Fast Positron Range Calculation in Heterogeneous Media for 3D PET Reconstruction

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1 Fast Positron Range Calculation in Heterogeneous Media for 3D PET Reconstruction László Sziray-Kalos, Milán Magdics, Balázs Tóth, Balázs Csébfalvi, Taás Uenhoffer, Judit Lantos, and Gergely Patay Abstract This paper presents a fast GPU-based solution to copensate positron range effects in heterogeneous edia for iterative PET reconstruction. We assue a factorized approach, where projections are decoposed to phases according to the ain physical effects. Positron range is the first effect in this chain, which causes a spatially varying blurring according to local aterial properties. In high-resolution sall anial PET systes, the average free path length of positrons ay be any ties longer than the linear size of s. This eans that positron range significantly coproises the reconstruction quality if it is not copensated, and also that the aterial dependent blurring should have a very large support so its space calculation would take prohibitively long. Frequency doain filtering does not have such coputational coplexity probles, but its direct for is ruled out by the fact that we need a spatially variant filtering in heterogeneous edia. To handle heterogeneous edia, we execute Fast Fourier Transfors for each aterial type and for appropriately odulated tracer densities and erge these partial results into a density that describes the coposed, heterogeneous ediu. Fast Fourier Transfor requires the filter kernels on the sae resolution as the tracer density is defined, so we also present efficient ethods for re-sapling the probability densities of positron range for different resolutions and basis functions. The algorith is ipleented on the GPU, built into the TeraToo TM syste, and requires just a few seconds on high resolution arrays. I. INTRODUCTION In Positron Eission Toography (PET), isotopes decay with positron eission. As a positron travels through the tissue it gives up its kinetic energy by Coulob interactions with electrons, which results in a chaotic path terinated by positron electron annihilation. The statistical properties of these paths heavily depend on the initial kinetic energy, i.e. the type of the isotope and on the electron density of the particular tissue, i.e. the type of the aterial (e.g. bone, flesh, air, etc.). The positron range, i.e. the translation between the isotope decay and the positron annihilation results in positional inaccuracies in toography reconstruction, which is one of the ost iportant liiting factors of the resolution in sall anial PETs [11]. This proble can be attacked by the inclusion of the positron range calculation [4], [3] in ML-EM reconstruction, which consists of the iteration This work has been supported by the TeraToo project of the National Office for Research and Technology, TÁMOP B-10/ , OTKA K , OTKA K , and OTKA K L. Sziray-Kalos, M. Magdics, B. Tóth, and T. Uenhoffer are with Budapest University of Technology and Econoics (e-ail: sziray@iit.be.hu). J. Lantos and G. Patay are with Mediso Ltd. (e-ail: Judit.Lantos@ediso.hu). of particle transport calculations, called forward projection, and corrective back projection [11]. The inputs of the reconstruction are the easured values in Lines of Responses or LORs: y =(y 1,y 2,...,y NLOR ),andtheaterial ap of the exained object, which is typically obtained by a CT scan. The output of the reconstruction ethod, which is siultaneously another input of a single iteration step, is the tracer density function x( v), which is approxiated in a finite function series for: x( v) = =1 x b ( v), (1) where x =(x 1,x 2,...,x N ) are the coefficients and b ( v) ( =1,...,N ) are pre-defined basis functions [1], which are typically defined on a grid. The correspondence between the coefficients of the tracer density function ( values) and the LOR hits is established by the syste sensitivity T ( v L) defining the probability that a radioactive decay happened in v is detected by LOR L. Forward projection coputes the expectation value of the nuber of hits in each LOR L: ỹ L = x( v)t ( v L)dv = =1 A L x (2) where is the doain of the reconstruction, i.e. the field of view of the toograph, and A L is the syste atrix: A L = b ( v)t ( v L)dv. (3) Taking into account that the easured hits follow a Poisson distribution, after each forward projection, the ML-EM schee executes a back projection correcting the estiates based on the ratios of easured and coputed LOR values [12]: L x = x A L yl ỹ L L A. (4) L The iteration will converge to a fix point where the scaling factors of this forula are equal to one, thus the ML- EM schee solves the following equation: y L A L = A L. ỹ L L L This equation can also written in atrix for A T y A x = AT 1, (5)

