# Finite Element Analysis of Mandibular Anterior Teeth with Healthy, but Reduced Periodontium
## 2 Materials and Methods
### Stages of periodontal ligament 3D modeling  
1. In the first stage, the 3D model with the dimensions obtained after copying the contours from the CBCT was scaled down to the real dimensions. In these contours, each pixel in the CBCT image represents one millimeter in the Catia V5R20 program. The resolution of the introduced images was 1920 \(\times\) 1080, so that each section was much more enlarged compared to reality. In order to convert the model to a realistic size, the real length was measured on the CBCT and then it was determined in the virtual model of the Catia V5R20 length. This resulted in a ratio of scaling factor F\({}_{1}\) = 0.05309038. This scaling factor was applied to the 3D model in order to ensure that the structure composed of the teeth, bone and PDL ligament reflected the real size. This factor was applied in turn to each component element (bone, teeth, periodontal ligament), and then the assembly was performed again in the _Assembly Design_ module. The overall positioning was not altered because the relative positions to the coordinate system were also taken into account when resizing. After scaling, the appearance does not change, only the dimensions. 2. In the second stage, the external dimensions of the ligament were built based on the geometry of the tooth. This scaling allowed the use of the actual dimensions of the ligament and the next step was the modeling of the PDL. For the construction of the ligament, an artifice frequently used in 3D modeling was used. Starting from the shape of the tooth, scaling was applied to obtain a duplicate of the tooth (its shape) at a distance of 0.4 mm from the exterior. For this, the maximum diameter of the tooth was measured in different places. Subsequently, the average value was calculated and the thickness of the ligament was added twice, because it appears on both sides of the tooth. This represented the outer diameter of the ligament. Following this, the ratio between the tooth's diameter and the ligament's diameter was made. The value obtained, F\({}_{2}\) = 1.023, was used as a scaling factor to obtain the duplicate of the tooth, necessary for the construction of the ligament. In the Catia V5 R20 program, the \"_scales_\" function was used. Two parameters are needed for this function: the scale factor, 1.023, and the invariant point of scaling, meaning the scaling center. In our case, the central point of the tooth was chosen. It was built using a line that connected the tip with the apex of the tooth. On this line, the middle of the segment was highlighted and it was selected as an invariant point (reference) for scaling. 3. In the last stage, the tooth geometry was displaced from the obtained ligament, and later, the geometry of the ligament was displaced from the mandibular bone. 4. The ligaments for the 3 teeth were similar for the two incisors. The thickness of the ligament resulting from this procedure was not constant, but, on average, it was around 0.4 mm. We believe that this approach and construction of the tooth-ligament-bone assembly is the one that best approximates the biological nature of tissues. The obtained periodontal ligaments of the three teeth are illustrated in Figure 7.  
After obtaining the geometric model of the assembly, we performed the finite element analysis to virtually test the response to various requests. The geometric model was discretized, and, with the help of the equations that describe the re-united phenomena in the form of a basic analytical model, the FEA was performed. The module used was Catia V5 R20--Generative Structural Analysis. The basic analytical model consists of a set of equations that describe the phenomenon studied and the behavior of the material under the action of external stresses. To these, boundary conditions were added. The boundary conditions describe the body's interaction with the environment. When the field sizes are variable over time, the analytical model must also include their initial conditions, meaning their condition at the beginning of the analysis.