Mechanical principles in orthodontic force control

MECHANICAL PRINCIPLES
IN ORTHODONTIC FORCE
CONTROL

Prepared by :
Alaa Abdelsalam
Yaser Basheer

Introduction
• Optimum orthodontic tooth movement is produced by
light, continuous force.

• It is particularly important that light forces do not
decrease rapidly, decaying away either because the material
itself loses its elasticity or because a small amount of tooth
movement causes a larger change in the amount of force
delivered.

ELASTIC MATERIALS
AND THE PRODUCTION
OF ORTHODONTIC
FORCE

The Basic Properties of Elastic Materials

1. The elastic behavior of any material is defined in terms of
its stress–strain response to an external load.

2. Stress: the internal distribution of the load, defined as
force per unit area.

3. Strain : is the internal distortion produced by the
load, defined as deflection per unit length.

orthodontic archwires and
springs can be considered as
beams, supported either
only on one end (e.g., a
spring projecting from a
removable appliance) or on
both ends (the segment of
an archwire spanning
between attachments on
adjacent teeth)

(A) Cantilever (B) and supported beams.

• If a force is applied to such a
beam, its response can be
measured as the deflection
(bending or twisting) produced
by the force)

• Force and deflection are
external measurements. Internal
stress and strain can be
calculated from force and
deflection by considering the
area and length of the beam.

• For orthodontic purposes, three major properties of beam
materials are critical in defining their clinical usefulness:
strength, stiffness (or its inverse, springiness), and range.
Each can be defined by appropriate reference to a force–
deflection or stress–strain diagram

The stiffness of the material is given by the slope of the linear
portion of the curve. The range is the distance along the X-axis
to the point at which permanent deformation occurs (usually
taken as the yield point, at which 0.1%permanent deformation
has occurred). Clinically useful springback occurs if the wire is
deflected beyond the yield point (as to the point indicated here as
―arbitrary clinical loading‖), but it no longer returns to its original
shape. At the failure point, the wire breaks. Stiffness and
springiness are reciprocal properties:

• Three different points on a stress–strain diagram can be
taken as representative of the strength of a material

•The first two points attempt to describe the elastic limit of
the material (the point at which any permanent deformation
is first observed).

• The first point is the proportional limit: the highest point
where stress and strain still have a linear relationship (this
linear relationship is known as Hooke's law). Precisely
determining this point can be difficult, so a more practical
indicator is the yield strength—the intersection of the
stress–strain curve with a parallel line offset at 0.1%strain.

• Typically, the true elastic
limit lies between these
two points.

• The third point is ultimate
tensile strength: The maximum
load the wire can sustain is
reached after some permanent
deformation, and is greater than
the yield strength.

• Since this ultimate strength
determines the maximum force the
wire can deliver if used as a
spring, it also is important
clinically, especially since yield
strength and ultimate strength differ
much more for the newer titanium
alloys than for steel wires.

• Range is defined as the distance
that the wire will bend elastically
before permanent deformation
occurs.

• For orthodontics, this distance is
measured in millimeters . If the
wire is deflected beyond this
point, it will not return to its
original shape, but clinically useful
springback will occur unless the
failure point is reached.

• These three major properties have an important relationship:

• Two other characteristics of some
clinical importance also can be
illustrated with a stress–strain
diagram: resilience and formability

• Resilience is the area under the stress–strain curve out to
the proportional limit. It represents the energy storage
capacity of the wire, which is a combination of strength
and springiness.

• Formability is the amount of permanent deformation that
a wire can withstand before failing. It represents the amount
of permanent bending the wire will tolerate (while being
formed into a clinically useful spring, for instance) before it
breaks.

The properties of an ideal wire material for
orthodontic purposes:
(1) high strength
(2) low stiffness (in most
applications)
(3) high range.
(4) high formability
(5) the material should be weldable
or solderable, so that hooks or
stops can be attached to the
wire.
(6) reasonable in cost.

• In the United States, orthodontic appliance
dimensions, including wire sizes, are specified in
thousandths of an inch
.016 inch = 16 mil

In Europe and many other areas of the world, appliance
dimensions are specified in millimeters. For the range of
orthodontic sizes, a close approximation of sizes in
millimeters can be obtained by dividing the dimensions in
mils by 4 and moving the decimal point one place to the left
(i.e., 16 mil = 0.4 mm; 40 mil = 1.0 mm).

ORTHODONTIC
ARCHWIRE MATERIALS

Precious Metal Alloys

• precious metal alloys were used routinely for
orthodontic purposes, because nothing else would tolerate
intraoral conditions.
• Gold itself is too soft for nearly all dental purposes, but
alloys (which often included platinum and palladium
along with gold and copper) could be useful
orthodontically.

• Currently, the only considerable
advantage to gold is the ease of
fabricating cast appliances, such as
custom-fit bonding pads used with fixed
lingual appliances

Stainless Steel and Cobalt–Chromium Alloys
• Stainless steel, or a cobalt–
chromium alloy (Elgiloy; Rocky
Mountain Co.) with similar
properties, replaced precious metal
in orthodontics because of
considerably better strength and
springiness with equivalent
corrosion resistance.

• Stainless steel's rust resistance results
from a relatively high chromium
content. A typical formulation for
orthodontic use has 18%chromium and
8%nickel (thus the material is often
referred to as an 18-8 stainless steel).

• During manufacture. Steel is
softened by annealing and hardened
by cold working.

• Fully annealed stainless steel
wires are soft and highly formable.

• The steel ligatures used to tie
orthodontic archwires into brackets
on the teeth are made from such
―dead soft‖ wire.

• Steel archwire materials are
offered in a range of partially
annealed states, in which yield
strength is progressively enhanced
at the cost of formability.

• The steel wires with the most
impressive yield strength
(―super” grades) are almost
brittle and will break if bent
sharply.

• The “regular” grade of
orthodontic steel wire can be bent to
almost any desired shape without
breaking.

• If sharp bends are not needed, the
super wires can be useful, but it is
difficult to show improved clinical
performance that justifies either their
higher cost or limited formability.

• Elgiloy, the cobalt–chromium
alloy, has the advantage that it can
be supplied in a softer and therefore
more formable state, and the wires
can be hardened by heat treatment
after being shaped.
• The heat treatment increases
strength significantly. After heat
treatment, the softest Elgiloy
becomes equivalent to regular
stainless steel, while harder initial
grades are equivalent to the ―super‖
steels.
• This material, however, had almost disappeared by the end of the
twentieth century because of its additional cost relative to stainless
steel and the extra step of heat treatment to obtain optimal properties.

Nickel–Titanium Alloys

Properties of Nickel–Titanium Alloys
• Archwires formed from nickel–titanium alloys are
extremely useful during initial orthodontic alignment due to
their exceptional ability to apply light force over a large
range of activations.
• The properties of NiTi alloys cannot be discussed without first
understanding that these alloys can exist in more than one
crystal structure.

• At lower temperatures and higher stress, the martensitic
form is more stable, while at higher temperatures and lower
stress, the austenitic form is more stable.

• The uniqueness of NiTi is that the transition between the
two structures is fully reversible and occurs at a remarkably
low temperature.

• This phase transition allows certain NiTi alloys to exhibit
two remarkable properties found in no other dental
materials— shape memory and superelasticity.

 Shape memory
refers to the ability of the material to ―remember‖ its
original shape after being plastically deformed while in the
martensitic form.
• In a typical application, a certain shape is set while the
alloy is maintained at an elevated temperature, above the
martensite–austenite transition temperature. When the alloy
is cooled below the transition temperature, it can be
plastically deformed, but the original shape is restored
when it is heated enough to regain an austenitic structure.

This temperature-induced change in crystal
structure (called thermoelasticity) was important
to the original nitinol use in the space program
but proved difficult to exploit in orthodontic
applications.

 Superelasticity

• Superelasticity refers to the very large reversible strains
that certain NiTi wires can withstand due to the martensite-
austenite phase transition.

• In engineering applications, it also is frequently
described as pseudoelasticity, due to the nonlinear stress–
strain curve, which is not typical of elastic behavior

• Materials displaying superelasticity are austenitic alloys
that undergo a transition to martensite in response to stress.

• Most archwire materials can be reversibly deformed only
by stretching interatomic bonds (which creates the linear
region of the stress–strain curve), while superelastic
materials can undergo a reversible change in internal
structure after a certain amount of deformation.

• This stress-induced martensitic transformation manifests
itself in the almost flat section of the load-deflection curve.
This means that an initial archwire could exert about the
same force whether it was deflected a relatively small or
large distance, which is a unique and extremely desirable
characteristic.

Section A-B represents purely elastic deformation of the austenitic phase (note that in
this phase A-NiTi is stiffer than M-NiTi). The stress corresponding to point B is the
minimum stress at which transformation to the martensitic phase starts to occur. At
point C, the transformation is completed. The difference between the slopes of A-B
and B-C indicates the ease with which transformation occurs. After the transformation
is completed, the martensitic structure deforms elastically, represented by section C-D
(but orthodontic archwires are almost never stressed into this region, and this part of
the graph usually is not seen in illustrations of the response of orthodontic archwires).
At point D, the yield stress of the martensitic phase is reached, and the material
deforms plastically until failure occurs at E.

If the stress is released before reaching point D (as at point C1 in the
diagram), elastic unloading of the martensitic structure occurs along the line C1-
F. Point F indicates the maximum stress on which the stress-induced
martensitic structure on unloading can exist, and at that point the reverse
transformation to austenite begins, continuing to point G, where the austenitic
structure is completely restored. G-H represents the elastic unloading of the
austenite phase.

•Although shape memory is a thermal reaction and
superelasticity is a mechanical one, they are inherently
linked.

• Due to the close interaction of these properties, wires
displaying martensite–austenite transitions are
subsequently referred to as A-NiTi. All other NiTi wires are
stabilized in the martensitic form and are subsequently
referred to as M-NiTi.

NiTi Wires in Clinical Orthodontics
• The original Nitinol wires marketed under that name in the
late 1970s by Unitek were M-NiTi wires, with no application
of phase transition effects.

• As supplied for orthodontic use, Nitinol is exceptionally
springy and quite strong but has poor formability.

• In the late 1980s, new nickel–titanium wires with an
austenitic grain structure (A-NiTi) appeared. These wires
(Sentinol, GAC; Copper NiTi, Ormco/Sybron; and several
other suppliers) exhibit superelasticity and/or shape
memory in various degrees.

• Part of the unusual nature of a
superelastic material like A-NiTi is
that its unloading curve differs
from its loading curve (i.e., the
reversibility has an energy loss
associated with it).

• This means the force that it delivers
is not the same as the force applied
to activate it. The different loading
and unloading curves indicate that
the force delivered by an A-NiTi
wire can be changed during clinical
use merely by releasing and retying
it .
• In contrast, the unloading stiffness for steel, beta-Ti, and M-
NiTi wires is the same for all activations.

For the orthodontist,
• wire bending in the classic sense is all but impossible with A-
NiTi wires because they do not undergo plastic deformation
until remarkably deformed . The wires can be shaped and their
properties can be altered, however, by heat treatment. This can
be done in the orthodontic office by passing an electric current
between electrodes attached to the wire or a segment of it.

