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ynam i c r el a t i o n sh i p s

an ter i o r segm en t

o f h e m a n d i b u l a r

S t a n l e y B r a u n D D S M M E a a n d W i ll ia m P . H n a t P h D b

I n d i a n a p o l i s I n d . a n d L o u i s v i l l e K y .

The hyperbo l ic cos i ne f unc t i on i s shown t o be an accu ra t e rep resen t a t i on o f the f o rm o f the

mand i bu l a r an t e r i o r t ee t h from t he can i ne / f i r s t p remo l a r con t ac t on one s i de a round t he pe r i me t e r

to the oppo si te s ide r = 0 .951). On the bas is o f th is math ema t ica l funct ion, the chang es in canine

width, anter ior segment depth, arch per imeter , and the i r re la ted inc isor angular a l terat ions are

f o recas t ab l e . Th i s know l edge w i l l a l l ow t he c l i n ic i an t o p red i c t t he e f f ec ts on va r i ous asp ec t s o f t he

an t e r i o r segm en t a rch f o rm as one o r m ore o f t hese v a r i ab l es a re a l te red w i t hou t reso r ti ng t o t r i a l

and e r ro r o r pe r f o rm i ng a wax -up . Fo r examp l e , the c l i n ic i an can p red i c t t he change in t he an t e r i o r

segme n t a rch dep t h and i nc i so r angu l a t ion t ha t wou l d o ccu r w i t h a l t e ra t ions in can i ne w id t h . Am J

Orthod Dentofac Orthop 1997;111:518-24. )

n u m b e r o f o r t h o d o n t i c t r e a t m e n t m o -

d a l i t i e s i n c o r p o r a t e i n t e r c a n i n e w i d t h e x p a n s i o n t o

r econc i le a r c h l eng th de f ic ienc ie s . 14 I nc i sor f l a r ing

h a s a l s o b e e n u s e d t o a c h i e v e t h is s a m e g o a l , as w e l l

a s to p r ov id e inc r ea sed l ip suppo r t . 6-9 I n add i t io n ,

a l t e r i n g t h e a r c h p e r i m e t e r t h r o u g h r e p r o x i m a t i o n

o f t h e a n t e r i o r t e e t h h a s b e e n u s e d t o r e c o n c i l e

s m a l l a r c h l e n g t h d e f i c i e n c i e s . T h e s e a l t e r a t i o n s i n

t h e f o r m o f t h e a n t e r i o r s e g m e n t m a y i n f l u en c e t h e

s t a b i l i t y o f t h e d e n t i t i o n . C l i n i c i a n s m a y t h e r e f o r e

c h o o s e t o i n t r o d u c e l o n g - t e r m r e t e n t i o n . 1-13 V a r i -

o u s i n v e s t i g a t o r s h a v e a l s o d o c u m e n t e d a n a t u r a l

c h a n g e i n t h e f o r m o f t h e a n t e r i o r s e g m e n t t h r o u g h

a d e c r e a s e i n t h e i n t e r c a n i n e w i d t h o v e r t i m e i n b o t h

t r ea ted and un t r e a te d p e r sons . 14-2

T h e s e c h a n g e s , w h e t h e r o c c u r r i n g n a t u r a l l y o r

t h r o u g h o r t h o d o n t i c t r e a t m e n t , a r e n o t i s o l a te d . F o r

e x a m p l e , w h e n t h e i n t e r c a n i n e w i d t h i s a l t e r e d ,

t h e r e i s a r e l a t e d c h a n g e i n t h e a r c h p e r i m e t e r a n d

in the po s i t ion o f the inc i sors . I n add i t ion , a s the a r ch

p e r i m e t e r i s c h a n g e d , t h e a n t e r i o r s e g m e n t d e p t h i s

a l t e r ed F ig . 1 ). The in te r r e la t ionsh ips be tween in te r -

can ine wid th , an te r io r s egm ent dep th , an te r io r a r ch

p e r i m e t e r, a n d i n c is o r a n g u l a ti o n a r e n o t k n o w n w i t h

any d egre e o f surety. Rick et ts e t a l . , 22 sugg ested th at

f o r each m i l l im e te r o f inc i sor advancem ent , the an te -

r io r a r ch pe r im e te r i s inc r eased by 2 m m , and tha t 1

m m o f i n t e r c a n i n e w i d t h i n c r e a s e c r e a t e s a 1 m m

incr ease in an te r io r a r ch pe r im e te r .

