How the sensitivity of TOF-PET depends on the interplay between the temporal and spatial detector resolutions and the resolution required for the imaging task

<i>Objective.</i>Sensitivity is a key feature of positron emission tomography (PET). Here, sensitivity can be defined as the reciprocal of the amount of injected radioactivity needed to obtain a sufficient image quality for a particular scan duration. PET sensitivity can not only be increased by increasing the solid angle covered by the detectors, but also by improving their spatial and the temporal resolution, which in turn determines the time-of-flight (TOF) resolution. This paper analyzes how the interplay between the spatial detector resolution, the TOF resolution and the required imaging resolution affects the sensitivity of a TOF-PET system.<i>Approach.</i>Two approaches are studied. The first computes the performance of the Hoteling observer for discriminating a small hot spot from a less small and less hot spot with the same total activity. This approach is flexible, and elegant closed form equations are obtained. In the second approach, analytical equations are derived for the variance of the reconstructed voxel values as a function of the TOF and spatial detector resolutions and of the reconstruction point spread function. To keep the mathematics tractable, the derivations are done for the center of a uniform sphere or cylinder, which is placed in the center of a cylindrical or spherical PET system. The results are verified with simulation experiments for 2D PET and 3D PET with septa.<i>Main results.</i>Both approaches confirm that the sensitivity of the PET system increases, when the spatial detector resolution is improved, in agreement with simulation results published by Muehllehner (1985<i>Phys. Med. Biol.</i><b>30</b>163). The same is true for improvements of the TOF resolution. Remarkably, when the TOF resolution (converted to a distance) approaches the spatial resolution, further improving it increases the sensitivity more than expected based on experience with current TOF-PET systems and on the analysis for moderate TOF resolution (Tomitani 1981<i>IEEE Trans. Nucl. Sci.</i><b>NS-28</b>4582-9). This agrees with simulation results published recently by Toussaint<i>et al</i>(2020<i>IEEE Trans. Radiat. Plasma Med. Sci.</i><b>5</b>729-37). Finally, our new Equations confirm that the value of a TOF-kernel is well characterized by the integral of its square, as was reported previously (Nuyts<i>et al</i>2022<i>IEEE Trans. Med. Imaging</i><b>42</b>1254-64; Nuyts<i>et al</i>2022<i>Phys. Med. Biol.</i><b>69</b>015011).<i>Significance.</i>This analysis explains how spatial detector resolution and TOF accuracy contribute to the sensitivity of PET systems, and predicts that pushing the TOF resolution well below 100 ps will produce a larger benefit than expected based on current rules of thumb.