# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review
## 4 Advanced Theories and Methods for TD-NIRS
### Perturbation Theory
#### 4.5.1 Formulation of the TD-Perturbation Based on the TD-DE  
Formulation of the TD-perturbation starts from the TD-DE (Equation (7)) for the unperturbed fluence rate, \(\phi_{0}(\textbf{r},\,t)\), under the unperturbed \(\mu_{a0}(\textbf{r})\) and \(D_{0}(\textbf{r})=1/3\mu_{s0^{\prime}}(\textbf{r})\), and the TD-DE for the perturbed fluence rate, \(\phi_{p}(\textbf{r},\,t)=\phi_{0}(\textbf{r},\,t)+\delta\phi(\textbf{r},\,t)\), under the perturbed coefficients, \(\mu_{ap}(\textbf{r})=\mu_{a0}(\textbf{r})+\delta\mu_{a}(\textbf{r})\) and \(D_{p}(\textbf{r})=D_{0}(\textbf{r})+\delta D(\textbf{r})\). By subtracting the unperturbed equation from the perturbed equation, the TD-DE for \(\delta\phi(\textbf{r},\,t)\) is obtained as,  
\[\bigg{\{}\frac{1}{c}\frac{\partial}{\partial t}-\
abla\cdot[D_{0}(\textbf{r} )\
abla]+\mu_{a0}(\textbf{r})\bigg{\}}\delta\phi(\textbf{r},\,t)=\{-\delta\mu _{a}(\textbf{r})+\
abla\cdot[\delta D(\textbf{r})\
abla]\}\phi_{p}(\textbf{r},\,t). \tag{30}\]  
If the right-hand side of Equation (30) is considered as a virtual source, \(Q_{v}(\textbf{r},\,t)=\{-\delta\mu_{a}(\textbf{r})+\
abla\bullet[\delta D( \textbf{r})\
abla]\}\phi_{p}(\textbf{r},\,t)\), and Equation (30) has the same form as the unperturbed equation. Then, if the Green's function of the unperturbed equation is known as \(G(\textbf{r},\,\textbf{r}^{\prime},\,t-t^{\prime})\), the solution of Equation (33) for the virtual source of \(Q_{v}(\textbf{r},\,t)\) is given by the Green's function method as,  
\[\delta\phi(\textbf{r},\,t)=\int_{V_{p}}d\tau\prime\int_{0}^{t}dttQ_{v}(\textbf {r},\,tt)G(\textbf{r},\,\textbf{r}^{\prime},t-t\prime), \tag{31}\]  
where \(V_{p}\) indicates the volume occupied by the perturbations. Substitution of \(Q_{v}(\textbf{r},\,t)\) into Equation (31) and some mathematical operations leads to the following equation,  
\[\phi_{p}(\textbf{r},\,t)=\phi_{0}(\textbf{r},\,t)+\delta\phi(\textbf{r},\,t)= \phi_{0}(\textbf{r},\,t)+\delta\phi_{a}(\textbf{r},\,t)+\delta\phi_{D}( \textbf{r},\,t), \tag{32}\]  
where  
\[\delta\phi_{a}(\textbf{r},\,t)=-\int_{V_{p}}d\tau\prime\int_{0}^{t}dtt\big{\{} \delta\mu_{a}(\textbf{r})\phi_{p}(\textbf{r},\,t\prime)G(\textbf{r},\,\textbf{ r},\,t-t\prime)\big{\}}, \tag{33}\]  
\[\delta\phi_{D}(\textbf{r},\,t)=-\int_{V_{p}}d\tau\prime\int_{0}^{t}dt\big{\{} \delta D(\textbf{r}^{\prime})[\
abla_{\textbf{r}}\phi_{p}(\textbf{r},\,t\prime )]\cdot[\
abla_{\textbf{r}}G(\textbf{r},\,\textbf{r},t-t\prime)]\big{\}}. \tag{34}\]  
Now, the key subject is how to calculate the integrals of Equations (33) and (34) including the unknown \(\phi_{p}(\textbf{r},\,t)\) (note: the integral equation composed of Equations (32), (33), and (34) is called the Fredholm equation of the second kind), and various assumptions and approximations have been employed. Usually, the perturbations are assumed constant inside the inclusions, i.e., \(\delta\mu_{a}(\textbf{r})=\delta\mu_{a}=\text{const}\) and \(\delta D(\textbf{r})=\delta D=\text{const}\), and the simplest approximation is to substitute \(\phi_{0}(\textbf{r},\,t)\) for \(\phi_{p}(\textbf{r},\,t)\) by assuming \(\phi_{p}(\textbf{r},\,t)\approx\phi_{0}(\textbf{r},\,t)\) resulting in the followings,  
\[\delta\phi_{a}^{(1)}(\textbf{r},\,t)=-\delta\mu_{a}\int_{V_{p}}d \tau\prime\int_{0}^{t}dtt\big{\{}\phi_{0}(\textbf{r},\,t\prime)G(\textbf{r}, \,\textbf{r}^{\prime},t-t\prime)\big{\}},\] (35) \[\delta\phi_{D}^{(1)}(\textbf{r},\,t)=-\delta D\int_{V_{p}}d\tau \prime\int_{0}^{t}dtt\big{\{}[\
abla_{\textbfThe third, fourth, and higher order perturbation can be formulated in a similar manner, and the cross talks between \(\delta\mu_{a}\) and \(\delta D\) begin to play a role. The first order perturbation, i.e., the Born approximation, is believed to be valid for inclusions with very small sizes and very small changes in \(\mu_{a}\) and \(D\) from those of the background. For large inclusions with large changes in \(\mu_{a}\) and \(D\) such as in the case of breast cancers, the second or third order perturbation may be necessary.