# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 3 Analytical Study on the New Proposed Model  
Once the morphological parameters provided by the algorithm are identified, the authors decided to carry out in vitro and numerical tests on a 10/1 scale to validate the results obtained with the algorithm.  
In the scientific literature, there are numerous mathematical methods for determining the resistance areas of a molar prosthetic abutment, assuming that the prosthetic abutment has the exact shape of a circular conical frustum [4; 5; 6; 17; 20; 28]. However, in clinical practice, this condition occurs only in a small number of cases and mainly where, due to dental disease or trauma, the amount of surviving dental material is minimal. In most cases, conversely, the shape given to the prosthetic restoration is similar to an elliptical conical frustum. The geometry is more complex than the one typically analyzed in theoretical studies, and depending on the tooth being treated, it is oriented according to the specific mesial-distal and vestibular-lingual references.  
This paper proposes an innovative design for the prosthetic abutment that optimally aligns with the dental reconstruction performed by the clinician. The proposed geometric configuration represents a significant advancement, combining two lateral circular conical frustums with a central rectangular conical frustum, following the geometric hypothesis of Figure 5, as illustrated in Figures 6-8.  
The analytical study was carried out considering, in addition to the geometric characteristics of the different teeth, the direction and point of application of the masticatory forces as a function of the position of the tooth in the arch.  
This novel design aims to enhance both functional and structural integration, ensuring superior performance in prosthetic applications.  
A novel algorithm capable of determining the axial resistance area by requesting geometric parameters as input data were developed. These parameters were the height of abutment (\(h\)), mesial-distal length (\(MD\)), vestibular-lingual length (\(VL\)), and \(\alpha\). The \(\alpha\) values taken into account were \(6^{\circ}\), \(12^{\circ}\), \(18^{\circ}\), \(24^{\circ}\), \(30^{\circ}\), and \(36^{\circ}\).  
This algorithm provides two distinct calculation procedures depending on the considered geometry: MD \(>\) VL or MD \(<\) VL (Figure 9).  
Once the geometry has been chosen, the algorithm is able to compare \(h\) and the height (\(h_{2}\)) derived from the intersection of the base circumference with the axial wall. Only when \(h>h_{2}\), an axial resistance area is obtained; otherwise, it is zero. The condition \(h<h_{2}\) occurs when \(MD>>h\). Then, the algorithm proceeds to the calculation of the effective axial resistance area (\(A_{m}\)) and the total axial area (\(A_{t}\)), also providing their percentage ratio (\(\rho\)). The calculation is repeated for all the default \(\alpha\) values, so as to evaluate the consequent variations in \(A_{t}\) for a given type of preparation. Figure 10 illustrates the key steps of the proposed general algorithm. However, as detailed in the case study, additional steps are necessary for its full implementation, depending on whether the condition MD \(\geq\) VL or MD \(<\) VL is met.