# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review
## 5 Studies toward Clinical Applications of TD-NIRS
### Measuring Optical Properties
#### 5.2.1 Homogenous Semi-Infinite Media or Infinite Slab  
Determination of \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) in homogeneous semi-infinite media by TD techniques is essentially based on fitting analytical or numerical solutions of the TD-DE to measured TR reflectances. The TR-reflectance, \(R(\rho,t)\), is expressed by Equation (15) for the TD-DE under the extrapolated boundary condition, and for simplicity the zero-boundary condition is sometimes employed as Equation (28). For the zero-boundary condition, the following two equations are derived [18],  
\[\lim_{t\rightarrow\infty}\frac{d}{dt}\ln R(\rho,T)=-\mu_{a}c,\ \mu_{s}f=\frac{1}{3 \rho^{2}}(4\mu_{a}c^{2}t_{\max}^{2}+10 ct_{\max})-\mu_{a}, \tag{43}\]  
where \(t_{max}\) is the time of maximum \(R(\rho,t)\). From these equations, it is possible to determine \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) of semi-infinite homogeneous media from the measured \(R(\rho,t)\). Especially, it should be noted that \(\mu_{a}\) is determined from the slope of \(\ln R(\rho,t)\) at very late time. This feature of \(R(\rho,t)\) is also stated by [PERSON] [136]. But \(R(\rho,t)\) measured at very late times are usually associated with noises, and it is difficult to determine \(t_{max}\) due to the IRF. Therefore, using Equation (43) is not appropriate for accurate determination of \(\mu_{a}\) and \({\mu_{s}}^{\prime}\), and fittings of the IRF convoluted analytical solutions of Equation (15) or Equation (28) to measured \(R(\rho,t)\) are often used to recover \(\mu_{a}\) and \({\mu_{s}}^{\prime}\). Note that deconvolution of measured \(R(\rho,t)\) by the IRF enhances measurement noises and instead the analytical solutions are convoluted by the IRF. Determination of \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) in homogeneous infinite slabs by TR transmittance measurements is conducted using the analytical equation for TR transmittance, which is derived similarly to that for TR reflectance using the mirror image source method as Equation (15) [18; 19]. [PERSON] et al. [137] made in vivo measurements of \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) of a piglet brain using a TD system at three wavelengths (759, 794, 824 nm) by the method stated above. To separate the contributions of different head layers, the measured \(R(\rho,t)\) were acquired at the surfaces of skin, skull, dura mater and brain, step by step. The values of \(\rho\) were chosen to assure the mean penetration depth within each layer. Measured \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) were compared with the other in vivo results reported in literatures, and the differences were discussed. [PERSON] et al. [138] estimated \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) of human foreheads in the wavelength range from 700 nm to 1000 nm. They found that the estimated \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) were close to those of the superficial (scalp and skull) layers by additional MC simulations for four-layered media simulating the structure of the human head. Using the optical mammography system shown in Figure 11c, which employed seven wavelengths of 635 nm, 685 nm, 785 nm, 905 nm, 930 nm, 975 nm, and 1060 nm, [PERSON] et al. conducted a pilot study for optical estimates of tissue components to differentiate malignant from benign breasts from the data of patients with 45 malignant and 39 benign lesions [54]. They measured TR reflectances in vivo through compressed breasts and obtained images of the differences in \(\mu_{a}\) from the background at the seven wavelengths, \(\Delta\mu_{a}(\lambda)\), using a time-gated perturbation analysis based on the microscopic Beer-Lambert law under the assumption that the compressed breasts were homogeneous slabs [54; 139]. The \(\Delta\mu_{a}(\lambda)\) images were converted to the changes in the contents of oxy-Hb, deoxy-Hb, water, lipid, and collagen using Equation (25), and from statistical analyses they concluded that the collagen content was the most important parameter for discriminating malignant and benign lesions. [PERSON] et al.