Biomechanical effect of en-masse versus substep extraction space closure on posterior teeth using clear aligners: a finite element study

Yiru Jiang; Li Mei; Li Xu; Jue Wang; Xiaoxia Feng; Xiaoyan Chen

doi:10.2478/aoj-2024-0029

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Biomechanical effect of en-masse versus substep extraction space closure on posterior teeth using clear aligners: a finite element study

Australasian Orthodontic Journal

Volume 40 (2024): Issue 2 (July 2024)

By: Yiru Jiang, Li Mei, Li Xu, Jue Wang, Xiaoxia Feng and Xiaoyan Chen

Open Access

|Dec 2024

Figures & Tables

FEA models, co-ordinate systems, and the measuring points. (A) Maxillary dentition with attachments^21,24 in white color, PDL in green color, and aligners in blue color; (B) Meshing process and discretization; (C) Co-ordinate systems and the measuring points.

Table I.

Number of nodes and elements of the components of finite element models

Component	Elements	Nodes
Teeth	133300	66734
PDL	72914	73866
Attachment	81836	40942
Clear aligners	76120	38814

Table II.

Material properties

Component	Young’s modulus (MPa)	Poisson’s ratio
Teeth	19600	0.3
PDL	0.67	0.45
Attachment	12500	0.36
Clear aligners	528	0.36

Grouping Diagrams. (A) Test 1: I₀ and II₀ represented en-masse TM; I_a and I_b represented substep anterior TM, while II_a and II_b represented substep posterior TM; (B) Test 2: three aligner models were established on each of the six aligner models of Test 1 according to different anti-tipping degrees (1°, 2°, 3°). All these 24 aligners in the above tests were made on tooth models for force loading; (C) To illustrate the combined effect of I_a and I_b, consider PM2. Initially, the tooth axis was aligned with the green dashed line. After I_a, the axis shifted to the red dashed line. Subsequently, I_b applied forces in two directions: an incisor retraction force that inclined PM2 mesially by z° and a crown distal tipping force from the original aligner that inclined PM2 distally by x’°. The net mesial inclination of PM2 after I_b was y°, in which y° = z° - x’°. If we neglect the impact of axial inclination on the anti-tipping design effect, y° can be calculated as shown in Figure 4B. After I_b, the tooth axis shifted to the blue dashed line.

Comparison between en-masse and substep space closure in Test 1. (A) Vector diagrams of tooth displacement; (B) The displacement tendencies of PM2, M1, and M2.

Table III.

Mesiodistal force and moment at the resistance centre of each tooth in the local co-ordinate system under en-masse and substep space closure