• Miura et al were the first to show that it is possible to
reposition the teeth on a dental cast to the desired
posttreatment occlusion, bond brackets to the setup, force an
A-NiTi wire into the brackets, and then heat treat the wire so
that it ―memorizes‖ its shape with the teeth in the desired
position. The wire then incorporates all of what would
otherwise be the ―finishing bends‖ usually required in the last
stages of

• The properties of A-NiTi have quickly made it the
preferred material for orthodontic applications in which a
long range of activation with relatively constant force is
needed (i.e., for initial archwires and coil springs).

M-NiTi remains useful, primarily in the later stages of
treatment when flexible but larger and somewhat stiffer
wires are needed.

• Rectangular A-NiTi wires, however, do not have enough
torsional stiffness to be effective torquing arches, so larger
rectangular wires used for more detailed positioning of
teeth perform better if made from M-NiTi.

Beta-Titanium

• In the early 1980s, after Nitinol but before A-NiTi, a quite
different titanium alloy, beta-titanium (beta-Ti), was
introduced into orthodontics.

• This beta-Ti material (TMA, Ormco/Sybron [the name is
an acronym for titanium-molybdenum alloy]), was
developed primarily for orthodontic use.

• It offers a highly desirable combination of strength and
springiness (i.e., excellent resilience), as well as
reasonably good formability.

• This makes it an excellent choice for auxiliary springs and
for intermediate and finishing archwires. Especially
rectangular wires for the late stages of edgewise treatment.

The relative strength, stiffness, and range for stainless
steel, TMA, and M-NiTi wires (which would be the same
for any wire size). Note that both TMA and M-NiTi have
half the strength of steel; M-NiTi has slightly less stiffness
but much more range than TMA.

Composite Plastics

• Orthodontic ―wires‖ of composite
materials have been shown in the
laboratory to have desirable properties
but have not yet come into clinical use.

• It was more than a decade before the
first NiTi wires went from clinical
curiosity to regular use, and a similar
time period may be needed to bring the
composite plastics into routine clinical
orthodontics.

Comparison of Contemporary Archwires
• Hooke's law (which defines the elastic behavior of
materials )applies to all orthodontic wires except
superelastic A-NiTi.

• For everything else, a useful method for comparing two
archwires of various materials, sizes, and dimensions is the
use of ratios of the major properties (strength, stiffness, and
range):

Mechanical principles in orthodontic force control

• The nomograms are particularly helpful in allowing one to
assess at a glance a whole set of relationships that would
require pages of tables.

• To compare 21 25 m-niti to 21 25 beta-ti in torsion
(the appropriate comparison if the wires would be used to
produce a torquing movement of the root of a tooth). 21
25 beta-ti has a torsional stiffness value of 6, while 21
25 m-niti has a value of 3, so the beta-ti wire would
deliver twice the force at a given deflection

EFFECTS ON ELASTIC
PROPERTIES OF BEAMS

Geometry: Size and Shape

• Changes related to size and shape are independent of the
material.

• In other words, decreasing the diameter of a steel beam
by 50%would reduce its strength to a specific percentage.

• Decreasing the diameter of a similarly supported TMA
beam by 50%would reduce its strength by exactly the
same percentage as the steel beam.

Cantilever Beams

• In orthodontic applications, this is the type of spring
often used in removable appliances, in which a wire
extends from the plastic body of the removable
appliance as a fingerspring.

• When a round wire is used as a fingerspring, doubling
the diameter of the wire increases its strength eight times
(i.e., the larger wire can resist eight times as much force
before permanently deforming or can deliver eight times
as much force).

Doubling the diameter, however, decreases springiness by a
factor of 16 and decreases range by a factor of two.

• Changing the diameter (d) of a beam, no matter how it is
supported, greatly affects its properties. As the figures below
the drawing indicate, doubling the diameter of a cantilever
beam makes it 8 times as strong, but it is then only1/16 as
springy and has half the range.

Supported Beams
• The situation is somewhat more complex for a beam
supported on both ends, as is the case for a segment of
archwire between two teeth. Supporting both ends makes the
beam stronger and less flexible, particularly if the ends are
tightly anchored as opposed to being free

The principle with any supported beam, however, is the
same as with a cantilever beam: as the beam size increases,
strength increases as a cubic function, while springiness
decreases as a fourth power function and range decreases
proportionately, not exponentially.

• useful steel wires are considerably smaller than the gold
wires they replaced.

• The titanium wires are much springier than steel wires of
equal sizes but not as strong. Their useful sizes therefore are
larger than steel and quite close to the sizes for gold.

Geometry: Length and Attachment

• Changing the length of a beam, whatever its size or the
material from which it is made, also dramatically affects its
properties .

• If the length of a cantilever beam is doubled, its bending
strength is cut in half, but its springiness increases eight
times and its range four times. More generally, when the
length of a cantilever beam increases, its strength decreases
proportionately, while its springiness increases as the cubic
function of the ratio of the length and its range increases as
the square of the ratio of the length.

• Length changes affect torsion quite differently from
bending: springiness and range in torsion increase
proportionally with length, while torsional strength is not
affected by length.

• Changing from a cantilever to a supported beam, though
it complicates the mathematics, does not affect the big
picture: as beam length increases, there are proportional
decreases in strength but exponential increases in
springiness and range.

Doubling the length of a cantilever beam cuts its strength in half but makes it 8 times
as springy and gives it 4 times the range. More generally, strength varies inversely
with length, whereas springiness varies as a cubic function of the length ratios and
range as a second power function.
Supporting a beam on both ends makes it much stronger but also much less springy
than supporting it on only one end. Note that if a beam is rigidly attached on both
ends, it is twice as strong but only one-fourth as springy as a beam of the same
material and length that can slide over the abutments. For this reason, the elastic
properties of an orthodontic archwire are affected by whether it is tied tightly or held
loosely in a bracket.

• The way in which a beam is attached also affects its
properties. An archwire can be tied tightly or
loosely, and the point of loading can be any point along
the span.
• a supported beam like an archwire is four times as springy
if it can slide over the abutments (in clinical use, through a
bracket into which it is loosely tied) rather than if the beam
is firmly attached (tied tightly).

• With multiple attachments, as with an archwire tied to
several teeth, the gain in springiness from loose ties of an
initial archwire is less dramatic but still significant.

CONTROLLING ORTHODONTIC
FORCE BY VARYING MATERIALS AND
SIZE–SHAPE OF ARCHWIRES

• The first consideration in spring
design is adequate strength: the wire
that is selected must not deform
permanently in use. As a general
rule, fingersprings for removable
appliances are best constructed using
steel wire. Great advantage can be
taken of the fact that fingersprings
behave like cantilever beams:
springiness increases as a cubic
function of the increase in length of
the beam, while strength decreases
only in direct proportion.
•Thus a relatively large wire, selected
for its strength, can be given the
desired spring qualities by increasing
its length.

In practice, this lengthening often
means doubling the wire back on
itself or winding a helix into it to
gain length while keeping the
spring within a confined intraoral
area . The same technique can be
used with archwires, of course; the
effective length of a beam is
measured along the wire from one
support to the other, and this does
not have to be in a straight line.
Bending loops in archwires can be
a time-consuming chairside
procedure, which is the major
disadvantage.

Another way to obtain a better combination of springiness and
strength is to combine two or more strands of a small and
therefore springy wire. Two 10 mil steel wires in tandem, for
instance, could withstand twice the load as a single strand
before permanently deforming, but if each strand could bend
without being restrained by the other, springiness would not be
affected. The genesis of the ―twin wire‖ appliance system was
just this observation: that a pair of 10 mil steel wires offered
excellent springiness and range for aligning teeth and that two
wires gave adequate strength, although one did not.

The properties of the multistrand wire depend both on the
characteristics of the individual wire strands and on how
tightly they have been woven together. Multistrand steel
wires offer an impressive combination of strength and
spring qualities but now have been displaced for most
applications by NiTi wires.

DESIGN FACTORS IN
ORTHODONTIC APPLIANCES

TWO-POINT CONTACT FOR CONTROL OF ROOT POSITION
• Definition of Terms

•Force: a load applied to an object that will tend to move it to a different
position in space.
Force, though rigidly defined in units of Newtons (mass × the acceleration of gravity), is usually
measured clinically in weight units of grams or ounces. In this context, for all practical purposes,
1.0 N = 100 gm (the actual value is between 97 and 98 gm).

•Center of resistance: a point at which resistance to movement can be
concentrated for mathematical analysis.
For an object in free space, the center of resistance is the same as the center of mass.
If the object is partially restrained, as is the case for a fence post extending into the earth
or a tooth root embedded in bone, the center of resistance will be determined by the
nature of the external constraints.
The center of resistance for a tooth is at the approximate midpoint of the embedded
portion of the root (i.e., about halfway between the root apex and the crest of the alveolar
bone;

THE CENTER OF RESISTANCE (CR) FOR ANY TOOTH IS AT THE APPROXIMATE MIDPOINT
OF THE EMBEDDED PORTION OF THE ROOT. IF A SINGLE FORCE IS APPLIED TO THE
CROWN OF A TOOTH, THE TOOTH WILL NOT ONLY TRANSLATE BUT ALSO ROTATE
AROUND CR (I.E., THE CENTER OF ROTATION AND CENTER OF RESISTANCE ARE
IDENTICAL) BECAUSE A MOMENT IS CREATED BY APPLYING A FORCE AT A DISTANCE
FROM CR. THE PERPENDICULAR DISTANCE FROM THE POINT OF FORCE APPLICATION
TO THE CENTER OF RESISTANCE IS THE MOMENT ARM. PRESSURE IN THE
PERIODONTAL LIGAMENT WILL BE GREATEST AT THE ALVEOLAR CREST AND
OPPOSITE THE ROOT APEX

• Moment: a measure of the tendency to rotate an object
around some point.
• A moment is generated by a force acting at a distance.
Quantitatively, it is the product of the force times the perpendicular
distance from the point of force application to the center of
resistance and thus is measured in units of gram-millimeter (or
equivalent). If the line of action of an applied force does not pass
through the center of resistance, a moment is necessarily created.
Not only will the force tend to translate the object, moving it to a
different position, it also will tend to rotate the object around the
center of resistance. This, of course, is precisely the situation when
a force is applied to the crown of a tooth . Not only is the tooth
displaced in the direction of the force, it also rotates around the
center of resistance—thus the tooth tips as it moves.

• Couple—two forces equal in magnitude and opposite in
direction.
• The result of applying two forces in this way is a pure moment, since the translatory
effect of the forces cancels out. A couple will produce pure rotation, spinning the
object around its center of resistance, while the combination of a force and a couple
can change the way an object rotates while it is being moved .
• Center of rotation—the point around which rotation actually occurs
when an object is being moved.
When two forces are applied simultaneously to an object, the center of rotation can be
controlled and made to have any desired location. The application of a force and a
couple to the crown of a tooth, in fact, is the mechanism by which bodily movement of a
tooth, or even greater movement of the root than the crown, can be produced.