~ C l i n i ca l p r o f e s s o r o f o r t h o d o n t i c s , V a n d e r b i l t U n i v e r s i t y a n d U n i v e r s i t y

o f I l l in o i s a t C h i c a g o .

b A s s o c i a t e p r o f e s s o r o f m e c h a n i c a l e n g i n e e r i n g , S p e e d S c i e n t i f i c S c h o o l ,

U n i v e r s i t y o f L o u i s v i l le .

R e p r i n t r e q u e s t s t o : D r . S t a n l e y B r a u n , 7 9 4 0 D e a n R d . , I n d i a n a p o l i s , I N

4 6 2 4 0 .

C o p y r i g h t 1 9 9 7 b y th e A m e r i c a n A s s o c i a t i o n o f O r t h o d o n t i s t s .

0 8 8 9 -5 4 0 6 / 9 7 / 5 . 0 0 + 0 8 / 1 / 7 1 0 7 9

5 1 8

Canine/F'LrSt remolax

N o rm a l Co n t a c t

I~ ~l In te rcan ine Width (W)

- - S eg me nt D e pt h (D )

M 3

t n = - -

9 . 4 + 4 . 8

o= t .

1 4 . 2 2

. : . I/

F i g . 1 . I ll us tra t ion o f d i mens i ons o f m and i bu l a r an t e r i o r

segm ent a nd of t yp ica l ma ndibu lar cent ra l inc isor . 21

T h i s i n v e s t i g a t i o n e s t a b l i s h e s t h e m a t h e m a t i c a l

r e l a t i o n s h i p s b e t w e e n t h e r e l a t e d v a r i a b l e s p r e v i -

o u s l y m e n t i o n e d s o t h a t t h e e f fe c t s o f a p a r t i c u l a r

t r e a t m e n t m o d a l i t y m a y b e f o r e c as t w i th i n c r e a s e d

accur acy .

M E T H O D S A N D M A T E R I A L S

Twenty-one mandibular cas ts of untreated pat ients

that showed zero to minimal anterior segment irregulari-

ties were selected. In addition, each cast showed a fully

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Volume 111, No 5

Fig 2 Brown and Sharp MicroVal Coordinate Measuring Machine

developed, complete dentit ion less third molars. Casts

exhibiting incisal or cuspal attrition, or fracture, or ectopi-

cally erupted teeth were not included.

Each cast was oriented in a Brown and Sharp MicroVal

Coordinate Measuring Machine (Brown and Sharp) as seen

in Fig. 2. This device is used extensively in the precision

machine tool industry. A frictionless air-bearing probe re-

solves the coordina tes of a point in space in each of the three

orthogonal axes to 0.001 mm. Linear accuracy is 0.006 mm,

and reproducibility is 0.004 ram. T he casts were each secured

to a fixed plane, and the touch trigger probe used to identify

each measurement point, recording the corresponding X, Y,

and Z coordinates automatically to a data file.

Measurements were made at each canine cusp tip, the

center of each incisor incisal edge, and the normal con-

tacts between the first premolars and canines. The coor-

dinate compo nents of the anterior dental arch form in

three-dimension al space for each of the 21 casts were

record ed in this manner. A planar pro jection of the

mandibular anterior segment was subsequently obtained

by reducing the Z coordinates of each point to zero. A

reproducibili ty study that used one cast was conducted to

evaluate the variation of these measurem ents. The results

revealed the me an deviation to be less than 0.5 . All

measurements were made by the same opera tor (W.H.) .

An analytical equation of the anterior segmen t shape

is required to describe the relationships between the

intercanine width, anterior segment depth, anterior arch

perimeter, and incisor angulation. This shape closely

resembles a hyperbolic cosine function as i l lustrated in

Fig.

3 24

Thus the generalized equation of the mandibula r

anterior segment shape may be expressed as

where W represents half the crossarch distance from the

normal canine/first premolar contact on one side to the

opposite side in millimeters, and D the perpendicular

distance from the most anterior point between the two

central incisors to the crossarch dimens ion in m illimeters.

