Point spread function for PET detectors based on the probability density function of the line segment

We propose a new approach to calculate the Point Spread Function (PSF) for PET detectors based on the probability density function (PDF) of the line segment connecting two detector elements. Positron Emission Tomography (PET) events comprise the detection and positioning of pairs of oppositely directed 511 keV photons. The most significant blurring effect in PET is the considerable size of the detector elements, which causes uncertainty in the detected positions of photons. Typically this physical blurring is modeled in the forward direction, following photons from the source to the detectors. This work presents an analytical framework for calculating this physical blurring, from the inverse approach, that is from the detector to the source. The kernel is derived from the parameterization of the line segment whose endpoints are random variables described by the intrinsic detector response function distribution. This kernel is calculated in a first order approximation, and when compared against a measured PSF profile yields less than 8% root mean square (RMS) differences. Also, from this kernel a PSF-FWHM function of the distance to the center of the scanner is derived. The ratio between the PSF-FWHM and the intrinsic detector resolution (FWHM0) agrees with the Monte Carlo simulations. For detectors whose intrinsic response functions are described by Gaussian profiles we calculated ratios 1/√2 and √5/8 at the center (R=0) and halfway from the center at ( R=system-radius/2) respectively in agreement with published values of 1/√2 and 0.85; similarly for uniform (rectangular) profiles we get 1/2 and 3/4 which are equal to published values.