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					                     Journal of Dental Research

3-D Force and Moment Analysis of Repulsive Magnetic Appliances to Correct Dentofacial Vertical
                      A.D. Vardimon, C. Bourauel, D. Drescher and G.P.F. Schmuth
                                       J DENT RES 1994 73: 67
                                 DOI: 10.1177/00220345940730010901

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J Dent Res 73(1):67-74, January, 1994

3-D Force and Moment Analysis
of Repulsive Magnetic Appliances
to Correct Dentofacial Vertical Excess
A.D. Vardimon, C. Bourauell, D. Drescher', and G.P.F. Schmuth'
The Maurice and Gabriela Goldschleger School of Dental Medicine, Department of Orthodontics, Tel Aviv University, Tel Aviv 69978, Israel;
and 'Poliklinik fur Kieferorthopadie, Rhein, Friedrich-Wilhelms-Universitat, Welschnonnenstr. 17, D-5300 Bonn 1, Germany

Abstract. Repulsive magnetic appliances can intrude poste-                                  Introduction
rior teeth, but create a lateral shift of the mandible and also
decrease in force as the mouth opens. To model their optimal                                A magnetic orthodontic appliance to intrude posterior teeth
use, the 3-D spatial force/displacement (F/D) and moment/                                   by means of repulsive permanent magnets was introduced by
displacement (M/D) diagrams of four magnetic repulsive ap-                                  Dellinger (1986). His appliance (the active vertical corrector =
pliances in diverse overlapping arrangements were character-                                AVC) consists of four juxtaposed upper and lower pairs of
ized and compared. In this orthodontic measurement and simu-                                magnets having the same polarity and located on the occlusal
lation system, only the medial eccentric magnetic arrange-                                  table of the posterior dental arches (Fig. 1).
ment, of the four compared, partially met the criteria of an                                    The improved intrusive effect over previously used spring-
optimal repulsive force system, i.e., keeping a constant intru-                             loaded bite blocks is well-documented in the literature (Kalra
ding force and excluding shearing force. The moment analysis                                et a ., 1989; Kiliardis et at., 1990; Barbe and Sinclair 1991; Wood
found that eccentric arrangements, however, developed high                                  and Nanda, 1988,1991). This improvement might be explained
Z-moment. Thus, a perplexing point was reached where the                                    by the application of the magnetic intrusive force along the
force analysis favored medial centric arrangement and the                                   longitudinal mid-axis of the tooth, in comparison with edge-
moment analysis favored centric arrangement. When the gap                                   wise mechanics, where the force is applied at the buccal
between juxtaposed magnets increased over 2 mm, the repul-                                  crown surface, lateral to the longitudinal mid-axis, thus pro-
sive force declined and the attractive force was favorably                                  ducing a moment which tends to tilt the crown buccally.
eliminated. At gap distances of 3 to 6 mm, the intrusive force                              However, the rapid onset of tooth intrusion with repulsive
was almost constant. These data suggest that centric arrange-                               magnets is overshadowed by the lateral shift of the mandible
ment is indicated clinically when the gap is minute and Mfiller                             (Kiliardis et a 1., 1990) due to the shearing f orces, and the rapid
prongs are used to prevent deleterious lateral shearing forces.                             decrease in the magnetic forces with increased mouth open-
                                                                                            ing (Tsutsui et at., 1979; Vardimon et al., 1989).
Key words. Orthodontic Appliances, Dental Models, Force                                         When a pair of magnets is displaced with their magnetic
and Moment Analyses.                                                                        axis along a straight line, a typical inverted relationship be-
                                                                                            tween the force and the square of the-displacement develops
                                                                                            (F - l/d2) (Tsutsui et al., 1979; Blechman and Smiley, 1978;
                                                                                            Blechman, 1985). A spatially dislocated translatory magnet
                                                                                            attracted by a stationary magnet moves in a hyperbolic course
                                                                                             toward a full-overlap contact with the stationary magnet,
                                                                                            when the attractive force is greater than the sum of all forces
                                                                                             acting on the system (e.g., friction) (Vardimon et al., 1991). This
 ReceivedJune 1,1993; Accepted September 8,1993                                              spatial convergence of attractive magnets, i.e., attraction even
 This investigation was supported in part by the German Research                             under a non-coplanar condition, is a major advantage in the
 Foundation (Deutsche Forschungsgemeinschaft), Grants Schm 232/                              correction of Cl II & III malocclusions (Vardimon et al., 1989,
 2-1 and Dr 196/1-1.                                                                         1990) or tooth impaction (Vardimon et al., 1991). In the latter,

