## 4 Discussion  
The DST results showed the \(\mu_{s}^{\prime}\)-values range from 11 to 70 cm\({}^{-1}\) for the alumina suspension at the volume fraction range 1-20% in Figure 4. These values are within the range for biological tissue volumes [51]. Meanwhile, the \(g\)-values are almost zero, different from a typical value of biological tissue volumes (\(g\lesssim 0.8\)) [51]. These results suggest that the alumina suspension is an appropriate tissue phantom modeling the reduced scattering coefficient rather than the scattering coefficient and anisotropy factor. Our comparative study of the RTE calculations with the PDE calculations showed the diffusion approximation holds for light propagation, meaning that \(\mu_{s}^{\prime}\) primarily characterizes light propagation. This result supports our previous suggestion for the applicability of the alumina suspension. According to the previous research works of time-dependent light propagation [20; 21; 22], the characteristic length where the diffusion approximation holds has been evaluated at approximately \(10/\mu_{t}^{\prime}\) with \(\mu_{t}^{\prime}=\mu_{s}^{\prime}+\mu_{a}\). In our current conditions, the characteristic lengths are given as 0.89 cm for the volume fraction of 1% and 0.15 cm for 20%, respectively. As shown in Figure 8a,b, the RTE results agree with the PDE results inside the distance of the characteristic length from the boundaries. Hence, the current results for steady-state light propagation are consistent with those for time-dependent light propagation. Our numerical study suggests that the PDE-DST is useful as a light propagation model for the alumina suspensions rather than the RTE-DST because the diffusion approximation is almost valid and the PDE calculations are faster than the RTE calculations. The numerical investigation of light propagation in the alumina suspension (liquid phantom) is essential for developing near-infrared spectroscopy and imaging toward their practical application, such as non-invasive cancer detection in the human body. Moreover, the RTE-DST and PDE-DST, developed here in the alumina suspensions, can be applied to other colloidal suspensions and research fields. For example, a titanium dioxide suspension is one of the photocatalytic systems. Because the evaluation of light propagation in a photocatalytic system is significant, the investigation in this paper probably provides valuable information in the research field of photocatalysis. Based on the following two reasons, we concluded that PDE-DST is beneficial over the RTE-DST in the alumina suspension at the volume fraction range of 1-20%, meaning the diffusion approximation is valid. The other is because the computational loads of the PDE-DST are smaller, and the numerical treatment is simpler. The RTE-DST provides accurate computation of light propagation but requires complicated treatments of the phase function using the DST and RMBC. As shown in Figure 8a,b, the difference between the numerical results mostly comes from the differences at the boundary regions (\(z=0.0\) cm and 2.5 cm). Hence, developing an efficient and accurate treatment of RMBC to a boundary condition of the PDE is crucial for future work. In future work, examining light propagation for the colloidal suspensions with larger particle sizes will be challenging. The scattering properties become larger with an increasing  
Figure 8: **(a,b)** Fluence rates calculated from the RTE and PDE at different positions. The RDs in the numerical results between the RTE and PDE are plotted in each figure’s bottom. (c) MAPD, \(D_{\Phi}(N,R;N,P)\), between the RTE and PDE at different volume fractions. The other details are the same as Figure 6.  
particle size. The large \(\mu_{s}\)-values indicate the diffusion approximation probably holds at a broader condition, such as the volume fraction compared to the suspension with a smaller particle size. Meanwhile, this result requires a finer spatial step size for the numerical calculation of the RTE-DST.