Optimum pure intrusion of a mandibular canine with the segmented arch in lingual orthodontics

Abstract

Background:

Approximately 50% patients with a deep bite possess anatomically extruded mandibular canines.

Objective:

The objective of this study was to specify an optimum toe (Θ) of the vertical segment of a cantilever from the distal aspect to achieve pure intrusion of a mandibular canine with the segmented arch in lingual orthodontics (LiO).

Methods:

The geometrical model of a mandibular canine was developed to compute the required values of Θ (positive values – toe-in, negative values – toe-out). Different positions of the cantilever attachment on the lingual bracket top (occlusal) surface were considered according to different heights (h) from the cusp of a canine i.e. 2, 3, 4 and 5 mm in each case of horizontal distance (x) from canine surface i.e. 2 and 3 mm. In $x = 2 mm$ case, the required values of Θ at $h = 2, 3, 4 and 5 mm$ were $- 5$ °, 0°, $+ 6$ ° and $+ 11$ ° respectively. While, in $x = 3 mm$ case, the required values of Θ at $h = 2, 3, 4 and 5 mm$ were 0°, $+ 5$ °, $+ 12$ ° and $+ 17$ ° respectively.

Results:

In finite element analysis (FEA), the computed values of Θ were verified by displaying the results of a mandibular canine movement in the form of vectors of nodal displacements along with undeformed and deformed models.

Conclusions:

In the cases of $x = 2 mm and 3 mm$ , no toe angles were required at heights $h = 3 mm and 2 mm$ respectively. Hence, these can be considered as better positions of the cantilever attachment than others.

Keywords

Mandibular canine pure intrusion segmented arch LiO geometrical model FEA

1. Introduction

The continuous orthodontic force system leads to tooth movement owing to remodeling of the alveolar bone structure. An optimal force system needs to be devised to predict and control the tooth movement [1–4]. To simplify the required force system, an accurate application of basic biomechanics is necessary to apply the best orthodontic treatment. It improves the efficiency of an orthodontic appliance by shortening the treatment time and avoiding any side effects [1].

Approximately 50% patients with a deep bite possess anatomically extruded mandibular canines [5,6]. Simultaneous intrusion of six mandibular anterior teeth is considered to be undesirable as it affects the anchorage of posterior teeth. Hence, the intrusion of mandibular canines should be achieved with the segmented arch to level the curve of spee [7].

Very few studies have stated the methods to achieve intrusion of a single canine. Ricketts et al. [8] used the utility arch as a stabilization arch and then tied an elastic band to a canine bracket from step down bypass segment in the utility arch. But, this method failed to explain the controlling of bucco-lingual tooth inclination. Burstone et al. [7] and Marcotte et al. [9] suggested that a cantilever should be used from auxiliary tube of the first molar to a canine bracket slot. In their study, the cantilever was inserted into the vertical (occlusal) slot of a canine bracket. But, insertion of a cantilever in a bracket slot produces an undesirable couple at bracket slot in addition to intrusion force. Caballero et al. [10] attached the cantilever arm on the top (occlusal) surface of a labial canine bracket to prevent the undesirable couple at canine bracket. As their study was focused on labial orthodontics (LaO), a significant amount of retraction (labiolingual) force was also applied to the canine bracket in addition to intrusion force.

In case of intrusive (vertical) forces, centre of resistance ( $C_{res}$ ) of the tooth is closer in lingual orthodontics (LiO) from point of force application ( $P_{f}$ ) on the bracket. Hence, lesser moment values are required in LiO keeping them more within the biological limit [11]. In LiO, $P_{f}$ is generally lingual, but sometimes may be labial with respect to $C_{res}$ of the tooth. Hence, depending on the position of $P_{f}$ , linguolabial or labiolingual component of force is required to apply in addition to intrusion force to pass their resultant force vector through $C_{res}$ . But, in LiO, the required amount of linguolabial or labiolingual force is very less as compared to intrusion force. Thus, in LiO, almost pure intrusion can be achieved while maintaining the bucco-lingual tooth inclination [11]. Therefore, our study is focused on the optimum pure intrusion of a mandibular canine with the segmented arch in LiO.

