt. When deff Rn ( + M), further ncreasng the aperture can ncrease the senstvty of conventonal parameters, but cannot ncrease the count of correspondng detector unts through pont source geometrc mappng. On the other hand, t would be expected to decrease the average mutual nformaton n the system because of the reducton n n A. Therefore, an optmal aperture should exst for the actual magng system. Smlarly, accordng to eq. (0), an optmal focal length should exst for the same magng object or the same FOV. From (5), the optmal aperture and optmal focal length can be obtaned under dfferent parameters. The openng angle of the pnhole collmator s determned by the focal length and detector sze. Fgure 4 Relatonshp between the average mutual nformaton and the focal length. (a) ξρ max t = 0 6 /mm ; (b) ξρ max t = 0 7 /mm. Analyss results In accord wth eq. (5), the effects of FOV, focal length, aperture and ntrnsc resoluton on the average mutual nformaton were ntally examned. The calculaton model was as follows: the pnhole collmators made of tungsten alloy were knfe-edge-shaped, the area of the NaI(Tl) crystal array detector was 0 mm 0 mm, and the radoactve source was 99m Tc, 40.5 kev radaton. In the calculaton, η = γ d /(γ d + γ p + γ s + γ b ) (d/d eff ), the values of ξρ max t were 0 6 /mm and 0 7 /mm respectvely. Fgure 4 shows the relatonshp between the average mutual nformaton and the focal length under dfferent magnfcatons, and dfferent condtons of FOV. The magnfcatons tested were,, 3 and 4 respectvely, the pnhole aperture was.0 mm, the ntrnsc resoluton of the detector was.0 mm, ξρ max t was 0 6 /mm n Fgure 4(a), and 0 7 /mm n Fgure 4(b). As shown n the fgure, an optmal focal length exsted, but ths ncreased wth the magntude of the ξρ max t ncrease. In addton, FOV truncaton caused substantal nformaton loss when the magng object was too bg. Fgure 5 shows the relatonshp between the average mutual Fgure 5 Relatonshp between the average mutual nformaton and the aperture. (a) ξρ max t = 0 6 /mm ; (b) ξρ max t = 0 7 /mm ; nformaton and the aperture under dfferent ntrnsc resolutons. The ntrnsc resolutons tested were,, 3 and 4 mm, whle the focal length was 00 mm, the source-to-collmator dstance was 30 mm, ξρ max t was 0 6 /mm n Fgure 5(a), and 0 7 /mm n Fgure 5(b). As shown n the fgure, an optmal aperture exsted, but t was related to the ntrnsc resoluton and the count detected by the detector. The above analyss ndcates that an optmal focal length, aperture, and ntrnsc resoluton of pnhole gamma camera exst n practcal applcatons of ths magng method, whch seems to conflct wth eq. (). Accordng to eq. (), the smaller the aperture and ntrnsc resoluton, the hgher the spatal resoluton, and the more nformaton can be obtaned from the mage. In fact, eq. () descrbes the spatal resoluton of the pnhole gamma camera under the deal state, not takng nto account the effects of statstcal fluctuaton, transmsson, scatterng or background radaton nose. As shown n Fgure 5, ncreasng the statstcal count (.e. de-

5 76 Da Q S Chnese Sc Bull September (0) Vol.56 No.5 creasng the statstcal nose), caused a gradual decrease n the optmal aperture and the ntrnsc resoluton. In practcal magng applcatons, the statstcal count s lmted and s not too bg due to the constrants mposed by radoactve drugs and magng tme. Thus, optmal values of aperture and ntrnsc resoluton exst, but vary under dfferent condtons of use. 3 Imagng results To verfy the conclusons descrbed above, a seres of pnhole gamma camera magng experments were desgned. The magng object conssted of four capllares, wth an nternal dameter of 0.3 mm, flled wth approxmately 0 mc/ml 99m Tc soluton. The dstances between capllares were mm, 3 mm and mm respectvely (Fgure 6). Imagng experments were completed wth a small gamma camera n the McroSPECT/CT laboratory of the Shangha Insttute of Appled Physcs (SINAP), CAS. The gamma camera conssted of a NaI(Tl) crystal array and a postonsenstve photo-multpler tube (PSPMT) wth crystal thckness of 5 mm and ntrnsc resoluton of.4 mm. The focal length was 0 cm. The pnhole collmators were made of tungsten alloy, wth apertures of 0.5 and.0 mm, channel heght of 0.5 mm, and acceptance angle of 60 degrees. We frst calculated the analyss values wth dfferent object dstances and dfferent acquston tmes (Fgure 7), n whch the object dstances were 0, 40, 60, 80 and 00 mm, respectvely, the acquston tmes were 30 and 300 s, respectvely, a max = ρπr R pn. As the FOV vared at dfferent object dstances, the nformaton densty was used for comparson, whch s the average mutual nformaton per unt area. Projecton mages were then obtaned at dfferent object dstances and dfferent acquston tmes. The profles of projecton mages are shown n Fgure 8. As shown n Fgure 8(a) and (c), a 0.5 mm aperture could dstngush a space nterval of mm when L = 0 mm Fgure 7 Analyss values of average mutual nformaton. Fgure 6 Imagng object consstng of four capllares. Fgure 8 Projecton mages of capllary and profles wth dfferent apertures, dfferent tmes and dfferent object dstances. (a) The aperture s 0.5 mm, acquston tme s 30 s; (b) the aperture s.0 mm, acquston tme s 30 s; (c) the aperture s 0.5 mm, acquston tme s 300 s; (d) the aperture s.0 mm, acquston tme s 300 s.