FORCES, MOMENTS, AND COUPLES IN TOOTH
MOVEMENT
• Consider the clinical problem posed by a protruding maxillary central incisor. If
a single force of 50 gm is applied against the crown of this tooth, as might
happen with a spring on a maxillary removable appliance, a force system will be
created that includes a 750 gm-mm moment (see Figure 9-18). The result will be
that the crown will be retracted more than the root apex, which might actually
move slightly in the opposite direction. (Remember that a force will tend to
displace the entire object, despite the fact that its orientation will change via
simultaneous rotation around the center of resistance.) If it is desired to
maintain the inclination of the tooth while retracting it, it will be necessary to
overcome the moment inadvertently created when the force was applied to the
crown.

A couple, as shown on the left, is defined as two forces equal in
magnitude and opposite in direction. The application of a
couple produces pure rotation. In clinical application, two
unequal forces applied to the crown of a tooth to control root
position can be resolved into a couple and a net force to move
the tooth. If a 50 gm force were applied to a point on the labial
surface of an incisor tooth 15 mm from the center of
resistance, a 750 gm-mm moment (the moment of the force, or
MF) would be produced, tipping the tooth. To obtain bodily
movement, it is necessary to apply a couple to create a
moment (the moment of the couple, or MC) equal in magnitude
and opposite in direction to the original movement. One way to
do this would be to apply a force of 37.5 gm pushing the incisal
edge labially at a point 20 mm from the center of resistance.
This creates a 750 gm-mm moment in the opposite direction, so
the force system is equivalent to a couple with a 12.5 gm net
force to move the tooth lingually. With this force system, the
tooth would not tip, but with so light a net force, there would be
only a small amount of movement. To achieve a net 50 gm for
effective movement, it would be necessary to use 200 gm
against the labial surface and 150 gm in the opposite direction
against the incisal edge. Controlling forces of this magnitude
with a removable appliance is very difficult, almost impossible—
effective root movement is much more feasible with a fixed
appliance.

• One way to decrease the magnitude of the moment is to
apply the force closer to the center of resistance. In
orthodontics, it is impractical to apply the force directly to
the root, but a similar effect could be achieved by
constructing a rigid attachment that projected upward from
the crown. Then the force could be applied to the
attachment such that its line of action passed near or
through the center of resistance. If the attachment were
perfectly rigid, the effect would be to reduce or eliminate the
moment arm and thereby the tipping (Figure 9-19). Since it
is difficult to make the arms long enough to totally eliminate
tipping, this procedure is only a partial solution at best, and
it creates problems with oral hygiene.

• Attachments extending toward the center of resistance, seen here as
hooks integrated into the canine brackets, can be used to shorten the
moment arm and thereby decrease the amount of tipping when elastics
or springs are used to slide teeth mesiodistally along an archwire. This
idea from the 1920s was reintroduced as part of the early straight-wire
appliance. Unfortunately, the longer the hook the more effective it is
mechanically but the greater the chance of oral hygiene problems
leading to gingival irritation and/or decalcification. Other methods for
controlling tipping are more practical.

• The usual orthodontic solution is a fixed attachment on the
tooth, constructed so that forces can be applied at two points.
With round wires in bracket slots, an auxiliary spring is needed to
produce a torquing couple (Figure 9-20). A rectangular archwire
fitting into a rectangular bracket slot on the tooth is most widely
used because the entire force system can be produced with a
single wire (Figure 9-21).

• A, Auxiliary root positioning springs and
auxiliary torquing springs were used
routinely with the Begg appliance, and
both can be seen in the maxillary arch of
this patient being treated with an early
(1980s) Begg-edgewise combination
appliance. The torquing spring contacts
the facial surface of the central incisors;
uprighting springs are present bilaterally
on the canines. Note that the base wires
are pinned in the Begg slot, while the
edgewise slot is not used at this point in
treatment. B, An auxiliary torquing spring
in use with the Tip-Edge appliance, the
current version of a combination Begg-
edgewise appliance. C, Root positioning
(side-winder) springs used with the Tip-
Edge appliance.

MC/MF RATIOS AND CONTROL OF ROOT
POSITION
• The previous analysis demonstrates that control of root position
during movement requires both a force to move the tooth in the
desired direction and a couple to produce the necessary
counterbalancing moment for control of root position. The
heavier the force, the larger the counterbalancing movement
must be to prevent tipping and vice-versa.

• A rectangular archwire fitting into a rectangular slot can generate
the moment of a couple necessary to control root position. The wire
is twisted (placed into torsion) as it is put into the bracket slot. The
two points of contact are at the edge of the wire, where it contacts
the bracket. The moment arm therefore is quite small, and forces
must be large to generate the necessary MC. Using the same tooth
dimensions indicated in Figure 9-18, a 50 gm net lingual force
would generate a 750 gm-mm moment. To balance it by creating an
opposite 750 gm-mm moment within a bracket with a 0.5 mm slot
depth, a torsional force of 1500 gm is required.

• Perhaps the simplest way to determine how a tooth will move is to consider the
ratio between the moment created when a force is applied to the crown of a
tooth (the moment of the force [MF]), and the counterbalancing moment
generated by a couple within the bracket (the moment of the couple [MC]). Then
it can be seen (Figure 9-22) that the following possibilities exist:

Pure tipping (tooth rotates around
MC/MF = 0
center of resistance)
Controlled tipping (inclination of
tooth changes but the center of
0 < MC/MF < 1 rotation is displaced away from the
center of resistance, and the root and
crown move in the same direction)
Bodily movement (equal movement of
MC/MF = 1
crown and root)
Torque (root apex moves further than
MC/MF > 1
crown)

• The ratio between the moment produced by the force applied to move a
tooth (MF) and the counterbalancing moment produced by the couple
used to control root position (MC) determines the type of tooth
movement. With no MC, (MC/MF = 0), the tooth rotates around the
center of resistance (pure tipping). As the moment-to-force ratio
increases (0 < MC/MF < 1), the center of rotation is displaced further
and further away from the center of resistance, producing what is called
controlled tipping. When MC/MF = 1, the center of rotation is displaced
to infinity and bodily movement (translation) occurs. If MC/MF > 1, the
center of rotation is displaced incisally and the root apex will move more
than the crown, producing root torque.

• The moment of the force is determined by the magnitude of the force
and the distance from the point of force application to the center of
resistance. For most teeth, this distance is 8 to 10 mm, so MF will be 8
to 10 times the force. In other words, if a 100 gm net force were used to
move such a tooth, a balancing moment of 800 to 1000 gm-mm would
be needed to obtain bodily movement. In the orthodontic literature, the
relationship between the force and the counterbalancing couple often
has been expressed in this way, as the “moment-to-force” ratio.

• In those terms, moment-to-force ratios of 1 to 7 would
produce controlled tipping, ratios of 8 to 10 (depending on
the length of the root) would produce bodily movement, and
ratios greater than that would produce torque. Because the
distance from the point of force application to the center of
resistance can and does vary, moment-to-force ratios must
be adjusted if root length, amount of alveolar bone
support, or point of force application differs from the usual
condition. MC/MF ratios more precisely describe how a
tooth will respond.

• Remember that when a force is applied to a bracket to slide
it along an archwire, as often is the case in clinical
orthodontics, the force felt by the tooth will be less than the
force applied to the bracket because of resistance to sliding
within the bracket. The net force (after resistance to sliding
is subtracted) and the moment associated with the net force
are what is important. In contrast, when a couple is created
within a bracket, friction rarely is a factor

• It is easy to underestimate the magnitude of the forces needed
to create the balancing couple. In the example presented
previously, if a 50 gm net force was used to retract a central
incisor, a 500 gm-mm moment would be needed to keep it from
tipping as the crown moved lingually. To produce a moment of
this magnitude within the confines of an 18 mil (0.45 mm)
bracket would require opposite forces of 1100 gm, derived
from twisting the archwire. These forces within the bracket
produce only a pure moment, so the periodontal ligament
(PDL) does not feel heavy force, but the necessary magnitude
can come as a considerable surprise. The wire must literally
snap into the bracket.

NARROW VERSUS WIDE BRACKETS IN FIXED
APPLIANCE SYSTEMS
• Control of root position with an orthodontic appliance is especially
needed in two circumstances: when the root of a tooth needs to be
torqued facially or lingually (as in the previous example) and when
mesiodistal root movement is needed for proper paralleling of teeth
when spaces are closed (as at extraction sites). In the former
instance, the necessary moment is generated within the bracket, and
the key dimensions are those of the archwire; in the latter
circumstance, the moment is generated across the bracket, and
bracket width determines the length of the moment arm.

• The wider the bracket, all other things being equal, the easier it
will be to generate the moments needed to bring roots together at
extraction sites or to control mesiodistal position of roots in
general. Consider retracting the root of a canine tooth into a first
premolar extraction site (Figure 9-23). With a retraction force of
100 gm and a 10 mm distance from the bracket to the center of
resistance, a 1000 gm-mm moment will be needed. If the bracket
on this tooth is 1 mm wide, 1000 gm of force will be needed at
each corner of the bracket, but if the bracket is 4 mm wide, only
250 gm of force at each corner will be necessary.

• The width of the bracket determines the length of the
moment arm (half the width of the bracket) for control of
mesiodistal root position. Bracket width also influences
the contact angle at which the corner of the bracket
meets the archwire. The wider the bracket, the smaller
the contact angle.

• This assumes even greater practical significance when
the extraction site is to be closed by sliding teeth along
an archwire, and binding between the wire and bracket
is encountered. Binding of the wire against the corners
of the bracket is affected by the force with which the
bracket contacts the archwire and the contact angle
between the wire and the bracket (see Figure 9-23).
The wider bracket reduces both the force needed to
generate the moment and the contact angle and is thus
advantageous for space closure by sliding.

• Despite their advantage when spaces are to be closed by
sliding teeth on an archwire, wide brackets have a partially
offsetting disadvantage. The wider the bracket on a tooth, the
smaller the interbracket span between adjacent teeth, and
therefore the shorter the effective length of the archwire
segments between supports. Reducing the span of the wire
segments in this way (reducing the length of the beam, in the
terminology of our previous discussion) greatly decreases both
the springiness of the archwire and its range of action. For this
reason, the use of extremely wide brackets is contraindicated.
The maximum practical width of a wide bracket is about half
the width of a tooth, and narrower brackets have an advantage
when teeth are malaligned because the greater interbracket
span gives more springiness to any type of wire.

EFFECT OF BRACKET SLOT SIZE IN THE
EDGEWISE SYSTEM
• The use of rectangular archwires in rectangular bracket slots was
introduced by Edward Angle in the late 1920s with his edgewise
arch mechanism (see Chapter 10). The original appliance was
designed for use with gold archwires, and the 22 × 28 mil bracket
slot size was designed to accommodate rectangular archwires of
approximately the same dimension. In Angle's concept of
treatment, sliding teeth along archwires to close extraction sites was
unnecessary because extractions for orthodontic purposes simply
were not done. Torquing movements, on the other hand, were
important, and a major goal of the appliance design was efficient
torque. The appliance was engineered to produce appropriate force
and a reasonable range of action in torsion when gold archwires of
22 × 28 dimension were used with narrow brackets.

• When steel archwires replaced gold, Angle's original
engineering calculations were no longer valid because
steel wire of the same size was so much stiffer. An
alternative was to redesign the edgewise
appliance, optimizing the bracket slot size for steel. A
reduction in slot size from 22 to 18 mil was advocated for
this purpose. Even with this smaller slot size, full-
dimension steel wires still produce slightly greater forces
than the original edgewise system did, but the properties of
the appliance system are close to the original. Good torque
is possible with steel wires and 18 mil edgewise brackets.

