## 3 Results
### Evaluation of PVE Recovery  
Figures 3-5 show the sphere-to-background counts concentration ratio as a function of the spheres dimension for the different situations above. However, in order to underline how the relationship between the dimension of lesions and the PSF of the imaging system impacts on PVE, the x-axes are reported as the ratio between the diameter of the six spheres and the \(\sigma\) of the system.  
The sphere-to-background counts concentration ratio was calculated as the ratio between the counts concentration in the spheres and in the background, which were measured as the average over four circular ROIs chosen in areas and slices far from the spheres.  
Figure 3 illustrates the comparison in concentration ratio (y-axis) before and after the application of the PVE recovery method at different acquisition times (1, 2, 4, and 8 min).  
The SBR and the reconstruction algorithm VPFXS were fixed.  
For each acquisition time, reported with the same colors, the smaller the lesion, the greater results the underestimation of the concentration ratio. In particular, the concentration underestimation ranges from a minimum of 16% for largest spheres to a maximum of 66% for the smallest sphere.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Spheres** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(a\) & 1.16 \(\pm\) 0.04 & 1.23 \(\pm\) 0.01 & 1.23 \(\pm\) 0.01 & 1.62 \(\pm\) 0.04 & 1.95 \(\pm\) 0.04 & 3.09 \(\pm\) 0.09 \\ \hline _Counts ratio for a resampling factor of 6_ & 1.16 \(\pm\) 0.01 & 1.23 \(\pm\) 0.01 & 1.22 \(\pm\) 0.01 & 1.63 \(\pm\) 0.01 & 1.96 \(\pm\) 0.01 & 3.07 \(\pm\) 0.02 \\ \hline \(b\) & 0.32 \(\pm\) 0.03 & 0.54 \(\pm\) 0.07 & 0.45 \(\pm\) 0.06 & 1.90 \(\pm\) 0.70 & 2.70 \(\pm\) 0.80 & 4.50 \(\pm\) 0.70 \\ \hline \(c\) & 0.50 \(\pm\) 0.03 & 0.42 \(\pm\) 0.07 & 0.50 \(\pm\) 0.07 & 0.30 \(\pm\) 0.10 & 0.30 \(\pm\) 0.10 & 0.46 \(\pm\) 0.07 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters obtained by the fit with the function f(x) = a \(-\) b \(\times\) c\({}^{\mathrm{x}}\). The \(a\) parameter represents the asymptote.  
Figure 3: Comparison between the concentration ratio for the raw image (dotted lines) and the PVE corrected image (continuous line) at different acquisition times, shown with different colors.  
Figure 4: Comparison of the concentration ratio for the same acquisition time (4 min) between the reconstruction with TOF correction, reported in red, and the one without TOF, reported in blue, before (dotted line) and after (continuous line) the application of PVE recovery method.  
Figure 5: Comparison for concentration ratio for SBR = 10.18 in light blue and SBR = 3.27 in violet. For both acquisitions, acquisition time was fixed at 4 min and the reconstruction algorithm was with TOF correction. The dotted lines represent the raw images, while the continuous ones the PVE corrected images.  
Table 5 shows the percentage of recovery for all the spheres of the NEMA phantom for all the acquisition time, highlighting the good PVE recovery for the biggest sphere, which for all the acquisition time results in the range of \(\pm\)3%.  
Figure 4 compares the concentration ratio with or without the TOF reconstruction for the 4 min acquisition, showing that every reconstruction underestimates the theoretical SBR, even if the TOF correction tends to have a proximity of 5% more than the NOTOE.  
After the recovery method application (continuous line), the PVE compensation is shown in Table 6.  
Finally, Figure 5 illustrates the change in SBRs with the theoretical concentration ratio of 3.27 and 10.19, maintaining unaltered the other parameters (acquisition time = 4 min and TOF reconstruction).  
For higher SBR, the underestimation of theoretical ratio concentration results almost always greater, as shown in Table 7.