Phys. Med. Biol. 39 (1994) 2023-2041. Printed in the UK An analytical approach to quantitative reconstruction of non-uniform attenuated brain SPECT Zhengrong Liang, Jinghan Ye and Donald P Harrington DepaNnent of Radiology, State University of New York, Stony Brook. NY 11794, USA Received 13 April 1994, in final form 13 July 1994 Abstract. An analytical approach to quantitative brain S P E ~ (single-photon-emission computed tomography) with non-uniform attenuation is developed. The approach formulates accurately the projection-transform equation as P summation of primary- and scatter-photon contributions. The scatter contribution can be estimated using the multiple-energy-window samples and removed from the primary-energy-window data by subtraction. The approach models the primary contribution as a convolution of the attenuated source and the detector-response kernel at a cowtant depth from the detector with the central-ray approximation. The attenuated Radon transform of the source wn be efficiently deconvolved using the depth-frequency relation. The approach inverts exactly the attenuated Radon transform by Fourier transforms and series expansions. The performance of the analytical approach was studied for both uniform- and non-uniform-attenuation cases, and compared to the conventional FBP (filtered-backprojection) method by computer simulations. A patient brain x-ray image was acquired by a CT (wmputed- tomography) scanner and converted to the object-specific attenuation map for 140 keV energy. The mathematical Hoffman brain phantom was used to simulate the emission source and was resized such that it was completely surrounded by the skull of the CT attenuation map. The detector-response kmel was obtained from measurements of a point source at several depths in air from a panllel-hole collimator of a SPECT camera. The projection data were simulated from the object-specific attenuating source including the depth-dependent detector response. Quantitative improvement (> 5%) in reconstructing the data wos demonsmted with the non- uniform attenuation compensation, as compared to the uniform attenuation correction and the conventional FBP reconstruction. The commuting time was less than 5 min on an HPn30 desktop computer for an image army of 12g2 x 32 from 128 projections of 128 x 32 size. 1. Introduction Quantitative reconstruction of brain SPECT (single-photon-emission computed tomography) has a wide clinical application, such as in neuroreceptor imaging (Ichise et al 1993, Verhoeff et al 1993) and cerebral perfusion analysis or cerebral blood-flow study (Nicolas et al 1993, Palestro et al 1993, Hoshi et al 1994, Tsuchida er al 1994). The currently available SPECT protocols for clinical studies support only qualitative reconstruction, in which photon attenuation by brain tissues, skull bone, and skin is either ignored or assumed to be uniform. If the attenuation is ignored, the reconstruction is usually performed by a conventional filtered backprojection (FBP) (Huesman et al 1977). If a uniform attenuation is assumed throughout the whole head, the reconstruction is practically carried out by multipiying the FBP result by attenuation-weight factors as described in the article by Chang (1978), although other mathematically more accurate methods have been suggested (Bellini et al 1979, Tretiak and Metz 1980, Gullberg and Budinger 1981, Clough and Barrett 1983, Kim et ~l 1984, Hawkins et al 1988, Inouye et al 1989). A quantitative reconstruction 0031-9155/94/112023+19$19.50 @ 1994 IOP Publishing Ltd 2023