The arch perimeter may then be calculated using

f + w ~ ( d ~ ) 2

L = 1 + d X

W

where L is the anterior segment arch perimeter from the

canine/first premolar normal contact along the arch to the

opposite side in millimeters. Taking the derivative of Y

with respect to X in the first equation and substituting it

into the second equation yields

lw . .2r

, rX c o sh - ' (D +

]]2

2 + [cosh_ l D + 1) j [ c snL W 1)

+w [co sh- l D + 1)]2

L = W 2 dX

w

The relationship between anterior arch depth (/7)), ante-

rior arch perimeter (L), and arch intercanine/first premolar

arch width (W), is thus described. Th e millimetric mea sure -

ments are obtained through numerical integration of the

coordinates of each point of the anterior segment previously

described with MathCad 5.0 for Windows (MathSoft Inc.).

The related angular change for a typical mandibular incisor

is given by the formula

\ 1 4 . 2 /

X I

coshE cosh 1,] D ,

where ZXD is the change in depth of the anterior segment

in millimeters (Fig. 1).

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May 1 9 9 7

Y

i 4 \

~ 3 ~1 _ _ Y h [ ~ c os h-1 ? .9 8 42 ) + ? .9 8 4 2 , r = 0. . .

-15 -12 -9 -6 -3 O 3 6 9 12 15

Intercanine Width 2W) mm)

Fig. 3. A nterior dental arch shape of hyperbo lic cosign function for cast no. 1. Typical of

21 casts.)

T a b l e I C a l c u l a t e d d a t a o f 2 1 c a s t s f r o m m e a s u r e d

c o o r d i n a t e s

Cast

Cast measurements

Intercanine nterioregment

width depth

mm ram)

Segment

perimeter

mm)

1 2 8 8 8 0 8 2 8 4 3 3 8 4 8

2 2 4 7 2 2 1 0 3 6 4 3 3 5 0 6

3 2 6 9 7 8 7 6 1 0 3 1 4 3 6

4 2 7 8 6 8 8 4 0 5 3 3 0 5 3

5 2 3 9 8 3 6 9 8 5 2 8 0 9 5

6 2 5 8 6 7 8 1 0 0 3 1 4 8 4

7 2 8 1 6 8 8 4 9 6 3 4 0 9 1

8 2 6 7 8 6 1 0 2 0 0 3 5 3 2 1

9 2 2 4 9 6 7 8 3 9 2 7 1 5 8

1 0 2 4 6 7 2 8 5 9 7 3 2 5 6 6

1 1 3 1 2 8 9 1 0 0 7 4 3 8 0 4 0

1 2 2 4 4 0 0 8 7 1 7 2 9 1 5 3

1 3 2 8 8 3 8 1 0 0 3 4 3 6 2 8 2

1 4 2 8 1 8 3 7 6 9 4 3 3 5 4 7

1 5 2 7 0 6 3 7 9 4 5 3 2 1 5 3

1 6 2 8 8 7 4 9 4 9 0 3 6 8 2 7

1 7 2 8 9 9 3 1 0 4 4 3 3 7 5 7 0

1 8 2 7 9 9 6 1 0 7 6 2 3 7 2 3 8

1 9 2 8 4 9 8 8 8 6 6 3 4 9 4 4

2 0 2 7 0 8 4 6 9 6 5 3 3 4 7 2

2 1 2 7 3 5 2 8 4 3 0 3 4 8 6 5

M e a n 2 7 0 9 5 8 7 7 6 3 3 5 5 5

S D 2 0 4 2 1 1 2 8 2 9 0 5

M a x i m u m 3 1 2 8 9 1 0 7 6 2 3 8 0 4 0

M i n i m u m 2 2 4 9 6 6 9 6 5 2 7 1 5 8

RESULTS

T h e c a l c u l a t e d d a t a o f e a c h o f t h e 2 1 c as t s a r e

s h o w n i n T a b l e I . E a c h o f th e i n t e g r a t i o n s w a s b a s e d

o n t h e X a n d Y c o o r d i n a t e s o b t a i n e d f r o m t h e

M i c r o V a l C o o r d i n a t e M e a s u r i n g M a c h i n e .

