# Optical Characterization of Homogeneous and Heterogeneous Intralipid-Based Samples
## 2 Light Propagation in Turbid Media
### Diffusion Approximation  
#### 2.2.1 Homogeneous Slab  
The light transport through a highly diffusing medium is well described by the diffusion approximation to the radiative transfer equation for the average diffuse intensity [20]:  
\[\left[D\
abla^{2}-\frac{1}{v}\frac{\partial}{\partial t}-\mu_{a}\right]\varphi \left(\overrightarrow{r},t\overrightarrow{r_{s}},t_{s}\right)=-\delta\left( \overrightarrow{r}-\overrightarrow{r_{s}}\right)\delta(t-t_{s}) \tag{12}\]  
Here, \(v\) is the speed of light in the medium, \(\mu_{s}^{\prime}\) is the reduced scattering coefficient, and \(\mu_{a}\) is the absorption coefficient. The term \(\delta\left(\overrightarrow{r}-\overrightarrow{r_{s}}\right)\delta(t-t_{s})\) is the point-like source term located at \(\overrightarrow{r_{s}}\) at time \(t_{s}\), and \(D=1/(3\mu_{s}^{\prime})\) is the photon diffusion coefficient. It should be taken into account that in some cases, \(D\) may also depend on the absorption coefficient [19]. Within the extrapolated boundary conditions, the average diffuse intensity \(\varphi\) vanishes at extrapolated surfaces outside the turbid medium at a distance \(z_{e}=2\Delta D\), where the coefficient A accounts for the refractive index mismatch between the slab and the external medium [37]. Generally, analytical solutions to Equation (12) are only given for simple geometry and homogeneous media, such as the case of the homogeneous slab. The basic solution to this problem for an isotropic source located at a depth \(z_{s}=1/\mu_{s}^{{}^{\prime}}\) and emitting a light pulse of unit energy at a time t = 0 is given as:  
\[\varphi\left(\overrightarrow{r},t;\overrightarrow{r_{s}},t_{s}\right)= \frac{e^{-\mu_{a}\varphi(t-t_{s})-\frac{\rho^{2}}{4D\varphi(t-t_{s})}}}{8\pi  
where \(\rho\) is the radial distance between positions \(\overrightarrow{r}\) and \(\overrightarrow{r_{s}}\). The quantities \(d_{e}=d+2z_{e}\) is the extrapolated thickness of the slab.  
Taking into account the extrapolated boundary conditions, the outward photon flux measured at the position \(\overrightarrow{r_{m}}\) on the surface of the slab is given by \(I\big{(}\overrightarrow{r_{m}},t;\overrightarrow{r_{s}},t_{s}\big{)}=(2\pi/ A)\varphi\big{(}\overrightarrow{r_{m}},t;\overrightarrow{r_{s}},t_{s}\big{)}\). The quantity \(I\) is the reflectance when the source and the detector are upon the same surface and is the transmittance when the source and the detector are upon opposite sides.  
In the present case, the quantity of interest for comparing models and experiments in TRT was the transmitted light in a specific position as a function of time. According to [PERSON] et al. [38], the expression describing the transmitted light, \(T(t)\), is obtained as a solution of the diffusion equation (Equation (12)) considering point-like sources and extrapolated boundary conditions  
\[T(t)=\frac{exp(-\mu_{a}vt)}{4\pi\;D\;A(n)d_{e}\;t}\sum\
olimits_{n=1}^{\infty} exp\bigg{(}-\frac{Dvt\pi^{2}n^{2}}{d_{e}^{2}}\bigg{)}\sin\bigg{(}\frac{n\pi(z_{s }+z_{e})}{d_{e}}\bigg{)}\sin\bigg{(}\frac{n\pi(d+z_{e})}{d_{e}}\bigg{)} \tag{14}\]