Introduction
An understanding of the basic principles of mechanics is essential to evaluate the force systems generated by an orthodontic appliance. These mechanical principles are found within a branch of engineering called ‘mechanics’. Mechanics describes the effects of forces on bodies. When these forces are applied to bodies in a biological system, such as teeth, the term used to describe the effects of these forces is called ‘biomechanics’.
An orthodontist would obviously like treatment to proceed as efficiently from point A to point B with the least amount of side effects and surprises. To this end, a thorough knowledge of the principles of biomechanics is essential. In order to design efficient orthodontic appliances, one needs to consider:
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1.
The type of force system required to produce a given centre of rotation, and
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The properties of materials so that one can use appropriate materials, wire sizes and loop configurations to produce optimum forces and moments.
Basics of biomechanics
Force
A force is defined as the action of one body on another that changes or tends to change the shape or motion of that second body. Forces act in a straight line and can be either a push or a pull. Although strictly speaking, force = m × a, where ‘m’ is mass and ‘a’ is acceleration, should be measured in Newtons, however, in orthodontics it is still widely measured in grams. 1 N = 101.97 g, however, in orthodontics, for purpose of ease one could consider 1 N equivalent to 100 g. Forces are vectors, having both magnitude and direction.
Vector addition
Two or more forces acting at a single point can be combined mathematically and be represented as a single resultant force at that point ( Fig. 50.1 ). The two forces are visualised as two sides of a parallelogram and the opposite sides are then drawn in to complete the parallelogram. The resultant force is represented by the line bisecting the parallelogram from end to end. A clinical application of this concept is shown in Fig. 50.2 .
Vector addition.
Two or more forces acting at a single point can be combined mathematically and be represented as a single resultant force at that point. The two forces (solid red lines) are visualised as two sides of a parallelogram, and the opposite sides are then drawn in (dotted lines) to complete the parallelogram. The resultant force (solid blue line) is represented by the line bisecting the parallelogram from end to end.
Vector addition.
In this example, the two forces of intrusion and retraction (green arrows) on the upper incisors generated by activation of a phase I ‘Intrusion Retraction arch wire’ move the incisors up and back in the direction of the resultant (red arrow). This allows the upper incisors to clear the lower incisors as they move back.
Vector resolution
Unless the force is oriented in truly horizontal or vertical direction in reference to a given plane, a single force can be resolved into two components—a horizontal component and a vertical component. In the example shown ( Fig. 50.3 ), the force from a class II elastic on a lower molar can be resolved into a mesial force and an eruptive force using the parallelogram method. A clinical application of this concept is shown in Fig. 50.4 .
Vector resolution.
A single force can be resolved into components and the components are visualised using the parallelogram method. For example, the force from a class II elastic (solid red line) on a lower molar can be resolved into a mesial force and an eruptive force (solid blue lines).
Vector resolution.
Clinical application of vector resolution—a patient treated with class II elastics.
Law of transmissibility
The effect of a force on a body is the same regardless of where it is applied along the line of action. For example, it does not matter if the force is applied on the labial of the tooth as a push or on the lingual as a pull. As long as the force is in the same direction and magnitude, the effect remains the same ( Fig. 50.5 ).
The law of transmissibility.
The effect of a force on a body is the same regardless of where it is applied along the line of action. For example, it does not matter if the force is applied on the labial of the tooth as a push or on the lingual as a pull, as long as the force is in the same direction and magnitude the effect remains the same.
Centre of mass
The centre of mass of a body is a point where mass distribution is uniform in all directions. It is the centre where all the mass of the body seems to be concentrated. This is the point to which a force maybe applied to cause a linear acceleration (bodily movement) without any angular acceleration (rotation).
Centre of gravity
On earth, a body is under the influence of the force of gravity, which somewhat alters the position of centre of mass of the body. The centre of gravity is based on the weight of the body. It is a point where weight is evenly distributed in all directions. Similar to centre of mass, centre of gravity is a point where the application of a single force would result in bodily movement of the body without any rotation.