			I₀			I_a			I_b			II₀			II_a			II_b
		F	M	M/F	F	M	M/F	F	M	M/F	F	M	M/F	F	M	M/F	F	M	M/F
0°	PM2	1.56	5.53	3.54	0.76	2.01	2.63	0.87	2.31	2.64	1.47	3.37	2.29	3.68	17.96	4.88	-1.7	-10.8	6.35
	M1	1.89	6.59	3.49	0.75	4.19	5.62	0.95	3.67	3.88	1.93	7.16	3.71	-1.61	0.19	-0.12	3.11	9.89	3.18
	M2	1.47	6.19	4.2	0.82	4.06	4.99	0.83	3.92	4.7	1.59	8.77	5.51	-0.43	0.09	-0.22	2.17	11.82	5.44
1°	PM2	0.63	-4.29	-6.78	-0.18	-7.63	43.41	0.03	-7.74	-238.72	0.92	-6.07	-6.61	3.07	6.3	2.05	-2.38	-21.66	9.1
	M1	2.37	-5.23	-2.2	1.24	-7.48	-6.03	1.51	-8.42	-5.57	2.31	-4.61	-1.99	-1.2	-12.27	10.19	3.45	-2.23	-0.65
	M2	1.35	-1.55	-1.15	0.71	-3.64	-5.09	0.75	-3.98	-5.28	1.14	2.16	1.89	-0.6	-8.07	13.54	1.93	4.21	2.18
2°	PM2	-0.22	-13.15	59.23	-1.13	-17.58	15.59	-0.77	-16.77	21.71	0.26	-14.2	-53.96	2.31	-3.91	-1.69	-3.2	-32.16	10.05
	M1	2.65	-15.76	-5.94	1.74	-19.18	-11.04	1.94	-19.49	-10.04	2.47	-16.48	-6.68	-0.42	-26.09	61.97	3.61	-12.53	-3.47
	M2	1.16	-7.06	-6.1	0.45	-9.67	-21.47	0.6	-9.74	-16.33	1.2	-6.49	-5.39	-0.91	-15.08	16.5	1.93	-2.67	-1.38
3°	PM2	-0.88	-21.76	24.79	-1.82	-26.58	14.64	-1.34	-25.03	18.74	-0.42	-22.57	54.12	1.74	-12.16	-6.99	-4.02	-42.77	10.64
	M1	2.55	-23.48	-9.2	1.69	-27.34	-16.17	1.89	-26.99	-14.29	2.46	-25.09	-10.2	-0.41	-35.37	85.94	3.67	-20.69	-5.63
	M2	1.08	-11.44	-10.58	0.42	-14.43	-34.51	0.59	-14.08	-23.95	1.19	-12.56	-10.6	-0.97	-20.72	21.39	2.01	-8.32	-4.14

1 Note. PM2, second premolar; M1, first molar; M2, second molar. +F, mesially; -F, distally; +M, mesially; -M, distally. F (N), M (N.mm).

Anti-tipping designs were added to the clear aligners of canines, PM2, M1, and M2 during Test 2. (A) Vector diagrams of tooth displacement; (B) Mesial tipping of posterior teeth; (C) The anti-tipping degrees for bodily movement of posterior teeth; (D) Schematic diagram of mechanical systems on M1: the red arrows indicated the force (F₀) and moment (M₀) produced by displacement of the aligner; the yellow arrow indicated the moment (M₁) produced by aligner’ anti-tipping design.

Effects of en-masse and substep space closure on lingual tipping of the central incisor.

Biomechanical effect for en-masse space closure versus substep space closure in extraction cases using clear aligners.

