## 3 Methodology  
Initially, the complex structure of the head is simplified into two layers to better understand and distinguish the hemodynamics of cerebral and extra-cerebral (skin and skull) tissues. For a two-layered semi-infinite medium (as shown in Figure 1), the measured boundary intensity at a detector at a distance \(\rho\) from a point-source can be written as [16,17]:  
\[I(\rho)=\frac{1}{2\pi}\int_{0}^{\infty}\phi(s)\;s\;J_{0}(sp)ds, \tag{9}\]  
where, \(s\) is the radial spatial frequency coordinate corresponding to the radial distance \(\rho\), \(J_{0}\) is the zeroth order Bessel function of first kind, and \(\phi\) is given by:  
\[\phi(s)=\frac{\sinh(\alpha_{1}(z_{b}+z_{0}))}{D_{1}\alpha_{1}}\times\frac{D_{1 }\alpha_{1}\cosh(\alpha_{1}L)+D_{2}\alpha_{2}\text{sinh}(\alpha_{1}L)}{D_{1} \alpha_{1}\cosh(\alpha_{1}(L+z_{b}))+D_{2}\alpha_{2}\text{sinh}(\alpha_{1}(L+ z_{b}))}-\frac{\sinh(\alpha_{1}z_{0})}{D_{1}\alpha_{1}} \tag{10}\]  
Here, L is the thickness of layer-1, the subscripts 1 and 2 indicate the properties for layer-1 and layer-2, \(\alpha_{i}=\sqrt{s^{2}+\frac{\mu_{i}}{D_{i}}}\), diffusion coefficient \(D_{i}=\frac{1}{3\left(\mu_{i_{1}}+\mu^{\prime}{}_{i_{1}}\right)}\), \(z_{0}=\frac{1}{\mu_{a_{1}}+\mu^{\prime}{}_{i_{1}}}\), \(z_{b}=\frac{1+R_{\text{eff}}}{1-R_{\text{eff}}}2D_{1}\). \(R_{\text{eff}}\) is the fraction of photons internally and diffusively reflected at the boundary. As shown in Section 2, for the measured CW intensity data on the boundary of a two-layered medium a unique solution exists for a parameter set \(\left\{\mu_{a_{1}},\mu^{\prime}_{b_{1}},\mu_{a_{2}},\mu^{\prime}_{b_{2}},\L\right\}\). To increase the number of independent measurements and therefore improve the recovery accuracy the problem is spectrally constrained by measuring the intensity data over a broad range of wavelengths. For biological tissues, since hemoglobin is the major absorbing chromophore in the near-infrared range, the wavelength range 650 to 850 nm at either side of the isosbestic region are considered (which is typically conventional in most systems, well away from 950 nm beyond which other molecules, such as Methemoglobin and water are the dominant absorbing chromophores). Although other chromophores have significant extinction coefficients in this wavelength range, their concentration in the tissue is relatively low, therefore oxy and deoxyhemoglobin can be considered as the main absorbers. The parameter set now becomes \(\mathbf{p}\equiv\left[HbO_{1},HbO_{2},Hb_{1},Hb_{2},Sa_{1},Sa_{2},Sp_{1},\,Sp _{2},L\right]\), and measurement data is the light intensity at source-detector distances \(\rho=10,\;20,\;30,\;40,\;50\) mm, and wavelengths \(\lambda=650\) to 850 nm, in steps of 10 nm. Instead of a conventional approach of recovering all parameters together using a least-square minimization method which generally exhibits a very high condition number, the recovery of parameters is achieved in two steps: (1) recovering layer-1 parameters from spectral intensity data at \(\rho=10\) mm i.e., a short source-detector distance, assuming a homogenous model as implemented by considering same parameters of layer-1 and layer-2 properties in Equation (10), the dependency on L is assumed to be minimal in such a case, and (2) utilizing the recovered layer-1 parameters in step-1, the layer-2 parameters and tissue thickness L, are recovered using spectral intensity data from \(\rho=10\) to 50 mm. This splitting of the parameter recovery is observed to greatly improve the inverse problem with decreasing the condition number by at least an order of magnitude (i.e., by a factor of 10).