# Dual-Tracer PET Image Separation by Deep Learning: A Simulation Study
## 3 2D Simulation Study
### Implementation of Compartment Model-Based Dual-Tracer Separation Method  
We compared the performance of the proposed DL-based method with the (voxel-wise) CM-based method for dual-tracer separation (a flowchart diagram can be found in Figure 1). In the CM-based method, dual-tracer TACs of each voxel (denoted with \(\hat{C}_{l}^{\text{Dual}}\)) were first extracted from the reconstructed dynamic dual-tracer MLEM images. The kinetic parameters were estimated for each tracer by fitting the dual-tracer kinetic model (2) to the measured dual-tracer TACs  
\[\tilde{\mathbf{k}}=\operatorname*{arg\,min}_{\mathbf{k}}\sum_{m=1}^{L}w_{m}(C _{t_{m}}^{\text{Dual}}(\mathbf{k})-\hat{C}_{t_{m}}^{\text{Dual}})^{2}, \tag{4}\]  
where \(L\) is the total number of frames, \(w_{m}\) is the weight factor of the \(m\)th frame, and \(t_{m}\) is the mid-time of the \(m\)th frame. We chose \(w_{m}=T_{m}/\hat{C}_{m}^{\text{Dual}}\) in accordance with the recommendation made in [30], where \(\hat{C}_{m}^{\text{Dual}}\) denotes the TAC value for a given voxel at frame \(m\), and \(T_{m}\) denotes the frame width for the \(m\)th frame. The trust-region-reflective algorithm [62; 63] (implemented in the Matlab function lsqonlin) was used to performed a voxel-wise weighted non-linear least squares (WNLS) fitting to solve (4). The initial values of each of the kinetic parameters contained in \(\mathbf{k}\) were set at 0.05 for all voxels, and the lower and upper bounds of each parameter were set to 0 and 1, respectively. Stopping criteria were set such that the optimization procedure terminated when the change of the objective function was less than \(1\times 10^{-6}\) or after 2000 iterations.  
Given the estimated kinetic parameter vector \(\tilde{\mathbf{k}}=\left[\tilde{\mathbf{k}}_{\text{FDG}},\tilde{\mathbf{k}}_{ \text{MET}}\right]\) and the AIFs of each tracer, the single-tracer TACs of each voxel can be calculated using Equation (1) (physical decay of each radioactive tracer was considered). According to the static single-tracer PET protocol described in Section 3.1, the recovered single-tracer TACs (the last 20 min for both tracers) were integrated and assigned to each voxel to recover the static single-tracer images,  
Figure 6: Brain phantom: an example of the simulated static single-tracer images (tracer activity distributions) for \({}^{[18]}\)FDG and \({}^{[11}\)C]MET: (**a**,**d**) ground-truth static images; (**b**,**e**) noise-free MLEM reconstructions; (**c**,**f**) noisy MLEM reconstructions.  
which were then normalised by the entire static acquisition time of 20 min to represent the mean activity distributions. Although, for \(\Delta t=2\) days, dynamic dual-tracer signals have been entirely separated due to a sufficiently long \(\Delta t\) (see the discussion in Section 3.1), the CM-based method is still valid by assuming either the \([^{18}\)F\(]\)FDG activity concentration or \([^{11}\)C\(]\)MET activity concentration to be zero in Equation (2). In this case, the CM-based method estimates the kinetic parameters of each tracer by fitting the single-tracer kinetic model (1) to the measured single-tracer TACs extracted from the reconstructed dynamic single-tracer MLEM images.