Anti-tipping design			0°			1°			2°			3°
Local coordinate system			X	Y	Z	X	Y	Z	X	Y	Z	X	Y	Z
En-masse retraction	CI	Crown (mm)	0.023	0.153	0.076	0.020	0.164	0.087	0.017	0.164	0.094	0.010	0.161	0.103
		Root (mm)	-0.016	-0.051	-0.048	-0.012	-0.062	-0.049	-0.010	-0.068	-0.045	-0.008	-0.075	-0.039
	LI	Crown (mm)	0.059	0.175	0.067	0.055	0.184	0.075	0.051	0.180	0.079	0.045	0.175	0.084
		Root (mm)	-0.015	-0.039	-0.098	-0.018	-0.046	-0.102	-0.021	-0.050	-0.098	-0.021	-0.056	-0.092
	Canine	Crown (mm)	0.060	0.193	0.040	0.125	0.113	0.011	0.196	0.021	-0.018	0.276	-0.069	-0.053
		Root (mm)	-0.008	-0.047	-0.091	-0.046	-0.009	-0.071	-0.087	0.031	-0.044	-0.132	0.074	-0.022
	PM2	Crown (mm)	0.029	-0.100	-0.050	0.001	-0.005	-0.076	-0.025	0.079	-0.095	-0.064	0.155	-0.097
		Root (mm)	-0.013	0.036	-0.032	0.003	-0.003	-0.076	0.019	-0.038	-0.109	0.041	-0.075	-0.124
	Ml	Crown (mm)	0.027	-0.106	-0.031	0.021	-0.019	-0.010	0.012	0.062	0.010	0.016	0.122	0.025
		Root (mm)	-0.003	0.015	-0.014	0.009	-0.028	-0.009	0.019	-0.066	-0.004	0.028	-0.097	0.002
	M2	Crown (mm)	0.027	-0.096	0.008	0.041	-0.005	0.069	0.067	0.064	0.120	0.099	0.121	0.163
		Root (mm)	-0.01 1	0.034	0.021	-0.009	-0.002	0.073	-0.008	-0.032	0.118	-0.010	-0.057	0.157
Canine distalisation	CI	Crown (mm)	0.021	-0.045	-0.010	0.020	-0.038	0.003	0.016	-0.037	0.013	0.010	-0.039	0.018
		Root (mm)	-0.015	0.014	0.025	-0.016	0.004	0.027	-0.016	-0.004	0.031	-0.014	-0.008	0.036
	LI	Crown (mm)	0.036	-0.024	0.006	0.034	-0.017	0.013	0.030	-0.016	0.016	0.026	-0.017	0.020
		Root (mm)	-0.012	-0.024	-0.002	-0.013	-0.029	-0.005	-0.013	-0.035	-0.004	-0.01 1	-0.037	-0.001
	Canine	Crown (mm)	-0.151	0.209	0.041	-0.085	0.121	0.009	-0.021	0.026	-0.024	0.054	-0.050	-0.055
		Root (mm)	0.067	-0.062	-0.072	0.026	-0.019	-0.051	-0.013	0.024	-0.024	-0.054	0.059	-0.01 1
	PM2	Crown (mm)	0.009	-0.051	-0.037	-0.014	0.045	-0.068	-0.043	0.138	-0.092	-0.086	0.214	-0.091
		Root (mm)	-0.004	0.017	-0.028	0.009	-0.021	-0.076	0.028	-0.058	-0.116	0.051	-0.096	-0.127
	M1	Crown (mm)	0.004	-0.066	-0.020	-0.001	0.020	-0.001	-0.014	0.108	0.023	-0.011	0.171	0.038
		Root (mm)	0.003	0.008	-0.012	0.014	-0.035	-0.008	0.024	-0.075	0.000	0.034	-0.107	0.006
	M2	Crown (mm)	-0.002	-0.060	-0.004	0.012	0.030	0.058	0.037	0.108	0.114	0.067	0.168	0.159
