Effect of maxillary incisor angulation and inclination on arch length

ad lay incisor angulatio~ on arch length am Hussels, Dr. Med. Dent.,* and Ram S. Nanda, D.D.S., MS., Ph.D.**

This article discusses the quantitative effects of angulation of incisors on dental arch length. The incisors are approximated mathematically to rectangular shapes, which enables the authors to calculate the change in arch length when teeth are tipped and to describe graphically the effect of other influencing parameters. The authors demonstrate that the height and the width of a tooth crown can enhance or diminish the effect of angulation on arch length. In addition to angulation, dental arch length is also influenced by torque. A mathematic formula has been derived and the results are demonstrated numerically and graphically. The authors show that vertical positioning of the brackets plays an important role because torquing is a rotational movement around the center of the bracket slot. Hence, in calculating the effect of torque on dental arch length, one must consider different angles and axes than those discussed by Andrews.4 The interaction between angulation an and its influence on dental arch length has been investigated. It has been found to cause little change in arch length. (AM J ORTHOD DENTOFAC ORTHOP 1987;9i :233-g.)

Key words: Xncisorangulation and inclination, space closure, bracket placement, mathematic evaluation

ssuming the presence of normal posterior occlusion with no overbite or overjet and no tooth-size arch length discrepancy, normalization of arch length may be influenced by angulation and inclination of the incisors. The purpose of this article is to explain mathematically the effect of inclination and angulation of maxillary incisors upon arch length. ANGULATION OF THE INCISORS-

Fig. 1 represents a front view of the crown of a maxillary incisor that is tipped relative to the occlusal plane 0. Assuming that this tooth has no axial rotation or lingual inclination, the mesiodistal width of the tooth occupies a measured amount of length within the arch. This length can be considered to be the sum of L, and L,. The amount of arch length this tooth will occupy is influenced by how much it is tipped; the length can be calculated by using one of the following equations: LA = L, + b

or

L, = b cos y f h sin y

In this equation L, is the total mesiodistal space occupied by the tooth L, is the space occupied by the incisal edge of the tipped crown (determined by dropping a perpendicular from the mesial edge to the oc-

*Visiting Fellow in Qthodontics. **Professor and Chairman, Department homa College of Dentistry.

clusal plane). L2 is additional space occupied if t&e tooth is tipped (determined by drawing a perpendicular from the distal cervical part of the tooth to the occlusal plane); b is the actual mesiodistal width of the crown; b isthe actual height of the crown; y is the angle of crown angulation. To render this equation independent of the crown width and to isolate the effect of angulation, both sides are divided by b: LA h - = cos y + ; sm y

b

Table I gives the calculated values of 2 for varying angles of crown tipping (y). The y values are listed on a scale of 1” within the range of 0” to 20”. Tbe arch length is influenced not only by the angle of inclination but also by the height/width ratio of the crown as indicated by the coefficient i in the above equation. Therefore, Table I is constructed to give calculated values of the additional arch length required by tipping an incisor from 0” to 20” for a varying heig~t/widt~ ratio. The i ranges from 0.8 to 1.8 in gra The calculated total arch length (LA) occupied by h the tipped tooth for a given b ratio and the crown an-

of Orthodontics,

University

of Ok&

gulation can be determined by finding the a~pro~~at~

Mussels and Nanda

Fig. 1. Diagrammatic representation showing increased space obtained by distal tipping of a tooth. The rectangular form represents the incisor tooth crown. h, Cervico-occlusal crown length. 6, Mesiodistal width of the crown. y, Angle of mesiodistal tip.

value from Table X, which is 2,

and multiplying it by

b. The results calculated by using Table I are accurate up to F 0.1 mm in actual arch length. The sources of error, if present, are in computations resulting from treating the crown form of an incisor as a rectangle. The varying + values given in Table I are graphically shown in Fig. 2. We find that the L, is highest when tbe incisors are tipped in the range of 40” to 60”. Fig. 2 indicates that larger crown height/width ratios will increase the L, even when the angle of crown tip is held constant. At a height/width ratio of 0.8, the curve is flatter compared with the ratio of 1.8. At an angulation of 20”, an incisor with a height/width ratio of 1.8 will occupy 28% more space in the arch than an incisor with a height/width ratio of 0.8, assuming both teeth have the same crown width. It is apparent, therefore, that both crown height and angulation can affect the arch length. Clinically, this information may be applied to eliminate spacing between the teeth resulting from variations in size of the teeth. Approximately 1 mm of space between two maxillary incisors can be closed by tipping a central incisor 10.4 mm 6” with a crown height/width ratio of 8,0 mm or by tipping both the central incisors 3” each. N OF THE MAXILLARY UAL T~~~~~~