Centre of resistance
The centre of resistance (C Res ) is a point analogous to centre of mass and centre of gravity where the application of a single force would result in bodily movement without rotation ( Fig. 50.6 ), except that this term is applied to bodies that are restrained. The tooth is not a free body but a restrained body. The root of the tooth is attached to the bone and is therefore restrained. When a force is applied to the tooth, the part of the tooth (crown) that is not restrained provides only a fraction of the resistance to movement as compared to the root which is embedded in bone. Therefore, C Res will not lie in the centre of the whole tooth but somewhere in the root.
Centre of resistance of a tooth is a point on the tooth where a single force would produce translation, that is, all points on the tooth moving in parallel straight lines. C Res Centre of resistance.
In a single-rooted tooth such as an incisor, the root is wider in the cervical area and provides more resistance to movement than the narrower part of the root towards the apex. Therefore, C Res will not be located halfway down the length of the root but closer to the cervical region, approximately two-fifths the length of root from the alveolar crest to the apex ( Fig. 50.7 ).
Centre of resistance of a healthy incisor is about two-fifths the length of the root from the alveolar crest to the apex. C Res Centre of resistance.
In a healthy upper central incisor, C Res is estimated to be located about 10–12 mm from the bracket. However, the distance of C Res from the bracket will vary from tooth to tooth depending upon the length and morphology of the root. For example, the distance from the bracket to C Res will be less for a molar and more for a canine as compared to an incisor ( Fig. 50.8 ).
Distance of centre of resistance from the bracket varies depending upon the length and morphology of the root of the tooth. (A) Central incisor. (B) Canine. (C) Molar.
In addition, as compared to a healthy tooth ( Fig. 50.9 A), C Res will move apically, away from the bracket if there is loss in alveolar bone height ( Fig. 50.9 B). The clinical implications in this situation, as compared to a healthy tooth, are:
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Lighter forces should be applied to move the tooth since the tooth provides less restraint and resistance to movement.
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A bigger moment of force is caused by a force at the bracket, therefore, a greater moment to force ratio needs to be applied to achieve bodily movement.
Distance of centre of resistance from the bracket changes if there is alveolar bone loss or root resorption. (A) Healthy tooth. (B) Distance from the bracket to centre of resistance increases when there is loss of alveolar bone height. (C) The distance decreases when the root is shortened due to resorption.
Similarly, in a tooth that has undergone root resorption, C Res will move towards the cervical region ( Fig. 50.9 C). The clinical implications in this case, as compared to a healthy tooth, are:
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1.
Lighter forces should be applied to move the tooth since the tooth provides less restraint and resistance to movement.
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A smaller moment of force is caused by a force at the bracket, therefore, smaller moment to force ratio needs to be applied to achieve bodily movement.
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Table 50.1 shows different positions of C Res as described by authors in the past. ,
TABLE 50.1
Positions of centre of resistance as described by various authors
Sl. Year Author Tooth/area Position of C Res 1 1997 Nanda Single-rooted tooth 25%–33% of root length 2 1995 Burstone Single-rooted tooth 33% of root length 3 1984 Smith and Burstone Single-rooted tooth 33%–50% of root length 4 -
1985
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2000
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Nikolai
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Proffit
Single-rooted tooth 50% of root length 5 Multi-rooted tooth Near the furcation of the root 6 1987 Bulcke Two central incisors Distal aspect of cuspid 7 1987 Bulcke Four incisor segment Between cuspid and bicuspid 8 2000 Proffit Four incisor segment Between lateral incisor and cuspid 9 1987 Bulcke Six teeth anterior segment (canine to canine) Distal to bicuspid 10 1990 Melsen Six teeth anterior segment (canine to canine) In line with the cuspid bracket -
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Centre of rotation
The centre of rotation (C Rot ) is any point around which the tooth rotates. Unlike the C Res of a tooth, which remains constant, the C Rot of the tooth can vary. In fact, it is this capacity to move the C Rot that allows the orthodontist to create controlled tipping, bodily movement or root movement of the tooth.
