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Mandibular autorotation in orthognathic surgery: a new method of locating the centre of mandibular rotation and determining its consequence in orthognathic surgery
Mandibular autorotation in orthognathic surgery: a new method of locating the centre of mandibular rotation and determining its consequence in orthognathic surgery Anders Nattestad, Poul Vedtofte
Department of Oral and Maxillofacial Surgery (Head: Prof. dr. E. Hjorting Hansen, DDS, PhD), Royal Dental College and University Hospital (Rigshospitalet), Copenhagen, Denmark.
S U M M A R Y. Rotation of the mandible is simulated through the centre of the condyle in planning orthognathic surgery. Previous studies have suggested that the initial mandibular movement is better characterized as multiple parts of the segments of a circle with the average centre located below and behind the centre of the condyle. This paper describes a method of locating the centre of mandibular rotation by computer-analysis of two lateral cephalograms with different degrees of opening. The method was used on 10 normal individuals showing an average centre of mandibular rotation 14.9 mm below and 5.0 mm behind the superior midsurface of the condyle for movements from occlusion to an opening of 10 ram. The implications of the located centre of rotation on orthognathic surgery was determined by simulating a closure of a 9.5 mm open bite. An error of up to 9.3 mm in the horizontal position of the maxilla would occur if the centre of the condyle was used. A retrospective study on l0 patients with increased anterior facial height was undertaken and a mean centre of rotation 8.4 mm behind and 25.1 mm below the centre of the condyle was found. Simulating the rotation of the mandible through the centre of the condyle would have resulted in a difference between the predicted and actual horizontal position of the maxilla ranging from 0.4-10.4 ram. It was concluded that using the centre of the condyle as the centre of mandibular rotation in the planning of superior maxillary movement with a Le Fort I osteotomy may cause considerable error in the horizontal position of the maxilla in most cases.
KEY W O R D S : Le Fort 1 0 s t e o t o m y -
Centre of mandibular r o t a t i o n - Prediction t r a c i n g - Articulator
used today. Fischer (1952, 1956) presented a method different from all others, using photographs taken of the moving mandible. The method, which was later modified by Chick (1960), involved taking a photograph of the mandible during opening with a camera using a long exposure time. The centre of mandibular rotation was calculated by a geometric principle, but due to inaccuracy in measurements on the picture and the time-consumption involved, the method never gained clinical application. Another method is the kinematic one, which has already been described by McCollum in 1939, but has recently been revived and modified. The method determ~-nes the centre of mandibular rotation by analysis of the pattern of mandibular movement on the individual patient and is claimed to be very precise (Bosman, 1974). The method requires the mounting of a rather extensive apparatus on the moving mandible, and an investigation has questioned the accuracy of the method (Winstanley, 1979). A recent biomechanical study by Baragar and Osborn (1984) presented a three-dimensional computer simulation model of the masticatory system and concluded that the mandible would rotate around a point 50 mm inferior and 25 mm posterior to the centre of the condyle, at an opening of a few mm. During greater opening movements the centre was found to move in an anterior and superior direction
INTRODUCTION
The centre of the condyle is one of the reference points registered in the planning of Le Fort I osteotomy by a face-bow recording (Hohl, 1978; Marko, 1986). A trial-operation is performed on dental casts in an articulator to establish the precise movement of the jaws necessary to obtain normal occlusion and the desired changes in facial morphology. The centre of the condyle is presumed to represent the centre of mandibular rotation (Bell et al., 1980; Turvey et al., 1982). Several methods of localizing the centre of mandibular rotation have been presented. The first method to gain overall acceptance was the arbitrary method which, based on a concept of human uniformity, used a point 13 mm in front of the external auditory meatus on a line to the lateral corner of the eye (Schallhorn, 1957). A later study has shown the arbitrary method to be quite inaccurate (Walker, 1980). Later, the method of choice was the palpatory method, where the centre of the condyle was located by palpation (Bosman, 1974). This method assumes that the centre of the condyle represents the centre of mandibular rotation, but other studies have shown this to be quite disputable (Grant, 1973; Brewka, 1981; Baragar and Osborn, 1984; Torii, 1989). Despite these findings, the palpatory method is the clinical method most widely 163
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Journal of Cranio-Maxillo-Facial Surgery
on a line ending about 10 mm in front of the original position of the centre of the condyle. The importance of accurate determination of the centre of mandibular rotation has been documented by Sperry et al. (1982) and by Rekow et al. (1985). Sperry et al. (1982) determined the centre of mandibular autorotation in 23 patients treated by Le Fort I osteotomy of the maxilla. The superior rotation of the mandible, due to maxillary impaction, occurred through a centre in the mastoid region, posterior and inferior to the centre of the condyle. Rekow et al. (1985) investigated the error in occlusion following splint removal and found an insignificant occlusal error, probably due to the small amount of rotation taking place on splint removal. Previous studies have shown a significant difference between the planned and immediate postoperative position of the maxilla after Le Fort I osteotomy, despite the introduction of new extraoral measurement techniques (Pospisil, 1987; Stanchina et al., 1988; Polido et al., 1990; Kahnberg et al., 1990). A study by Nattestadet al. (1991) has demonstrated, that a discrepancy between the centre of the condyle and the centre of mandibular rotation in the range 10-20 mm can have a clinically significant influence on the postoperative result of orthognathic surgery. The significance of this finding is however dependent upon the ability to locate the centre of mandibular rotation in the individual patient undergoing surgery. The purpose of this study was to develop a method for localizing the centre of mandibular rotation and to determine the implication for orthognathic surgery.
MATERIAL AND METHOD Equipment An IBM ~ compatible 386 computer with Super VGA display and a Summagraphics Supergrid ® (with backlight) was used for the measurements. The resolution of the digitizer was 0.025 mm, with an accuracy of 0.1 mm on repeated measurements. The software used was Autocad ® version 9.01n and SAS ® Statistical Analysis System.
Method-accuracy and description of method A natural size sketch of the mandible was firmly mounted in the middle of the digitizer. The sketch had 5 points marked with crosses placed within the body of the mandible. The centre of rotation of the sketch was secured by placing a needle through this point. With the sketch in a neutral position, the 5 points and the centre of rotation (the needle) were traced. The sketch was rotated 5, 10, 15, 20 or 25 ° and the 5 points in the body of the mandible were traced again. The centre and amount of rotation was determined by the principle of least squares. The appendix explains the calculations to determine the centre and amount of rotation, that brought the points before rotation closest possible to the points after rotation. The calculations were performed in the statistical analysis
program SAS. The registration procedure and the calculations were performed 5 times for each rotation of 5, 10, 15, 20 and 25 °.
Clinical study 10 normal individuals with no history of TMJ symptoms participated in the study. Initially, the palpatory centre of the condyle was marked with lead (a' x ' on the right side and a ' + ' on the left side were fixed with tape). Each participant had two lateral cephalograms taken in immediate succession: one in active retruded occlusion and one at 10 mm opening, also in an active retruded position of the condyles. The participants were placed in the natural head posture in the cephalostat with ear pins and a light cross attempting a left-right symmetry of the face. The cephalograms were placed similarly on the digitizer, using superimposition on the cranial base and calvarium. On each cephalogram the superior midsurface of the condyles, the palpatory centres and 10 reference points were digitized. The reference points were distributed over the entire cephalogram and transferred to the second cephalogram by superimposing on easily recognizable structures in the mandible. The centre of rotation, size of movement and superimpositional error was determined for each individual using equations 1-18 (see Appendix). The superior mid-surface of the condyle is described by Turvey et al. (1982) and was chosen as the reference-point in the condyle because it was a more specific reference point than the usual geometric centre of the condyle. The point, which is later referred to as the radiographic centre of the condyle, was determined as the middle of the superior surface of the condyle, determined by two parallel tangents to the anterior and posterior outline of the condylar head. The superimposition, registration and calculation procedure were made twice on each individual in order to establish the overall accuracy of the method. The entire procedure was repeated with new cephalograms for 5 of the individuals in order to establish the reproducibility. The condylar movement during opening was calculated from the registered positions of the superior midsurface of the condyle. A cephalometric analysis was performed on the 10 individuals comprising: sagittal position of the maxilla (s-n-ss) and mandible (s-n-pg), inclination of the maxilla ( N S L / N L ) and mandible ( N S L / M L ) relative to the cranial base (NSL), maxillary inclination relative to mandibular inclination ( N L / M L ) , flexion of cranial base (n-s-ba), gonial angle (ML/RL), mandibular t-angle (MBL/ML), mandibular basis (pg-tgo) and ramus length (cd-tgo).
Retrospective study This material comprised of 10 patients with increased anterior facial height treated by Le Fort I osteotomy without concomitant osteotomy in the mandibular ramus. The criteria for inclusion in the study was the
Mandibular autorotation in orthognathic surgery 165 existence of lateral cephalograms taken immediately before and 2-5 months after surgery. The splint had in all cases already been removed at this time. The Le Fort I osteotomy was performed through a horizontal incision in the buccal sulcus extending posterior to the first molars. The maxilla was downfractured, mobilized and placed in the preplanned position using a composite splint and intraoral vertical and horizontal reference lines on the anterior wall of the maxilla. The centre of mandibular rotation was determined retrospectively by computer-analysis of the pre- and postoperative lateral cephalograms. The possible consequence of the difference between the calculated centre of rotation and the centre of the condyle for the horizontal position of the maxilla at surgery was simulated using a set of equations, which has recently been described (Nattestad et al., 1991).