		Root (mm)	-0.003	0.025	0.003	-0.002	-0.010	0.056	0.000	-0.043	0.105	-0.002	-0.069	0.146
Incisor retraction	CI	Crown (mm)	0.028	0.194	0.064	0.024	0.207	0.081	0.020	0.211	0.091	0.015	0.209	0.101
		Root (mm)	-0.013	-0.046	-0.081	-0.012	-0.060	-0.079	-0.010	-0.067	-0.076	-0.008	-0.074	-0.068
	LI	Crown (mm)	0.067	0.220	0.049	0.064	0.231	0.059	0.057	0.232	0.068	0.050	0.225	0.081
		Root (mm)	-0.009	-0.018	-0.134	-0.008	-0.025	-0.1 35	-0.008	-0.032	-0.131	-0.009	-0.042	-0.119
	Canine	Crown (mm)	0.064	-0.034	0.023	0.151	-0.090	-0.010	0.230	-0.149	-0.040	0.317	-0.204	-0.074
		Root (mm)	-0.019	-0.008	0.027	-0.067	0.027	0.027	-0.110	0.059	0.032	-0.157	0.093	0.032
	PM2	Crown (mm)	0.032	-0.049	-0.024	0.004	0.049	-0.048	-0.026	0.135	-0.061	-0.069	0.206	-0.059
		Root (mm)	-0.017	0.013	-0.017	0.000	-0.029	-0.058	0.019	-0.067	-0.086	0.044	-0.103	-0.096
	Ml	Crown (mm)	0.025	-0.062	-0.014	0.017	0.025	0.010	0.005	0.108	0.032	0.006	0.165	0.046
		Root (mm)	0.000	0.004	-0.003	0.012	-0.041	0.003	0.022	-0.080	0.011	0.032	-0.110	0.016
	M2	Crown (mm)	0.016	-0.056	0.001	0.027	0.035	0.064	0.050	0.107	0.117	0.080	0.161	0.159
		Root (mm)	-0.007	0.023	0.010	-0.004	-0.012	0.062	-0.003	-0.043	0.109	-0.005	-0.068	0.148
Mesialization of the posterior teeth	CI	Crown (mm)	0.014	0.168	0.073	0.010	0.176	0.084	0.005	0.185	0.093	0.002	0.185	0.103
		Root (mm)	-0.008	-0.048	-0.057	-0.007	-0.058	-0.057	-0.005	-0.065	-0.057	-0.003	-0.072	-0.050
	LI	Crown (mm)	0.025	0.180	0.071	0.021	0.188	0.075	0.018	0.201	0.081	0.013	0.200	0.089
		Root (mm)	-0.001	-0.050	-0.098	0.000	-0.054	-0.102	0.003	-0.058	-0.107	0.003	-0.067	-0.103
	Canine	Crown (mm)	0.014	0.182	0.048	0.091	0.102	0.009	0.177	0.045	-0.021	0.262	-0.022	-0.055
		Root (mm)	0.008	-0.056	-0.075	-0.035	-0.010	-0.063	-0.080	0.023	-0.061	-0.125	0.060	-0.054
	PM2	Crown (mm)	0.064	-0.082	-0.060	0.018	0.000	-0.073	-0.015	0.089	-0.077	-0.055	0.166	-0.080
		Root (mm)	-0.029	0.023	-0.048	-0.001	-0.016	-0.075	0.019	-0.050	-0.095	0.039	-0.085	-0.110
	Ml	Crown (mm)	0.058	-0.114	-0.027	0.040	-0.025	-0.001	0.021	0.067	0.015	0.01 1	0.135	0.030
		Root (mm)	-0.003	0.010	-0.005	0.012	-0.031	0.002	0.023	-0.073	0.001	0.032	-0.103	0.004
	M2	Crown (mm)	0.059	-0.122	0.007	0.082	-0.037	0.059	0.058	0.059	0.119	0.061	0.134	0.167
		Root (mm)	-0.022	0.042	0.027	-0.025	0.010	0.070	-0.012	-0.024	0.117	-0.007	-0.052	0.157