• On the other hand, using undersized archwires in edgewise
brackets is a way to reduce the frictional component of
resistance to sliding teeth along an archwire, which was an
important consideration by the time steel wire replaced
gold. As a practical matter, sliding teeth along an archwire
requires at least 2 mil of clearance to minimize friction, and
even more clearance may be desirable. The greater
strength of an 18 mil archwire compared with a 16 mil wire
can be an advantage in sliding teeth. The 18 mil wire
would, of course, offer excellent clearance in a 22-slot
bracket but fit tightly in an 18-slot bracket. The original 22-
slot bracket therefore would have some advantage during
space closure but would be a definite disadvantage when
torque was needed later.

• With steel archwires of 21 mil as the smaller
dimension (close enough to the original 22 mil
bracket slot size to give a good fit), springiness
and range in torsion are so limited that effective
torque with the archwire is essentially impossible.
Using wide brackets to help with space closure
would make the torque problem worse.
Exaggerated inclinations of smaller rectangular
wires, for example, 19 × 25, are one alternative,
but torquing auxiliaries (see Figure 9-20) are often
necessary with undersized steel wires in 22-slot
edgewise brackets.

• In this situation, a role for the new titanium archwires
becomes clearer. If only steel wires are to be used, the 18
mil slot system has considerable advantage over the larger
bracket slot size. With their excellent springback and
resistance to permanent deformation, A-NiTi wires
overcome some of the alignment limitations of steel wires
in wide 22 mil slot brackets, while rectangular NiTi and
beta-Ti wires offer advantages over steel for the finishing
phases of treatment and torque control. In short, titanium
archwires greatly help overcome the major problems
associated with continued use of the original edgewise slot
size.

MECHANICAL ASPECTS OF ANCHORAGE
CONTROL
• FRICTION VERSUS BINDING IN RESISTANCE TO SLIDING
• When teeth slide along an archwire, force is needed for two
purposes: to overcome resistance created by contact of the wire
with the bracket and to create the bone remodeling needed for
tooth movement. As we have pointed out in Chapter 8, controlling
the position of anchor teeth is accomplished best by minimizing the
reaction force that reaches them. Use of unnecessarily heavy force
to move teeth creates problems in controlling anchorage.
Unfortunately, anchor teeth feel the reaction to both the force
needed to overcome resistance to sliding and the additional force
necessary to create tooth movement, so controlling and minimizing
resistance to sliding is an important aspect of anchorage control.

FRICTION IN FIXED APPLIANCE TREATMENT
• When one object moves relative to another, friction at their
interface produces resistance to the direction of movement.
Friction ultimately is derived from electromagnetic forces
between atoms—it is not a fundamental force that can be
defined independently of local conditions. It is proportional to
the force with which the contacting surfaces are pressed
together and is affected by the nature of the surface at the
interface (e.g., rough or smooth, chemically reactive or
passive, modified by lubricants). Interestingly, friction is
independent of the apparent area of contact.

• This is because all surfaces, no matter how smooth, have
irregularities that are large on a molecular scale, and real
contact occurs only at a limited number of small spots at the
peaks of the surface irregularities (Figure 9-24). These
spots, called asperities, carry all the load between the two
surfaces. Even under light loads, local pressure at the asperities
may cause appreciable plastic deformation of those small areas.
Because of this, the true contact area is to a considerable extent
determined by the applied load and is directly proportional to it.

• When two solid surfaces are pressed together or one slides over
the other, real contact occurs only at a limited number of small
spots, called asperities, that represent the peaks of surface
irregularities. These junctions shear as sliding occurs, and the
force to produce this plastic deformation of the surface
irregularities is the frictional resistance.

• When a tangential force is applied to cause one material to slide
past the other, the junctions begin to shear. The coefficient of
friction then is proportional to the shear strength of the junctions
and is inversely proportional to the yield strength of the
materials (because this determines the extent of plastic
deformation at the asperities). At low sliding speeds, a ―stick-
slip‖ phenomenon may occur as enough force builds up to shear
the junctions and a jump occurs, then the surfaces stick again
until enough force again builds to break them.

• Two other factors affect friction: the interlocking of surface
irregularities, which obviously becomes more important when the
asperities are large or pointed, and the extent to which asperities
on a harder material plow into the surface of a softer one. Thus
the total frictional resistance will be the sum of three
components: (1) the force necessary to shear all junctions, (2)
the resistance caused by the interlocking of roughness, and (3)
the plowing component of the total friction force.In practice, if the
two materials are relatively smooth and not greatly dissimilar in
hardness, friction is largely determined by the shearing
component.

THE CONCEPT THAT SURFACE QUALITIES ARE
AN IMPORTANT VARIABLE IN DETERMINING
FRICTION HAS BEEN EMPHASIZED BY
EXPERIENCE IN RECENT YEARS WITH BOTH
TITANIUM WIRES AND CERAMIC OR PLASTIC
BRACKETS. STAINLESS STEEL BRACKETS SLIDE
REASONABLY WELL ON STEEL WIRES, BUT THE
SITUATION IS NOT SO FORTUNATE WITH SOME
OTHER POSSIBLE COMBINATIONS.

SURFACE QUALITIES OF WIRES
• When NiTi wires were first introduced, manufacturers claimed
that they had an inherently slick surface compared with
stainless steel, so that all other factors being equal, there
would be less interlocking of asperities and thereby less
frictional resistance to sliding a tooth along a NiTi wire than a
stainless steel one. This is erroneous—the surface of NiTi is
rougher (because of surface defects, not the quality of
polishing) than that of beta-Ti, which in turn is rougher than
steel. More importantly, however, there is little or no
correlation for orthodontic wires between the coefficients of
friction and surface roughness (i.e., interlocking and plowing
are not significant components of the total frictional
resistance).

• Although NiTi has greater surface roughness, beta-Ti has
greater frictional resistance. It turns out that as the titanium
content of an alloy increases, its surface reactivity
increases, and the surface chemistry is a major influence.
Thus beta-Ti, at 80%titanium, has a higher coefficient of
friction than NiTi at 50%titanium, and there is greater
frictional resistance to sliding with either than with steel.
With beta-Ti, there is enough titanium reactivity for the wire
to ―cold-weld‖ itself to a steel bracket under some
circumstances, making sliding all but impossible.

• A possible solution to this problem is alteration of the
surface of the titanium wires by implantation of ions into the
surface. Ion implantation (with nitrogen, carbon, and other
materials) has been done successfully with beta-Ti and has
been shown to improve the characteristics of beta-Ti hip
implants. In clinical orthodontics, however, implanted NiTi
and beta-Ti wires have failed to show improved
performance in initial alignment or sliding space closure
respectively.

SURFACE QUALITIES OF BRACKETS
• Bracket surfaces also are important in friction. Most modern
orthodontic brackets are either cast or milled from stainless
steel and, if properly polished, have relatively smooth surfaces
comparable with steel wires. Titanium brackets now are coming
into use, primarily because they eliminate the chance of an
allergic response to the nickel in stainless steel. Fortunately,
many individuals who show cutaneous sensitivity to nickel do
not have a mucosal reaction, but the increasing number of
allergic patients is becoming a problem.

• At best, the surface properties of titanium brackets are
like those of titanium wires, and polishing the interior of
bracket slots is difficult enough that these critical areas
may be rougher than wires. Sliding with titanium
brackets therefore may be problematic, particularly if
titanium archwires also are used.

• Ceramic brackets became quite popular in the 1980s
because of their improved esthetics, but their surface
properties are far from ideal. The ones made from
polycrystalline ceramics have considerably rougher
surfaces than steel brackets. The rough but hard
ceramic material is likely to penetrate the surface of
even a steel wire during sliding, creating considerable
resistance, and of course, this is worse with titanium
wires. Although single crystal brackets are quite
smooth, these brackets also can damage wires during
sliding, and so they also have increased resistance to
sliding.12 Recently, ceramic brackets with metal slots
have been introduced, a rather explicit recognition of the
problems created by friction against ceramic surfaces

• It is quite likely that composite plastic brackets will
begin to be used in orthodontics within the next few
years. They have the advantages of being tooth colored
and non allergenic, and at least in theory, should have
surface properties . that would not be as troublesome
as ceramics. As with composite plastic
wires, however, their fabrication is difficult and their
advantages relative to metal may not be worth the
additional expense.

ELASTIC AND INELASTIC BINDING IN
RESISTANCE TO SLIDING
• The amount of force between the wire and the bracket
strongly influences the amount of resistance to sliding.
This resistance is determined primarily by two things:
friction as the wire contacts the walls or bottom of the
bracket and elastic or inelastic binding as the wire
contacts the corners of the bracket. As we will
see, binding, not friction, is the major component of
resistance to sliding.

• In theory, a wire could move through a bracket or tube
with no friction whatever, if it were small relative to the
bracket and did not contact any part of the bracket
(Figure 9-25, A). Even in the laboratory, this is very
difficult—unless everything is aligned perfectly, the wire
will contact the bottom of the bracket or some other
area, but friction will be small if nothing forces the wire
against the bracket.

IT IS POSSIBLE IN THE LABORATORY TO ORIENT A SMALL
WIRE WITHIN A BRACKET (OR TUBE) SO THAT IT DOES NOT
TOUCH ANY OF THE WALLS, AND THEN THERE WOULD BE
NO FRICTIONAL RESISTANCE TO MOVING THE WIRE
RELATIVE TO THE BRACKET. B, IN THE MOUTH, A CURVED
ARCHWIRE PASSING THROUGH A SERIES OF BRACKETS
WILL INEVITABLY CONTACT THE BOTTOM OF THE
BRACKET, SO THERE WILL BE SOME FRICTION.

• This, however, has no resemblance to what happens in
the mouth. When a wire shaped to arch form fits
through multiple brackets, contact with the base of the
bracket and/or bracket walls is inevitable (Figure 9-
25, B). If a tooth is pulled along an archwire, the
resistance to sliding will be only friction until the tooth
tips enough to bring the corners of the bracket into
contact with the wire. The tooth tips, of course, because
the force is applied to a bracket on its crown, and the
center of resistance is halfway down the root. As soon
as the corners of the bracket engage the wire, which
happens after a very small movement of the tooth

• , a moment is generated that opposes further tipping
(Figure 9-26). This generates elastic binding between
the bracket and the wire, which is different from friction.
The greater the angle at which the wire contacts the
corners of the bracket, the greater the force between
the wire and bracket—so, as we noted previously, there
is greater resistance to sliding with narrow than wide
brackets. As can be seen from Figure 9-27, resistance
to sliding includes elastic binding almost immediately
when tooth movement begins and goes up rapidly as
the angle between the bracket and the wire increases.

A AND B, THE FORCE TO MOVE A BRACKET ALONG AN
ARCHWIRE INITIALLY WILL BE RESISTED ONLY BY FRICTION
DUE TO CONTACT OF THE WIRE WITH THE BOTTOM OR SIDES
OF THE BRACKET SLOT. C, BECAUSE THE ROOT OF A TOOTH
RESISTS MOVEMENT, THE TOOTH TIPS UNTIL THE CORNERS
OF THE BRACKET CONTACT THE WIRE, AND AT THAT
POINT, ELASTIC BINDING OF THE WIRE AGAINST THE CORNER
OF THE BRACKET ADDS TO THE RESISTANCE TO SLIDING.

