Material and methods

The FE model was created from tomography images of a dry skull of an adolescent using Mimics software (version 15.01; Materialise, Leuven, Belgium). Visual-mesh software (version 7.0; ESI Group, Paris, France) was used to produce a tetrahedral FE mesh. The maxilla, dentition, and alveolar bone in the FE mesh were sectioned into 1-mm tetrahedrons. The remaining skull bones including their sutures were sectioned into 5-mm tetrahedrons. The model consisted of 1088171 elements and 226458 nodes ( Fig 1 ). The dentition, alveolar bone, and periodontal ligament were considered homogenous and isotropic. The thickness of the periodontal ligament was considered to be 0.2 mm, and the thickness of the midpalatal sutures was considered to be 0.5 mm. All circumaxillary sutures were modeled with a width of approximately 0.5 mm divided into 2 layers. Thus, the size of the suture mesh along its thickness was 0.25 mm with an aspect ratio 0.25. The thickness of cortical bone was determined according to the study of Farnsworth et al. The mechanical properties of the different structures of the model are shown in Table I .

Finite element mesh of the skull showing that the maxilla and the teeth were made of 1-mm tetrahedrons, and the rest of the skull was made of 5-mm tetrahedrons.

Foramen magnum was fixed and set as the origin point. The forehead was in contact with a fixed band to mimic the effect of the FM. The 3-dimensional coordinates were x (anteroposterior plane), y (transverse plane), and z (vertical plane). Positive values for x, y, and z indicated forward, outward, and upward displacements.

Four models were made: FM with labiolingual appliance (type A), FM with bone-borne expander supported by 4 miniscrews (C-implants; Cimplant, Seoul, Korea) with a diameter of 1.8 mm and a length of 8.5 mm placed in the paramedian area (type B), FM with bone-borne expander supported by 4 miniscrews placed in the palatal slopes (type C), and FM with conventional hyrax expander (type D). Each rapid palatal expansion appliance had an expansion screw for 0.25 mm of widening per turn ( Figs 2 and 3 ).

A, The skull model showing the headband of the facemask and B, its labiolingual appliance; C, cross-sectional image of the bone-screw interface.

Occlusal views of the 4 models: type A, labiolingual appliance of facemask connected with transpalatal archwire; type B, labiolingual appliance and paramedian bone-borne expander; type C, labiolingual appliance and palatal-slope bone-borne expander; type D, labiolingual appliance connected transpalatally using conventional tooth-borne expander.

In type B, the miniscrews were placed 3 mm lateral to the midpalatal suture and connected to the expander via 0.9-mm stainless steel wires. In type C, the miniscrews were placed at the palatal slope, 8 mm beneath the alveolar ridge: 2 between the canines and first premolars, and 2 between the second premolars and first molars. The miniscrews were connected to the expander through an acrylic resin cover reinforced with 0.9 mm stainless steel wire. This wire shared some nodes with the miniscrews, and no relative motion was allowed between them.

The first premolars and first molars were banded (bands were meshed using shell elements connected to teeth using tied interface) and connected with 0.9-mm stainless steel wire (modeled using beam elements) at both the buccal and lingual sides. A protraction hook was attached to the band of the first premolar with the distal end 2 mm apical to the alveolar crest between the canines and first premolars. Bands were connected to the contralateral sides using an expander in type D (or stainless steel transpalatal archwires in type A).

The expanders were activated transversely for 0.25 mm at the level of the expansion screw in the y direction and unfixed in the z and x directions. A protraction force of 500 g was engaged at the hooks of the appliances. To minimize anticlockwise rotation, a 30° downward vector of the maxillary occlusal plane was applied. The displacement and von Mises stress distribution were assessed at the maxilla and the maxillary dentition. The geometric nonlinear theory was used for model analysis with PAM-MEDYSA software (version 2011; ESI Group) for solving and Visual-Viewer for postprocessing (version 7.0; ESI Group). The load step size was 0.1 (each step was 0.025 mm), displacement convergence tolerance was 0.0001, and force convergence tolerance was 0.01.

Results

Tooth displacement is shown in Table II and Figures 4 through 7 . In type A, FM only, the application of protraction forces showed anterior displacement of the maxillary dentition. The posterior teeth exhibited downward displacement of 0.02 mm, and the central incisor showed a minimal upward displacement of 0.006 mm.

Dental and skeletal displacements on the transverse plane after activation of the types B, C, and D expanders.

Dental displacement in all planes after activation of the FM in type A, and the types B, C, and D expanders.

