Mechanical Efficacy Equivalence of W-Arch and Quad-Helix Orthodontic Arch Expansion Appliances: A Finite Element Study

Abstract

The quad-helix and W-arch are commonly used appliances for expanding the dental arch in orthodontic treatment. However, differences in performance between these two expanders remain unclear, and no guidelines exist for selecting one over the other. The purpose of this study was to investigate whether there were differences in dental arch expanding ability between these appliances. Maxillary arch expansions were simulated using the finite element method. The expander was assumed to be an elastic beam, while the teeth and alveolar bone were treated as rigid bodies. The periodontal ligament (PDL) was modeled as a nonlinear elastic material. The teeth moved in the same direction as the initial movement caused by the elastic deformation of the PDL. The right and left canines, premolars, and first molars were expanded symmetrically in either parallel or fan shapes. When the wire diameter of the W-arch was set to 0.032 inches its stiffness became equivalent to that of a quad-helix with a wire diameter of 0.036 inches. Canines and premolars were expanded through tipping movements. The molars initially tipped buccally, then became upright and moved bodily. Both expanders expanded the arch in almost the same manner. There was no difference in arch expansion ability between the W-arch made of 0.032-inch wire and the quad-helix made of 0.036-inch wire. The W-arch may be preferred as the first choice due to its simpler structure compared to the quad-helix.

1. Introduction

In orthodontic treatment, lateral expansion of the dental arch can be accomplished using either a rapid or a slow expander. Rapid expanders separate the midpalatal suture and require several kilograms of force. In contrast, slow expanders apply a light force of a few hundred grams to the teeth, expanding the arch through the remodeling of the alveolar bone, that is, through orthodontic tooth movement.

The quad-helix is one of the most commonly used slow expanders [1,2,3,4,5,6]. By modifying its shape, the expansion force can be easily controlled, allowing for either parallel or fan-shaped expansion. Due to its ease of use, the quad-helix has been widely utilized in clinical treatments.

The quad-helix acts as an elastic spring, and its mechanics have been studied. Honme et al. measured the orthodontic forces produced by the quad-helix during parallel and fan-shaped expansions [7]. Jones et al. calculated the relationship between the shape and the mechanical properties of the quad-helix [8,9]. Santos et al. measured the forces generated by two types of stainless-steel quad-helices [10]. These studies focused on the force systems immediately after the quad-helix was placed on the arch.

The quad-helix is constructed by incorporating four helices into a W-shaped spring (W-arch) [1]. These helices reduce the stiffness of the W-arch, allowing for larger expansion with a light force. In addition, they minimize force decay during expansion, leading to more efficient expansion.

Instead of using helices, the stiffness of the W-arch can be reduced by decreasing the wire size (diameter of the cross-section). By selecting the appropriate wire size, the stiffness of the W-arch can be made equivalent to that of the quad-helix. If the W-arch can achieve the same expansion as the quad-helix, it could serve as a viable alternative. The W-arch without helices is easier to fabricate and more comfortable for patients. Currently, differences in performance between these two expanders remain unclear, and no guidelines are available for selecting one over the other.

The purpose of this study was to examine whether there was a difference in the expansion ability of the quad-helix and the W-arch. If no difference is found, the W-arch could be used instead of the quad-helix. Using the finite element method (FEM), we simulated expansion with both the quad-helix and the W-arch. The present study is valuable not only for clarifying the mechanisms of expanders, but also for selecting an appropriate expander in clinical settings.

2. Materials and Methods

2.1. FEM Model

The upper canines, premolars, and first molars on both sides were expanded in parallel or in a fan shape (Figure 1). Assuming the dental arch and the expander to be symmetrical, their right half was modeled (Figure 2). A FEM software (ANSYS 16.1, ANSYS Inc., Canonsburg, PA, USA) was used for the present study.

A three-dimensional model of the dentition was created using cross-sectional images of a dental study model (i21D-400C, Nissin Dental Products Inc., Kyoto, Japan) taken by cone beam computed tomography (CBCT). The surfaces of the teeth were divided into triangular shell elements (SHELL181: element name in ANSYS). The teeth were assumed to be rigid bodies, and their surface nodes were constrained to be rigid. The periodontal ligament (PDL), with a constant thickness of 0.2 mm, was constructed over the root and divided into three-dimensional solid elements (SOLID45). The PDL was assumed to be a nonlinear elastic material with a piecewise linear stress–strain curve [11]. The alveolar bone and teeth were assumed to be rigid bodies because their Young’s moduli (approximately 10 GPa) were significantly higher than that of the PDL (approximately 0.1 MPa). Under this assumption, we could exclude the alveolar bone from the FEM model.

