Human Masticatory Muscle Forces during Static Biting by sanmelody

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      Human Masticatory Muscle Forces during Static Biting
J.C. Nickel, L.R. Iwasaki, R.D. Walker, K.R. McLachlan and W.D. McCall, Jr.
                         J DENT RES 2003 82: 212
                     DOI: 10.1177/154405910308200312

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                                       International and American Associations for Dental Research
 RESEARCH REPORTS
 Clinical

J.C. Nickel1,2, L.R. Iwasaki1,2,3,
R.D. Walker4, K.R. McLachlan5,                                           Human Masticatory Muscle
and W.D. McCall, Jr.6
                                                                         Forces during Static Biting
1 University of Nebraska Medical Center College of
Dentistry, Departments of Growth and Development and
2 Oral Biology, 40th and Holdrege Streets, PO Box

830740, Lincoln, NE 68583-0755, USA; 4 Private
Practice, 3200 North Dobson Rd., Building A, Chandler,
AZ 85224, USA; 5University of Manitoba, Faculty of
Engineering, Department of Civil Engineering, Winnipeg,
MB R3T 2N2, Canada; and 6University at Buffalo School
of Dental Medicine, Department of Oral Diagnostic
Sciences, 355 Squire Hall, Buffalo, NY 14214-3008,                       INTRODUCTION
USA; 3corresponding author, liwasaki@unmc.edu

J Dent Res 82(3):212-217, 2003                                           In humans, muscle recruitmentperiodontalal.,and chewing depends in part
                                                                                                       during biting
                                                                          on bite force direction (van Eijden et
                                                                         information is provided by
                                                                                                                       1990, 1993). Directional
                                                                                                                   ligament mechanoreceptors
                                                                         (Trulsson and Johansson, 1994, 1996). However, it is unknown whether
                                                                         there is a governing neuromuscular objective for the mix of masticatory
ABSTRACT                                                                 muscle forces produced for a given biting situation. Data from living
Muscle forces determine joint loads, but the                             subjects are sparse. The mix of muscle forces determines the magnitudes
objectives governing the mix of muscle forces                            and directions of temporomandibular joint (TMJ) loads (Throckmorton et
involved are unknown. This study tested the                              al., 1990; Trainor et al., 1995), and thus, is clinically important, especially
hypothesis that masticatory muscle forces exerted                        since the likelihood of degenerative joint disease increases with increased
during static biting are consistent with objectives                      TMJ loads (Iwasaki et al., 1997). Recent analysis of the mix of muscle
of minimization of joint loads (MJL) or muscle                           outputs measured in post-orthognathic surgery patients suggested that the
effort (MME). To do this, we compared numerical                          neuromuscular objectives of minimization of joint load (MJL) or
model predictions with data measured from six                            minimization of muscle effort (MME) determined the magnitudes of joint
subjects. Biting tasks which produced moments on                         loads (Nickel et al., 2002).
molar and incisor teeth were modeled based on                                 The aim of this project was to elucidate neuromuscular control of
MJL or MME. The slope of predicted vs.                                   craniomandibular mechanics by testing the hypothesis that masticatory muscle
electromyographic (EMG) data for an individual                           forces, in response to static biting which produces moments on molars and
was compared with a perfect match slope of 1.00.                         incisors, can be predicted on the basis of MJL or MME, or both. To test this
Predictions based on MME matched best with                               hypothesis, we compared data from living subjects with numerical model
EMG activity for molar biting (slopes, 0.89-1.16).                       predictions of muscle forces calculated from the three-dimensional geometry
Predictions from either or both models matched                           of the individual and an objective function of MJL or MME.
EMG results for incisor biting (best-match slopes,
0.95-1.07). Muscle forces during isometric biting
appear to be consistent with objectives of MJL or                        MATERIALS & METHODS
MME, depending on the individual, biting                                 One female and five male subjects, aged 23 to 32 yrs, participated. Informed
location, and moment.                                                    consent was obtained, and the project was approved by the appropriate
                                                                         Institutional Review Board.
KEY WORDS: numerical modeling, TMJ, loading,
electromyography.                                                        Anatomical Geometry
                                                                         We developed a three-dimensional geometry for each individual using
                                                                         standardized lateral and postero-anterior cephalometric radiographs to
                                                                         determine the relative positions of the masticatory muscles, condyles, and
                                                                         tooth row (Fig. 1A). The x-, y-, and z-coordinates were determined for
                                                                         centroids of the origins and insertions of 5 muscle pairs (masseter, anterior
                                                                         temporalis, medial pterygoid, lateral pterygoid, and anterior digastric
                                                                         muscles), the center supero-anterior point on the mandibular condyles, and
                                                                         positions of the mandibular central incisors, canines, and first molars.
                                                                         Preliminary studies utilizing cadaver material aided protocol development
                                                                         for soft- and hard-tissue coordinate identification in living subjects, with the
                                                                         use of these radiographs. Maximum measurement errors for all coordinates,
Received December 5, 2001; Last revision October 30,
                                                                         from serial radiographic tracings, were + 3.5 mm. This geometry was used
2002; Accepted November 7, 2002
                                                                         in computer models for the prediction of effective sagittal TMJ eminence
A supplemental appendix to this article is published                     morphology and the calculation of muscle and joint forces for unilateral
electronically only at http://www.dentalresearch.org.                    molar and centered incisor biting.