Premolar mesialisation	CI	Crown (mm)	0.018	0.052	0.012	0.013	0.068	0.028	0.010	0.085	0.039	0.005	0.088	0.052
		Root (mm)	-0.01 1	-0.001	-0.020	-0.009	-0.014	-0.022	-0.007	-0.023	-0.026	-0.005	-0.032	-0.021
	LI	Crown (mm)	0.027	0.065	0.027	0.023	0.081	0.034	0.018	0.096	0.040	0.012	0.097	0.049
		Root (mm)	-0.007	-0.026	-0.044	-0.004	-0.032	-0.052	-0.002	-0.036	-0.058	-0.003	-0.044	-0.054
	Canine	Crown (mm)	0.016	0.058	0.022	0.098	-0.007	-0.015	0.177	-0.063	-0.052	0.258	-0.1 26	-0.088
		Root (mm)	0.001	-0.019	-0.020	-0.044	0.020	-0.016	-0.085	0.055	-0.019	-0.128	0.091	-0.016
	PM2	Crown (mm)	0.076	-0.246	-0.026	0.035	-0.144	-0.047	-0.004	-0.044	-0.061	-0.054	0.033	-0.064
		Root (mm)	-0.037	0.085	0.018	-0.01 1	0.037	-0.023	0.015	-0.005	-0.056	0.042	-0.041	-0.071
	Ml	Crown (mm)	-0.009	0.014	0.009	-0.024	0.110	0.034	-0.045	0.208	0.065	-0.051	0.280	0.081
		Root (mm)	0.008	-0.009	0.005	0.020	-0.053	0.012	0.033	-0.100	0.026	0.043	-0.133	0.031
	M2	Crown (mm)	-0.031	0.012	-0.015	-0.021	0.108	0.048	-0.002	0.198	0.108	0.022	0.268	0.159
		Root (mm)	0.006	0.008	-0.017	0.008	-0.029	0.036	0.012	-0.066	0.087	0.012	-0.096	0.133
Molar mesialisation	CI	Crown (mm)	0.014	0.122	0.053	0.008	0.127	0.064	0.004	0.128	0.070	0.000	0.125	0.080
		Root (mm)	-0.009	-0.032	-0.041	-0.007	-0.041	-0.038	-0.005	-0.046	-0.034	-0.003	-0.052	-0.027
	LI	Crown (mm)	0.026	0.136	0.058	0.021	0.142	0.065	0.018	0.145	0.070	0.012	0.140	0.079
		Root (mm)	-0.004	-0.043	-0.075	-0.002	-0.049	-0.076	-0.001	-0.053	-0.074	-0.001	-0.061	-0.068
	Canine	Crown (mm)	0.015	0.134	0.043	0.095	0.058	0.010	0.180	-0.009	-0.018	0.263	-0.072	-0.051
		Root (mm)	0.004	-0.044	-0.050	-0.041	-0.004	-0.037	-0.086	0.030	-0.027	-0.1 31	0.065	-0.021
	PM2	Crown (mm)	-0.030	0.117	-0.033	-0.051	0.212	-0.056	-0.077	0.303	-0.066	-0.107	0.382	-0.068
		Root (mm)	0.012	-0.045	-0.053	0.026	-0.089	-0.093	0.043	-0.1 30	-0.116	0.060	-0.169	-0.1 30
	Ml	Crown (mm)	0.064	-0.160	-0.049	0.047	-0.071	-0.024	0.032	0.007	-0.005	0.028	0.069	0.012
		Root (mm)	-0.003	0.020	-0.021	0.010	-0.024	-0.015	0.022	-0.060	-0.01 1	0.030	-0.091	-0.004
	M2	Crown (mm)	0.074	-0.167	0.003	0.089	-0.079	0.063	0.076	0.001	0.115	0.083	0.068	0.163
		Root (mm)	-0.026	0.056	0.030	-0.024	0.021	0.079	-0.019	-0.005	0.121	-0.016	-0.031	0.162

