# Deep-Learning Based Positron Range Correction of PET Images
## 4 Discussion  
This paper presents the use of a deep convolutional neural network to provide an accurate PRC in PET. The method was evaluated in simulations in preclinical studies and its performance characterized. To the best of our knowledge, this is the first work to successfully combine deep learning and PRC in a coherent framework. Our results indicate that overall, the image quality produced by the learned model is comparable to that of the reference images, with recoveries going from around 60% to more than 95% while maintaining low noise levels. One interesting question that we wanted to address in this work is the amount of input data required for these types of algorithms. With PR being a three-dimensional effect (i.e., it affects not only a particular single slice) and being dependent on both the PET activity and the material distribution (CT), it may be reasonable to expect that many PET and CT input slices will be needed to generate an accurate output slice. On the other hand, many works have shown that the information in PET and CT images is not independent, and that a deep neural network may be able to estimate to some extent images of one modality from the other. This fact may indicate that using only the PET image as input may be enough. The results obtained with all methods considered were good, as shown by the L1 loss function in the validation cases. Nevertheless, it seems that the training with just the PET images as input, and with large enough axial slices was the best option. It is important to note that in our case, each slice was two times larger than the pixel size in the transverse direction. Therefore, considering five slices corresponded to using 4 mm in the axial direction, which seemed enough to consider the effects of surrounding slices for a specific slice. In this work the number of slices were limited by the amount of memory available in the GPU. Nevertheless, this is something that will be easily solved with new GPU models. The training was based on minimization of the L1 norm between the reference images and the estimated ones. The L1 norm is known to be more stable and robust to the presence of noise than the L2 norm [50]. In any case, other loss functions could be explored in this context, including a loss term from an adversarial network (GAN). A detailed comparison of the performance of the proposed method with previously proposed ones (described in Section 1.2) is outside the scope of this work, but we plan to perform this detailed comparison in a future work. In any case, the fact that the proposed Deep-PRC had no significant impact on the noise level of the images is a clear advantage compared to previous approaches [25]. It is important to note that although we have proposed the method as a postprocessing step, the same neural network architecture could be used to generate a PR model that could be applied in the forward projection within an image reconstruction (simply by inverting the input and outputs of the NN). This line of research will be explored in future works. In this work, the axial FOV of the scanner was large enough for mice, so no significant axial truncation of the PET activity was present. If there is significant truncation (as it usually happens in whole-body PET acquisitions in which only a section of the body is examined in each bed axial position), it may be advisable to perform this PRC on the final 3D whole-body volume to avoid possible truncation artefacts (from activity out of the axial FOV in each particular bed position). This can be argued to be an advantage of the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Recovery (\%)**} & \multicolumn{3}{c}{**Noise (\%)**} \\ \hline  & **Heart** & **Bladder** & **Tumor** & **Heart** & **Bladder** & **Tumor** \\ \hline \({}^{18}\)F & 100.00 & 100.00 & 100.00 & 2.00 & 3.75 & 3.39 \\ \({}^{68}\)Ga & 67.41 & 53.49 & 60.50 & 4.03 & 4.96 & 4.08 \\ \({}^{68}\)Ga Deep-PRC & 95.19 & 96.16 & 97.46 & 2.52 & 3.96 & 3.02 \\ \hline \hline \end{tabular}
\end{table}