S T A T I S T I C A L ANALYSIS
Method-accuracy The overall variation in determining the centre of rotation in the non-clinical situation was evaluated by mean and standard error of the mean for each of the 5 rotations (5, 10, 15, 20 and 25°). A Wilcoxon nonparametric signed rank test was used to evaluate whether the position of the calculated centre of rotation was significantly different from the actual centre of rotation.
Clinical study A Wilcoxon non-parametric signed rank test was performed to evaluate if the calculated centre of mandibular rotation deviated significantly from the radiographic- or palpatory centre of the condyle. The condylar movement, measured at the superior midsurface of the condyle, was compared to the position of the calculated centre of rotation with a correlation and regression analysis. The statistical significance of the calculated consequence for orthognathic surgery was evaluated with a Wilcoxon non-parametric signed rank test. The results of the cephalometric analysis were compared to the calculated consequence for orthognathic surgery with a Wilcoxon non-parametric signed rank test. Retrospective study A Wilcoxon non-parametric signed rank test was performed to evaluate if the calculated centre of mandibular rotation deviated significantly from the
radiographic centre of the condyle. The condylar movement, measured at the superior midsurface of the condyle, was compared to the position of the calculated centre of rotation With a correlation and regression analysis.
RESULTS
Method-accuracy The method accuracy was described for rotations of 5, 10, 15, 20 and 25 ° as a horizontal and a vertical difference in mm between the actual and calculated centre of rotation. The mean absolute horizontal difference was found to range from 0.05-0.47 mm with a standard error of mean ranging from 0.07-0.25 ram. The mean absolute vertical difference ranged from 0.01 0.32 mm with a standard error of the mean ranging from 0.08-0.32 ram. The mean distinct between the two centres ranged from 0.36-0.71 mm with a standard error of the mean ranging from 0.03 0.23 mm. Generally the precision of the method improved with larger rotations. No statistically significant difference was found between the actual and calculated centre for any of the 5 rotations (Table 1).
Clinical study The 10 individuals showed an average centre of rotation 14.9ram below and 5.0ram behind the radiographic centre of the condyle (Table 2A, Fig. 1). The inter-individual variation was large both in the horizontal and vertical direction with a standard error of the mean of 3.5 ram. No significant horizontal difference was found between the coordinate position of the calculated centre of rotation and the radiographic centre of the condyle (p = 0.08). The vertical difference however showed a significant lower position of the calculated centre of rotation (p --- 0.004). The difference between the radiographic and the palpatory centre of the condyle was found to be more uniform with the average palpatory centre of the condyle located 9.3 mm in front of and 11.4 mm below the radiographic centre. The standard error of the mean was 0.6 and 0.9 mm respectively. This divergence was highly significant both in the horizontal (p = 0.002) and vertical (p = 0.002) direction. When the calculated centre of mandibular rotation was described relative to the palpatory centre of the condyle, a position 14.3 mm behind the 3.5 mm below the palpatory centre of the condyle was found. The standard error of the mean was 3.2 and 3.9 mm respectively, which was significant for the horizontal
Table 1- Analysis of method error: difference between the known centre of rotation and the calculated centre. The horizontal deviation is measured on the x-axis, the vertical deviation on the y-axis. The distance is the direct span between the two centres Rotation degrees
N
Horizontal deviation mm s.e. p
Vertical deviation mm s.e. p
Distance mm s.e.
p
5 10 15 20 25
5 5 5 5 5
0.05 -0.37 -0.47 -0.11 -0.23
-0.21 0.01 0.32 0.25 0.11
0.71 0.67 0.65 0.48 0.36
0.06 0.06 0.06 0.06 0.06
0.25 0.19 0.10 0.14 0.07
0.63 0.13 0.06 0.63 0.06
0.32 0.25 0.12 0.18 0.08
0.55 1.00 0.13 0.31 0.13
0.23 0.14 0.03 0.10 0.03
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Journal of Cranio-Maxillo-Facial Surgery
Table 2A - Results of the study involving 10 normal individuals showing the distances between the radiographic centre of the condyle
(RCOC), the palpatory centre of the condyle (PCOC), the calculated centre of mandibular rotation (CCR) and the degree of opening and the condylar movement during opening. The resulting horizontal error in the position of the maxilla was calculated from a set of equations previously presented (Nattestad et al., 1991). The result of a Wilcoxon non-parametric signed rank test is shown at the bottom line, where relevant
Cases
RCOC to COR (ram) X Y
1 2 3 4 5 6 7 8 9 10
30.2 6.7 -14.0 0.0 8.4 1.9 9.1 3.5 3.1 1.2
Average Std. error P
--37.2 -9.9 --19.4 --17.1 --22.2 -11.2 -15.7 3.7 -16.2 -3.8
5.0 - 14.9 3.5 3.5 0.08 0.004
PCOC to CCR (ram) X Y 36.3 --25.8 17.3 6.6 --4.4 -6.4 9.6 -8.1 16.7 -14.0 10.6 3.4 17.6 -6.1 15.7 16.1 10.1 - 9 . 2 13.6 8.8 14.3 - 3 . 5 3.2 3.9 0.004 0.49
RCOC to PCOC (ram) X Y
Opening deg.
Condylar movement X Y
Horizontal error (mm)
-6.2 -10.6 --9.6 --9.6 --8.3 --8.7 --8.5 -12.2 --7.1 -12.3
6.4 7.1 5.5 6.0 7.1 8.8 10.9 6.7 7.0 7.4
-3.9 -0.7 -1.9 -1.8 -2.4 -1.8 -2.6 0.2 -1.1 -0.1
9.3 0.7 0.4 1.8 3.5 0.5 2.5 -0.9 2.1 0.0
- 9 . 3 - 11.4 0.6 0.9 0.002 0.002
Table 2B - Results of the retrospective study involving 10 surgical cases showing the distance between the radiographic centre of the condyle (RCOC) and the calculated centre of mandibular rotation (CCR), and the degree of opening and the condylar movement during opening. The resulting horizontal error in the position of the maxilla was calculated from a set of equations previously presented (Nattestad et al., 1991). The result of a Wilcoxon non-parametric signed rank test is shown in the bottom line, where relevant
Case
RCOC to CCR Condylar (mm) Open- movement X Y ing deg. X Y
Horizontal error (mm)
1 2 3 4 5 6 7 8 9 I0 Average Std. error P
ll.l 27.3 --17.6 27.7 19.9 10.0 -2.8 21.0 7.4 -19.7 8.4 5.4 0.13
6.6 10.4 0.4 7.6 3.9 3.3 5.1 0.8 1.4 0.8 4.0 1.1 0.002
-30.9 -41.8 -18.7 -17.7 -15.7 -35.4 -44.9 -4.6 -9.3 -32.0 -25.1 4.4 0.002
-5.4 -3.8 -4.4 -4.4 -3.7 -2.8 -5.0 -1.2 -3.6 -2.8 -3.7 0.4
2.8 2.9 1.4 1.3 0.9 1.8 3.9 0.1 0.6 1.6 1.7 0.3 0.002
0.8 1.7 -1.3 2.0 1.1 0.4 -0.3 0.5 0.6 -1.1 0.4 0.4 0.24
-11.3 --16.5 --13.0 -9.0 -8.3 --14.6 -9.6 -12.4 -7.0 -12.6
d i r e c t i o n (p = 0.004), b u t n o t f o r t h e v e r t i c a l d i r e c t i o n (p = 0.49 ( T a b l e 2 A ) . T h e o p e n i n g o f t h e j a w s w a s i n t e n d e d t o b e 10 m m at the incisors corresponding to an opening of a p p r o x i m a t e l y 7.4 ° ( w i t h a v e r t i c a l o v e r j e t o f 3 m m ) . T h e a v e r a g e c a l c u l a t e d o p e n i n g f r o m e q u a t i o n 11 w a s 7.3 ° w i t h a s t a n d a r d e r r o r o f t h e m e a n o f 0.5 ° ( T a b l e 2A). T h e c o n d y l a r m o v e m e n t w a s r e c o r d e d as 1.6 m m h o r i z o n t a l l y ( a n t e r i o r ) a n d 1 . 0 m m v e r t i c a l l y (inferior), both with a standard error of the mean of 0.4 m m ( T a b l e 2 A ) . A l l r e c o r d i n g s w e r e r e p e a t e d a n d s h o w e d a n a v e r a g e a b s o l u t e a c c u r a c y o f t h e regist r a t i o n o f 3.6 m m h o r i z o n t a l l y a n d 2.2 m m v e r t i c a l l y , with a numerical range of 1.1--6.4mm and 0.35.1 m m r e s p e c t i v e l y . T h e e n t i r e p r o c e d u r e w a s repeated with new cephalograms for 5 individuals and showed an average absolute change in the calculated
7.3 0.5
--3.9 -0.5 0.6 0.1 -1.9 -0.7 -2.3 0.0 -0.9 -0.1
- 1.6 -- 1.0 0.4 0.4 0.005 0.05
2.0 0.9 0.03
c e n t r e o f r o t a t i o n o f 6.8 m m h o r i z o n t a l l y a n d 7.5 m m v e r t i c a l l y w i t h a s t a n d a r d e r r o r o f t h e m e a n o f 2.4 a n d 2.5 m m r e s p e c t i v e l y . The simulation of surgical closure of an anterior o p e n b i t e o f 9.5 m m s h o w e d a n a b s o l u t e h o r i z o n t a l positioning error of the maxilla in the range 0-9.3 mm, w i t h a m e a n o f 2.0 m m a n d a s t a n d a r d e r r o r o f t h e m e a n o f 0.9 m m . A W i l c o x o n n o n - p a r a m e t r i c s i g n e d rank test showed this to be significantly larger than 0 (p = 0.03) ( T a b l e 2A). T h e 10 n o r m a l i n d i v i d u a l s w e r e s e p a r a t e d i n t o 2 e q u a l sized g r o u p s a c c o r d i n g t o t h e surgical consequence of the calculated rotation centres
t
Figure 1 - Results of the clinical and retrospective part of the study. The centre of the coordinate system was chosen in the radiographic centre of the condyle with the Frankfort horizontal as x-axis. The centres found in the study comprising 10 normal individuals with an active opening of 10 mm are shown as dots. The centres found in the study comprising 10 retrospective surgical cases are shown as triangles. The mean centre is shown as a bold ~ for the 10 normal patients and as a thin x for the 10 retrospective surgical cases.
