SPECIAL ARTICLES

Rigid-body kinematics and single-tooth displacements

Robert J. Nikolai, PhD ~ St. Louis, Mo.

T h e branch of physics known as mechanics has influenced analyses and designs of orthodontic appli- ances and has contributed to the prediction of tooth movement. The form of the orthodontic displacement of the tooth is related to the characteristics of the mechani- cal action delivered by the appliance; the biomechanical relationship, although not fully understood, is a source of sustained'interest in the mechanics of physics applied to orthodontics. The focus of this article excludes the force- displacement relationship; rather, the emphasis is on a response to the mechanical action: the displacement and its description. The branch of mechanics (of physics) that pertains to the movements of particles or bodies, without references to causes, is known as kinematics.1

Orthodontists were describing the movements of teeth "by" their appliances before they realized that mechanics, the applied science, could offer a means of advancing knowledge within the specialty. As the under- standing of the mechanics of appliances and the relation- ships to tooth movement developed within the discipline, the language of mechanics became intertwined with orthodontic jargon in the descriptions of displacements. This mixing of terminology unfortunately has led to incompletely descriptive, often vague, and sometimes mechanically incorrect usage of words and phrases. In submitting this manuscript to the Journal, the need for a review of kinematics applied to orthodontics, specifically, with reference to the description of the displacement of a single tooth is suggested.

To begin a review of single-tooth kinematics, several concepts and reasonable assumptions are presented. First, the individual tooth is essentially nondeformable; crown and root(s) cannot move independently. The dis- tance, then, between any two specific points of the tooth is a constant, independent of time, tooth position, or orientation. Second, a reference frame is necessary from which to describe any displacement. The appropriate reference is the familiar, "local" framework for a single tooth consisting of three mutually perpendicular and intersecting axes; faciolingual, mesiodistal, and occluso- apical describe their directions. The location and angu- lation of the frame should define the ideal position and

"Professor, Biomechanics in Orthodontics, Graduate Department of Orthodontis, Saint Louis University Health Sciences Center. Am J Orthod Dentofac Orthop 1996;110:88-92. Reprint requests to: Dr. Robert J. Nikolai, Professor of Biomechanics in Orthodontics, Graduate Department of Orthodontics, Saint Louis Uni- versity Health Sciences Center, 3556 Caroline St., St. Louis, MO 63104- 1085. Copyright © 1996 by the American Association of Orthodontists. 0889-5406/96/$5.00 + 0 8/1/59441

88

orientation of the tooth, and this reference frame is fixed in basal bone. Third, to determine a displacement, asso- ciated beginning and ending states must be defined. These states refer to successive locations and orienta- tions of the tooth, and the second state follows the first in time.

DISPLACEMENTS OF POINTS/PARTICLES, LINE SEGMENTS, AND RIGID BODIES

Consider a single particle or a point of a body labeled "P," and subscripts "1" and "2" denote successive loca- tions of the particle. Assume that the particle moves within an x-y plane. Fig. 1 provides a graphic description of the two states of the particle and the displacement, independent of time. The displacement magnitude is the straight-line distance between the two state points (xl, Yl) and (x2, Y2); the "path" followed by the particle is irrelevant. The displacement direction has a two-part expression: (1) the line between the two states has an orientation with respect to the reference frame; and (2) the displacement proceeds from state 1 to state 2 (and not the opposite; often termed sense). A vector has been drawn between the two states; the characteristics of the vector (magnitude [dp], orientation or angulation with respect to the x-y frame [direction], and sense) collec- tively describe the particle displacement.

Suppose that points P (of Fig. 1) and Q define the ends of a specific line segment. Assume that the segment cannot "stretch," shorten, or bend. Consider the dis- placement of point Q relative to point P; if P is stationary and the line does not deform, point Q can move only along a circular arc. The displacement of Q relative to P overlays the chord extending from the beginning of the arc to its end, and is shown in Fig. 2; the magnitude of the relative displacement (vector) is, symbolically, dp/o. If the line PQ translates, i.e., if its orientation does not change, point Q relative to point P does not move; hence, the displacement of Q relative to P is associated with a change in angulation of the line segment. In Fig. 2, state 1 of the line PQ is dashed (PQ1), but, notably, PQ2 is generally not state 2 of the line.

