Archwires are the vital and instrumental components of fixed orthodontic treatment. They store and deliver power through the brackets and bands to the teeth and surrounding tissues. Improperly shaped archwires create and contribute to many posttreatment problems. Unplanned contraction or expansion, especially in the canine or molar regions, produces instability.

With the advent of highly elastic nickel-titanium preformed archwires, the clinician can often introduce larger cross-section rectangular wires in the early stages of leveling and alignment. Nickel-titanium archwires exhibit exceptional springiness, but wire bending in the classical sense (or the formability of the wire) is poor because they do not undergo plastic deformation until high forces are applied.

The initial nickel-titanium archwires (0.014 or 0.016 in) used for leveling and alignment exert light forces and are usually tied for short periods. They can be expected to have little influence on the arch form, since their forms are often temporarily distorted from tying on malaligned teeth. As the teeth align and the treatment progresses, heavier rectangular nickel-titanium or heat-activated nickel-titanium followed by stainless steel archwires are used. They have greater tensile strength and are used for longer times, resulting in more influence on the arch form.

The rectangular nickel-titanium archwires of 0.017 × 0.025 in and 0.019 × 0.025 in can permanently alter the arch form if tied for longer periods. This could be a significant factor in determining posttreatment retention and stability. The mandibular intercanine and intermolar widths are accurate indexes of the muscular balance inherent in each patient and dictate the limits of arch expansion during treatment. They are important to establish and maintain proper functional balance as an aid in treating and retaining orthodontic patients.

The dental arch form is initially shaped by the configuration of the supporting bone. After the eruption of teeth, the circumoral musculature and intraoral functional forces have significant influences over the arch form. The principal clinical factors that affect a patient’s arch form and dimensions are dental perimeter, cross-arch width, and arch depth ( Fig 1 ). An analytical equation of dental arch shape is necessary to describe the relationship between arch width, depth, and perimeter, which influence the arch form.

Dental arch shape with variables of arch width, depth, and perimeter that determine the arch form.

The occlusal views of dental arches in an Angle Class I occlusion have been described by the mathematical beta function with average correlation coefficients of 0.98 for the mandible and 0.97 for maxilla. The beta function is shown to be an accurate representation of planar projections of natural arch forms defined by spatial coordinates of labial bracket interfacing surfaces in maxillary and mandibular arches.

This study was undertaken to correlate the arch forms of Angle Class I normal occlusion and various archwire-bracket assemblies derived by the beta function in their form and dimension. The intercanine and intermolar widths of natural arch forms were compared with those of rectangular nickel-titanium archwire-bracket assemblies in the maxillary and mandibular arches of both sexes.

Material and methods

Orthodontic models of maxillary and mandibular arches of 20 male and 20 female subjects with normal occlusions were selected as the control groups from approximately 2500 students in Sawangi, Wardha, India. The control groups belonged to different ethnic groups of the Indian population, with various arch forms and sizes. The inclusion criteria were Angle Class I molar and canine relationships, normal overbite and overjet, and well-aligned arches with the second molars in occlusion. Models with attrition, fractured teeth, ectopically erupted teeth, deciduous teeth, and arch length deficiency greater than 3 mm were excluded. The control groups were female maxillary, female mandibular, male maxillary, and male mandibular arches.

A bracket height gauge was used to mark the bracket placement points on the labial surfaces of the teeth. Points were marked at the center of the tooth incisogingivally and mesiodistally along the long axis ( Fig 2 ). Each cast was oriented in a Microval Co-ordinate Measuring Microscope (Nikon, Tokyo, Japan ) for determining the coordinates of bracket placement points ( Fig 3 ). This device is used in the machine tool industry to measure the precision of parts with an accuracy of 0.001 mm. The linear accuracy of the machine is 0.006 mm, and reproducibility is 0.004 mm.

Marking of bracket placement points.

Oriented model in the measuring microscope.

The casts were secured to a fixed plane, and an optical beam was used to identify the 14 points. The corresponding x, y, and z coordinates were stored automatically in a computer data file. Coordinates of a point in space in 3 orthogonal axes were measured up to 1 μm. The curve fitting program (table curve 2-dimensional version 5.1; Systat Software, Chicago, Ill) was used to apply mathematical beta function (with the least squares method) to process the measured x and y coordinates. The z coordinates were reduced to zero to obtain planar projections of each point. The values were fed into the software, and the beta function was selected from the equation list for data processing and to derive uniform bilaterally symmetrical arch forms representing natural arch forms.