2 where T represents transpose, y is the vector of easured LOR values, vector division is defined as eleent-wise division, and 1 is a vector containing 1 for every. Syste sensitivity can be decoposed according to the ain physical effects, like positron range, geoetric projection and attenuation, scattering in the tissues, scattering in the detector crystals, and finally the process of detection. In this respect, positron range is odeled by conditional probability density P ( v p v a ) of positron annihilation in v a provided that a positron was born in point v p. The annihilation density x a ( v a ) is obtained fro the tracer density x( v p ) applying the blurring caused by the positron range: x a ( v a )= x( v p )P ( v p v a )dv p. Substituting the finite eleent approxiation of the tracer density, this convolution is expressed by a discrete filtering operation: x a ( v a )= x b ( v p )P ( v p v a )dv p. =1 The finite eleent coefficient of the annihilation density is coputed by ultiplying both sides with the adjoint basis function b, that are orthonoral to the original basis functions, i.e. b b dv =1 if = and zero otherwise. We use two options, piece-wise constant basis functions when the adjoints are also piece-wise constant basis functions, and tri-linear basis functions when the adjoints are Dirac-delta functions selecting the corners. The result of the scalar product is x a = b ( v a )x a ( v a )dv a = =1 x b ( v a )b ( v p )P ( v p v a )dv p dv a = =1 P, x, where the discrete filter kernel is P, = b ( v a )b ( v p )P ( v p v a )dv p dv a. If piece-wise constant basis functions are used, then atrix eleent P, is the probability that a positron is annihilated in provided that it was born in. Syste atrix A is also often factored according to the ain physical effects and is approxiated by a product of atrices, each representing a physical phenoenon: A = D (G + S) P, where D is the atrix representing detector effects, G is the non-square atrix of geoetric projection and phanto attenuation, S is the non-square atrix of scattered projection, and P is an N N size atrix of the positron range. Unfortunately, atrix P is huge and depends both on the radiotracer isotope, e.g. 18 F, 15 Oor 82 Rb, and on the tissue type of different s. For isotope and aterial types where the positron range is sall, like 18 F in bones, the atrix is sparse because the probability that a positron gets far is approxiately zero. However, less dense aterials like water and air, or isotopes eitting high kinetic energy positrons, like 82 Rb, correspond atrices of uch fewer zero eleents. II. METHODS In order to siulate positron range, we need effective techniques to ipleent the filtering operator. We consider two cases, a special one when the aterial is hoogeneous and the filtering becoes spatially invariant, and a general case when the aterial is inhoogeneous, and the filtering is spatially varying. A. Hoogeneous aterial If the tissue is hoogeneous, we could exploit the translational syetry of the positron range, i.e. its probability atrix depends just on the distance of positron generation and annihilation: P ( v p v a )=P( v a v p ) which akes positron range calculation equivalent to a convolution x a ( v a )= x( v p )P ( v a v p )dv p. The coputation of the discrete version of the convolution kernel can also benefit fro the spatial invariance. Matrix eleent P, = b ( v a )b ( v p )P ( v a v p )dv p dv a = P O,r (6) depends just on the relative location r of with respect to, and thus it is enough to copute it for a reference O and all s r. A convolution can be evaluated both in spatial doain, i.e. space, and in frequency space having applied Fourier transforation. As the coputational coplexity of filtering in spatial doain is proportional to the product of the nubers in the positron density volue and the filter kernel, spatial filtering gets prohibitively expensive for large kernels. Note that the linear size of sall anial PETs ay be about , while the FWHM of the positron range effect in water is about 1 4 depending on the isotope [2], thus the required size of the 3D filter kernel is greater than s. Approxiating the filter kernel by a separable approxiation like the Gaussian filter can speed up the process, but the Gaussian would be a rather poor approxiation of the positron range phenoenon [6]. Fortunately, the convolution can also be evaluated in frequency doain having applied 3D Fast Fourier Transfors F, and

The noise of Monte Carlo siulation is filtered out by fitting the siulation data on functions of for P ( v) = aαe α v + bβe β v, 2π v which is based on [4], [10], [6] stating that the probability