INCISORS

From an occlusal view, the incisal edges of normally aligned maxillary incisors form an arc. The radius of this arc determines the width of the arch form. The

cervical part of normally inclined crowns describes an arc smaller than the one formed at the incisal edges, so there is less arch length at the cervical part of the crowns in comparison with the incisal edges. Hence, lingual crown inclination of the maxillary anterior teeth will result in space consolidation. The degree of space consolidation depends upon: Angle of lingual inclination (lingual root torque) The radius of the arc formed by the teeth. In patients under orthodontic treatment, it will be the radius of an arc formed and described by the bracket slots. The vertical position of the point of rotationin this instance, the center of the bracket slot Examination of Fig. 3 shows that the lingual inclination of the labial cervical part of the crown will depend on its distance k from the center of the bracket, the angle IS, which is formed by k and the line perpendicular to the occlusal plane and passing through the center of the bracket slot. If “R” is the radius of an arc described through the center of the bracket slots and “r” the radius of an arc passing through the center of the cervical part of the incisor crowns? it is obvious that radius R is equal to r + p as shown in Fig. 3. p can be described as k sin p. Using the general equation for the length of an arc 27rCX of a circle, L = __ r, z equal to 3.14, and (Y rep360 resenting the opening angle of the arc of the circle, the original equation, p = R - r, can be substituted to L=

s(R-k

sin p). L, is the reduced length

of the arch after lingual root tor ue and consolidation of spaces between the incisors. For normalization this equation can be trall§fo~ed to:

Table II has been formulated to show normalized arch length based on the mathematic equation for varying degrees of p angle and k/R ratios. Fig. 4 is a graphic representation of the same data. Absolute arch length can be calculated by multiplying

the FR

by

2CXnR If length k is longer and 6 angle is the same, 360 ’ the lingual root torque will be more effective in consolidating the maxillary incisors. From the right side of the normalized equation, it can be concluded that the size of R influences the change in arch length at the level of the cervical part of the teeth when the maxillary

Effect of maxillary

incisor angulation

and inclination

h, I

0

Fig. 2. Effect (normalization The

curves

0.1 and

5

10

15

of mesiodistal angulation on dental in respect to the width b of the tooth are

calculated

angulation

within

for the the range

height/width

ratio

20

Y

degrees

on arch length

b

i*

25

arch length. Ordinate is normalized arch length crown). Abscissa is the angle of distal inclination. % varying

of 0” to 25” to illustrate

between the range

0.8 and of common

1.8 in gradations clinical

of

deviations.

Table I. Normalized values of dental arch length as a result of mesiodistal angulation (y) and height~widtb of the tooth crown

y 1 2 3 4 5 6 I 8 9 10 11 12 13 14 I5 16 17 18 19 20

I

1.0

1.1

1.2

1.3

1.015

1.017

1.030 1.045

1.034 1.050

1.019 1.037

1.060 1.074 1.088 1.102 1.115 1.128 1.141 1.153 1.165 1.176 1.188

1.067

1.074

1.083 1.099 1.114 1.129 1.144 I.158 1.172 1.186 1.199 1.212 1.224 1.236 1.248 1.260 1.271 1.281

1.092 1.109

1.020 1.041 1.061 1.081 1.100 1.119 1.138 1.157 1.175 1.193 1.210 1.227 1.244 1.260 1.276 1.292 1.307 1.321 1.336 1.350

0.8

0.9

1.013 1.027 1.040 1.053 1.065 1.078 1.090 I.101 1.112 1.123 1.134 1.144 1.154 1.163 1.172 1.181 1.190 1.198

I.198 1.209 1.219

1.205

1.229 1.238

1.213

1.247

1.056

1.126 1.143 1.159 1.175 1.191 1.206 1.221 1.236

1.250 1.264 1.277 1.290

1.303 1.315

incisors are torqued. In cases with a large R (which is the diameter of the anterior segment of the dental arch), torquing is iess effective compared with cases with a . This is true because R is the denominator on