• Because binding creates most of the resistance to sliding as
the angle of contact between the wire and the corner of the
bracket increases, resistance to sliding in very early alignment
is the sum of elastic binding (BI) and friction (FR), but almost
immediately the proportion of resistance from binding exceeds
friction by so much that the frictional component can be
disregarded—for all practical purposes the resistance to
sliding is due just to binding

• A series of experiments in Kusy's laboratory help to put
into perspective the importance of binding versus
friction as components of resistance to sliding. In the
experiments, 21 × 25 M-NiTi and steel wires were tied
into a 22-slot steel twin bracket with 200 gm ligature
force, so there was a lot of friction. Then the resistance
to sliding was evaluated as a function of the angle of
contact between the wire and the bracket. As Figure 9-
28 shows, with a 3-degree contact angle, most of the
resistance to sliding was binding with a steel wire, and
nearly half was binding with a M-NiTi wire. At a 7-
degree or greater angle, almost all the resistance was
due to binding with both wires.

• The conclusion: for very early alignment of
teeth, resistance to sliding is due to a combination of
friction and binding, but almost immediately, unless the
tooth is allowed to tip to keep the contact angle low, the
frictional component becomes so low that it is
negligible, and resistance to sliding is due almost totally
to elastic binding.

• During orthodontic tooth movement, inelastic binding
also is likely to be encountered. This occurs when
notching of the edge of the wire occurs (Figure 9-29).
When a notch encounters the edge of the bracket, tooth
movement stops until displacement of the tooth during
function releases the notch released). Given the
presence of both types of binding, it is not surprising
that observations of tooth movement show that it
happens almost entirely in a series of steps, not in a
smooth flow.

• Scanning electron microscope images show that sliding a
bracket along a wire (or a wire through a bracket) causes
surprising amounts of distortion of the wire surface. A, 16 mil
steel wire after sliding through a bracket—note the significant
tear in the wire. B, 21 × 25 steel wire after sliding—note the
series of indentations. When the corner of the bracket
catches on damaged areas like these, tooth movement stops
until the notching is released by masticatory function.

MAGNITUDE OF RESISTANCE TO SLIDING
• Perhaps the most important information to be gained from a
consideration of resistance to sliding is an appreciation of its
magnitude, even under the best of circumstances. If a canine
tooth is to slide along an archwire as part of the closure of an
extraction space, and a 100 gm net force is needed for tooth
movement, approximately another 100 gm will be needed to
overcome the effects of binding and friction (Figure 9-30).
The total force needed to slide the tooth therefore is twice as
great as might have been expected. Because this resistance
is due mostly to binding, replacing an elastomeric ligature tie
with a bracket cap so that the wire is not forced against the
bottom of the bracket does not lead to faster space closure.

• To retract a canine tooth by sliding it along an archwire, an unknown
amount of resistance created largely by binding of the wire against the
edges of the bracket must be overcome. A general guideline from
laboratory studies is that the combination of binding and friction will be
approximately equal to the amount of force needed for tooth
movement, so one would expect to need approximately 200 gm force to
slide the canine along the wire and the posterior anchorage would feel that
amount of force. Clinically, problems in controlling anchorage arise largely
because the true resistance to sliding is unknown. A generous amount of
force beyond what is needed to move the tooth is added to ensure clinical
effectiveness, but the excess force increases undesired movement of the
anchor teeth.

• In terms of the effect on orthodontic anchorage, the problem
created by resistance to sliding is not so much its presence as
the difficulty of knowing its magnitude. To slide a tooth or teeth
along an archwire, the clinician must apply enough force to
overcome the resistance and produce the biologic response. It
is difficult to avoid the temptation to estimate the resistance to
sliding generously and add enough force to be certain that tooth
movement will occur. The effect of any force beyond what was
really needed to overcome resistance to sliding is to bring the
anchor teeth up onto the plateau of the tooth movement curve
.Then either unnecessary movement of the anchor teeth occurs
or additional steps to maintain anchorage are necessary (such
as headgear or bone screws).

• If a springy loop is bent into the archwire, activated to
produce tooth movement, and then tied tightly, archwire
segments move, taking the teeth with them instead of the
teeth moving relative to the wire. Springs of this type are
called retraction springs if they attach to only one tooth or
closing loops if they connect two archwire segments
(Figure 9-31). Incorporating springs into the archwire
makes the appliance more complex to fabricate and to
use clinically but eliminates the difficulty in predicting
resistance to sliding.

• A closing loop is being used to retract the maxillary
incisors, while a spring to slide the lower teeth along the
archwire is being used in the lower arch. In this Class II
patient, the closing loop eliminates resistance to sliding as a
factor in maintaining the position of the maxillary posterior
teeth, while the sliding space closure in the lower arch and the
light Class II elastic both serve to move the lower posterior teeth
forward as part of overjet reduction.

METHODS TO CONTROL ANCHORAGE
• Reinforcement:
• The extent to which anchorage should be reinforced (by
adding teeth to the anchorage unit) depends on the tooth
movement that is desired. In practice, this means that
anchorage requirements must be established individually
in each clinical situation. Once it has been determined that
reinforcement is desirable, however, this typically involves
including as many teeth as possible in the anchorage. For
significant differential tooth movement, the ratio of PDL
area in the anchorage unit to PDL area in the tooth
movement unit should be at least 2 to 1 without sliding and
4 to 1 with it. Anything less produces something close to
reciprocal movement, especially if force levels are not well
controlled.

• Satisfactory reinforcement of anchorage may require the
addition of teeth from the opposite dental arch to the
anchor unit. For example, to close a mandibular
premolar extraction site, it would be possible to stabilize
all the teeth in the maxillary arch so that they could only
move bodily as a group and then to run an elastic from
the upper posterior to the lower anterior, thus pitting
forward movement of the entire upper arch against distal
movement of the lower anterior segment (Figure 9-32).
This addition of the entire upper arch would greatly alter
the balance between retraction of the lower anterior
teeth and forward slippage of the lower posterior teeth.

• Reinforcement of anchorage can be obtained by adding
additional teeth within the same arch to the anchor unit, or
by using elastics from the opposite arch to help produce
desired tooth movement, as with the interarch elastic
shown here. Additional reinforcement can be obtained with
extraoral force, as with addition of a facebow to the upper
molar to resist the forward pull of the elastic.

• This anchorage could be reinforced even further by having
the patient wear an extraoral appliance (headgear) placing
backward force against the upper arch. The reaction force
from the headgear is dissipated against the bones of the
cranial vault, thus adding the resistance of these structures to
the anchorage unit. The only problem with reinforcement
outside the dental arch is that springs within an arch provide
constant forces, whereas elastics from one arch to the other
tend to be intermittent, and extraoral force is likely to be even
more intermittent. Although this time factor can significantly
decrease the value of cross-arch and extraoral
reinforcement, both can be quite useful clinically.

SUBDIVISION OF DESIRED MOVEMENT
• A common way to improve anchorage control is to pit the resistance
of a group of teeth against the movement of a single tooth, rather
than dividing the arch into more or less equal segments. In our
same extraction site example, it would be perfectly possible to
reduce the strain on posterior anchorage by retracting the canine
individually, pitting its distal movement against mesial movement of
all other teeth within the arch (Figure 9-33). After the canine tooth
had been retracted, one could then add it to the posterior
anchorage unit and retract the incisors. This approach has the
advantage of dissipating the reaction force over a large PDL area in
the anchor unit—but only if the retraction force is kept light, as
discussed in Chapter 8. Its disadvantage is that closing the space in
two steps rather than one would take nearly twice as long.

• Subdivision of tooth movement improves the
anchorage situation regardless of whether sliding
is involved and where a space in the arch is
located. If it is desired to slip all the posterior teeth
forward (in which case the anterior teeth are the
anchor unit), bringing them forward one at a time
is the most conservative way to proceed. Moving
them one at a time with a loop, of course, will put
less strain on anchorage than sliding them one at
a time.

RETRACTION OF THE CANINE BY ITSELF, AS THE
FIRST STEP IN A TWO-STAGE SPACE
CLOSURE, OFTEN IS USED TO CONSERVE
ANCHORAGE, PARTICULARLY WHEN SLIDING TEETH
ALONG AN ARCHWIRE.

TIPPING/UPRIGHTING
• Another possible strategy for anchorage control is to tip the teeth and
then upright them, rather than moving them bodily. In our familiar
extraction site example, this would again require two steps in treatment.
First, the anterior teeth would be tipped distally by being pitted against
mesial bodily movement of the posterior segment . Allowing the teeth to
tip as they slide along an archwire keeps the contact angle between the
wire and bracket small, which reduces binding and therefore keeps
resistance to sliding small. As a second step, the tipped teeth would be
uprighted, moving the canine roots distally and torquing the incisor roots
lingually, again with stationary anchorage in the posterior segments. It
would be extremely important to keep forces as light as possible during
both steps, so that the teeth in the posterior segment were always
below the optimum force range, while the anterior teeth received
optimum force.

ANCHORAGE CONTROL IN SPACE CLOSURE

• At this point, it is interesting to consider a relatively
typical extraction situation, in which it is desirable
to close the extraction space 60%by retraction of
the anterior teeth and 40%by forward movement of
the posterior segments (Figure 9-34). This outcome
would be expected from any of three possible
approaches: closing loop approach,

• (1) one-step space closure with no sliding (via closing loops
so that segments of wire are moved with the teeth
attached, rather than sliding); (2) a two-step closure sliding
the canine bodily along the archwire, then retracting the
incisors (as in the original Tweed technique); or (3) two-step
space closure, tipping the anterior segment with some
friction, then uprighting the tipped teeth (as in the Begg
technique).

• Closure of a premolar extraction site often is desired in a ratio of
60%retraction of incisors, 40%forward movement of molar and second
premolar. This result can be obtained straightforwardly in three ways: (1)
one-step space closure with no sliding (closing loop); (2) two-step space
closure with sliding mechanics, retracting the canine individually, and
then retracting the four incisors in a second step (the classic Tweed
approach); or (3) two-step sliding with distal tipping of the canine and
incisors initially, followed by uprighting of these teeth (the classic Begg
approach). Good clinical results can be obtained with all three methods.
The cost of resistance to sliding in space closure, with well-managed
orthodontic appliances, is paid more in increased treatment time than in
decreased quality of result.

• The example makes the cost of binding and friction in a
clinical setting more apparent: the greater strain on
anchorage when brackets slide along an archwire must
be compensated by a more conservative approach to
anchorage control. The price therefore is paid in
increased treatment time. The though more difficult to
fabricate and manipulate, will result in the same space
closure significantly faster.

• Note that strategies for anchorage control are
associated with particular orthodontic
appliances, indeed are literally built into the
appliance in many instances. The approach to
anchorage control that is implicit in the appliance
design is sometimes called the appliance
philosophy, not quite so strange a term when
viewed in this way.