3 the coputational coplexity of frequency doain filtering is independent of the kernel size: x a ( v) =F 1 [F[x( v)] F[P ( v)]]. The actual for of kernel P ( v a v p )=P( v) depends on the aterial isotope pair. We calculate P ( v) on high resolution off-line with GATE [5] siulations. The noise of Monte Carlo siulation is filtered out by fitting the siulation data on functions of for P ( v) = aαe α v + bβe β v, 2π v which is based on [4], [10], [6] stating that the probability density of positron range projected onto Cartesian axis X can be well approxiated by p X (X) = ae αx + be βx where paraeters a, α, b, β depends on the aterial isotope pair. During fitting, we also ipose the requireent that p X is a probability density, thus it integrates to 1. Matrix eleents P, also depend on the discretization, i.e. the basis functions and on the size of s. So, having the probability density of the positron range in a continuous analytical for or defined as histogras of easured data, the positron range effect copensation would require the resapling of these functions according to the resolution and the basis functions of the grid (equation 6), and then a convolution operation with the currently estiated tracer density. As in pre-clinical research PETs, the size is also a user controllable paraeter, the re-sapling is executed onthe-fly applying nuerical quadrature when the input data are loaded. B. Inhoogeneous aterial Fig. 1. Positron range is a spatially variant blurring operator. The path of a positron depends on the aterial where it is born, on the aterial where it is annihilated, and also on the aterial that is visited between the generation and the annihilation. In inhoogeneous objects, blurring kernel P ( v p v a ) also depends on the aterial (e.g. bone, air, soft tissue) distribution of the easured object, i.e. the aterial of every (Fig. 1). The precise treatent of this phenoenon would require the consideration of all possible positron paths, which would lead to a high-diensional integral for every point pair, and would pose prohibitive coputational requireents in PET systes. However, assuing that the aterial is hoogeneous, i.e. the blurring kernel is independent of the aterial type in points v a, v p and elsewhere in the object, would be the other extree approach that would ignore the significantly different probability densities associated with different aterials. Fig. 2. Intuitive explanation of the siplified ethod when the aterial of the positron generation is used. Blurring each with the filter kernel associated with the aterial in this eans the replaceent of a single spatial variant filtering by one spatial invariant filtering for each aterial and a suation. This paper proposes a practical coproise that is sufficiently accurate for PET reconstruction and can be coputed in reasonable tie with the support of highly parallel GPU hardware. The basic idea is that instead of considering the aterial in all points, we take into account the aterial type only at one end of the positron path. This eans that we blur each with the filter kernel associated with the aterial in this and ignore the fact that there ight be a aterial boundary nearby. This siplification replaces a spatially variant filtering by several spatially invariant convolutions and a suation. Selecting the aterial of the positron generation location (Fig. 2) and denoting the index of the aterial at point v p by ( v p ), we obtain: x a ( v a ) x( v p )ξ ( v p )P ( v a v p )dv p (7) where ξ ( v p ) is an indicator function that is 1 if there is aterial of index in point v p and zero otherwise, and P ( v) is the probability density of the positron translation between its generation and annihilation in hoogeneous aterial of index. Coputing Fourier transfor F for each ter and then a single inverse Fourier transfor, we get: [ ] x a ( v) F 1 F[x( v)ξ ( v)] F[P ( v)]. Note that this coputation requires the Fourier transfors of the blurring functions coputed during pre-processing for each aterial, the Fourier transforation of the positron density once for each aterial type (usually two or three), and a single inverse Fourier transforation. Instead of using the kernel associated with the aterial of the positron generation location, we can also apply the kernel

