## 5 Discussion  
It is widely known that TOF accelerates contrast recovery and that the acceleration increases as CTR decreases. However, acceleration concerns also noise; therefore, at the same number of iterations, images reconstructed with TOF are noisier than those reconstructed without TOF [4; 13]. For this reason, for a center that installs a new scanner, it might be difficult to set up the OSEM reconstruction protocol that best exploits an improved CTR. Generally, experimental stopping rules are derived on phantoms. As we showed, however, these are not necessarily applicable to patients, due to the convergence speed dependence on background characteristics. The biggest risk is generally that the high levels of CR measured on phantoms like the NEMA-IQ (which is small) with high contrast ratios (e.g., contrasts as high as 10:1 have been recommended in some guidelines [14]) cannot be replicated in patient studies, especially in large districts with multiple high-activity background structures, like the pelvis or the abdomen. To find the way to best exploit early stopped OSEM in TOF scanners, we theoretically revised MLEM convergence properties. We found that signal convergence speed increases with signal contrast and dimension. Without TOF, signal convergence decreases with increasing background diameter, i.e., patient dimension. TOF instead, in an ideal setting without random and scatter coincidences, is able to fully eliminate this dependence, thus making early stopped reconstructions robust to variations in patient size and background composition. An increased fraction of random and scatter coincidences generally decrease the convergence speed. Importantly, TOF significantly reduces random coincidences impact  
Figure 6: Reconstructions of the hot rods section of the phantom in different configurations. Upper rows: Single acquisitions (**a**–**c**). Lower rows: double acquisitions (**d**–**f**), only HCR shown on Signa for clarity. In each panel the upper row is TOF reconstruction, lower row is non-TOF. **Left (a,d)**: D690, 48 updates. **Center (b,e)**: Signa, 48 Updates. **Right (c,f)**: Proposed stopping rule. on convergence; the scatter coincidences impact is instead not modified by TOF. As to noise behaviour with iterations, higher frequencies converge more slowly than the lower ones and TOF makes noise convergence faster. At convergence, TOF reduces noise with respect to non-TOF by a factor \(\sqrt{D_{eff}/D}\)[1]; however, if a fixed stopping criteria is used, TOF actually increases noise by the inverse of this factor. For this reason, until regularization algorithms become routine in clinical applications, criteria to properly reduce the iteration number while maintaining quantitative accuracy are needed to fully exploit TOF benefits. In this paper, it was theoretically found that modifying the iteration number by a factor \(\
icefrac{{\Delta f}}{{\Delta t}}\) matches contrast recovery between two scanners with different CTR. As high frequency noise increases approximately linearly with iterations, this rule achieves, in early stopped reconstructions, the same noise reduction expected at convergence (factor \(\sqrt{D_{eff}/D}\)). It is widely known from the literature [3; 15] that OSEM and MLEM behaviour are slightly different at convergence, since OSEM converges to a limit cycle rather than to a unique solution. This however does not impact on the convergence trend at first iterations, which are the ones concerned by the proposed rule. The rule has been validated on digital simulations and on real phantoms acquired on two TOF systems with different CTRs. Without TOF, contrast matching on larger background requires to perform more iterations compared to smaller backgrounds and this inevitably increases noise. With TOF, better CTRs allow reducing the update number, thus maintaining accuracy while minimizing noise. When deriving the theoretical expression few approximations were required.