Strengths and weaknesses of energy-based scatter estimation using three basis functions

<i>Objective.</i>Quantitative imaging in positron emission tomography (PET) requires accurate, precise, and efficient scatter correction techniques. Conventional scatter estimation typically relies on single-scatter simulation (SSS) combined with a tail-fitting strategy. However, the accuracy of tail-fitted SSS is limited, for example, by mismatches between the attenuation image and the PET emission data or by the presence of activity outside the field of view (FOV). These shortcomings can be addressed using energy-based scatter estimation (EBSE), as recently proposed by Efthimiou<i>et al</i>(2022<i>Phys. Med. Biol.</i><b>67</b>095010); Hamill<i>et al</i>(2024<i>Med. Phys.</i><b>51</b>54-69). The aim of this work is to (1) investigate the accuracy of EBSE by accounting for the line-of-response dependence of the energy spectrum of unscattered photons, (2) improve the computational speed of EBSE through better initialization and a more efficient optimization algorithm, and (3) validation and characterization of EBSE using a three-basis model across different object sizes and activity distributions.<i>Approach.</i>The proposed improved EBSE method models the energy spectrum of scattered photons with two probability density functions (PDFs), and incorporates a position-dependent (local) energy PDF for unscattered photons. These energy PDFs form the basis of a forward model-a linear nine-parameter model -used for scatter estimation based on 2D energy histograms. The performance of the EBSE was evaluated using GATE Monte Carlo simulations and a NEMA phantom acquisition on a GE SIGNA PET/MR scanner. Furthermore, we assessed the stability of EBSE across the forward model by varying the number of counts in the 2D energy histograms via data mashing.<i>Main results.</i>EBSE reduced artifacts caused by out-of-FOV activity and demonstrated performance comparable to tail-fitted SSS in other regions. Incorporating a local unscattered PDF improved off-center quantification, and NEGML with improved initialization plus histogram down-sampling substantially reduced computation without compromising accuracy. Limitations were observed: the proposed basis-function model for scattered-photon energy spectra lacks full generality across attenuation and activity distributions.<i>Significance.</i>This study improves the accuracy and computational efficiency of EBSE for clinically realistic activity and attenuation conditions, while clarifying scattered basis function limitations.