Experimental quantification of the mechanical forces and moments applied on three adjacent orthodontic brackets

Abstract

Orthodontic appliances deliver forces and moments that will determine movement of teeth. To analyze this latter, we developed an experimental setup to measure the mechanical forces applied on the teeth and to calculate, through a simplified theoretical analysis, the reactive forces and corresponding moments onto the brackets of three adjacent teeth. To validate the theoretical and experimental results, we use a simplified clinical situation of a maxillary canine in infraclusion and surrounded by its corresponding upper lateral incisor and first premolar. Forces are then measured experimentally and compared with the calculated results. From this, we show the specific dissymmetry of the mechanical forces on each side of the maxillary canine due to the applied mechanical forces and the undesirable induced generated moments occurring on each tooth that will directly impact the bone remodeling process and the final tooth repositioning.

Keywords

Orthodontics dental arch periodontium bone remodeling

1. Introduction

In orthodontics, fixed appliances made of principally brackets and wires are usually used to correct malocclusions. A wire is inserted into the bracket slots in order to apply forces and moments generating three-dimensional movements of the teeth [1]. In this work, we focus on the quantification of the applied forces and moments that are not accessible to conventional measuring devices [2] to establish the resulting tooth movements adjacent to the maxillary canine. These applied mechanical forces are initiating biological periodontal cellular responses and are responsible for bone remodeling [3–9]. Teeth will then move through a process of bone resorption and apposition. Knowledge of these forces and moments provide useful inputs for new theoretical numerical models [10–23], that are able to predict bone remodeling on the long term and are more specifically patient dependent. Here, we experimentally measure the applied mechanical forces on the orthodontic wire in an often encountered clinical situation of an upper infracluded canine neighbored by its first premolar and lateral incisor. Simplified 2D calculations applying the principle of static mechanical equilibrium, allow us to evaluate the reactive forces and moments on each of the three brackets.

2. Methods

A full dental arch is composed of fourteen teeth in a 3D space. However, little knowledge exists in the orthodontic literature [2,5,24–26] to precisely quantify both the forces and moments present on more than two adjacent teeth. The reference literature usually quantifies the applied mechanical forces and moments only on two (see Burstone [5]) or three (see Badawi [2]) adjacent teeth. We propose here a simplified theoretical and experimental model to determine both the forces and moments on any teeth of a dental arch and we focus more specifically on three adjacent teeth being the canine, first premolar and lateral incisor. This model is then compared to experimental results extracted from an in-house experimental setup. From this comparison, interpretation of the possible clinically predicted tooth movements are determined.

2.1. Theory

We used the torsor formulation to be able to transport any forces and moments between any given points of the considered studied geometry. A classical formulation of a torsor ${T_{A}}_{x, y, z}$ applied at a given point A and in the referential $(x, y, z)$ is provided by $\begin{matrix} {T_{A}}_{x, y, z} = {\vec{F_{A}} ∣ \vec{M_{A}}}_{x, y, z} = \{\begin{matrix} X_{A} \\ Y_{A} \\ Z_{A} \end{matrix} {|\begin{matrix} L_{A} \\ M_{A} \\ N_{A} \end{matrix}\}}_{x, y, z} \end{matrix}$ with $X_{A}$ , $Y_{A}$ and $Z_{A}$ being the components of the force $\vec{F_{A}}$ and, $L_{A}$ , $M_{A}$ and $N_{A}$ being the components of the moment $\vec{M_{A}}$ .

We study the case of three consecutive teeth being the right maxillary lateral incisor, canine and first premolar, as shown on Fig. 1(a). Points A, B and C in Fig. 1 are respectively the centers of brackets positioned on the lateral incisor, canine and the first premolar. D, E and F are the corresponding equilibrium points in the root of the lateral incisor, canine and first premolar. We suppose the three teeth to be coplanar defined in an $(x, y)$ 2D plane with negligible effect of the dental arch curvature and surrounded by a similar periodontal environment. In addition, we assume that the periodontium reacts in an opposite and equal way to the system of forces and moments applied at the teeth brackets. Weight of materials is also neglected due to its weak mechanical impact compared to the mechanical forces applied onto the brackets.