212
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                                                             International and American Associations for Dental Research
J Dent Res 82(3) 2003                                         Muscle Forces During Static Biting                                                                213




Figure 1. Mechanical schemes. (A) Force vectors involved in numerical
models of isometric biting in humans. Forces on the mandible (e.g.,
vertical bite force), at the joints (F condyle, R = right, L = left), and
representing 5 muscle pairs (m 1,2 = masseter, m 3,4 = anterior
temporalis, m5,6 = lateral pterygoid, m7,8 = medial pterygoid, m9,10 =
anterior digastric muscles), plus the axis system, are shown. (Modified
from Smith et al., 1986). (B) Equivalent moments for in vivo (1) and
modeled (2) molar ( + M x ) and incisor ( + M z ) biting tasks. Line
diagrams illustrate mesial views of mandibular right first molar (top)
and left central incisor (bottom) with acrylic crowns in place, where: CR
is the center of resistance of the tooth/teeth, ¡ is the moment arm
                                                   r
vector, y is the angle away from vertical, and xz is in a plane
parallel to the occlusal plane.


Modeled Eminence Shape                                                                          We used an optimization strategy of either MJL or MME to
                                                                                                calculate muscle and joint forces for any prescribed mandibular
The effective sagittal TMJ eminence morphology was defined
                                                                                                position, and position and direction of mandibular load. The
as the shape of hard- and soft-tissue structures articulating with
                                                                                                optimizing process was achieved by means of quadratic
the mandibular condyle in sagittal view. We used a custom
                                                                                                programming. Companion algorithms were constructed for data
program with optimization based on unconstrained MJL and the
                                                                                                management and systematic and sequential testing of possible
subject's geometry to predict the effective sagittal eminence
                                                                                                force combinations until the designated objective was satisfied for
morphology (Trainor et al., 1995). In brief, it operated as
                                                                                                the given biting situation (Trainor et al., 1995).
follows. Joint load directions were calculated for a series of
                                                                                                     Modeled conditions produced moments on the molar and
bilateral vertical bite forces applied in 20 steps from first
                                                                                                incisor teeth that were similar to those tested in vivo (see below).
molars to central incisors. The mandibular position, consistent
                                                                                                Bite forces of 100 units, at angles from vertical ( y) of 0 to 30º,
with bite force position for each calculation, ranged from
                                                                                                were loaded unilaterally on mandibular first molars at azimuth
relatively retruded (molar biting) to protruded (incisor biting).
                                                                                                angles ( xz) of -90 and 90º, and centrally on the incisors at xz of 0
For equilibrium, the joint load must be directed perpendicular to
                                                                                                and 180º (Fig. 1B). The mandibular position during each biting task
the effective eminence. Therefore, the effective eminence shape
                                                                                                was simulated based on in vivo data (e.g., Appendix Tables A1, A2).
for an individual was delineated by a series of 20 lines, each
perpendicular to the predicted joint load direction for a given
                                                                                                Experimental Eminence Shape
step, to which a cubic polynomial was fitted.
                                                                                                In vivo effective sagittal eminence morphology was determined by
Modeled Muscle Forces                                                                           a simplified method of tracking the condylar pathway using a
Muscle forces were predicted by 2 three-dimensional models with                                 modified mandibular orthodontic retainer with a facebow attached.
different objective functions: (a) minimization and equalization of                             Colored markers on the facebow were positioned opposite palpated
right and left TMJ loads and (b) minimization of the sum of                                     lateral poles of both mandibular condyles, and recorded by means
squared muscle force magnitudes. In both models, the sagittal                                   of video cameras set up directly lateral to the subject. Graph paper
eminence shape was determined as described above. This                                          affixed medially to the markers provided a reference system and
eminence shape constrained the sagittal component ( x , y) of the                               orthogonal axes for determining scale.
joint load directions (perpendicular to the predicted eminence),                                     A set of 10 protrusion/retrusion movements, with upper and
while maintaining an independent mediolateral component ( z).                                   lower teeth separated by 1 to 3 mm, was recorded at each of 2

