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Incisor uprighting: Mechanism for late secondary crowding in the anterior segments of the dental arches
Incisor uprighting: Mechanism for late secondary crowding in the anterior segments of the dental arches Raymond
E. Siatkowski,
D.M.D*
Essex, Conn.
T
he orthodontic literature abounds with speculation on the etiology of incisor crowding as a late (postpubertal) growth phenomenon. Lombardil cites most of the prominently considered factors in his survey of the literature: tooth discrepancies, arch size, interference of canines in lateral excursive movements, overbite, overjet, tooth removal, incisor-mandibular plane angle, type of original occlusion, third molars, and muscular forces. In their growth studies Bj6rk,2 Subtelny and Sakuda,3 and van der Linden4 report or present data revealing lingual uprighting of the incisor segments during late growth. Weinstein,5 citing Bjijrk’s study, speculated on the possible etiologic role of this uprighting in incisor-segment crowding during the typical postorthodontic-treatment growth period. LundstrijmG mentions the possible role of net lingual crown movement of the anterior teeth in decreases in arch circumference. The present study is a mathematical investigation of incisor uprighting, conducted in two parts. In the first part a mathematical method is derived to serve as a vehicle for the interpretation of and correlation between the data available from two growth studies, those of Sillman’ and Bj6rk.2 The method is used to predict the anterior crowding to be expected from the changes in mean anterior arch dimension reported by Sillman during a postpubertal growth period. The changes in mean incisor axial inclination during late growth reported by Bjiirk arc then related to the changes in mean anterior arch dimension of Sillman to determine the possible contribution of the former to the latter. The above investigation is based on mean dimensions and can therefore provide only a typical indication of the actual events. Therefore, the second part of the Supported by tional Institute *Resident
398
United States Public of Dental Research,
in Orthodontics,
University
Health Service Grant National Institutes of Connecticut
Health
DE of
00283 Health,
Center.
from the Bethesda,
NaMd.
Fig. 1. Assumed anterior ference. W, Canine width.
arch form 1, Anterior
and mathematical parameters. arch depth. R, Radius of circle.
D, Anterior
arch
circum-
CUSP
Fig.
2. Assumed
anterior
arch
form
with
mathematical
parameters
study tests the analysis developed in the first part by applying untreated subjects longitudinally. PART Methods
rearranged.
it to individual
I and
material
A mathematical method for calculating anterior arch circumference is derived first. The following assumptions are made for mathematical simplicity: oc-
400 Table
Am. J. Orthod. October1974
Siatkowski 1. Summary
of
selected
mean
anterior
arch
width
and
depth
data
Mule Maxilla
Age (yrs.1 Anterior Anterior Change Change
14
arch width, W (mm.) arch depth, I, (mm.) in width (mm.) in depth (mm.)
Sillman
Female Mandible
14
21
from
21
36.69
36.75 30.06
14.55
13.77
Mandibk
MaxiZZa 13
21
13
21
29.83 34.26 34.66 28.29 27.93 IO.12 9.30 13.73 12.99 9.96 9.04 -0.23 +0.40 -0.36 -0.82 -0.74 -0.92
+O.OS -0.78
clusally, the in&al edges and cusp tips of the maxillary and mandibular anterior six teeth fall on arcs of circles of appropriate radii, and the uprighting occurs in such a way that the mean position of the incisal edges and cusp tips also falls on arcs of circles of appropriate radii. For comparison, calculations are also carried out for catenary and parabolic anterior arch forms. E’ig. 1 illustrates the assumed anterior arch form and identifies the mathematical parameters of interest. The mathematical parameters arc shown in a more convenient form in Fig. 2. From geometry WCknow that the arc length of a circle subtended by the angle, 8, is equal to the product of the radius of the circle and the subtended angle, where the units of the angle are in radians rather than degrees. Applying this to the geometry illustrated in Fig. 2 : D ;;” = RL9 or, D = 2RO
(1)
We must now calculate R and 8 in terms of W and 1~. Applying rean theorem to the lower right triangle in Fig. 2 :
or,
R2 - 2RL
or,
Applying Fig. 2 :
the definition
+ 1~’ +
wz
IZ =
81,
JJ
+ 2
(2)
of a sine function
sin therefore,
9 =
the I’ythago-
w
T/R B =
= sin-’
to the lower right
triangle
in
‘1V ~~
(3)
If we are given anterior arch width, W, and anterior arch depth, I,, we can substitute their values into equation 2 to calculate the radius, R. The values of R and W can then be substituted into equation 3 to calculate the subtended angle, 0.
Volume
66
Table
II.
Nunabel’
Incisor
4
Calculated
mean
anterior
arch
uprighting
circumferences
Female
Male
Maxilla
Age (vs.1 Arch circumference, D (mm.) Change in arch circumference (mm.) Per cent change in arch circumference
Table
III.
anterior
Changes
arch
in anterior
arch
Male
(ages
401
/ Mandible
1
14 21 14 21 50.53 49’.25 38.45 37.04 -1.28 -1.41 -2.53 -3.67
circumferences
for
Maxilla
1 Mandible
13 21 47.43 46.45 -0.98 -2.07
catenary,
circular,
13 21 36.87 35.17 -1.70 -4.61
and
parabolic
forms
Catenary Circular Parabolic
14-U)
I
Female
(ages
13-H)
Maxilla
Mandible
Maxilla
llfandible
(mm.1
(mm.)
(mm.)
(mm.)
