IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 8, AUGUST 2002 867 Efficient Fully 3-D Iterative SPECT Reconstruction With Monte Carlo-Based Scatter Compensation Freek J. Beekman*, Senior Member, IEEE, Hugo W.A.M. de Jong, Member, IEEE, and Sander van Geloven Abstract—Quantitative accuracy of single photon emission computed tomography (SPECT) images is highly dependent on the photon scatter model used for image reconstruction. Monte Carlo simulation (MCS) is the most general method for detailed modeling of scatter, but to date, fully three-dimensional (3-D) MCS-based statistical SPECT reconstruction approaches have not been realized, due to prohibitively long computation times and excessive computer memory requirements. MCS-based reconstruction has previously been restricted to two–dimensional approaches that are vastly inferior to fully 3-D reconstruction. Instead of MCS, scatter calculations based on simplified but less accurate models are sometimes incorporated in fully 3-D SPECT reconstruction algorithms. We developed a computationally efficient fully 3-D MCS-based reconstruction architecture by combining the following methods: 1) a dual matrix ordered subset (DM-OS) reconstruction algorithm to accelerate the reconstruction and avoid massive transition ma- trix precalculation and storage; 2) a stochastic photon transport calculation in MCS is combined with an analytic detector modeling step to reduce noise in the Monte Carlo (MC)-based reprojection after only a small number of photon histories have been tracked; and 3) the number of photon histories simulated is reduced by an order of magnitude in early iterations, or photon histories calcu- lated in an early iteration are reused. For a 64 64 64 image array, the reconstruction time required for ten DM-OS iterations is approximately 30 min on a dual processor (AMD 1.4 GHz) PC, in which case the stochastic nature of MCS modeling is found to have a negligible effect on noise in reconstructions. Since MCS can calculate photon transport for any clinically used photon energy and patient attenuation distribution, the proposed methodology is expected to be useful for obtaining highly accurate quantitative SPECT images within clinically acceptable computation times. Index Terms—Dual matrix reconstruction, image reconstruc- tion, single photon emission computed tomography, monte carlo methods, scattering. I. INTRODUCTION S INGLE photon emission computed tomography (SPECT) images are degraded by attenuation of photon flux, detector and collimator blurring, and the improper detection of scattered photons. These image degrading factors can have a large im- pact on quantitative accuracy and clinical diagnosis. For ex- ample, photon attenuation is one of the major causes of false Manuscript received November 15, 2001; revised May 20, 2002. The Asso- ciate Editor responsible for coordinating the review and recommending its pub- lication was Z. Liang. Asetrisk indicates corresponding author. *F. J. Beekman is with the Department of Nuclear Medicine, Image Sciences Institute, University Medical Center Utrecht, Room E02.222, Heidelberglaan 100, 3584 CX, Utrecht, The Netherlands (e-mail: f.beekman@azu.nl). H. W. A. M. de Jong and S. van Geloven are with the Department of Nuclear Medicine, Image Sciences Institute, University Medical Center Utrecht, 3584 CX, Utrecht, The Netherlands. Digital Object Identifier 10.1109/TMI.2002.803130 positive cardiac SPECT perfusion images [1]. When correction for nonuniform attenuation is performed, the diagnostic accu- racy of cardiac SPECT for the detection and localization of coronary artery disease can be improved [2]. Collimator blur- ring results in degraded resolution of SPECT images. Correc- tion for this effect by using accurate models of distance-depen- dent camera blurring during iterative reconstruction improves signal-to-noise ratio (SNR) and resolution [3]–[5]. Detection of scattered photons results in haziness and degraded quantitative accuracy. When accurate scatter models are used during iter- ative reconstruction, the SNR of small cold lesions and quan- titative accuracy are improved (e.g., [6]–[10]). Several experi- ments have shown that attenuation, blurring, and scatter all have to be corrected for in order to achieve optimal clinical SPECT imaging [9], [11]–[14]. Statistical reconstruction algorithms like maximum-like- lihood expectation-maximization (ML-EM, [15], [16]) are attractive, for example because they place nonnegativity constraints on the reconstructions, they have attractive noise properties [17], [18], and they enable to account for Poisson noise in the projection data. Furthermore, realistic models that include effects of scatter detection, camera blurring, and attenuation can be incorporated in the transition matrix during reconstruction. Ideally, a transition matrix element equals the expectation value of the fraction of the number of photons being emitted by a volume element (voxel) that is detected in a projection pixel . A specific part of a matrix column that represents the probabilities of photons emitted from a single voxel and being detected in the pixels on a specific projection image closely represents a point spread function (PSF). The exact matrix (or a complete set of PSFs) depends on acquisition parameters, the gamma camera system, and the density distribution of the patient. This allows correction for effects of detector blurring (e.g., [4], [5], and [19]–[21]). When a transmission scan registered with the emission data is available to add patient specific information to the transition matrix, accurate correction for nonuniform attenuation (e.g., [22] and [23]) and scatter (e.g., [6] and [24]–[26]) is possible. In order to correct the SPECT images accurately for photon crosstalk between trans-axial slices, one needs fully three-di- mensional (3-D) reconstruction [3], [4], [6], [27], [28]. In con- trast with 2-D (slice-by-slice) reconstruction, fully 3-D recon- struction uses a large matrix which enables to take into account photons that are detected in out-of-slice projection pixels. This further improves the accuracy of the reconstructions, albeit at the cost of a relatively high computational burden. In addition to providing improved quantitative accuracy, fully 3-D iterative reconstruction with accurate modeling results in significant im- 0278-0062/02$17.00 © 2002 IEEE