# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review
## 4 Advanced Theories and Methods for TD-NIRS
### Multi-Layered Media  
Most biological organs can be categorized in multi-layered media from an optical point of view. Two typical examples are the human head and the tissues above the skeletal muscle, the former consisting of five layers of scalp (skin), skull, cerebrospinal fluid (CSF), gray matter, and white matter layers, the latter consisting of three layers of skin, fat, and muscle layers. The analytical solutions of the equations describing light propagation in multi-layered media will be very useful to recover the optical properties of each layer from measurements of TR reflectance. No analytical solution for the TD-RTE for multi-layered media has been reported so far, but the analytical solutions for the TD-DE have been obtained as follows.  
[PERSON] et al. [98] reported the analytical solution of TR reflectance for two-layered semi-infinite media based on the TD-DE to recover \(\mu_{a}\) and \(\mu_{s}{}^{\prime}\) of the layers. But the analytical solutions derived using the Fourier-transform approach were not explicitly given in TD. [PERSON] et al. [99] reported another analytical solution of TR reflectance for two-layered semi-infinite media based on the TD-DE by applying the mirror image method for semi-infinite homogeneous media to two-layered media using distributed sources. Further, [PERSON] et al. extended [PERSON] et al. study to multi-layered media and obtained explicitly the asymptotic solution of TR reflectance in the very late time when the asymptotic solution is dependent on \(\mu_{a}\) and \(\mu_{s}{}^{\prime}\) of the deepest layer [98,100]. [PERSON] and [PERSON] extended the method for two-layered media to general \(N\)-layered media although explicit analytical solutionswere not given [98; 101]. The TD solutions were obtained using a fast Fourier transform from the FD solutions for \(N\)-layered media at many (512 for example) frequencies.  
The analytical solutions of TR reflectance using the eigenfunction method were reported in series by [PERSON] et al. [96; 97; 102; 103]. The geometries of the objects were a two-layered semi-infinite medium [102], two-layered parallelepiped [96], two-layered finite cylinder [97], and three-layered finite cylinder [103]. Using the separation of variables for space (**r**) and time (\(t\)) based on the microscopic Beer-Lambert law, the analytical solution of the TD-DE for a three-layered finite cylinder shown in Figure 14 are obtained as [103],  
\[\Phi(\textbf{r},t)=\sum_{l,n=1}^{\infty}cJ_{0}(K_{l}\rho)\sin(K_{lni}z+\gamma_ {lni})\sin^{*}(K_{lni}z_{0}+\gamma_{ln1})\exp[-(K_{i}^{2}D_{i}+\mu_{ai})ct]/N_{ ln}^{2} \tag{40}\]  
where \(i=1,2,3\) denote the first, second, and third layer, respectively, \(J_{0}\) is the 0 th order Bessel function, \(K_{l}\) are the roots of \(J_{0}(K_{l}L)=0\), and for \(K_{lni}\), \(\gamma_{lni}\), and \(N_{ln}\) refer to the reference [103]. The TR reflectance, \(R(\rho,t)\), is obtained from Equation (40) by use of [PERSON]'s law at \(z=0\), and the TR partial mean pathlength inside each layer, \(<\!\!l_{i}(\rho,t)\!\!>\), is given analytically by the following equation,  
\[<l_{i}(\rho,t)>=-\frac{1}{R(\rho,t)}\frac{\partial R(\rho,t)}{\partial\mu_{ai}}. \tag{41}\]  
The results are shown for the human head model having three layers consisting of the combined scalp and skull layer, the CSF layer, and the brain. The mean partial pathlength, given by temporal integration of Equation (41), of the third (brain) layer is also shown to vary from 0.3 mm to about 50 mm with \(\rho\) varying from 10 mm to 50 mm, respectively. Software for the two-layered finite cylinders is given in the Reference [32] (pp. 109-125).  
Another analytical solution of the TD-DE for finite cylindrical stratified media was reported by [PERSON] based on the separation of variables [104]. This method was verified by comparing the results for two-layered media with those of [PERSON] et al. [99], and reportedly the computation was fast enough for application to DOT.