Positron Range Modeling for Statistical PET Image Reconstruction † Bing Bai, ‡ Ananya Ruangma, ‡ Richard Laforest, ‡ Yuan-Chuan Tai and † Richard M. Leahy † Signal and Image Processing Institute, University of Southern California, Los Angeles, CA 90089 ‡ School of Medicine, Washington University, St. Louis, MO, 63110 Abstract— Positron range is one of the factors that fundamen- tally limits the spatial resolution of PET images. With the higher resolution of small animal imaging systems and increased interest in using higher energy positron emitters, it is important to consider range effects when designing image reconstruction methods. The positron range distribution can be measured experimentally or calculated using approximate analytic formulae or Monte Carlo simulations. We investigate the use of this distribution within a MAP image reconstruction framework. Positron range is modeled as a blurring kernel and included as part of the forward projection matrix. We describe the use of a 3D isotropic shift-invariant blur kernel, which assumes that positrons are propagating in a homogeneous medium and is computed by Monte Carlo simulation using EGS4. We also propose a new shift-variant blurring model for positron range that accounts for spatial inhomogeneities in the positron scatter properties of the medium. Monte Carlo simulations, phantom, and animal studies with the isotopes Cu-60 and Cu-64 are presented. I. I NTRODUCTION Positron range is one of the factors that fundamentally limits the spatial resolution of PET images [1]. A positron travels a short distance before positron-electron annihilation. The range of the positron depends on its energy as well as the effective atomic number and atomic weight of the medium. Major interactions between positrons and the surrounding medium include Coulomb elastic collisions with atomic nuclei and inelastic collisions with atomic electrons [1]. In most cases a positron first loses all its energy and then annihilates with an electron [2]. In each inelastic collision, the positron only loses a small part of its energy [1], as a result many collisions will happen before annihilation and the trajectory of each positron is tortuous. Positron range in water has been measured experimentaly for several medically important isotopes [3], [4], [5]. These results show considerable variation, primarily because the resolution of the detectors were comparable to the positron range. Palmer and Brownell [6] proposed a 3D-Gaussian model for the anni- hilation point distribution, assuming the positrons behave diffu- sively. Their calculation is based on an empirical range-energy formula. Difficulties in experimental range measurements have also lead to the recent use of Monte Carlo simulation to calculate positron range [1], [7]. This work was supported in part by Grant R01 EB000363 from the National Institute of Biomedical Imaging and Bioengineering. The effect of positron range is a blurring of the recon- structed image. Based on the measured positron annihilation point distribution, Derenzo and Haber proposed a method to remove the blurring by spatial deconvolution [8], [9]. While this method can partly recover the resolution loss, by decoupling the deconvolution from image reconstruction we lose the ability to optimally handle noise amplification through the use of an accurate likelihood function. We have developed a 3D MAP reconstruction method in which a factored system model is used [10]. In our previous 3D MAP reconstructions, positron range has been ignored. Recently, the development of new detector technology has reduced crystal size so that 1mm spatial resolution is poten- tially achievable with small animal PET scanners such as the microPET II [11]. The spatial resolution of these scanners is comparable to the positron range of the isotopes that are commonly used (e.g., the mean positron range of F-18 in water is 0.5mm). High-energy isotopes with longer positron range have also been used in small animal PET studies [7]. In this paper we describe positron range modeling in our system model using blurring operators in the image space. We describe two approaches. The first uses a shift-invariant blurring operator that implicitly assumes homogeneous range throughout the subject. The second approach uses a sequence of convolutions to account for the effects of inhomogeneities in the subject. Preliminary results using high-energy isotopes show significant improvements in the spatial resolution of the reconstructed images compared with methods without positron range modeling. II. MAP I MAGE RECONSTRUCTION Maximum a Posteriori (MAP) image reconstruction is a Bayesian approach that can combine accurate statistical and physical models for the data with a prior on the unknown image [10]. PET data are modeled as a collection of independent Poisson random variables with mean y = Px + r + s (1) where r is the mean of the randoms, and s is the mean of the scattered events. P is the system matrix describing the probability that an unscattered event is detected, which we factor as [10] P = P norm P blur P attn P geom P range (2) 0-7803-8257-9/04/$20.00 © 2004 IEEE. 2501