Since bone remodeling is a slow kinematic evolution, it is reasonable to assume that at any given time t, the structure is in a state of mechanical equilibrium. This enables to calculate the static equilibrium of forces and moments on the three brackets as shown on Fig. 1(a) (blue arrows) and Fig. 1(b).

Fig. 1.

Definition of the geometry and mechanics of the three-tooth clinical situation.

We apply the fundamental principles of static equilibrium on the wire given by the following equations $\sum_{i = 1}^{n} \vec{F_{i}} (\overline{S} \to S) = \vec{0}$ and $\sum_{i = 1}^{n} \vec{M_{i}^{P}} (\overline{S} \to S) = \vec{0}$ . Here, S is the wire segment between the three adjacent teeth and $\overline{S}$ the outer system. $\vec{F_{i}}$ represents the resulting force at point i and $\vec{M_{i}^{P}}$ is the resulting moment of forces of the bracket on the wire at any given point P of the system. n equals 3 for the three studied teeth. Finally, using the experimentally measured forces and knowing the exact geometry of the studied system enables us to calculate the unknown moments at any given point of each bracket and hence to determine the possible rotation of each tooth.

2.2. Experimental model

Once the forces and moments available at all points of the geometry were calculated, an experimental setup was developed to measure experimentally these values on each of the three bracket positions (three teeth) located on a full dental arch (see Fig. 2). This experimentation allowed us to quantify the forces applied onto the middle and sides of each bracket slot, indicated as N^o4 for the first premolar, N^o5 for the canine and N^o6 for the lateral incisor. The forces are measured on each tooth once the dental wire is placed on site for different displaced positions of the canine N^o5 and assuming a symmetric opposite response from the base onto which the teeth are fixed.

Fig. 2.

Experimental set-up of a full maxillary dental arch presenting aligned brackets to the exception of the bracket of an upper right infracluded canine. Forces delivered by a straight orthodontic wire were measured.

The typodont represents the dental arch with the configuration described in Fig. 1. The teeth are represented only by their corresponding brackets. A nickel-titanium wire (NiTi, 0.3556 mm diameter) is inserted into the brackets (3M Victory – Ricketts prescription) and secured with standard elastic ligatures on each tooth. The set-up is held on a fixed base and the measurements are carried out using an electronic dynamometer also fixed on the base. Measurements are made in tension of the wire (pulling perpendicular to the teeth principal alignment). Accuracy of measurement is $\pm 0.01 N$ as defined by the manufacturer. The moments applied on each tooth are found by calculation according to the above described theory.

3. Results

All results are presented and compared in Table 1. The experimentally applied initial force is set on the canine (bracket N^o5) and measured to a value of 9.34 N for an apical canine displacement of 5 mm. The corresponding average experimentally measured forces defined at the bracket center of each adjacent tooth are respectively 3.48 N and 7.49 N for bracket N^o4 and N^o6 and display a good accuracy (average error of 2.5%).

Table 1
Quantification of forces and moments on each of the orthodontic brackets (x axis is defined following the wire main axis from brackets 4 to 6. y axis is defined vertical perpendicular to wire main axis and following the upwards displacement of the canine – see Fig. 1(b))

Maxillary first premolar (bracket N^o4) Maxillary canine (bracket N^o5) Maxillary lateral incisor (bracket N^o6)

Average measured vertical force (y axis) at bracket center (N) $- 3.48$ ( $\pm 0.05$ ) $+ 9.34$ ( $\pm 0.32$ ) $- 7.49$ ( $\pm 0.21$ )

Calculated vertical force (y axis) with static equilibrium at bracket center (N) (wire supposed anchored within brackets) $- 4.59$ $- 4.74$

Calculated horizontal force (x axis) with static equilibrium (N) (wire supposed anchored within brackets) $- 677.25$ $+ 715.73$

Average measured horizontal force (y axis) at center of bracket (N) (wire sliding within brackets) $- 1.1$ ( $\pm 0.2$ ) $+ 4.4$ ( $\pm 0.9$ )

Calculated moment on brackets from experimental forces (N.m) $- 2.19 \times 10^{- 3}$ $+ 3.85 \times 10^{- 3}$ $+ 1.14 \times 10^{- 3}$

