# Finite Element Method (FEM) Modeling of Laser-Tissue Interaction during Hair Removal
## 3 Results
### Temperature and Thermal Damage Distributions--Protocol #1  
Figure 6 shows temperature distribution maps for FEM Model #1 immediately after irradiation for all three combinations of fluence and pulse duration, both without and with forced cooling. As expected, the maximal temperature (Table 3) occurred at the very tip of the hair shaft due to the hair having a larger absorption coefficient than the epidermis and dermis. The largest gradient was observed along the hair shaft. Interestingly, the temperature in the hair follicle was higher than in the hair shaft just above it, which can be attributed to the hair follicle absorbing a much larger number of scattered photons than the tiny hair shaft. All these observations are congruent with those of [PERSON] et al. (2014). The combination of fluence and pulse duration had a non-intuitive effect, as the maximum temperature increased with increasing fluence but decreased with increasing pulse duration. The maximum temperature at the tip of the hair shaft was thus higher for a fluence of 50 J/cm\({}^{2}\) and a pulse duration of 25 ms than for a fluence of 60 J/cm\({}^{2}\) and a pulse duration of 50 ms. This happened because the difference in irradiation power normalized to the surface area of the laser beam (2000 W/cm\({}^{2}\) in the first case and 1200 W/cm\({}^{2}\) in the second case) was not compensated by delivering 17% more energy in twice the amount of time in the second case. The temperature was highest when irradiating the model with a fluence of 80 J/cm\({}^{2}\) and a pulse duration of 50 ms. In this case, the irradiation power, normalized  
Figure 5: Dependence of the average temperature of the hair follicle immediately after the first, second, and third pulses, respectively, on the number of tetrahedral FEM elements. to the surface of the laser beam, equaled 1600 W/cm\({}^{2}\), but the higher energy and longer pulse were sufficient to deliver the highest temperature among the three combinations. These observations were corroborated by the maximum temperatures calculated within the follicle (Table 3). We observed that cooling had no impact on the follicle temperature. Figure 7 shows distribution maps of thermal damage \(\Omega\) for FEM Model #1 immediately after irradiation for three combinations of fluence and pulse duration, for modes without forced cooling and with forced cooling. As expected, the area of thermal necrosis (where \(\Omega\) exceeds 1) increased with increasing fluence, while cooling led to the area of necrosis in the epidermis being substantially smaller compared to that without cooling, particularly for the fluence of 50 J/cm\({}^{2}\) with a pulse duration of 25 ms and the fluence of 60 J/cm\({}^{2}\) with a pulse duration of 50 ms. It is notable that the area of thermal necrosis along the hair shaft toward the hair follicle was not connected, reflecting the temperature gradient along the hair shaft and the fact that the temperature was higher in the follicle than in the hair shaft just above it (Figure 6). \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{5}{c}{**Maximum Follicle Temperature after Pulse [\({}^{\circ}\)C]**} \\ \hline Fluence of 50 J/cm\({}^{2}\) & Fluence of 60 J/cm\({}^{2}\) & Fluence of 80 J/cm\({}^{2}\) \\ Pulse Duration of 25 ms & Pulse Duration of 50 ms & Pulse Duration of 50 ms \\ \hline No Cooling & Cooling & No Cooling & Cooling & No Cooling & Cooling \\ \hline
79.3 & 79.3 & 73.6 & 73.6 & 86.8 & 86.8 \\ \hline \end{tabular}
\end{table} 
Table 3: Maximum follicle temperature in FEM Model #1 immediately after irradiation for three combinations of fluence and pulse duration, for modes without forced cooling and with forced cooling. Figure 6: Temperature distribution maps shown in the \(y=0\) plane for FEM Model #1 immediately after irradiation for three combinations of fluence, pulse duration, and modes without forced cooling and with forced cooling.