1 CI, central incisor; LI, lateral incisor; +X, right; -X, left; +Y, posteriorly; -Y, anteriorly; +Z, extrusion; -Z, intrusion.

Anti-tipping design			0°			1°			2°			3°
Local co-ordinate system			X	Y	Z	X	Y	Z	X	Y	Z	X	Y	Z
En-masse retraction	CI	Crown (mm)	-0.007	-0.171	0.018	-0.013	-0.1 86	0.015	-0.017	-0.189	0.008	-0.024	-0.190	-0.001
		Root (mm)	-0.004	0.005	0.013	0.002	0.007	0.009	0.004	0.008	0.002	0.007	0.009	-0.007
	LI	Crown (mm)	-0.009	-0.187	0.061	-0.016	-0.196	0.060	-0.019	-0.195	0.054	-0.023	-0.192	0.045
		Root (mm)	0.005	0.094	0.051	0.005	0.102	0.050	0.003	0.103	0.044	0.004	0.104	0.035
	Canine	Crown (mm)	-0.112	-0.163	0.060	-0.006	-0.161	0.052	0.114	-0.157	0.039	0.240	-0.160	0.033
		Root (mm)	0.058	0.063	0.058	0.001	0.065	0.055	-0.062	0.066	0.047	-0.128	0.071	0.045
	PM2	Crown (mm)	0.107	-0.004	0.043	0.010	-0.005	0.076	-0.077	-0.006	0.100	-0.160	0.010	0.109
		Root (mm)	-0.036	-0.001	0.035	0.009	-0.008	0.075	0.049	-0.016	0.105	0.091	-0.028	0.116
	Ml	Crown (mm)	0.111	-0.011	0.018	0.024	-0.018	0.005	-0.059	-0.022	-0.007	-0.119	-0.035	-0.017
		Root (mm)	-0.014	-0.002	0.015	0.030	-0.005	0.006	0.068	-0.006	-0.004	0.100	-0.009	-0.013
	M2	Crown (mm)	0.098	-0.008	-0.016	0.007	-0.038	-0.071	-0.061	-0.075	-0.117	-0.114	-0.117	-0.156
		Root (mm)	-0.037	0.005	-0.018	-0.006	0.010	-0.072	0.020	0.016	-0.120	0.041	0.024	-0.160
Canine distalisation	CI	Crown (mm)	0.029	0.039	-0.015	0.025	0.026	-0.022	0.021	0.021	-0.030	0.015	0.021	-0.036
		Root (mm)	-0.019	-0.004	-0.014	-0.018	-0.002	-0.021	-0.018	0.000	-0.029	-0.015	-0.001	-0.035
	LI	Crown (mm)	0.041	0.002	-0.013	0.037	-0.006	-0.014	0.033	-0.008	-0.017	0.029	-0.007	-0.021
		Root (mm)	-0.003	0.022	-0.015	-0.001	0.028	-0.016	0.000	0.032	-0.020	0.003	0.030	-0.023
	Canine	Crown (mm)	-0.256	-0.008	0.052	-0.142	-0.001	0.044	-0.023	0.011	0.032	0.087	0.003	0.029
		Root (mm)	0.110	0.006	0.039	0.047	0.007	0.037	-0.017	0.005	0.031	-0.072	0.015	0.034
	PM2	Crown (mm)	0.054	0.003	0.034	-0.042	-0.004	0.071	-0.138	-0.004	0.101	-0.223	0.015	0.107
		Root (mm)	-0.016	-0.003	0.029	0.027	-0.009	0.074	0.071	-0.019	0.110	0.115	-0.031	0.117
	Ml	Crown (mm)	0.068	0.005	0.014	-0.020	-0.003	0.002	-0.111	-0.003	-0.012	-0.173	-0.015	-0.023
		Root (mm)	-0.005	-0.006	0.012	0.038	-0.009	0.003	0.078	-0.009	-0.009	0.110	-0.012	-0.019
	M2	Crown (mm)	0.058	0.013	-0.001	-0.032	-0.016	-0.056	-0.108	-0.055	-0.106	-0.166	-0.094	-0.148
		Root (mm)	-0.026	-0.001	-0.001	0.004	0.004	-0.056	0.034	0.010	-0.108	0.055	0.018	-0.151
Incisor retraction	CI	Crown (mm)	-0.009	-0.200	0.051	-0.016	-0.218	0.043	-0.021	-0.227	0.037	-0.026	-0.230	0.026
		Root (mm)	0.001	0.007	0.044	0.004	0.009	0.035	0.007	0.009	0.028	0.009	0.009	0.018
	LI	Crown (mm)	-0.016	-0.210	0.104	-0.024	-0.224	0.102	-0.031	-0.228	0.095	-0.036	-0.229	0.079