-1.57 -1.28 -1.01
-1.58 -1.41 -1.21
-1.29 -0.98 -0.69
-1.91 -1.70 -1.47
Finally, the values for R and 8 can be substituted in equation 1 to calculate the anterior arch circumference. Sillman’s longitudinal study includes data on the changes in anterior arch depth and width from birth to age 21 for forty subjects. The mean values and their increments for the late growth age span (male, ages 14 to 21; female, ages 13 to 21) are presented in Table I. The values in Table I reflect some interesting phenomena. Both of the arches for both male and female subjects show a decrease in anterior arch depth. The arch-depth decreases arc greater in the mandible than in the maxilla. The changes in anterior arch width arc increases in the maxilla and decreases in the mandible. Therefore, in the maxilla the width changes tend to relieve any arch circumference decrease due to depth decreases, whereas the width changes tend to aggravate the circumference decrease in the mandible. Results
The values from Table I for anterior arch width, W, and anterior arch depth, L, are substituted in equations 1, 2, and 3 to calculate the mean anterior arch circumferences. The results are shown in Table II. All arch circumference changes arc deercases. The absolute and per cent decreases are greater in the mandibular arch for both male and female subjects. The greatest absolute amI per cent decrease is in the female mandibular arch. Note that small increments in anterior arch depth and width (Table I) result in relatively larger changes in arch circumference. For comparison, the calculations were repeated for catenary and parabolic anterior arch forms, and the results are presented in Table III. The
contribution
of changes
in the incisor
axial
inclination
with
growth,
as
Am. J. Orthod. October 1974
A I = RI al A? : R2dz a, = CL2
IF , ,
Fig.
Table reported
3. Two
forms
IV. Mean by
Bitirk
of incisor
interincisal and
rotation.
angles
AZ ) A,,
A, Pure
and
tipping.
change
in anterior
plus
arch
bodily
depth
R2 > R,
movement.
(based
on
values
Sillman’) Male,
Interincisal angle (degrees) Change in interincisal angle (degrees)
Change in anterior
8, Tipping
SINCE
arch depth (mm.)
age
Hale,
18
128.45
ages
21-M
137.44
+8.99
I
Male (age 12-81) Maxi1 la -1.16
I
Ma.n4Jible -0.74
reported by Bjbrk, to changes in the anterior arch depth is evaluated next. The mean values from BjBrk’s cross-sectional study as well as the corresponding values from Sillman’s longitudinal data are shown in Table IV. Since there is no known cranial plane fixed with growth with which to relate changes in individual incisor inclination, no attempt is made here to assign a given amount of the net change in interincisal angle to either incisor. Bjiirk does measure the incisal angles to occlusal plane and occlusal plane to several cranial reference lines, none of which one can use as an absolute reference during growth. A change in incisal angle can be produced by pure tipping (rotation about the center of resistance of the root) or by a combination of tipping and bodily movement (rotation about a point other than the center of resistance of the root). Mechanistically, the former is simpler, requiring only a net lingually directed force on the crown. The latter, however, if it produces a center of rotation apical to the center of resistance of the root, will produce a greater arch depth change per degree of rotation than pure tipping, as illustrated in Fig. 3. Now, with certain assumptions, it is possible to calculate the axial inclination
zwcisor u~prigh’ti7ly
Fig. 4. Calculations The
angles
are
of maxillary and mandibular exaggerated in the drawing for
arch clarity.
depth
decreases
with
pure
403
tipping.
changes that would he associated with pure tipping to account for the arch-depth tlecreascs in the Sillman sample. On the basis of Wheeler’s” mean dimensions for incisors and Haack’s and Burstone’s!’ calculation of the center of resistance for a parabolically shaped root (Fig. 1), the changes in axial inclination resulting from pure Gpping for the arch-depth decreases reported by Sillman t’or the age range covered by Bjiirk are calculate<1 in Fig. 1. al + cut2 = ?l.23o + 3.03o = 7.23o which is less than Bjiirk’s value of 8.99 degrees. Thcrcfore, the incisor uprighting, as reported by BjBrk, is sufficient to account for the decreases in anterior arch depth as reported by Sillman, without requiring such mechanisms as mcsial drift of the canines or bodily movement of the incisors. Bccansc of the inherent dangers of comparing differ& populations and of comparing data from cross-sectional and longitudinal studies, the anal.vsis is applied longitudinally to intliviclual untreated subjects in Part 11.
404
Am. J. Orthod. October 1974
Xiatkowski A
APRIL
APRIL 1971
1966
AW = -0lmm AL=-0.9mm
+
+ AW = +0.3mm At=-0.6mm
-50
mm
-4Omm -30
mln
-20
mm
_ IO mm -0
5. Data set for male measurement points made anterior segments. C, Copy
Fig.
PART Methods
Subject 7. A, Tracings of maxillary anterior segments and from photograph negative. 6, Same as A except mandibular of tracings made from lateral cephalometric radiographs.
II and
materials
Forsyth Dental Center longitudinal records, consisting of yearly study models and lateral cephalometric radiographs, were examined to obtain subjects who exhibited unmutilated, nonorthodontically treated occlusions that were + 1 mm. Angle Class I. Of 107 sets of records examined, eighteen met the selection criteria; the patients consisted of ten males and eight females, all Caucasian. Two sets of data were collected for each subject, the first at an age when the permanent dentition mesial to the second molars was fully erupted, usually at age
Incisor
upri@ting
405
13, and the second at the last available age, usually at age 18. Each set of data included the following : 1. A tracing from the lateral cephalometric radiograph from which interincisal angle and central incisor crown and root lengths could be measured (Fig. 5. C) . A millimeter rule located at the midsagittal plane appeared on the radiographs and provided a means of correcting for magnification. 2. One-to-one photographs of maxillary and mandibular study models using the “occlusogram” camera system”’ with occlusal plane oriented parallel to the plane of the film. A millimeter rule located in the plane of the photographed casts appeared on the negatives and provided verification of life-size reproduction. For each set of data the following measurements were taken : 1. Interincisal angle. 2. Crown and root lengths of maxillary and mandibular central incisors. 3. Intercanine width measured between canine cusp tips and measured directly on the photograph negatives with vernier calipers. 4. Anterior arch depth measured from a point at the midincisal edge between the central incisors perpendicular to a line drawn between the canine cusp tips and measured directly on the photograph negatives with vernier calipers. All measurements were repeated twice, and the average value was used. One data set is shown in Fig. 5. The analyses of Part I were carried out for each individual subject. Anterior arch widths and depths were substituted in equations 1, 2, and 3 to calculate anterior arch circumferences and their changes with age. The change in maxillary and mandibular anterior arch depth with age was used with incisor root and crown length measurements, corrected for magnification, to calculate changes in maxillary and mandibular central incisor axial inclination expected if the teeth rotated via pure tipping (rotation about the center of resistance), The sum of the expected change in maxillary and mandibular central incisor axial inclination was then compared to the net measured interincisal angle change with age. Results
The measurements and calculated results are tabulated in Table V. An increase in interincisal angle with age occurred in all subjects. The range was 1 to 15.75 degrees, with a mean of 7.4 degrees for the male subjects and 4.6 degrees for the female subjects. All subjects except one male exhibited decreases in maxillary anterior arch circumference, and all but one female showed decreases in mandibular anterior arch circumference. For every individual subject the predicted total interincisal angle cha,nge (based upon pure tipping of the central incisors, about their centers of resistance, needed to match the measured anterior arch depth changes) agreed with the measured interincisal angle change to within t 2 degrees, except for one male with a deviation of 2.22 degrees from the predicted value. Considering the poten-
Am. J. Orthod. October 1974
13 13 14 12 13 13 12 13 13.0 12 14
3 4 5 6 7 8 9 10 Mean Minimum Maximum
Table
V,
‘i 8 Mean
Minimum Maximum
B.
