## 2 Materials and Methods
### Hyperspectral Data Analysis
#### 2.3.2 Influence of Tissue Thickness Underneath the Measured Surface on Measured Spectrum  
The penetration depth of light varies with tissue composition and wavelength [10; 14; 22; 23]. In tissue with a thickness less than the penetration depth, some of the wavelengths can penetrate through the tissue and be absorbed by the rubber or polymethylene underneath the tissue. Therefore, less light will be diffusely reflected and collected by the camera. Since the thickness of the slices is less than the lumpectomy specimens, we expect spectral differences at wavelengths that penetrate the tissue deeper than the minimum tissue slice thickness. To accurately compare spectra from the tissue slices dataset and the lumpectomy dataset, these wavelengths should be excluded from the analysis.  
To estimate the penetration depth, we used the optical properties from the lumpectomy measurements, which we estimated using an analytical fit model based on diffusion theory, given as [24]:  
\[R_{d}=\frac{\alpha^{\prime}}{1+2k(1-\alpha^{\prime})+(1+\frac{2k}{3})\sqrt{3(1- \alpha^{\prime})}}, \tag{2}\]  
where \(\alpha^{\prime}=\frac{\mu_{s}^{\prime}}{\mu_{s}^{\prime}+\mu_{a}}\) and \(k=\frac{1+\mu_{d}}{1-\mu_{d}}\). \(r_{d}\) is the internal reflection coefficient for diffuse light and depends on the refractive index of the sample, which was set to be 1.33. \(\mu_{s}^{\prime}\) and \(\mu_{a}\) are, respectively, the scattering and absorption coefficients and here represented as [25]:  
\[\mu_{s}^{\prime}=\mu_{s,800}^{\prime}\left(f_{mie}\left(\frac{\lambda}{800\; \text{nm}}\right)^{-b}+(1-f_{mie})\left(\frac{\lambda}{800\;\text{nm}}\right)^{ -4}\right), \tag{3}\]  
\[\mu_{a}=C.v_{blood}.\mu_{a,blood}+v_{WL}\left(\left(1-\left(\frac{[lipid]}{[ lipid]+[water]}\right)\right)\right).\mu_{a,water}(\lambda)+\left(\frac{[lipid]}{[lipid]+[water]}\right).\mu_{a, lipid}(\lambda)\right), \tag{4}\]  
with  
\[\mu_{a,blood}=(1-StO_{2})\mu_{a,Hb}(\lambda)+StO_{2}\mu_{a,HbO_{2}}(\lambda), \tag{5}\]  
\[C(\lambda)=\frac{1-exp(-R.\mu_{a,blood})}{R.\mu_{a,blood}}, \tag{6}\]  
where \(\mu_{s,800}^{\prime}\) is the reduced scattering at 800 nm, \(b\) the scatter power and \(f_{mie}\) the fraction Mie scattering with respect to Rayleigh scattering. \(\mu_{a,water}\) and \(\mu_{a,lipid}\) are the absorption coefficients of water and lipid, respectively, and \([lipid]\) and \([water]\) correspond to the concentration of lipid and water, respectively. \(v_{WL}\) is the fraction of water and lipid in the tissue and assumed to be 100% in the near-infrared wavelength region and nearly 0% in the visual wavelength range. \(v_{blood}\) and \(StO_{2}\) correspond to the blood volume fraction and the level of hemoglobin saturation by oxygen, respectively. \(\mu_{a,Hb}\) and \(\mu_{a,HbO_{2}}\)) are the absorption coefficients of hemoglobin and oxyhemoglobin, respectively, and \(R\) corresponds to the effective vessel diameter. The measured spectra were fitted using a nonlinear least-squares inversion algorithm, and for each estimated value, a 95% confidence interval was computed that was used to assess the reliability for each fit parameter.  
By applying the analytical fit model to the measured spectra on the lumpectomy specimens, we obtained the absorption and scattering coefficients that can be used to estimate the penetration depth, which is given by:  
\[d_{eff}=\frac{1}{\sqrt{3\mu_{a}(\mu_{a}+\mu_{s}^{\prime})}}, \tag{7}\]  
The minimum thickness of the tissue slices is 2.5 mm [15]. Therefore, by excluding wavelengths with a penetration depth above 2.5 mm, the spectra obtained on the tissue slices should be comparable with the spectra obtained on the lumpectomy specimens.