The aim of this study was to specify an optimum toe angle (toe-in or toe-out) of the vertical segment of a cantilever from the distal aspect to achieve pure intrusion of a mandibular canine with the segmented arch in LiO. Different positions of the cantilever attachment on lingual bracket top (occlusal) surface were considered according to different heights (h) from the cusp of a canine i.e. 2 mm, 3 mm, 4 mm and 5 mm in each case of horizontal distance (x) from canine surface i.e. 2 mm and 3 mm. For this purpose, the geometrical model of a mandibular canine was developed and values of optimum toe angles were computed numerically for all the positions. Now, finite element analysis (FEA) has been established as an accurate and cost-effective tool to simulate and analyze tooth structures [12–15]. Hence, determined optimum values of toe angles were verified with FEA.

Fig. 1.

Appliance setup for the pure intrusion of a mandibular canine with the segmented arch in LiO. (A) Lingual aspect showing three segments of a cantilever and posterior teeth anchorage unit. (B) Toe-in and toe-out of vertical segment of a cantilever from distal aspect of a mandibular canine.

2. Materials and methods

Generally, a cantilever has three segments [10] as shown in Fig. 1(A). The distal segment of a cantilever ( $W X$ ) fits into the first molar auxiliary tube and is extended mesially up to the area surrounding the interproximal contact point between canine and first premolar. At this point (X), the cantilever is bent by 90° occlusally as observed from the lingual aspect of a mandibular canine. This forms a vertical segment ( $X Y$ ) which ended up to the level of top (occlusal) surface of a canine bracket. Finally, another 90° bend is given to make a final segment ( $Y Z$ ) which is parallel to the occlusal plane of a mandibular canine. This final segment is tied to the top surface of a mandibular canine bracket and acts as a cantilever attachment point ( $P_{f}$ ). An anchorage is provided to the posterior teeth (first premolar to second molar).

From the distal aspect of a mandibular canine, vertical segment ( $X Y$ ) also makes an angle of 90° with the occlusal plane. This study suggests that if this vertical segment has given a toe i.e. toe-in (towards the lingual surface of a canine) or toe-out (away from the lingual surface of a canine) as shown in Fig. 1(B), then force vector by a cantilever will pass along this vertical segment and eventually through $C_{res}$ of a mandibular canine. Thus, the pure intrusion of a mandibular canine will be achieved with the segmented arch. To determine the required toe angle of a vertical segment, the geometrical model of a right mandibular canine was developed. Then, the magnitudes of required toe-in or toe-out were computed numerically for different positions of cantilever attachment.

2.1. Geometrical model of a mandibular canine

The basic anatomic and geometric features of a right mandibular canine were taken from Wheeler’s Dental Anatomy; Physiology and Occlusion [16]. The length of the crown and root along long axis was taken as 11 mm and 15 mm respectively [16]. The normal inclination of the long axis of a mandibular canine was considered to be equal to that of a maxillary canine which is 19° with respect to a vertical line through $C_{res}$ [1]. The position of $C_{res}$ was taken approximately 0.77 times the root length from apex on the long axis [17]. The force system was applied to the top surface of a canine bracket. For cantilever attachment, four values of the heights from the cusp of a canine were considered i.e. $h = 2, 3, 4 and 5 mm$ in two cases of the horizontal distance from canine surface as $x = 2 and 3 mm$ . Thus, total eight positions of cantilever attachment were considered in this study. Figs 2(A) and 2(B) show geometrical model of a right mandibular canine as viewed from the distal aspect for $x = 2 and 3 mm$ respectively.

Fig. 2.

Geometrical model of a right mandibular canine as viewed from the distal aspect with four different heights of cantilever attachment ( $P_{f}$ ) on the bracket top surface from the cusp of a canine i.e. $h = 2, 3, 4 and 5 mm$ in two cases of horizontal distance (x) from canine surface. (A) $x = 2 mm$ case, (B) $x = 3 mm$ case (all length dimensions are in mm).