Females,
13 14 13 13 14 13 13 14 13.4 13 14
17 19 18 18 18 18 18 18 18.1 17 19
34.0 31.9 34.1 34.7 34.6 34.75 33.9 33.5 33.6 30.0 34.75
maxillary
18 18 18 20 18 18 18 18 18.2 18 20
32.5 32.4 33.85 32.6 34.4 34.2 32.9 32.3 33.14 32.3 34.4
33.5 31.9 34.3 34.45 34.5 33.6 33.5 32.9 33.4 30.0 35.0
-0.05 0 0.2 -0.25 -0.1 -1.15 -0.4 -0.6 -0.21 0.7 -1.15
dimensional
32.55 33.3 34.1 32.2 32.7 33.5 32.7 31.7 32.84 31.7 34.1
10.75 11.55 8.2 8.85 9.4 10.0 11.25 10.25 9.66 8.0 11.55
8.25 10.95 7.0 6.35 8.5 8.5 8.7 9.4 8.31 6.35 10.95
-2.5 -0.6 -1.2 -2.5 -0.9 -1.5 -2.55 -0.85 -1.35 -0.35 -2.55
42.44 42.09 39.14 40.44 41.05 41.98 43.11 41.32 40.65 35.4 43.11
38.68 41.13 37.99 37.49 39.84 39.07 39.23 39.65 38.75 34.65 41.13
-3.76 -0.96 -1.15 -2.95 -1.21 -2.91 -3.88 -1.67 -1.9 +0.20 -3.76
9.7 91.3 9.0 9.2 11.7 9.5 10.0 7.5 9.49 7.5 11.7
-0.45 -0.55 -0.9 -1.55 0 -0.9 -0.6 -0.15 -0.64 0 -0.9
40.38 39.87 41.11 41.35 44.18 42.09 41.36 36.93 40.91 36.93 44.18
39.78 39.84 40.12 38.81 42.92 40.28 40.33 36.24 39.79 36.24 42.92
-0.6 -0.03 -0.99 -2.54 -1.26 -1.81 -1.03 -0.69 -1.12 -0.03 -2.54
changes
0.55 0.9 0.35 -0.4 -1.7 -0.7 -0.2 -0.6 -0.29 0.9 -1.7
10.15 9.85 9.9 10.75 11.7 10.4 10.6 7.65 10.13 7.65 11.7
tial sources of error in the study, t 2 degrees represents experimental error.
a gratifyingly
small
Discussion
The evidence presented shows that a late growth change, the lingual uprighting of incisors via pure tipping, can fully account for decreases in anterior
Incisor Table
V, C. Males,
P 2 6 1 2 3 4 5 6 7 8 9 10 Mean Minimum Maximum Table
mandibular
hi pl B
1 2 3 4 5 6 7 8 Mean Minimum Maximum
18
13 13 13 14 12 13 13 12 13 13.0 12 14
19 17 19 18 18 18 18 18 18 18.1 17 19
13 14 13 13 14 13 13 14 13.4 13 14
407
changes
=r Q, 7
14
V, D. Females,
dimensional
uprighting
mandibular
18 18 18 20 18 18 18 18 18.2 18 20
25.4 22.9 25.35 23.85 25.1 26.75 24.5 26.3 24.45 26.4 25.1 22.9 26.75
25.2 22.9 24.7 23.65 25.1 26.0 24.8 25.35 23.75 25.65 24.7 22.9 26.0
dimensional
23.5 24.5 24.65 24.2 25.2 25.7 25.2 23.85 24.6 23.5 25.7
23.0 25.4 24.85 23.75 25.2 24.3 25.4 23.85 24.47 23.0 25.4
-0.2 0
-0.65 -0.2 0 -0.75 0.3 -0.95 -0.7 -0.75 -0.39 0.3 -0.9'5
6.0 5.65 7.25 8.2 5.45 5.9 5.8 5.9 6.8 7.65 6.46 5.45 8.2
5.6 5.35 6.3 7.55 4.95 4.85 5.2 5.5 5.2 7.3 5.78 4.85 7.55
-0.4 -0.3 -0.95 -0.65 -0.5 -1.05 -0.6 -0.4 -1.6 -0.35 -0.68 -0.3 -1.6
29.03 26.45 30.56 30.77 28.15 30.09 28.01 29.7 29'.22 31.96 29.39 26.45 31.96
28.4 26.1 27.49 29.63 27.63 28.35 27.61 28.42 26.68 30.87 28.12 26.1 30.87
-0.63 -0.35 -3.07 -1.14 -0.52 -1.74 -0.4 -1.28 -2.54 -1.09 -1.28 -0.35 -3.07
4.8 6.0 6.2 5.65 5.5 6.8 5.4 4.8 5.64 4.8 6.8
-1.2 -0.8 -0.3 -0.2 0 -0.1 -0.45 -0.6 -0.46 0 -1.2
27.39 29.26 28.99 27.81 28.29 30.39 28.68 26.99 28.48 26.99 30.39
25.58 29.03 28.79 27.19 28.29 29.09 28.36 26.35 27.84 25.58 29.09
-1.81 -0.23 -0.2 -0.62 0 -1.3 -0.32 -0.64 -0.64 0 -1.81
changes
-0.5 0.9 0.2 -0.45 0 -1.4 0.2 0 -0.13 0.9 -1.4
6.0 6.8 6.5 5.85 5.5 6.9 5.85 5.4 6.1 5.4 6.9
arch depth and, therefore, decreases in anterior arch circumference. If the anterior segments are aligned at the circumpubertal ages, the decrease in anterior arch circumference resulting from incisor uprighting will produce anterior crowding, greater on the average in the mandibular arch than in the maxillary arch. Changes in canine widths will tend to relieve the crowding in the maxillary arch and aggravate it in the mandibular arch, on the average. Small increments
Am.
