# Dual-Tracer PET Image Separation by Deep Learning: A Simulation Study
## 2 Materials and Methods  
### Dual-tracer Single-Acquisition PET Measurement  
Dynamic PET scanning can capture the spatiotemporal distribution of a radiotracer within a living organism. The activity concentration of a radiotracer of an image voxel at time \(t\) (i.e., TAC) can be described by the following model [49]  
\[C_{t}(\mathbf{k})=V_{B}S_{t}^{\mathrm{wb}}+(1-V_{B})A_{t}\otimes R_{t}( \mathbf{k}), \tag{1}\]  
where \(\mathbf{k}\) represents the kinetic parameters that determine the tracer uptake in the tissue, \(S_{t}^{\mathrm{wb}}\) is the tracer concentration in whole blood (we assume that the tracer concentration in whole blood \(S_{t}^{\mathrm{wb}}\) is known and is equal to \(A_{t}\) for simplicity), \(A_{t}\) is the AIF representing the whole blood activity concentration, \(R_{t}(\mathbf{k})\) is the impulse response function, specified by a certain compartment model and corresponding kinetic parameters \(\mathbf{k}\), \(V_{B}\in[0,1]\) is the fractional volume of blood in the tissue, and \(\otimes\) denotes the convolution operator. In our work, we consider a reversible two-tissue compartment model for both [18F]FDG and [11C]MET.  
Since in dual-tracer PET, the PET scanner measures the sum of the PET signals of two tracers, the dual-tracer activity concentration can be modelled as the sum of the activity concentrations of each tracer  
\[C_{t}^{\mathrm{Dual}}(\mathbf{k})=C_{t}^{\mathrm{FDG}}(\mathbf{k}_{\mathrm{FDG}})e^{-\lambda_{t}[\mathrm{sg}]t}+C_{t}^{\mathrm{MET}}( \mathbf{k}_{\mathrm{MET}})e^{-\lambda_{t}[11C]t}, \tag{2}\]  
Figure 1: Flowchart diagram of the DL-based method and the CM-based method for dual-tracer [18F]FDG and [11C]MET separation.  
where \(\lambda_{[18]\mathrm{F}]}\) is the rate of radioactive decay of \({}^{18}\)F, \(\lambda_{[1^{11}\mathrm{C}]}\) is the rate of radioactive decay of \({}^{11}\)C, and the vector \(\mathbf{k}=[\mathbf{k}_{\mathrm{FDC}},\mathbf{k}_{\mathrm{MET}}]\) contains the kinetic parameters of each tracer. The single-tracer activity concentration \(C_{t}(\mathbf{k})\) is multiplied by the decreasing exponential term \(e^{-\lambda t}\) in the right-hand side of (2), which models the radioactive decay of each tracer. A dynamic PET scan measures the activity of both tracers at multiple consecutive time frames. In each frame, the coincidence events were recorded from the start of the frame until the end of the frame. The intensity of the dual-tracer image at pixel \(j\) in the \(m\)th frame can be expressed as [50]  
\[x_{m}(\mathbf{k}_{j})=\int_{t_{m,s}}^{t_{m,s}}C^{\mathrm{Dual}}(\tau,\mathbf{k }_{j})d\tau,\]  
where \(t_{m,s}\) and \(t_{m,s}\) represent the start and end points of frame \(m\), respectively. The expectation of the projection data \(\hat{y}_{m}\) in time frame \(m\) can be expressed as  
\[\hat{y}_{m}(\mathbf{k})=Px_{m}(\mathbf{k})+r_{m},\]  
where the (\(i\), \(j\))th element of the system matrix \(P\in\mathbb{R}^{M\times N}\) is the probability of detecting an event originating in voxel \(j\) by detector pair \(i\), \(r_{m}\in\mathbb{R}^{N}\) is the expectation of scattered and random events in the \(m\)th frame, and \(M\) and \(N\) are the total number of detector pairs and image voxels, respectively.