The displacement of point Q with respect to an established reference frame now may be described as a combination of two components. One component is equal to the displacement of point P as viewed from the frame (and is the only component if the line segment translates). The second component is the displacement of Q relative to P. The two components, each having magnitude, direction, and sense, are vectors and are not necessarily located in a common plane. Because two

American Journal of Orthodontics and Dentofacial Orthopedics Nikolai 89 Volume 110, No. 1

P2

P I ~ ( x 2 , Y 2 )

(Xl,Y I)

x

Fig. 1. Coplanar displacement of point or particle labeled "P" from state 1 to state 2 within x-y reference frame.

distinct points of a line define that line (perhaps without bounding it), the displacements of two points of that line effectively describe the displacement of the line. Shown in Fig. 3 are example (coplanar) displacements of point P (a repetition of Fig. 1) and point Q (do: the "vector sum" of the components dp and do/p). The displace- ments from state 1 of the two specific points on the line determine state 2 of the line segment (P2Q2) and the displacement of the segment. State 2 of the segment may be obtained by translating it as point P is displaced to its new location (P2) (and each point on the line experiences the displacement dp), then rotating the segment about P2 until point Q reaches its new position (Q2) (concurrently generating the chord of length/magnitude do/p).

A rigid body is a model of mechanics, 1 and any specific line in the rigid body is nondeformable; however, lines of the body are displaced as the body as a whole is moved. Several forms of rigid-body displacements are specified. The body translates if the displacement vectors of all points of the body are alike in magnitude, direction, and sense. If the body moves, but does not translate, it rotates. In axial rotation all points on one line (the "axis") remain stationary, but all other points of the body move along circular paths with the centers of those circles on that line. The body undergoes point-wise rotation if just one point of the body remains stationary, and the displacement vector of each other point overlays the chord of the arc of a circle with its center coinciding with the stationary point. If the body experiences a coplanar, generalized rotation, the magnitudes and direc- tions of the displacement vectors generally differ, but all such vectors are contained in mutually parallel planes.

THE INDIVIDUAL TOOTH: ITS KINEMATIC REFERENCES

The displacement of the single tooth may be de- scribed in terms of the displacements of several points of and/or lines within the tooth. Absolute and relative displacements of such points and lines demand establish- ment of kinematic references within the tooth (similar to point P in Figs. 2 and 3). Two familiar points that may

Q1

Fig. 2. Coplanar displacement of point Q on (nondeformable) line segment as if point P, also on line, is stationary within x-y frame; thus described graphically is displacement of Q on line-segment PQ relative to P.

Q2

Pz

QI

Fig, 3. Line-segment PQ moves within x-y reference frame from state 1 to state 2. Displacement of segment is obtained as if dashed line (P1Q1) first translates to the dash-dot posi- tions, defined by dp, and then rotates about P2 (creating dam). Individual displacements of points P and Q define displace- ment of line.

serve as in-tooth references are the center of the crown (CC) and the point of apical extent of the tooth (A); point A coincides with the root-apex if the tooth is single rooted and is an average of the locations of the apices for the multirooted tooth. These two points lie on, and at any instant of time define the position and orientation of, the long axis of the tooth.

To describe some, but not all, single-tooth displace- ments, either point CC or A and the long axis are sufficient. Additional useful in-tooth references are the point on the extension of the crown-surface at the center of the site of force delivery from the appliance, often adjacent to the facial surface (point S), and the line through this point and intersecting the long axis orthogo- nally. Two other possible reference points are the center of resistance (CRE; on the long axis at approximately the center of the volume of the root[s]; analogous to the center of mass of an unsupported body) 2 and the point

90 Nikolai American Journal of Orthodontics and Dentofacial Orthopedics July 1996

where the long axis pierces the occlusal surface or the incisal edge of the tooth (I). Another potential reference line is also perpendicular to and intersecting the long axis at CC, but is oriented mesiodistally with the tooth in its ideal position. These eight reference points and lines are shown collectively in two views of a maxillary canine in Fig. 4. Seemingly, all of these points and lines lie on a reference frame embedded in the tooth. Such a frame- work is kinematically useful, however, only in describing at an instant of time the orientation of the tooth; it is not the local reference frame described earlier in this article, and no point or line of the tooth moves relative to the in-tooth framework.