Mandibular autorotation in orthognathic surgery 167 Table 3 - Cranial morphology of the 10 individuls participating in the clinical part of the study grouped by sex. The following variables
were measured: Sagittal position of the maxilla (s-n-ss) and mandible (s-n-pg), inclination of the maxilla (NSL/NL) and mandible (NSL/ML) relative to the cranial base (NSL), maxillary inclination relative to mandibular inclination (NL/ML), flexion of cranial base (n-s-ba), gonial angle (ML/RL), mandibular t-angle (MBL/ML), mandibular basis (pg-tgo) and ramus length (cd-tgo). The measurements are grouped by sex and and by surgical consequence of the calculated rotation centre (CCR). Group 1 showed the largest surgical error (~> 1.8 mm) and group 2 showed the smallest error (< 1.8 mm). Group 1 and 2 were compared with a Wilcoxon nonparamatric signed rank test. The standard-values (CSV) are from lngerslev and Solow (1975) (2) and Solow and Tallgren (1976) (~)
~ s.e. CSV d' ~ s.e. CSV 2 Group 1 s.e. Group 2 s.e. Significance
s-n-ss
s-n-pg
NSL/NL NSL/ML NL/ML
n-s-ba
ML/RL
MBL/ML pg-tgo
cd-tgo
81.83 1.54 82.22 85.63 3.18 81.53 82.30 2.13 84.40 2.50 0.54
82.41 0,84 81,42 83.25 2.86 80.45 82.60 1.91 82.90 1.55 0.91
5.50 1.06 8.04 5.75 1.27 7.13 4.50 1.27 6.70 0.66 0.16
130.67 1.51 129.22 130.63 2.84 130.37 131.60 2.30 129.70 1.59 0.52
116.67 2.42 120.82 119.25 5.54 121.17 112.70 1.50 122.70 3.65 0.03
30.17 0.83 27.59 26.50 2.18 -30.40 0.40 27.00 1.97 0.16
68.17 0.75 68.35 62.50 0.87 63.58 65.80 1.56 66.00 1.64 0.93
24.67 2.22 27.56 27.13 2.95 29.64 21.30 0.94 30.30 1.72 0.002
(group 1 ~ e r r o r t > 1.8mm, group 2~error< 1.8 ram). The cephalometric analysis showed a significantly smaller i n c l i n a t i o n (p = 0.002) a n d gonial angle (p = 0.03) of the m a n d i b l e in g r o u p 1 (Table 3). The analysis of the c o n d y l a r m o v e m e n t showed a significant correlation between the vertical m o v e m e n t a n d the p o s i t i o n of the r o t a t i o n centre b o t h horizontally (r = - 0 . 9 1 , p = 0.0003) a n d vertically (r = 0.70, p = 0.02) (Table 4). The h o r i z o n t a l m o v e m e n t of the condyle was correlated to the vertical position of the r o t a t i o n centre (r = 0.93, p = 0.0001), b u t n o t to the h o r i z o n t a l p o s i t i o n (r = - 0 . 5 4 , p = 0.11) (Table 4A). The regression e q u a t i o n s are s h o w n in T a b l e 4A. Retrospective
19.17 1.73 19.52 21.38 2.18 21.95 16.80 1.12 23.00 1.17 0.004
85.67 1.91 83.02 79.25 2.17 -82.60 3.30 83.60 1.50 0.78
m o v e m e n t was f o u n d to vary considerably within the cases, with a m e a n m o v e m e n t of 1.7 m m posteriorly a n d 0.4 m m superiorly. The possible h o r i z o n t a l error in the p o s i t i o n of the maxilla was calculated to vary between 0.4 a n d 10.4 ram, with a n average of 4.0 m m , which was f o u n d to be significantly larger t h a n 0 (p = 0.002) (Table 2 B). The correlation analysis showed a significant relationship between the vertical c o n d y l a r m o v e m e n t a n d the h o r i z o n t a l p o s i t i o n of the centre of r o t a t i o n ( r = 0.96, p = 0 . 0 0 0 1 ) , a n d between the h o r i z o n t a l m o v e m e n t of the condyle a n d the vertical position of the centre of r o t a t i o n (r = - 0 . 9 2 , p =
0.OOO2). DISCUSSION
study
The 10 retrospective surgical cases showed a n average centre of r o t a t i o n 8.4 m m b e h i n d a n d 25.1 m m below the r a d i o g r a p h i c centre of the condyle, which was significantly different from the r a d i o g r a p h i c centre of the condyle in the vertical direction (p = 0.002), b u t n o t in the h o r i z o n t a l direction (p = 0.13) (Table 2B). A large i n t e r - i n d i v i d u a l v a r i a t i o n was found, larger in the h o r i z o n t a l (s.e. = 5 . 4 r a m ) t h a n in the vertical direction (s.e. = 4.4 ram). The average r o t a t i o n of the m a n d i b l e was calculated to be 3.7 °, c o r r e s p o n d i n g to a closure at the incisors of 6 ram. The c o n d y l a r
Previous studies have s h o w n a significant difference between the p l a n n e d a n d i m m e d i a t e postoperative p o s i t i o n of the maxilla after Le F o r t I osteotomy, despite the i n t r o d u c t i o n of new extraoral measurem e n t techniques (Pospisil, 1987; Stanchina et al., 1988; Polido et al., 1990; Kahnberg et al., 1990). A recent study (Nattestad et al., 1991) has s h o w n that a n error in d e t e r m i n i n g the correct centre of m a n d i b u l a r r o t a t i o n could cause considerable error in the position of the maxilla, when the r o t a t i o n of the m a n d i b l e is used to guide the position of the maxilla d u r i n g
Table 4 - Statistical comparison of condylar movement during opening and position of the calculated centre of rotation (RCOC to CCR). The results of a correlation and regression analysis is shown (A) 10 normal individuals Correlation analysis Regression analysis
Condylar movement Condylar movement
X Y X Y
(B) 10 retrospective surgical cases Correlation analysis Regression analysis
Condylar movement Condylar movement
X Y X Y
RCOC to CCR X
Y
- 0.54 (p = 0.11) -0.91 (p = 0.0003) not significant Xoor = - 7.2. Yconclyle
0.93 (p = 0.0001) 0.70 (19= 0.02) Ycor = 8.3 "Xcon,iyie-1.7 Y0cr = 5.6. Yoondyae--9.6
--
1.9
RCOC to CCR X
Y
-0.14 (p = 0.71) 0.96 (p = 0.0001) not significant Xocr = 15.I/Yoondyle+ 1.8
-0.92 (p = 0.0002) 0.10 (p = 0.79) Yoor = - 10"9"Xcondyle-- 6.2 not significant
168 Journal of Cranio-Maxillo-Facial Surgery surgery with a composite or intermediate splint. The present study describes a method to determine the centre of mandibular rotation and to calculate the surgical consequence of the deviation between this centre and the centre of the condyle. The method was applied to 10 normal individuals and an average centre of mandibular rotation was found 14.9 mm below and 5 mm behind the superior midsurface of the condyle. A large inter-individual variation in the centres of rotation was found, with some centres close to the condyle and one centre as far as 37 mm below and 30 mm behind the centre of the condyle. A retrospective study of 10 surgical cases showed a centre of rotation on average 25.1mm below and 8.4 mm behind the centre of the condyle and with a large inter-individual variation. The average calculated centres of rotation found in this study was in accordance with previous studies by Grant (1973) and Baragar and Osborn (1984). Grant (1973) examined the function of the human jaw in a laboratory set-up and described a centre of mandibular rotation 7 mm below and 25 mm behind the centre of the condyle. The rationality for this observation was, according to Grant (1973), the complex condylar movement with a combination of roll, translation and rotation. Baragar and Osborn (1984) examined the three-dimensional structure of the masticatory system and described an initial rotation centre located 50 mm below and 25 mm behind the centre of the condyle. An analysis of the method developed in a laboratory set-up showed an error of 0.36-0.71mm dependent upon the amount of rotation. Winstanley (1979) described the error of the kinematic method in a similar non-clinical situation and found an error of up to 2.4 mm with an opening of 10 mm inter-incisally. The method developed was concluded to be acceptably accurate, compared to the precision of previous methods. The mean error of the method in the study comprising the 10 normal individuals was described in absolute numbers by repeating all measurements and found to be 3.6ram horizontally and 2.1 mm vertically. The reproducibility of the method was analyzed by repeating measurements in 5 of the 10 normal individuals and a mean difference of 6.8 mm horizontally and 7.5 mm vertically was found. The size of this error is most likely due to the inaccuracy of superimposing on the mandible. The specific inaccuracy of the superimposition was found to range from 0.3-0.4mm and this error was magnified 16 times in the calculations of the rotation centre (equation 18). The large variation in rotation centres between the patients and by repeating the registration has also been reported by Nevakari (1954, 1956), who found a difference between two registrations on the same individual of 7.1 mm. The larger difference by taking new cephalograms as opposed to repeating measurements on the same cephalograms was caused by a difference in mandibular movement. The condylar movement was shown to be highly correlated to the position of the calculated centre of rotation and it was found that even a very small movement of the