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68        Vardimon et al.                                                                                                                            J Dent Res 73(l) 1994
the translatory magnet is attached to the impacted tooth and is
guided to its normal anatomic position by an intra-oral sta-
tionary magnet. In contrast to attractive f orce systems, the 3-D
F/D behavior of the AVC repulsive magnets has not yet been
studied. This lack of data restricts our ability to take full
advantage of intrusive magnetic force and to control its two
by-products, the shearing force and the rapid decline in intru-
sive force with an increase in the magnetic gap. Full manage-
ment of the applied force is especially required in light of the
potential hazard of root resorption in uncontrolled tooth in-                                                         A
trusion (Reitan, 1974; Melsen et al., 1991).
   The objectives of the present study were to determine and
compare the spatial force/displacement (F/D) and moment/
displacement (M/D)diagrams of four magnetic repulsive appli-
ances, characterized by diverse overlapped arrangements of the
fourjuxtaposed pairs of magnets. This comparison was aimed at
defining the optimum arrangement where minimum shearing
forces and maximum intrusive forces are exerted at a broad
range of jaw movements.
                                                                                                                                        M+                         A
Materials and methods
The spatial F/D and M/D diagrams were determined by an
orthodontic measurement and simulation system (OMSS)
(Drescher et al., 1991; Bourauel et al., 1992) (Fig. 2a). The OMSS
consists of two force-torque-transducers which are capable of
picking up forces as well as torques(moments) simultaneously
in all spatial directions. Each sensor is mounted on a set of three
translatory and three rotational stages driven by stepping                                   Figure 1. Dellinger's active vertical corrector (AVC), consisting of
motors. Thus, scan measurements can be perf ormed along any                                  four disc-shaped rare earth magnets (samarium-cobalt). juxtaposed
                                                                                             upper and lower magnets (M) have the same polarity (e.g., +) and are
combination of the six translation/rotation axes, or at an arbi-                             located on the occlusal table ofthe posterior dental arches. The two
trary intersection plane in space, or along a pre-selected spatial                           right and left pairs of magnets in each jaw are embedded in an acrylic
area. The force-torque-sensors have a measuring range of 15 N                                wafer (AW) and connected by a rigid stainless steel lingual bar (LB).
(450 Nmm). Each force-torque-sensor and each positioning
table are commanded by a single board microcomputer
(MC68008). The data acquisition system is controlled by a                                    2c). Each of the presented 3-D diagrams was constructed out of
main computer (PC486/50). Further technical details are de-                                  441 measurement points. Every point was an average of three
scribed in Drescher et al. (1991a,b) and Bourauel et al. (1990,                              independently recorded measurements, with a mean mea-
1992). The maxillary jaw was simulated by four magnets at-                                   surement error of ± 0.02 N. The data were then processed by
tached to the first force-torque-transducer. The maxillary                                   the Plotit version 1.6 software and plotted on a 31 x 31 meshed
translatory and rotational stages remained fixed. Mandibular                                 diagram (Akima, 1978).
jaw movements were simulated by four magnets linked to the                                       Four types of magnetic arrangements were tested (Fig. 3).
second set of force-torque-transducer and positioning tables                                 The centric arrangement refers to four maxillary and four man-
(Fig. 2b). These tables performed the scan measurements. The                                 dibular disc-shaped magnets (CoSm5, 8mm in diameter x 2mm
translatory mandibular stage was displaced step-wise 10 mm                                   high), which form four pairs of magnets in fully-overlapped
apart from full contact with the stationary maxillary stage                                  configuration, i.e., with no displacement at the zero position.
along the vertical X axis, which refers to open-/closed-mouth                                This arrangement corresponds to the magnetic arrangement of
movements. For every 2 mm of vertical displacement, the                                      the AVC. The medial eccentric arrangement refers to four max-
translatory stage was then traversed in the horizontal plane                                 illary magnets, medially displaced with respect to their four
(20 mm x 20 mm). The horizontal coplanar movement con-                                       mandibular counterparts at the zero position. The diagonal
sisted of ± 10 mm transverse displacement along the Y axis                                   eccentric arrangement refers to the situation where the four
and ± 10 mm sagittal displacement along the Z axis, where                                    maxillary magnets are converged mediodiagonally with re-
"plus" stands for right lateral jaw movements (+Y) or anterior                               spect to the mandibular magnets at the zero position. The last
sagittal displacement (+Z), and "minus" denotes lef t lateraljaw                             configuration, the centric combined, has a fully-overlapped
movements (-Y) or posterior sagittal displacement (-Z) (Fig.                                 arrangement similar to that of the first configuration, but with
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j Dcnt Res 7.30) 1994                        3-D Force and Moment Analysis of RepulIsive Magnets                                                                             69

                                                                                                                            centric                     medial eccentric
                                                                                                  10    I
                                                                                                    0                                                      (             )
                                                                                                 -1 0

                                                                                                                 diagonal eccentric centric combined
                                                                                                         rS                                                O         0
                                                                                                  10     -              .                J0                              0
                                                                                                                            10 0 10                            10 0 10       y
                                                                                                Figure 3. The four testecl naginetic arrangemrents cli I ct- in the cxtent
                                                                                                and location of ovcrlap withiin the t'out- pairs ol magnLets at thc zcro
                                                                                                position:tentrici lull ovcrlapf mnedlial cuectiti mnnaxillaiy mnagnietscdis
                                                                                                placed 4 mTn medially with respect to their mnandihulat- couLtCrpalrts
                                                                                                diagonal eccent ri iaxillary magnets converged 4 mmi mccio-diago
                                                                                                nally with iespect to the mandibular magnets, and clitt i( ( tonhilecd,
                                                                                                full overlapped paii of magnets with large maxillary (8 mmll diameter)
                                                                                                and small mandibular magnets(4 mim diamileter)

                                                                                                large maxillary andsmall mandibular miagniets.