1.4

1.5

1.6

1.7

1.8

1.022

1.024

1.044

1.048

1.027 1.055

1.066 1.088 1.109 1.130 1.150 1.171 1.191 1.210 1.229

1.071 1.095 1.118 1.140 1.163 1.185 1.206 1.227 1.248

1.026 1.051 1.077 1.102 1.126 1.151 1.175 1.199 1.222

1.248 1.266

1.269

1.029 1.058 1.087 1.116 1.144 1.172 I.199 1.226 1.253 1.250 1.306 1.331 1.356 1.381 1.405 1.429 1.453 1.476 1.498 1.521

1.03I 1.062 1 .a92 1.123 1.153 1.182 1.211 1.240 1.269 1.297 1.325 I.352 1.379 1.405 1.431 1.457 1.482 1.507 1.531 1.555

1.284 1.302 1.319 1.336

1.352 1.368 1.384

1.289 1.308 1.328 1.347

1.365 1.383 1.401 1.418

1.245 1.267 1.290

1.311 1.333 1.354 1.374 1.394 1.414 1.433 1.452

1.082 1.109 1.135

1.161 1.187 1.212 I .237 1.262 1.286 1.310 1.334 1.357 1.380 1.402 1.424 1.445 1.466 1.486

the right side of the equation. So in a case with a very large R, the right side of the equation will be close to one. This type of case can be described as possessing a broad anterior arch form in which the incisor teeth

Nussels and Nanda

I IL Fig. 3. Long axis of the crown to show inclination. R, Radius described by the arc formed by the points at the center of the brackets. r, Radius of the arc formed through the center of the cervical part of the crown. p, Difference between the two radii. k, Line connecting the center of the bracket with the center of the cervical part of the crown. p, Angle between a line perpendicular to the occlusal plane and line k. Table

II. Normalized values of dental arch length as they vary with the lingual inclination /3 and the ratio k

0.025 0.050 0.975 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 O.A75 0.500 0.525 0.550 0.575 0,600

0.995 0.991 0.986 0.982 0.978 0.973 0.969 0.965 0.960 0.956 0.952 0.947 0.943 0.939 0.934 0.930 0.926 0.921 0.917 0.913 0.908 0.904 0.900 0.895

0.991 0.982 0.974 0.965 0.957 0.948 0.940 0.931 Q.923 0.914 0.905 0.897 0.888 0.880 0.871 0.863 0.854 0.846 0.837 0.828 0.820 0.811 0.803 0.794

0.987 0.975 0.962 0.950 0.937 0.925 0.912 0.900 0.887 0.875 0.862 0.850 0.837 0.825 0.812 0.800 0.787 0.775 0.762 0.750 0.737 0.725 0.712 0.700

0.983 0.967 0.951 0.935 0.919 0.903 0.887 0.871 0.855 0.839 0.823 0.807 0.791 0.775 0.758 0.742 0.726 0.710 0.694 0.678 0.662 0.646 0.630 0.614

are lined up almost on a straight line. In contrast to this extreme example, insert a small R (less than k) into this equation, which leads to a large negative expression on the right side of the normalized equation. This means

0.980 0.96! 0.942 0.923 0.904 0.885 0.865 0.846 0.827 0.808 0.789 0.770 0.751 0.731 0.712 0.693 0.674 0.655 0.636 0.616 0.597 0.578 0.559 0.540

0.978 0.956 0.935 0.913 0.891 0.870 0.848 0.826 0.805 0.783 0.761 0.740 0.718 0.696 0.675 0.653 0.631 0.610 0.588 0.566 0.545 0.523 0.502 0.480

0.976 0.953 0.929 0.906 0.882 0.859 0.835 0.812 0.788 0.765 0.741 0.718 0.694 0.671 0.647 0.624 0.600 0.577 0.553 0.530 0.506 0.483 0.459 0.436

