*2.5. Quantitative Analysis*

The difference between Ga-68 PET images corrected by CNN-based positron range correction (IPRC) and the corresponding gamma source images (Igamma) was quantified by calculating the RMSE and PSNR:

$$\text{RMSE} = \sqrt{\frac{\sum\_{i=1}^{V} \left(\mathbf{I}\_{\text{gamma}} - \mathbf{I}\_{\text{PRC}}\right)^2}{\mathbf{V}}} \tag{1}$$

where V is the number of voxels within the whole image,

$$\text{PSNR} = 20 \log\_{10} \frac{\text{l}\_{\text{max}}}{\text{RMSE}} \tag{2}$$

where Imax is the maximum intensity value of the image. RMSE and PSNR provide a measure of image quality over the whole image.

The ability of Ga-68 PET images (IGa68), Igamma, and IPRC to recover contrast in small targets was quantified by calculating the recovery coefficient (RC), which was defined as:

$$\text{RC} = \frac{\text{AVG}\_{\text{target}}}{\text{AVG}\_{\text{uniform}}} \tag{3}$$

where AVGtarget is the average of a small target, and AVGuniform is the average of a uniform region. The coefficient of variation of RC (CVRC) was defined as:

$$\text{CVRC} = \sqrt{\left(\frac{\text{SD}\_{\text{target}}}{\text{AVG}\_{\text{target}}}\right)^2 + \left(\frac{\text{SD}\_{\text{uniform}}}{\text{AVG}\_{\text{uniform}}}\right)^2} \tag{4}$$

where SDtarget is the standard deviation of a small target, and SDuniform is the standard deviation of a uniform region. To calculate AVGtarget and SDtarget, the image slices of PHAN5rod over the central 50 mm length were averaged to obtain one average image, which was used to determine the voxel coordinate with maximum intensity for each rod. The pixel coordinates were then used to create a line profile along the axial direction. The AVGtarget and SDtarget were the average and standard deviation of pixel values in the line profile. As for AVGuniform and SDuniform, the image slices of PHAN1sphere over the central 50 mm length were averaged to obtain one average image. A circular region-of-interest (ROI) with 10 mm diameter was placed on the cylinder of the average image to calculate AVGuniform and SDuniform, corresponding to the average and standard deviation within the circular ROI, respectively.

The spill-over of activity in IGa68, Igamma and IPRC was quantified by calculating the spill-over ratio (SOR), which was defined as:

$$\text{SOR} = \frac{\text{AVG}\_{\text{cold}}}{\text{AVG}\_{\text{hot}}} \tag{5}$$

where AVGcold is the average of a cold spot, and AVGhot is the average of a hot spot. The coefficient of variation of SOR (CVSOR) was defined as:

$$\text{CV}\_{\text{SOR}} = \sqrt{\left(\frac{\text{SD}\_{\text{cold}}}{\text{AVG}\_{\text{cold}}}\right)^2 + \left(\frac{\text{SD}\_{\text{hot}}}{\text{AVG}\_{\text{hot}}}\right)^2} \tag{6}$$

where SDcold is the standard deviation of a cold spot, and SDhot is the standard deviation of a hot spot. A 10-mm-diameter ROI was placed on the cold sphere of PHAN1sphere in the slice of the sphere center to calculate AVGcold and SDcold, corresponding to the average and standard deviation within the cold ROI, respectively. For the same image slice, a 10-mmdiameter ROI was placed on the cylinder of PHAN1sphere to calculate AVGhot and SDhot, corresponding to the average and standard deviation within the hot ROI, respectively.