                                                                            E                   Spatial diagrams of F[(Y, Z) for the f'our tested itiaglietic ar
                                                                                                rangements, i.e., the repulsive f'orce (F), were obtained when
                                                                                                the translatory stage traversed along the first horizontal plane
                                                                                                (Y, Z) (Fig. 4). The horizontal plane was coinposed of' trans
                                                                                                verse (Y) and sagittal (7) displacements. The centric arrange-
                                                                                                ment (Fig. 4a) had a tapered mountain f'orm, with a peak of
                                                                                                F ,(Y, Z) of 8.4 N (Newton) at no transverse displacement (Y =
   C                                                                                            0 mm) and no anteroposterior displacement (Z = mim). The
                                 A                                                              medial eccentric arrangement (Fig. 4b) demonstrated a table-
                                                                                                mountain form, with a plateau of F\(Y, Z) = 4.3 N at Z = 0 mm
                                                                                                and between Y = +3 mm and -3 mmi The diagonal eccentric
                                                       Mz                 +Fz                   (Fig. 4c) had a round-mountain form, with a peak of F,(Y, Z) =
                                                                   ,      (sagittal)            4,5 N at Y, Z = 0 mm. The last arrangement, the centric com-
(aey                                                                                            bined (Fig. 4d), had a round mountain f'orm, with the lowest
(transversal)                                                                                   peakof F,(Y,Z)= 3.8NatY, Z =0mm
                                                                                                    The 3-D F,(Y, Z) diagramsof the lateral shearing force (FE) at
                                                                                                the first horizontal plane (Y, Z) for the four tested magnetic
                                                                                                arrangements are shown in Fig. 5, The mountain-valley forn
                                                                                                should not be misinterpreted as an attractive/repulsive force
                                k      Ix
                                                                                                but refersto repulsive force at right (+F,) vs. left (-F) lateral shift
                                                                                                of the mandible, respectively, The F(Y, Z) diagram for the
                                 X   +Fx (vertical)                                             centric arrangement (Fig 5a) had a dominant mountain/valley
                                                                                                form,with twopeaksof 4,8N at Y =+3mmandZ= O mm andol'
                                                                                                -4.9 N at Y = -3 mm and Z = 0 mm The medial eccentric
Figure 2. (a, top) An orthodontic measurement and si mulation sys-                              arrangement (Fig. 5b) demonstrated an area where basically no
tem (OMSS)i A translatory/stationary stage. The mnaxillary jaw
              (b)                                                                               lateral shearing force [F,(Y, Z)] prevails between Y = +2 mm and
was simulated by four magnets (M) attached to a statio nary stage ( a                           -2 mm, and Z = +10 mm and -lO mm. A maximum value of F,(Y,
in Fig 2a) Mandibular jaw movements were simulated bylfour mag-
nets linked to atranslatory stage ("b" in Fig 2a). (c) 3-P force (F) and                        Z) + 2.5 N was obtained at Y = 6 mm and Z = 0 mm. The
moment (M) coordinates oriented according to the righit-hand rule.                              diagonal eccentric arrangement (Fig. 5c) had maximum values
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 70        Vardimon et al.                                                                                                                                JDent Res 73(l) 1994

   a                                                                                      ~:I

                                  2i                                                       I

                                                                                      [ I
                                                                                       . L'

                                             -11                                      a

Figure 4. The spatial diagrams of F (Y,Z)f orthe four tested magnetic                               Figure 5. The 3-D F,(Y, Z) diagrams lor the four tested magnietic
arrangements: (a) centric, (b) medial eccentric (c) diagonal eccentric                              arrangements: (a) centric, (b) medial cccentflici (c) diacgonal ti e/1tt ic,
and (d) centric combined. F (Y, Z) refers to the repulsive force (F,)                               and(d)centriccombined F (Y Z)relerstothe lateral shca-ingloi-ce(F
obtained when the translatory stage traverses along the first horizon-                              at the first horizontal plane (Y,Z) with 0.2 mm gap letween thc maxil
tal plane (Y, Z) at the 0.2-mm gap between the maxillary and mnan                                   lary and mandibular magnets A positive lateral shearing l'orce (+F,)
dibular magnets The horizontal plane is composed of transverse (Y)                                 standsforshearingLorce in theright i-ansversedirection,anida negative
and sagittal (Z) displacements By deflinition, a positive force along the                          lateralshearingforce(-F )i-eflerstoashearinig iorcein thelet tttaus%serse
X axis (+F ) denotes a repulsive force, while a negative force along the                           direction Note that the mountain-vallcy foirin shouldc not he misinter
X axis ( F,) is assigned for an attractive force.                                                  preted asattractive/repulsive force but i-et'erstorepulsiveItorceat right
                                                                                                   vs lel t lateral shif t ol the mandible, respectively

