## 2 Materials and Methods  
### Radiative Transfer Equation  
Estimation of radiation attenuation and multiple scattering effects requires knowledge of the radiative transfer of optical energy through the medium [1]. In the field of optical diagnostics, the radiative transfer equation is commonly used to describe the photons transport, specifying a balance of optical energy distributed between the incident, outgoing, absorbing, and scattering radiation propagated through the medium. The radiative transfer equation is presented below:  
\[\frac{1}{c}\frac{\partial I(\overrightarrow{\mathbf{r}},\overrightarrow{ \mathbf{s}},\mathbf{t})}{\partial t}=-\mu_{e}I(\overrightarrow{\mathbf{r}},\overrightarrow{\mathbf{s}},\mathbf{t})+\mu_{s}I(\overrightarrow{\mathbf{r} },\overrightarrow{\mathbf{s}},\mathbf{t})\underset{4\pi}{\overset{\rightarrow} {\mathbf{s}},\mathbf{t})}\underset{4\pi}{\overset{\rightarrow}{\mathbf{s}}, \mathbf{t})}f(\overrightarrow{\mathbf{s}},\overrightarrow{\mathbf{s}})I( \overrightarrow{\mathbf{r}},\overrightarrow{\mathbf{s}},\mathbf{t})\mathrm{d} \Omega\prime \tag{1}\]  
where \(I\) is a radiance in the particular direction (either \(s\) or \(s^{\prime}\)), \(t\) is time, \(\overrightarrow{\mathbf{r}}\) is the vector position, \(\overrightarrow{\mathbf{s}}\) is the incident direction of propagation, \(\mu_{e}\) is an extinction coefficient [38; 39], and \(\mu_{s}\) is a scattering coefficient. This is a value inverse proportional to the distance on which the collimated light flux will be reduced due to the scattering [40; 41]. \(f(\overrightarrow{\mathbf{s}},\overrightarrow{\mathbf{s}})\) is the scattering phase function derived from the appropriate scattering theory (e.g., Mie or [PERSON] theory), \(\mathrm{d}\Omega\prime\) is the solid angle spanning \(\overrightarrow{\mathbf{s}}^{\rightarrow^{\prime}}\), and \(c\) is the speed of the light in the surrounding medium.  
In other words, Equation (1) can be explained as follows: The resultant radiance along the direction \(\overrightarrow{\mathbf{s}}\) is a sum of the scattered radiance coming from all directions \(\overrightarrow{\mathbf{s}}^{\rightarrow^{\prime}}\) to the direction \(\overrightarrow{\mathbf{s}}\) and the radiance loss due to the absorption. The total extinction represented by the first right-side term of Equation (1) describes two processes that contribute to the total resultant radiance, namely the radiance loss due to the scattering process and the radiance loss due to the absorption process. Though the radiative transfer equation can be applied for a wide variety of media, the analytical solution of this equation is available for rather simple circumstances with different assumptions. Thus, techniques such as the method of path integrals, diffusion approximation, light scattering by Brownian particles [42], and the method of the small-angle approximation [43] were developed.  
Nevertheless, the techniques mentioned above are supposed to be used under different assumptions and simplifications. For example, the methods of light scattering by Brownian particles, as well as the diffusion approximation, only suit the small anisotropy factor-scattering media. The method of small-angle approximation does not consider the diffusively scattered light and cannot be applied to our task because the idea of the research is to analyze the influence of both quasi-ballistic and diffusive components of the scattered radiation. The method of path integrals has two variations: analytical and stochastic. Analytical method is basically the modified method of Brownian particles, whereas stochastic method is Monte Carlo method. As is known, Monte Carlo technique [44; 45] is the most versatile and widely used numerical technique that provides an approximate solution of the radiative transfer equation for the vast majority of conditions.