condyle could result in a marked positional change in the rotation centre. It is therefore suggested, that the large inter-individual variation is explained by differences in opening movement of the individual patient caused by differences in morphology of the masticatory system. The influence of projection errors as described by Ahlqvist et al. (1988) was not evaluated. It was found, that the correlation and regression equations between the condylar movement and the calculated centre of rotation in the two parts of the study was equally significant, but with an opposite sign. A probable cause for this finding is the different direction of mandibular movement in the two parts of the study, where the 10 normal individuals performed an inferior movement and the 10 post-surgical cases had a superior movement of the mandible. These two different directions of movement resulted in an opposite path of condylar movement. The movement of the condyle in the current study was on average 1.6mm anteriorly and 1.0mm inferiorly in the 10 normal individuals and 1.7 mm posteriorly and 0.4ram superiorly in the 10 retrospective surgical cases. The movement of the condyle during orthognathic surgery has been described previously by Herbosa et al. (1990), who found an average posterior movement of 0.6 mm in 29 patients, with closure of an open bite of 3-7 mm. Viteporn et al. (1990) showed an average movement of 2.2mm posteriorly and 1.3 mm inferiorly in 15 patients with different deformities corrected by a Le Fort I osteotomy alone. The possible resulting error in the horizontal position of the maxilla by a surgical correction of a 9.5 mm open bite was simulated in the 10 normal individuals and found to range from 0 to 9.3 ram, with a mean of 2.0 ram. The cephalometric analysis showed a significantly smaller mandibular inclination and gonial angle in individuals with the largest resulting error in the position of the maxilla. A similar analysis was performed in the retrospective study involving 10 surgical cases. Simulating the rotation of the mandible through the centre of the condyle would have resulted in a difference ranging from 0.4-10.4 ram, with a mean of 4.0 mm between the predicted horizontal position of the maxilla and the actual position obtained.
CONCLUSIONS The present study supports the hypothesis, that a centre of mandibular rotation outside the condylar body can exist. A large inter-individual variation was found in the position of the centre of mandibular rotation, which it is suggested can be caused by interindividual difference in craniofacial morphology. In some cases, a very large discrepancy between the centre of the condyle and the calculated centre of mandibular rotation was found, which would cause considerable error in the position of the maxilla at surgery. This investigation has shown that the method described is not optimal in locating the centre of mandibular rotation. Further work needs to be put
Mandibular autorotation in orthognathic surgery
into the description of precise techniques to locate the pre- and peroperative centre of mandibular rotation for a specific mandibular movement. However until refined methods are available, it is suggested, that this method is superior to the kinematic and palpatory methods and should be applied in cases, where a large rotation of the mandible is anticipated. Appendix
Initially a gravity point of the N points before (x(1), y(1)) and after ((~(2),y(2)) rotation is defined: 1 N
X(1) = ~ x(1)~.=
y(1) =
y(1)~ y
(Equation 3~4)
Each of the N points (x, y) are redefined (s, t) relative to these centres of gravity (x, ~): t(1)~ = y(1)~-y(1) (Equation 5 6) s(2)~ = x(2)~-2(2)
(Equation 7-8) Thereafter the following variables are defined: N
A = Y', [(s(1)i*s(2)~) + (t(1)~*t(2)~)]
(Equation 9)
i-1 N
B = ~ [(s(1)~*t(2)~) + (s(2)~*t(1)~)]
(Equation 10)
i=l
The amount of rotation (0) can now be determined by: /n\
0 = Tan-1 tA )
(Equation 11)
The centre of rotation (X, Y) can be determined by:
2
(Cot~)*(~(1)-x(2))
2
2 (Equation 13)
0 where the variable Cot ~ can be substituted by A and B since: A B Cos(0) - - Sin(0) ~/A ~+ B 2 ~/)l 2+ B 2 (Equation 14-15) and thereby Cot 0 = ~/A 2+ B 2+ A B
The error of the determination (ft) could be assessed using the error in superimpositioning (ao) and the amount of rotation (0) by the following equation:
(0)
f~ = ao*Cot ~
(Equation 18)
The authors wish to extend their gratitude to Professor U. Haagerup, University of Odense, for his help with the mathematical model and to Professor S. Kreiborg, Department of Pedodontics, Royal Dental College, Copenhagen, Denmark for the loan of the computer equipment.
References errors on angular meaurements in cephalometry. Eur. J. Orthod. 10 (I988) 353 Baragar, F. A., J. W. Osborn : A model relating patterns of human jaw movement to biomechanical constraints. J. Biomechanics 10 (1984) 757 Bell, W. H., W. R. Proffit, R. P. White: Surgical correction of dentofacial deformities. Saunders, Philadelphia 1980 Bosman, A. E. : Hinge axis determination of the mandible. Thesis, University of Utrecht, 1974 Brewka, R. E. : Pantographic evaluation of cephalometric hinge axis. Am. J. Orthod. 79 (1981) 1 Chick, A. : The rotary nature of some mandibular movements. J. Prosthet. Dent. 10 (1960) 857 Fischer, R. : Beitrag zum Artikulationsproblem. Schweiz. Mschr. Zahnheilk. 62 (1952) 317 Fischer, R. : Die Artikulationslehre. In: Hfiupl K. (ed.): Die Zahn-, Mund-, und Kieferheilkunde. Ein Handbuch ffir die zahnn/irztliche Praxis 4 (1956) 95-176 Grant, P. G. : Biomechanical significance of the instantaneous centre of rotation : The human temporomandibular joint. J. Biomechanics 6 (1973) 109 Herbosa, E. G., If. S. Rotskoff , B. F. Ramos, H. S. Ambrookian :
(Equation 12) y__ ~(1)+y(2))
(Equation 17)
Ahlqvist, J., S. Eliasson, U. Welander : The effect of projection
t(2)~ = y(2)~-y(2)
2
1 N Ao = ~ ..~ [s(1)~ + t(1)~ + s(2)~ + t(2)~] - 2~/A 2+ B ~
Acknowledgement
1=1
s(1)~ = x(1)~-x(1)
occur due to measurement error. The size of this error can be determined by the following equation:
(Equation 1-2)
/=1
1 N x(2) = N~--~t.-x(2)~ y ( 2 ) =
169
(Equation 16)
As the mandible is not deformed during movement, but performs a rotation (by definition), all the points measured should be transformed into the corresponding points after rotation. However, this will not
Condylar position in superior maxillary repositioning and its effect on the temporomandibular joint. J. Oral Maxillofac. Surg. 48 (1990) 690 Hohl, T. H. : The use of an anatomical articulator in segmental orthognathic surgery. Am. J. Orthod. 73 (1978) 428 [ngerslev, C. H., B. Solow." Sex differences in craniofacial morphology. Acta Odontol. Scand. 33 (1975) 85 Kahnberg, K. E., B. Sunzel, P. Astrand: Planning and control of vertical dimension in Le Fort I osteotomies. J. Cranio-Max.Fac. Surg. 18 (1990) 267 Marko, J. V. : Simple hinge and semi-adjustable articulators in orthognathic surgery. Am. J. Orthod. Dentofac. Orthoped. 90 (1986) 37 McCollum, B. B. : Fundamentals involved in prescribing restorative dental remedies. Dent. Items Interest 61 (1939) 522, 641,724, 852, 942 Nattestad, A., P. Vedtofte, E. Mosekilde : The significance of an erroneous recording of the mandibular centre of rotation in orthognathic surgery. J. Cranio-Max.-Fac. Surg. 19 (1991) 254 Nevakari, K. : A new 'triangular transfer' method for studying mandibular movements on the basis of cephalometric roentgenograms. Acta Odontol. Scand. 12 (1954) 293 Nevakari, K. : An analysis of the mandibular movement from rest to occlusal position. Acta Odontol. Scand. 14 (1956) suppl. 19 Polido, W. D., E. Ellis, D. P. Sinn: An assessment of predictability of maxillary surgery. J. Oral Maxillofac. Surg. 48 (1990) 697
170
Journal of Cranio-Maxillo-Facial Surgery
Pospisil, O. A.. Reliability and feasibility of prediction tracing in orthognathic surgery. J. Cranio-Max.-Fac. Surg. 15 (1987) 79 Rekow, E. D., F. W. Worms, A. G. Erdman, T. M. Speidel : Treatment induced errors in occlusion following orthognathic surgery. Am. J. Orthod. 88 (1985) 425 Schallhorn, R. G..' A study of the arbitrary centre and the kinematic centre of rotation for face-bow mountings. J. Prosthet. Dent. 7 (1957) 162 Solow, B,, A. Tallgren : Head posture and craniofacial morphology. Am. J. Phys. Anthrop. 44 (1976) 417 Sperry, T. P., M. J. Steinberg, B. J. Gans." Mandibular movement during autorotation as a result of maxillary impaction surgery. Am. J. Orthod. 81 (1982) 116 Stanehina, R., E. Ellis, W. J. Gallo, R. J. Fonseea: A comparison of two measures for repositioning the maxilla during orthognathic surgery. Int. J. Adult Orthod. Orthognat. Surg. 3 (1988) 149 Torii, K. : Analysis of rotation centers of various mandibular closures. J. Prosthet Dent. 61 (1989) 285
Turvey, T. A., D. J. Hall, L. C. Fish, B. N. Epker : Surgicalorthodontic treatment planning for simultaneous mobilisation of the maxilla and mandible in correction of dentofacial deformity. Oral Surg. 54 (1982) 491 Viteporn, S., B. Melsen, M. Bundgaard: Postsurgical change of mandibular position in patients following Le Fort I osteotomy. Int. J. Adult Orthod. Orthognat. Surg. 5 (1990) 91 Walker, P. M. : Discrepancies between arbitrary and true hinge axes. J. Prosthet. Dent. 43 (1980) 279 Winstanley, R. B. : Hinge axis location on the articulator. J. Prosthet. Dent. 42 (1979) 135
A. Nattestad, DDS, PhD, Department of Oral and Maxillofacial Surgery, Royal Dental College, Norre Alle 20, DK-2200 Copenhagen, Denmark. Paper received 31 May 1991 Accepted 28 October 1991
Department of Oral and Maxillofacial Surgery (Head: Prof. dr. E. Hjorting Hansen, DDS, PhD), Royal Dental College and University Hospital (Rigshospitalet), Copenhagen, Denmark.