SKELETAL ANCHORAGE
• Skeletal anchorage is derived from implants, miniplates
attached with screws to basal bone of the maxilla or
mandible, or just a screw with a channel for attaching a
spring that is placed into the alveolar process (Figure 9-35).
Collectively, these devices are referred to as temporary
anchorage devices (TADs). their use to accomplish tooth
movement that was very difficult or impossible previously is
presented in Chapters 14 to 18. With properly designed
skeletal anchorage, there is no concern about moving teeth
that were not intended to be moved, but the amount of force
to teeth that are to be moved still must be determined with
resistance to sliding in mind.

Bone anchors retained by screws or
screws with a head that extends into the
mouth can be placed in both the
mandibular and maxillary arches to provide
skeletal anchorage for tooth movement.
This makes it possible to produce tooth
movement that otherwise would be
impossible. A, Placement of screws to hold
a bone anchor in the mandible. B, Anchors
in place bilaterally. C, Surgical placement
of a palatal anchor. D, Anchor (Straumann
Orthosystem) in position. E, Stabilizing
lingual arch attached to the anchor, in
preparation for retraction of the protruding
maxillary incisors. F, Removal of a small
area of mucosa over the site where a bone
screw is to be placed in the maxillary
alveolar process. G, TOMAS screw
(Dentaurum) with a stabilizing wire
attached from a channel in the screw head
to the molar headgear tube being used to
prevent movement of the maxillary first
molar as the second molar is moved
distally.

DETERMINATE VERSUS INDETERMINATE FORCE
SYSTEMS
• The laws of equilibrium require not only that for every force
there is an equal and opposite reactive force but also that the
sum of the moments around any arbitrary point are equal to
zero. In other words, the moments, as well as the
forces, generated by an orthodontic appliance system must be
balanced, in all three planes of space. It can be very difficult to
visualize the total force system when multiple teeth are
involved, but unexpected and unwanted tooth movement easily
can result when an important component of the system is
overlooked.

• Force systems can be defined as statically
determinate, meaning that the moments and forces can
readily be discerned, measured, and evaluated, or as
indeterminate. Statically indeterminate systems are too
complex for precisely calculating all forces and moments
involved in the equilibrium. Typically, only the direction of
net moments and approximate net force levels can be
determined.

• This is more of a problem in orthodontics than in most engineering
applications because the eventual action of an orthodontic appliance
is affected by the biologic response. For instance, the amount of tooth
movement will be determined to a large extent by the magnitude of the
forces felt by anchor teeth and teeth whose movement is intended, not
just by the differential between those forces. If the force applied to the
anchor teeth is high enough to pull them up onto the plateau of the
pressure–response curve, reciprocal tooth movement will occur
despite a difference in PDL pressures .Similarly, whether intrusion of
incisor teeth or extrusion of posterior teeth occurs is almost totally a
function of the magnitude of intrusive versus extrusive forces, not their
direction or the difference between them. Determinate force systems
therefore are particularly advantageous in orthodontics when control of
force magnitudes is necessary to produce the desired biologic
response.

• For all practical purposes, determinate systems in orthodontics are
those in which a couple is created at one end of an attachment, with
only a force (no couple) at the other (i.e., a one-couple system).
This means that a wire that will serve as a spring can be inserted
into a tube or bracket at one end but must be tied so that there is
only one point of contact on the other (Figure 9-36). When the wire
is tied into a bracket on both ends, a statically indeterminate two-
couple system has been created.

• An intrusion arch made from rectangular wire,
which fits into a rectangular tube on the molars
and is tied to one point of contact on the incisor
segment, is an example of a determinate one-
couple system. If the archwire is activated by
pulling it down and tying it to the incisor segment
so that it delivers 40 gm intrusion force (10 gm
per tooth, 20 gm per side), and if the distance
from the molar tube to the point of attachment is
30 mm, each molar will feel a 20 gm extrusive
force in reaction and a 600 gm-mm moment to tip
the crown distally. At the incisor segment, the
force will create a 200 gm-mm moment to rotate
the incisor crowns facially. At each molar, the
extrusive force also would create a moment to roll
the crown lingually. If the buccal tube were 4 mm
buccal to the center of resistance, the magnitude
of this moment would be 80 gm-mm.

ONE-COUPLE SYSTEMS
• In orthodontic applications, one-couple systems are
found when two conditions are met: (1) a cantilever
spring or auxiliary archwire is placed into a bracket or
tube(s) and typically attaches to a tooth or teeth that
are part of a stabilized segment (i.e., reinforced
anchorage is being used) and (2) the other end of the
cantilever spring or auxiliary archwire is tied to a tooth
or group of teeth that are to be moved, with a single
point of force application.

• For analysis, the teeth in the anchor unit are considered as if
stabilization had created a single large multirooted
tooth, with a single center of resistance. It is important to tie
teeth in an anchor unit tightly together with as rigid a
stabilizing wire segment as possible. Often the posterior
teeth on both sides are tied together with a rigid lingual
arch, so that a single posterior stabilizing segment is
created. If the goal is to move more than one tooth, the tooth
movement segment similarly must be tied so the teeth
become a single unit.

CANTILEVER SPRING APPLICATIONS
• Cantilever springs are used most frequently to bring severely displaced (impacted)
teeth into the arch (Figure 9-37). These springs have the advantage of a long range of
action, with minimal decrease in force as tooth movement proceeds and excellent
control of force magnitude. There are two disadvantages: (1) as with most devices
with a long range of action, cantilever springs do not fail safely—if they are distorted
by the patient, significant tooth movement in the wrong direction is quite possible—
and (2) the moment of the force on an unerupted tooth rotates the crown lingually as
the tooth is brought toward the occlusal plane, which is undesirable if the tooth is
already lingual to its proper position (as most unerupted teeth are). Although an
additional force can be added to overcome this, the system rapidly becomes
extremely complex enough that it is hard to know what forces and couples exist. If the
cantilever spring is tied into a bracket on the unerupted tooth so that a couple can be
created for better control, the force system becomes statically indeterminate and force
magnitudes are no longer known with certainty.

• A cantilever spring, made from a rectangular
wire that fits into a rectangular tube (or
bracket) on one end and is tied to one point of
contact on the other, produces a determinate
one-couple system in which the forces and
moments can be known precisely. A, Lateral
view of the force system created by a
cantilever spring to extrude an impacted
maxillary canine. If the distance between the
molar tube and a button on the canine to which
the spring is tied is 20 mm, placing a 50 gm
extrusive force on the canine creates a 50 gm
intrusive force on the molar and also a 1000
gm-mm moment to rotate the molar crown
forward around its center of resistance. If the
molar tube is 4 mm in length, the moment
would be created by a couple with 250 gm
force upward on the mesial end of the tube and
250 gm downward on the distal end.

• B, Frontal view of the same force system. Consider the
buccolingual (torque) moments created by the force on
the molar and canine. If the center of resistance of the
canine is 5 mm lingual to the button on its crown, a 50 gm
extrusive force creates a 250 gm-mm moment to rotate
the crown lingually (which usually is not desired). At the
molar, if the center of resistance is 4 mm lingual to the
tube on the buccal surface, the 50 gm intrusive force
creates a 200 gm-mm moment to rotate the crown
facially. But if the impacted canine is 10 mm lingual to the
buccal surface of the molar, activating the spring also
twists it, creating a 500 gm-mm torquing moment to rotate
the molar crown lingually. The result at the molar is a net
300 gm-mm moment to torque the molar crown lingually
and roots buccally. If the rectangular spring were tied into
a bracket on the canine, a moment to torque its root
facially could be generated, but the resulting two-couple
system would be indeterminate—it would no longer be
possible to know the forces and moments with certainty.

AUXILIARY INTRUSION OR EXTRUSION ARCHES
• The major use of one-couple systems is for intrusion, typically of
incisors that have erupted too much. For this purpose, light
force against the teeth to be intruded is critical. An intrusion arch
typically employs posterior (molar) anchorage against two or
four incisors (Figure 9-38). Because the intrusive force must be
light, the reaction force against the anchor teeth also is
light, well below the force levels needed for extrusion and
tipping that would be the reactive movements of the anchor
teeth. Tying the molar teeth together with a rigid lingual arch
prevents buccal tipping of the molars. In adults, usually the
premolar teeth also are added to the anchor unit.

• Two factors in the action of an intrusion arch are the relationship of the point of force application
relative to the center of resistance of the incisor segment and whether the incisor teeth are free to tip
facially as they intrude or whether the arch is cinched back to produce lingual root torque. A, An
intrusion arch can be tied at any point along the incisor segment. If it is tied behind the lateral incisor
bracket, the force is applied in line with the center of resistance, and there is no moment to rotate the
incisors faciolingually. The effect on the anchor molar would be the same as if the intrusion arch were
tied in the midline (see Figure 9-36). B, If the intrusion arch were tied in the midline and cinched back
so it could not slide forward in the molar tube, the effect would be lingual root torque on the incisors as
they intruded. Equilibrium requires that both moments and forces be balanced, so the moment on the
incisors would be balanced by a similar moment on the anchor molars. Each would receive a 100 gm-
mm moment to bring the crown mesially, which would require a 10 gm force at the distal of the molar
tube if the distance from the tube to the molar's center of resistance is 10 mm.

• It would be easy enough to activate an auxiliary archwire to
produce extrusion of incisors rather than intrusion. This is rarely
done clinically, however. The force needed for extrusion is four
to five times higher than intrusion, so the reactive force against
the anchor teeth also would be higher and the anchor teeth
would not be as stable. Perhaps more importantly, the precise
control of force magnitude that is the major advantage of a one-
couple system is less critical when extrusion is desired. The
additional complexity of stabilizing segments and an auxiliary
archwire may not be cost-effective if extrusion is the goal

TWO-COUPLE SYSTEMS
• Utility Arches for Intrusion
• An easy way to see the effect of changing from a determinate one-couple
to an indeterminate two-couple system is to observe the effect of tying an
intrusion arch into brackets on incisor teeth, rather than tying it with one-
point contact.The utility arch, popularized by Ricketts and recommended
for incisor intrusion, makes just this change. It is formed from rectangular
wire so that it will not roll in the molar tubes, bypasses the canine and
premolar teeth, and (unlike a one-couple intrusion arch) is tied into the
incisor brackets (i.e., it is a 2 × 4 archwire attached to two molars and
four incisors). The resulting long span provides excellent load deflection
properties so that the light force necessary for intrusion can be created.
One-couple intrusion arches can look quite similar. The difference comes
when the utility arch is tied into the incisor brackets, creating a two-
couple system.

• When the utility arch is activated for intrusion, the
moment of the intrusive force tips the crowns facially
(Figure 9-39). One way to prevent the facial tipping is to
apply a force to retract the incisors, which would create
a moment in the opposite direction. This could be done
by cinching or tying back the intrusion utility arch.
Although the retraction force could be light, any force to
bring the anchor teeth mesially is likely to be
undesirable.

