# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 1 Introduction  
Modern dentistry has developed numerous techniques allowing the complete restoration of the chewing function. Where a tooth is damaged due to trauma, the current trend is to adopt conservative techniques, and, only as extrema ratio, the tooth is removed. A generic conservative technique involves a restoration process of the tooth through two successive phases: (a) modeling with the bur (to eliminate the injured part) and (b) encapsulation on the machined tooth (abtment) of a prosthetic crown (dental capsule) using cement. The prosthetic restoration thus obtained can be subjected to breakage due to different causes: lack of retention, bad occlusions, caries, uncemented restorations, and other mechanical problems. [PERSON] et al. found that biological factors influence only 20% of failures, while mechanical (30%), biomechanical (27%), and esthetic (11%) reasons account for approximately 70%. The authors also demonstrated that the most common type of problem in fixed dental prostheses was mechanical failure (over 50%), which was also significantly associated with the presence of crown restorations for a period of 1-5 years of service. These failures were mostly due to insufficient tooth preparation [1]. Abtument preparation is one of the most important steps for the tooth restoration as highlighted in [2]. Moreover, as discussed in [3], the abtument height can influence several parts of the chewing systems. Therefore, a correct design of the shape and size of the abtument-prosthetic crown coupling, based on the assessments of the clinical case in question, is necessary in order to adequately counteract the destabilizing effect of any occlusal load [4; 5; 6; 7]. The abtument-crown system under working conditions is subjected to the destabilizing action of chewing loads (alternating and variable in both intensity and direction), which can cause partial or total loss of the cement with consequent loss of coupling [8; 9]. The design of fixed prosthetic crowns must satisfy the following mechanical principles: retention, resistance, and prevention of deformation; the abtument selection is an important criterion for the success and longevity of the prosthesis [10]. Retention refers to the ability of a preparation to prevent restoration displacements under the action of forces parallel to the positioning path (axial forces) [11; 12]. To design a good retention form, the magnitude of the axial forces must first be taken into account. A fundamental role is played by the geometry of the preparation and, particularly, by height and taper of the preparation [13; 14]. A greater height increases the retention due to a higher friction between the walls. Moreover, limiting the taper of the abtument effectively reduces the number of directions in which a crown can be moved. Theoretically, maximum retention is achieved if a tooth preparation has parallel walls [15]. However, the main problem is given by the lateral forces (horizontal and oblique), generated during chewing, which tend to dislodge the restoration through rotation around the gingival margin. The attitude of the preparation to oppose such rotation is called resistance. The analyzed preparation reduces the tensil value increasing stability in order to prevent rotation. They are called resistance areas and constitute the resistance form of the preparation. Resistance depends on the taper, diameter and height of the preparation [13; 14; 15]. In particular, it decreases with increasing taper or diameter and increases with increasing height of the preparation [6; 16]. The relationship between the height or diameter of the preparation and the resistance to displacement has been found to be approximately linear [13]. [PERSON]. [PERSON] proposed that the abtument of a fixed partial denture should have a minimum total occlusal convergence (TOC), also called taper, to ensure adequate retention of the prosthesis that will be made for the abtument, given the height of the latter [17]. Hereafter, TOC will be indicated as \(\alpha\).