Several points were defined on the geometrical model i.e. points along the long axis and points along lines parallel to the occlusal plane (OP) which pass through lingual bracket top surface. The points defined on the long axis were, $P = cusp of a canine$ , $J = point joining crown and root$ , $C_{res} = centre of resistance$ , $A = root apex$ . The points on the lines parallel to OP are shown in Table 1.

Table 1

Nomenclature of points on the lines parallel to OP through lingual bracket top surface

Height (h) of lingual bracket top surface from cusp (mm)	Points on the lines parallel to OP

	Point on bracket top surface ( $P_{f}$ )	Point on canine surface in LiO	Point intersecting vertical line through $C_{res}$	Point intersecting long axis
2	a	$a^{'}$	$z_{2}$	$x_{2}$
3	b	$b^{'}$	$z_{3}$	$x_{3}$
4	c	$c^{'}$	$z_{4}$	$x_{4}$
5	d	$d^{'}$	$z_{5}$	$x_{5}$

In a geometrical model,

Length of crown, $L_{crown} = P J = 11 mm$ ;

Length of root, $L_{root} = A J = 15 mm$ .

The vertical line through $C_{res}$ i.e. line $E - C_{res}$ cuts the plane OP at point E. Points a, b, c and d are the points of cantilever attachment ( $P_{f}$ ) for the heights (h) of 2 mm, 3 mm, 4 mm and 5 mm respectively. As stated earlier, the horizontal distance between $P_{f}$ on the bracket top surface and canine surface was considered to be $x mm$ in LiO. $\begin{matrix} {a a}^{'} = {b b}^{'} = {c c}^{'} = {d d}^{'} = x mm . \end{matrix}$

The horizontal distances within canine i.e. between canine surface and long axis at different heights (h) were determined graphically as, $\begin{matrix} a^{'} x_{2} = 1 mm, b^{'} x_{3} = 1.6 mm, c^{'} x_{4} = 2.4 mm, d^{'} x_{5} = 2.6 mm . \end{matrix}$

The basic trigonometric equations were applied to calculate the required toe of the vertical segment of a cantilever. $\begin{array}{l} (1) & {P x}_{2} = \frac{2}{cos (19^{\circ})}, P x_{3} = \frac{3}{cos (19^{\circ})}, P x_{4} = \frac{4}{cos (19^{\circ})}, P x_{5} = \frac{5}{cos (19^{\circ})}, \\ (2) & A C_{res} = 0.77 \times L_{root}, \\ (3) & P C_{res} = (L_{crown} + L_{root}) - A C_{res}, \\ (4) & \begin{aligned} x_{2} C_{res} = P C_{res} - P x_{2}, x_{3} C_{res} = P C_{res} - P x_{3}, \\ x_{4} C_{res} = P C_{res} - P x_{4}, x_{5} C_{res} = P C_{res} - P x_{5} . \end{aligned} \end{array}$ Horizontal distances of $x_{2}$ , $x_{3}$ , $x_{4}$ and $x_{5}$ from vertical line $E - C_{res}$ are given respectively as: $\begin{matrix} (5) & \begin{matrix} x_{2} z_{2} = x_{2} C_{res} \times sin (19^{\circ}), x_{3} z_{3} = x_{3} C_{res} \times sin (19^{\circ}), \\ x_{4} z_{4} = x_{4} C_{res} \times sin (19^{\circ}), x_{5} z_{5} = x_{5} C_{res} \times sin (19^{\circ}) . \end{matrix} \end{matrix}$ In LiO, horizontal distances of a, b, c and d from vertical line $E - C_{res}$ are given respectively as: $\begin{matrix} (6) & \begin{matrix} a z_{2} = a a^{'} + a^{'} x_{2} - x_{2} z_{2}, {b z}_{3} = b b^{'} + b^{'} x_{3} - x_{3} z_{3}, \\ {c z}_{4} = c c^{'} + c^{'} x_{4} - x_{4} z_{4}, {d z}_{5} = d d^{'} + d^{'} x_{5} - x_{5} z_{5} . \end{matrix} \end{matrix}$ In LiO, vertical distances between $P_{f}$ and $C_{res}$ for different heights (h) are given as: $\begin{matrix} (7) & \begin{matrix} z_{2} C_{res} = [P C_{res} \times cos (19^{\circ})] - 2, z_{3} C_{res} = [P C_{res} \times cos (19^{\circ})] - 3, \\ z_{4} C_{res} = [P C_{res} \times cos (19^{\circ})] - 4, z_{5} C_{res} = [P C_{res} \times cos (19^{\circ})] - 5 . \end{matrix} \end{matrix}$

