## 4 Discussion  
In this work, we developed a post-reconstruction method for the correction of the PVE in PET images. Quantitative PET images are increasingly required for diagnostic \begin{table}
\begin{tabular}{c c c c c c} \hline
**I/\(\sigma\)** & **3.6** & **4.7** & **6.2** & **8.0** & **10.1** & **13.4** \\ \hline \(\text{TOF}_{raw}\) & \(-55\pm 2\%\) & \(-42\pm 1\%\) & \(-30.5\pm 0.9\%\) & \(-22.2\pm 0.7\%\) & \(-19.3\pm 0.6\%\) & \(-16.7\pm 0.5\%\) \\ \hline \(\text{TOF}_{raw}\) & \(-30\pm 2\%\) & \(-13\pm 1\%\) & \(-2\pm 1\%\) & \(+1.0\pm 0.8\%\) & \(+0.4\pm 0.7\%\) & \(-2.5\pm 0.6\%\) \\ \hline \(\text{NOTOF}_{raw}\) & \(-55\pm 2\%\) & \(-45\pm 1\%\) & \(-33.3\pm 0.9\%\) & \(-26.5\pm 0.7\%\) & \(-21.4\pm 0.6\%\) & \(-20.1\pm 0.5\%\) \\ \hline \(\text{NOTOF}_{cor}\) & \(-31\pm 2\%\) & \(-19\pm 1\%\) & \(+7\pm 1\%\) & \(+4.6\pm 0.8\%\) & \(+3.3\pm 0.7\%\) & \(+6.8\pm 0.6\%\) \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of percentage of recovery for the same acquisition time (4 min) between the reconstruction with TOF correction and the one without TOF. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**I/\(\sigma\)** & **3.6** & **4.7** & **6.2** & **8.0** & **10.1** & **13.4** \\ \hline \(\text{SBR3}_{raw}\) & \(-55\pm 2\%\) & \(-42\pm 1\%\) & \(-30.5\pm 0.9\%\) & \(-22.2\pm 0.7\%\) & \(-19.3\pm 0.6\%\) & \(-16.7\pm 0.5\%\) \\ \hline \(\text{SBR3}_{corr}\) & \(-30\pm 2\%\) & \(-13\pm 1\%\) & \(-2\pm 1\%\) & \(+1.0\pm 0.2\%\) & \(+0.4\pm 0.7\%\) & \(-2.5\pm 0.6\%\) \\ \hline \(\text{SBR10}_{raw}\) & \(-67.5\pm 0.6\%\) & \(-42.5\pm 1\%\) & \(-35.1\pm 0.2\%\) & \(-26.1\pm 0.2\%\) & \(-20.6\pm 0.1\%\) & \(-14.21\pm 0.07\%\) \\ \hline \(\text{SBR10}_{corr}\) & \(-28.7\pm 0.7\%\) & \(+9.4\pm 1\%\) & \(+4.9\pm 0.3\%\) & \(+6.7\pm 0.2\%\) & \(+5.5\pm 0.1\%\) & \(+6.27\pm 0.07\%\) \\ \hline \end{tabular}
\end{table} 
Table 7: Comparison of percentage of recovery for the different SBR. For both acquisitions, acquisition time was fixed at 4 min and the reconstruction algorithm was with TOF correction. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**I/\(\sigma\)** & **3.6** & **4.7** & **6.2** & **8.0** & **10.1** & **13.4** \\ \hline \(1\)\(min_{raw}\) & \(-66\pm 2\%\) & \(-48\pm 2\%\) & \(-27\pm 2\%\) & \(-20\pm 1\%\) & \(-17.8\pm 0.8\%\) & \(-16.3\pm 0.5\%\) \\ \hline \(1\)\(min_{cor}\) & \(-56\pm 2\%\) & \(-24\pm 2\%\) & \(+4\pm 2\%\) & \(+3\pm 1\%\) & \(+1.4\pm 0.9\%\) & \(-1.9\pm 0.6\%\) \\ \hline \(2\)\(min_(SUV) and therapeutic (committed dose in metabolic radiotherapy) purposes and radiomics applications [21]. In general, post-processing corrective methods offer multiple advantages with respect to those integrated into the reconstruction algorithm [19]:  
1. they do not need access to the reconstruction-code, which is practically impossible for the standard user, and they can be easily used;
2. they allow their application independently of the acquisition machine and, consequently, can be used to standardize data from different sites, for example for multi-center studies. Our method is based on the theoretical calculation of the PVE effect caused by a uniformly distributed source on a uniform background. The hypotheses on which it is \"exactly\" valid are:  
1. uniform lesion on a uniform background
2. ideal segmentation of the ROI (Region Of Interest), or by means of CT imaging Its application is very simple, as it only needs:
3. to measure the contribution of the background directly on the image;
4. to know the PSF (Point Spread Function) of the acquisition system, acquiring a linear or point source or taking the data directly from the NEMA post installation quality assurance;
5. to implement a simple analytical formula in a spreadsheet.