J. Orthod.
October1974
Table
V, E. Males,
1
4 5 6 I 8 9 10 Mean Minimum Maximum
14 13 13 13 14 12 13 13 12 13 13.0 12 14
angular
changes
18 19 17 19 18 18 18 18 18 18 18.1 17 19
126 146 126 136.75 130 133.5 l”6I . 75 124.75 131.25 126.5 130.75 124.75 146
I
127 151.25 141.5 140.25 134.5 147.25 132.25 131 147 7 29.5 138.15 127 151.25
5.25 15.5 3.5 4.5 13.i5 5.5 6.25 15.75 3 7.4 1 15.75
15.3 16.0 15.5 15.4 17.5 15.7 16.6 16.2 17.7 17.1 16.3 15.3 17.7
14.1 14.4 13.5 13.1 16.3 14.7 15.5 14.9 14.9 14.9 14.6 13.1 16.3
1.31 2.15 9.25 2.23 3.93 9.12 3.11 5.3 8.26 2.85 4.75 1.31 9.25
1.62 1.19 4.03 2.84 1.76 4.1 2.22 1.54 6.15 1.35 2.68 1.19 6.15
2.93 3.34 13.28 5.07 5.69 13.22 5.33 6.84 14.41 4.2 7.43 2.93 14.41
+1.93 -1.91
-2.22 +1.57 +1.19 PO.53 -0.17 +0.59 -1.34 +1.2 +0.03
-2.22 +I.93
in anterior arch depth and width result in relatively larger changes in arch circumference. Mechanistically, the uprighting occurs via pure tipping about the center of resistance of the root which is caused by a net lingually directed force on the incisor crowns, giving support to the muscle force theories of late crowding etiology. There could be a differential growth pattern between the facial skeleton and its overlying soft tissue, with the soft-tissue growth decelerating faster than the skeletal, resulting in lingually directed force against the incisors. Another possibility is an increase in stiffness, with age, of the oral soft-tissue drape during late growth. There are several clinical implications from the findings of the study. Later relapse of anterior arches put into alignment at the circumpubcrtal ages may be an expression of the natural late growth process described herein. With a decrease in available arch circumference with growth, the anterior teeth may tend to move toward their previous maloccluded positions because of some inherent hysteresis in the supporting tissues rather than assume new malrelatetl positions. The findings suggest re-evaluation of present retention techniques in order to accommodate the described growth changes. If maintaining the alignment of the mandibular incisors is a prime posttreatment goal, a retention device that would allow their lingual uprighting with growth as a unit should be used until terminal growth is completed or nearly completed. One such device has been in successful clinical use.ll It consists of a soldered lingual arch from lateral incisor
Table
V,
F. Females,
angular
changes
. B 0 Tf -cl 3p gs--. GE 2 .$ ‘E 2 2& 2 .C 22 z;P x3 .r,s? kgjb 6r -G ‘“H“H tE 5 s I’d
2 2 c, s e T2 .Q a’ s,w kcl +ti$ I Lu’-
i 8 Mean Minimum Maximum
13 14 13 13 14 13 13 14 13.4 13 14
18 18 18 20 18 18 18 18 18.2 18 20
139.5 128 144.25 123 127.25 136.75 119.5 148.25 133.31 119.5 148.25
148 132.25 149.75 130 128.5 138.25 124.25 152.25 137.91 124.25 152.25
8.5 4.25 5.5 7 1.25 1.5 4.75 4 4.59 1.25 8.5
13.1 14.4 14.2 13.2 13.5 14.4 15.1 13.2 13.89 13.1 15.1
15.3 13.7 16.8 16.1 15.1 18.2 15.8 14.8 15.73 13.7 18.2
1.68 2.3 3.07 5.51 0 2.83 2.18 0.58 2.27 0 5.51
-1.57 +1.27 -1.22 -0.62 -1.25 +1.73 -0.86 -0.82 - 0.42 -1.5T +1.73
to lateral incisor with a soldered labial bar from each lateral incisor to its adjacent canine. This approach permits lingual uprighting of the incisors with growth as a unit, with the concomitant loss of anterior arch circumference being compensated for via lingual displacement of the canines (the premolar-caninclateral incisor contacts occurring at a shorter mesiodistal cross section of the canine). Conclusions
1. During late growth, changes in anterior arch width and depth result in a decrease of anterior arch circumference, greater on the average in the mandibular arch than in the maxillary arch. 2. The decrease in anterior arch depth can be fully accounted for on a longitudinal basis by lingual uprighting of the incisors via pure tipping. 3. Such uprighting lends credence to muscle force theories as the cause of anterior arch crowding during terminal growth. 4. Re-evaluation of present retention techniques in order to accommodate the described growth changes is recommended. The author wishes to thank Coenraad longitudinal records at the Forsyth Dental cwcourqement.
F. A. Moorrees Center and Sam
for kindly making available Weinstein for his kind advice
the and
REFERENCES
1. Lombardi, A. R.: Mandibular 374.383, 1972. 2. Bjsrk, A. : The face in profile,
incisor Lund,
crowding 1917,
Berlingska
in
completed Boktryckeriet.
cases,
AM.
J. ORTHOIL
61:
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3. Sul)trlny, :\n~.
J. I)., :~nd S:tku~l:t,
.T. OKTIIOIL
52:
-N5-51i,
M. : Muwlt~
fuwt
ion,
0~11 ~I~:~lfo~n1:~tio~~,
ant1 growth
c~hngw,
1066.
dcr I,indcn, E'. 1'. CT. M.: lnterrehtcd fuctors development of the dentition ant1 cr&of:lri:rl growth, 518.528, lY70. 5. \V~~instcin, S. : Thirtl mohr implications iu trrthotlontiw, 1. van
in tllcl morphogenesis Schrrriz. Monatsschr .I.
