## 4 Conclusions  
The presented 3D TMCE model is developed as a flexible and accurate approach for predicting light propagation in skin tissues. Body-fitted tetrahedra are generated to avoid photon reflection and refraction errors in the interface cells. A direct comparison shows that the results of TMCE are consistent with the energy deposition map in GMC. Figure 16: Effect of vessel cross section on energy deposition in the axis of the deep blood vessel at \(H_{\textit{chad}}\) = 150 \(\upmu\)m and \(\alpha_{0}\) = 30\({}^{\circ}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**GS/\(\upmu\)m**} & \multicolumn{3}{c}{\(H_{\textit{chad}}\) = 150 \(\upmu\)m} & \multicolumn{3}{c}{\(H_{\textit{chad}}\) = 250 \(\upmu\)m} & \multicolumn{3}{c}{\(H_{\textit{chad}}\) = 350 \(\upmu\)m} & \multicolumn{3}{c}{\(H_{\textit{chad}}\) = 450 \(\upmu\)m} \\ \cline{2-13}  & **WS** & **OS** & **Ratio** & **WS** & **OS** & **Ratio** & **WS** & **OS** & **Ratio** & **WS** & **OS** & **Ratio** \\ \hline
80 & — & — & — & — & 87 & 123 & 71\% & 69 & 99 & 70\% & 53 & 78 & 68\% \\
120 & 99 & 136 & 73\% & 75 & 107 & 70\% & 57 & 85 & 67\% & 44 & 67 & 66\% \\
160 & 90 & 127 & 71\% & 67 & 99 & 68\% & 53 & 80 & 66\% & 43 & 64 & 67\% \\
200 & 81 & 118 & 69\% & 62 & 95 & 65\% & 49 & 77 & 64\% & 41 & 60 & 68\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Energy deposition (J\(\cdot\)cm\({}^{-3}\)) in deep vessels with different \(H_{\textit{chad}}\) values and geometric shadows (GSs). The shading effect of two parallel blood vessels irradiated with 585 nm pulsed dye laser was investigated using the TMCE method. The results showed that the shading effect is more significant with an increase in shading angle \(\theta_{shad}\) or a decrease in vessel distance \(H_{shad}\). When \(\theta_{shad}\) = 90\({}^{\circ}\) and \(H_{shad}\) = 150 \(\upmu\)m, the energy deposition of deep vessels after shading decreases to 63% compared with that without shading, and the peak deposition is 80 J/cm\({}^{3}\), which is lower than that of the shallow vessel. The findings show that the penetration depth of a 585 nm laser is shallow and photon deposition in deep vessels is insufficient. A deeper penetration laser wavelength (e.g., 1064 nm Nd:YAG laser) is suggested to treat deep PWS vessels. This study on the shading effect of cross-bridge blood vessels shows that the cross angle, vessel distance, and GS of vessels determine the peak characteristics of photon deposition in the deep vessel by affecting the relative deposition of collimated and diffused light. In particular, the axis distribution of energy deposition of the deep vessel changes from a gentle single peak to a steep double peak due to an increase in shading angle and a decrease in vessel distance and vessel GS. This condition benefits the injury of deep vessels. By contrast, an increase in vessel distance and GS shades and attenuates a portion of collimated light, making the treatment of deep and shallow vessels using only one laser pulse difficult. Therefore, two pulses should be considered to damage deep and shallow vascular lesions, resulting in an increase in therapy expense and time. The numerical results of this work can be used for individualized and precise treatment based on the morphology and embedding characteristics of vascular lesions. Conceptualization, H.J. and B.C. ; methodology, H.J. ; software, H.J. ; validation, D.L; formal analysis, B.C. ; investigation, H.J. ; resources, B.C. ; data curation, H.J. ; writing--original draft preparation, H.J. ; writing--review and editing, B.C. ; visualization, D.L. ; supervision, B.C. ; project administration, H.J. ; funding acquisition, H.J. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (51727811), Natural Science Foundation of Zhejiang Province (LZ15E090002) and Science Foundation of Zhejiang Sci-Tech University (ZSTU) (19022101-Y, 11130231711926). Not applicable. Data available upon request.