A Solution for Scaling Problem in Joint Estimation of Activity and Attenuation

In time-of-flight PET reconstruction, the algorithms for joint estimation of activity and attenuation (JEAA), such as maximum-likelihood activity and attenuation reconstruction and maximum-likelihood attenuation correction factors, cannot provide quantitative attenuation information due to the scaling problem. Then, for quantitative PET reconstruction, some post-processing is essentially required to make the estimated non-quantitative attenuation information quantitative. In this paper, we propose a post-processing method to estimate a quantitative attenuation image from attenuation factors (AFs) scaled by an unknown factor. The proposed method is based on the mathematical relationship between a scaled AFs and an attenuation image, and estimates then corrects the non-uniform offset (NUO) in the attenuation image caused by the AF scaling. The NUO correction assumes that the supports of activity and attenuation distributions are identical and the attenuation coefficient of the subject is partially known. To prove the concept of the NUO correction, we evaluated through 3-D brain PET simulations the quantitative accuracy of activity corrected for attenuation with using the NUO-corrected attenuation image. The attenuation coefficient of the brain tissue can be approximated well by that of the water, then transmission-less brain PET imaging would be a suitable application of the proposed method. Under a wide range of conditions of counting statistics, the median of the absolute relative error between the reference and estimated activity images was less than 5% within the brain regions. These results show that, under the above-mentioned assumptions, the scaling problem in the JEAA problem was solved with practical accuracy.