Moment
If a force is applied at any point on the tooth other than the C Res the tooth will have a tendency to rotate (although C Res will also move in the direction of the force). This rotational tendency is called the ‘moment of the force’ ( Fig. 50.10 ). The rotational tendency can be either in a clockwise (Cw) or counterclockwise (Ccw) direction. For example, if a distally directed force is applied to the bracket of a maxillary central incisor, the tooth will have a tendency to rotate in a Cw direction.
Moment of the force.
If a force is applied at any point on the tooth other than the centre of resistance, the tooth will have a tendency to rotate, although centre of resistance will also move in the direction of the force. This rotational tendency is called the moment of the force. C Res Centre of resistance.
The magnitude of the moment is determined by the amount of force applied, multiplied by the perpendicular distance between the point of force application and C Res of the tooth, M = F × D ( Fig. 50.11 ). As mentioned earlier, in a healthy maxillary central incisor C Res of the tooth is about 10 mm apical to the bracket. Therefore, 100 g of distally directed force applied at the bracket will produce 100 g × 10 mm = 1000 g/mm Cw moment.
Magnitude of the moment is determined by the amount of force applied, multiplied by the perpendicular distance between the point of force application and centre of resistance of the tooth. M = F × D (F = force and D = distance). C Res Centre of resistance.
It is important to remember that the magnitude of the moment is dependent on both the amount of force and the perpendicular distance from C Res . Therefore, we can at times dramatically increase or decrease the magnitude of the moment by altering the distance of the point of force application.
Moment to force ratio
If a single force is applied on the crown, the tooth will rotate in a way that the C Rot is very close to and just apical to the C Res of the tooth. In this type of movement, the crown moves in the direction of the force and the apex of the root moves in the opposite direction. This type of movement, with the root moving in the opposite direction, however, is rarely desirable. If we do not want the root apex to move in the opposite direction, then we need to apply a moment on the tooth that will counteract the tipping moment that is being generated by force. The ratio of the moment (torque or mesiodistal angulation of the wire) that we apply to counteract the tipping and the force we apply on the crown gives the moment to force ratio (M:F).
Types of tooth movement
Uncontrolled tipping
In uncontrolled tipping, a single force is applied at the crown, whereby the tooth rotates around a point just apical to the C Res . The crown moves in the direction of the force and the apex of the root moves in the opposite direction. The C Res of the tooth also moves in the direction of the force ( Fig. 50.12 A).
Uncontrolled tipping.
In uncontrolled tipping, only a single force is applied at the crown, whereby the tooth rotates around a point just apical to the centre of resistance. The crown moves in the direction of the force and the apex of the root moves in the opposite direction. The centre of resistance of the tooth also moves in the direction of the force. C Rot Centre of rotation.
Controlled tipping
In our previous example of a healthy maxillary central incisor, we stated that C Res is about 10 mm apical to the bracket. Therefore, a force of 100 g applied in a distal direction at the bracket results in a 1000 g/mm clockwise tipping moment. If we apply a Ccw moment of 600 g/mm (M:F = 6:1), then although this counter moment is not sufficient to totally negate the effect of the clockwise tipping moment, it is, however, sufficient enough to decrease the amount of tipping by moving the C Rot to the apex of the root. Therefore, in controlled tipping, the crown tips back in the direction of the force but the root apex does not move forward in the opposite direction ( Fig. 50.12 B). The C Res also moves in the direction of the force. This kind of movement may be indicated when retracting excessively proclined maxillary incisors.
Controlled tipping.
A force of 100 g applied in a distal direction at the bracket results in a 1000 g/mm clockwise (Cw) tipping moment. If we apply a counterclockwise (Ccw) moment of 600 g/mm (M:F = 6:1), then, although this counter moment is not sufficient to totally negate the effect of the Cw tipping moment. However, it is sufficient to decrease the amount of tipping by moving the centre of rotation to the apex of the root. The centre of resistance also moves in the direction of the force. C Rot , Centre of rotation; Cw , Clockwise; Ccw , Counterclockwise.