4 Fig. 3. Intuitive explanation of the siplified ethod when the aterial of the positron annihilation is used. Blurring each with the filter kernel associated with the aterial in this eans the execution of a spatial invariant filtering for each aterial and a suation after asking according to the aterial. of the aterial at the position of the annihilation (Fig. 3), which leads to the following forula: x a ( v a )= x( v p )P ( vp)( v a v p )dv p = ξ ( v a ) x( v p )P ( v a v p )dv p. The convolutions in the su can also be coputed via Fourier transforations: x a ( v) ξ ( v)f 1 [F[x( v)] F[P ( v)]]. This second option is ore expensive coputationally since here both the nuber of Fourier transfors and the nuber of inverse Fourier transfors are equal to the nuber of aterials. The accuracy of the two techniques depends on whether or not the aterial including ost of the radioisotopes occupies a larger part of the object. For typical aterials and isotopes, the difference of the reconstructed volues is negligible. III. POSITRON RANGE IN BACK PROJECTION The ML-EM back projector considers the ratios of easured and estiated LOR values and executes the steps of forward projection backwards in reverse order to update the estiates. The positron range operator can be reversed, the only difference is that if we define kernels according to the aterial at the location of positron generation in the forward projection, then kernels should correspond to the aterial at the position of annihilation in the back projection, or vice versa. The geoetric back projection is essential in the iterative reconstruction algorith but positron range and even the detector response ay be skipped in this phase, aking forward and back projections unatched [13]. The application of unatched back projectors can reduce the tie needed by a single iteration and can iprove convergence, but generally akes the syste less stable and accurate for low-dose and therefore low-statistics easureents. Ignoring positron range in back projection, however, does not reduce accuracy in the liiting case, i.e. this odification does not distort the fixed point of the iteration. To show why this is true, let us consider a atched EM iteration using syste atrix A both in the forward and back projections, and an unatched one where the syste atrix is odified to B in back projection, and suppose that the issing factor is the space blurring atrix P that is responsible for positron range, i.e. A = B P. The atched iteration will converge to values x that satisfy A T y A x = AT 1. (8) Suppose that this is a well defined reconstruction proble, i.e. it has a unique solution, which requires A T A to be nonsingular, which eans that neither P can be singular. The liiting case of the unatched projection satisfies B T y A x u = BT 1. As ultiplying both sides by P T, this equation becoes siilar to equation 8 and since P is an invertible square atrix, the two equations are equivalent, thus their solutions are equal: x = x u. I. RESULTS The proposed ethods are ipleented in CUDA [9], integrated into the TeraToo TM syste [7], and run on NIDIA GeForce GTX 480 GPUs. 3D Fourier transforations are coputed with the NIDIA cufft library. In this syste the positron range calculation for 3 aterials at and resolutions take 0.6 sec and 2 sec, respectively. We note that the nuerical accuracy of cufft is iproved if the array is padded to have resolutions that are powers of two. During padding, we use the values at the borders siilarly to the clap texturing ode of GPUs. To deonstrate the potential of the proposed algoriths, we used Mediso s nanoscan PET/CT [8], which has 12 detector odules consisting of crystals of size In the first set of experients, we considered siulation data obtained by GATE. We reconstructed the ring phanto of hoogeneous activity put into water and bone aterials with and without positron range copensation (Fig. 4). Note that the hoogeneous activity inside the ring could be well reconstructed for 18 Fand 15 O isotopes. The ring geoetry also shows up nicely for 82 Rb, but the hoogeneous activity is coproised at aterial boundaries. The reason of this artifact is our approxiate odel which uses the aterial at the position of the annihilation and assues that the positron was also born in the sae aterial. We also exained the Micro Derenzo phanto with rod diaeters 1.0, 1.1,...,1.5 in different segents, which was reconstructed at resolution ( s)

120 100 Ring 1000s - Fluor 18 POSRANGE=ON, ATT=ON POSRANGE=OFF, ATT=ON POSRANGE=ON, ATT=OFF POSRANGE=OFF,

POSRANGE=OFF, ATT=OFF 120 100 Ring 1000s - Rubidiu 82 POSRANGE=ON, ATT=ON POSRANGE=OFF, ATT=ON POSRANGE=ON,

60 70 80 90 Iteration 20 0 10 20 30 40 50 60 70 80 90 Iteration 20 0 10 20 30 40 50 60 70 80 90 Iteration 18

The upper row contains the L2 error curves and the aterial ap.

respectively. Coluns correspond to different isotopes. Fig. 5.

copensation. with and without positron range copensation.

5 Ring 1000s - Fluor 18 POSRANGE=ON, ATT=ON POSRANGE=OFF, ATT=ON POSRANGE=ON, ATT=OFF POSRANGE=OFF, ATT=OFF Ring 1000s - Oxygen 15 POSRANGE=ON, ATT=ON POSRANGE=OFF, ATT=ON POSRANGE=ON, ATT=OFF POSRANGE=OFF, ATT=OFF Ring 1000s - Rubidiu 82 POSRANGE=ON, ATT=ON POSRANGE=OFF, ATT=ON POSRANGE=ON, ATT=OFF POSRANGE=OFF, ATT=OFF Error (L2) Error (L2) Error (L2) Iteration Iteration Iteration 18 F 15 O 82 Rb line profiles Fig. 4. Reconstruction of the ring. The upper row contains the L2 error curves and the aterial ap. The iages of the iddle and lower rows show the reconstructions without and with positron range copensation, respectively. Coluns correspond to different isotopes. Fig. 5. Micro Derenzo phanto reconstruction and the line profiles showing the difference produced by positron range copensation. with and without positron range copensation. The transversal slice and the line profiles are shown by Fig. 5. Fig. 6 shows the reconstruction of a sall anial IQ phanto and the line profiles obtained with and without positron range. Fig. 6. Sall anial IQ phanto reconstruction and line profiles coparing to the reconstruction without positron range copensation.

