# Finite Element Method (FEM) Modeling of Laser-Tissue Interaction during Hair Removal
## 2 Materials and Methods
### Laser-Tissue Interaction  
The transport of photons in the tissue is the first of the two main physical processes occurring during laser hair removal. In the first step, we simulated the irradiation of each of our FEM models and, to that end, used the mesh-based Monte Carlo method (MMCM) [22] as an implementation algorithm for solving the transport equation. The MMCM algorithm assumed homogeneity of the optical properties of the tissue within each tetrahedral element and allowed for parallelization via several processor cores or a graphics card, which significantly accelerated the simulations [23]. Our software enabled varying the type, dimension, and direction of the laser beam. The optical properties of different tissue types used in simulations are summarized in Table 1. While a substantial amount of work has been carried out in determining the optical properties of human skin and hair [24; 25], it has been noted [26] that the tabulated values of optical properties may have a range of values and may be determined only for specific wavelengths; additionally, the absorption coefficients of epidermis and hair have proven to be particularly difficult to assess because they contain melanin, which exhibits pronounced inter-subject variability [27; 28; 29; 30; 31; 32]. For that reason, we have developed, verified, and piloted a protocol for in vivo measurement of absorption coefficients of epidermis and hair in human subjects, which is described in detail in Appendix A.  
The second modeled physical process is heat diffusion within the tissue. The absorbance calculated by MMCM for each tissue type and each FEM element was converted into a heat source (W/mm\({}^{3}\)). The calculated heat source was then fed into the heat diffusion equation, subject to the initial conditions, the boundary conditions between the tissue surfaces, and the convective thermal boundary condition with a constant heat transfer coefficient h, as specified in Appendix B [14]. Again, homogeneity of the thermal properties of the tissue within each tetrahedral FEM element was assumed. We solved the heat diffusion equation in the MATLAB software environment (R2019b) using the PDE toolbox. The \"createpde\" function was employed with the parameters \"thermal\" and \"transient\" to simulate the time-varying temperature distribution in a model. To simulate different therapeutic protocols, we varied the laser fluence and pulse duration, or forced cooling  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Tissue** & \(\mu_{a}\) [mm\({}^{-1}\)] & \(\mu_{a}\) [mm\({}^{-1}\)] & \(g\) & \(n\) \\ \hline Epidermis & 0.07 [1] & 26.1 [2] & 0.89 [3] & 1.42 [4] \\ Dermis & 0.038 [5] & 5.25 [2] & 0.72 [3] & 1.37 [4] \\ Subcutis & 0.008 [6] & 2.27 [6] & 0.78 [3] & 1.44 [7] \\ Hair & 0.56 [1] & 79.7 [8] & 0.9 [8] & 1.37 [9] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optical properties of different tissue types used in simulations along with references; \(\mu_{a}\) and \(\mu_{a}\) are absorption and scattering coefficients, respectively; \(n\) is the refractive index; and \(g\) is an anisotropy factor.  
Figure 3: Finite element method (FEM) Model #3 of skin and hair with an array of hair. (**A**) 3D representation of the model; (**B**) cross-section through the model center. An array of hair had the shape of a cross, with the central hair at x = y = 0 mm and each neighboring hair at 3 mm from the central one. All five hair shafts and follicles were identical to the ones presented in Figure 1.  
parameters. The thermal properties of the tissues used in the simulations are summarized in Table 2.