Biomechanics in stage_1

BIOMECHANICAL
CONSIDERATIONS
IN BEGG STAGE I
AND STAGE II
www.indiandentalacademy.com

CONTENTS


Introduction
 What is Biomechanics
 Center of gravity
 Center of resistance
 Center of rotation
 Various Terminologies and laws
FORCE
MOMENT
COUPLE
MOMENT TO FORCE RATIO
STATEOF EQUILIBRIUM

 Begg Mechanotherapy
 INTRODUCTION
 Objectives of Stage-I
o Biomechanics of incisor intrusion
Degree of anchor bend
Role of Class II elastics
o Biomechanics of Incisor Tipping
 Objectives of Stage-II
o Biomechanics of space closure
 Conclusion
 Bibliography

INTRODUCTION

The physical concepts that form the
foundation of orthodontic mechanics
are the key in understanding how
orthodontic appliances work .
The principles are not unique to
orthodontics but are basic to the
science of static mechanics.

With the objective of achieving
predictable results based on
predetermined treatment goals, the
basic mechanics underlying
orthodontic appliance activation must
be thoroughly understood.

BIOMECHANICS

Biomechanics is the study of
mechanics as it affects the biologic
systems. It is the application of
mechanics to the biology of tooth
movement.
Biology + Mechanics = Biomechanics

Physical properties such as distance,
weight, temperature and force are
treated mathematically as either
SCALARS or VECTORS.

SCALARS include temperature and
weight, they have a definite
magnitude but do not have a direction.
They are completely described by
their magnitude.www.indiandentalacademy.com

VECTORS include force, these have
both magnitude and direction. In case
of force, along with magnitude and
direction, point of application of force
must be taken into account.

Various terminologies and laws:
FORCE
MOMENT
COUPLE
MOMENT TO FORCE RATIO

FORCE:
It is defined as an act upon a body that
changes or tends to change the state of
rest or motion of the body.
Force is a vector it has both magnitude
and direction.

The forces are indicated by straight
arrows

In case of understanding the magnitude
and direction of tooth movement, point
of application of force is important

CENTER OF MASS
Each body has a point in its mass,
which behaves as if the whole mass is
concentrated at that single point, which
we call the CENTER OF MASS in a
gravity free environment.

Center of Gravity:
The same is called center of Gravity in
an environment where gravity is
present.

The center of gravity of the tooth is
located more towards the crown of the
tooth as the mass of the tooth is
concentrated more coronally

Since the tooth is partially restrained
as its root is embedded in bone its
center of gravity moves apically and
this is known as CENTER OF
RESISTANCE (Cres)

Center of Resistance
Center of Gravity
Center of Resistance

 In case of single rooted tooth center of
resistance is on the long axis of tooth
between one third and one half of the
root length apical to alveolar crest.

 For a multirooted root, the center of
resistance is probably between the
roots 1-2 mm apical to furcation.

 Center of Resistance Varies (Cres):
Length of root: Maxillary canine has a
longer root than maxillary lateral incisor,
thus center of resistance of canine will
be more apically placed as compared
with center of resistance of lateral
incisor.www.indiandentalacademy.com

Periodontal status: The center of
resistance shifts apically in
periodontally compromised patients.
Alveolar bone height: Center of
resistance shifts apically as with the
alveolar bone loss.

The center of resistance for a single
rooted tooth estimated by different
authors is;
 At 50% of root length – Proffit, Nikoli
 Between 50% to 30% of root length –
Smith and Burstone.
 At 33% of root length – Burstone
 Between 25% to 33% root length –
Nanda

Moment
Is defined as a tendency to rotate

Moments can be symbolized by
curved arrows.

MOMENT is the product of the force times
the perpendicular distance from the point of
force application to the center of resistance.
M = F x d
It is measured in grams – millimeters.
F
d

MOMENT OF FORCE:
Moment is a measure of the turning
tendency produced by a force.
When a force is applied at any point
other than through the center of
resistance in addition of moving the
center of resistance in direction of the
force, a moment is created.

In case of tooth, since it is embedded in
the alveolar bone, we cannot apply
force directly on Cres, but can apply
force on the exposed part of the tooth,
which is at a distance from Cres.
Therefore with a single force we
invariably create a moment called as
moment of force.

A MOMENT may be referred as
Rotation
Tipping
Torquing.