Dental and skeletal displacements in the sagittal plane after activation of FM.

Dental displacements in all planes after activation of FM.

Type B, the paramedian miniscrew expander, demonstrated posterior and outward displacement of the whole dentition along with downward displacement anteriorly and upward posteriorly after the application of the expansion forces. Then, after protraction, the posterior teeth showed anterior displacement, and the central incisors had a minimal backward displacement. The upward displacement of the posterior region changed to a downward one.

Type C, the palatal-slope miniscrew expander, showed a similar displacement tendency as type B but at smaller magnitudes.

For type D, the tooth-borne expander, expansion forces resulted in anterior and outward displacement of the maxillary dentition with downward displacement of posterior teeth but upward displacement of the incisors. Then, after protraction, there was an anterior displacement of the whole dentition (0.05-0.07 mm). The amount of expansion decreased posteriorly and become almost zero at the central incisor. There was an increase in the vertical displacement.

The amount of alveolar bone displacement was similar to the corresponding dental landmarks except for the vertical direction of type D in which the posterior alveolar bone showed a small upward displacement, whereas the teeth showed downward displacement ( Table III ).

Table III

Displacement of alveolar bone landmarks after expansion and then after adding the protraction force (in mm)

	Type A			Type B						Type C						Type D
	After protraction			After expansion			After protraction			After expansion			After protraction			After expansion			After protraction
	x	y	z	x	y	z	x	y	z	x	y	z	x	y	z	x	y	z	x	y	z
Central incisor lin alv crest	0.028	−0.001	0.002	−0.038	0.071	−0.049	−0.010	0.071	−0.048	−0.008	0.013	−0.011	0.020	0.013	−0.010	0.012	0.001	0.017	0.039	0.000	0.020
Bone at central incisor apex	0.022	0.000	0.001	−0.039	0.035	−0.052	−0.018	0.035	−0.053	−0.008	0.005	−0.012	0.014	0.005	−0.011	0.008	0.002	0.017	0.029	0.002	0.019
Central incisor lab alv crest	0.027	0.000	0.005	−0.039	0.047	−0.048	−0.013	0.047	−0.045	−0.008	0.008	−0.011	0.018	0.008	−0.007	0.011	0.001	0.019	0.036	0.001	0.025
First premolar midway palatal	0.027	−0.001	−0.004	−0.020	0.084	−0.017	0.006	0.084	−0.022	−0.004	0.016	−0.004	0.023	0.016	−0.008	0.011	0.001	0.013	0.037	0.000	0.010
First premolar lin alv crest	0.028	−0.001	−0.003	−0.018	0.089	−0.013	0.010	0.089	−0.017	−0.004	0.017	−0.003	0.025	0.017	−0.006	0.012	0.001	0.014	0.039	0.000	0.012
Bone at first premolar apex	0.022	0.000	−0.003	−0.007	0.048	0.005	0.015	0.048	0.001	−0.001	0.008	0.002	0.021	0.008	−0.002	0.007	0.002	0.013	0.029	0.002	0.011
First premolar buc alv crest	0.028	−0.001	−0.001	−0.005	0.076	0.011	0.023	0.076	0.009	−0.001	0.015	0.003	0.027	0.015	0.002	0.011	0.001	0.014	0.037	0.000	0.014
First molar midway palatal	0.027	−0.001	−0.010	−0.017	0.106	−0.008	0.010	0.105	−0.019	−0.003	0.021	−0.001	0.025	0.021	−0.011	0.011	0.001	0.008	0.036	0.000	0.000
First molar lin alv crest	0.029	−0.001	−0.010	−0.012	0.117	0.001	0.016	0.117	−0.009	−0.002	0.024	0.000	0.027	0.023	−0.009	0.012	0.000	0.009	0.040	−0.001	0.001
Bone at first molar palatal apex	0.024	0.000	−0.010	−0.009	0.087	0.004	0.015	0.087	−0.007	−0.002	0.016	0.001	0.023	0.016	−0.009	0.009	0.001	0.009	0.032	0.000	0.001
Bone at first molar palatal apex	0.025	0.000	−0.009	−0.001	0.084	0.017	0.023	0.084	0.008	0.000	0.016	0.004	0.025	0.016	−0.004	0.009	0.001	0.009	0.032	0.000	0.002
First molar buc alv crest	0.029	−0.001	−0.009	0.003	0.109	0.028	0.032	0.109	0.018	0.001	0.022	0.007	0.030	0.022	−0.002	0.012	0.001	0.009	0.039	−0.001	0.002

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