0.975 0.950 0.926 0.901 0.876 0.852 0.827 0.803 0.778 0.753 0.739 0.704 0.679 0.655 0.630 0.606 0.581 0.556 0.532 0.501 0.482 0.458 0.433 0.409

0.975 0.950 0.925 0.900 0.875 0.850 0.825 0.800 0.775 0.750 0,725 0.700 0.675 0.650 0.625 0.600 0.575 0.550 0.525 0.500 0.475 0.450 0.425 0.400

that in patients who have a narrow anterior arch form torquing decreases the dental arch length at the cervical level of the teeth. The analysis suggests that in patie&with a broad flat arch form of the incisor teeth torquing

Eflect of maxillary

incisor angulation

and inclination

on arch length

. 4. Effect of labiolingual inclination (torque) on dental arch length. Ordinate is the normalized arch length. Abscissa is the angle of lingual inc!ination (p). The curves are calculated for i

ratios

graded

between

0.05

and

0.6.

will be less effective in closing the anterior spaces. It can also be concluded that the placement of brackets is critical to achieve effective labiolingual inclination of the incisors. Brackets positioned more incisally will produce more effective torquing and potential for closing all the incisor spaces. For small teeth, like lateral incisors, the brackets are to be placed closer to the cervical region because of their shorter crown height. In these instances, the p is large and may even exceed 90”. The attempt to torque these teeth may result in actually flaring them. This explains why clinicians have for a long time put less torque in the wire or bracket of lateral incisors than in central incisors.* Fig. 5 illustrates the effect of bracket position on lingual root torque. EEN MESIODISTAL ANNUAL INCLINATION 360 1- -kcos y sm p L -= 2nctR R

The effect of rnesiodistal angulation at a given /3 and i ratio on dental arch length is shown in Fig. 6. The calculated values obtained show that with increasing mesiodistai angulation, the effect of lingual inclination on dental arch length at the level of the cervical part of the maxillary incisor crowns is reduced. In the above equation, increasing angulation y reduces the size of the numerator, thus increasing the whole expression on the right side of the equation even if lingual inclination /3 is kept constant. So lingual root torque and distal a~gulati~n interact in respect to dental arch length. An increase of 10” in angirlation y reduces the effect of lingual inclination on dental arch length by 1.5%. It

Fig. 5. Point of rotation related to bracket position. Large k and smaller p values are more effective for lingual torquing of the cervical part of the crown. More cervical placemerit of the bracket reduces the k value and the torquing procedures may result in unwanted further flaring of the incisal edges.

Fig. 6. interaction between mesiodistal angulation and labiolingual inclination (torque). Ordinate is normalized arch length expected after torquing. Abscissa is the angle of lingual inclination p. The effect of mesiodistal angulation y at values of 0 and 30” are shown

on k curves

of 0.2, 0.4, and 0.6.

may be concluded, therefore, that the interaction between mesiodistal angulation and lingual i~cli~at~o~ is minimal, especially when k/R is small and y is less than 20”. DISCUSSION

Normalizing occlusion involves many factors that include normalizing the mesiodistal angulation and labiolingual inclination of the teeth. These a~angeme~ts are crucial in closing and consolidating the i~t~~~en~a~