• Another strategy to control the facial tipping is immediately
apparent: place a twist in the anterior segment of the utility arch, to
torque the incisors lingually, and accentuate the torque by cinching
the end of the wire at the molar tube. Let us examine the effect of
doing this (see Figure 9-39, B and C). An effect of the couple within
the bracket is to increase the intrusive force on the incisors and also
the reactive extrusive force on the molars. Although one can be
sure that the magnitude of the intrusive force would increase, it is
impossible to know how much. An increase in the magnitude of the
intrusive force often is not anticipated from such an apparently
unrelated change in the archwire. In addition, now the magnitude of
the reactive forces is not known with certainty, which makes it
impossible to accurately adjust the archwire even if you do
anticipate the increase. Both effects help explain why utility arches
often produce disappointing amounts of incisor intrusion relative to
molar extrusion.

• A utility arch often is an intrusion arch in a two-couple configuration, created by tying
the rectangular intrusion arch into the brackets on the incisors. When this is done,
the precise magnitude of forces and couples cannot be known, but the initial
activation of the arch should be to provide about 40 gm to the incisor segment for
intrusion. A, Activating the utility arch by placing it in the brackets creates the
intrusion force, with a reactive force of the same magnitude on the anchor molar and
a couple to tip its crown distally. At the incisors, a moment to tip the crowns facially
(MF) is created by distance of the brackets forward from the center of resistance, and
an additional moment in the same direction is created by the couple within the
bracket (MC) as the inclination of the wire is changed when it is brought to the
brackets. The moment of this couple cannot be known, but it is clinically important
because it affects the magnitude of the intrusion force

• . B, Placing a torque bend in the utility arch creates a moment to bring the crown
lingually, controlling the tendency for the teeth to tip facially as they intrude, but it
also increases the magnitude of the intrusive force on the incisor segment and
the extrusive force and couple on the molar. C, Cinching back the utility arch
creates a force to bring the incisors lingually, and a moment of this force opposes
the moment of the intrusion force. At the molar, a force to bring the molar mesially
is created, along with a moment to tip the molar mesially. Especially if a torque
bend still is present, it is difficult to be certain which of the moments will prevail or
whether the intrusion force is appropriate. With this two-couple system, the
vertical forces easily can be heavier than desired, changing the balance between
intrusion of the incisors and extrusion of the molars.

SYMMETRIC AND ASYMMETRIC BENDS
• When a wire is placed into two brackets, the forces of the equilibrium
always act at both brackets. For analysis, two groups of teeth that have
been tied together to create the equivalent of a single multirooted tooth
can be treated as if there were just one bracket for each group. There
are three possibilities for placing a bend in the wire to activate it:

• •Symmetric V-bend, which creates equal and opposite couples at
the brackets (Figure 9-40). The associated equilibrium forces at
each bracket also are equal and opposite, and therefore cancel
each other out. A symmetric V-bend is not necessarily halfway
between two teeth or two groups of teeth; the important quality is
that it generates equivalent couples at both ends. These couples
are affected by both bracket width and bracket alignment, so care
must be taken if placing symmetric V-bends before the teeth are
well aligned. Additionally, if a symmetric V-bend is to be placed
between posterior and anterior teeth, studies have shown that the
bend must be placed closer to the posterior segment due to the
curve of the archwire. Finally, equal and opposite couples have the
benefit of no net reaction forces, but these equal couples will not
generate equivalent tooth movement if the anchorage of one
section is much greater.

• A, When a symmetric V-bend is placed halfway between two units with
equal resistance to movement, it creates equal and opposite
moments, and the intrusive/extrusive forces cancel each other. B, To
create equal and opposite couples between two units with unequal
resistance to movement, a V-bend must be displaced toward the unit
with greater resistance, so a symmetric V-bend between an incisor
and molar would be offset toward the molar. One must know the
approximate anchorage value of teeth or units of the dental arch to
calculate the appropriate location of symmetric or asymmetric V-
bends.

• •Asymmetric V-bend, which creates unequal and opposite
couples, and net equilibrium forces that would intrude one unit and
extrude the other (Figure 9-41). Although the absolute magnitude of
the forces involved cannot be known with certainty (this is, after
all, an indeterminate system), the relative magnitude of the
moments and the direction of the associated equilibrium forces can
be determined. The bracket with the larger moment will have a
greater tendency to rotate than the bracket with the smaller
moment, and this will indicate the direction of the equilibrium forces.
Placing the short segment of the wire in the bracket is a good way
to visualize the direction of the equilibrium forces. As the bend
moves closer to one of two equal units, the moment increases on
the closer unit and decreases on the distant one, while equilibrium
forces increase.

• An asymmetric V-bend creates a greater moment on one tooth or
unit than the other. As the bend moves toward one tooth, the
moment on it increases and the moment on the distant tooth
decreases. When the bend is one-third of the way along the
interbracket span, the distant tooth (on the right in this drawing)
receives only a force, with no moment. B, If the V-bend moves
closer than the one-third point to one of the teeth, a moment in the
same direction is created on both teeth, instead of opposite
moments. A V-bend placed to parallel the roots of the adjacent teeth
would not do so if the bend were too close to one of the teeth.

• In most studies of asymmetric V-bends, a location is found where no
moment is felt at the distant bracket, only a single force. When the
bend moves closer than this to one bracket, moments at both brackets
are in the same direction, and equilibrium forces increase further. The
location of this point, however, varies in different studies, ranging from
one-third of the distance along the wire to not being found even when
the bend is placed right by one anchor unit. The difficulty of locating
this point and therefore of confidently predicting the effect of
placement of the bend is another illustration of why it is important to
closely examine what happens clinically for undesired side effects.

• •Step bend, which creates two couples in the same direction
regardless of its location between the brackets (Figure 9-42).
The location of a V-bend is a critical variable in determining its
effect, but the location of a step bend has little or no effect on
either the magnitude of the moments or the equilibrium forces.

• A step bend between two teeth
produces intrusive force on one tooth,
extrusive force on the other, and
creates couples in the same direction.
In contrast to V-bends, there is little
effect on either the force or the
couples when the step bend is moved
off-center.

• The general relationship between bend location and the
forces and moments that are produced is shown in
Table 9-3. Note that for V-bends, the force increases
steadily as the beam moves off-center. For step
bends, since both couples are in the same
direction, the force is increased over what a symmetric
V-bend would produce.

TABLE 9-3 FORCE SYSTEMS FROM STEP AND V-
BENDS
Data from experiment 16
Percentage of Moment far steel, 7 mm span, 0.35 mm bend
Force general
total span to tooth/moment
condition Moment (gm-
closest bracket near tooth Force (gm)
mm)
Step Bend
All 1.0 XX 347 1210/1210
V-Bend
0.5 -1.0 None 0 803/803
0.4 -0.3 X
0.33 0 XX
0.29 353 2210/262
0.2 0.3 XXX
0.14 937 4840/1720
0.1 0.4 XXXX

FORCES AND COUPLES CREATED BY
INTERBRACKET BENDS
• Under laboratory conditions, the forces and couples
created in a two-couple system by interbracket bends
can be evaluated experimentally.18 With a 16 mil steel
wire and an interbracket distance of 7 mm (about what
would be found between central incisors with twin
brackets or between narrow canine and premolar
brackets), a step bend of only 0.35 mm would produce
intrusive/extrusive forces of 347 gm and 1210 gm-mm
couples in the same direction (see Table 9-3).
Permanent distortion of the wire would occur with a

• The heavy vertical forces produced by what orthodontists
would consider modest bends in a light archwire like 16
mil steel explain why extrusion is the response to step
bends in continuous archwires. An asymmetric V-bend
that places the apex of the bend 0.35 mm above the
plane of the brackets produces 803 gm-mm couples with
no net intrusive/extrusive forces at the one-third position.
At the one-sixth position, intrusive/extrusive forces over
900 gm occur, with very large moments (see Table 9-
3), so the result here also would be extrusion in addition
to root movement.

• The moments and forces are greatly reduced as interbracket distances increase. For
instance, the same 0.35 mm step bend that produced 347 gm with a 7 mm interbracket
span produces only 43 gm with a 14 mm span (which is still too high for intrusion). Even
with flexible archwires, an interbracket span equivalent to the distance from the first molar
to the lateral incisor usually is needed to obtain the light force necessary for intrusion
(which is why intrusion arches are designed to bypass the premolars and canines).

• Longer spans also make the location of V-bends less critical. With a 7 mm interbracket
span, moving a V-bend only 1.2 mm from a centered position would put it at the one-third
position that totally eliminates the moment on the distant bracket. With a 21 mm span, the
same error would be almost negligible. It is much easier therefore to control two -couple
systems when the distances between attachments are relatively large, as they are when
wires connect only to molars and incisors in a 2 × 4 arrangement, or to anterior and
posterior segments.

• Still another level of complexity exists for a 2 × 4 two-couple wire because three-
dimensional (3-D) effects are produced when the wire goes around the arch from the
molars to the incisors. This makes the analysis of torque bends particularly difficult. Using
finite analysis modeling, Isaacson et al have shown that the general principles of two -
dimensional (2-D) analysis remain valid when 3-D analysis is done.19 In a long-span wire
like a utility arch, however, a V-bend at the molar produces significantly less moment and
associated equilibrium forces than the same V-bend located at the same distance from
the incisor segment. In addition, the reversal of moments so that the moment is in the
same direction on the molar and the incisor does not occur in the 3-D analysis when the
V-bend moves closer than one-third of the distance to the molar or incisors. This makes
the effect of utility arches with complex bends even less predictable.

TWO-COUPLE ARCHWIRES TO CHANGE INCISOR
INCLINATION
• A two-couple system to change the inclination of incisors can be
arranged to produce either tipping or torque. The change in
inclination is the same for tipping or torque, the difference is crown
movement or root movement. If a wire spanning from the molars to
the incisors is activated to rotate incisors around their center of
resistance, the crowns will move facially when the wire is free to
slide through the molar tube (Figure 9-43, A). Occasionally, this
provides a convenient way to tip maxillary incisors facially to
correct anterior crossbite in the mixed dentition

• If the wire is cinched back (Figure 9-43, B), the effect
will be to torque the incisor roots lingually, and a
reaction force to bring the molar mesially is created.
The incisors also will extrude, while the molars will
intrude and roll lingually. For incisor root torque, the
long range of action provided by a 2 × 4 two-couple
system is not necessarily an advantage, particularly
when there is nothing to control the vertical side effects
on the incisors. In patients with severely upright
maxillary central incisors (as in Class II division 2
malocclusion), a one-couple torquing arch with all the
other teeth as anchorage can be used to advantage.

• An asymmetric V-bend in a rectangular wire spanning from the first
molars to the incisor segment produces a moment to rotate the incisors
faciolingually, with an intrusive force but no moment on the molars and
an extrusive force on the incisors. A, If the archwire is free to slide
forward through the molar tube, the result is anterior tipping and
extrusion of the incisors. Occasionally, this is desirable in the correction
of anterior crossbite in the mixed dentition. B, If the archwire is cinched
behind the molar so that it cannot slide, the effect is lingual root torque
and extrusion for the incisors and a mesial force on the molars.

• For torque of very upright maxillary central incisors (as in Class II division 2
malocclusion), a one-couple torquing arch designed by Burstone can be very
effective. A, A heavy stabilizing arch is placed in all the teeth but the central incisors,
contoured so that it steps below the brackets on the central incisors and contacts the
facial surface of these teeth, and tied back against the molars. A wire tied into the
central incisor brackets and activated by bending it down and hooking it between the
first molar and second premolar then produces the desired moment. B, Because the
stabilizing archwire prevents facial tipping and extrusion of the central incisors, the
result is lingual root torque with optimum force over a long range. The reaction force
to intrude the remaining teeth and bring them anteriorly is distributed over all the
other teeth, minimizing the reaction.