DEFINED ORTHODONTIC, SINGLE-TOOTH DISPLACEMENTS

At least six "fundamental" orthodontic displace- ments of the single tooth are recognized; each is a coplanar displacement, and each is designated according to either the crown force system that produces it or the location (or nonexistence) of the center of rotation, 2 an end view of the line about which the tooth seems to have rotated. Tooth points move transversely (faciolingually and/or mesiodistally) in four of these displacements. One displacement is observed from a locally occlusal perspec- tive. Three displacements are translations, and three are rotations. In no particular order, these displacements may be described as (1) bodily movement (a transverse translation), (2) transverse rotation (produced by a me- chanical crown-couple; center of rotation coincides with the CRE), (3) "simple" tipping (produced by a transverse force acting through the crown-center point), (4) extru- sion (a translation; sense of point-wise displacements locally occlusal), (5) intrusion (a translation; tooth points move apically), and (6) long-axis rotation (action is a mechanical couple in a plane perpendicular to the long axis; all points on the long axis are stationary and centers of rotation)?

MORE GENERAL DISPLACEMENTS OF A SINGLE TOOTH: COMBINATIONS OF THE SiX FUNDAMENTAL DISPLACEMENTS

Each whole-tooth displacement is governed by the action delivered by the appliance to the crown, and the delivery site is typically several millimeters off the long axis: e.g., point S in Fig. 4. Crown action at the facial surface may produce one of the fundamental transverse displacements if that action is directed perpendicular to and through the long axis, assuming a tooth/periodon- tium model exhibiting anatomic symmetry about the central, mesiodistal plane. If the delivered force system adjacent to the facial (or lingual) crown-surface is di- rected mesiodistally or parallel to the long axis, the resulting displacement is a combination of bodily move- ment and long axis rotation or extrusion (or intrusion) combined with transverse rotation, respectively.

The fact that the orthodontic appliance can deliver to a facial (or lingual) crown-surface site as many as six

independent, force-system components (three forces and three [mechanical] couples) suggests that an extensive variety of whole-tooth displacements is possible; notably, all are combinations of the six cited fundamental displacements.

A GENERAL DESCRIPTION OF TOOTH MOVEMENT

The "rigid body" movement of a single tooth may be fully described by the displacements of one line in the tooth and one point on that line, but judicious selection of both line and point is necessary for the specific tooth movement. The collective characteristics of the displace- ment of the in-tooth reference point describe an "aver- age" movement as if the entire tooth was concentrated at that point. If the displacement is nontranslational, the chosen reference line in the tooth must undergo an angular change and thereby represent the rotational part of the displacement. Although the displacement of a line may be defined by the displacements of two separate points of that line (as in Fig. 3), a whole-body rotational displacement generally is not conveniently expressed in vector form. In describing some rotational displacements, particularly in discussions when precise magnitudes are unnecessary, those descriptions may be more vivid if referenced to a representative tooth surface rather than a line within the tooth.

Two facets in a proposed description of tooth move- ment pertain to (1) the form of the displacement and (2) the desire to relate it to the local reference frame. First, if the whole-body displacement is translational, or if the displacement is rotational with a definable center or axis of rotation (akin to the coplanar displacement shown in Fig. 2 if segment PQ actually rotates about a stationary point P), the displacement may be described in a single "phase." On the other hand, if the displacement is one of generalized rotation for which the center or axis of rotation is unclear or undefinable, the displacement is proposed to be described in two, sequential "phases": a combination of a translation followed by a rotation (in the manner of the displacement of line PQ in Fig. 3). Second, if a single tooth displacement is skewed to the local reference frame (and the displacement as a whole may or may not be coplanar), that displacement may be described as successive "component" movements, each of these movements coplanar, in one or two phases, and associated with the local reference-frame, i.e., viewed from the faciolingual, mesiodistal, and occlusoapical per- spectives.