 similar to those of the medial eccentric arrangement [F t(Y,Z)=}+
 2.5N at Y =+ 4mmand Z= 0 nm].Thecentriccombined (Fig.5d)                                          no transverse displacement occurred (Y = 0 mm in Fig. 4) is
 had a round-mountain form similar to the centric arrangement                                     demonstrated in Fig 7b. Except for the medial eccentric ar-
 but withalowpeakof F (Y, Z)= 2.lNatY =3mmandZ =0mm.                                               rangement, all other arrangements showed a slightly wider
    The 3-D F (Z, Y) diagrams of the sagittal shearing force (F.)                                  bell-shaped curve in comparison with F,(Y) (Fig. 7a)
at the first horizontal plane are presented in Fig 6. Essentially,                                    The lateral shearing f'orce F\(Y), which is the force that
the F/D diagrams of F. were very similar to F,. The visual                                        shiftsthemandible transverselyalong the Yaxis, isdepicted in
difference between Fig. 5 [F)y(Y, Z) and Fig. 6 [F7(Z Y)] is due to                               Fig. 8a. This force reached maximium force values F ,(Y) of
diverse projection of the Y and Z axes along the abscissa and                                     about half the magnitude of the vertical fiorce F, 1j,(Y). For
ordinate coordinates of the horizontal plane.                                                     example, for the centric arrangement, F ,,j (Y)was 8.4 N (Fig.
    Fach of the 3-D diagrams (Figs. 4-6) was bisected once                                        7a), and F,,,, ,x(Y) was 4.7N (Fig. 8a). However, when F, ,,,,,(Y)
along the Y = 0 plane and once along the Z = 0 plane, producing                                   reached its peak at 3 mm lateral shift (Fig. 8a). F,(Y) had
2-D diagrams of force (F,, F,, F.) vs sagittal (Z) or vs. transverse                              decreased beyond this level to 3.1 N (Fig. 7a).
(Y) displacements (Figs. 7-10).                                                                       The lateral shearing force F,(Z) in the anteroposterior di-
    The repulsive force vs. transversedisplacement[F,(Y)]when                                     rection when no transverse displacement occurred is delin-
no anteroposterior movement is performed (Z = 0 mm in Fig. 4)                                     eated in Fig. 8b.There wasa substantial reduction in the lateral
is illusrated in Fig. 7a. The intrusive force for the centric ar-                                 shearing force in the anteroposterior direction, FJ(Z), coi-
rangement declined from its peak [FF,, (Y) = 8.4 N] at no                                         pared with the lateral shearing f'orce in the transverse direc-
transverse displacement (Y = 0 mm) to F,(Y) = 0 N at 4 mm                                         tion, F (Y). For example, for the centric arrangement, F ,(Z)        _,,
latero-excursion. In contrast, the medial eccentric arrange-                                      was 0.7 N (Fig. 8b) and F, 1 l,x(Y) was 4.7 N (Fig. 8a).
ment had an initial lower F, ...(Y) of 4.3 N, which dropped to                                        The sagittal shearing force vs. transverse displacement
zeroonly after 8mm of lateral movement, thus forming a wide                                       [F,(Y)] when no anteroposterior movement was performed is
plateau of steady force. An additional feature was the inver-                                     outlined in Fig. 9a. Basically, for all arrangements, the value of
sion of the magnetic force from repulsion into attraction,                                        F,(Y) was almost null.
when the lateral movement continued beyond the F,(Y) = 0 N                                            The sagittal shearing force vs. sagittal displacement [F (Z)[
point. This attractive force was very weak, with maximum                                          when no transverse displacement occurred is described in Fig.
values of -1 N for the centric arrangement, and even weaker for                                   9b. The shape and amplitude of F (Z) curves (Fig. 9b) were
all other arrangements.                                                                           similar to those of Fv(Y) (Fig. 8a). For example, for the centric
    The repulsive force vs. sagittal displacement             when  [F,(Z)I                       arrangement, Fz,a-(Z) was -5.0 N at Z = 3 mm (Fig. 9b) and
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    D3-DForce and Moment Analysis ofRepulsive Magnets
j Dent Res 73(i) 1994                                                                                                                                                                71

                                      Ib                                                               a             10.0

                                                                                                             -'        8.0

                                                                                                             £         6.0

                                       d'                                                                    .2        2.0
                                                                                       I                     jD        0.0

                                                                                      t: i
                                                                                                                         -10.0                -5.0         0.0         5.0    10.0
                                                                                                                                            Y transversal displacement [mm]

