## 2 Materials and Methods
### Validation of TMCE  
To validate the TMCE algorithm, we initially compared TMCE and TMC with GMC [10] by recording the energy deposition in the skin model (Figure 5), which is composed of two layers of media: 60 \(\upmu\)m-thick epidermis and 940 \(\upmu\)m-thick dermis underneath. In the dermis, a single cylindrical PWS vessel with a diameter of 120 \(\upmu\)m is buried at 250 \(\upmu\)m. The laser spot with top hat profile was irradiated at the center of the incident plane with a diameter of 1.0 mm and energy fluence of 1 J/cm\({}^{2}\). A 1.4 mm \(\times\) 1.4 mm \(\times\) 1.0 mm domain was used because it was sufficiently large and no evident fluence variation occurred at the tissue boundary when the computation space was expanded. The total photon number was 10\({}^{8}\). In the model, melanin is assumed to be homogeneously distributed throughout the epidermis. The epidermal absorption coefficient, \(\mu_{a,epi}\), is calculated using the following equations [20]:  
\[\mu_{a,epi}=f_{mel}\mu_{a,mel}+(1-f_{mel})\mu_{a,base} \tag{4}\]  
\[\mu_{a,mel}=6.6\times 10^{10}\;\text{mm}^{-1}\bigg{(}\frac{\lambda}{\text{nm}} \bigg{)}^{-3.33} \tag{5}\]  
\[\mu_{a,base}=0.0244\;\text{mm}^{-1}+8.53\;\text{mm}^{-1}\exp\bigg{(}-\frac{ \lambda-154\;\text{nm}}{66.2\;\text{nm}}\bigg{)} \tag{6}\]  
where \(\mu_{a,mel}\) is the absorption coefficient of melanosome, and \(\mu_{a,base}\) is the average baseline value for the skin. In this study, the laser wavelength \(\lambda\) is 585 nm and \(f_{mel}\) is 5%. These values correspond to Asian skin. Other optical properties corresponding to the wavelength of 585 nm was used similar with that in Table 2 in [15]. Figure 5: Diagram of the skin model with single port-wine stain (PWS) vessel. In the TMCE and TMC methods, a tetrahedron mesh is generated using the commercial software GAMBIT 2.4. Body-fitted tetrahedra are generated along the vessel-tissue interface. The total number of cells is 144,783. In the simulation, energy deposition is recorded with respect to depth at the beam center. The energy deposited in the column of cubes directly below the center of the beam (0.7 mm) via GMC, TMCE, and TMC is illustrated in Figure 6. The line graph includes four parts: epidermis, dermis, blood vessel, and dermis (Figure 6). The result of TMCE agrees well with that of GMC. By contrast, errors appear at the epidermis-dermis and vessel-dermis boundaries in TMC. This finding is attributed to the accurate boundary description of TMCE. To further compare the accuracy of the three methods, we assessed photon deposition distribution in the central cross-section of the vascular region in the blood vessel (Figure 7). Figure 7 shows that in GMC the photon that transmits to the vessel from the top half should be mostly deposited near the vessel wall, forming the crescent of high deposition region. The results computed by TMCE show a more similar crescent shape to Figure 7 than that by TMC. Deposition errors in the annular cylinder close to the inner wall of the vessel was segmented by \(\Delta r\) of 5 \(\upmu\)m and \(\theta\) of 1\({}^{\circ}\) (Figure 8a). The deposition in the 1\({}^{\circ}\) annular cylinder computed by GMC, TMC, and TMCE was collected. The relative error \(K_{\text{TMC}}\) and \(K_{\text{TMCE}}\) of TMC, and TMCE are calculated with GMC as a benchmark, as follows:  
\[K_{\text{TMC}}=\left|\frac{E_{TMC}-E_{GMC}}{E_{GMC}}\right|\times 100\%,\ K_{ TMCE}=\left|\frac{E_{TMCE}-E_{GMC}}{E_{GMC}}\right|\times 100\% \tag{7}\]  
where \(E\) is the photon deposition in each 1\({}^{\circ}\) annular cylinder. Figure 8b presents the relative error distribution in each 1\({}^{\circ}\) annular cylinder. We can see a \(K_{\text{TMC}}\) larger than 8% was observed at 0\({}^{\circ}\)-60\({}^{\circ}\) and 330\({}^{\circ}\)-360\({}^{\circ}\), whereas \(K_{\text{TMCE}}\) maintained less than 7.5%, not sensitive to angles. Figure 6: Line graph of energy deposition vs. depth at the beam center for simulations of skin with a single blood vessel.