Magnitude or extent of tooth movement is often not the first characteristic mentioned in describing the whole-body displacement. For example, a displacement of a single rooted tooth initially might be termed "me- siodistal," meaning that the whole-body movement is coplanar and no point of the tooth moves other than mesially or distally. If this displacement is described as mesiodistal tipping, the displacement magnitudes of crown points are implied to be larger than those of root

American Journal of Orthodontics and Dentofacial Orthopedics Volume 110, No. 1 N i ko la i 91

dA / h~\

dCC/A

Fig. 4. Views of maxillary canine from facial (left) and mesial perspectives. Five potential reference points in the tooth (and \ extension of it) are labeled. Point S seems not to be "of" tooth, \ but in kinematic analyses affixed crown attachment (e.g., orthodontic bracket) is assumed integral with tooth. From facial perspective, points S and CC are coincident.

points, and the whole-body tooth movement is rotational. The displacement of the root apex or point A, may be smaller than that of the crown center (CC), but has this whole-body movement resulted in a more or less favor- able orientation of the long axis? This question is an- swered by determination of the sense of the displace- ment of point A (i.e., whether the center of rotation is occlusal or apical of the root apex). A sketch that depicts a version of this example displacement is presented in Fig. 5; the state 1 configuration of the maxillary canine is dashed. As drawn, the center of rotation (CRO) for the displacement is occlusal of the root apex. To complete the description of the displacement, the specific location of the CRO and the displacement of one point (e.g., CC), or the displacements of two points (e.g., CC and A) are required.

As noted early in this article, beginning and ending states, but not the "paths" of points, are necesary to the description of a displacement. An alternative to the description in the previous paragraph is the depicting of the movement of the right side maxillary canine in two phases: translation of the tooth toward the mesial (dA) combined with a rotation about the root apex (charac- terized by dcc/A) that yield a net distal displacement of the crown center point (dcc). Because the three vectors in this example are nearly parallel, the relationship among them is virtually algebraic: dcc = dcc/A -- dA, where dcc/A in magnitude is approximately the product of the distance between points CC and A and the change in angulation of the long axis. (In Fig. 5, both the absolute displacements of CC and A and the relationship among the three vectors are shown; the latter is divided, part mesial and part distal to the tooth sketch, because the vectors are so nearly parallel.)

Most actual, whole-tooth displacements may be par- titioned into three categories. In the "transverse" cat- egory the tooth points move with the long axis, and no

dc C

Fig. 5. Transverse, mesiodistal (coplanar) rotation of right, maxillary canine. State 1 is represented by dashed, partial outline. Absolute displacements of apical and crown-center points are shown. Separately and partitioned because of near coincidence, vector relationship is shown: vector sum of dA and dcc/A equals dcc.

occlusoapical displacement components of substance oc- cur. This indicates that the tooth moves distally or lingually, for example, is sufficient to describe only a translational displacement (bodily movement). If the long-axis angulation changes (from any perspective), the transverse displacement is a rotation. (Fig. 5 shows such a displacement.) Depending on the relative magnitudes of displacements of crown and root points, the tooth movement is termed tipping (defined previously) or torquing. Crown lingual tipping and root distal torquing are examples that contribute direction and sense to the descriptions of rotational movements or to the rotational parts of "two-phase" whole-body displacements. Terms such as crown tipping or root torquing are unsatisfactory to the purists, however; the whole tooth, not just the crown or root(s), undergoes the displacement. The ortho- dontic community should create a description for the co- planar, transverse, single-tooth movement, in which root- point displacements are generally larger than displace- ments of crown points, to replace the word torquing. Torquing is a mechanical action, not a response, and refers to a specific force system (a mechancial couple), not a displacement?

Tooth points move relative to the long axis, but occlusoapical displacement components are again negli- gible, in a second category of orthodontic tooth move- ments. The whole-tooth displacement in two phases is a combination of bodily movement and long-axis rotation. From a locally occlusal perspective, when the long-axis angulation is ideal, several potential reference points appear coincident (points I, CC, CRE, and A in Fig. 4), and either of the two transverse lines in the tooth of Fig.

