# Dual-Tracer PET Image Separation by Deep Learning: A Simulation Study
## 4 Results  
### DL-Based Dual-Tracer Separation Results  
The whole brain region \(\Omega_{\text{brain}}\), where the activity concentration is non-zero in the ground-truth images of each tracer, was assessed for testing the separation quality at the pixel level. Figure 7 shows the NRMSE averaged over the entire test set (10 test datasets) as a function of the number of training pairs for the DL-based method. More specifically, the figure shows the NRMSE with respect to three different reference images: the static single-tracer ground-truth images \(y_{\text{GT}}^{*}\), the reconstructed static single-tracer noisy MLEM images \(y_{\text{Noisy}}^{*}\) and the reconstructed static single-tracer noise-free MLEM images \(y_{\text{NP}}^{*}\), where \(*\) denotes either \([^{18}\)F\(]\)FDG or \([^{11}\)C\(]\)MET; see Figure 6 for an illustration of the reference images. For each \(\Delta t\), the separation error of the DL-based method in terms of the average NRMSE with respect to \(y_{\text{GT}}^{*}\) was higher than when \(y_{\text{Noisy}}^{*}\) and \(y_{\text{NF}}^{*}\) were used as reference images. This is due to the fact that noise-free or noisy MLEM will not give perfect resultswith respect to the ground truths. Comparing our DL-based separation results to MLEM is therefore helpful as the perfectly separated MLEM images are not achieving the ground truths. While the proposed network was trained with noisy labels \(y_{\text{Noisy}}^{*}\), the separation results achieved better average NRMSE values across all \(\Delta t\) when noise-free images \(y_{\text{NF}}^{*}\) were used as references. This is a consequence of using MSE as a loss function in our task. We note that the arithmetic mean of the observations \(z\), i.e., \(y=\mathbb{E}_{z}\{z\}\), is found to be the minimum of the \(L_{2}\) loss \(L(y,z)=(y-z)^{2}\) in MSE. This indicates that a neural network learns to output the average of all plausible explanations when trained with MSE loss. Therefore, our proposed network can be trained using noisy labels \(y_{\text{Noisy}}^{*}\) with MSE loss to learn to separate the dual-tracer signals and output the estimation of the mean images of each tracer, and as a result, the separated images are closer to \(y_{\text{NF}}^{*}\). For the remainder of the paper, we use the reconstructed static single-tracer noise-free MLEM images \(y_{\text{NF}}^{*}\) as the reference images to calculate the error metrics in all cases. Figure 7 shows for each \(\Delta t\) that the average NRMSE decreases and that a smaller SD is obtained in the test set as the number of training pairs increases from 1 to 80. This is consistent with the visual impression of the separations obtained by the DL-based method, as shown in Figures 8 and 9. The first to fifth columns display the separated static single-tracer images (activity distributions) based on different numbers of training pairs (for plotting only, we limit the colour scale of the figures to fall within the same range as the reference images in Figure 6b for \([^{18}\text{F}]\)FDG and Figure 6e for \([^{11}\text{C}]\)MET, respectively). Although all cases successfully recover the separated images, using more training pairs results in higher separation performance in terms of recovering the tumour regions, shapes and low contrast of the white and grey matter for \([^{11}\text{C}]\)MET. Figure 10a,c also suggest that the more training pairs used, the better the average NRMSE, which is in line with the results shown in Figure 7. With 80 training pairs, the DL-based method achieved average NRMSE values for \([^{18}\text{F}]\)FDG and \([^{11}\text{C}]\)MET of \(\sim\)15% and \(\sim\)12%, respectively, for the test set across all \(\Delta t\) (see the magenta lines in Figure 10a,c). The impact of \(\Delta t\) for the DL-based method using the same number of training pairs is shown in Figure 10a,c.