Our objective was to evaluate the relationship between subjective classification of dental-arch shape, objective analyses via arch-width measurements, and the fitting with the fourth-order polynomial equation.

1. Subjective classification and objective analysis of
the mandibular dental-arch form of orthodontic
patients
Kazuhito Araia
and Leslie A. Willb
Tokyo, Japan, and Boston, Mass
Introduction: Our objective was to evaluate the relationship between subjective classification of dental-arch
shape, objective analyses via arch-width measurements, and the fitting with the fourth-order polynomial
equation. Methods: Twenty-seven pretreatment mandibular dental casts (from 13 males and 14 females;
ages, 12-31 years) were selected. Standardized photographs of the arches were serially organized from tapered
to square by 10 examiners. The mean position in the ranking of each cast was calculated as a rank of each arch
form. The dental casts were analyzed with a 3-dimensional laser scanning system. Dental-arch widths at the
canines and molars were measured, and then a fourth-order polynomial equation was fit to each arch.
Correlations between the rank of arch shape and the objective measurements were statistically tested.
Results: The arch forms having the greatest variations among the examiners were those with an intermediate
(ovoid) ranking. Statistically significant correlations were found between the ranks of arch shape, arch dimen-
sions, and the polynomial equation analyses. Conclusions: Subjective clinical assessments were generally
in agreement at the extremes of tapered and square arch forms; the exceptions were arches with an ovoid
shape. There were statistically significant correlations between subjective dental-arch classifications and
dental-arch dimensions, as well as the ratio determined from these variables and polynomial equation
analyses. Therefore, fourth-order polynomial equations might be an important factor in the quantitative
analysis of dental-arch form in orthodontic patients. (Am J Orthod Dentofacial Orthop 2011;139:e315-e321)
O
ne goal of orthodontic treatment is to create an
individualized dental arch that is ideal for the pa-
tient.1-3
The original dental-arch form of the pa-
tient is mimicked to achieve stable treatment results
because an arch form that has been orthodontically
modified has a tendency to return to its original width.4,5
Therefore, reliable evaluation and accurate analysis of
the patient’s pretreatment dental-arch form are essential
steps for orthodontic diagnosis.
In general, subjective classification methods with 3
or 5 simple shape categories have been commonly ap-
plied to evaluate the initial dental-arch form during
the orthodontic diagnostic process.6-8
For example,
the subjective classification method, which uses 3
recommended shapes of tapered, ovoid, and square
forms, has been widely used in the clinic to select
prefabricated orthodontic archwires for a specific
patient.8
However, general human error can be expected in
subjective analysis; therefore, the intraoperator and inter-
operator reproducibility of these evaluations might be in-
accurate. Additionally, although a mathematical definition
of these square, ovoid, and tapered dental-arch forms has
been proposed, clinical application of this evaluation
method for computer analysis is still rather limited.9
Another common analysis of the dental-arch form is
measuring canine and molar widths for clinical and re-
search purposes.10
These transverse dimensions are usu-
ally measured at the cusp tips or other anatomic
structures of the tooth crown. One considerable advan-
tage of this method is the numeric analysis of the
dental-arch form. Additionally, determination of the
canine-molar width ratio has been also used as a simple
quantification method and has been widely used in a
clinical setting.11
a
Professor and chair, Department of Orthodontics, Nippon Dental University,
Tokyo, Japan.
b
Chair and Anthony A. Gianelly Professor of Orthodontics, Boston University
Goldman School of Dental Medicine, Boston, Mass.
The authors report no commercial, proprietary, or financial interest in the prod-
ucts or companies described in this article.
Reprint requests to: Kazuhito Arai, Department of Orthodontics, Nippon Dental
University, 1-9-20 Fujimi, Chiyoda-ku, Tokyo 102-8159, Japan; e-mail, drarai@
tky.ndu.ac.jp.
Submitted, April 2009; revised and accepted, December 2009.
0889-5406/$36.00
Copyright Ó 2011 by the American Association of Orthodontists.
doi:10.1016/j.ajodo.2009.12.032
e315
ONLINE ONLY

2. In comparison, objective numeric analysis methods,
which use particular geometric and mathematical
models, have been developed to quantitatively describe
the dental-arch forms of orthodontic patients.12-27
For
instance, a parabola or second-order polynomial,19
beta function,18
cubic spline function,20
and fourth-
order or larger polynomial equations have all been ap-
plied.13,21-25
Recently, AlHarbi et al26
compared these
functions and concluded that the fourth-order polyno-
mial function (y 5 ax4
1 bx3
1 cx2
1 dx 1 e) was
the most reasonable equation for analysis when the ob-
jective was to describe the general smooth curvature of
the dental arch.
