## 5 Spatially-Resolved Spectroscopic Technique for Optical Property Measurement
### Principle and Mathematical Theory  
Light propagation in turbid media can be described by the Boltzmann transport equation [105], and solutions to the transport equation for the light fluence rate can be an intractable problem. However, by assuming that scattering is dominant in the medium (i.e., \(\mu_{\text{s}}^{\prime}\gg\mu_{\text{a}}\)) and that the light source is isotropic, we can reduce the Boltzmann transport equation to a second-order partial differential equation [106]:  
\[D\
abla^{2}\phi(r,t)-\frac{1}{c}\frac{\partial\phi(r,t)}{\partial t}-\mu_{ \text{a}}\phi(r,t)=-S(r,t) \tag{7}\]  
where \(D=\frac{1}{3(\mu_{\text{a}}+\mu_{\text{s}}^{\prime})}\) is the diffusion coefficient, \(\
abla^{2}\) is the Laplace operator, \(c\) is the speed of light travelling in the medium, \(r\) is the distance from the light source (Figure 19), \(t\) is the time, \(\phi\) is the fluence rate and \(S\) is the spherical harmonic representing the isotropic source. At the steady-state condition, Equation (7) can be simplified as follows:  
\[\
abla^{2}\phi(r,t)-\frac{\mu_{\text{a}}}{D}\phi(r,t)=-\frac{S(r,t)}{D} \tag{8}\]  
This steady-state diffusion approximation equation provides a theoretical basis for different spatially-resolved or spatial frequency domain techniques. For the homogeneous semi-infinite turbid medium, an analytical solution can be obtained by using an extrapolated boundary condition (i.e., the fluence rate is forced to zero at a height \(z_{b}\) above the medium surface) and a photon dipole source [97]:  
\[\phi(r,z,z_{0})=\frac{1}{4\pi D}\bigg{[}\frac{\exp(-\mu_{\text{eff}}r_{1})}{ r_{1}}-\frac{\exp(-\mu_{\text{eff}}r_{2})}{r_{2}}\bigg{]} \tag{9}\]  
where,  
\[\mu_{\text{eff}}=\big{[}3\mu_{\text{a}}(\mu_{\text{a}}+\mu_{\text {s}}^{\prime})\big{]}^{1/2} \tag{10}\] \[r_{1}=\Big{[}(z-z_{0})^{2}+r^{2}\Big{]}^{1/2} \tag{11}\]\[r_{2}=\left[\left(z+z_{0}+2z_{b}\right)^{2}+r^{2}\right]^{1/2} \tag{12}\]  
and \(z\) is the depth (Figure 19), \(z_{b}=2 AD\) where \(A\) is a constant that can be determined by an empirical approach [107]. Then the Green's function for the diffuse reflectance can be calculated as the gradient of the fluence rate (i.e., flux):  
\[ \begin{array}{ll}R_{\rm flux}(r,z_{0})&=-D\
abla\phi(r,z,z_{0})\big{|}_{z=0} \\ &=\frac{1}{4\pi}\Bigg{[}z_{0}(\mu_{\rm eff}+\frac{1}{r_{1}})\frac{ \exp(-\mu_{\rm eff}r_{1})}{r_{1}^{2}}+(z_{0}+2z_{b})(\mu_{\rm eff}+\frac{1}{r_ {2}})\frac{\exp(-\mu_{\rm eff}r_{2})}{r_{2}^{2}}\Bigg{]}\end{array}  \tag{13}\]  
where \(r_{1}\) and \(r_{2}\) are given by Equations (11) and (12), respectively, with \(z\) set to 0 (i.e., at the tissue surface). From Equation (13), it can be seen that the flux depends only on the distance and the two parameters \(\mu_{\rm a}\) and \(\mu_{\rm s}^{\prime}\). Finally, the radically dependent diffuse reflectance for a single scatter source is given by integrating the Green's function as follows [97]:  
\[ \begin{array}{ll}R(r)&=\int\limits_{0}^{\infty}R_{\rm flux}(r,z_{0})a^{ \prime}\delta(z_{0}-\frac{1}{\mu_{t}^{\prime}})dz_{0}\\ &=a^{\prime}R_{\rm flux}(r,z_{0})\big{|}_{z_{0}=1/\mu_{t}^{\prime}}\end{array}  \tag{14}\]  
where \(a^{\prime}=\mu_{\rm s}^{\prime}/(\mu_{\rm a}+\mu_{\rm s}^{\prime})\) is the transport albedo, \(\mu_{t}^{\prime}=\mu_{\rm a}+\mu_{\rm s}^{\prime}\) is the total attenuation coefficient and \(\delta\) is the Dirac Delta function.