Rotation
Tipping
Torquing
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If a line of action does not pass
through the center of resistance the
force will produce some rotation. The
potential for rotation is called as
moment.

The direction of a moment can be
determined by continuing the line of
action of the force around the center
of resistance.
F x d(X) = M(X)www.indiandentalacademy.com

CENTER OF ROTATION:
It may be defined as a point about
which a body appears to have rotated
as determined from its initial to final
position.

A simple method of determining a
Center of rotation - Draw the long axis
of the tooth in its initial and final
positions; we will see that both these
lines intersect at a point.
This is the point around which the
tooth rotates and is called Center of
rotation.

Center of Rotation

Center of rotation could be at the center
of resistance, apical or Incisal to Cres
or at infinity. Its position will
determine the type of tooth
movement.
The moment to force ratio controls the
center of rotation for the intended
tooth movement.

TYPES OF TOOTH
MOVEMENT
POSITION OF THE
CENTER OF ROTATION
A. Translation
B. Uncontrolled tipping
C. Controlled tipping
D. Root movement or Torquing
Lies at infinity
Slightly apical to center of
resistance
Apex of root
Incisal or occlusal edge

 Uncontrolled tipping: In this situation,
when force is applied the crown
moves in one direction and root
moves in the opposite direction. Here
Center of rotation lies near to center
of resistance. This is referred as
uncontrolled tipping.

Uncontrolled tippingUncontrolled tipping

 Controlled Tipping: In this situation,
crown moves in the direction of force
but the root position remains the same
or get minimally displaced. Here
Center of rotation lies at apex of the
root.

Controlled tippingControlled tipping

 Translation : In this situation tooth
moves bodily i.e. both crown and root
portion of tooth moves bodily in the
direction of force. Here Center of
rotation lies at infinity. All the points in
the tooth move by same distance in
the same direction in translation.

Bodily MovementBodily Movement
Center of
Rotation
at infinity

 Root movement: In this situation,
root moves in the direction of force but
the crown position remains the same
or get minimally displaced. Here
Center of rotation lies at incisal edge
of the crown.

RootRoot
MovementMovement

COUPLE:
Two equal and opposite, non -
collinear forces are called a couple.
Couple consists of two forces of equal
magnitude, which are parallel to each
other but not coincident and they face
in opposite direction.

The moment of this couple is equal to
the magnitude of one of the forces
multiplied by the perpendicular
distance between the two lines of
action of force.

If the two forces of the couple act on
opposite sides of the center of
resistance, their effect is additive.
However, if they are on the same side
of the center of resistance, their effect
is subtractive

AdditiveAdditive
+
F1
F2
F1XDM1= M2=F2XD
D
M=F X D

SubtractiveSubtractive
-
F1
F2
M1=F1XD1 M2=F2XD2
D1
D2
M= F X (D1-D2)

MOMENT TO FORCE
RATIO:

In terms of direction, the counter-
balancing moment is always going to
be in the direction opposite the
moment of force.

It seems that type of movement
exhibited by a tooth is determined by
the ratio of the counter-balancing
moment produced to the net force that
is applied to a tooth .
This is called as the moment to
force ratio .

Moment of force
Force
Counter-balancing moment

The ratio of the counter-balancing
moment to the force applied
determines the type of tooth
displacement, brought about by the
combined application of a force and
counter-balancing moment.
As the counter-balancing moment
increases, the center of rotation
moves apically.

At one specific level of M/F, the
moment which arises from the force
and the applied counter-balancing
moment cancel out each other i.e.
there is no rotational component, and
hence only a translation takes place
under the effect of force .

 M/F Ratio values normally quoted of various
types of displacements are
M/F ratio less than 5:1 causes uncontrolled
tipping in which the crown and the root apex
move in opposite directions.
M/F ratio between 5:1 and 8:1 causes
controlled tipping in which the root apex
remains stationary and only the crown
moves.

M/F ratio of 10:1 causes translation.
The crown and the root apex move to
same extent in the same direction of
force.
M/F ratio of 12:1 causes root
movement. The crown remains
stationary while only the root moves.

It is important to note that the
differences between the M/F Ratio for
controlled tipping, translation and root
movement are small.

In other words, even small alterations
in the magnitude of the applied force
or the counter-balancing moment will
alter the type of tooth movement
brought about.