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I)ent.
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of tcetll, Zahnheilk 82:
the 80:
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IYil.
9. Burstonc, (editors) 10. GxrncI., 50:
11.
C. J. : The biomcclmnics o-1’ tooth movemcwt. In KKIUS, H. S., and : Vistas in orthotlontics, Philatic~lpllia, 1962, Lea & Fchiger. 1,. 1). : 0wlusogr:rm: 1 ts :lppliwrion to c:ldt lxllatc~ ~clr:r~)ilit:ltioll,
1286.1289,
1971.
Hicc,
T. L. : Personal
Jfnin
St.
(06409)
cornrriullic~:ltioll.
Heidel,
K. A.
.T. Ihtlt.
lies.
E. Siatkowski,
D.M.D*
Essex, Conn.
T
he orthodontic literature abounds with speculation on the etiology of incisor crowding as a late (postpubertal) growth phenomenon. Lombardil cites most of the prominently considered factors in his survey of the literature: tooth discrepancies, arch size, interference of canines in lateral excursive movements, overbite, overjet, tooth removal, incisor-mandibular plane angle, type of original occlusion, third molars, and muscular forces. In their growth studies Bj6rk,2 Subtelny and Sakuda,3 and van der Linden4 report or present data revealing lingual uprighting of the incisor segments during late growth. Weinstein,5 citing Bjijrk’s study, speculated on the possible etiologic role of this uprighting in incisor-segment crowding during the typical postorthodontic-treatment growth period. LundstrijmG mentions the possible role of net lingual crown movement of the anterior teeth in decreases in arch circumference. The present study is a mathematical investigation of incisor uprighting, conducted in two parts. In the first part a mathematical method is derived to serve as a vehicle for the interpretation of and correlation between the data available from two growth studies, those of Sillman’ and Bj6rk.2 The method is used to predict the anterior crowding to be expected from the changes in mean anterior arch dimension reported by Sillman during a postpubertal growth period. The changes in mean incisor axial inclination during late growth reported by Bjiirk arc then related to the changes in mean anterior arch dimension of Sillman to determine the possible contribution of the former to the latter. The above investigation is based on mean dimensions and can therefore provide only a typical indication of the actual events. Therefore, the second part of the Supported by tional Institute *Resident
398
United States Public of Dental Research,
in Orthodontics,
University
Health Service Grant National Institutes of Connecticut
Health
DE of
00283 Health,
Center.
from the Bethesda,
NaMd.
Fig. 1. Assumed anterior ference. W, Canine width.
arch form 1, Anterior
and mathematical parameters. arch depth. R, Radius of circle.
D, Anterior
arch
circum-
CUSP
Fig.
2. Assumed
anterior
arch
form
with
mathematical
parameters
study tests the analysis developed in the first part by applying untreated subjects longitudinally. PART Methods
rearranged.
it to individual
I and
material
A mathematical method for calculating anterior arch circumference is derived first. The following assumptions are made for mathematical simplicity: oc-
400 Table
Am. J. Orthod. October1974
Siatkowski 1. Summary
of
selected
mean
anterior
arch
width
and
depth
data
Mule Maxilla
Age (yrs.1 Anterior Anterior Change Change
14
arch width, W (mm.) arch depth, I, (mm.) in width (mm.) in depth (mm.)
Sillman
Female Mandible
14
21
from
21
36.69
36.75 30.06
14.55
13.77
Mandibk
MaxiZZa 13
21
13
21
29.83 34.26 34.66 28.29 27.93 IO.12 9.30 13.73 12.99 9.96 9.04 -0.23 +0.40 -0.36 -0.82 -0.74 -0.92
+O.OS -0.78
clusally, the in&al edges and cusp tips of the maxillary and mandibular anterior six teeth fall on arcs of circles of appropriate radii, and the uprighting occurs in such a way that the mean position of the incisal edges and cusp tips also falls on arcs of circles of appropriate radii. For comparison, calculations are also carried out for catenary and parabolic anterior arch forms. E’ig. 1 illustrates the assumed anterior arch form and identifies the mathematical parameters of interest. The mathematical parameters arc shown in a more convenient form in Fig. 2. From geometry WCknow that the arc length of a circle subtended by the angle, 8, is equal to the product of the radius of the circle and the subtended angle, where the units of the angle are in radians rather than degrees. Applying this to the geometry illustrated in Fig. 2 : D ;;” = RL9 or, D = 2RO
(1)
We must now calculate R and 8 in terms of W and 1~. Applying rean theorem to the lower right triangle in Fig. 2 :
or,
R2 - 2RL
or,
Applying Fig. 2 :
the definition
+ 1~’ +
wz
IZ =
81,
JJ
+ 2
(2)
of a sine function
sin therefore,
9 =
the I’ythago-
w
T/R B =
= sin-’
to the lower right
triangle
in
‘1V ~~
(3)
If we are given anterior arch width, W, and anterior arch depth, I,, we can substitute their values into equation 2 to calculate the radius, R. The values of R and W can then be substituted into equation 3 to calculate the subtended angle, 0.
Volume
66
Table
II.
Nunabel’
Incisor
4
Calculated
mean
anterior
arch
uprighting
circumferences
Female
Male
Maxilla
Age (vs.1 Arch circumference, D (mm.) Change in arch circumference (mm.) Per cent change in arch circumference
Table
III.
anterior
Changes
arch
in anterior
arch
Male
(ages
401
/ Mandible
1
14 21 14 21 50.53 49’.25 38.45 37.04 -1.28 -1.41 -2.53 -3.67
circumferences
for
Maxilla
1 Mandible
13 21 47.43 46.45 -0.98 -2.07
catenary,
circular,
13 21 36.87 35.17 -1.70 -4.61
and
parabolic
forms
Catenary Circular Parabolic
14-U)
I
Female
(ages
13-H)
Maxilla
Mandible
Maxilla
llfandible
(mm.1
(mm.)
(mm.)
(mm.)