Application of fourth-order polynomials to represent
the dental-arch form has several advantages for dental-
arch form analysis.13,21-25
According to Lu,13
1 advan-
tage of the fourth-order polynomial curve-fitting
method is that the coefficients of each term can be sim-
ply associated with specific aspects of the arch form. The
coefficients of the fourth (x4
or quartic) and second (x2
or quadratic) terms describe the square and tapered
shapes of dental-arch forms, respectively. Although
a large fourth-order coefficient “a” indicates a square
arch form, a large second-order coefficient “c” describes
a tapered arch form. Consequently, the fourth-order
polynomial equation has been applied extensively on
the basis of this hypothesis.13,21-25
For example,
Hayama et al23
evaluated the relationship between max-
illary and mandibular dental-arch forms and found
statistically significant positive correlations for all coef-
ficients. Richards et al16
applied the fourth-order poly-
nomial equation to analyze the correlation between
twin subjects and concluded that this equation can ac-
curately represent dental-arch shape. Additionally, Fer-
rario et al17
investigated dental-arch size differences
between the sexes using this equation. However, little
research has been conducted to compare the mathemat-
ical description of the dental-arch form with the subjec-
tive evaluations and the objective width measurements
made by clinicians as orthodontic diagnostic tools.
The purpose of this study was to evaluate the rela-
tionship between the results of both a subjective classi-
fication by human judgment and 2 objective analytic
methods by using dental-arch dimension measurements
and the fourth-order polynomial equation for analysis of
the dental-arch form of orthodontic patients.
MATERIAL AND METHODS
A total of 27 pretreatment casts from 13 male and 14
female subjects, ranging in age from 12 to 31 years
(mean age, 16.5 6 5.0 years), were selected from 720
records at the Harvard School of Dental Medicine
Teaching Clinic in Boston, Mass. The following inclusion
criteria were required: (1) complete dentition, excluding
third molars; (2) no prosthetic crowns and minimal resto-
rations; (3) minimal signs of occlusal attrition; and (4) min-
imal spacing. Subjects with severe crowding (.10 mm)
and transposed teeth were excluded from the analysis.
This sample size was first analyzed and determined to
provide an adequate statistical power (80%, b 5 0.2) for
detection of a correlation coefficient of 0.5 for a 5 0.05
and 0.6 for a 5 0.01.
All selected subjects signed informed consent
forms granting permission for their records to be used
in research. A detailed protocol of this project was ap-
proved by the Committee on Human Studies of Harvard
Medical School and Harvard School of Dental Medicine
(X040501-1).
Standardized occlusal photographs of the mandibu-
lar dental casts of the 27 mandibular arches were taken
with a digital camera (DSC-F55V, Sony, Japan) and
printed in actual size (Fig 1, A). Photographs of the
mandibular dental arches were organized in a series
from tapered to square shapes by 10 members of the
Fig 1. A, Digital photograph of a dental cast (cast 11); B,
reference points determined on the cloud data in shaded
display (cast 11).
e316 Arai and Will
April 2011 Vol 139 Issue 4 American Journal of Orthodontics and Dentofacial Orthopedics

3. Department of Orthodontics (5 teaching staff members
and 5 fourth-year and third-year residents) from the De-
partment of Growth and Development, Harvard School
of Dental Medicine, Boston, Mass, and Department of
Orthodontics, Nippon Dental University, Tokyo, Japan.
Each examiner evaluated the 27 dental arches and
ranked them from most tapered (1) to most square
(27). The average rank given to a particular cast by all
10 examiners was determined to generate a mean rank
for the dental arch. The standard deviation of the mean
rank was also calculated for each dental arch. For exam-
ple, a dental cast photograph ranked as most tapered (1)
by 7 examiners and the second most tapered (2) by the
other 3 examiners would receive an overall mean rank
of 1.3 6 0.5. This dental arch was then placed into the
shape continuum near the most tapered side. The exam-
iners were asked to judge the dental-arch form using their
usual clinical evaluation method, and no discussion or
calibration sessions occurred before the evaluations.