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                                                               International and American Associations for Dental Research
214                                                                            Nickel et al.                                                              J Dent Res 82(3) 2003

                                                                                                                                                  sessions, at least 3 days apart.
                                                                                                                                                  Recordings were viewed frame-
                                                                                                                                                  by-frame, and the condylar
                                                                                                                                                  marker position was traced on
                                                                                                                                                  acetate mounted on the video
                                                                                                                                                  screen. Points along the path of
                                                                                                                                                  the marker were quantified in two
                                                                                                                                                  dimensions with occlusal plane
                                                                                                                                                  and a perpendicular line as axes.
                                                                                                                                                  The occlusal plane was chosen as
                                                                                                                                                  a reference axis, given its
                                                                                                                                                  importance in the mechanics of
                                                                                                                                                  biting. A custom computer pro-
                                                                                                                                                  gram was used to correct for
                                                                                                                                                  scale, to record x- and y-coordin-
                                                                                                                                                  ates for each point, and to
                                                                                                                                                  calculate a best-fit third-order
                                                                                                                                                  polynomial which delineated the
                                                                                                                                                  effective eminence shape. Intra-
                                                                                                                                                  and inter-session measurement
                                                                                                                                                  errors in condylar position were,
                                                                                                                                                  on average, + 0.3 mm or + 10%
                                                                                                                                                  of full scale.
                                                                                                                                                  Experimental Biting Tasks
                                                                                                                                                  Biting tasks were performed
                                                                                                                                                  during 2 or 3 recording sessions,
                                                                                                                                                  at least 3 days apart. Static biting
                                                                                                                                                  occurred on a small steel ball
                                                                                                                                                  between acrylic crowns cemented
                                                                                                                                                  to maxillary and mandibular first
                                                                                                                                                  molars and central incisors. Each
                                                                                                                                                  acrylic crown had a flat
                                                                                                                                                  occlusal/incisal surface that was
                                                                                                                                                  approximately parallel to the
                                                                                                                                                  occlusal plane. For mandibular
                                                                                                                                                  crowns, these surfaces each had
                                                                                                                                                  three small depressions, 3 to 4
                                                                                                                                                  mm apart, which ensured that
                                                                                                                                                  bolus position was consistent
                                                                                                                                                  relative to the center of resistance
                                                                                                                                                  of loaded teeth, and a range of
                                                                                                                                                  mechanical moments was tested
                                                                                                                                                  (Fig. 1B). The depressions were
                                                                                                                                                  parallel to the z-axis on man-
                                                                                                                                                  dibular molar crowns, and the x-
                                                                                                                                                  axis on mandibular incisor
                                                                                                                                                  crowns. Incisor crowns had extra-
                                                                                                                                                  oral reference markers attached,
                                                                                                                                                  so that changes in mandibular
                                                                                                                                                  position could be monitored
                                                                                                                                                  directly or from video images.
                                                                                                                                                  Each subject produced a moderate
                                                                                                                                                  static bite force for 5 sec, 5 times
                                                                                                                                                  at each of 3 positions on molars
Figure 2. Effective TMJ eminence results. (A) Eminence height (mm), predicted by the model vs.                                                    and incisors, with 20 seconds' rest
measured, for a given postero-anterior position from 0 to 5.0 mm anterior to a retruded condylar                                                  between bite positions.
position, in 0.5-mm increments, for subject m1 ("worst-case", N‚) and subject m4 ("best-case", .). (B)
                                                                                                                                                        Bipolar surface electro-
Sagittal morphology of the effective TMJ eminence in each of the six subjects. The vertical axis
represents the eminence height (mm) relative to a retruded condylar position and perpendicular to the
                                                                                                                                                  myographic (EMG) data were
occlusal plane, and the horizontal axis represents condylar protrusion (mm) from the retruded position.                                           measured as surrogates for forces
Function beyond 5 mm of protrusion was unlikely, since this placed mandibular incisors anterior to                                                during biting tasks. The muscle
maxillary incisors in all subjects.                                                                                                               bulk center was located by