2

Hrnssels and Nanda

spaces and obtaining an ideal overbite-overjet relationship. empster, Adams, and Duddles’ studied the inclination of the long axis of the teeth in normal dentitions. They found that all teeth are arranged at an angle to the occlusal plane and each has an optimum inclination labiolingually to best perform its individual and collective functions. Andrews3 studied angulations and inclinations of the long axis of untreated ideal occlusions and a large sample of orthodontically treated occlusions and found that there are six keys or quantities in the arrangement and occlusion of teeth. He defined the terms “angulation” as mesiodistal tip of the crown and ‘“inclination” as labiolingual or buccolingual inclination of the tooth crown. Andrews’ definition of torque is an angle between the tangent to the middle of the clinical crown and a perpendicular line dropped on the occlusal plane. The point of intersection of those lines is positioned on the occlusal plane close to the incisal edge. However, the clinician places the brackets much more cervically. This is true especially when bonding brackets on short lateral incisors in which the center of rotation (bracket slot) quite often is closer to the tooth neck than the incisal edge. Therefore, when torque is applied with an edgewise wire, one must consider different angles p compared with those torque angles discussed by Andrews.4 The reason is that the center of rotation is not close to the incisal edge but somewhere above the occlusal plane. Therefore, torquing, which is in physical terms a rotation around the center of the bracket slot, may even result in a forward movement of the incisal edge. We have considered the same definition for the terms used in this article. Ideally, the center of the cervical part of the crown is distal to the center of the incisal edge and the angulation varies with the individual tooth types. Arch length is clearly influenced by the degree of inclination and angulation of the maxillary incisors. Tuverson’ stated that distal angulation would increase dental arch length by more than 2 mm and lingual inclination would add another 1 mm. ut, as shown in this article, increase in dental arch length depends on various parameters, so the numbers given by Tuverson are more or less subjective. Achieving normal arch length in patients receiving orthodontic treatment sometimes can present difficulties that may not be completely perceived by an orthodontist. At times problems involving arch length are caused by tooth size discrepancy.6 However, there are instances when other factors are equally important. The focus of this article is to examine how angulation and inclination of the maxillary incisors may affect arch length.

For purposes of this discussion, an incisor tooth form has been considered as a rectangle. In view of the variation in size and form of incisor teeth, a rectangle was considered to be a fairly representative model. This has facilitated an analysis of the space closure problems. Our analysis shows that arch length is affected by the amount of angulation in combination with the height/width ratio of the incisor crowns. The large angulations and height/width ratios have a p~opo~iouately greater effect on arch length. The quantitative effect of angulation and height/width ratios on arch length is illustrated in Fig. 2 and Table I. Stripping of mesial and distal surfaces of incisors is often attempted to relieve dental crowding, reduce overjet, or stabilize treated occlusions. Injudicious and heavy stripping can set up conditions that may be difficult to rectify when the problems prima~ly resulted from incorrect angulation. In instances where the size of the incisors on the right and left sides is different and imbalance is inherent, the problems may be minimized by increasing the angulation of the smaller teeth. When changing crown angulation, the position of roots must also be considered. Labiolingual inclination of maxillary incisors also influences arch length. From an occlusal view, the arc formed by the incisal edges of the maxillary teeth should have a larger radius than the arc formed by the cervical parts of the crowns. If the incisal arc is not larger than the cervical arc, there will be adverse arch length, and esthetic and functional implications. Lingual torquing of the cervical part of the incisor crown is essential to produce favorable esthetic and functional relationships. Bracket placement on the incisor crown is an important determining factor in idealizing the labiolingual crown inclination. The most effective torquing is produced by placement of the bracket closest to the incisal edge because the center of the bracket receiving the edgewise wire is the center of rotation. The smaller p angle and larger k length are most conducive and effective in producing proper labiolingual inclinations. When the bracket placement is closer to the cervical part of the crown, attempts at torquing are least effective. Diminutive lateral incisors with a small incisorgingival crown height necessitate higher (cervical) bracket placement. Flaring and incorrect inclination will invariably result in these cases. A better approach to “peg” lateral incisors may be to restore their size with crowns before the final space closure and finishing procedures so that the bracket can be at a proper height.

Effect of maxillary incisor angulation and inclination

1. Thurow RC. Edgewise orthodontics. 2nd ed. St. Louis: The C. V. Mosby Company, 1966:193. 2. Dempster WT. Adams WJ, Duddles RA. Arrangement in the jaws of the roots of the teeth. J Am Dent Assoc 1963;67:779-97. 3. Andrews LF. Six keys to normal occlusion. AM J ORTHOD !972;62:296-309. 4. Andrews LF. The diagnostic system: occlusal analysis. Dent Clin N Am 1976;20:671-90. 5. Tuverson DL. Anterior interocclusal relations. AM J ORTHOD 1980;78:361-70.

BOUND

VOLUMES

AVAILABLE

on arc/z lengtlr

6. Boiton WA. The clinical application of a tooth-sire analysis. AM J ORTHOD 1962:48:504-29.

Reprinr

requeslstG:

Dr. Ram Nanda University of Oklahoma Health Sciences Center College of Dentistry P.O. Box 26901, 1001 Stanton L. Young Blvd. Oklahoma City, OK 73190

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