The vertical segment toe of a cantilever as viewed from the distal aspect of a mandibular canine was denoted by angle Θ. Hence, angles $a - C_{res} - z_{2}$ , $b - C_{res} - z_{3}$ , $c - C_{res} - z_{4}$ and $d - C_{res} - z_{5}$ are represented by $Θ_{2}$ , $Θ_{3}$ , $Θ_{4}$ and $Θ_{5}$ respectively. Suffix of Θ represents the height (h) of $P_{f}$ . $\begin{matrix} (8) & \begin{matrix} Θ_{2} = {tan}^{- 1} \frac{[a z_{2}]}{[z_{2} C_{res]}}, Θ_{3} = {tan}^{- 1} \frac{[{b z}_{3}]}{[z_{3} C_{res]}}, \\ Θ_{4} = {tan}^{- 1} \frac{[{c z}_{4}]}{[z_{4} C_{res]}}, Θ_{5} = {tan}^{- 1} \frac{[{d z}_{5}]}{[z_{5} C_{res]}} . \end{matrix} \end{matrix}$ Table 2 shows numerically computed values of vertical segment toe (Θ) of a cantilever from the distal aspect of a mandibular canine for different positions of cantilever attachment ( $P_{f}$ ) based on h and x value. All values of Θ were rounded off to nearer integer values to make them suitable to apply clinically. Positive values of Θ represent toe-in while negative values of Θ represent toe-out. In the cases of $x = 2 and 3 mm$ , no toe was required for the heights $h = 3 and 2 mm$ respectively as the horizontal distance between $P_{f}$ and vertical line $E - C_{res}$ was zero for these positions of cantilever attachment. To verify the optimum toe angles (Θ), FEA was performed.

Table 2

Magnitude of vertical segment toe (Θ) of a cantilever from distal aspect of a mandibular canine for different positions of cantilever attachment ( $P_{f}$ ) based on h and x value

Height (h) of lingual bracket top surface from cusp of a canine (mm)	Magnitude of vertical segment toe of a cantilever from distal aspect (Θ)

	$x = 2 mm$	$x = 3 mm$
2	−5°	0°
3	0°	+5°
4	+6°	+12°
5	+11°	+17°

2.2. Finite element analysis (FEA) of a mandibular canine

When a cantilever is tied to the bracket top surface, it applies a single force on the bracket at a particular point ( $P_{f}$ ). Hence, the cantilever considered in this study was not included in FEA. Only the force and its direction of application due to the toe of the vertical segment of a cantilever was considered in FEA.

In this study, three-dimensional computer aided design (CAD) models of all the structures were prepared using CREO Parametric (version 2.0, PTC, Needham, Massachusetts, USA). The CAD model of a mandibular canine was prepared from the dimensions mentioned in Wheeler’s Dental Anatomy; Physiology and Occlusion [16]. CAD models of surrounding tissue structures, i.e. PDL and alveolar bone were also prepared. The width of PDL was taken as 0.2 mm [11]. The lingual canine bracket was modeled as a simplistic rectangular block to apply force by a cantilever at an offset from a canine surface according to the value of x. All the structures were assembled in CREO Parametric. Eight assembly models were prepared representing different positions of cantilever attachment as mentioned in Table 2. Figure 3 shows an assembly model with all the structures, i.e. bracket, mandibular canine, PDL and alveolar bone. The assemblies in CREO Parametric were saved in a STEP file format and imported into ANSYS.