Shapes of the quad-helix and W-arch were determined by referring to Proffit’s text [1]. In the quad-helix, the diameters of the anterior and posterior helices were 2.8 mm and 3.2 mm, respectively. The W-arch was made by eliminating the helices from the quad-helix. Both expanders were made of stainless-steel wires with a Young’s moduli of 200 GPa. They were divided into three-dimensional beam elements (BEAM188). The expanders were assumed to be soldered on the molar bands and fixed on the crowns with rigid beam elements (MPC184). Contact elements were overlaid on the expanders and crown surfaces with a frictional coefficient of 0.15.

Figure 2 shows that the total numbers of shell elements and solid elements were 932 and 14,361, respectively. Similarly, the total numbers of beam elements in the W-arch and quad-helix were 96 and 120, respectively. The total number of nodes in all elements became 15,978 for the W-arch and 16,026 for the quad-helix.

2.2. Activation of Spring

The distance between the right and left arms was reduced to place the expander into the maxillary arch. This is called activation, and the amount of reduced distance is defined as activation amount δ. At activation, denoting a force acting on the first molar as W, δ/W is defined as the flexibility of the expander. Its inverse, W/δ, is the stiffness of the expander. The forces and movements acting on the teeth are called the force system produced by the expander.

The wire size of the quad-helix was assumed to be 0.036 inches (0.91 mm). On the other hand, the size of the W-arch was selected in such a way that its stiffness became about the same as that of the quad-helix. We targeted two types of expansion: parallel and fan-shaped expansions. In each expansion type, the amount of activation was adjusted so that the activation force was the same for both expanders.

2.3. Simulation of Long-Term Tooth Movement

In the present study, tooth movement due to the dentoalveolar process was simulated. We used the same method as in a previous article [11]. It was assumed that the tooth moved in the same direction and the same amount as the initial tooth movement, which was produced by elastic deformation of the PDL. The simulation procedure was constructed using three steps (Figure 3). First, an expander was activated. Second, the initial tooth movement was calculated. Third, the alveolar socket was moved by the initial tooth movement. By iterating the latter two steps, the teeth were moved step by step. The force system was updated at each step. The number of iterations, N, was equivalent to the elapsed time after activation, but N could not be converted to an actual time. This method could not simulate tooth movement due to skeletal change or orthopedic effect.

In each iterative calculation, static equilibrium nonlinear equations for obtaining the initial movement were solved using the Newton–Raphson method. Convergence criteria were automatically set to default values by the FEM software.

3. Results

To examine the effects of helices, a 0.036-inch quad-helix and W-arch were activated in parallel by 2 × 2 mm (Figure 4). The forces required for activation were 2.4 N (quad-helix) and 4.0 N (W-arch). This means that the stiffness of the W-arch decreased by 60% due to the helix. A similar decrease in stiffness was obtained by reducing the wire diameter of the W-arch from 0.036 to 0.032 inches. When the 0.032-inch W-arch was activated, the force required was almost the same as that of the quad-helix (Figure 4). The maximum equivalent stress on the 0.032-inch W-arch (614 MPa) became higher than that on the quad-helix (437 MPa).

By iterating the calculation up to N = 500, the arch was expanded. In each iterative calculation, 5 to 20 iterations of the Newton–Raphson method were required for convergence. As a result, including these iterations, the total number of iterations was approximately 2500 up to N = 500.

The movement patterns of the teeth were the same in all cases: the molar moved bodily, while the canine and premolars tipped (Figure 5A). In response to these movement patterns, compressive strain (−) occurred on the buccal side of the molar root and near the cervical region, while tensile strain (+) occurred near the apex on the buccal side of the canine and premolars. Additionally, the molar slightly tipped distally, which occurred because the mesial end of the arm slid along the lingual surface of the canine (Figure 5B).

In the case of parallel expansion, the buccal tipping angle of the molar crown increased and then decreased to zero. This change suggested the molars initially tipped, and then became upright and moved bodily (Figure 6). Changes in buccal movement and the tipping angle of the molar were nearly identical between the quad-helix and W-arch.

Using both the W-arch and the quad-helix, the dental arch was expanded with parallel and fan-shaped expansions (Figure 7). The shapes of the expanders returned to their initial shapes, and the forces acting on the teeth became very small. In the fan-shaped expansion, no force acted on the canine at N = 500, because the arm of the expander was not in contact with the crown.

At N = 500, the tipping angles of the teeth and the deformed shape of the expander were almost identical between the quad-helix and W-arch (Figure 8 and

Table 1. Buccal tipping angle of teeth at N = 500.