S U M M A R Y. Rotation of the mandible is simulated through the centre of the condyle in planning orthognathic surgery. Previous studies have suggested that the initial mandibular movement is better characterized as multiple parts of the segments of a circle with the average centre located below and behind the centre of the condyle. This paper describes a method of locating the centre of mandibular rotation by computer-analysis of two lateral cephalograms with different degrees of opening. The method was used on 10 normal individuals showing an average centre of mandibular rotation 14.9 mm below and 5.0 mm behind the superior midsurface of the condyle for movements from occlusion to an opening of 10 ram. The implications of the located centre of rotation on orthognathic surgery was determined by simulating a closure of a 9.5 mm open bite. An error of up to 9.3 mm in the horizontal position of the maxilla would occur if the centre of the condyle was used. A retrospective study on l0 patients with increased anterior facial height was undertaken and a mean centre of rotation 8.4 mm behind and 25.1 mm below the centre of the condyle was found. Simulating the rotation of the mandible through the centre of the condyle would have resulted in a difference between the predicted and actual horizontal position of the maxilla ranging from 0.4-10.4 ram. It was concluded that using the centre of the condyle as the centre of mandibular rotation in the planning of superior maxillary movement with a Le Fort I osteotomy may cause considerable error in the horizontal position of the maxilla in most cases.
KEY W O R D S : Le Fort 1 0 s t e o t o m y -
Centre of mandibular r o t a t i o n - Prediction t r a c i n g - Articulator
used today. Fischer (1952, 1956) presented a method different from all others, using photographs taken of the moving mandible. The method, which was later modified by Chick (1960), involved taking a photograph of the mandible during opening with a camera using a long exposure time. The centre of mandibular rotation was calculated by a geometric principle, but due to inaccuracy in measurements on the picture and the time-consumption involved, the method never gained clinical application. Another method is the kinematic one, which has already been described by McCollum in 1939, but has recently been revived and modified. The method determ~-nes the centre of mandibular rotation by analysis of the pattern of mandibular movement on the individual patient and is claimed to be very precise (Bosman, 1974). The method requires the mounting of a rather extensive apparatus on the moving mandible, and an investigation has questioned the accuracy of the method (Winstanley, 1979). A recent biomechanical study by Baragar and Osborn (1984) presented a three-dimensional computer simulation model of the masticatory system and concluded that the mandible would rotate around a point 50 mm inferior and 25 mm posterior to the centre of the condyle, at an opening of a few mm. During greater opening movements the centre was found to move in an anterior and superior direction
INTRODUCTION
The centre of the condyle is one of the reference points registered in the planning of Le Fort I osteotomy by a face-bow recording (Hohl, 1978; Marko, 1986). A trial-operation is performed on dental casts in an articulator to establish the precise movement of the jaws necessary to obtain normal occlusion and the desired changes in facial morphology. The centre of the condyle is presumed to represent the centre of mandibular rotation (Bell et al., 1980; Turvey et al., 1982). Several methods of localizing the centre of mandibular rotation have been presented. The first method to gain overall acceptance was the arbitrary method which, based on a concept of human uniformity, used a point 13 mm in front of the external auditory meatus on a line to the lateral corner of the eye (Schallhorn, 1957). A later study has shown the arbitrary method to be quite inaccurate (Walker, 1980). Later, the method of choice was the palpatory method, where the centre of the condyle was located by palpation (Bosman, 1974). This method assumes that the centre of the condyle represents the centre of mandibular rotation, but other studies have shown this to be quite disputable (Grant, 1973; Brewka, 1981; Baragar and Osborn, 1984; Torii, 1989). Despite these findings, the palpatory method is the clinical method most widely 163
164
Journal of Cranio-Maxillo-Facial Surgery
on a line ending about 10 mm in front of the original position of the centre of the condyle. The importance of accurate determination of the centre of mandibular rotation has been documented by Sperry et al. (1982) and by Rekow et al. (1985). Sperry et al. (1982) determined the centre of mandibular autorotation in 23 patients treated by Le Fort I osteotomy of the maxilla. The superior rotation of the mandible, due to maxillary impaction, occurred through a centre in the mastoid region, posterior and inferior to the centre of the condyle. Rekow et al. (1985) investigated the error in occlusion following splint removal and found an insignificant occlusal error, probably due to the small amount of rotation taking place on splint removal. Previous studies have shown a significant difference between the planned and immediate postoperative position of the maxilla after Le Fort I osteotomy, despite the introduction of new extraoral measurement techniques (Pospisil, 1987; Stanchina et al., 1988; Polido et al., 1990; Kahnberg et al., 1990). A study by Nattestadet al. (1991) has demonstrated, that a discrepancy between the centre of the condyle and the centre of mandibular rotation in the range 10-20 mm can have a clinically significant influence on the postoperative result of orthognathic surgery. The significance of this finding is however dependent upon the ability to locate the centre of mandibular rotation in the individual patient undergoing surgery. The purpose of this study was to develop a method for localizing the centre of mandibular rotation and to determine the implication for orthognathic surgery.
MATERIAL AND METHOD Equipment An IBM ~ compatible 386 computer with Super VGA display and a Summagraphics Supergrid ® (with backlight) was used for the measurements. The resolution of the digitizer was 0.025 mm, with an accuracy of 0.1 mm on repeated measurements. The software used was Autocad ® version 9.01n and SAS ® Statistical Analysis System.
Method-accuracy and description of method A natural size sketch of the mandible was firmly mounted in the middle of the digitizer. The sketch had 5 points marked with crosses placed within the body of the mandible. The centre of rotation of the sketch was secured by placing a needle through this point. With the sketch in a neutral position, the 5 points and the centre of rotation (the needle) were traced. The sketch was rotated 5, 10, 15, 20 or 25 ° and the 5 points in the body of the mandible were traced again. The centre and amount of rotation was determined by the principle of least squares. The appendix explains the calculations to determine the centre and amount of rotation, that brought the points before rotation closest possible to the points after rotation. The calculations were performed in the statistical analysis
program SAS. The registration procedure and the calculations were performed 5 times for each rotation of 5, 10, 15, 20 and 25 °.