Figure 6. The 3 D F,(Z, Y) diagrams of the sagittal shearing I orce (F.)
for the four tested magnetic arrangements: (a) centric, (b) media,l                                    b
eccentric, (c) diagonal eccentric, and (d) centric combitned, F (Y, Z)
refers to the sagittal shearing force (F )at the first horizontal plane (Y,
Z). A positive sagittal shearing f'orce (+F ) stands for shearing I orce in
the anterior transverse direction, and a negative lateral shearinig force
(-F) refers to a shearing force in the posterior transverse direction,                                       tB        6.0
Note that the mountain-valley fuorm should not be misinterpreted as                                          LLX
attractive/repulsive torce but refers to repulsive force at anterior vs.                                               4.0
posterior sagittal shift of the mandible, respectively
                                                                                                             .>        2.0
F, jl,,,(Y) was -5.0N at Y = 3 mm (Fig.8a).                                                                  2?       0.0
    Fig. 10 exhibits six F(Y) curves of the same magnetic ar-
rangement (centric combined), where each curve corresponds                                                                                            4-             XI
                                                                                                                     -2.0 X-.                   *II,
to a different gap (X) between the maxillary and mandibular                                                            -10.0                  -5.0          0.0         5.0   10.0
magnets, i.e., the total plot presentsvariation in Fl asaf unction                                                                            Z sagittal displacement [mm]
of the displacement X.
    The inversion of the repulsive force into attractive force                                      Figure 7. (a) F (Y) ftor the four magnetic arrangements F,(Y) stands
                                                                                                    t'or the repulsive force vs transverse displaceiient when no sagittal
holds true only for gaps (X) less than 2 mm, at Y < -4 mm and                                       displacement occurred (Z = 0 mm in Fig. 4) (b) F (Z) tor the tour
Y > +4 mm. At greater gaps (X > 2 mm), a nullification of the                                       magnetic arrangements, where F (Z) refers to the repulsive t'orce vs
repulsive force developed at Y < -5 mm or Y > +5 mm.                                                sagittal displacement when no transverse displacement occurred (Y
                                                                                                    0 mm in Fig. 4).
    From the three moments that developed in the system (M.,
M, M) (Fig. 11), only the moment M (Fig. lic), which acts
around the Z axis, affected the system. This moment will cause                                     very sharply to 0 N level after 4 mm of transverse displace-
in vivo deleterious unilateral intrusion on one side of the jaw.                                   ment (Fig. 7a). In contrast, the medial-eccentric arrangement,
Magnetsin centric arrangements produced favorably low val-                                         though possessing a lower F,,, of half the magnitude of the
ues of Mz up to 20 Nmm, whereas eccentric magnetic arrange-                                        centric arrangement, maintained this force level over a 6-mm
ments developed unfavorably high moment values up to 100                                           range of transverse displacement (Y = - 3 mm) (Figs. 4b, 7a).
Nmm.                                                                                               However, this plateau of repulsive force was preserved only
                                                                                                   along theY axis; any anteroposteriordisplacement along the Z
Discussion                                                                                         axis initiated a rapid decline in Fx (Fig. 7b).
                                                                                                      For the centric arrangement, if the force distribution of
An optimum magnetic arrangement to intrude teeth should                                            Fj(Y) is "best fitted" by a quartic polynomial regression equa-
demonstrate steadiness of high intrusive force (F.) at a wide                                      tion [F. (Y)], it will yield
range of vertical, transverse, and sagittal displacements, low
magnitude of the shearing forces (Fy F), and negligible mo-                                              F                 =   6.952 - 0.122Y - 0.289Y2 + 0.001Y I+ 0.002Y4,
ments (M., M,, M). However, of the four arrangements tested,
none could meet all these criteria.                                                                with a correlation coefficient of 0.905.
    The centric arrangement demonstrated the highest level of                                         For the medial eccentric arrangement, the "best f it" quartic
FX ma at Y, Z = 0 mm (Figs. 4a, 7a). However, this force declined                                  polynomial regression equation F, (Y)] yields
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72         Vardimon et aR. 73(l) 1994
                          j Dcnt Res

  a                                                                                                  a             2.0 -
                                                                                                                   1.6 -
                                                                                                                   1.2 -
                                                                                                                   0.8 -
                                                                                                         ,LN       0.4 -   -- -- -- --  ---   w-- -- --   -- ---
                                                                                                                                                                   -1--          ---
                                                                                                                                                                          -- -- --
                                                                                                                                                                                       r-   - - -   ------

