A rewriting of the relation between the acolinearity of annihilation photons and their energy in the context of positron emission tomography

TL;DR

This paper clarifies the statistical distribution of photon acolinearity in PET, emphasizing the importance of precise interpretation of the distribution's nature and revisiting previous proofs with a 3D geometric approach.

Contribution

It provides a detailed analysis of the acolinearity distribution in PET and corrects previous interpretations by employing a 3D geometric framework.

Findings

Acolinearity follows a Gaussian distribution in PET.

The interpretation of acolinearity as an amplitude or deviation affects the distribution model.

Revised proof of the acolinearity distribution using 3D sphere geometry.

Abstract

Acolinearity of the annihilation photons observed in Positron Emission Tomography (PET) is described as following a Gaussian distribution. However, it is never explicitly said if it refers to the amplitude of the acolinearity angle or its 2D distribution relative to the case without acolinearity (herein defined as the acolinearity deviation). Since the former is obtained by integrating the latter, a wrong interpretation would lead to very different results. The paper of Shibuya et al. (2007), differs from the previous studies since it is based on the precise measurement of the energy of the annihilation photons. They also show that acolinearity follows a Gaussian distribution in the context of PET. However, their notation, which relies on being on the plane where the two annihilation photons travel, could mean that their observation refers to the amplitude of the acolinearity angle. If that understanding is correct, it would mean that acolinearity deviation follows a 2D Gaussian distribution divided by the norm of its argument. Thus, we revisited the proof presented in Shibuya et al. (2007) by using an explicit description of the acolinearity in the 3D unit sphere.