Clinical study 10 normal individuals with no history of TMJ symptoms participated in the study. Initially, the palpatory centre of the condyle was marked with lead (a' x ' on the right side and a ' + ' on the left side were fixed with tape). Each participant had two lateral cephalograms taken in immediate succession: one in active retruded occlusion and one at 10 mm opening, also in an active retruded position of the condyles. The participants were placed in the natural head posture in the cephalostat with ear pins and a light cross attempting a left-right symmetry of the face. The cephalograms were placed similarly on the digitizer, using superimposition on the cranial base and calvarium. On each cephalogram the superior midsurface of the condyles, the palpatory centres and 10 reference points were digitized. The reference points were distributed over the entire cephalogram and transferred to the second cephalogram by superimposing on easily recognizable structures in the mandible. The centre of rotation, size of movement and superimpositional error was determined for each individual using equations 1-18 (see Appendix). The superior mid-surface of the condyle is described by Turvey et al. (1982) and was chosen as the reference-point in the condyle because it was a more specific reference point than the usual geometric centre of the condyle. The point, which is later referred to as the radiographic centre of the condyle, was determined as the middle of the superior surface of the condyle, determined by two parallel tangents to the anterior and posterior outline of the condylar head. The superimposition, registration and calculation procedure were made twice on each individual in order to establish the overall accuracy of the method. The entire procedure was repeated with new cephalograms for 5 of the individuals in order to establish the reproducibility. The condylar movement during opening was calculated from the registered positions of the superior midsurface of the condyle. A cephalometric analysis was performed on the 10 individuals comprising: sagittal position of the maxilla (s-n-ss) and mandible (s-n-pg), inclination of the maxilla ( N S L / N L ) and mandible ( N S L / M L ) relative to the cranial base (NSL), maxillary inclination relative to mandibular inclination ( N L / M L ) , flexion of cranial base (n-s-ba), gonial angle (ML/RL), mandibular t-angle (MBL/ML), mandibular basis (pg-tgo) and ramus length (cd-tgo).
Retrospective study This material comprised of 10 patients with increased anterior facial height treated by Le Fort I osteotomy without concomitant osteotomy in the mandibular ramus. The criteria for inclusion in the study was the
Mandibular autorotation in orthognathic surgery 165 existence of lateral cephalograms taken immediately before and 2-5 months after surgery. The splint had in all cases already been removed at this time. The Le Fort I osteotomy was performed through a horizontal incision in the buccal sulcus extending posterior to the first molars. The maxilla was downfractured, mobilized and placed in the preplanned position using a composite splint and intraoral vertical and horizontal reference lines on the anterior wall of the maxilla. The centre of mandibular rotation was determined retrospectively by computer-analysis of the pre- and postoperative lateral cephalograms. The possible consequence of the difference between the calculated centre of rotation and the centre of the condyle for the horizontal position of the maxilla at surgery was simulated using a set of equations, which has recently been described (Nattestad et al., 1991).
S T A T I S T I C A L ANALYSIS
Method-accuracy The overall variation in determining the centre of rotation in the non-clinical situation was evaluated by mean and standard error of the mean for each of the 5 rotations (5, 10, 15, 20 and 25°). A Wilcoxon nonparametric signed rank test was used to evaluate whether the position of the calculated centre of rotation was significantly different from the actual centre of rotation.
Clinical study A Wilcoxon non-parametric signed rank test was performed to evaluate if the calculated centre of mandibular rotation deviated significantly from the radiographic- or palpatory centre of the condyle. The condylar movement, measured at the superior midsurface of the condyle, was compared to the position of the calculated centre of rotation with a correlation and regression analysis. The statistical significance of the calculated consequence for orthognathic surgery was evaluated with a Wilcoxon non-parametric signed rank test. The results of the cephalometric analysis were compared to the calculated consequence for orthognathic surgery with a Wilcoxon non-parametric signed rank test. Retrospective study A Wilcoxon non-parametric signed rank test was performed to evaluate if the calculated centre of mandibular rotation deviated significantly from the
radiographic centre of the condyle. The condylar movement, measured at the superior midsurface of the condyle, was compared to the position of the calculated centre of rotation With a correlation and regression analysis.
RESULTS
Method-accuracy The method accuracy was described for rotations of 5, 10, 15, 20 and 25 ° as a horizontal and a vertical difference in mm between the actual and calculated centre of rotation. The mean absolute horizontal difference was found to range from 0.05-0.47 mm with a standard error of mean ranging from 0.07-0.25 ram. The mean absolute vertical difference ranged from 0.01 0.32 mm with a standard error of the mean ranging from 0.08-0.32 ram. The mean distinct between the two centres ranged from 0.36-0.71 mm with a standard error of the mean ranging from 0.03 0.23 mm. Generally the precision of the method improved with larger rotations. No statistically significant difference was found between the actual and calculated centre for any of the 5 rotations (Table 1).
Clinical study The 10 individuals showed an average centre of rotation 14.9ram below and 5.0ram behind the radiographic centre of the condyle (Table 2A, Fig. 1). The inter-individual variation was large both in the horizontal and vertical direction with a standard error of the mean of 3.5 ram. No significant horizontal difference was found between the coordinate position of the calculated centre of rotation and the radiographic centre of the condyle (p = 0.08). The vertical difference however showed a significant lower position of the calculated centre of rotation (p --- 0.004). The difference between the radiographic and the palpatory centre of the condyle was found to be more uniform with the average palpatory centre of the condyle located 9.3 mm in front of and 11.4 mm below the radiographic centre. The standard error of the mean was 0.6 and 0.9 mm respectively. This divergence was highly significant both in the horizontal (p = 0.002) and vertical (p = 0.002) direction. When the calculated centre of mandibular rotation was described relative to the palpatory centre of the condyle, a position 14.3 mm behind the 3.5 mm below the palpatory centre of the condyle was found. The standard error of the mean was 3.2 and 3.9 mm respectively, which was significant for the horizontal
Table 1- Analysis of method error: difference between the known centre of rotation and the calculated centre. The horizontal deviation is measured on the x-axis, the vertical deviation on the y-axis. The distance is the direct span between the two centres Rotation degrees
N
Horizontal deviation mm s.e. p
Vertical deviation mm s.e. p
Distance mm s.e.
p
5 10 15 20 25
5 5 5 5 5
0.05 -0.37 -0.47 -0.11 -0.23
-0.21 0.01 0.32 0.25 0.11
0.71 0.67 0.65 0.48 0.36
0.06 0.06 0.06 0.06 0.06
0.25 0.19 0.10 0.14 0.07
0.63 0.13 0.06 0.63 0.06
0.32 0.25 0.12 0.18 0.08
0.55 1.00 0.13 0.31 0.13
0.23 0.14 0.03 0.10 0.03
166
Journal of Cranio-Maxillo-Facial Surgery
Table 2A - Results of the study involving 10 normal individuals showing the distances between the radiographic centre of the condyle
(RCOC), the palpatory centre of the condyle (PCOC), the calculated centre of mandibular rotation (CCR) and the degree of opening and the condylar movement during opening. The resulting horizontal error in the position of the maxilla was calculated from a set of equations previously presented (Nattestad et al., 1991). The result of a Wilcoxon non-parametric signed rank test is shown at the bottom line, where relevant
Cases
RCOC to COR (ram) X Y
1 2 3 4 5 6 7 8 9 10
30.2 6.7 -14.0 0.0 8.4 1.9 9.1 3.5 3.1 1.2
Average Std. error P
--37.2 -9.9 --19.4 --17.1 --22.2 -11.2 -15.7 3.7 -16.2 -3.8
5.0 - 14.9 3.5 3.5 0.08 0.004
PCOC to CCR (ram) X Y 36.3 --25.8 17.3 6.6 --4.4 -6.4 9.6 -8.1 16.7 -14.0 10.6 3.4 17.6 -6.1 15.7 16.1 10.1 - 9 . 2 13.6 8.8 14.3 - 3 . 5 3.2 3.9 0.004 0.49
RCOC to PCOC (ram) X Y
Opening deg.