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                                                             International and American Associations for Dental Research
J Dent Res 82(3) 2003                                       Muscle Forces During Static Biting                                                           215

palpation for masseter, anterior
temporalis, and anterior digastric
muscles. The skin over each
muscle was debrided, and paired
electrodes (E5SH Standard
Silver®, 7 mm, Grass Instrument
Company, Quincy, MA, USA)
with conducting paste (Liqui-
Cor ® , Burdock Corp., Milton,
WI, USA) were held in place
with adhesive pads. Surface
impedances were less than 30
k , applied input impedance was
20 M , and band width was 0.1
to 3 kHz. Muscle activity was
amplified (P511 AC Grass
Preamplifiers®, Astor-Med Inc.,
West Warwick, RI, USA),
viewed in real-time with
commercial software (PCScan
MKII PCIF250NI Real-time
Data Transfer System ® , Sony
Magnescale America Inc.,
Farmington Hills, MI, USA), and
stored digitally (Sony PC-216A
16 Channel Recorder®, Spectris
Technologies Inc., Decatur, GA,
USA). Signal-to-noise ratios
were better than 8 to 1.
     Muscle output data were
replayed and analyzed with
commercial software (Microsoft
Excel ® , Microsoft Corp.,
Seattle, WA, USA). Muscle
activities over a two-second
period were sampled at 1.7 kilo-
samples/second/channel and
expressed as root-mean-square
(RMS) values (mV).
Eminence Shape
Comparisons
Predicted and measured effective
eminence shapes for each subject
were compared by means of the
polynomial equations for deter-
mination of y-axis coordinates,
representing eminence height
                                         Figure 3. Examples of model results for biting tasks. (A) Predicted I/C muscle force ratios for molar biting
(mm), and for x-axis coordinates,
                                         tasks at a range of bite force angles that created moments equivalent to in vivo biting at lateral, center, and
representing anteroposterior             medial positions. MJL model predictions for masseter ( ) and anterior temporalis ( ) muscles and MME
condylar positions over the range        model predictions for masseter ( ) and anterior temporalis (X) muscles are shown for subject f1. Please note
x = 0 to 5 mm in 0.5-mm increm-          the discontinuity of scale in the vertical axis. (B) Predicted normalized muscle forces for incisor biting tasks at
ents. Predicted and measured y-          a range of bite force angles that created moments equivalent to in vivo biting at posterior, center, and
                                         anterior positions. Masseter muscle force predictions from MJL model ( subject f1, N subject m5) and from
axis coordinates were plotted for        MME model ( subject f1, L subject m5), and anterior temporalis muscle force predictions from MJL model
each subject, and a linear correla-      ( subject f1, I subject m5) and from MME model ( subject f1, G subject m5).
tion coefficient was calculated.

Muscle Output Comparisons                                                                     use of ratios obviated the need for bite force measurements and
Predicted and measured muscle outputs for the biting tasks were                               force-EMG calibration for individual muscles. At each of 3 molar
compared. We calculated ipsilateral-to-contralateral (I/C) muscle                             bite positions, RMS values from 5 bites were averaged, and I/C
output ratios to compare molar biting results quantitatively. This                            muscle output ratios were calculated. For example, the I/C


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                                                             International and American Associations for Dental Research
216                                                                                 Nickel et al.                                                           J Dent Res 82(3) 2003