6 Da Q S Chnese Sc Bull September (0) Vol.56 No and acquston tme s 300 s, ndcatng that the amount of nformaton ncreases wth an ncrease n acquston tme. From Fgure 8(a) and (b), the sgnal to nose rato (SNR) of the 0.5 mm aperture decayed more quckly wth ncreasng object dstance compared to when the aperture was.0 mm. When L = 80 mm, a 0.5 mm aperture could not dstngush a 3 mm space nterval, whereas an.0 mm aperture could. Ths fndng ndcates that an.0 mm aperture s better than a 0.5 mm aperture under these magng condtons. Comparng Fgure 7 and Fgure 8 ndcates that the expermental results were consstent wth the theoretcal predctons. The results descrbed above ndcate that the proposed formula was able to predct the experment results, and that an optmal aperture exsts n ths magng system under dfferent magng condtons. Therefore, an optmzed desgn of pnhole gamma camera system should be based on the characterstcs of the magng object, ncludng ts sze, the desred magng tme, and the concentraton and specfcty of radoactve substances. 4 Concluson Although further studes are requred, the current results revealed that the channel model of nformaton theory s approprate for descrbng the pnhole gamma camera. In addton, the fndngs ndcated that average mutual nformaton can replace resoluton, senstvty and FOV n provdng descrptons of the propertes of magng systems. The formula used to approxmate the pnhole gamma camera presented here was able to determne optmzed parameters under dfferent magng condtons, provdng valuable nformaton for optmal desgn and use. Future studes wll examne the channel model of the pnhole SPECT magng system on the bass of ths theory. In addton, the current method can be used for optmzng the desgn of other magng systems, such as photon countng computed tomography. The author would lke to thank Yujn Q of the Shangha Insttute of Appled Physcs for the experment nstruments. Mekle S R, Kench P, Kassou M, et al. Small anmal SPECT and ts place n the matrx of molecular magng technologes. Phys Med Bol, 005, 50: R45 R6 Madsen M T. Recent advances n SPECT magng. J Nucl Med, 007, 48: Schramm N U, Ebel G, Engeland U, et al. Hgh-resoluton SPECT usng multpnhole collmaton, IEEE Trans Nucl Sc, 003, 50: Furenld L R, Wlson D W, Chen Y C, et al. Fast SPECT II: A second-generaton hgh-resoluton dynamc SPECT mager. IEEE Trans Nucl Sc, 004, 5: Beekman F J, van der Have F, Vastenhouw B, et al. U-SPECT-I: A novel system for submllmeter-resoluton tomography wth radolabeled molecules n mce. J Nucl Med, 005, 46: van der Have F, Vastenhouw B, Ramakers R M, et al. U-SPECT-II: An ultra-hgh-resoluton devce for molecular small-anmal magng. J Nucl Med, 009, 50: Song T Y, Cho Y, Chung Y H, et al. Optmzaton of pnhole collmator for small anmal SPECT usng Monte Carlo smulaton. IEEE Trans Nucl Sc, 003, 50: Beekman F J, Vastenhouw B. Desgn and smulaton of a hgh-resoluton statonary SPECT system for small anmals. Phys Med Bol, 004, 49: Cao Z, Bal G, Accors R, et al. Optmal number of pnholes n mult-pnhole SPECT for mouse bran magng a smulaton study. Phys Med Bol, 005, 50: Rentmeester M C M, van der Have F, Beekman F J. Optmzng mult-pnhole SPECT geometres usng an analytcal model. Phys Med Bol, 007, 5: Vunckx K, Beque D, Defrse M, et al. Sngle and multpnhole collmator desgn evaluaton method for small anmal SPECT. IEEE Trans Med Imagng, 008, 7: Vanhove C, Defrse M, Lahoutte T, et al. Three-pnhole collmator to mprove axal spatal resoluton and senstvty n pnhole SPECT. Eur J Nucl Med Mol Imagng, 008, 35: Zhang X Z, Q Y J. Development of fully 3D mage reconstructon technques for pnhole SPECT magng. Chnese Sc Bull, 0, 56: Yuan S, Dong Z L, Mao L, et al. Research on the reconstructon of fast and accurate AFM probe model. Chnese Sc Bull, 00, 55: Lu D T, Zhang T, Yang J Q, et al. A reconstructon method of porous meda ntegratng soft data wth hard data. Chnese Sc Bull, 009, 54: Zhao X, Lu J, Gong A D, et al. Operatonal spectrum reconstructon of data from the Fourer transform hyperspectral mager onboard HJ- A satellte, Chnese Sc Bull, 00, 55: Anger H O. Radosotope Cameras n Instrumentaton n Nuclear Medcne. New York: Academc press, Da Q S, Q Y J. Investgaton of the spatal resoluton of pnhole SPECT magng (n Chnese). Acta Phys Sn, 00, 59: Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts any use, dstrbuton, and reproducton n any medum, provded the orgnal author(s) and source are credted.