# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 3 Analytical Study on the New Proposed Model
### Case Study  
One of the primary advantages of the proposed algorithm is its ability to analyze any dental geometry. In the specific case under consideration, a lower molar was analyzed, as this type of prosthetic restoration exhibits the highest likelihood of dislodgement when subjected to an external force parallel to the occlusal plane, as reported in the scientific literature [22]. The analysis of the proposed model reveals that, for fixed MD and VL values, increasing heights (h = 3, 4, 5 mm) result in significant variations as the taper changes. Referring to the results in Table 2, when MD = 8.5 mm and VL = 7.5 mm or MD = 7.5 mm and VL = 8.5 mm, it was observed that A\({}_{\text{ra}}\) decreases with increasing taper for all values of h. It can also be seen that, for values of h that are reduced relative to MD and VL and for high taper, the axial resistance contribution is zero. Figure 11 summarizes the percentage ratio trend in the areas as a function of abutment height, considering variations in taper for both cases: MD > VL (a) and MD < VL (b). Notably, the differing trends indicate that the MD < VL case yields greater axial resistance area values compared to the other case. Therefore, in this study, the MD < VL configuration was selected. For similar reasons, an abutment height of 5 mm was chosen. It should be noted that the axial resistance area does not provide uniform resistance. Specifically, as illustrated in Figure 12, the resistance contribution depends on the pressure exerted on the axial resistance area, which varies with the angle \(\varphi\) according to a cosine law. Consequently, the region perpendicular to the VL direction offers the most significant contribution to resistance. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Abutment Height [mm]**}} & \multicolumn{6}{c}{**Total Occlusal Convergence (TOC)**} \\ \hline \multicolumn{1}{c}{\multirow{2}{*}{**Abutment Height [mm]**}} & \multirow{2}{*}{**MD \textgreater{}VL**} & \multirow{2}{*}{\(6^{\circ}\)} & \multirow{2}{*}{\(12^{\circ}\)} & \multirow{2}{*}{\(18^{\circ}\)} & \multirow{2}{*}{\(24^{\circ}\)} & \multirow{2}{*}{\(30^{\circ}\)} & \multirow{2}{*}{\(36^{\circ}\)} \\ \cline{3-3} \cline{5-8}  & & & & & & & \\ \hline \multirow{4}{*}{3} & \(A_{ra}\) & 23.70 & 13.26 & 0 & 0 & 0 & 0 \\ \cline{2-8}  & \(A_{t}\) & 34.03 & 33.04 & 32.32 & 31.78 & 31.37 & 30.98 \\ \cline{2-8}  & \(\rho\) & 69.65 & 40.14 & 0 & 0  
In light of the above, the axial area lost due to the presence of the cusp corresponds to the portion of the axial resistance area that contributes the least to the overall resistance. Nevertheless, it was observed that the contribution of the axial wall to the overall resistance is greater than that of the cusp wall under both conditions (MD > VL and MD < VL). Consequently, a second algorithm was developed to specifically evaluate the cusp resistance area in the optimal scenario of MD < VL (Figure 13). The application of load on half of the prosthetic crown, inclined at 10\({}^{\circ}\) relative to the normal direction of the occlusal plane, represents the most common loading condition for a lower molar. This results from the ZA-8 of the mandible during typical chewing motions, which generates oblique forces with respect to the occlusal plane [29]. To select the optimal value for the \(\alpha\)-angle that maximizes the axial resistance area and aligns with the no-dislodging rotation described in Section 2 (case a), an angle of \(\alpha=12^{\circ}\)--slightly greater than the load inclination--was chosen. With \(\alpha\) being fixed, the algorithm performs calculations as the cusp wall slope, \(\gamma\), varies. However, the presence of the cusps must not affect the structural strength of the abutment. Therefore, a superior limit of 0.25 for the ratio (HR) between the heights of the cusp sulcus (h-h\({}_{c}\)) and the abutment (h) was set. Table 3 shows the values of the cusp resistance area as the angle \(\gamma\) varies, once \(\alpha=12^{\circ}\) and h = 5 mm were set.