# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review
## 5 Studies toward Clinical Applications of TD-NIRS
### Time-Domain Diffuse Optical Tomography (TD-DOT)
#### 5.3.1 General Concept of TD-DOT  
Some studies employed the reconstruction algorithm for X-ray CT to reconstruct the absorption images using the temporally extrapolated absorbance method [158; 159] or using the time-gated measurements at early times [160; 161; 162; 163]. These methods aimed to detect the ballistic-like components of transmitted ultra-short pulse light, but their applications were limited to small-sized objects because transmitted light was unmeasurable with high signal-to-noise ratios for objects thicker than a few centimeters. Standard algorithms of TD-DOT for thick or large objects are categorized as a model-based iterative image reconstruction (MOBIIR) which employs a forward model of light propagation and a non-linear iterative inversion process. [PERSON] et al. proposed an algorithm of MOBIIR based on the DE [164]. They derived the sensitivities (Jacobians) for the mean time-of-flight, \(<\!\!t>\), as the data type, from the perturbation theory, and formulated the inverse process to reconstruct \(\mu_{a}\) and \(\mu_{s}\)\({}^{\prime}\) images. The software developed in this framework was named as \"TOAST\" and its revised version \"TOAST++\" was presented as an open-source software [165]. A general algorithm of MOBIIR is summarized below. Figure 15 shows a standard process of MOBIIR mainly consisting of measurements and an inverse process. The measured TD data (TOF distributions), \(\Gamma(t)\), can be used fully or converted to featured TD data. Several types of featured TD data characterizing the TOF distributions are often used as the following [4],  
\[\text{Time}-\text{gated intensity}:\;M_{DC(T_{a},T_{b})}=\int_{T_{a}}^{T_{b}} \Gamma(t)dt, \tag{45}\]  
\[n-\text{th temporal moment}:\;M_{n}=\int_{0}^{\infty}t^{n}\Gamma(t)dt, \tag{46}\]  
\[\text{Mellin-Laplace transform}:\;M_{ML}(n,p)=\int_{0}^{\infty}t^{n}e^{-pt} \Gamma(t)dt, \tag{47}\]  
Amplitude and phase of frequency-domain data:  
\[M_{amp}=\left|\hat{\Gamma}(\omega)\right|=\left|\int_{0}^{\infty}\Gamma(t)e^{ -i\omega t}dt\right|,\;M_{\theta}=\text{arg}\hat{\Gamma}(\omega)=\text{arg}( \int_{0}^{\infty}\Gamma(t)e^{-i\omega t}dt). \tag{48}\]The time-gated intensity with \(T_{a}\) = 0 and \(T_{b}\) = \(\infty\) corresponds to the CW intensity, and multiple time-gated intensities with a small time-step of \(T_{b}-T_{a}\) = \(\Delta T\) dividing the whole TOF distributions are equivalent to the use of full TOF distributions. The \(n\)-th temporal moment with \(n\) = 0 is also identical to the CW intensity, and those with \(n\) = 1, 2, 3 are the mean, \(M_{1}\) = \(<\)\(t\)\(>\), variance, \(M_{2}\), and skew, \(M_{3}\), of the TOF distribution, respectively. The Mellin-Laplace transform with \(n\) = 0 or \(p\) = 0 corresponds to the simple Laplace or Mellin transform, respectively. Equation (48) describes the transformation of the TD data to the FD data at a frequency of \(\omega\). Several updating methods are available, and Newton-type non-linear reconstruction is a standard method. The objective function, \(\Psi\), defined below is minimized,  
\[\Psi=\|\boldsymbol{\Gamma}_{cal}-\boldsymbol{\Gamma}_{mes}\|^{2}+\lambda\| \mathbf{x}\|^{2}, \tag{49}\]  
where \(\boldsymbol{\Gamma}_{cal}\) and \(\boldsymbol{\Gamma}_{mes}\) are the vectors of the measured and calculated (featured) TD data with the components of \(\boldsymbol{\Gamma}_{cal}\)(\(\mathbf{r}_{sm}\), \(\mathbf{r}_{dm}\);\(\mu_{s}\)(\(\mathbf{r}_{n}\)), \(D\)(\(\mathbf{r}_{n}\))), and \(\boldsymbol{\Gamma}_{mes}\)(\(\mathbf{r}_{sm}\), \(\mathbf{r}_{dm}\)), respectively, the index \(m\) (=1,..., \(M\)) specifies the TD data number with \(M\) being the number of the TD data, the index \(n\) (=1,..., \(N\)) specifies the voxel number with \(N\) being the number of voxels in the medium, respectively, x = [\(\mu_{a}\)(\(\mathbf{r}_{n}\)), \(D\)(\(\mathbf{r}_{n}\))]\({}^{T}\) is a vector consisting of the optical properties at the \(n\)-th voxel, and \(\lambda\) is the regularization parameter. Here, the measurement errors are neglected for simplicity.