POSTERIOR CROSSBITE CORRECTION: TRANSVERSE
MOVEMENT OF POSTERIOR TEETH
• Dental posterior crossbite, requiring expansion or constriction of
molars, can be approached with two-couple archwires. Then the anterior
segment becomes the anchorage and movement of one or both first
molars is desired (Figure 9-45). Incorporating the canines into the anchor
segment is a necessity (i.e., this requires a 2 × 6 rather than a 2 × 4
appliance). A long span bypassing the premolars still is needed for
appropriate force levels and control of moments. Asymmetric expansion
or constriction to correct unilateral crossbite is quite feasible and often is
the indication for using this method. As with other applications of two-
couple systems, the large range of the appliance means that teeth can be
moved a considerable distance with a single activation of the appliance.
The corresponding disadvantage, of course, is that the system has poor
fail-safe properties.

• A 2 × 6 appliance can be used to produce
transverse movement of first permanent molars.
In this circumstance, the anterior segment
becomes the anchorage and it is important to add
the canines to the anchor unit, but the premolars
cannot be tied to the archwire without destroying
its effectiveness. The long span between the
canine and molar is needed to produce the
desired forces and moments in this two-couple
system. A, An outward bend a few millimeters
behind the canine bracket results primarily in
expansion of the molar, with little or no rotation
(with the unequal segments, this approximates
the one-third position between the units of the
two-couple system). B, An outward bend behind
the canine combined with a toe-in bend at the
molar results in expansion and mesial-out
rotation of the molar.

LINGUAL ARCHES AS TWO-COUPLE SYSTEMS
• Still another example of a two-couple appliance system is a transpalatal
lingual arch (or a mandibular lingual arch that does not contact the
anterior teeth).22 Lingual arches often are employed to prevent tooth
movement rather than create it. The need for a lingual arch to stabilize
posterior segments in many situations has been noted previously. When
a lingual arch is used to move teeth, spring properties are
required, which means that either a different wire size or different
material is needed for an active rather than a stabilizing lingual arch.
Whatever a lingual arch is made of and however it is attached, its two-
couple design predicts the effect of symmetric V, asymmetric V, and step
bends. Often, it is desirable to rotate maxillary first molars so that the
mesiobuccal cusp moves facially. This can be accomplished bilaterally
with symmetric bends or unilaterally with an asymmetric bend (Figure 9-
46). An asymmetric activation tends to rotate the molar on the side
closest to the bend and move it mesially, while the molar on the other
side is displaced distally.

• A, Bilateral toe-in bends at the first molars create equal and opposite couples, so the
mesiodistal forces cancel and the teeth are rotated to bring the mesiobuccal cusp
facially. When space has been lost in the maxillary arch or when a Class II molar
relationship exists, this type of rotation often is desired, but a flexible rather than a
rigid lingual arch is needed to obtain it. B, A unilateral toe-in bend rotates the molar on
the side of the bend, and creates a force to move the other molar distally. Although
mesial movement of the molar on the side of the bend is limited by contact with the
other teeth, mesial movement may occur. Although net distalization of both molars has
been claimed by bends of this type on first one side, then the other, significant distal
movement of both teeth is unlikely.

• It is tempting to think that net distal movement of an upper
molar can be accomplished routinely with this type of
activation of a transpalatal lingual arch, and it has been
suggested that a clinician can distalize one molar while
rotating the other, then reverse the process, moving both of
them back. However, the evidence indicates that significant
distal movement beyond rotation of the buccal cusps is
unlikely—mesial movement of the anchor molar is entirely
possible and indeed quite likely.

• A lingual arch also can be activated to torque roots facially or
lingually (Figure 9-47). Symmetric torque when molars are
expanded provides bodily movement rather than tipping. An
interesting approach to unilateral crossbite is the use of a
lingual arch with buccal root torque (lingual crown torque) on
one side pitted against buccal tipping on the other side. As
Ingervall and coworkers have shown quite convincingly,
significant expansion on the tipping side can be produced,
perhaps more effectively if the appliance is converted to a one-
couple device by placing a round rather than rectangular wire in
the bracket on the tipping side

• A somewhat unusual application of a lingual arch would
be to tip one molar distally, uprighting it. The reciprocal,
of course, would be mesial tipping of the opposite
molar. This activation would require a twist in the
lingual wire. The location of this twist bend is not
critical. The relative moments on the molar teeth will be
equal and opposite, wherever the twist bend is placed.

• A, Bilateral expansion of molars can be
created by expansion of a transpalatal
arch, which typically is achieved by
opening a loop in the middle. The moment
of the expansion force tips the crowns
facially. B, Placing a twist in the wire
creates a moment to torque the roots
facially. The moment of the couple must be
greater than the moment of the force for
this to occur. Unless a flexible wire is used
for the lingual arch, it can be difficult to
insert the activated lingual arch with
enough twist to produce the desired torque.
C, A twist in the wire on one side can be
used to create stationary anchorage to tip
the opposite molar facially. This is
particularly effective if the wire is rounded
on the movement side, so that a one-
couple rather than two-couple system
exists in the faciolingual plane of space.

SEGMENTED ARCH MECHANICS
• What is often called segmented arch mechanics is best considered
an organized approach to using one-couple and two-couple
systems for most tooth movements, so that an engineering analysis
can provide an approximation of both forces and moments. This
would allow use of more favorable force levels and potentially
provide better control. The essence of the segmented arch system
is the establishment of well-defined units of teeth, so that
anchorage and movement segments are clearly defined. The
desired tooth movement is accomplished with cantilever springs
where possible, so that the precision of the one-couple approach is
available, or with the use of two-couple systems through which at
least net moments and the direction of equilibrium forces can be
known (as they cannot be with the multicouple force system
created by a continuous rectangular archwire tied into brackets on
all teeth in a dental arch).

• In segmented arch treatment, lingual arches are used for
stabilization in a majority of the patients, and stabilizing wire
segments in the brackets of teeth in anchor units also are used
routinely. The requirements for stabilization, of course, are just
the opposite of those for tooth movement: the heaviest and
most rigid available wires are desired. For this reason, the 22-
slot edgewise appliance is favored for segmented arch
treatment. The wires used for stabilizing segments usually are
21 × 25 steel, which is far too stiff for tooth movement. The
stabilizing lingual arches usually are 36 steel, soldered to the
molar bands or with doubled-over ends that fit into rectangular
sheaths.

• Typical segmented arch treatment would call for initial alignment
within anterior and posterior segments, the creation of
appropriate anchorage and tooth movement segments, vertical
leveling using intrusion or extrusion as needed, space closure
with differential movement of anterior and posterior segments,
and perhaps the use of auxiliary torquing arches. Sliding
archwires through brackets almost always is avoided in this
technique because resistance to sliding hampers efforts to
control anchorage and introduces uncertainties into the
calculation of appropriate force levels. Continuous archwires,
particularly rectangular wires, would be reserved for the final
stages of treatment when quite small but precise movements
are required.

• The advantages of the segmented arch approach are the control
that is available and the possibility of tooth movements that
would be impossible with continuous archwires. The
disadvantages are the greater complexity of the orthodontic
appliance and the greater amount of the doctor's time needed to
install, adjust, and maintain it. It is an interesting paradox that
simplifying the engineering analysis of the appliance, by dealing
insofar as possible with identifiable one- and two-couple
systems, complicates the appliance rather than making it
simpler.

• An excellent example of the segmented arch approach is the design
of an appliance to simultaneously retract and intrude protruding
maxillary central incisors. This is difficult to accomplish because
lingual tipping of the incisors tends to move the crown downward as
the tooth rotates around its center of resistance. Intrusion of the root
apex is necessary to keep the crown at the same vertical level relative
to the lip and other teeth. This problem can be solved by creating
anterior and posterior segments, using a rigid bar to move the point of
force application distal to the center of resistance of the incisor
segment, and applying separate intrusion and retraction forces
(Figure 9-48).25 This could be done much more easily now, however,
by using TADs as illustrated in Chapter 15. Skeletal anchorage has
the potential to replace many of the more complex applications of
segmented arch treatment.

• A segmented arch approach allows simultaneous retraction and
intrusion of an anterior segment. A rigid bar in the anterior segment
can be extended posteriorly so that the point of application of an
intrusive force is at or distal to the center of resistance of the
incisor segment. If a cantilever spring is used to apply an intrusive
force at that point, the tendency of a retraction force to elongate the
anterior segment can be overcome.

• Complex segmented arch treatment carries with it two
other potential disadvantages that must be kept in
mind. First, even with the most careful engineering
analysis, it can turn out that something was overlooked
in the determination of the likely outcome. The
application of engineering theory to orthodontics is
imperfect enough that a unique force system for an
individual patient may not produce the expected
outcome.

• Second, most segmented arch mechanisms contain
little or nothing to control the distance that teeth can be
displaced if something goes wrong. If precisely
calibrated springs with a long range of action encounter
something that distorts them (such as a sticky candy
bar), major problems can occur. The mechanical
efficiency of a segmented appliance can be both an
advantage and a disadvantage.

CONTINUOUS ARCH MECHANICS
• Engineering analysis of the effects of a continuous archwire, one that is tied
into the brackets on all the teeth, is essentially impossible. All that can be said
is that an extremely complex multicouple force system is established when the
wire is tied into place. The initial result is a small movement of one tooth. As
soon as that occurs, the force system is changed, and the new system causes
a small movement of another tooth (or a different movement of the first tooth).
Either way, the result is still another complex force system, which causes
another movement, leading to another change in the system, and so on.
Sometimes, orthodontic tooth movement is conceived as a slow, smooth
transition of the teeth from one arrangement to another, but the force systems
involved, particularly those with continuous arch mechanics, make it plain that
this is far from the case. If it were possible to take time-lapse photographs of
teeth being moved into position, we undoubtedly would see ―the dance of the
teeth,‖ as the complex force systems formed and changed, producing varied
effects in sequence, and Hayashi recently has shown just this sort of
movement.26 It is a saving grace that a continuous archwire usually does not
allow the teeth to move very far from the desired endpoint.

• The advantages and disadvantages of the continuous arch
approach are just the reverse of those with the segmented arch
approach. Continuous arch treatment is not as well-defined in terms
of the forces and moments that will be generated at any one time
and certainly is less elegant from an engineering perspective. But
continuous archwires often take less chair time because they are
simpler to make and install, and they have excellent fail-safe
properties in most applications. In modern orthodontics, often the
clinician must evaluate the trade-off between segmented and
continuous arch approaches to specific problems. For those who
use primarily the segmented approach, some use of continuous
archwires simplifies life. For those who use primarily continuous
archwires, some use of the segmented approach is necessary to
meet specific objectives. Quite literally, you consider the benefits
versus the cost (time) and risks, and take your choice.

• The development of contemporary fixed appliances and
their characteristics are discussed in Chapter 10.
Clinical applications of the mechanical principles
reviewed in this chapter and further information about
the use of specific treatment methods are provided in
some detail in Chapters 14 to 18.

Mechanical principles in orthodontic force control

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