Condylar movement X Y
Horizontal error (mm)
-6.2 -10.6 --9.6 --9.6 --8.3 --8.7 --8.5 -12.2 --7.1 -12.3
6.4 7.1 5.5 6.0 7.1 8.8 10.9 6.7 7.0 7.4
-3.9 -0.7 -1.9 -1.8 -2.4 -1.8 -2.6 0.2 -1.1 -0.1
9.3 0.7 0.4 1.8 3.5 0.5 2.5 -0.9 2.1 0.0
- 9 . 3 - 11.4 0.6 0.9 0.002 0.002
Table 2B - Results of the retrospective study involving 10 surgical cases showing the distance between the radiographic centre of the condyle (RCOC) and the calculated centre of mandibular rotation (CCR), and the degree of opening and the condylar movement during opening. The resulting horizontal error in the position of the maxilla was calculated from a set of equations previously presented (Nattestad et al., 1991). The result of a Wilcoxon non-parametric signed rank test is shown in the bottom line, where relevant
Case
RCOC to CCR Condylar (mm) Open- movement X Y ing deg. X Y
Horizontal error (mm)
1 2 3 4 5 6 7 8 9 I0 Average Std. error P
ll.l 27.3 --17.6 27.7 19.9 10.0 -2.8 21.0 7.4 -19.7 8.4 5.4 0.13
6.6 10.4 0.4 7.6 3.9 3.3 5.1 0.8 1.4 0.8 4.0 1.1 0.002
-30.9 -41.8 -18.7 -17.7 -15.7 -35.4 -44.9 -4.6 -9.3 -32.0 -25.1 4.4 0.002
-5.4 -3.8 -4.4 -4.4 -3.7 -2.8 -5.0 -1.2 -3.6 -2.8 -3.7 0.4
2.8 2.9 1.4 1.3 0.9 1.8 3.9 0.1 0.6 1.6 1.7 0.3 0.002
0.8 1.7 -1.3 2.0 1.1 0.4 -0.3 0.5 0.6 -1.1 0.4 0.4 0.24
-11.3 --16.5 --13.0 -9.0 -8.3 --14.6 -9.6 -12.4 -7.0 -12.6
d i r e c t i o n (p = 0.004), b u t n o t f o r t h e v e r t i c a l d i r e c t i o n (p = 0.49 ( T a b l e 2 A ) . T h e o p e n i n g o f t h e j a w s w a s i n t e n d e d t o b e 10 m m at the incisors corresponding to an opening of a p p r o x i m a t e l y 7.4 ° ( w i t h a v e r t i c a l o v e r j e t o f 3 m m ) . T h e a v e r a g e c a l c u l a t e d o p e n i n g f r o m e q u a t i o n 11 w a s 7.3 ° w i t h a s t a n d a r d e r r o r o f t h e m e a n o f 0.5 ° ( T a b l e 2A). T h e c o n d y l a r m o v e m e n t w a s r e c o r d e d as 1.6 m m h o r i z o n t a l l y ( a n t e r i o r ) a n d 1 . 0 m m v e r t i c a l l y (inferior), both with a standard error of the mean of 0.4 m m ( T a b l e 2 A ) . A l l r e c o r d i n g s w e r e r e p e a t e d a n d s h o w e d a n a v e r a g e a b s o l u t e a c c u r a c y o f t h e regist r a t i o n o f 3.6 m m h o r i z o n t a l l y a n d 2.2 m m v e r t i c a l l y , with a numerical range of 1.1--6.4mm and 0.35.1 m m r e s p e c t i v e l y . T h e e n t i r e p r o c e d u r e w a s repeated with new cephalograms for 5 individuals and showed an average absolute change in the calculated
7.3 0.5
--3.9 -0.5 0.6 0.1 -1.9 -0.7 -2.3 0.0 -0.9 -0.1
- 1.6 -- 1.0 0.4 0.4 0.005 0.05
2.0 0.9 0.03
c e n t r e o f r o t a t i o n o f 6.8 m m h o r i z o n t a l l y a n d 7.5 m m v e r t i c a l l y w i t h a s t a n d a r d e r r o r o f t h e m e a n o f 2.4 a n d 2.5 m m r e s p e c t i v e l y . The simulation of surgical closure of an anterior o p e n b i t e o f 9.5 m m s h o w e d a n a b s o l u t e h o r i z o n t a l positioning error of the maxilla in the range 0-9.3 mm, w i t h a m e a n o f 2.0 m m a n d a s t a n d a r d e r r o r o f t h e m e a n o f 0.9 m m . A W i l c o x o n n o n - p a r a m e t r i c s i g n e d rank test showed this to be significantly larger than 0 (p = 0.03) ( T a b l e 2A). T h e 10 n o r m a l i n d i v i d u a l s w e r e s e p a r a t e d i n t o 2 e q u a l sized g r o u p s a c c o r d i n g t o t h e surgical consequence of the calculated rotation centres
t
Figure 1 - Results of the clinical and retrospective part of the study. The centre of the coordinate system was chosen in the radiographic centre of the condyle with the Frankfort horizontal as x-axis. The centres found in the study comprising 10 normal individuals with an active opening of 10 mm are shown as dots. The centres found in the study comprising 10 retrospective surgical cases are shown as triangles. The mean centre is shown as a bold ~ for the 10 normal patients and as a thin x for the 10 retrospective surgical cases.
Mandibular autorotation in orthognathic surgery 167 Table 3 - Cranial morphology of the 10 individuls participating in the clinical part of the study grouped by sex. The following variables
were measured: Sagittal position of the maxilla (s-n-ss) and mandible (s-n-pg), inclination of the maxilla (NSL/NL) and mandible (NSL/ML) relative to the cranial base (NSL), maxillary inclination relative to mandibular inclination (NL/ML), flexion of cranial base (n-s-ba), gonial angle (ML/RL), mandibular t-angle (MBL/ML), mandibular basis (pg-tgo) and ramus length (cd-tgo). The measurements are grouped by sex and and by surgical consequence of the calculated rotation centre (CCR). Group 1 showed the largest surgical error (~> 1.8 mm) and group 2 showed the smallest error (< 1.8 mm). Group 1 and 2 were compared with a Wilcoxon nonparamatric signed rank test. The standard-values (CSV) are from lngerslev and Solow (1975) (2) and Solow and Tallgren (1976) (~)
~ s.e. CSV d' ~ s.e. CSV 2 Group 1 s.e. Group 2 s.e. Significance
s-n-ss
s-n-pg
NSL/NL NSL/ML NL/ML
n-s-ba
ML/RL
MBL/ML pg-tgo
cd-tgo
81.83 1.54 82.22 85.63 3.18 81.53 82.30 2.13 84.40 2.50 0.54
82.41 0,84 81,42 83.25 2.86 80.45 82.60 1.91 82.90 1.55 0.91
5.50 1.06 8.04 5.75 1.27 7.13 4.50 1.27 6.70 0.66 0.16
130.67 1.51 129.22 130.63 2.84 130.37 131.60 2.30 129.70 1.59 0.52
116.67 2.42 120.82 119.25 5.54 121.17 112.70 1.50 122.70 3.65 0.03
30.17 0.83 27.59 26.50 2.18 -30.40 0.40 27.00 1.97 0.16
68.17 0.75 68.35 62.50 0.87 63.58 65.80 1.56 66.00 1.64 0.93
24.67 2.22 27.56 27.13 2.95 29.64 21.30 0.94 30.30 1.72 0.002
(group 1 ~ e r r o r t > 1.8mm, group 2~error< 1.8 ram). The cephalometric analysis showed a significantly smaller i n c l i n a t i o n (p = 0.002) a n d gonial angle (p = 0.03) of the m a n d i b l e in g r o u p 1 (Table 3). The analysis of the c o n d y l a r m o v e m e n t showed a significant correlation between the vertical m o v e m e n t a n d the p o s i t i o n of the r o t a t i o n centre b o t h horizontally (r = - 0 . 9 1 , p = 0.0003) a n d vertically (r = 0.70, p = 0.02) (Table 4). The h o r i z o n t a l m o v e m e n t of the condyle was correlated to the vertical position of the r o t a t i o n centre (r = 0.93, p = 0.0001), b u t n o t to the h o r i z o n t a l p o s i t i o n (r = - 0 . 5 4 , p = 0.11) (Table 4A). The regression e q u a t i o n s are s h o w n in T a b l e 4A. Retrospective
19.17 1.73 19.52 21.38 2.18 21.95 16.80 1.12 23.00 1.17 0.004
85.67 1.91 83.02 79.25 2.17 -82.60 3.30 83.60 1.50 0.78
m o v e m e n t was f o u n d to vary considerably within the cases, with a m e a n m o v e m e n t of 1.7 m m posteriorly a n d 0.4 m m superiorly. The possible h o r i z o n t a l error in the p o s i t i o n of the maxilla was calculated to vary between 0.4 a n d 10.4 ram, with a n average of 4.0 m m , which was f o u n d to be significantly larger t h a n 0 (p = 0.002) (Table 2 B). The correlation analysis showed a significant relationship between the vertical c o n d y l a r m o v e m e n t a n d the h o r i z o n t a l p o s i t i o n of the centre of r o t a t i o n ( r = 0.96, p = 0 . 0 0 0 1 ) , a n d between the h o r i z o n t a l m o v e m e n t of the condyle a n d the vertical position of the centre of r o t a t i o n (r = - 0 . 9 2 , p =
0.OOO2). DISCUSSION
study
The 10 retrospective surgical cases showed a n average centre of r o t a t i o n 8.4 m m b e h i n d a n d 25.1 m m below the r a d i o g r a p h i c centre of the condyle, which was significantly different from the r a d i o g r a p h i c centre of the condyle in the vertical direction (p = 0.002), b u t n o t in the h o r i z o n t a l direction (p = 0.13) (Table 2B). A large i n t e r - i n d i v i d u a l v a r i a t i o n was found, larger in the h o r i z o n t a l (s.e. = 5 . 4 r a m ) t h a n in the vertical direction (s.e. = 4.4 ram). The average r o t a t i o n of the m a n d i b l e was calculated to be 3.7 °, c o r r e s p o n d i n g to a closure at the incisors of 6 ram. The c o n d y l a r
Previous studies have s h o w n a significant difference between the p l a n n e d a n d i m m e d i a t e postoperative p o s i t i o n of the maxilla after Le F o r t I osteotomy, despite the i n t r o d u c t i o n of new extraoral measurem e n t techniques (Pospisil, 1987; Stanchina et al., 1988; Polido et al., 1990; Kahnberg et al., 1990). A recent study (Nattestad et al., 1991) has s h o w n that a n error in d e t e r m i n i n g the correct centre of m a n d i b u l a r r o t a t i o n could cause considerable error in the position of the maxilla, when the r o t a t i o n of the m a n d i b l e is used to guide the position of the maxilla d u r i n g
Table 4 - Statistical comparison of condylar movement during opening and position of the calculated centre of rotation (RCOC to CCR). The results of a correlation and regression analysis is shown (A) 10 normal individuals Correlation analysis Regression analysis
Condylar movement Condylar movement
X Y X Y
(B) 10 retrospective surgical cases Correlation analysis Regression analysis
Condylar movement Condylar movement
X Y X Y
RCOC to CCR X
Y
- 0.54 (p = 0.11) -0.91 (p = 0.0003) not significant Xoor = - 7.2. Yconclyle
0.93 (p = 0.0001) 0.70 (19= 0.02) Ycor = 8.3 "Xcon,iyie-1.7 Y0cr = 5.6. Yoondyae--9.6
--
1.9
RCOC to CCR X
Y
-0.14 (p = 0.71) 0.96 (p = 0.0001) not significant Xocr = 15.I/Yoondyle+ 1.8
