# Dual-Tracer PET Image Separation by Deep Learning: A Simulation Study
## 4 Results
### Bias and Standard Deviation Analysis (Pixel-Level)  
To capture the variation of the separation caused by the random initialisation of the network parameters in network training, we retrained the proposed network for each scenario. Specifically, for each \(\Delta t\), the proposed network was independently retrained three times on the same dataset (with same number of training pairs) using different randomised weight initialization and random shuffling of the data per epoch during training. Armed with the retraining process, we can now compute the uncertainty of the separated image values of the retraining outcomes. The bias and standard deviation averaged from the three training reruns were used for analysis. The bias and SD of the separation results for the whole brain region \(\Omega_{\text{brain}}\) of one of the test data cases are shown in Figure 11. The scatter plots show that using more training pairs results in lower bias of the DL-based separation, which is referred to as the reducible epistemic uncertainty. However, their SD values are at a similar level for each \(\Delta t\), indicating that this uncertainty is irreducible by introducing more training data. We refer to this uncertainty as the irreducible aleatoric uncertainty, which is caused by the inherent noise of the dynamic input image data. We also evaluated the performance of the DL-based method (with 80 training pairs) applied to the reconstructed dynamic dual-tracer noise-free MLEM images, i.e., we trained the proposed networks (with 3 training reruns and 80 noisy training pairs), but the reconstructed dynamic dual-tracer noise-free MLEM images were used as the network inputs in the prediction stage. The separation results for the noise-free case reached \(\sim\)14% bias for \([^{18}\)F]FDG and \(\sim\)11% bias for \([^{11}\)C]MET, which are higher than the bias for the noisy case using the same trained networks. The reason for this is that the reconstructed dynamic dual-tracer noise-free MLEM images are out-of-domain data that were not observed during training, and hence the trained networks do not perform as well for out-of-domain predictions. For the CM-based method, a longer \(\Delta t\) helps to decrease bias and SD in the separation results for both tracers when the reconstructed dynamic dual-tracer noisy MLEM images separation was considered. The bias decreases gradually for \(\Delta t=5,10\) and 15 min after a sharp decline simultaneous to a 5 min delay; see the grey diamonds in Figure 11. When \(\Delta t=2\) days, the CM-based method achieved \(\sim\)5% bias of the separated images for both tracers. High SD of the separation results (above 30% for both \([^{18}\)F]FDG and \([^{11}\)C]MET) were found using the CM-based method for \(\Delta t=0,5,10\) and 15 min. The SD drastically decreased to \(\sim\)8% for both \([^{18}\)F]FDG and \([^{11}\)C]MET when the dual-tracer signals were entirely separated by \(\Delta t=2\) days. The separation of the reconstructed dynamic dual-tracer noise-free MLEM images achieved a reduced bias for each \(\Delta t\) when compared to  
Figure 11: Brain phantom (whole brain region): bias and SD of separated static \([^{18}\)F]FDG images (**a**) and \([^{11}\)C]MET images (**b**) using DL-based and CM-based methods for the whole brain region \(\Omega_{\text{Brain}}\). the separation of noisy MLEM images using the CM-based method. In the noise-free case, the bias changed rapidly from 0 min (simultaneous) to 5 min, decreased gradually for \(\Delta t\) = 5, 10 and 15 min and attained zero when \(\Delta t\) = 2 days. The aforementioned results are consistent with the findings in Section 4.2, suggesting longer \(\Delta t\) results in better separation using the CM-based method. We now compare the bias and SD of the separation results between the DL-based method and the CM-based method on the reconstructed dynamic dual-tracer noisy MLEM images. The DL-based method achieved much smaller SD over all \(\Delta t\), even using a small number of training pairs.