## 4 Discussion
The common issues found in these numerical studies [19; 20; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72;  
and displacements, and the use of linear elasticity and the isotropy approach may not be correct [2; 3; 9; 16; 48]. A study of linearity vs. non-linearity was conducted by [PERSON] et al. [32; 33] in PDL of upper incisor subjected to 0.2-1 N of intrusion and tipping, which employed the S1 and S3 (brittle failure) criteria. The authors reported that up to 20-50% less quantitative applied force is needed for non-linearity vs. linearity. However, the employed failure criterion was of brittle-like material, while PDL is a ductile-like material; thus, their reports have accuracy issues. The use of failure criteria, which are specially designed for non-homogenous materials (as Tresca), deals with the homogeneity/non-homogeneity issue. The main limit of an FEA numerical analysis is related to the fact that it cannot accurately reproduce clinical conditions. Clinically, there are no pure movements, but rather an association and combination; thus, the amount of stress displayed at tissular level could be smaller than the herein results. We foresee this limit and compensate through data interpretation (especially those close to the physiological limit). Nevertheless, the main advantage of numerical analysis is related to the fact that it can produce individual analyses of each tissular component (the only available method), while by changing the boundary conditions, it requires a small sample size. This is why most FEA studies [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44] employed an acceptable sample size of one (one patient, one model, few simulations). To obtain correct and valid results, we approached this issue by using a larger sample size of nine (nine patients, eighty-one models, and 810 simulations), which was found to be superior to other numerical studies. A limit could also be seen to the sample size of nine (i.e., nine patients). However, it must be emphasized that all of the previous numerical studies used a sample size of one (one patient with one FEA 3D model and only few simulations), while this study used 81 3D models and 810 numerical simulations. The FEA method allows for multiple changes in the physical properties and boundary conditions of the models, allowing for many simulations over the same models; thereby supplying reliable results even if a low number of models is used. A good example of these changes is represented in our study by the tissular reconstruction of missing tissues (bone and PDL) and by the reduction of 1 mm to simulate various levels of bone loss (small enough to be numerically quantifiable and clinically relevant, since 1 mm is considered to be clinically relevant in both periodontics and orthodontics). It must be emphasized that if a large amount of bone loss occurs, the chances of keeping the tooth's functionality reduces, and thus the viability of keeping it in the oral cavity. Most FEA analyses fasten the process by employing anatomical simplified models. Our approach was the manual reconstruction segmentation process (automated detection software missed some areas). Thus, our intact periodontium models had 5.06-6.05 million C3D4 tetrahedral elements, 0.97-1.07 million nodes, and a global element size of 0.08-0.116 mm, no error element, and only a limited number of elements. When compared with other FEA analyses, a lower number of elements and nodes and higher global element size were used, with influence over results accuracy: 142,305 elements [44], 23,565-32,812 elements [43], 30,838 nodes and 167,089 elements [42], 148,097 elements and 239,666 nodes [32; 33], 11,924 elements and 20,852 nodes [29], and higher global element size of 1.2 mm [43] and 0.25-1 mm [44]. To confirm numerical simulations, correct results must be compared and correlated with both physiological constants, with clinical data, and other studies.