                                                                                                                   0.0 i
                                                                                                          0)      -0.4 -I -----------
      0)                                                                                                                              ----------------------
                                                                                                                  -0.8 -
                                                                                                          ou      -1.2 -}                                  diag. eccentric
                                                                                                                                      '-. med. eccentric
                                                                                                                  -1.6 - ---------- ----------- -- -E centric, combi
                                                                                                                  -2.0.-              ,,                                  I,
                                                                                                                                                                               e-e centric
             -10.0        -5.0          0.0         5.0                         10.0                                 -10.10               -5.0         0.0         5.0                                  10.0
                         Y transversal displacement [mm]                                                                                Y transversal displacement [mm]

  b         2.0
            0.4                                                                                          ,LN
            0.0                                                                                          c2
      .2   -0.4                                                                                           CD
           -0.8                                                                                           Co
           -1.2    .---          -diag. eccentric
                                               .--med. eccentric
           -1.6 ----------X---------------     E centric, combi
           -2.0                                e-0 centric
                                                                                                                     -10.0                 -5.0          0.0         5.0                                10.0
             -10.0       -5.0          0.0         5.0         10.0
                                                                                                                                           Z sagittal displacement [mm]
                         Z sagittal displacement [mm]

Figure 8. (a) F (Y) for the four magnetic arrangements. F (Y) is                               Figure 9. (a) F (Y) for the tour magnetic arrangen3ents. F (Y) stands
assigned to the lateral shearing force vs. transverse displacement,                            for the sagittal shearing f'orce vs. transverse displacement when no
when no anteroposterior movement is perftormed (Z= 0mm in Fig. 5).                             anteroposterior movement is performed (Z = 0 mm in Fig 6) (b) F (7)
(b) Fy(Z) for the four magnetic arrangements. Fy(Z) is the lateral                             forthe fourmagnetic arrangements. F ,(Z) represents thesagittal shear
shearing force vs. sagittal displacement when no transverse displace-                          ing f'orce vs. sagittal displacement when no transverse displacement
ment occurred (Y 0 in Fig 5).                                                                  occurred (Y = 0 mm in Fig. 6).

F46Y),__c= 0.058Y
         4.427                     0.103Y +0.0006Y3+0.0005Y',                                  shifts the mandible anteroposteriorly along the Z axis at Y = 0
                                                                                               mm, reached maximum values of about half the magnitude of
with a correlation coefficient of 0.977.                                                       the vertical force F,(Figs. 8a, 9b). Theoretically, one can con-
    Thus, f'or Y < 5 mm, the behavior of F, (Y) was dictated                                   clude that because F XJ,1~11. > F and F > F     y IliY
                                                                                                                                                           the mandible
                                                                                                                                                                   .III        -IliS
mainly by the free constant (6.952 vs. 4.627) and by the second-                               tends to dislodge in opening movement along the vertical axis
power variable (-0.289Y2 vs. -0.103Y2).                                                        rather than shift aside along the Y or Z axis. However, the
    The inversion of Fx from repulsive into attractive force at                                intrusive force and the shearing forces reached their peaks at
extreme latero-excursions (Fig. 7a) seems initially insignifi-                                 a different spatial location. In the case of the centric arrange-
cant, due to the low attractive force level (< 1 N). However,                                  menttii F ,xcorresponded to Y, Z =0mm, whereas F , 1 corre
when the tendency of the jaw to move from a high to a low                                      sponded to Y = + 3 mm and Z = 0 mm; consequently, F,, , 3
repulsive magnetic field is taken into consideration, then the                                 O) > FX( + 3 Z -) (Figs. 4a, 7a vs. 5a, 8a). Thus, at any right or left off-
attractiveforcemightcontributetotheimplementationof the                                        center position greater than 3 mm, the tendency of the jaw
undesired side-effect by locking the mandible in maximum                                       toward lateral movement is greater than the jaw's opening
latero-excursion.                                                                              movement.
   The lateral shearing force [Fv(Y)], which is the force that                                      Only the medial eccentric arrangement showed some fa-
shif ts the mandible transversely along the Y axis at Z = 0 mm,                                vorable delay in commencement of the lateral shearing force,
and the sagittal shearing force [F (Z)I, which is the force that                               which was more pronounced in the anteroposterior direction
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j Dent Rcs 73(l) 1994                      3-D Force and Moment Analysis ofRepulsive Magnets                                                                                                                            73

                                                                                                  a                100.0
                                                                                                            E       80.0-                             -----------        -----------
Z      3.0                                                                                                  E       60.0-                             ----------- -----------
£            -1                                                                                             a)
                                                                                                                    40.0-                                                                          L-

LLx    2.0                                                                                                  0       20.0-
 a)                                                                                                          E                                                           ----------------------

                                                                                                                -20.0-                                -                  -                             -

                                                                                                                -60.0-                   diag. eccentric                                           r
                                                                                                             x                           med. eccentric
                                                                                                                -80.0- G-u               centric, combi
                                                                                                                       a-n               centric,
      -1.0 -                                                                                                       -10.0                        -5.0         0.0         5.0                                       10.0
        -10.0          -5.0         0.0         5.0                            10.0                                                           Y transversal displacement [mm]
                     Y transversal displacement (mm]

