# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review
## 5 Studies toward Clinical Applications of TD-NIRS
### Measuring Optical Properties
#### 5.2.2 Multi-Layered Media
One-hundred to 500 training data with different noises were input into the NN, and the NN estimated \(\mu_{a,brain}\) under the condition that the optical properties of the five layers except \(\mu_{a,brain}\) and the thicknesses of the upper four layers are known with some uncertainties. Resultantly, \(\mu_{a,brain}\) was estimated with errors less than 5% even if there are uncertainties of 20% in the other optical properties and the thicknesses. Using the eigen-function analytical solutions of the TD-DE [97], [PERSON] et al. employed optimal estimation method of a Bayesian approach to incorporate prior information in the process of recovering the optical properties of the two-layered media [150; 151]. The method was verified by recovering of \(\mu_{a1}\), \(\mu_{a2}\), \(\mu_{s1}{{}^{\prime}}\), and \(\mu_{s2}{{}^{\prime}}\) from the TR reflectances provided by MC simulations and phantom experiments with better performances than the standard non-linear least-squares methods. The thickness of the top layer, \(s_{1}\), and the time origin of the TR reflectance, \(t_{0}\), were also included in the unknowns. The values of \(\mu_{a2}\) were very accurately estimated because the TR partial pathlength of the bottom layer, \(l_{2}(t)\), becomes much longer than that of the top layer, \(l_{1}(t)\), after 2 ns in the decaying period. Even if the thickness of the top layer, \(s_{1}\), is unknown the values of \(\mu_{a2}\) are obtained with errors less than 10%, and this method will be well suited for applications to human heads where the effect of the CSF is mainly to decrease \(\mu_{s2}{{}^{\prime}}\). [PERSON] et al. and [PERSON] et al. utilized the analytical formulations of the partial pathlength of Equation (24) with the analytical solutions of the TD-DE for two-layered media to estimate perturbations of \(\mu_{a2}\)[32] (pp. 109-125) [152; 153]. When the spectroscopic TR reflectance, \(R(t,\,\lambda)\), is divided into many time gates, the ratio of the perturbed and unperturbed \(R(t,\,\lambda)\) averaged over the \(g\)-th time-gate is derived from Equation (24) as,  
\[\ln\frac{R_{g}(\lambda)}{R_{0g}(\lambda)}=-\sum_{i=1}^{N}\Delta\mu_{a,i}( \lambda)l_{g,i}(\lambda) \tag{44}\]  
where \(R_{0g}(\lambda)\) and \(R_{g}(\lambda)\) are the unperturbed and perturbed \(R(t,\,\lambda)\) averaged over the \(g\)-th time-gate, respectively, \(\Delta\mu_{a,i}(\lambda)\) is the perturbation of \(\mu_{a,i}\) in the \(i\)-th layer, \(N\) is the number of layers, and \(l_{g,i}(\lambda)\) is the partial pathlength of the \(i\)-th layer averaged over the \(g\)-th time-gate, which is given by the analytical solution of the TD-DE in case of two-layered media (\(N=2\)). Then a system of linear equations of Equation (44) with the number of the time-gates is constructed and \(\Delta\mu_{a,i}(\lambda)\) is obtained by inverting the system of the linear equations. The optical properties of the human heads were estimated by in vivo measurements using multi-wavelength TD spectroscopy system at a group of laboratories [154]. The analytical solution of the TD-DE for two-layered media was fitted to the measured \(R(t,\,\lambda)\)[103], and five parameters, \(\mu_{a1}\), \(\mu_{a2}\), \(\mu_{s1}\)\({}^{\prime}\), \(\mu_{s2}\)\({}^{\prime}\), and \(s_{1}\) were estimated. Although the inter-laboratory differences are rather large, a definite trend was found after comparisons with the results using \(R(t,\,\lambda)\) generated by MC simulations. [PERSON] et al. estimated the changes in \(\mu_{a}\) of the brain and superficial layers after the intravenous administration of indocyanine-green from multi-distance TR reflectance measurements and MC simulations for a two-layered model [155]. In the estimation process, the information of the partial pathlengths of the two layers was important for calculating the sensitivities to absorption changes in the layers.