## 5 Discussion  
A hyperspectral continuous wave NIRS parameter recovery method is presented to distinguish extra-cereb and cerebral tissue hemodynamics, by separating the tissue absorption and scattering parameters and also by simultaneously fitting for the superficial layer thickness. Major challenges of continuous wave NIRS system have been the recovery of scatter (i.e., separating absorption and scattering information from the measured NIRS signal), and removing/separating the extra-cereb tissue contamination from the NIRS signal to estimate the cerebral hemodynamics. It is shown theoretically in Section 2 that, while for a given spectral derivative of intensity measurements (any derivative measurements) have non-unique solution sets of layered tissue parameters, the absolute intensity measurements have a unique solution set of layered tissue parameters. A two-step approach is presented to recover the two-layered properties along with tissue thickness. In the case of data generated on a two-layered semi-infinite model, the proposed two step approach using a semi-infinite model in the inverse problem is shown to recover the tissue parameters within 16.6% errors. The recovery error of layer-1 properties are reduced with increasing thickness, this is because of the assumption in step-1 that the short distance measurement is only influenced by layer-1. This increase in accuracy of the estimation of layer-1 properties in step-1 which increase with thickness, should improve the recovery of layer-2 properties in step-2 but with increased thickness there is a drop in the sensitivity of layer-2 and therefore a similar trend of improvement in accuracy is not seen for layer-2. The opposite sign of mean-relative errors for absorption (i.e., HbO and Hb) and scattering parameters (scattering amplitude) shows that the cross-talk between these parameters is negatively related, any over-estimation of absorption is accompanied by an under-estimation of scattering amplitude and vice versa.  
In the case of data generated on head models, a semi-infinite recovery model is shown to recover results with mixed accuracy. This is due to the complex structure i.e., the spatially varying curvature of external medium boundary and internal regional boundary. This is evident from the recovery accuracy of tissue thickness in Figure 9 for some head models where a correct two-layered semi-infinite model approximation wasn't found. Therefore, to tackle this problem, using an appropriate boundary structure with the similar two-step parameter recovery approach, it has been demonstrated that the accuracy is greatly improved and similar to the initial case where the data is generated and recovered on semi-infinite model.  
The practical implementation of this method has been shown in Section 4.3. The results presented in Table 4 show that the variation in tissue oxygenation levels recovered over three different days is less than 5%, and the variation in the recovery of extra-cereb tissue thickness was not more than \(\sim\)1 mm, which clearly demonstrates the repeatability of the recovery algorithm.  
It is important to highlight that the human head consists of multiple layers, such as skin, skull, CSF and brain tissue. Therefore, although the proposed methodology only assumes a two-layer model, to account for superficial signal contamination, the results indicate that this is much more suitable and robust as compared to the conventional homogenous assumptions. Future work should be directed towards extending this methodology to account for multiple layers, which may require the utilization of more boundary measurements and data-types to better extract information about pathlength [10].