# Finite Element Method (FEM) Modeling of Laser-Tissue Interaction during Hair Removal
## 2 Materials and Methods  
### FEM (Finite Element Method) Modelling  
We developed our skin and hair modeling for the Nd:YAG laser using the finite element method (FEM), building upon previous research [14; 15]. To ensure accurate results, we adapted the model's geometric shape to a cylinder instead of the conventional rectangular shape with sharp corners. This modification reduced the need for a larger number of volumetric elements while maintaining comparable accuracy. SALOME (v7) software was used to create mesh models. The process began by generating the geometry of the mesh, with careful consideration given to achieving a seamless connection between the surfaces to eliminate any gaps. Subsequently, the NETGEN [16] algorithm was employed to establish a surface mesh, which served as a foundation for generating the volumetric mesh.  
Overall, we created three types of models: FEM Model #1, FEM Model #2, and FEM Model #3.  
The FEM Model #1 had a radius of 10 mm, a height of 8.5 mm, and included the epidermis [14; 17], dermis [14; 18], subcuits [19], hair shaft [20], and follicle [21], as shown in Figure 1. The thicknesses of the epidermis, dermis, and subcuits were 0.07 mm, 3.43 mm, and 5 mm, respectively. The hair was placed at the center of the model cylinder. The hair shaft was modelled as a tiny cylinder with a radius of 0.03 mm and a length of 2.5 mm,oriented along the long axis of the larger cylinder, while the follicle had a radius of 0.1 mm and was located at a depth of 2.5 mm [19]. In simulations, we applied laser beams with a radius of no more than 4 mm; therefore, the width of the model was deemed sufficient to account for the multiple scatterings of photons [14].  
Hair typically exits the skin at an angle, and for that purpose, FEM Model #2 was built, identical to FEM Model #1, with the only difference being that the hair shaft grew out of the follicle at different angles. In Figure 2, an example is depicted for the angle of 45\({}^{\circ}\). In that specific case, the hair was 3.53 mm long, exiting the skin at the coordinates x = 2.5 mm and y = 0 mm (instead of x = 0 mm and y = 0 mm when considering straight hair). The exit point of the hair from the skin was thus still covered by a laser beam with a radius of 4 mm.  
The surface density of hair on human skin has on average 14-32 hair follicles per square centimeter [20], which we have taken into account with the even more realistic FEM Model #3, with an array of hair arranged in the shape of a cross (Figure 3); this arrangement corresponded to a density of 18 follicles per square centimeter.  
Figure 1: Finite element method (FEM) Model #1 of skin and hair. The model includes the epidermis, dermis, subcutils, hair shaft, and follicle and has a radius of 10 mm and a height of 8.5 mm; the thickness of the epidermis, dermis, and subcutils is 0.07 mm, 3.43 mm, and 5 mm, respectively; the cylindrical hair shaft has a radius of 0.03 mm and a length of 2.5 mm. The follicle has a radius of 0.1 mm and is located at a depth of 2.5 mm. (**A**) 3D representation of the whole model; (**B**) cross-section through the model center; (**C**) detailed view of the hair shaft and follicle.  
Figure 2: Finite element method (FEM) Model #2 of skin and hair with the hair exiting the skin at an angle of 45\({}^{\circ}\). (**A**) Cross-section through the model center; (**B**) detailed view of the hair shaft and follicle.