6 . CONCLUSIONS This paper presented an efficient positron range copensation algorith. The positron range calculation in heterogeneous aterial is decoposed to a series of positron range calculations in hoogeneous aterials, once for each aterial type of the exained object, and a final copositing step. Positron range in hoogeneous aterial, in turn, is evaluated in the frequency doain applying FFTs. The coplete reconstruction progra is ipleented in CUDA and runs on the GPU, providing positron range siulations in 0.6 sec and 2 sec at and resolutions, respectively. We also investigated the need of positron range calculation in the back projection of the ML-EM schee, and concluded that it can be safely skipped, aking the reconstruction faster. REFERENCES [1] J. Cabello and M. Rafecas. Coparison of basis functions for 3D PET reconstruction using a Monte Carlo syste atrix. Physics in Medicine and Biology, 57(7): , [2] J. Cal-González, J.L. Herraiz, S. Espana, M. Desco, J.J. aquero, and J.M. Udias. Positron range effects in high resolution 3d pet iaging. In IEEE Nuclear Science Syposiu Conference Record (NSS/MIC), pages , [3] J. Cal-González, J.L. Herraiz, S. Espana, E. icente, E. Herranz, M. Desco, J.J. aquero, and J.M. Udias. Study of CT-based positron range correction in high resolution 3D PET iaging. Nuclear Instruents and Methods in Physics Research Section A: Accelerators, Spectroeters, Detectors and Associated Equipent, 648(Suppleent 1):S172 S175, [4] S. E. Derenzo. Matheatical reoval of positron range blurring in high resolution toography. IEEE Transactions on Nuclear Science, 33: , [5] S. Jan and et al. GATE: A siulation toolkit for PET and SPECT. Physics in Medicine and Biology, 49(19): , [6] C. S. Levin and E. J. Hoffann. Calculation of positron range and its effect on the fundaental liit of positron eission toography syste spatial resolution. Physics in Medicine and Biology, 44: , [7] M. Magdics, L. Sziray-Kalos, B. Tóth, D. Légrády, Á. Cserkaszky, L. Balkay, B. Doonkos, D. ölgyes, G. Patay, P. Major, J. Lantos, and T. Bükki. Perforance Evaluation of Scatter Modeling of the GPUbased Tera-Too 3D PET Reconstruction. In IEEE Nuclear Science Syposiu and Medical Iaging, pages , [8] Mediso. Nanoscan-PET/CT. [9] NIDIA. In The CUDA Hoepage, [10] M. R. Paler and G. L. Brownell. Annihilation density distribution calculations for edically iportant positron eitters. IEEE Transactions on Medical Iaging, 11: , [11] A. J. Reader and H. Zaidi. Advances in PET iage reconstruction. PET Clinics, 2(2): , [12] L. Shepp and Y. ardi. Maxiu likelihood reconstruction for eission toography. IEEE Transactions on Medical Iaging, 1: , [13] G. Zeng and G. Gullberg. Unatched projector/backprojector pairs in an iterative reconstruction algorith. IEEE Transactions on Medical Iaging, 19(5): , 2000.

Ming-Wei Lee 1, Wei-Tso Lin 1, Yu-Ching Ni 2, Meei-Ling Jan 2, Yi-Chun Chen 1 * National Central University

Ming-Wei Lee 1, Wei-Tso Lin 1, Yu-Ching Ni 2, Meei-Ling Jan 2, Yi-Chun Chen 1 * National Central University Rapid Constructions of Circular-Orbit Pinhole SPECT Iaging Syste Matrices by Gaussian Interpolation Method Cobined with Geoetric Paraeter Estiations (GIMGPE Ming-Wei Lee, Wei-Tso Lin, Yu-Ching Ni, Meei-Ling