## 3 Numerical Results
### Fluence Rate  
We investigated the fluence rates at different positions on the line parallel to the \(z\)-axis with \((x,y)=(0,0)\) inside the medium (blue dash line in Figure 1a). Here, the distances between source and calculating positions (SC distances) range from 0.05 to 2.50 cm at equal intervals of 0.05 cm. Before discussion of the RTE results, we investigated the PDE results. In Figure 6a,b, we compared the numerical solutions (NSs) of the PDE with the analytical solutions (ASs) of the PDE for the infinite slab (Equation (4)) with the boundaries at \(z=0.0\) cm and 2.5 cm for different volume fractions of 1% and 20%. The results are normalized by the values at the SC distance of 1.25 cm. The numerical results nicely agree with the analytical solutions at all the SC distances, verifying our numerical calculations of the PDE. The agreement suggests that the boundaries except those at \(z=0.0\) cm and 2.5 cm less influence the numerical results on the line inside the medium, because the NSs are for the cubic medium while the ASs for the infinite slab. In Figure 6a,b, we plot the AS for the semi-infinite medium at the \(z=0\) boundary, obtained from Equation (4) with \(s=0\). We observe differences in the ASs between the infinite slab and semi-infinite medium in the SC distances from 2.3 cm to 2.5 cm due to the boundary effects at \(z=2.5\) cm. We plot the relative differences (RDs) between the NS and AS for the infinite slab at each figure's bottom. The small RDs are observed near the boundaries of \(z=0.0\) cm and 2.5 cm. We evaluated the differences in the normalized fluence rates between the NS and AS for the infinite slab by the mean absolute percentage difference (MAPD):  
\[D_{\Phi}(N,P;A,P)=\text{Mean}\left[\left|\frac{\dot{\Phi}^{c}_{AS,PDE}-\dot{ \Phi}^{c}_{NS,PDE}}{\dot{\Phi}^{c}_{NS,PDE}}\right|\right], \tag{8}\]  
where \(\dot{\Phi}^{c}_{AS,PDE}\) and \(\dot{\Phi}^{c}_{NS,PDE}\) represent the AS and NS at the \(c\)-th calculating point, respectively. As shown in Figure 6c, the MAPDs are less than 3% at all the volume fractions. Next, we investigated the diffusion approximation at the boundary by comparing the numerical results for the RTE with the AS (Equation (6)) for the infinite slab at different volume fractions. The AS of the RTE (Equation (6)) is based on the diffusion approximation to the boundary region. Meanwhile, the numerical calculations of the RTE treat the RMBC. As shown in Figure 7a,b, the numerical result nicely agrees with the AS except the boundary region (the SC distances from 2.3 cm to 2.5 cm) at the volume fraction of 1%. The agreements suggest the verification of our numerical calculations of the RTE. The difference at the boundary region (\(z=2.5\) cm) in Figure 7a is probably due to the fact that the diffusion approximation is invalid at the volume fraction of 1%. We plot the AS of the RTE for the semi-infinite medium at the \(z=0\) boundary obtained from Equation (6) with \(s=0\). Similar to the AS of the PDE, we observe the difference between the infinite slab and semi-infinite medium in the SC distance region from 2.3 cm to 2.5 cm. As shown in Figure 7c, the MAPDs, \(D_{\Phi}(N,R;A,R)\), between the NS of the RTE and AS of the RTE for the infinite slab are less than 5% except for the case for the volume fraction of 1%. The small MAPD-values  
Figure 5: Normalized phase functions calculated from the DST with the scattering angle at different volume fractions of (**a**) 1% (\(g=0.046\)) and (**b**) 20% (\(g=0.003\)). The results using the Rayleigh theory and the Henyey–Greenstein type are compared. indicate that the diffusion approximation is valid at the boundary at the volume fraction range higher than 1%. Finally, we investigated the diffusion approximation inside and at the boundary by comparing the numerical results for the RTE with those for the PDE. Figure 8a shows that at the volume fraction of 1%, the numerical results agree with each other except the boundary regions at \(z=0.0\) cm and 2.5 cm.