92 Nikolai American Journal of Orthodontics and Dentofacial Orthopedics July 1996

4 could be the reference line. The displacement of points S might be expressed as the vector sum of dcc and ds/co representing the translational and rotational phases of the tooth movement, respectively. The amount of rota- tion approximately equals the magnitude of ds/cc divided by the distance between points CC and S. The sense of the rotation may be indicated through relating sequential orientations of a tooth surface to local reference planes (perpendicular to corresponding reference axes): e.g., distolingual rotation means that a surface is reorienting from the distal and toward the lingual position. In the opposite direction, or with the opposite sense, the move- ment phase could be termed mesiolingual or distofacial rotation.

The third category is characterized by occlusoapical displacements of tooth points. At least a component of the translational phase is seemingly intrusion or extru- sion, but care must be taken in describing the whole-body movement when the state 1 orientation of the long axis is substantially different from that of the occlusoapical axis of the local reference frame, and when this dif- ference changes during tooth movement. The reference point and line within the tooth for this displacement category are proposed as the CRE and the long axis (assuming no accompanying long-axis rotation); i.e., extrusion or intrusion is part of the whole-tooth move- ment if the CRE displacement has an occlusoapical component. Once again referring to Fig. 4, exemplifying this category might be the vector addition of the dis- placements dcR E and dcc/cmz, representing the trans- lational and rotational phases and summing to dcc. The rotation that occurs is approximately the magnitude of dcc/cRE divided by the distance from the center of resistance to crown center.

The three whole-tooth displacement categories are individually distinctive, associated with tooth point dis- placements that are transverse, "around" the long axis, and occlusogingival, respectively. Note, though, that in some instances the overall tooth movement may overlap categories, e.g., combined long axis rotation, transverse tipping, and extrusion. In such instances, however, in- dividual component perspectives associated with the local reference frame often will enable "viewing" the tooth movement with the component displacements separated by category. (In the example cited, the ro- tation, tipping, and extrusion might be viewed from the occlusal, the facial, and a mesiodistal perspectives, re- spectively.)

CLOSURE

The kinematics of orthodontic tooth movement have been reviewed in this article. Intentionally, the systems of mechanical forces associated with the displacements have generally been excluded from the discussion. In general, if the whole-body displacement is not transla- tional and in the absence of an obvious center or axis of rotation location, a suggestion offered is the partitioning of that displacement into two phases, a translation and a rotation, characterized by the (vector) displacement of an in-tooth reference point and the change in orienta- tion/angulation of a reference line in the tooth. To the extent it is deemed necessary, the whole-tooth movement may be decomposed into two or three (component) parts, coinciding with views along the three axes of the local, orthodontic reference frame.

A description of the force system delivered by the appliance to the crown generally will not alone deter- mine the orthodontic displacement of a tooth. Rather, it is necessary to undertake the complete description of a tooth movement by examinations of both the initial and final states (positions, orientations) of the tooth- infor- mation that the practitioner often has at hand, in plaster and/or x-ray f i lms- then consideration of reaching state 2 from state 1 through a single translational or rotational movement or a combination of the two. When the displacement is a rotation (and the vast majority of tooth movements are rotations), describing the movement in terms of the characteristics of the displacement (vector) of a single point of the tooth is insufficient. The general, three-dimensional nature of tooth movement makes the description potentially complex, but categorizing the whole-tooth movement, representing the movement through the displacements of a reference point and a reference line in the tooth, expressing the whole-tooth displacement in terms of translational and rotational phases, and, as warranted, decomposing the description into local-frame:plane components, serve to organize and, hopefully, clarify the kinematic analysis.

REFERENCES

1. Beer FP, Johnston ER Jr. Vector mechanics for engineers. 4th ed. New York: McGraw-HiU, 1984:2, 55, 417.

2. Graber TM, Vanarsdall RL Jr. Orthodontics: current principles and techniques. 2nd ed. St Louis: Mosby-Yearbook, 1994:238.

3. Nikolai RJ. Bioengineering analysis of orthodontic mechanics. Philadelphia: Lea & Febiger, 1985:151-69.

4. Popov EP. Introduction to mechanics of solids. Englewood, Cliffs, New Jersey: Prentice-Hall, 1968.