The average correlation coefficient between the dimensions measured on the models and the mathematical arch forms expressed by the beta function was 0.97 (SD 0.03) ( Tables I and II ). By using the average correlation coefficient as a measure of fit, the natural dental arch form is accurately represented mathematically by the beta function as also seen in previous studies. Therefore, the beta function was used to derive accurate planar projections of dental arch forms by using the coordinates of the bracket placement points. These arch forms were used for correlation with arch forms of 0.019 × 0.025-in rectangular nickel-titanium wire-bracket assemblies derived by the beta function.

Table I

Correlation of intercanine and intermolar widths and arch depth of bracket coordinates in the maxillary and mandibular arches of female subjects with the beta function

Sample	Measured values (mm)			Curve fit values (mm)			Correlation
	Intercanine width	Intermolar width	Depth	Intercanine width	Intermolar width	Depth	Coefficient
Maxilla
1	32.007	57.415	42.019	32.319	58.021	42.21	0.97
2	31.216	55.616	41.987	31.645	56.104	42.707	0.96
3	32.412	57.853	41.843	32.716	58.757	42.201	0.99
4	32.179	55.813	40.758	32.948	55.915	41.201	0.98
5	33.769	57.314	42.719	34.143	57.925	43.371	0.96
6	35.016	60.879	39.716	35.216	61.425	40.371	0.97
7	31.142	54.119	42.014	31.416	54.724	42.864	0.96
8	34.207	57.614	38.713	34.916	58.124	39.971	0.98
9	36.715	59.143	40.219	37.103	60.215	41.371	0.98
10	31.204	59.463	39.142	31.716	60.224	39.864	0.96
11	32.312	57.216	43.714	32.977	58.106	44.154	0.96
12	34.109	57.515	41.649	34.716	58.216	42.054	0.97
13	28.757	54.719	41.463	29.157	55.216	42.865	0.98
14	32.216	60.763	39.012	32.849	61.931	39.864	0.98
15	34.191	57.419	41.613	34.716	58.345	42.054	0.97
16	34.012	58.213	41.253	34.571	58.941	42.058	0.96
17	31.012	55.814	40.257	31.679	56.745	40.864	0.96
18	34.107	60.713	40.531	34.519	61.025	41.371	0.98
19	31.218	56.771	42.143	31.847	57.214	42.701	0.97
20	31.129	54.121	41.693	31.519	54.781	42.154	0.96
Average	32.65	57.42	41.12	33.13	58.10	41.81	0.97
SD	1.83	2.05	1.30	1.83	2.14	1.19	0.03
SEM	0.41	0.46	0.29	0.41	0.48	0.27	0.01
Maximum	36.71	60.88	43.71	37.10	61.93	44.15	0.99
Minimum	28.76	54.12	38.71	29.16	54.72	39.86	0.96
Mandible
1	24.213	51.718	38.027	24.816	52.241	38.514	0.98
2	24.131	51.415	36.857	24.619	52.247	37.956	0.98
3	25.017	55.614	36.219	25.413	56.104	36.701	0.96
4	24.867	51.713	39.016	25.013	52.147	39.167	0.96
5	25.105	54.721	36.845	25.419	55.021	37.228	0.97
6	26.213	58.137	39.147	26.701	59.051	39.824	0.98
7	24.315	51.713	37.916	24.859	52.051	38.686	0.97
8	26.191	54.619	38.153	26.512	55.051	38.824	0.96
9	26.314	54.859	34.819	26.814	55.284	35.239	0.97
10	27.131	55.613	36.913	27.456	56.121	37.199	0.95
11	25.814	51.783	33.857	26.219	53.184	34.439	0.96
12	26.138	54.783	35.716	26.598	55.284	36.239	0.96
13	21.949	51.649	36.413	22.017	52.213	37.236	0.97
14	25.513	57.413	38.71	26.014	58.741	39.239	0.97
15	27.716	54.615	36.014	28.151	55.787	37.839	0.97
16	25.312	53.714	37.914	25.819	54.159	38.139	0.98
17	26.413	54.106	37.013	26.713	54.253	37.647	0.98
18	27.013	56.213	38.814	27.431	57.105	38.641	0.97
19	22.813	53.141	37.615	22.859	53.841	38.849	0.98
20	23.817	50.014	32.143	24.314	50.147	33.167	0.98
Average	25.30	53.88	36.91	25.69	54.50	37.54	0.97
SD	1.45	2.19	1.79	1.52	2.32	1.70	0.03
SEM	0.33	0.49	0.40	0.34	0.52	0.38	0.01
Maximum	27.72	58.14	39.15	28.15	59.05	39.82	0.98
Minimum	21.95	50.01	32.14	22.02	50.15	33.17	0.95

 

SEM , Standard error of the mean.

