# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review
## 4 Advanced Theories and Methods for TD-NIRS
### Advanced Monte Carlo Simulations  
[PERSON] and [PERSON] proposed a method to produce results of MC simulations for many combinations of \(\mu_{a}\) and \(\mu_{s}\) by correcting the result of one MC simulation for the reference combination of \(\mu_{a}=0\) and \(\mu_{s}=\mu_{s0}\)[105]. When the TR reflectance from a semi-infinite homogeneous medium with \(\mu_{a}=0\) and \(\mu_{s}=\mu_{s0}\) is \(R_{0}(\rho,t)\), the TR reflectance, \(R(\rho,t)\), for a medium with \(\mu_{a}\) and \(\mu_{s}\) is given by,  
\[R(\rho,t)=\left(\frac{\mu_{s}}{\mu_{s0}}\right)^{3}R_{0}(\rho\frac{\mu_{s}}{ \mu_{s0}},t\frac{\mu_{s}}{\mu_{s0}})\exp(-\mu_{a}ct). \tag{42}\]  
Figure 14: Geometry of a three-layered finite cylinder simulating the human head. Modified from Reference [103].  
Many results of \(R(\rho,t)\) produced from a single MC simulation were used for estimating \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) from the measured TR reflectance. A similar MC method was employed by [PERSON] et al. [106] where they called the method white Monte Carlo.  
[PERSON] et al. [107] proposed another correction method to produce many results from a single Monte Carlo simulation, called perturbation Monte Carlo (pMC) method. At the first step, a standard MC simulation is performed for homogeneous and non-absorbing medium with \(\mu_{a}=0\) and \(\mu_{s}=\mu_{s0}\), and all the coordinates of scattering positions only for detected photons are stored. The weights of the detected photons are unity at this step. At the second step, another medium with an inhomogeneity is considered with \(\mu_{ai}\) and \(\mu_{si}\) for the inhomogeneity and with \(\mu_{a0}\) and \(\mu_{s0}\) for the surroundings. Then the corrections of the weights of the detected photons are given by the following two scaling relationships, \(w_{a}\) for absorption and \(w_{s}\) for scattering,  
\[w_{a}=\exp(-\mu_{ai}l_{i}-\mu_{a0}l_{0})w_{s}=\left(\frac{\mu_{si}}{\mu_{s0}} \right)^{K_{i}}\exp[-(\mu_{si}-\mu_{s0})l_{i}]\] (4-14)  
where \(l_{i}\) and \(l_{0}\) are the pathlength of the photons inside and outside of the inhomogeneity, respectively, and \(K_{i}\) is the number of scattering events inside the inhomogeneity. The two scaling relationships were combined into one equation by [PERSON] et al. [108]. By this method the computation time is significantly reduced. Later, [PERSON] proposed a faster perturbation MC method by storing the seed values of random number series only for the detected photons [109]. This can speed up computation by 1000 times under some conditions.  
Standard MC methods are inappropriate for inverse processes due to long computation times, but the pMC method can be used in retrieving the optical properties from measured TR reflectance [110], in functional imaging in small animals using time-gated technique [111], and in computing the Jacobians for DOT [112]. However, still faster MC methods are desirable for TD-DOT, and an MC method using a graphics processing unit (GPU) and two parallel random number generators was developed to report 300 times faster calculations than using conventional CPUs [113]. GPU-based MC methods are developed to simulate light propagation in teeth where the optical properties vary in a wide range due to different tissues such as enamel and tubules in dentin [114].  
The equivalence between MC simulations and solving the RTE is further studied by [PERSON] et al. [115] to compare the results of MC simulations with those of the Maxwell equation in light propagation with polarization.