## 2 Materials and Methods
### Simulation Overview  
For our Time Domain Diffuse Optical Tomography (TD-DOT) simulations in reflectance geometry we used a fixed geometry of sources and detectors placed in a rectangular pattern (30 mm \(\times\) 20 mm). On each long side of the rectangle 4 pairs of sources and detectors are placed, with overlapping source and detector positions. Spacing between two nearest source-detector pairs is of 10 mm. The simulated probed volume is a parallelepiped (60 mm side and 30 mm) and the rectangular pattern of sources and detectors is located on its top surface. A localized spherical heterogeneity (~5 mm radius, \(\mu_{a}\) = 0.04 mm\({}^{-1}\)) is moved within the simulated volume (\(\mu_{a}\) = 0.01 mm\({}^{-1}\)), thus probing the system performances in the region underlying the sources and detectors arrangement. Figure 2 depicts the phantom and measure geometry. Figure 2: Simulated phantom system. Red circles and blue crosses identify sources and detectors, respectively. The black sphere is the heterogeneity that is moved to probe the entire simulated volume. Both sources and detectors are simulated using realistic signal and noise (e.g., source power, Instrument Response Function, detector area and quantum efficiency, and signal noise), mimicking the typical performances of a TD instrument. We modeled the breast as a semi-infinite medium, with a local absorbing perturbation with respect to the background. To provide a full characterization of our system we performed a thorough series of simulations studying the main parameters that could affect system performances both on the hardware side (e.g., temporal gating) and the software side (e.g., reconstruction parameters). The set of simulations is summarized in Table 1.  
Table 2 lists the values of the parameters that have been studied. Quantification of the simulations outcome is reached through the definition of FoMs:  
* The error on the localization of the reconstructed inclusion along the three axes \(x,y\) and \(z\) with respect to the simulated position according to the center of mass. \[\epsilon_{loc}=\int_{ROI}r_{i}^{rec}-\int_{ROI}r_{i}^{true}\quad i=x,y,z,\] (5)
* The volumetric error on \(\Delta\mu_{a}\) is calculated as: \[\epsilon_{vol}=\frac{\int_{ROI}\Delta\mu_{a}^{rec}dV-\int_{ROI}\Delta\mu_{a}^{ true}dV}{\int_{ROI}\Delta\mu_{a}^{true}dV},\] (6) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Simulation** & \multicolumn{5}{c}{**Parameter \#**} \\ \hline
**Name** & **1** & **2** & **3** & **4** & **5** & **Added Poisson Noise** \\ \hline A & Xp & Yp & Zp & Regularization & Voxel Size & No \\ B & Xp & Yp & Zp & Time Channel & Number of Temporal Steps & No \\ C & Xp & Zp & Number of Time Windows & Regularization & - & Yes \\ D & Xp & Zp & Number of Delays & Regularization & - & Yes \\ E & Xp & Zp & Counts per Second & Regularization & - & Yes \\ F & Xp & Yp & Zp & Regularization & - & Yes \\ \hline \end{tabular}
\begin{tabular}{c c c c} \hline \hline
**Parameter Name** & **Range** & **Default Value** & **Unit** \\ \hline Xp & \(-20\):5:20 & - & mm \\ Yp & \(-20\):5:20 & 0 (C, D, E) & mm \\ Zp & \(5\):5:25 & - & mm \\ Counts per Second & \(10^{5},10^{6},10^{7},10^{8}\) & \(10^{6}\) & cps \\ Number of Delays & \(1,3,5,10\) & 1 (A, B), 3 (C, D, E, F) & - \\ Regularization & \(10^{-4},10^{-3},10^{-2},10^{-1},1\) & \(10^{-4}\) (B) & - \\ Voxel size (1 side of a cube) & \(1,2,5\) & 2 & mm \\ Time Channel & \(6,12,24,48\) & 12 & ps \\ Number of Temporal Steps & \(200,400,600\) & 600 & - \\ Number of Time Windows & \(5,10,20,40\) & 20 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ranges and default values of the parameters under investigation.