Accurate Attenuation Correction in PET Without Accurate Transmission Measurements A. Welch 1 , C. Campbell 1 , R. Clackdoyle 2 , F. Natterer 3 , P. Mikecz 1 , F. Chillcot 1 , M. Dodd 1 , P. Hopwood 1 , S. Craib 1 , G.T. Gullberg 2 , and P. Sharp 1 1 Aberdeen PET Unit, Dept. of Biomedical Physics and Bioengineering, Foresterhill, Aberdeen, AB25 2ZD, Scotland 2 Medical Imaging Research Lab, Dept. of Radiology, University of Utah, Salt Lake City, UT84132, USA 3 Institut für Numerische and Instrumentelle Mathematik, Universität Münster, Germany Abstract In this study we use the consistency conditions of the Radon transform to aid attenuation correction in PET. The conditions are used both for estimating the boundary of a uniform attenuation distribution (without any transmission measurements) and for correcting for patient motion between the transmission and emission acquisitions. The results show that, for a uniform elliptical attenuation distribution, the image reconstructed with an attenuation distribution estimated using the consistency conditions is almost indistinguishable from an image reconstructed using the true attenuation coefficient map. The method is shown to be fairly tolerant to the effects of photon counting statistics and to small non-uniformities in the attenuation distribution (such as skull attenuation). The results also show that the consistency conditions may be useful in correcting for patient motion. The method is shown to accurately compensate for shifts in two dimensions using both simulated and experimental data. I. INTRODUCTION Previously we have shown that the consistency conditions of the attenuated Radon transform can be used to aid attenuation correction in SPECT [1]. In this study we investigate some of the potential uses of consistency information in PET. In particular we will consider: a) Attenuation coefficient map estimation (using the consistency conditions of the PET data, in isolation, to estimate the boundary of a uniform elliptical attenuation distribution), b) The effect of photon counting statistics on the accuracy of the estimated map, c) The effect of small non-uniformities in the attenuation distribution, and d) Transmission-emission misregistration (using the consistency conditions to correct for patient motion between the transmission and emission acquisitions). II. THEORY A. Consistency Conditions (Radon Transform) The consistency conditions (or moment conditions) are a set of mathematical conditions which, in the absence of noise, the measured data should satisfy. An intuitive understanding of the first two conditions (zeroth and first) for unattenuated data can be gained by considering a physical object where the intensity at each location corresponds to the mass of the object at that location. The projection operation can now be viewed as crushing the two-dimensional object into a one-dimensional projection (see Fig. 1). Object Projection x y s Figure 1: Schematic showing how, for a physical object where the intensity at each location corresponds to the mass of the object, the projection operation can be viewed as crushing the two-dimensional object into a one-dimensional projection. Zeroth Condition. The zeroth moment is simply the sum of the data in each projection. It should be clear that this sum (the total mass of the object) will be a constant, independent of the projection angle First Condition. The first moment is the sum of the data in each bin multiplied by the distance, s, of that bin from the origin. In our analogy this moment gives the centre of mass of the projection, which is simply the projection of the centre of mass of the object onto the detector. Since the centre of mass is a single point we know that its projection will be a sine wave (of period one) in the sinogram Higher Order Conditions. Higher moments give combinations of sine waves with higher periods. The full set of consistency (or moment) conditions are often called the Helgason-Ludwig conditions. B. Consistency Conditions (PET) PET data can be corrected for the effect of attenuation (yielding an unattenuated Radon transform) by multiplying by a simple exponential attenuation correction factor. These corrected data should satisfy the consistency conditions described earlier.