STATE OF
EQUILIBRIUM

When an appliance is fitted in the
mouth, it assumes a state of equilibrium.
The active elements in the appliance
generate certain forces or moments.
Other forces or moments arise
automatically in the system to balance
these forces or moments. Some of them
may be beneficial while others may be
undesirable.

Whenever state of equilibrium is
established in the system the sum of all
forces and moments (together) present
must be zero in all three planes.

For example, tip back bend (like the
bite opening bend in Begg appliance)
generates a moment which tends to
tip the molar tooth crown distally. This
is balanced by an automatic creation
of another moment in the overall
system in opposite direction
comprising of two forces an intrusive
force at the anterior end and on
extrusive force on the molar.

BEGG
MECHANOTHERAPY

Begg mechanotherapy is very efficient
in opening the deep anterior
overbites. It is generally agreed that
Begg mechanics bring about bite
opening by a combination of molar
extrusion (especially of lower molars)
and some intrusion of lower anteriors.

Upper anteriors may not change in
their position in vertical direction (i.e.
they are prevented from erupting) or
may intrude slightly or may even
extrude slightly.

There are three basic movements in
the Begg mechanotherapy
 Incisor intrusion
 Tipping of teeth
 Root uprighting.

The mechanism of intrusion is
understood by considering the net
intrusive force magnitude and direction
in relation to Centre of resistance of
tooth.
While tipping of teeth and root
uprighting is explained on the basis of
M/F ratio.

OBJECTIVES OF
STAGE I:

 Open the anterior bite :
Proper amount of bite opening bends
or curves in the arch wire.
Continuous wearing of Class II
(intermaxillary) elastics as required.

 Eliminate anterior crowding :
Vertical loops between crowded
anterior teeth, with bracket areas
modified for desired overcorrection.
Loop arch wire
NiTi wire

 Close anterior spaces :
Plain arch wire with latex elastic or
elastomeric chain from cuspid to
cuspid.
Closure of Anterior spaces by cuspid tie

 Over correct rotated cuspids and
bicuspids :
Rotating springs
Elastomeric traction into the arch
wire

Rotating spring
Rotating spring on Premolar
Elastomeric traction

ANTI ROTATIONAL ADJUSTMENT
Incisor
Premolar

 Over correct the mesiodistal
relationship of the buccal segments
Continuous wearing of class II
elastics.
Proper bite opening bends in both
upper and lower arch wires.

BIOMECHANICS OF STAGE I

As we understand today the Begg
appliance is a good example of single
couple system.
Stage I arch wire

The orthodontic environment created
during stage I is conducive to rapid
movement of anterior teeth under the
light forces generated by the arch
wires and intermaxillary elastics

MECHANISM OF INTRUSION:

Lack of true intrusion of the maxillary
incisors was one of the major
weaknesses of traditional Begg.
Bite opening occurred mainly on
account of molar extrusion and some
intrusion of the lower incisors.

Whether the upper incisors are
intruded is a debated issue.
The round archwire derives bite
opening force from the anchor bends.

A clockwise moment generated by the
anchor bend in the molar tube (upper)
is automatically balanced by the
generation of anticlockwise moment in
the anterior segment along with
intrusive force on the anterior and
extrusive force on the molars in order
to establish state of equilibrium.

This anticlockwise moment generated
in the anterior segment bring about
labial flaring of the upper anteriors.

This flaring tendency of upper incisors
can be resisted by using Class II
elastics during stage I.
But class II force along with horizontal
component have vertical component
of force which reduces the magnitude
of the intrusive force of the arch wire
on the upper anteriors.

Thus the interplay between the
intrusive force from the archwire and
the retractive force from the elastics
determines both the magnitude and
direction of the net resultant force
acting on anterior teeth.

THE INTERPLAY BETWEEN THE
ANCHOR BEND AND CLASS II
ELASTICS
CLASS II ELASTIC
FORCE
INTRUSIONFORC

VARIOUS TYPES OF BITE
OPENING BENDS:
 The Anchor bend the conventional
bite opening bend causes more
intrusion of canines while the lateral
and central incisors progressively lag
behind.

 A Gable bend causes a progressively
more intrusion of central and lateral
incisor, as compared to canine
 Mollenhouer’s bite opening curve –
Mollenhouers especially recommends
it with use of 0.018 wire.

 Swain modification: Mild gingival curve
is incorporated in the anterior section,
from mesial of cuspid to mesial of other
side cuspid.