The dental casts were also scanned and analyzed with
a 3-dimensional dental cast-measuring system. This
system consisted of a laser-scanning unit (Surflacer
VMS- 100F/ UNISN, Osaka, Japan) and a computer-
aided design (CAD) software program (Surfacer version
9.0. Imageware Inc., Ann Arbor, Mich).27,28
The reference points digitized on the image of each
cast were the midpoints of the incisal edges, the canine
cusp tips, and the buccal cusps of the premolars and the
first and second molars (Fig 1, B). The midpoint between
the mesiobuccal and distobuccal cusp tips of the first
and second molars was then computed and used to de-
fine the molar position. Only 1 point was used for each
molar to eliminate the effects of rotation, resulting in
a single point for the position of the tooth.
X and y coordinate data were extracted, and the dis-
tances between bilateral reference points for the canines
and the first and second molars were calculated as ca-
nine width, first molar width, and second molar width,
respectively. The canine-first molar and canine-second
molar ratios (percentages) were also calculated. Then
the means and standard deviations of canine width, first
molar width, second molar width, canine-first molar ra-
tio, and canine-second molar ratio were calculated.
Additionally, these coordinate sets were used to fit
a fourth-order polynomial equation (y 5 ax4
1 bx3
1
cx2
1 dx 1 e) to the 14 reference points for a dental
arch by using the least squares method (Fig 2). Then,
the coefficients of fourth-order and second-order terms
(“a” and “c,” respectively) could be determined from this
equation for each dental arch. The means and standard
deviations of the “a” and “c” terms were also calculated.
The nonparametric Spearman rank correlations (rs)
between the mean rankings of the dental-arch forms,
the canine and molar widths and ratios, and the “a”
and “c” coefficients derived from the equations were sta-
tistically analyzed. The Pearson correlation coefficients
(r) between the dental-arch width and ratios, as well as
the “a” and “c” terms, were also calculated and statisti-
cally analyzed with the Fisher z-transformation.29
Two statistical analysis methods for correlations were
used in this research: (1) the nonparametric Spearman
rank correlation to rank data and (2) the Pearson coeffi-
cient of correlation for continuous data.
To evaluate the reliability of the dental-arch form
ranking, photographs of 10 casts were randomly selected
from the sample. These standardized photographs were
ranked twice from tapered to square shapes by the
same examiner (K.A.), with an interval of 2 weeks
between evaluations, to determine intraexaminer
reliability. The same determination was made once by
another examiner (L.A.W.) to evaluate interexaminer re-
liability. Nonparametric Spearman rank correlations of
intraexaminer and interexaminer reliability were 0.71
(P 5 0.03) and 0.93 (P 0.01), respectively.
To evaluate the reliability of landmark location, 10
dental casts were randomly selected from the sample.
The following 10 reference points were determined: in-
cisal edges of the central incisors, canine cusp tips, buc-
cal cusp tips of the first premolars, and mesial and distal
buccal cusp tips of the first molar on the left side of the
dental arch. The same examiner (K.A.) determined 100
reference points in duplicate to evaluate intraexaminer
reliability, with 2 weeks between evaluations. The same
determination was made once by another examiner
(L.A.W.) to evaluate interexaminer reliability. The mean
differences between the 2 determinations by the same
examiner were 0.21 mm (SD, 0.21 mm) in the sagittal
plane and 0.17 mm (SD, 0.19 mm) in the transverse
plane. The mean differences between the 2 examiners
were 0.19 mm (SD, 0.22 mm) in the sagittal plane and
0.21 mm (SD, 0.19 mm) in the transverse plane.
RESULTS
The correlations between the means and standard
deviations of the subjective evaluation rankings for all
dental-arch forms are shown in Figure 3. The dental-
arch forms ranked as most tapered or most square had
small standard deviations, and interexaminer agreement
among examiners was obtained. In contrast, the arch
forms exhibiting relatively greater standard deviations, in-
dicating interexaminer variation in the evaluation, were
those with an intermediate ranking (ovoid arch form).
Statistically significant positive correlations were
found between the mean rankings of the dental-arch
forms, the widths and ratios of the canine and the first
and second molars, and the fourth-order term “a” of
Arai and Will e317
American Journal of Orthodontics and Dentofacial Orthopedics April 2011 Vol 139 Issue 4

4. the equation. Statistically significant negative correla-
tions were found between the mean rankings, the
canine-molar ratios, and the second-order term “c” of
the equation (Table I). The highest statistically signifi-
cant positive correlation was observed between the
mean rank and canine width. In contrast, the highest
negative correlation was obtained between the mean
rank and the second-order term “c” (Table I).
