# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review
## 2 Theoretical Background of TD-NIRS
### Time-Domain Sensitivity Functions of Optical Signals  
Assume that the TR fluence rate, \(\phi_{0}(\textbf{r},t)\), is a solution of the TD-DE in a homogeneous medium with the absorption and diffusion coefficients, \(\mu_{a0}\) and \(D_{0}\), for the impulse point source at the origin and time zero. If \(\mu_{a0}\) and \(D_{0}\) in a small volume of the medium, \(V_{p}\), are perturbed to \(\mu_{a}=\mu_{a0}+\delta\mu_{a}\) and \(D=D_{0}+\delta D\), the perturbations in the fluence rate due to the perturbations of \(\delta\mu_{a}\) and \(\delta D\), \(\delta\phi_{a}(\textbf{r},t)\), and \(\delta\phi_{D}(\textbf{r},t)\), respectively, are derived by the first order perturbation theory (explained in detail in a later section), and the TD sensitivities (weights or Jacobians) of the fluence rate to the changes in \(\mu_{a}\) and \(D\), defined as the limits of \(\delta\mu_{a}\to 0\) and \(\delta D\to 0\) of the absolute values of \(\delta\phi_{a}(\textbf{r},t)/\delta\mu_{a}\) and \(\delta\phi_{D}(\textbf{r},t)/\delta D\), respectively, are given as,  
\[W_{a}^{(\phi)}(\textbf{r},t)=\int_{V_{p}}dr\prime\int_{0}^{t}dt\{\phi_{0}( \textbf{r},tt)G(\textbf{r},\textbf{r},t-tt)\}, \tag{17}\]  
\[W_{D}^{(\phi)}(\textbf{r},t)=\int_{V_{p}}dr\prime\int_{0}^{t}dtt\{[\
abla_{ \textbf{r}}\phi_{0}(\textbf{r},tt)]\cdot[\
abla_{\textbf{r}}G(\textbf{r}, \textbf{r},t-tt)]\} \tag{18}\]  
where \(G(\textbf{r},\textbf{r}^{\prime},t-t^{\prime})\) is the [PERSON]'s function which is the solution of \(\phi(\textbf{r},t)\) in the TD-DE for the impulse point source at position \(\textbf{r}^{\prime}\) and time \(t^{\prime}\). The right-hand sides of Equations (17) and (18) are calculated analytically using the analytical solutions of TD-DE for simple geometries or numerically for complex geometries. Equations (17) and (18) are the TD sensitivities of the fluence rate to the changes in \(\mu_{a}\) and \(D\), respectively, but the TD sensitivities of any optical signals such as the reflectance and transmittance can be defined. For example, when the reflectance, \(R(\textbf{r}_{d},\textbf{r}_{s},t)\) (\(\textbf{r}_{d}\) and \(\textbf{r}_{s}\) are the detector and source positions, respectively) is considered, the TD sensitivity of the reflectance to the change in \(\mu_{a}\) is expressed approximately as,  
\[W_{a}^{(R)}(\textbf{r}_{d},\textbf{r}_{s},t)=\int_{V_{p}}dr\prime\int_{0}^{t} dt\prime\Big{\{}G^{(R)}(\textbf{r}_{d},\textbf{r},tt)G(\textbf{r},\textbf{r}_{s},t -tt)\Big{\}}, \tag{19}\]  
where \(G^{(R)}(\textbf{r}_{d},\textbf{r}^{\prime},t)\) is the Green's function of the reflectance measured at \(\textbf{r}_{d}\) for the impulse point source at position \(\textbf{r}^{\prime}\) and time \(t^{\prime}\), and \(G(\textbf{r}^{\prime},\textbf{r}_{s},t-t^{\prime})\) is the Green's function of the fluence rate measured at position \(\textbf{r}^{\prime}\) and time \(t-t^{\prime}\) for the impulse point source at position \(\textbf{r}_{s}\) and time \(t^{\prime}\). Now the physical meaning of Equation (19) should be considered. By integrating the right-hand side of Equation (19) for infinitesimal volume of \(V_{p}=\delta(\textbf{r}^{\prime}-\textbf{r})\) around **r**, and after some mathematical manipulations, a three-point function of \(\textbf{r}_{d},\textbf{r}_{s},\) and **r** can be defined as Equation (20),  
\[H(\textbf{r}_{d},\textbf{r}_{s},\textbf{r},t)=\alpha\int_{0}^{t}\{G(r,r_{s}, tt)G(r_{d},r,t-tt)\}dt=\alpha\int_{0}^{t}\{G(r,r_{s},tt)G(r,r_{d},t-tt)\}dt, \tag{20}\]  
where \(\alpha\) is a proportionality constant and the second equality is derived by the reciprocity between the position of the detector and the position of interest, or the adjoint formulation.