Axial wall angulation for rotational resistance in a theoretical‐maxillary premolar model

Abstract Objectives The aim of this study was to determine the influence of short base lengths and supplemental grooves on surface area and rotational resistance in a simulated‐maxillary premolar. Materials and Methods Trigonometric calculations were done to determine the total surface area with and without supplemental grooves. Additional computations were done to determine the maximum wall angle needed to resist rotation displacement in a premolar‐sized model. Wall heights of 3.0, 4.0, and 5.0 mm were used in the surface area and rotational axis computations. The rotational axis was located on the lingual restoration margin to produce a buccal‐to‐lingual rotational displacement. Results Total surface area decreased with increasing four‐wall taper levels from 2° to 18° and decreasing preparation heights from 5 to 3 mm. Significant surface area improvements were found with the supplemental use of mesial and distal axial grooves compared with the same condition without grooves in all taper levels and preparation height categories. Resistance to rotational displacement was determined to occur at only at very low levels of opposing wall taper angles. The use of supplemental grooves on mesial and distal axial walls significantly improved both total surface area and rotational resistance. Conclusions The vertical wall taper angles, preparation heights, and supplemental grooves play a role in resistance form and restoration stability.

One of the most basic considerations that influences the maximal axial wall taper to provide rotational resistance is the vertical height of the tooth preparation. Taller tooth preparation height provides greater resistance to rotation at larger angles of taper compared with shorter tooth preparation height for the same tooth size (Bowley & Kieser, 2007). Surface area of the tooth preparation has also been shown to be indirectly associated with resistance to rotation (Bowley & Lai, 2007). As the axial wall taper angle increases, the total surface area decreases, presumably, reducing the area for luting agent-restoration interaction. Supplemental grooves have been recommended to improve poor or marginal resistance form (Goodacre et al., 2001) with one study demonstrating these adjuncts increase the surface area of the preparation (Bowley & Lai, 2007).
An additional modifier associated with tooth size is the influence of base width of the base; a recent investigation (Bowley et al., 2017) has shown that a larger distance from the resisting wall to the rotational axis requires significantly lower wall taper angulations for resistance compared with the single molar-sized tooth form. Lastly, the relative height of the preparation rotational axis has been shown to significantly influence preparation resistance to rotation. Two studies (Bowley et al., 2017;Bowley, Sun, & Barouch, 2004) have shown that shortening the height of the rotational axis relative to the height of the resisting wall requires much lesser limited taper angles compared with the same preparation in which the opposing finish line and the rotational axis are at the same, even height levels.
As cited above, vertical preparation height and base widths have been investigated in previous studies (Bowley et al., 2017;Bowley & Kieser, 2007) in the molar-and Fixed Partial Denture (FPD)-sized restorations. However, these factors have not been assessed in the smaller, rectangular-tooth form of maxillary premolars. The purpose of the present study is to determine the contribution of smaller tooth base size of the premolar model as well as vertical height to axial wall angulations needed to provide resistance to rotation.

| Simulated-premolar tooth form, geometric model
A geometric figure served as the simulated-tooth form in a theoretical experimental model system; the experimental-tooth form with axial wall preparations was a truncated pyramid, and the base lengths and widths approximated the size of a maxillary premolar. In the preoperative condition, the tooth form prior to axial wall preparation was represented by a rectangular cube. This experimental cubic tooth form was manipulated to simulate a crown preparation with four-angled vertical walls and a flat occlusal surface; the experimental-premolar tooth can be seen in Figure 1.

| Height categories and base widths
The horizontal base-width and base-length dimensions were 5 mm in mesial-distal width (M-D B 5 mm ) and 8 mm in bucco-lingual length (B-Li B 8 mm ). The model had three vertical height categories-3.0, 4.0, and 5.0 mm (H 3, 4, 5 mm )-with trial manipulations within the experimental model system. These three height categories, H 3, 4, 5 mm , served as an independent variable in this investigation. According to the literature (Goodacre et al., 2001), all three-tooth height categories, H 3, 4, 5 mm , used in this investigation would be considered acceptable height levels for this premolar-sized tooth with these dimensions.