Table. Slopes and R2 Values for the Relationship between Predicted and                             eminence shapes for each subject showed slopes from 0.87 to
Measured Masseter and Anterior Temporalis Muscle Outputs for                                       1.13, and a range of correlations, 0.92 < R2 < 0.99 (Fig. 2A).
Equivalent Biting Moments (e.g., Appendix Fig.)                                                    The mean slope was 0.94 with a standard deviation (SD) of
                                                                                                   0.12. An independent t test showed no significant difference
                  Model vs. Measured Muscle Outputs for:                                           between predicted and experimental results for eminence
                MJL Model              MME Model         Number of                                 shape. However, in vivo eminence shapes showed inter-subject
Subject       Slope     R2           Slope       R2      Data Points                               variability, where the individual slopes of fitted polynomials
                                                                                                   differed by up to 3 to 1 (Fig. 2B).
                                  Molar Biting
                                                                                                   Muscle Outputs
m1a           1.88         0.33              1.03b            0.96              6                  Anterior digastric muscle force predictions and RMS values
m2            6.10         0.74              0.93b            0.96             12                  were similar for the biting tasks investigated, and near or at
m3           10.20         0.69              0.92b            0.91             17                  baseline levels. The 2 models predicted quite different I/C
f1            1.60         0.96              0.89b            0.92             12                  masseter and anterior temporalis muscle force ratios for
m4            0.19         0.52              0.92b            0.92             12                  moments resulting from lateral and medial molar biting angles
m5            2.30         0.52              1.16b            0.94             12                  (Fig. 3A). Plots of predicted vs. measured muscle output during
                                                                                                   molar biting were compared for the models (Table). All
                                  Incisor Biting                                                   subjects' molar biting I/C muscle output ratios were predicted
                                                                                                   better according to the MME model (mean slope, SD: 0.98,
m1a           0.95c        0.58              0.95c            0.58              6                  0.10) compared with the MJL model (mean slope, SD: 3.71,
m2            1.04c        0.63              0.79             0.76             12                  3.74). Similarly, plots of predicted vs. measured normalized
m3            0.10         0.09              1.00c            0.73             12                  masseter and anterior temporalis muscle output values for
f1            0.95c        0.78              0.95c            0.78             12                  incisor biting were compared for the two models (Table). The
m4            0.20         0.07              0.96c            0.51             12                  predictions from the MJL and MME models were the same for
m5            1.07c        0.73              0.69             0.65             12                  two subjects (e.g., f1, Fig. 3B) and matched well with the EMG
a
                                                                                                   data for these subjects. Predictions from the 2 models were not
      Data from biting task session prior to facial trauma.                                        the same for the other four subjects (e.g., m5, Fig. 3B). The
b,c   Best-match model results with experimental data.
                                                                                                   MJL model results matched best with the EMG data from two
                                                                                                   of these subjects, while the MME model results matched best
masseter muscle output ratio for lateral biting on right and left first
                                                                                                   with the EMG data from the two remaining subjects. Slopes
molars was calculated as:
                                                                                                   from plots of predicted vs. measured results for molar and
                                                                                                   incisor biting (Table) were compared with perfect-match slopes
                 Imasseter Imasseter x Imasseter                                                   of 1.00. Separate independent t tests showed no significant
                 ______ = __________________
                                    R            L

                 Cmasseter Cmasseter x Cmasseter                                                   differences between best model-predicted and experimental
                                         R                L
                                                                                                   data for each biting condition (p > 0.05). However, post hoc
                                                                                                   power analyses indicated that sample sizes were too small for
where I and C were the ipsilateral and contralateral muscle output
                                                                                                   statistically significant differences between predicted and
for lateral molar biting, and R and L indicated right and left sides.
                                                                                                   measured values to be excluded.
In vivo intra- and inter-sessional ratio variabilities were, on
average, + 13%. Predicted vs. measured I/C muscle output ratios
for biting moments were plotted for each subject. Linear                                           DISCUSSION
regressions were calculated and slopes were tested for significant                                 Computer-generated models based on human anatomy render
differences from a perfect-match slope of 1.00.                                                    solutions based on an optimal strategy or objective function,
      Incisor biting tasks were centered on the mandible, so RMS                                   which is of biological importance and thus represents a theory
values for right and left muscles were averaged for 5 bites at each                                of underlying neuromuscular control (Pruim et al., 1980;
of 3 bite positions. These data for each subject and recording                                     Osborn and Baragar, 1985, 1992; Smith et al., 1986; Koolstra
session were normalized relative to the peak RMS value for a                                       et al., 1988; Herzog and Binding, 1994; Koolstra and van
given muscle. Similarly, predicted muscle forces for incisor biting                                Eijden, 1995, 1997, 1999; Osborn, 1995; Trainor et al., 1995;
were normalized relative to the peak predicted muscle force for the                                Buchanan and Shreeve, 1996; Itoh et al., 1997; Raikova, 1999).
subject and muscle. Intra- and inter-sessional normalized RMS                                      A neuromuscular objective of MJL is of interest because
value variabilities were, on average, + 14%. Predicted vs.                                         cartilage has limited repair potential, and theoretically, this
measured normalized muscle output values for biting moments                                        objective reduces the likelihood of mechanical fatigue.
were plotted for each subject. Linear regressions were calculated                                  However, MJL tends to increase muscle forces for given biting
and slopes were tested for significant differences from a perfect-                                 tasks, theoretically increasing the likelihood of muscle fatigue.
match slope of 1.00.                                                                               In contradistinction, an objective of MME reduces muscle
                                                                                                   forces for a biting task, but increases joint loading, especially in
RESULTS                                                                                            the TMJ contralateral to a molar bite force. The data presented
                                                                                                   here are the first, to our knowledge, to test model predictions of
Eminence Shapes                                                                                    masticatory muscle output for changes in biting moments in
The maximum vertical difference between predicted and                                              individual subjects. Variance seen between predicted and
experimental results for the group was 0.4 mm for protrusion                                       experimental data is likely due to factors such as the use of
up to 5 mm. Plots of predicted vs. measured effective sagittal                                     synthetic crowns, a static biting task rather than routine

