## 2 Background  
Pulse oximetry principles rely on light absorbance by different chromophores in blood. For instance, at high blood oxygen saturations (>80%), the red light (600 to 750 nm) is absorbed more than the infrared light (750 to 1000 nm), while at low saturations (<80%), the absorption is inverted. The pulse oximetry technique utilizes the pulsatile (AC) PPG components at both wavelengths to estimate the ratio of absorbances, limiting the use of the DC component for normalization only. The ratio of ratios (\(R\)) is then related to arterial oxygen saturation by empirical curves obtained from controlled hypoxia studies on healthy volunteers. A typical relation of \(R\) and \(SpO_{2}\) is shown in Equation (1) [18; 19]. \[SpO_{2}=110-25\cdot R \tag{1}\]  
\[\text{where, }R=\frac{AC_{R}/DC_{R}}{AC_{IR}/DC_{IR}}\]  
Equation (1): Oxygen saturation (\(SpO_{2}\)), Ratio of ratios (\(R\)), pulsatile component of the red wavelength (\(AC_{R}\)), non-pulsatile component of the red wavelength (\(DC_{R}\)), pulsatile component of the infrared wavelength (\(AC_{IR}\)), and non-pulsatile component of the infrared wavelength (\(DC_{IR}\)). NIRS does not include the pulsatile component of the signal and focuses on the analysis of the slow-changing DC components. Following the modified Beer-Lambert law (\(\Delta A=\Delta\mu_{a}*l\)), changes in chromophores absorption coefficients (\(\Delta\mu_{a}\)) can be calculated from changes in light attenuation (\(\Delta A\)) and the differential path length (\(l\)). Due to \(\mu_{a}=\varepsilon\cdot c\), change in light attenuation allows the calculation of changes in chromophores concentrations once the light travelling path (\(l\)) is known. However, the modified Beer-Lambert law method measures only the changes in concentrations from an arbitrary point. Therefore, spatially resolved spectroscopy (SRS) is implemented to measure an absolute tissue oxygenation index. The Tissue Oxygen Index (TOI) is expressed in percentage and represents the hemoglobin oxygenation saturation of the sampled volume of tissue. This method calculates the relative chromophores concentrations by measuring the relative change in light attenuation (\(A\)) along the distance (\(\rho\)). In NIRS, two distal photodetectors are employed--close together (8 mm), and far from the light source, so that the measurement is less affected by the head geometry or surface non-uniformity. Moreover, due to the small distance between the photodetectors (<10 mm), the change of A along \(\rho\) can be assumed linear, which is a prerequisite for SRS. Furthermore, the multi-distance approach makes SRS less susceptible to the superficial structures within the light path. The value of \(\frac{\partial A}{\partial\rho}\) is measured by the probe as the change of light attenuation along the photodetectors' distance. Light attenuation is related to light intensity (voltage measurements) by applying the Beer-Lambert Law as follows:  
\[\frac{\partial A(\lambda_{\text{j}})}{\partial\rho}=\frac{log\left(\frac{t_{ photodiode}(\lambda_{\text{j}})}{t_{photodiode}(\lambda_{\text{j}})}\right)}{\rho_{photodiode}-\rho_{photodiode}1} \tag{2}\]  
Equation (2): Specific wavelength (\(\lambda_{\text{j}}\)), change of light attenuation along the photodetectors' distance (\(\partial A/\partial\rho\)), light intensity at each photodetector (\(I_{photodiode}\)), and source-detector distance of each photodetector (\(\rho_{photodiode}\)).