						Unit:
		Canine	1st Premolar	2nd Premolar	Molar
Parallel expansion	Quad-helix	6.4	7.8	8.3	−0.01
	W-arch	6.3	7.6	8.1	−0.08
Fan-shaped expansion	Quad-helix	9.7	12.1	7.6	−1.3
	W-arch	9.5	11.5	6.7	−1.1

Minus sign (−) indicates lingual tipping.

).

4. Discussion

4.1. Mechanical Equivalence of Quad-Helix and W-Arch

The stiffness of the expander, which is equivalent to the bending stiffness of the wire, is proportional to the fourth power of the wire diameter [12]. When changing the wire diameter of the W-arch from 0.036 to 0.032 inches, its stiffness decreased to 62%, as calculated by (0.032/0.036)⁴ = 0.62. This reduction in stiffness was comparable to that caused by incorporating the helix. Thus, the stiffness of the 0.032-inch W-arch was almost identical to that of the 0.036-inch quad-helix (Figure 5).

On the other hand, the stress induced in the W-arch (614 MPa) was greater than that in the quad-helix (437 MPa), meaning that the safety margin for plastic deformation was reduced in the W-arch. However, the maximum equivalent stress in the W-arch (614 MPa) was still less than half of the yield stress of stainless steel (1500 MPa), and therefore, the W-arch could be activated without causing plastic deformation [13].

The arm of the expander requires sufficient stiffness to expand the canine and premolars. If the stiffness is too low, the arm will be elastically deformed by the force applied from the teeth, impeding the desired expansion. Although the arm of the 0.032-inch W-arch was less stiff than that of the 0.036-inch quad-helix, it maintained enough stiffness to allow the desired expansion.

The expansion patterns achieved by the 0.032-inch W-arch were almost identical to those of the 0.036-inch quad-helix (Figure 7 and Table 1). The deformation of the 0.032-inch W-arch was nearly the same as that of the 0.036-inch quad-helix (Figure 8). In the present simulations, there was no difference in arch-expanding ability between the two expanders. The W-arch can therefore be used in the same way as the quad-helix. From a mechanical standpoint, the W-arch, which has a simpler structure and is easier to fabricate, may be the preferred choice for slow expansion in clinical settings. However, the findings of the present FEM study should be validated by future clinical studies.

4.2. Simulation of Tooth Movement

In the present simulation, a tooth was assumed to move in the same direction as its initial displacement, which occurred due to the elastic deformation of the PDL. This assumption was based on clinical observations, which indicate that movement patterns during orthodontic tooth movement resemble those of the initial displacement. Additionally, an in vivo experiment has demonstrated that the initial displacement could serve as a predictor of long-term movement [14].

The movement pattern of a tooth varies depending on the movement–force (MF) ratio acting on it. The canine and premolars were in contact with the expander arm, and no movement acted on these teeth, meaning the MF ratio was zero. Hence, these teeth tipped during expansion. On the other hand, the molars were fixed to the expanders, and a movement that prevented tipping acted on it. Initially, the MF ratio acting on the molar was so small that it tipped. As the molar moved buccally, the MF ratio increased, causing it to become upright. Consequently, the molar moved bodily. This change in the MF ratio was accounted for through iterative calculations, which is a feature of the present simulation method.

It is difficult to predict long-term movement patterns based on the initial force system alone. Changes in the force system and movement pattern have been observed in other types of treatments, such as those using wire brackets and aligners [15,16].

The number of iterative calculations corresponded to the treatment period. In each iteration, the tooth moved by the amount of the initial displacement. However, the exact period required for this process has not yet been clarified in clinical settings, making it impossible to convert the number of iterations, N, into a treatment duration. In the present study, predicting the treatment duration required for arch expansion was not possible. This was a limitation of the present FEM simulation. Despite this limitation, the arch expansion patterns could be predicted, allowing for a comparison of the expansion abilities of the W-arch and the quad-helix.

Figure 5B shows that the molar tipped distally due to the sliding of the arm on the lingual surface of the canine crown. In the present simulation model, the frictional coefficient between the arm and the crown surface was assumed to be 0.15. If the coefficient of friction were smaller, the arm would slide more easily on the crown surface, which would result in increased tipping of the molar and decreased labial movement of the canine. The effect of the frictional coefficient on arch expansion needs to be investigated in future studies.

5. Conclusions

By reducing the wire size of the W-arch, its stiffness became the same as that of the quad-helix. There was no difference in the ability to expand the dental arch found for these expansion appliances. The W-arch, which has a simpler shape, can be used instead of the quad-helix.