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                                                                  International and American Associations for Dental Research
J Dent Res 82(3) 2003                                       Muscle Forces During Static Biting                                                                     217

masticatory activity, and the limitations of the numerical                                    Koolstra JH, van Eijden TM (1997). The jaw open-close movements
model, which necessitated that the in vivo biting tasks be                                        predicted by biomechanical modelling. J Biomech 30:943-950.
represented by bite force positions that would produce                                        Koolstra JH, van Eijden TM (1999). Three-dimensional dynamical
equivalent moments, but not identical force vectors.                                              capabilities of the human masticatory muscles. J Biomech 32:145-
Experimental evidence comparable with model predictions                                           152.
from the current study has been reported (van Eijden et al.,                                  Koolstra JH, van Eijden TM, Weijs WA, Naeije M (1988). A three-
1993). Human I/C masseter muscle peak EMG ratios calculated                                       dimensional mathematical model of the human masticatory system
from unilateral gum-chewing data ranged from 1.18 to 2.66. In                                     predicting maximum possible bite forces. J Biomech 21:563-576.
the current study, mean predicted I/C masseter muscle force                                   Nickel JC, Yao P, Spalding PM, Iwasaki LR (2002). Validated
ratios were 1.5 (SD 0.7) for non-vertical bite forces.                                            numerical modeling of the effects of combined orthodontic and
    In summary, predictions of muscle recruitment were tested                                     orthognathic surgical treatment on TMJ loads and muscle forces.
with experimental data. The mix of masticatory muscle outputs                                     Am J Orthod Dentofacial Orthop 121:78-83.
during isometric unilateral molar biting and centered incisor                                 Osborn JW (1995). Biomechanical implications of lateral pterygoid
biting was generally consistent with MJL or MME for the                                           contribution to biting and jaw opening in humans. Arch Oral Biol
individuals investigated. The specific objective function                                         40:1099-1108.
involved appeared to depend on the individual's anatomy and                                   Osborn JW, Baragar FA (1985). Predicted pattern of human muscle
the specific location and direction of bite force. These factors                                  activity during clenching derived from a computer assisted model:
are the focus of further inquiry.                                                                 symmetric vertical bite forces. J Biomech 18:559-612.
                                                                                              Osborn JW, Baragar FA (1992). Predicted and observed shapes of
ACKNOWLEDGMENTS                                                                                   human mandibular condyles. J Biomech 25:967-974.
                                                                                              Pruim GJ, de Jongh HJ, ten Bosch JJ (1980). Forces acting on the
Mr. Kim Theesen, Illustrator, Learning Resources, UNMC                                            mandible during bilateral static bite at different bite force levels. J
College of Dentistry, is acknowledged for his help with the                                       Biomech 13:755-763.
figures. The authors thank the subjects who participated in the                               Raikova R (1999). About weight factors in the non-linear objective
study. This project was supported by the UNMC Orthodontic                                         functions used for solving indeterminate problems in
Development Fund. This paper is based on a thesis submitted                                       biomechanics. J Biomech 32:689-694.
to the Faculty of Graduate Studies, University of Nebraska, in                                Smith DM, McLachlan KR, McCall WD Jr (1986). A numerical model
partial fulfillment of the requirements for the Master of Science                                 of temporomandibular joint loading. J Dent Res 65:1046-1052.
degree for R.D. Walker.                                                                       Throckmorton GS, Groshan GJ, Boyd SB (1990). Muscle activity
                                                                                                  patterns and control of temporomandibular joint loads. J Prosthet
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