Figure IO. F (Y) as a unction of thedistancc X forthe'same magnetic                                         E
arrangement (centric com bi ned)iEach curve corresponds to a given X
distance, e, a specific gap between the maxillary and mandibular                                            E
magnets.                                                                                                    E
(+-O-to -10 mm)thanin the lateral direction (+2 to -2mm)(Figs.
5b, 8). In contrast, the increase in lateral shearing force from
zero to maximum values was most dramatic and undesirable
for the centric arrangement.                                                                                          D.0     - diag. eccentric               ----     -----------                     -----

                                                                                                                             -. med. eccentric
    The likelihood of the jaw to slide in the anteroposterior or
lateral direction is similar in magnetic arrangements with a                                                          1.0   [3£          centric, combi
unif'orm Z,Y bilateral symmetry, as in the case of magnetic                                                           -10.0                     -5.0        0.0         5.0                                       10.0
arrangements with a square geometry of' right angles among                                                                                   Y transversal displacement [mm]
the f'our magnets of eachjaw. [Bilateral symmetry is defined as
a situation in which similar parts are arranged on opposite                                        C
sides of a median axis (Y or 7 in our case), dividing the body                                                 100.0
                                                                                                             E  80.0-
into identical halves.] However, the trapezoid geometry of the
                                                                                                            z 60.0-
present intrusive magnetic arrangements has a right- left sym-                                              C                 --   - -
                                                                                                                                          -   -   -   -----        -             -- --   --    -   L-------        --

metry only, i.e., along the Z-symmetrical line at Y = 0 mm (Fig.                                            a)  40.0-
                                                                                                            E                         -\-     ----------
3). Consequently, when each of the shearing forces (F, F,) is                                               E 20.0- k-A~~~~
                                                                                                            0           d.g    ceti---- -X--
                                                                                                                                      --/                    ---

separated toitsplanarcomponents[Fy(Y), F (Z) vs. F(Z),: (Y)],
                                                            F                                                    0.0 n- cetrc Eombi --A~---~ ~ ~ ~ r-~~
                                                                                                                            ,j      -----             ------

then F,(Y)at Z= 0mm isequal in magnitude toF (Z)at Y= 0mm                                                      -20.0
(Figs. 8a vs. 9b), but F (Z) at Y = 0 mm is greater in magnitude                                             C,) -40.0        ----------.                 ----------,-!


than F_(Y) at Z = 0 mm (Figs. 8b vs. 9a). Although the values of                                                   -60.0 " diag. eccentric                     -------
                                                                                                                            -" med. eccentric                            :\       /
F,(Z) are very low per se, it is possible that, due to this asym-                                                  -80.0 &-Ej centric, combi
                                                                                                                                                                          ~~x~/,---                        ---------