-1.57 -1.28 -1.01
-1.58 -1.41 -1.21
-1.29 -0.98 -0.69
-1.91 -1.70 -1.47
Finally, the values for R and 8 can be substituted in equation 1 to calculate the anterior arch circumference. Sillman’s longitudinal study includes data on the changes in anterior arch depth and width from birth to age 21 for forty subjects. The mean values and their increments for the late growth age span (male, ages 14 to 21; female, ages 13 to 21) are presented in Table I. The values in Table I reflect some interesting phenomena. Both of the arches for both male and female subjects show a decrease in anterior arch depth. The arch-depth decreases arc greater in the mandible than in the maxilla. The changes in anterior arch width arc increases in the maxilla and decreases in the mandible. Therefore, in the maxilla the width changes tend to relieve any arch circumference decrease due to depth decreases, whereas the width changes tend to aggravate the circumference decrease in the mandible. Results
The values from Table I for anterior arch width, W, and anterior arch depth, L, are substituted in equations 1, 2, and 3 to calculate the mean anterior arch circumferences. The results are shown in Table II. All arch circumference changes arc deercases. The absolute and per cent decreases are greater in the mandibular arch for both male and female subjects. The greatest absolute amI per cent decrease is in the female mandibular arch. Note that small increments in anterior arch depth and width (Table I) result in relatively larger changes in arch circumference. For comparison, the calculations were repeated for catenary and parabolic anterior arch forms, and the results are presented in Table III. The
contribution
of changes
in the incisor
axial
inclination
with
growth,
as
Am. J. Orthod. October 1974
A I = RI al A? : R2dz a, = CL2
IF , ,
Fig.
Table reported
3. Two
forms
IV. Mean by
Bitirk
of incisor
interincisal and
rotation.
angles
AZ ) A,,
A, Pure
and
tipping.
change
in anterior
plus
arch
bodily
depth
R2 > R,
movement.
(based
on
values
Sillman’) Male,
Interincisal angle (degrees) Change in interincisal angle (degrees)
Change in anterior
8, Tipping
SINCE
arch depth (mm.)
age
Hale,
18
128.45
ages
21-M
137.44
+8.99
I
Male (age 12-81) Maxi1 la -1.16
I
Ma.n4Jible -0.74
reported by Bjbrk, to changes in the anterior arch depth is evaluated next. The mean values from BjBrk’s cross-sectional study as well as the corresponding values from Sillman’s longitudinal data are shown in Table IV. Since there is no known cranial plane fixed with growth with which to relate changes in individual incisor inclination, no attempt is made here to assign a given amount of the net change in interincisal angle to either incisor. Bjiirk does measure the incisal angles to occlusal plane and occlusal plane to several cranial reference lines, none of which one can use as an absolute reference during growth. A change in incisal angle can be produced by pure tipping (rotation about the center of resistance of the root) or by a combination of tipping and bodily movement (rotation about a point other than the center of resistance of the root). Mechanistically, the former is simpler, requiring only a net lingually directed force on the crown. The latter, however, if it produces a center of rotation apical to the center of resistance of the root, will produce a greater arch depth change per degree of rotation than pure tipping, as illustrated in Fig. 3. Now, with certain assumptions, it is possible to calculate the axial inclination
zwcisor u~prigh’ti7ly
Fig. 4. Calculations The
angles
are
of maxillary and mandibular exaggerated in the drawing for
arch clarity.
depth
decreases
with
pure
403
tipping.
changes that would he associated with pure tipping to account for the arch-depth tlecreascs in the Sillman sample. On the basis of Wheeler’s” mean dimensions for incisors and Haack’s and Burstone’s!’ calculation of the center of resistance for a parabolically shaped root (Fig. 1), the changes in axial inclination resulting from pure Gpping for the arch-depth decreases reported by Sillman t’or the age range covered by Bjiirk are calculate<1 in Fig. 1. al + cut2 = ?l.23o + 3.03o = 7.23o which is less than Bjiirk’s value of 8.99 degrees. Thcrcfore, the incisor uprighting, as reported by BjBrk, is sufficient to account for the decreases in anterior arch depth as reported by Sillman, without requiring such mechanisms as mcsial drift of the canines or bodily movement of the incisors. Bccansc of the inherent dangers of comparing differ& populations and of comparing data from cross-sectional and longitudinal studies, the anal.vsis is applied longitudinally to intliviclual untreated subjects in Part 11.
404
Am. J. Orthod. October 1974
Xiatkowski A
APRIL
APRIL 1971
1966
AW = -0lmm AL=-0.9mm
+
+ AW = +0.3mm At=-0.6mm
-50
mm
-4Omm -30
mln
-20
mm
_ IO mm -0
5. Data set for male measurement points made anterior segments. C, Copy
Fig.
PART Methods
Subject 7. A, Tracings of maxillary anterior segments and from photograph negative. 6, Same as A except mandibular of tracings made from lateral cephalometric radiographs.