Fig. 3.

An assembly model consisting of bracket, mandibular canine, PDL and alveolar bone.

ANSYS is computer-aided analysis software which performs FEA. ANSYS Workbench (version 14.5, ANSYS Inc., Canonsburg, Pennsylvania, USA) was used in this study. To mesh each assembly model in ANSYS, 10-node tetrahedron and 20-node hexahedron elements (connected by nodes) were used. On an average, one assembly model consisted of 76,825 nodes and 44,334 elements. The optimum range of intrusive force is 10–20 g [18]. In FEA, the magnitude of force was kept within this biological limit. To represent the force applied by a cantilever, a force was applied to the top surface of a canine bracket at an angle Θ for different positions of cantilever attachment as mentioned in Table 2. Boundary conditions were applied by fixing lower and lateral surfaces of the alveolar bone region. Finally, solution command was executed to obtain the results of FEA.

All of the materials were assumed to be isotropic and linear elastic in nature. Canine bracket was considered to be made of stainless steel. Table 3 shows material properties of all the structures [11].

Table 3

Material properties of mandibular canine (tooth), PDL, alveolar bone and bracket

No.	Material	Young’s modulus [MPa]	Poisson’s ratio
[1]	mandibular canine (tooth)	19,600	0.30
[2]	PDL	0.667	0.45
[3]	alveolar bone	13,700	0.26
[4]	bracket (stainless steel)	200,000	0.30

3. Results

In FEA, a force was applied on the bracket top surface which was equivalent to that applied by a cantilever. The inclination of force was kept as that of the vertical segment toe of a cantilever. In FEA, the results were displayed in the form of movement of a mandibular canine after application of a force.

Fig. 4.

Vectors of nodal displacements along with undeformed ( $d_{1}$ ) and deformed ( $d_{2}$ ) models for different heights (h) of cantilever attachment ( $P_{f}$ ) in case of horizontal distance of 2 mm from canine surface ( $x = 2 mm$ ). (A) $h = 2 mm$ , (B) $h = 3 mm$ , (C) $h = 4 mm$ , (D) $h = 5 mm$ .

Fig. 5.

Vectors of nodal displacements along with undeformed ( $d_{1}$ ) and deformed ( $d_{2}$ ) models for different heights (h) of cantilever attachment ( $P_{f}$ ) in case of horizontal distance of 3 mm from canine surface ( $x = 3 mm$ ). (A) $h = 2 mm$ , (B) $h = 3 mm$ , (C) $h = 4 mm$ , (D) $h = 5 mm$ .

Figure 4 shows vectors of nodal displacements along with undeformed ( $d_{1}$ ) and deformed ( $d_{2}$ ) models for different heights (h) of cantilever attachment ( $P_{f}$ ) in case of horizontal distance of 2 mm from canine surface ( $x = 2 mm$ ). Similarly, Fig. 5 shows vectors of nodal displacements along with undeformed ( $d_{1}$ ) and deformed ( $d_{2}$ ) models for different heights (h) of cantilever attachment ( $P_{f}$ ) in case of horizontal distance of 3 mm from canine surface ( $x = 3 mm$ ).

From the directions of vectors of nodal displacements as well as the positions of undeformed and deformed models, it was clear that the desired pure intrusion of a mandibular canine model was achieved in each case of cantilever attachment. No tipping of a mandibular canine was observed in FEA. Thus, the values of the required vertical segment toe of a cantilever from the distal aspect of a mandibular canine were verified with FEA.