metrical force dispersion, the probability that the lateral shear-                                                -100.0
ingforce will dictatejaw side movement isgreater than that of                                                         -10. 0           -5.0                  0.0         5.0                                       10.0
the sagittal shearing force. This view is also supported by the                                                                               Y transversal displacement [mm]
clinical findingsof Kiliardis et al. (1990), whofound unilateral                              Figure 11. The three moments that developed in the system for the
posterior cross-bite adjunct with magnetic intrusive treat-                                   four magnetic arrangements: (a) M(Y), which is the moment that
ment. Nevertheless, it is also plausible that the higher inci-                                develops around the X axis at diverse transverse displacement- (b)
                                                                                              M (Y), which is the moment produced around the Y axis at diverse
dents of mandibular lateral shift with AVC (Kiliardis et al.,                                 transverse displacement; (c) M (y), which is the moment that is gener-
1990) over anterior shift (Kalra et al., 1989) are related to di-                             ated around the Z axis at diverse transverse displacement
verse resistance of the facial musculature, i.e., muscles that
counteract jaw protrusion are more resistable than those that                                 force was maintained over a 6-mm range of transverse dis-
counterbalance latero-excursion jaw movements.                                                placement (Figs. 4b, 7a), and no shearing force developed over
    From the four examined magnetic arrangements, only the                                    a 4-mm range of transverse displacement (Figs. 5b, 8). How-
medial eccentric met the criteria of an optimum repulsive                                     ever, the moment analysisfound eccentric arrangements to be
force system to a very limited extent, i.e., a constant intrusive                             more prone tohigh Z-moment than centric arrangements(Fig.
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74        Vardimon et al.                                                                                                                            J Dent Res 73(l) 1994
llc). Considering the fact that Mz is the moment which causes                                Blechman AM (1985). Magnetic forces systems in orthodontics. AmJ
an asymmetrical unilateral intrusion of ipsilateral upper and                                    Orthod 87:201-210.
lower dental quadrants and consequently tilting the occlusal                                 Blechman AM, Smiley M (1978). Magnetic forces in orthodontics. Am
plane, a perplexing point was reached where the force analysis                                  J Orthod 74:435-443.
was in favor of medial eccentric arrangement and the moment                                  Bourauel C, Drescher D, Thier M (1990). Kraft-Momenten-Aufnehmer
analysis was in favor of centric arrangements.                                                   fur die Kieferorthopadie. Feinwerktechni k & Mej?technik 98:419-
    The second disadvantage of repulsive magnetic forces is                                      422.
the rapid decrease of the intrusive force with mouth opening.                                Bourauel C, Drescher D, Thier M (1992). An experimental apparatusfor
Although the decline of FX(Y) as a function of X is polynomial,                                  the simulation of three-dimensional movements in orthodontics.
it drops less rapidly as in previous permanent magnets, i.e., not                               J Biomed Eng 14:371-378.
proportional to the square of the distance (Fig. 10), probably                               Dellinger E (1986). A clinical assessment of the active vertical correc-
due to a geometric magnetic configuration. The fact that, at                                     tor-a nonsurgical alternative for skeletal openbite treatment. Am
gap distances greater than 2 mm, no attractive force devel-                                     J Ort hod 89:428-436.
oped, and that almost a constant intrusive f orce was generated                              Drescher D, Bourauel C, Thier M (1991a). Application of the orthodon-
at gap distances between 3 and 6 mm might suggest a better                                       tic measurement and simulation system (OMSS) in orthodontics.
performance of the repulsive force system when minute spac-                                      Eur] Ort hod 13:169-178.
ing between juxtaposed magnets exists.                                                       Drescher D, Bourauel C, Thier M (1991b). Orthodontisches Meg- und
    Clinically, a repulsive force system is also applied to                                      Simulations-System zur statischen und dynamischen Analyse der
distalized teeth (Gianely et al., 1988,1989), whereby a sectional                                Zahnbewegung. Fortschr Kieferorthop 52:133-140.
arch wire is incorporated into the appliance to eliminate shear-                             Gianely AA, Vaitas AS, Thomas WM, Berger DG (1988). Distalization
ing forces. Correspondingly, it is recommended that Muller                                       of molars with repelling magnets.J Clin Orthod 20:40-44.
prongs be used (Muller, 1962) in conjunction with a magnetic                                 Gianely AA, Vaitas AS, Thomas WM (1989). The use of magnets to
appliance in centric arrangement to intrude posterior teeth and                                  move molars distally. AmJ Ort hod Dentofac Ort hop 96:161-167.
thus bar the influence of the shearing forces. The two Muller                                Kalra V, Burstone C}, Nanda R (1989). Effects of a fixed magnetic
prongs are linked to the maxillary appliance on the palatal side                                 appliance on the dento facial complex. Am J Orthod Dentofac
of the right and left posterior alveolar ridges. They extend                                     Ort hop 95:467-478.
vertically inferiorly reaching contact with the lingual surface                              Kiliaridis S, Egermark I, Thilander B (1990). Anterior open bite treat-
of the mandibular appliance at the the posterior alveolar ridge                                  ment with magnets. EurJOrt hod 12:447-457.
upon closure of the mouth. If the mandibular acrylic is ground                               Melsen B, McNamaraJAJr, Hoenie CD (1991). Effect of biteblocks and
selectively, grooves can be assembled to eliminate both the                                      repelling magnets on root formation of unerupted premolars in
lateral and sagittal shearing forces or only the lateral shearing                                Macaca monkeys. Proc Finn Dent Soc 87:109-114.
force. Thus, in a proper design, repulsive magnetic appliances                               Muller GH (1962). Die Doppelplatte mit Oberkiefer-Spornfuhrung.
not only correct vertical excess (open bite) but also simulta-                                   FortschrKieferorthop 23:243-250.
neously can treat sagittal discrepancies (Cl. II malocclusion).                              Reitan K (1974). Initial tissue behavior during apical root resorption.
                                                                                                 Angle Orthod 44:68-82.
Acknowledgments                                                                              Tsutsui H, Kinouchi Y, Sasaki H, Shiota M, Ushita T (1979). Studies on the
We gratefully acknowledge Engineer M. Beck, Fine Instru-                                     Vardimon AD, Stutzmann JJ, Graber TM, Petrovic AM (1989). Func-
ment Shop, Department of Physiology, University of Bonn,                                        tional orthopedic magnetic appliance (FOMA)II-modusoperandi.
Germany; Engineer A. Bar-Sever Israel; and Mrs. R. Lazar, Sci-                                  Am] Orthod Dentofac Orthop 95:371-387.
entific Editor, School of Dental Medicine, Tel Aviv University,                              Vardimon AD, Graber TM, Voss LR, Muller TP (1990). Functional
Israel.                                                                                         orthopedic magnetic appliance (FOMA) III-modusoperandi. Am
                                                                                               J Ort hod Dentofac Ort hop 97:135-148.
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AkimaHA(1978).Algorithm526.Bivariatedinterpolationandsmooth                                    512.
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   Math Software 4:160-164.                                                                    An experiment in growing baboons. Angle Orthod 58:136-150.
Barbe RE, Sinclair PM (1991). A cephalometric evaluation of anterior                         Wood MG, Nanda RS (1991). Intrusion of posterior teeth with magnets:
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   Angle Orthod 61:93-102.                                                                     Orthop 100:393-400.

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