## 2 Materials and Methods
### Simulation Overview  
We performed simulations using diffusion theory [22; 32] to evaluate the performance of the hybrid FD/CW DOI technique and compare it with the conventional topographic reflectance DOI. For the forward problem, the FEM numerical solver of the open source DOT software, NIRFAST [30; 31], was used to generate tissue reflectance for tumor-containing breast tissue for the desired source-detector configuration. A modulation frequency of 200 MHz was used for FD data. Afterwards, random Gaussian noise of 1% complex amplitude and 1% reflectance was added to the simulated FD and CW data, respectively. We verified that this noise level is typical for our DOI imaging system, characterized from one-hour continuous measurements (121 iterations) of a homogeneous phantom of known optical properties. Finally, the tissue optical properties were recovered at each wavelength in the range of 650 to 1000 nm to simulate the hybrid DOI technique. As our goal was to compare the hybrid DOI method to the conventional topographic DOI, we performed two different inversions. For the conventional topographic DOI method, a semi-infinite analytical diffusion model [33] that assumes optical homogeneity in the medium was utilized to extract 2D topographic maps of tissue optical properties (absorption \(\mu_{a}\) and reduced scattering \(\mu_{s}^{\prime}\)) for each wavelength ranging from 650 to 1000 nm at every 5 nm. The optical properties corresponding to normal and tumor containing tissue were extracted at each wavelength by averaging the properties over regions of interest (ROIs) selected on the 2D maps. The tumor and normal ROI were manually chosen over the 2D spatial maps of reconstructed properties. These homogeneous model based results also served as the initial guess for the hybrid DOI hyperspectral recovery. Figure 2 depicts the simulation overview. Figure 1: Hybrid frequency-domain (FD)- and CW-DOT (Continuous Wave-Diffuse Optical Tomography) method (**a**) overview and (**b**) example. To generate simulated data, a 3D rectangular slab mesh consisting of 139,435 nodes corresponding to 710,944 tetrahedron elements was used. A second 3D slab mesh with a total of 45,688 nodes corresponding to 215,430 tetrahedron elements was generated to perform reconstructions to avoid inverse crime [34]. Our tissue model mimicked a human breast tissue containing an ellipsoidal tumor. The 20 mm \(\times\) 30 mm \(\times\) 20 mm ellipsoidal tumor was investigated at depths varying from 0 to 10 mm between the surface of the tissue (i.e., skin) and top surface of the tumor. This represents a typical range of palpable breast tumor depths studied with reflectance-based DOI [35]. The source and detector geometry represents the manual translation of a simple single source-detector separation (SDS) probe with a separation of 28 mm over a 1 cm spaced rectangular 2D grid pattern [5], similar to that used in the ACRIN 6691 multicenter trial of diffuse optical spectroscopy in breast cancer therapy [36]. We initially simulated an array of multiple SDSs, but because of use of hard spatial a priori information for normal and tumor regions within the heterogeneous tissue, the additional SDSs did not increase the overall accuracy. The overall geometry is illustrated in Figure 3. The source detector arrangement lies in such a way that the long axis of the ellipse was taken along the line between the source and detector. This was an arbitrary choice that is not expected to influence the simulations. Figure 3: Simulated heterogeneous system. Blue circles and red crosses identify sources and detectors, respectively. The gray ellipsoid is the heterogeneity, which was varied in depth. Figure 2: Simulation framework. Table 1 lists the concentration of the absorbing components and scattering parameters specified in the heterogeneous tissue model. Optical properties were chosen based on chromophore concentrations (oxy-, deoxy-, water, and bulk lipids) and scattering parameters (a and b) of human breast tumors measured by DOT [37].