# Optical Characterization of Homogeneous and Heterogeneous Intralipid-Based Samples
## 2 Light Propagation in Turbid Media
### Diffusion Approximation
#### 2.2.2 Heterogeneous Slab  
In order to describe the effects of the presence of an inclusion hidden in an otherwise homogenous slab (Figure 1), a perturbation approach to the diffusion equation was needed. We make the assumption that the difference between the optical coefficient of the inclusion and the host media is radially described by a Gaussian law, as expressed by the form. \[\mu(\rho)=\Delta\mu\;exp\bigg{(}-ln2\bigg{(}\frac{\rho}{R}\bigg{)}^{2}\bigg{)} \tag{15}\]  
The quantity \(\Delta\mu\) is the perturbation intensity, and it is observed at the centre of the inclusion. The quantity \(R\) is defined as the radius of the inclusion and corresponds to the width at half maximum of the Gaussian curve. It can be shown that at first-order perturbation approximation, the outward photon flux \(I\) transmitted in presence of a defect can be written as the sum of three terms, as follows  
\[I_{pert}(t;z_{incl})=I(t)+\delta I_{\mu_{a}}(t;z_{incl})+\delta I_{\mu_{s}} (t;z_{incl}) \tag{16}\]  
The first term describes the photon flux transmitted through the homogeneous slab. The second and third term, the quantities \(\delta I\), describe, at first approximation, the change in the outward photon  
Figure 1: Schematic arrangement of a heterogeneous slab with an absorptive inclusion. flux due to the presence of an absorptive and a scattering inclusion, respectively. Here, we present the analytical forms obtained for a Gaussian absorptive and scattering inclusion:  
\[\delta I_{\mu_{a}}(t;z_{incl}) = \frac{\Delta\mu_{a}\textit{exp}(-\mu_{a}\textit{v}t)}{2\pi^{2}A(n )D^{2}d_{e}^{2}t}\sum\
olimits_{k,l=1}^{\infty}R_{k,l}(R,t)\cdot Z_{k,l}(h,z_ {incl},t) \tag{17}\] \[\delta I_{\mu_{s}^{\prime}}(t;z_{incl}) = \frac{\Delta\mu_{s}^{\prime}\textit{exp}(-\mu_{a}\textit{v}t)}{8 \pi^{2}d_{e}^{2}A(n)t}\sum\
olimits_{k,l=1}^{\infty}R_{k,l}^{\prime}(R,t)\cdot Z _{k,l}^{\prime}(h,z_{incl},t) \tag{18}\]  
In both cases, a double series characterized by a factor and the product of two functions is present. The former is a function of the radius of the inclusion, the latter depends on inclusion longitudinal extension. So, the parameters that characterize the perturbation induced by the inclusion are its radius, height, and the perturbation intensity value \(\Delta\mu_{a}\) and \(\Delta\mu_{s}\)'. The accuracy of the proposed perturbation model can be investigated by considering the time-resolved contrast function, that is defined as follows  
\[C\Big{(}t,z_{pc},\Delta\mu\Big{)}=\frac{I_{pert}\Big{(}t,z_{pc},\Delta\mu\Big{)} -I(t)}{I(t)} \tag{19}\]  
So, the contrast function is the relative change in the outward photon flux due to the presence of the inclusion. As defined, the contrast depends on the size of the inclusion and its perturbation intensity value. In order to investigate the accuracy of the Gaussian perturbation model, we numerically solved the diffusion equation by using a Finite Element Method (FEM) for a slab with a thickness of \(d\) = 40 mm, absorption coefficient \(\mu_{a}\) = 0.01 mm\({}^{-1}\), reduced scattering coefficient \(\mu^{{}^{\prime}}s\) = 1.0 mm\({}^{-1}\), assuming that the inclusion had a diameter equal to the height. In Figure 2a, the absorptive contrast (Equation (19)) measured in transmittance configuration as a function of time for an absorptive inclusion of radius \(R\) = 5 mm and \(\Delta\mu_{a}/\mu_{a}\) = 0.2 located at different depths inside the slab is shown. The solid curves are the FEM simulations, i.e., the exact solution, the dashed curves are the perturbation model predictions. The contrast function increased as the inclusion moved toward the boundary surfaces. For each considered depth of the inclusion, the contrast increased as time increased. The accuracy of perturbation model prediction was better than 7% for a relative perturbation intensity of less than 20%.