Statistically significant positive correlations were
found between the canine-first molar ratio and the
second-order term “c,” and between the canine-second
molar ratio and the fourth-order term “a.” Statistically
significant negative correlations were found between ca-
nine width, first molar width, second molar width, and
canine-second molar ratio and “c” (Table II).
DISCUSSION
Several subjective methods of classification have
been used to evaluate the characteristics of a patient’s
pretreatment dental-arch form for orthodontic diagno-
sis and treatment planning.6-8
In this study, arch forms
were subjectively ranked from tapered to square by 10
examiners, according to their own clinical evaluation
methods without any calibration sessions among
examiners. The arches ranked closest to either end of
the continuum varied the least among examiners,
suggesting that dental-arch forms that were distinctively
tapered or square were easier to classify. This result indi-
cates that the visual cognitive ability of the trained clini-
cian can almost instantaneously recognize the
differences between characteristics of complex dental-
arch shapes that are subtly distinct without a measuring
device, resulting in consistent classifications of arches
with the simple description of “tapered” or “square.”
These results also indicate that this type of subjective
classification with ranking becomes relatively more dif-
ficult for dental-arch forms that are intermediate, such
as the ovoid shape. Therefore, this difficulty in classify-
ing intermediate arch forms might result in unreliable
classification of the ovoid shape and suggests that cali-
bration should be performed among examiners before
classification and that quantitative analysis can be im-
portant, especially for the boundaries between tapered
and ovoid and between ovoid and square shapes.7
Addi-
tionally, relatively high interevaluator and intraevaluator
reliabilities of a subjective classification of dental-arch
forms by using a set of standard arch form templates
was recently reported.30
Therefore, additional study
might be necessary to compare these classification
methods.
There were statistically significant positive correla-
tions between rank and dental-arch widths (Table I).
This result suggests that wider dental arches had a ten-
dency to be ranked as square shapes. Also, there were
statistically significant negative correlations between
the ranks and ratios of the canines and the first and sec-
ond molars. Therefore, dental-arch forms with canines
that are wide relative to both the first and second molars
had a tendency to be square. In comparison, the highest
positive correlation was obtained between rank and ca-
nine width. These results indicate that the examiners
might predominantly focus on individual variations in
Fig 2. Example of dental-arch form analyses. Reference
points for each tooth and the polynomial equation with the
curve on the dental-arch form are shown (cast 11). The
results of the analyses were (1) mean rank 6 SD, 21.6
6 2.7; (2) canine width, 28.04 mm; (3) first molar width,
46.12 mm; (4) canine-first molar ratio, 59.56%; (5) second
molar width, 51.53 mm; (6) canine-second molar width ra-
tio, 54.41%; (7) “a,” 6.970 3 10À5
; and (8) “c,” 0.013.
Fig 3. Correlation between rank of dental-arch form and
standard deviation of the ranking.
e318 Arai and Will
April 2011 Vol 139 Issue 4 American Journal of Orthodontics and Dentofacial Orthopedics

5. canine width during classification and also support
a clinical system of preformed archwires that varies the
canine widths only.8
Additionally, canine width mea-
surements made with calipers might serve as a quantita-
tive analysis of dental-arch form for preformed
orthodontic archwire selection in the clinic. For example,
based on this study, the mean 6 standard deviation of
canine width was 25.98 6 3.02 mm. Therefore, the
ovoid arch shape can be mathematically defined as a ca-
nine width between 22.96 and 29.00 mm. To apply this
range of canine width for ovoid dental-arch shapes as
a standard in preformed archwire selection,8
further
studies in Class I normal occlusion and other Angle clas-
sifications7
might be required.
Statistically significant positive and negative correla-
tions between the subjective rankings of dental-arch
forms and objective quantitative analyses by using the
coefficients “a” and “c” of the fourth-order polynomial
equation were found, respectively (Table I). These find-
ings indicate that a tapered dental-arch form has a large
“c,” and a square dental-arch form has a large “a,” sup-
porting the hypothesis of Lu.13
Therefore, dental-arch
forms can be quantitatively analyzed, including the ref-
erence points on every tooth, by the result of the fitted
fourth-order term “a” and the second-order term “c”
of the dental arch, even in intermediate ovoid arches.