4. Discussion

The orthodontic appliances produce an accurate and desired tooth movement only when they are simplistic from the biomechanics point of view [19]. Orthodontic appliances such as cantilever, spring, etc. deliver relatively constant forces owing to large inter-bracket distance between two points of attachment. In a two-tooth system, if an appliance is engaged in the bracket slots of both the teeth, it generates a force and a couple at both the brackets resulting in a two-couple statically indeterminate system [19]. But, if an appliance is engaged in the bracket slot of one tooth and tied as a point of contact on the bracket of other tooth, then this force system is called as one-couple system as a couple acts only at the bracket slot where an appliance is engaged. One-couple system is statically determinate as equal and opposite forces act at both the attachment sites (engaged and tied) to stabilize an appliance [19].

In the study of individual intrusion of a canine by Marcotte [9], insertion of a cantilever into both accessory tubes of the first molar and canine brackets slot created a two-couple statically indeterminate system. Additionally, bucco-lingual inclination of a canine was not controlled in their study. In the study of Burstone [7], despite the better control of bucco-lingual inclination, a cantilever was inserted into the vertical (occlusal) slot of canine bracket creating a two-couple statically indeterminate system. But, Caballero et al. [10] applied a one-couple statically determinate system by tying a cantilever on the top (occlusal) surface of a canine bracket in LaO.

Jost-Brinkmann et al. [20] and Hong et al. [21] stated that the tooth movement is more effective in case of lingual vertical forces as $C_{res}$ is closer in LiO. Hence, in this study, an intrusion of a mandibular canine with the segmented arch in LiO was suggested. To achieve this intrusion, one end of a cantilever was considered to be engaged with the first molar auxiliary tube and the other end was considered to be tied to the top (occlusal) surface of a canine bracket. At the tied end, intrusive force was considered to be acted by a cantilever. This creates a one-couple statically determinate system. Moreover, LiO is a more aesthetic orthodontic technique than LaO.

In one couple system as stated in this study, a reactive force as well as a couple will be produced on molar tube. Couple and force on the molar tube will try to displace the molar tooth. This can be minimized by engaging an archwire from second molar tube to first molar tube and extending further through brackets slots of first and second premolars to generate posterior teeth anchorage segment as shown in Fig. 1(A). Thus, posterior teeth will act as a one complete anchorage unit, making the effects of couple and reactive force significantly lower than the effect of force on the canine. Additionally, undesirable couples and forces on molars can be restricted by using independent counteracting force systems such as elastics, transpalatal archwires (TPA), etc. [19].

This study was limited to only an initial movement of a mandibular canine. But, the nature of tooth movement in initial displacement can be assumed as equivalent to total tooth displacement due to bone resorption (apposition) [3]. Thus, if this study is established as a new treatment method, it is anticipated that the pure intrusion of a mandibular canine with the segmented arch in LiO will be achieved quiet efficiently and rapidly.

5. Conclusion

This study suggests that there is a need of vertical segment toe (Θ) of a cantilever from the distal aspect to achieve pure intrusion of a mandibular canine with the segmented arch in LiO. Different positions of cantilever attachment on lingual bracket top (occlusal) surface were considered according to different heights (h) from the cusp of a canine i.e. 2 mm, 3 mm, 4 mm and 5 mm in each case of horizontal distance (x) from canine surface i.e. 2 mm and 3 mm. Toe-in and toe-out were represented by positive and negative angles respectively.

In $x = 2 mm$ case, the required vertical segment toe angles at 2 mm, 4 mm and 5 mm heights (h) were $- 5$ °, $+ 6$ ° and $+ 11$ ° respectively. Only at 3 mm height (h), no toe was required. Hence, for $x = 2 mm$ , $h = 3 mm$ can be considered as better position of a cantilever attachment than others.

In $x = 3 mm$ case, the required vertical segment toe angles at 3 mm, 4 mm and 5 mm heights (h) were $+ 5$ °, $+ 12$ ° and $+ 17$ ° respectively. Only at 2 mm height (h), no toe was required. Hence, for $x = 3 mm$ , $h = 2 mm$ can be considered as better position of a cantilever attachment than others.

Conflict of interest

The authors have no conflict of interest to report.

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