CONSIDERATION OF
THE MAGNITUDE
OF INTRUSIVE
FORCE.

OPTIMAL INTRUSIVE FORCE VALUE
Many authors have suggested
optimum intrusive force values
ranging from 15-30 grams per upper
incisor and slightly higher values for
upper canines.

For active intrusion the upper
anteriors should receive
approximately 60 grams net force in
the midline, after negating the
extrusive component of Class II
elastics.

ROLE OF LIGHT
CLASS II ELASTICS:

Hocevar stated that 120 grams of
intrusive force generated by arch wire
in conjunction with 60 grams of Class
II elastics pull on either side is
“efficient for intrusion”

According to Dr.Jayade net intrusive
force of 60 grams can be obtained by
a combination of 75 grams of intrusive
force generated by arch wire and
some modification in wearing of
elastics that is by using light elastic
forces for longer periods from 2-5
days. Very light Class II force is
delivered as the elastic force
diminishes rapidly in oral environment.

Sims states the use of 3/8” ultra light
elastics instead of routinely used
5/16” light elastics. He said continue
the same elastic for 4-5 days till they
break.

Role of Class I Elastic
Forces

Modifying the force system to achieve
simultaneous intrusion and retraction
using Class I elastic instead of Class
II elastics was first illustrated by Shin
Yang Liu (1981).

 He summarized that the direction of
resultant force should pass through
the center of resistance of anterior
teeth (or close to it).
 Therefore, substituting Class II elastic
forces by Class I elastic forces would
orient the resultant force more
vertically passing nearer to the center
of resistance of anterior teeth.

 In traditional begg technique the
direction of the intrusive vector of the
maxillary arch wire and the extrusive
vector of the class II elastics are
opposite. This accounts for the
difficulty in obtaining anterior maxillary
teeth intrusion.

 In this technique modification, of
using Class I elastics, it solve the
problem of lack of intrusion of the
maxillary anteriors.

 In this arrangement the vectors are in
the same direction as the elastic pull
and the archwire force are
unidirectional and hence synergistic.

Dr. Jyothindra Kumar introduced
concept of power arms as a point of
attachment high up in the vestibule for
the engagement of Class I elastics.
CLASS I ELASTIC FORCE

Dr. Jayade has been using Class I
elastics, which were worn from
transpalatal arch (TPA) to the canine
hooks/loops.

It is impossible to precisely calculate
the required intrusive force every time,
for every patient, since it is dependent
on various variables.
 Different root sizes and tooth
inclination.
 Different arch sizes, which affect the
length of the wire spans and stretch of
the elastics.

 Individual biomechanical response
 Difference in the archwire sizes.
Normally .018” wire will produce more
intrusive force as compared to 0.016”
wire when some degree of anchor
bend is given.

THE CONCLUSION IS, “TO USE
HIGHER INTRUSIVE FORCES IN
COMBINATION WITH VERY
LIGHT CLASS II ELASTIC
FORCES FOR ACTIVE UPPER
INCISOR INTRUSION”

CONSIDERATION OF
THE DIRECTION OF THE
RESULTANT FORCE:

Teeth respond only to the resultant
of the forces which are applied and
not to the individual components of
the force system.

During Stage I, the upper anteriors are
subjected to two forces i.e. the
retractive force of class II elastics and
the intrusive force generated by the
anchor bend in the arch wire. The
resultant of these two will determine
how the upper anterior teeth respond
to the intrusion.

The direction and magnitude of
resultant force both depend upon the
interplay between.
 Magnitude of Intrusive Force: Whose
direction remains constant i.e.
tangential to the arc that the anterior
segment of the archwire would
subscribe, if released from the
brackets.
 Magnitude and the direction of the
elastic force.

Different inclinations of the anterior
teeth would require different
combinations of the intrusive and
elastic forces.
Hocevar states, that the teeth are not
affected by the magnitudes of
various components of force
systems, they experience only the
total resultant force

For example, in case of severely
proclined upper anteriors a low
magnitude of intrusive force along
with high class II force would give a
desired resultant force, passing
palatal to Cres, this will help
correcting the proclination of incisors .
Once the inclination of upper incisors
is corrected then the class II elastics
force is reduced helping in keeping
the resultant force close to Cres .