II and
materials
Forsyth Dental Center longitudinal records, consisting of yearly study models and lateral cephalometric radiographs, were examined to obtain subjects who exhibited unmutilated, nonorthodontically treated occlusions that were + 1 mm. Angle Class I. Of 107 sets of records examined, eighteen met the selection criteria; the patients consisted of ten males and eight females, all Caucasian. Two sets of data were collected for each subject, the first at an age when the permanent dentition mesial to the second molars was fully erupted, usually at age
Incisor
upri@ting
405
13, and the second at the last available age, usually at age 18. Each set of data included the following : 1. A tracing from the lateral cephalometric radiograph from which interincisal angle and central incisor crown and root lengths could be measured (Fig. 5. C) . A millimeter rule located at the midsagittal plane appeared on the radiographs and provided a means of correcting for magnification. 2. One-to-one photographs of maxillary and mandibular study models using the “occlusogram” camera system”’ with occlusal plane oriented parallel to the plane of the film. A millimeter rule located in the plane of the photographed casts appeared on the negatives and provided verification of life-size reproduction. For each set of data the following measurements were taken : 1. Interincisal angle. 2. Crown and root lengths of maxillary and mandibular central incisors. 3. Intercanine width measured between canine cusp tips and measured directly on the photograph negatives with vernier calipers. 4. Anterior arch depth measured from a point at the midincisal edge between the central incisors perpendicular to a line drawn between the canine cusp tips and measured directly on the photograph negatives with vernier calipers. All measurements were repeated twice, and the average value was used. One data set is shown in Fig. 5. The analyses of Part I were carried out for each individual subject. Anterior arch widths and depths were substituted in equations 1, 2, and 3 to calculate anterior arch circumferences and their changes with age. The change in maxillary and mandibular anterior arch depth with age was used with incisor root and crown length measurements, corrected for magnification, to calculate changes in maxillary and mandibular central incisor axial inclination expected if the teeth rotated via pure tipping (rotation about the center of resistance), The sum of the expected change in maxillary and mandibular central incisor axial inclination was then compared to the net measured interincisal angle change with age. Results
The measurements and calculated results are tabulated in Table V. An increase in interincisal angle with age occurred in all subjects. The range was 1 to 15.75 degrees, with a mean of 7.4 degrees for the male subjects and 4.6 degrees for the female subjects. All subjects except one male exhibited decreases in maxillary anterior arch circumference, and all but one female showed decreases in mandibular anterior arch circumference. For every individual subject the predicted total interincisal angle cha,nge (based upon pure tipping of the central incisors, about their centers of resistance, needed to match the measured anterior arch depth changes) agreed with the measured interincisal angle change to within t 2 degrees, except for one male with a deviation of 2.22 degrees from the predicted value. Considering the poten-
Am. J. Orthod. October 1974
13 13 14 12 13 13 12 13 13.0 12 14
3 4 5 6 7 8 9 10 Mean Minimum Maximum
Table
V,
‘i 8 Mean
Minimum Maximum
B.
Females,
13 14 13 13 14 13 13 14 13.4 13 14
17 19 18 18 18 18 18 18 18.1 17 19
34.0 31.9 34.1 34.7 34.6 34.75 33.9 33.5 33.6 30.0 34.75
maxillary
18 18 18 20 18 18 18 18 18.2 18 20
32.5 32.4 33.85 32.6 34.4 34.2 32.9 32.3 33.14 32.3 34.4
33.5 31.9 34.3 34.45 34.5 33.6 33.5 32.9 33.4 30.0 35.0
-0.05 0 0.2 -0.25 -0.1 -1.15 -0.4 -0.6 -0.21 0.7 -1.15
dimensional
32.55 33.3 34.1 32.2 32.7 33.5 32.7 31.7 32.84 31.7 34.1
10.75 11.55 8.2 8.85 9.4 10.0 11.25 10.25 9.66 8.0 11.55
8.25 10.95 7.0 6.35 8.5 8.5 8.7 9.4 8.31 6.35 10.95
-2.5 -0.6 -1.2 -2.5 -0.9 -1.5 -2.55 -0.85 -1.35 -0.35 -2.55
42.44 42.09 39.14 40.44 41.05 41.98 43.11 41.32 40.65 35.4 43.11
38.68 41.13 37.99 37.49 39.84 39.07 39.23 39.65 38.75 34.65 41.13
-3.76 -0.96 -1.15 -2.95 -1.21 -2.91 -3.88 -1.67 -1.9 +0.20 -3.76
9.7 91.3 9.0 9.2 11.7 9.5 10.0 7.5 9.49 7.5 11.7
-0.45 -0.55 -0.9 -1.55 0 -0.9 -0.6 -0.15 -0.64 0 -0.9
40.38 39.87 41.11 41.35 44.18 42.09 41.36 36.93 40.91 36.93 44.18
39.78 39.84 40.12 38.81 42.92 40.28 40.33 36.24 39.79 36.24 42.92
-0.6 -0.03 -0.99 -2.54 -1.26 -1.81 -1.03 -0.69 -1.12 -0.03 -2.54
changes
0.55 0.9 0.35 -0.4 -1.7 -0.7 -0.2 -0.6 -0.29 0.9 -1.7
10.15 9.85 9.9 10.75 11.7 10.4 10.6 7.65 10.13 7.65 11.7
tial sources of error in the study, t 2 degrees represents experimental error.
a gratifyingly
small
Discussion
The evidence presented shows that a late growth change, the lingual uprighting of incisors via pure tipping, can fully account for decreases in anterior
Incisor Table
V, C. Males,
P 2 6 1 2 3 4 5 6 7 8 9 10 Mean Minimum Maximum Table
mandibular
hi pl B
1 2 3 4 5 6 7 8 Mean Minimum Maximum
18
13 13 13 14 12 13 13 12 13 13.0 12 14
19 17 19 18 18 18 18 18 18 18.1 17 19
13 14 13 13 14 13 13 14 13.4 13 14
407
changes
=r Q, 7
14
V, D. Females,
dimensional
uprighting
mandibular
18 18 18 20 18 18 18 18 18.2 18 20
25.4 22.9 25.35 23.85 25.1 26.75 24.5 26.3 24.45 26.4 25.1 22.9 26.75
25.2 22.9 24.7 23.65 25.1 26.0 24.8 25.35 23.75 25.65 24.7 22.9 26.0
dimensional
23.5 24.5 24.65 24.2 25.2 25.7 25.2 23.85 24.6 23.5 25.7
23.0 25.4 24.85 23.75 25.2 24.3 25.4 23.85 24.47 23.0 25.4
-0.2 0
-0.65 -0.2 0 -0.75 0.3 -0.95 -0.7 -0.75 -0.39 0.3 -0.9'5
6.0 5.65 7.25 8.2 5.45 5.9 5.8 5.9 6.8 7.65 6.46 5.45 8.2
5.6 5.35 6.3 7.55 4.95 4.85 5.2 5.5 5.2 7.3 5.78 4.85 7.55
-0.4 -0.3 -0.95 -0.65 -0.5 -1.05 -0.6 -0.4 -1.6 -0.35 -0.68 -0.3 -1.6
29.03 26.45 30.56 30.77 28.15 30.09 28.01 29.7 29'.22 31.96 29.39 26.45 31.96
28.4 26.1 27.49 29.63 27.63 28.35 27.61 28.42 26.68 30.87 28.12 26.1 30.87
-0.63 -0.35 -3.07 -1.14 -0.52 -1.74 -0.4 -1.28 -2.54 -1.09 -1.28 -0.35 -3.07
4.8 6.0 6.2 5.65 5.5 6.8 5.4 4.8 5.64 4.8 6.8
-1.2 -0.8 -0.3 -0.2 0 -0.1 -0.45 -0.6 -0.46 0 -1.2
27.39 29.26 28.99 27.81 28.29 30.39 28.68 26.99 28.48 26.99 30.39
25.58 29.03 28.79 27.19 28.29 29.09 28.36 26.35 27.84 25.58 29.09
-1.81 -0.23 -0.2 -0.62 0 -1.3 -0.32 -0.64 -0.64 0 -1.81
changes
-0.5 0.9 0.2 -0.45 0 -1.4 0.2 0 -0.13 0.9 -1.4
6.0 6.8 6.5 5.85 5.5 6.9 5.85 5.4 6.1 5.4 6.9
arch depth and, therefore, decreases in anterior arch circumference. If the anterior segments are aligned at the circumpubertal ages, the decrease in anterior arch circumference resulting from incisor uprighting will produce anterior crowding, greater on the average in the mandibular arch than in the maxillary arch. Changes in canine widths will tend to relieve the crowding in the maxillary arch and aggravate it in the mandibular arch, on the average. Small increments
Am.