		Root (mm)	0.005	0.099	0.093	0.008	0.104	0.090	0.010	0.106	0.083	0.012	0.107	0.067
	Canine	Crown (mm)	0.057	-0.040	-0.032	0.162	-0.066	-0.023	0.263	-0.086	-0.019	0.367	-0.112	-0.009
		Root (mm)	-0.015	0.010	-0.029	-0.070	0.029	-0.016	-0.121	0.043	-0.008	-0.175	0.062	0.005
	PM2	Crown (mm)	0.057	-0.018	0.019	-0.042	-0.021	0.049	-0.132	-0.018	0.068	-0.212	0.003	0.074
		Root (mm)	-0.016	0.011	0.019	0.032	0.004	0.057	0.075	-0.006	0.081	0.117	-0.021	0.087
	Ml	Crown (mm)	0.066	-0.015	0.005	-0.023	-0.020	-0.010	-0.108	-0.020	-0.024	-0.165	-0.030	-0.033
		Root (mm)	-0.004	-0.001	0.004	0.042	-0.004	-0.008	0.081	-0.006	-0.020	0.111	-0.009	-0.029
	M2	Crown (mm)	0.058	-0.006	-0.006	-0.034	-0.032	-0.062	-0.105	-0.067	-0.110	-0.157	-0.106	-0.148
		Root (mm)	-0.025	0.003	-0.007	0.006	0.008	-0.062	0.033	0.013	-0.112	0.053	0.020	-0.152
Mesialisation of the posterior teeth	CI	Crown (mm)	-0.019	-0.181	0.028	-0.025	-0.192	0.024	-0.032	-0.204	0.020	-0.036	-0.209	0.011
		Root (mm)	0.004	0.003	0.022	0.006	0.005	0.017	0.010	0.005	0.013	0.012	0.006	0.004
	LI	Crown (mm)	-0.043	-0.183	0.056	-0.050	-0.189	0.057	-0.058	-0.201	0.059	-0.062	-0.204	0.051
		Root (mm)	0.021	0.097	0.047	0.024	0.102	0.048	0.029	0.107	0.050	0.031	0.111	0.042
	Canine	Crown (mm)	-0.135	-0.124	0.044	-0.018	-0.128	0.046	0.086	-0.154	0.051	0.198	-0.173	0.055
		Root (mm)	0.068	0.049	0.040	0.006	0.054	0.049	-0.047	0.071	0.059	-0.103	0.084	0.067
	PM2	Crown (mm)	0.101	-0.042	0.051	0.010	-0.022	0.071	-0.085	-0.017	0.082	-0.169	0.000	0.092
		Root (mm)	-0.027	0.017	0.051	0.020	0.000	0.074	0.059	-0.01 1	0.090	0.100	-0.021	0.102
	Ml	Crown (mm)	0.124	-0.039	0.009	0.032	-0.035	-0.007	-0.063	-0.031	-0.013	-0.133	-0.032	-0.021
		Root (mm)	-0.010	0.000	0.007	0.032	-0.006	-0.006	0.075	-0.009	-0.010	0.106	-0.011	-0.017
	M2	Crown (mm)	0.130	-0.035	-0.019	0.046	-0.072	-0.065	-0.057	-0.066	-0.115	-0.134	-0.082	-0.158
		Root (mm)	-0.048	0.014	-0.023	-0.021	0.024	-0.068	0.012	0.018	-0.118	0.036	0.020	-0.161
Premolar mesialisation	CI	Crown (mm)	0.008	-0.053	0.018	0.000	-0.074	0.013	-0.007	-0.093	0.012	-0.013	-0.102	0.004
		Root (mm)	-0.009	-0.004	0.017	-0.005	-0.002	0.011	-0.002	0.000	0.010	0.002	0.000	0.001
	LI	Crown (mm)	0.001	-0.072	0.022	-0.008	-0.087	0.025	-0.018	-0.100	0.029	-0.025	-0.104	0.021
		Root (mm)	0.005	0.048	0.018	0.010	0.056	0.021	0.013	0.063	0.024	0.015	0.067	0.016
	Canine	Crown (mm)	-0.038	-0.051	0.009	0.071	-0.067	0.016	0.172	-0.087	0.028	0.279	-0.104	0.036
		Root (mm)	0.020	0.016	0.009	-0.037	0.029	0.021	-0.086	0.043	0.037	-0.139	0.056	0.048
	PM2	Crown (mm)	0.259	-0.005	0.009	0.151	0.004	0.037	0.045	0.011	0.060	-0.043	0.037	0.070
		Root (mm)	-0.093	0.013	-0.011	-0.037	-0.002	0.025	0.013	-0.017	0.054	0.055	-0.034	0.065
	Ml	Crown (mm)	-0.016	0.008	-0.007	-0.115	0.008	-0.021	-0.218	0.014	-0.041	-0.291	0.009	-0.051