| Axial wall taper categories
The rectangular cube had four levels of axial wall inclinations to simulate a tooth preparation with a narrowing of the occlusal surface as the axial wall angulations increased. The simulated preparation had four categories of axial wall inclination, 2 , 6 , 12 , 18 , in the axial walls mesial (M), distal (D), buccal (B), and lingual (Li). These levels of axial wall inclination (M,D,B,and Li 2 ,6 ,12 ,18 ) served as an independent variable throughout the investigation.

| α 1 Total surface area
Each of the angulation categories with four-axial wall inclinations transformed the rectangular cube into a truncated pyramid with known dimensions. A series of trigonometric analyses were conducted to determine the total surface area of the simulated preparations at each taper angulation, 2 , 6 , 12 , 18 , within each height category, H 3, 4, 5 mm . The calculation of the total surface area in square millimeters, four-axial walls and an occlusal surface, served as a dependent variable α 1 , formula derivations in molar-sized tooth model published in Bowley and Lai (2007). The total surface area data, α 1 , at each of the four-axial wall angulation categories in three height categories H 3, 4, 5 mm can be seen in Table 1.

| α 2 Surface area gain supplemental M-and Dgrooves
Additional surface area calculations, represented as α 2 values in square millimeters, were the surface area of two supplemental grooves in the M-and D-axial walls as a second dependent variable.
The two supplemental grooves were experimentally placed M-and Daxial walls in all four 2 , 6 , 12 , 18 taper categories in all three H 3, 4, 5 mm height categories. The total surface area of both supplemental grooves was determined by trigonometric methods. The final surface area gained was the total surface area of both grooves minus the occlusal and axial wall surface area lost in groove placement; these net gain values were α 2 data and can be seen in Table 2; α 2 formula derivations in the molar-sized tooth model have been published in Bowley and Lai (2007).

| α 3 Maximal buccal axial wall rotational resistance
The α 3 values, as the third dependent variable shown in Table 3, represented the trigonometric calculation of the maximum axial wall angulation needed to provide rotational resistance around the lingual axis.
The α 3 values were calculated according to the formula: These three rotational resistance values in were done for the truncated pyramid in each H-category without axial groove supplements; α 3 formula derivations have been published in Parker, Gunderson, Gardner, and Calverley (1988).

| α 4 Rotational resistance M-and D-grooves
The fourth dependent variable, α 4 , was the level of rotational resistance provided by the supplemental axial grooves; α 4 formula derivations have been published in Parker et al. (1988). The α 4 values for each H-category were calculated with the same formula as α 3 above but a shorter base length B-Li B 3.41 mm : T A B L E 1 Total surface area, as α 1 , four vertical walls and occlusal surface of simulated prepared premolar tooth form B 5 mm M-D × B 8 mm B-Li widths: Total surface area with increasing axial wall taper angulations compared with unaltered rectangular cube as α 1 in mm 2 at 2 , 6 , 12 , and 18 for categories H 3, 4, 5 mm

| Stepwise computations with trigonometric formula
The total surface area and two-groove area calculations can be seen in the stepwise formulas with illustrations in Appendix

| α 1 Total surface area
The simulated-maxillary premolar model demonstrated a decreasing total surface area α 1 values as preparation M-, D-, B-, and Li-wall tapers increase from 2 to 18 and as vertical preparation heights decreased from H 5.0 mm to H 3.0 mm , as can be seen in Table 1. Total surface area α 1 values of the uncut rectangular blocks in each height category were as follows: F I G U R E 2 Large and small cones related to calculation of the 172-tapered bur's diameter of the preparation at the occlusal surface level; known values are bur taper 6 convergence angle for 3 groove wall angulation and bur tip diameter 1.16 mm T A B L E 3 Premolar 5 mm M-D × 8 mm B-Li widths with maximal rotational resistance wall opposite rotational axis for three height categories H 3, 4, 5 mm (α 3 ) and two M-and D-grooves improvement (α 4 )