45gms
60gms
60 gms
30gms

In Class II Division 2 cases , where the
upper centrals are retroclined , only
intrusive force should be used
(Avoiding the Class II elastics) The
intrusive force acts labial to Cres and
corrects the retroclination. Once the
inclination is corrected then we can use
Class II elastics .

60gms
45gms
20gms

MECHANISM OF
ROTATION

Rotating spring works on the principal
of couple.
Since in rotating spring the couple
generated is acting on one side of
Cres of tooth so it is less effective as
compared to couple acting on either
side of Cres

OCCLUSAL VIEWwww.indiandentalacademy.com

MECHANISM OF
TIPPING:

CLASS II ELASTIC
FORCE
The concept of tipping back the
teeth in the first stage & further in
stage II …
INTRUSIONFORC

Generally, uncontrolled tipping is
undesirable because it leads to root
resorption as stated by Reitan. There
is more resorption when uncontrolled
tipping is in labio-lingual direction.

Intrusion and tipping are intimately
related not only because they are
carried out simultaneously but also,
when both are balanced judiciously it
help in overcoming uncontrolled
tipping of incisors.

This is achieved by manipulating the
intrusive force generated by wire and
retractive component of force from the
Class II elastics.

BOTH THE ANCHOR BEND IN THE
WIRE AND CLASS II ELASTICS
PRODUCE MOMENTS IN THE
SAME LABIO-LINGUAL PLANE
BUT ACT IN OPPOSITE
DIRECTIONS.

The intrusive force produces crown
labial-root lingual moment i.e.
anticlockwise moment on the upper
anteriors. While the retractive force
produced the Class II elastics
generates clockwise moment i.e.
crown lingual-root labial moment

The moment from the intrusive force
can act as the counter balance
moment against the moment
produced by the elastic force.
The ratio of the former to the
retraction component of the elastic
force is the M/F ratio which governs
the type of tipping while retracting the
anterior teeth.

The most important consideration is to
keep light Class II elastic and use
adequate amount of intrusive force
so that correct M/F ratio (8:1) is
obtained to have a controlled
tipping.

PREVENTING
UNCONTROLLED
TIPPING OF
UPPER INCISORS

In the refined Begg mechanics, use of
MAA (Mollenhauer’s Aligning
Auxillary) which provides a moment in
the labio-lingual plane by creating a
couple. This couple moment is an
anti-clockwise moment.

CLASS II ELASTIC
FORCE
INTRUSIONFORC

The moment produced by the anchor bend is
in the anticlockwise direction in the Y – Axis.
In case of MAA, the moment of couple
generated again is in anticlockwise direction
but in X – Axis.
Both the moments generated by the anchor
bend and the MAA are in the anticlockwise
direction thus gets summed up.

Once the bite is opened in the first
stage, the intrusive force level is
reduced which inturn reduces M/F
ratio. This leads to greater likelihood
of uncontrolled tipping of upper
anterior teeth during later part of the
first stage and whole of second stage.

Thus the anticlockwise moment
produced by anchor bend on anterior
is supplemented by the moment of
couple produced by MAA

Flaring occurs as lower incisors are
subjected to crown labial root-lingual
moment from the intrusive force
generated in arch wire, while there is
no restraining force on these teeth as
similar to Class II elastic force on
Upper incisors.

PREVENTING
UNCONTROLLED
TIPPING OF LOWER
INCISORS:

The flaring can be avoided by two
means;
 Minimizing the clockwise force
moment by reducing the intrusive
force.
 Secondly, cinching tightly the distal
ends of the arch wire.

 Lastly by producing counter moment
using a MAA for labial root torque or a
reverse torquing auxiliary (Udder
arch)

In case of severely lingually tipped
lower anteriors, Cres will be lying
buccal to the point of application of
the intrusive force generated by the
anchor bend so there is more chances
to tip the lower anteriors more
lingually.

So in that case we give a By pass
arch wire in order to upright the lower
incisors .

BEGG STAGE II

Among the traditionally described
stages of Begg technique, the second
stage of treatment involves closure of
extraction spaces. This is thought to
be the easiest phase of treatment.

During Stage II all the corrections
achieved during Stage I should be
maintained.

 Maintain Edge to Edge relationship
of anterior teeth
 Maintain anterior space closure
 To maintain overcorrected or normal
mesiodistal molar relationship

 In addition to the above, the stage II of
the refined Begg aims are
Controlled tipping of the incisors,
when space closure is to be mainly
achieved by the anterior retraction.