T h e m e a n v a l u e o f t h e c a n in e /f i rs t p r e m o l a r

n o r m a l c o n t a c t c r o s s a r c h d i m e n s i o n W ) is 2 7 .0 m m

w i t h a s t a n d a r d d e v i a t i o n o f _+2 m m . T h e m a x i m u m

d i m e n s i o n i s 3 1. 3 m m a n d t h e m i n i m a l d i m e n s i o n

2 2.5 m m . T h e m e a n v a l u e o f t h e a n t e r i o r s e g m e n t

d e p t h D ) i s 8 .8 m m w i t h a s t a n d a r d d e v i a t i o n o f _+

r a m . T h e m a x i m u m d e p t h is 10 .8 m m a n d t h e

m i n i m u m 7 m m . T h e m e a n a n t e ri o r s e g m e n t pe r i m -

e t e r L ) , is 3 3 .6 m m w i t h a s t a n d a r d d e v i a t i o n o f

_+3 m m . T h e m a x i m u m p e r i m e t e r i s 38 m m a n d t h e

m i n i m u m 2 7 . 2 m m .

T h e r e l a t io n s h i p o f t he a n t e r i o r s e g m e n t a r c h

d e p t h D ) t o v a r i o u s i n t e r c a n i n e w i d t h s f o r t h e

m e a n a r c h p e r i m e t e r i s g i ve n b y th e f o r m u l a D =

- 0 . 6 5 5 W . T h e c o r r e l a t i o n c o e f f i c i e n t o f t h is r e l a -

t i o n s h i p i s r = 0 . 92 2 . T h i s r e l a t i o n s h i p a l t e r s t o D =

- 0 . 6 2 2 W r = 0 . 9 9 5) f o r a n a r c h l e n g t h o f + 1

s t a n d a r d d e v i a t i o n f r o m t h e m e a n , a n d t o D =

- 0 . 7 1 4 W r = 0 .9 8 3 ) f o r - 1 s t a n d a r d d e v i a t io n .

T h e s e r e l a t i o n s h i p s a r e s h o w n g r a p h i c a l l y in

F ig . 4 .

T h e r e l a t i o n s h i p o f t h e in c i s o r a n g u l a r c h a n g e

A ) t o t h e i n t e r c a n i n e w i d t h W ) is s h o w n

g r a p h i c a l l y i n F ig . 5 , a n d i s e x p r e s s e d b y t h e

f o r m u l a A = - 2 . 6 0 9 W r = 0 .9 9 2) f o r t h e m e a n

a r c h p e r i m e t e r o f 3 3 .5 m m . T h i s a l t e rs t o A =

- 2 . 4 8 0 W r = 0 . 99 5 ) f o r + 1 s t a n d a r d d e v i a t i o n in

a r c h p e r i m e t e r , a n d t o A = - 2 . 8 3 4 W r = 0 .9 8 4 )

f o r - 1 s t a n d a r d d e v i a t i o n in a r c h p e r i m e t e r .

T h e a l t e r a t i o n o f i n c i s or a n g l e w i t h r e s p e c t t o

a r c h p e r i m e t e r c h a n g e i s g iv e n by t h e e q u a t i o n

2~ = 2 .874 2xL) - 1 .678 r = 0 .993) fo r the

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Volume 111 No 5

i r ' -

o ~ ~ ~ 1 1 + 1 S .D .

S e g m e n t P e r i m e t e r

= 3 6 . 5 m m

, ~ ' - O ~ ~ ~ ~ ~

M e a n S l o p e

= - 0 . 6 2 2

" ' ' ' m ' - ~ " 1 " " " " " . ' ~ / ' /, r = 0 . 9 9 5

' - . . .

' ' - . . , .

\

- 1 S . D . S e g m e n t P e r i m e t e r = 30 .5 mm ~ . , , ' - . \

M e a n S e g m e n t P e r i m e t e r = 3 3 . 6 m m

M e a n S l o p e

= - 0 . 6 5 5

r = 0 . 9 9 2

0 ] I I I ] I I I I I I I I I

2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3 4

I n t e r e a n i n e W i d t h ( r a m )

Fig 4 Rela t ionsh ip betw een anter io r segm ent dep th to in tercan ine wid th fo r g iven arch

per imeters

5

o

6

5

=