Additionally, the curve created by the equation results
in a smooth flexible curve, which represents all potential
tooth positions for the dental arch and, therefore, can be
used as an archwire template for each patient.13,26
There were statistically significant correlations be-
tween all arch-width measurements and canine-molar
ratios and the second-order term “c” (Table II). For ex-
ample, wider canines had smaller “c” values in the poly-
nomial equation fitted to the dental arch. On the other
hand, only the canine-second molar ratio was significantly
correlated with the fourth-order term “a” (Table II). These
results suggest that the more posterior teeth exert
a greater influence on the polynomial curve by the
fourth-order term “a.” Therefore, the second-order
term “c” is the main element for consideration in the
mathematical analysis with the polynomial equation
curve and of greater significance than the fourth-order
term “a” for the anterior region. Additionally, the
fourth-order term “a” can serve as a modifier for the
curve fitting in the posterior area of the dental-arch
form. Although we analyzed dental-arch forms without
much crowding, mathematical curve fitting for the
Table I. Means and standard deviations of the canines, first and second molar widths, canine-first and canine-second
molar ratios, and fourth-order and second-order terms of coefficients of the equation (“a” and ”c”) of the dental arch
forms
Canine
width (mm)
First molar
width (mm)
Canine-firstmolar
ratio (%)
Second molar
width (mm)
Canine-second molar
ratio (%)
Fourth-order
term “a”
Second-order
term “c”
Mean 25.98 45.55 60.67 50.50 51.44 4.942E-05 0.030
SD 3.02 3.49 6.37 3.45 4.77 2.649E-05 0.019
rs 0.820 0.582 À0.535 0.445 À0.651 0.656 À0.840
Z-transformation 4.183 2.969 À2.728 2.269 3.318 3.344 À4.283
P value 0.0001y
0.0030y
0.0064y
0.0233* 0.0009y
0.0008y
0.0001y
Results of the statistical analyses by nonparametric Spearman rank correlations between the mean rank and canine width, first and second molar
widths, canine-first molar and canine-second molar ratios, and “a” and “c.”
*P 0.05; y
P 0.01.
Table II. Pearson coefficient of correlation (r) between the dental arch width and ratios and “a” and “c” were also
calculated and statistically analyzed with the Fisher z-transformation
“a” “c”
r P value r P value
Canine width 0.364 0.0619 À0.683 0.0001y
First molar width 0.091 0.6541 À0.638 0.0002y
Canine-first molar ratio À0.050 0.8050 0.596 0.0008y
Second molar width À0.214 0.2880 À0.392 0.0422*
Canine-second molar ratio 0.616 0.0004y
À0.593 0.0008y
*P 0.05; y
P 0.01.
Arai and Will e319
American Journal of Orthodontics and Dentofacial Orthopedics April 2011 Vol 139 Issue 4

6. dental arch with crowding has also been reported.31
Therefore, how crowding and asymmetries affect the
arch form and whether this observation has an implica-
tion on the ranking method can be studied by using the
fourth-order polynomial equation in a future study.
The disadvantage of the fourth-order polynomial
equation is the current lack of simplicity for daily use
in a clinical setting to fit the curve onto the dental
arch. However, a software program fitting polynomial
equations to the dental arch is available,32
and some
computer systems with 3-dimensional technology for
dental cast analysis, diagnosis, and treatment planning
have recently been introduced and are rapidly spreading
in orthodontics.33-35
Soon, it is likely that this
technology will be readily available in orthodontic
offices. Based on the results of this study, the fourth-
order polynomial equation can be applied to estimate
the dental-arch forms of orthodontic patients with an
accurate and a flexible mathematical expression.
CONCLUSIONS
1. Subjective clinical assessments were generally in
agreement at the extremes of tapered and square
dental-arch forms, but the exceptions were arches
with an ovoid shape.
2. There were statistically significant correlations be-
tween subjective dental-arch classifications and
dental-arch dimensions, as well as the ratio deter-
mined from these variables and polynomial equa-
tion analyses.
3. Coefficients of fourth-order polynomial equations
were significantly correlated with individual vari-
ations in the size and shape of dental-arch forms
and might be a reliable tool for quantitative
analysis of dental-arch form in orthodontic
patients.
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