		Root (mm)	0.010	-0.005	-0.007	0.054	-0.008	-0.019	0.101	-0.010	-0.038	0.135	-0.013	-0.047
	M2	Crown (mm)	-0.016	0.028	0.017	-0.114	0.001	-0.039	-0.203	-0.033	-0.092	-0.272	-0.069	-0.137
		Root (mm)	-0.005	-0.007	0.018	0.027	-0.002	-0.038	0.060	0.003	-0.093	0.085	0.009	-0.141
Molar mesialisation	CI	Crown (mm)	-0.010	-0.131	0.021	-0.017	-0.141	0.014	-0.022	-0.144	0.009	-0.026	-0.146	0.000
		Root (mm)	0.000	0.000	0.017	0.002	0.002	0.010	0.005	0.001	0.004	0.007	0.001	-0.005
	LI	Crown (mm)	-0.026	-0.142	0.039	-0.033	-0.149	0.037	-0.037	-0.154	0.034	-0.041	-0.154	0.024
		Root (mm)	0.015	0.079	0.032	0.019	0.083	0.030	0.022	0.085	0.027	0.024	0.087	0.017
	Canine	Crown (mm)	-0.099	-0.097	0.026	0.015	-0.108	0.025	0.125	-0.129	0.024	0.233	-0.149	0.028
		Root (mm)	0.050	0.038	0.024	-0.011	0.046	0.028	-0.067	0.060	0.032	-0.121	0.074	0.040
	PM2	Crown (mm)	-0.118	-0.007	0.040	-0.213	-0.015	0.069	-0.307	-0.017	0.085	-0.391	-0.010	0.094
		Root (mm)	0.049	-0.003	0.050	0.098	-0.007	0.086	0.144	-0.013	0.106	0.187	-0.019	0.117
	Ml	Crown (mm)	0.172	-0.039	0.026	0.080	-0.036	0.012	-0.001	-0.033	0.001	-0.064	-0.039	-0.011
		Root (mm)	-0.018	-0.003	0.022	0.026	-0.008	0.011	0.064	-0.011	0.003	0.095	-0.013	-0.007
	M2	Crown (mm)	0.177	-0.041	-0.019	0.088	-0.071	-0.072	0.003	-0.073	-0.117	-0.065	-0.091	-0.160
		Root (mm)	-0.062	0.016	-0.024	-0.032	0.021	-0.077	-0.009	0.022	-0.120	0.013	0.024	-0.163

1 +x, mesicilly; -x, distally; +y, buccally; -y, lingually; +z, intrusion; -z, extrusion.

References

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Articles in this issue

DOI: https://doi.org/10.2478/aoj-2024-0029 | Journal eISSN: 2207-7480 | Journal ISSN: 2207-7472

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Language: English

Page range: 149 - 171

Submitted on: Aug 1, 2024

Accepted on: Nov 1, 2024

Published on: Dec 30, 2024

Published by: Australian Society of Orthodontists Inc.

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Related subjects:

Medicine,

Basic medical science,

Basic medical science, other

© 2024 Yiru Jiang, Li Mei, Li Xu, Jue Wang, Xiaoxia Feng, Xiaoyan Chen, published by Australian Society of Orthodontists Inc.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 40 (2024): Issue 2 (July 2024)

Biomechanical effect of en-masse versus substep extraction space closure on posterior teeth using clear aligners: a finite element study

Figures & Tables

Figure 1.

Table I.

Number of nodes and elements of the components of finite element models

Table II.

Material properties

Figure 2.

Figure 3.

Table III.

Mesiodistal force and moment at the resistance centre of each tooth in the local co-ordinate system under en-masse and substep space closure

Figure 4.

Figure 5.

Figure 6.

Paradigm

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