OBJECTIVES OF
STAGE II:

When all the objectives of Stage I are
met stage II mechanics can be
instituted.
The sole or main purpose of II stage is
closure of extraction spaces.

The extraction space can be closed by
either retraction of the anteriors or
protraction of the posteriors or
combination of both.

BIOMECHANICS OF
STAGE II

The anchor bend should be sufficient
enough as to produce a counter
clockwise moment slightly less than
the clockwise moment produced by
the Class I elastics in anterior section
of upper arch.

The M/F ratio should be sufficient
around 8/1 so as to have a controlled
tipping movement.

INTRUSIONFORC
At the end of Stage II

Same way in lower arch the clockwise
moment should be slightly lesser than
anticlockwise moment produced by
Class I elastics, so as to have
controlled tipping movement.

Normally 0.016 upper and lower arch
wires with reduced bite opening bends
are used. Some authors say use of
heavy arch wire 0.020 as it will
function as retainers to maintain arch
form and bite opening achieved during
stage I.

Dr. Swain advocated the use of lingual
attachments on molars and cuspids to
allow the use of lingual space closing
elastics to aid the traditionally used
buccal vector of intra maxillary elastic
force during stage II known as half
strength elastics.

Two distinct advantages in using intra
maxillary (Half strength) space closing
elastics
 It gives a better positional control over
the anchor molar thus obviating the
need for a mandatory compensate toe
in bend when using elastic force only
from buccal side.
 Closure of extraction spaces becomes
easier.

USE OF BRAKING
MECHANICS

When further retraction of anterior
teeth into the remaining extraction
space is deemed undesirable
clinically, then the posterior teeth are
brought forward, that is posterior
teeth are mesialized.

To achieve mesialization of posterior
teeth heavy elastic forces are
required with concurrent use of brakes
in the anterior region.

Various brakes are:
 Using uprighting springs (passive
springs)
 Reverse torque to incisor roots (Udder
arch and MAA)
 Using T pins

Passive Uprighting Springs T PINS
MAAUDDER ARCH

The brakes reverse the anchorage
site from the posterior to anterior
segment by allowing bodily movement
rather than the tipping of anterior
teeth, this bodily movement provides
more resistance hence acting as a
anchorage.

CONCLUSION

The importance of biomechanics is
well understood in clinical
orthodontics. Application of
biomechanical principles improve
the efficacy of our appliance
system as well as orthodontic
technique.

A common misconception is that the
application of biomechanical
properties would make the technique
too cumbersome. On the contrary,
biomechanically designed appliance
gives a predictable tooth movement,
optimal biologic tissue response and
minimal side effects.

In the lighthearted note - One can say
that on the average, an orthodontist
spends half the treatment time on
problems presented by the patient
and other half on problems resulting
from treatment side effects .

ORTHODONTICS COULD BE IN
OUR HAND IF WE USE
EFFICIENT BIOMECHANICS

THANK YOU

BIBLIOGRAPHY

 Nanda Ravindra. Biomechanics in
clinical orthodontics.Philadelphia: W.B
Saunders Company ;1997
 Begg, P. R.: Begg orthodontic theory
and technique, Philadelphia, 1965, W.
B. Saunders Company.
 Swain, B. F., and Ackerman, J. L.: An
evaluation of the Begg technique, AM.
J. ORTHOD. 55: 668-687, 1969.

 Hocevar RA: Orthodontic force systems:
Technical refinements for increased
efficiency. AM J ORTHOD 81: 1-11, 1982.
 Hocevar RA: Understanding, planning, and
managing tooth movement: Orthodontic
force system theory. AM J ORTHOD 80:
457-477, 1981.
 Reitan K: Tissue behavior during
orthodontic tooth movement. AM J
ORTHOD 46: 881-900, 1960.

 Cadman, G. R.: Nonextraction treatment of
Class II, Division 1 malocclusion with the
Begg technique, AM. J. ORTHOD. 68: 481-
498, 1975.
 Sims, M. R.: Anchorage variation with the
light wire technique, AM. J. ORTHOD. 59:
456-469, 1971.
 Marcotte MR: Prediction of orthodontic tooth
movement. AM J ORTHOD 69: 511-523,
1976.

 Thompson, W. J.: Current application of
Begg mechanics, AM. J. ORTHOD. 62:
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Biomechanics in stage_1

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