Bodily movement
In our example, if we were to increase the Ccw moment to 1000 g/mm then this would exactly offset the tipping moment, and the tooth would translate back bodily, without any tipping ( Fig. 50.12 C). The C Rot in this situation is at infinity. This kind of movement is often required when retracting a canine into an extraction site.
Bodily movement.
If we were to increase the counterclockwise moment to 1000 g/mm, then this would exactly offset the tipping moment, and the tooth would translate bodily without any tipping. The centre of rotation is at infinity. Cw , Clockwise; Ccw , Counterclockwise.
Root movement
If we increase the moment to force ratio beyond 12:1, the C Rot moves to the crown. First, it moves to the incisal edge, and any further increase in the moment to force ratio moves it to the bracket. In this situation, the apex of the root moves back in the direction of the applied force, and the crown may actually move forward in the opposite direction ( Fig. 50.12 D). This type of movement is often utilised if the maxillary central incisors are retroclined following retraction in an extraction case.
Root movement.
If we increase the moment to force ratio beyond 12:1, the centre of rotation moves to the crown. First, it moves to the incisal edge, and any further increase in the moment-to-force ratio moves it to the bracket. In this situation, the apex of the root moves back in the direction of the applied force, and the crown may actually move forward in the opposite direction. Cw , Clockwise; Ccw , Counterclockwise.
Static equilibrium
Newton’s third law of motion states that for every action, there must be an equal and opposite reaction. Since an active orthodontic appliance in the mouth does not move instantaneously, the net force system produced by the appliance as a whole must be zero. To meet the requirements of equilibrium, the sum of all forces present in the horizontal direction must be zero; the sum of all forces present in the vertical direction must be zero and the sum of all moments present must be zero. In other words, for a body to be in equilibrium:
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SFx=0
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SFy=0
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SM=0
However, one should bear in mind, although from a mechanical perspective, there may be an equal and opposite reaction at the two ends of an appliance; the clinical response in terms of tooth movement is rarely the same at the two ends. As seen in Fig. 50.13 , the compressed coil spring is exerting 100 g of distally directed force on the canine and an equal and opposite 100 g of mesially directed force on the central incisor. Clinically, the central incisor will move mesially since it has space mesial to it. However, the canine, which has a number of large teeth distal to it, is not likely to move distally. Similarly, 100 g of intrusive force produced by an intrusion arch provides optimum force for the intrusion of the anterior teeth. However, the heavy forces of occlusion, especially in a low angle deep bite case, are likely to prevent extrusion of the molars by the 100 g of extrusive force on them. Other factors that come into play include inter-digitation of cusps of teeth, especially in adults that might have a ‘locked’ in occlusion.
The compressed coil spring is exerting equal and opposite forces on the canine and central incisor.
However, the clinical effects will not be equal and opposite. The central incisor will move mesially as it has space mesial to it but the canine, which has a number of large teeth distal to it will not move much.
Statically determinate and indeterminate force systems
In a statically determinate force system, the forces and moments generated can be readily measured and calculated. On the other hand, statically indeterminate systems are too complex for precisely measuring and calculating all the forces and moments involved in the equilibrium.
A one couple force system, where there is a force at one end and a couple at the other is considered a statically determinate force system. An example of this system is the Burstone’s intrusion arch wire where there is a single point contact at the incisors and a couple at the molar ( Fig. 50.14 ). However, in a two couple force system where the wire is engaged into the brackets on both ends, thereby generating couples on either end the system becomes too complex to accurately determine the forces and moments generated in the system. An example of this is Rickett’s utility intrusion arch in which the wire is engaged into the molar tube on one end and the incisors brackets on the other end thereby generating couples on both ends ( Fig. 50.15 ). In a continuous arch wire system, where the wire is engaged into the brackets of several continuous teeth, the force system becomes even more complex and indeterminate.