Table II

Correlation of intercanine and intermolar widths and arch depth of bracket coordinates in the maxillary and mandibular arches of male subjects with the beta function

Sample	Measured values (mm)			Curve fit values (mm)			Correlation
	Intercanine width	Intermolar width	Depth	Intercanine width	Intermolar width	Depth	Coefficient
Maxilla
1	34.103	59.814	42.919	34.639	60.714	43.169	0.98
2	33.219	57.396	42.349	33.714	58.153	43.614	0.98
3	32.413	56.514	43.215	32.749	57.214	44.674	0.98
4	36.747	60.819	42.267	37.143	61.216	43.674	0.98
5	34.103	61.413	45.213	34.913	61.874	46.716	0.98
6	32.219	58.014	41.945	32.839	58.241	43.064	0.98
7	32.314	57.894	44.394	32.914	58.357	45.641	0.98
8	32.105	58.716	45.218	32.216	59.315	46.314	0.99
9	31.716	58.013	43.129	32.082	58.837	44.214	0.98
10	34.716	58.127	39.142	35.216	59.537	40.674	0.97
11	36.107	59.418	46.746	36.419	60.061	46.917	0.99
12	34.213	58.231	43.149	34.713	58.968	44.215	0.97
13	35.357	59.213	44.314	35.813	60.219	45.016	0.96
14	32.814	58.761	42.371	33.256	59.662	42.914	0.97
15	32.747	56.649	42.764	33.143	57.214	43.641	0.98
16	33.216	55.213	43.214	33.813	56.153	44.215	0.98
17	35.014	59.613	43.974	35.649	60.153	44.687	0.99
18	32.987	58.713	46.461	33.413	59.147	47.214	0.97
19	32.659	57.216	46.741	33.143	58.143	46.971	0.97
20	31.807	57.103	42.641	32.219	57.419	42.914	0.98
Average	33.53	58.34	43.61	34.00	59.03	44.52	0.97
SD	1.45	1.48	1.85	1.48	1.46	1.70	0.03
SEM	0.33	0.33	0.41	0.33	0.33	0.38	0.01
Maximum	36.75	61.41	46.75	37.14	61.87	47.21	0.99
Minimum	31.72	55.21	39.14	32.08	56.15	40.67	0.96
Mandible
1	27.945	56.719	38.103	28.143	57.413	38.413	0.97
2	25.341	55.614	35.917	25.817	56.419	36.416	0.98
3	28.164	57.107	36.914	28.716	57.654	37.814	0.98
4	27.264	54.715	37.213	27.614	55.025	37.819	0.96
5	27.647	55.915	39.214	28.103	56.314	40.215	0.96
6	26.814	53.613	37.814	27.417	53.707	38.814	0.96
7	27.914	53.719	37.614	28.513	54.959	38.271	0.97
8	27.641	53.615	33.716	28.115	54.819	34.291	0.97
9	28.146	57.412	38.416	28.716	58.012	39.145	0.95
10	27.846	57.615	40.215	28.315	58.214	40.697	0.97
11	27.761	54.197	39.614	28.149	54.942	40.215	0.96
12	27.209	55.012	40.716	27.769	55.553	41.678	0.95
13	27.461	53.007	38.214	28.144	53.009	39.845	0.98
14	23.016	55.127	40.159	23.412	55.713	41.023	0.98
15	25.541	57.614	37.2154	26.017	58.149	37.841	0.98
16	22.914	57.293	34.974	23.213	57.913	35.417	0.97
17	24.612	57.614	41.697	25.597	58.916	42.614	0.97
18	27.106	59.615	39.146	27.763	60.521	39.941	0.98
19	25.314	56.549	38.645	25.897	57.743	39.745	0.98
20	24.136	58.715	37.415	24.913	58.917	38.413	0.98
Average	26.54	56.04	38.15	27.02	56.70	38.93	0.97
SD	1.75	1.87	1.93	1.70	1.95	2.04	0.03
SEM	0.39	0.42	0.43	0.38	0.44	0.46	0.01
Maximum	28.16	59.62	41.70	28.72	60.52	42.61	0.98
Minimum	22.91	53.01	33.72	23.21	53.01	34.29	0.94

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