J. Orthod.
October1974
Table
V, E. Males,
1
4 5 6 I 8 9 10 Mean Minimum Maximum
14 13 13 13 14 12 13 13 12 13 13.0 12 14
angular
changes
18 19 17 19 18 18 18 18 18 18 18.1 17 19
126 146 126 136.75 130 133.5 l”6I . 75 124.75 131.25 126.5 130.75 124.75 146
I
127 151.25 141.5 140.25 134.5 147.25 132.25 131 147 7 29.5 138.15 127 151.25
5.25 15.5 3.5 4.5 13.i5 5.5 6.25 15.75 3 7.4 1 15.75
15.3 16.0 15.5 15.4 17.5 15.7 16.6 16.2 17.7 17.1 16.3 15.3 17.7
14.1 14.4 13.5 13.1 16.3 14.7 15.5 14.9 14.9 14.9 14.6 13.1 16.3
1.31 2.15 9.25 2.23 3.93 9.12 3.11 5.3 8.26 2.85 4.75 1.31 9.25
1.62 1.19 4.03 2.84 1.76 4.1 2.22 1.54 6.15 1.35 2.68 1.19 6.15
2.93 3.34 13.28 5.07 5.69 13.22 5.33 6.84 14.41 4.2 7.43 2.93 14.41
+1.93 -1.91
-2.22 +1.57 +1.19 PO.53 -0.17 +0.59 -1.34 +1.2 +0.03
-2.22 +I.93
in anterior arch depth and width result in relatively larger changes in arch circumference. Mechanistically, the uprighting occurs via pure tipping about the center of resistance of the root which is caused by a net lingually directed force on the incisor crowns, giving support to the muscle force theories of late crowding etiology. There could be a differential growth pattern between the facial skeleton and its overlying soft tissue, with the soft-tissue growth decelerating faster than the skeletal, resulting in lingually directed force against the incisors. Another possibility is an increase in stiffness, with age, of the oral soft-tissue drape during late growth. There are several clinical implications from the findings of the study. Later relapse of anterior arches put into alignment at the circumpubcrtal ages may be an expression of the natural late growth process described herein. With a decrease in available arch circumference with growth, the anterior teeth may tend to move toward their previous maloccluded positions because of some inherent hysteresis in the supporting tissues rather than assume new malrelatetl positions. The findings suggest re-evaluation of present retention techniques in order to accommodate the described growth changes. If maintaining the alignment of the mandibular incisors is a prime posttreatment goal, a retention device that would allow their lingual uprighting with growth as a unit should be used until terminal growth is completed or nearly completed. One such device has been in successful clinical use.ll It consists of a soldered lingual arch from lateral incisor
Table
V,
F. Females,
angular
changes
. B 0 Tf -cl 3p gs--. GE 2 .$ ‘E 2 2& 2 .C 22 z;P x3 .r,s? kgjb 6r -G ‘“H“H tE 5 s I’d
2 2 c, s e T2 .Q a’ s,w kcl +ti$ I Lu’-
i 8 Mean Minimum Maximum
13 14 13 13 14 13 13 14 13.4 13 14
18 18 18 20 18 18 18 18 18.2 18 20
139.5 128 144.25 123 127.25 136.75 119.5 148.25 133.31 119.5 148.25
148 132.25 149.75 130 128.5 138.25 124.25 152.25 137.91 124.25 152.25
8.5 4.25 5.5 7 1.25 1.5 4.75 4 4.59 1.25 8.5
13.1 14.4 14.2 13.2 13.5 14.4 15.1 13.2 13.89 13.1 15.1
15.3 13.7 16.8 16.1 15.1 18.2 15.8 14.8 15.73 13.7 18.2
1.68 2.3 3.07 5.51 0 2.83 2.18 0.58 2.27 0 5.51
-1.57 +1.27 -1.22 -0.62 -1.25 +1.73 -0.86 -0.82 - 0.42 -1.5T +1.73
to lateral incisor with a soldered labial bar from each lateral incisor to its adjacent canine. This approach permits lingual uprighting of the incisors with growth as a unit, with the concomitant loss of anterior arch circumference being compensated for via lingual displacement of the canines (the premolar-caninclateral incisor contacts occurring at a shorter mesiodistal cross section of the canine). Conclusions
1. During late growth, changes in anterior arch width and depth result in a decrease of anterior arch circumference, greater on the average in the mandibular arch than in the maxillary arch. 2. The decrease in anterior arch depth can be fully accounted for on a longitudinal basis by lingual uprighting of the incisors via pure tipping. 3. Such uprighting lends credence to muscle force theories as the cause of anterior arch crowding during terminal growth. 4. Re-evaluation of present retention techniques in order to accommodate the described growth changes is recommended. The author wishes to thank Coenraad longitudinal records at the Forsyth Dental cwcourqement.
F. A. Moorrees Center and Sam
for kindly making available Weinstein for his kind advice
the and
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incisor Lund,
crowding 1917,
Berlingska
in
completed Boktryckeriet.
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AM.
J. ORTHOIL
61:
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M. : Muwlt~
fuwt
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