| α 2 Surface area gain two supplemental M-and D-grooves
The α 2 data (see Table 2) showed a net gain of total surface area in all taper angulations and all height groupings with two supplemental grooves; the two supplemental grooves were introduced to the M-and D-axial walls to the dimensions of a 172-tapered fissure bur. Each groove preparation produced an axial and occlusal surface area loss from the groove surface preparation procedure; these initial surface area losses were H 5 mm 2 as

| α 4 Two-groove rotational resistance
The placement of two supplemental grooves with the tapered bur this process would be expected to contribute to a longer restoration functional lifetime.
A summary of the various factors to influence restorations stability in function include total surface area, axial wall inclination, vertical preparation height, tooth base width, preparation supplemental adjuncts, luting agents, dentine or core materials, and masticatory forces. The current investigation looked at axial angulation, vertical height, base width, and grooves in the absence of these other factors.
The objective of theoretical rotational resistance form studies has been to maximize the preparation attributes to minimize or offset the effects of other negative factors such as masticatory forces of occlusion and luting agent function.
It is difficult to compare the current investigation to any of these studies due to use of luting agent, differing levels of axial wall angulation, variability of different tooth model modulus of elasticity, and so on. The potential importance of the contribution of the tooth's modulus of elasticity can be seen in two finite element analysis (FEA) studies (Bowley et al., 2013;Wiskott, Krebs, Scherrer, Botsis, & Belser, 1999); both studies revealed a rotational axis located midway between buccal and lingual axial walls below the restoration in the root area. This axis location in both studies was different than the current investigation and would be expected to be more favorable to restoration stability. Although neither investigation stressed the system to failure, the axis location is an important consideration.
In contrast to the above cited in vitro laboratory studies, two FEA studies have demonstrated that the restoration-tooth combination under angular load caused a bending of the system. The bending of the tooth-restoration system increased with increasing axial wall angulation levels due to decreasing amounts of tooth structure overall. Although this FEA investigation did not generate restoration failure, the loading level of 200 N was significantly higher than the 2 kg of the in vitro investigation. This consideration, tooth system rigidity and flexure under load, may be a significant factor and was not considered in the current investigation or the in vitro laboratory studies.
The most significant findings of the present investigation were the As has been shown in other investigations (Bowley et al., 2017;Bowley & Kieser, 2007), the present study's α 1 data showed decreases in total preparation surface area with increasing axial wall taper levels; however, the present investigation's differences between axial wall angulation categories at the same vertical height were more moderate compared with previous investigations (Bowley et al., 2017;Bowley & Kieser, 2007). The trigonometric analyses showed gradual reductions of surface area with each increment of taper increase from 2 to 18 , at the same vertical preparation heights. The α 1 data comparison between height categories, H 3, 4, 5 mm , revealed substantial differences with changes in vertical height.
As in previous investigations (Bowley et al., 2017;Bowley & Kieser, 2007), this independent variable, preparation height was a significant factor in preparation resistance form. Dependent variable α 1 , as the total surface area of a simulated tooth preparation, has been previously investigated in simulated molar-sized and FPD-sized tooth form as truncated pyramids with larger base widths (Bowley et al., 2017;Bowley & Kieser, 2007).
Dependent variable α 2 , the net gain of preparation surface area with groove supplementation, is a new factor in the literature and was derived from standard geometric and trigonometric calculations with cones, right triangles, and other components. Dependent variables α 1, 3, 4 were also determined with geometric and trigonometric methods to determine rotational resistance of the axial wall opposite the axis of rotation. The three dependent variables have appeared in the literature in other studies (Bowley & Kieser, 2007;Parker et al., 1988).

| CONCLUSIONS
The maxillary premolar tooth preparation has the smallest base widths of restorations on posterior teeth, but the preparation design features of vertical height, axial wall angulation, and groove supplementation can improve the rotational resistance form of this tooth form.