M/F ratio (mm) $+ 0.628$ $+ 0.41$ $- 0.15$

	Maxillary first premolar (bracket N^o4)	Maxillary canine (bracket N^o5)	Maxillary lateral incisor (bracket N^o6)
Average measured vertical force (y axis) at bracket center (N)	$- 3.48$ ( $\pm 0.05$ )	$+ 9.34$ ( $\pm 0.32$ )	$- 7.49$ ( $\pm 0.21$ )
Calculated vertical force (y axis) with static equilibrium at bracket center (N) (wire supposed anchored within brackets)	$- 4.59$		$- 4.74$
Calculated horizontal force (x axis) with static equilibrium (N) (wire supposed anchored within brackets)	$- 677.25$		$+ 715.73$
Average measured horizontal force (y axis) at center of bracket (N) (wire sliding within brackets)	$- 1.1$ ( $\pm 0.2$ )		$+ 4.4$ ( $\pm 0.9$ )
Calculated moment on brackets from experimental forces (N.m)	$- 2.19 \times 10^{- 3}$	$+ 3.85 \times 10^{- 3}$	$+ 1.14 \times 10^{- 3}$
M/F ratio (mm)	$+ 0.628$	$+ 0.41$	$- 0.15$

When theoretically calculating the corresponding vertical resulting forces (with fundamental principles of equilibrium) at the center of bracket N^o4 and N^o6, we find forces on brackets N^o4 and N^o6 of $- 4.59 N$ and $- 4.74 N$ respectively. These two values are compared with the measured values of $- 3.48 N$ and $- 7.49 N$ . We observe that forces have similar amplitude but are not distributed symmetrically on each side of the canine since they display different values.This is due to the fact that on the side of bracket N^o4, only three teeth remain distal to the upper first premolar whereas on the side of bracket N^o6, eight teeth extend from the upper right lateral incisor to the upper left second molar. Having more teeth on the incisor side makes stronger fixture due to friction between the wire and the brackets. In the experimental setup, friction effects are present within the experimental measurements made on brackets N^o4 and N^o6 but resulting theoretical forces on brackets N^o4 and N^o6 are calculated from the fundamental principle of equilibrium using only the applied force on bracket N^o5 providing different results since no friction effect is included.

We measured the horizontal forces on brackets N^o4 and N^o6 (friction effect is considered and wire is sliding) and found horizontal resulting forces of $- 1.1 N$ and $+ 4.4 N$ respectively on each bracket. Calculating forces without sliding of the wire (the wire is supposed anchored within the brackets and we use linear elasticity and Young’s modulus $E = 40 GPa$ for the wire elongation), the calculated forces were $- 677.25 N$ and $+ 715.73 N$ respectively. We observe that the wire sliding within brackets is predominant and has a very important effect in the static equilibrium of the system. In addition, we also observe a bigger effect on the incisor side (with eight teeth) than on the premolar side (with only three teeth). This shows clearly its importance and necessity to be included in all results interpretations.

Finally, the Moment to Force ratio (M/F) applied on one tooth is important in orthodontics since it gives an indication on the type of displacement of the tooth under mechanical load. However, moments are not measurable with conventional techniques. Hence, we calculated them over the three studied teeth using the measured experimental forces and known geometry. The calculated moments resulting values are of $- 2.19 \times 10^{- 3} N.m$ for the first premolar in a clockwise direction, $+ 3.85 \times 10^{- 3} N.m$ for the canine, and $+ 1.14 \times 10^{- 3} N.m$ for the lateral incisor both in a counter clockwise direction. Similar opposite intensities are found for the moments applied on left and right brackets N^o4 and N^o6, as expected.

An important undesirable effect for orthodontic application is the uncontrolled tipping of the teeth [13] when the critical level of M/F is below the value of 8 mm. Here, all three teeth show an uncontrolled tipping which will influence the overall equilibrium of the teeth realignment since these two teeth will come to close the space available for the canine. Also, although we applied a vertical (y axis) controlled force on the canine in order to extrude it, this force was asymmetrically distributed along the bracket slot and undesired tipping of the canine was observed during the realignment process.

4. Conclusion

We presented here a quantification of the applied forces and moments applied on three adjacent orthodontic brackets. The presented results show important effects regarding the bone remodeling process and highlighted some undesirable uncontrolled effects. The proposed methodology showed efficiency in collecting the required information. Although simplifying assumptions were made, the methodology showed good potential for the prediction of orthodontic tooth movements.

Conflict of interest

None to report.

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