-0.92 (p = 0.0002) 0.10 (p = 0.79) Yoor = - 10"9"Xcondyle-- 6.2 not significant
168 Journal of Cranio-Maxillo-Facial Surgery surgery with a composite or intermediate splint. The present study describes a method to determine the centre of mandibular rotation and to calculate the surgical consequence of the deviation between this centre and the centre of the condyle. The method was applied to 10 normal individuals and an average centre of mandibular rotation was found 14.9 mm below and 5 mm behind the superior midsurface of the condyle. A large inter-individual variation in the centres of rotation was found, with some centres close to the condyle and one centre as far as 37 mm below and 30 mm behind the centre of the condyle. A retrospective study of 10 surgical cases showed a centre of rotation on average 25.1mm below and 8.4 mm behind the centre of the condyle and with a large inter-individual variation. The average calculated centres of rotation found in this study was in accordance with previous studies by Grant (1973) and Baragar and Osborn (1984). Grant (1973) examined the function of the human jaw in a laboratory set-up and described a centre of mandibular rotation 7 mm below and 25 mm behind the centre of the condyle. The rationality for this observation was, according to Grant (1973), the complex condylar movement with a combination of roll, translation and rotation. Baragar and Osborn (1984) examined the three-dimensional structure of the masticatory system and described an initial rotation centre located 50 mm below and 25 mm behind the centre of the condyle. An analysis of the method developed in a laboratory set-up showed an error of 0.36-0.71mm dependent upon the amount of rotation. Winstanley (1979) described the error of the kinematic method in a similar non-clinical situation and found an error of up to 2.4 mm with an opening of 10 mm inter-incisally. The method developed was concluded to be acceptably accurate, compared to the precision of previous methods. The mean error of the method in the study comprising the 10 normal individuals was described in absolute numbers by repeating all measurements and found to be 3.6ram horizontally and 2.1 mm vertically. The reproducibility of the method was analyzed by repeating measurements in 5 of the 10 normal individuals and a mean difference of 6.8 mm horizontally and 7.5 mm vertically was found. The size of this error is most likely due to the inaccuracy of superimposing on the mandible. The specific inaccuracy of the superimposition was found to range from 0.3-0.4mm and this error was magnified 16 times in the calculations of the rotation centre (equation 18). The large variation in rotation centres between the patients and by repeating the registration has also been reported by Nevakari (1954, 1956), who found a difference between two registrations on the same individual of 7.1 mm. The larger difference by taking new cephalograms as opposed to repeating measurements on the same cephalograms was caused by a difference in mandibular movement. The condylar movement was shown to be highly correlated to the position of the calculated centre of rotation and it was found that even a very small movement of the
condyle could result in a marked positional change in the rotation centre. It is therefore suggested, that the large inter-individual variation is explained by differences in opening movement of the individual patient caused by differences in morphology of the masticatory system. The influence of projection errors as described by Ahlqvist et al. (1988) was not evaluated. It was found, that the correlation and regression equations between the condylar movement and the calculated centre of rotation in the two parts of the study was equally significant, but with an opposite sign. A probable cause for this finding is the different direction of mandibular movement in the two parts of the study, where the 10 normal individuals performed an inferior movement and the 10 post-surgical cases had a superior movement of the mandible. These two different directions of movement resulted in an opposite path of condylar movement. The movement of the condyle in the current study was on average 1.6mm anteriorly and 1.0mm inferiorly in the 10 normal individuals and 1.7 mm posteriorly and 0.4ram superiorly in the 10 retrospective surgical cases. The movement of the condyle during orthognathic surgery has been described previously by Herbosa et al. (1990), who found an average posterior movement of 0.6 mm in 29 patients, with closure of an open bite of 3-7 mm. Viteporn et al. (1990) showed an average movement of 2.2mm posteriorly and 1.3 mm inferiorly in 15 patients with different deformities corrected by a Le Fort I osteotomy alone. The possible resulting error in the horizontal position of the maxilla by a surgical correction of a 9.5 mm open bite was simulated in the 10 normal individuals and found to range from 0 to 9.3 ram, with a mean of 2.0 ram. The cephalometric analysis showed a significantly smaller mandibular inclination and gonial angle in individuals with the largest resulting error in the position of the maxilla. A similar analysis was performed in the retrospective study involving 10 surgical cases. Simulating the rotation of the mandible through the centre of the condyle would have resulted in a difference ranging from 0.4-10.4 ram, with a mean of 4.0 mm between the predicted horizontal position of the maxilla and the actual position obtained.
CONCLUSIONS The present study supports the hypothesis, that a centre of mandibular rotation outside the condylar body can exist. A large inter-individual variation was found in the position of the centre of mandibular rotation, which it is suggested can be caused by interindividual difference in craniofacial morphology. In some cases, a very large discrepancy between the centre of the condyle and the calculated centre of mandibular rotation was found, which would cause considerable error in the position of the maxilla at surgery. This investigation has shown that the method described is not optimal in locating the centre of mandibular rotation. Further work needs to be put
Mandibular autorotation in orthognathic surgery
into the description of precise techniques to locate the pre- and peroperative centre of mandibular rotation for a specific mandibular movement. However until refined methods are available, it is suggested, that this method is superior to the kinematic and palpatory methods and should be applied in cases, where a large rotation of the mandible is anticipated. Appendix
Initially a gravity point of the N points before (x(1), y(1)) and after ((~(2),y(2)) rotation is defined: 1 N
X(1) = ~ x(1)~.=
y(1) =
y(1)~ y
(Equation 3~4)
Each of the N points (x, y) are redefined (s, t) relative to these centres of gravity (x, ~): t(1)~ = y(1)~-y(1) (Equation 5 6) s(2)~ = x(2)~-2(2)
(Equation 7-8) Thereafter the following variables are defined: N
A = Y', [(s(1)i*s(2)~) + (t(1)~*t(2)~)]
(Equation 9)
i-1 N
B = ~ [(s(1)~*t(2)~) + (s(2)~*t(1)~)]
(Equation 10)
i=l
The amount of rotation (0) can now be determined by: /n\
0 = Tan-1 tA )
(Equation 11)
The centre of rotation (X, Y) can be determined by:
2
(Cot~)*(~(1)-x(2))
2
2 (Equation 13)
0 where the variable Cot ~ can be substituted by A and B since: A B Cos(0) - - Sin(0) ~/A ~+ B 2 ~/)l 2+ B 2 (Equation 14-15) and thereby Cot 0 = ~/A 2+ B 2+ A B
The error of the determination (ft) could be assessed using the error in superimpositioning (ao) and the amount of rotation (0) by the following equation:
(0)
f~ = ao*Cot ~
(Equation 18)
The authors wish to extend their gratitude to Professor U. Haagerup, University of Odense, for his help with the mathematical model and to Professor S. Kreiborg, Department of Pedodontics, Royal Dental College, Copenhagen, Denmark for the loan of the computer equipment.
References errors on angular meaurements in cephalometry. Eur. J. Orthod. 10 (I988) 353 Baragar, F. A., J. W. Osborn : A model relating patterns of human jaw movement to biomechanical constraints. J. Biomechanics 10 (1984) 757 Bell, W. H., W. R. Proffit, R. P. White: Surgical correction of dentofacial deformities. Saunders, Philadelphia 1980 Bosman, A. E. : Hinge axis determination of the mandible. Thesis, University of Utrecht, 1974 Brewka, R. E. : Pantographic evaluation of cephalometric hinge axis. Am. J. Orthod. 79 (1981) 1 Chick, A. : The rotary nature of some mandibular movements. J. Prosthet. Dent. 10 (1960) 857 Fischer, R. : Beitrag zum Artikulationsproblem. Schweiz. Mschr. Zahnheilk. 62 (1952) 317 Fischer, R. : Die Artikulationslehre. In: Hfiupl K. (ed.): Die Zahn-, Mund-, und Kieferheilkunde. Ein Handbuch ffir die zahnn/irztliche Praxis 4 (1956) 95-176 Grant, P. G. : Biomechanical significance of the instantaneous centre of rotation : The human temporomandibular joint. J. Biomechanics 6 (1973) 109 Herbosa, E. G., If. S. Rotskoff , B. F. Ramos, H. S. Ambrookian :
(Equation 12) y__ ~(1)+y(2))
(Equation 17)
Ahlqvist, J., S. Eliasson, U. Welander : The effect of projection
t(2)~ = y(2)~-y(2)
2
1 N Ao = ~ ..~ [s(1)~ + t(1)~ + s(2)~ + t(2)~] - 2~/A 2+ B ~
Acknowledgement
1=1
s(1)~ = x(1)~-x(1)
occur due to measurement error. The size of this error can be determined by the following equation:
(Equation 1-2)
/=1
1 N x(2) = N~--~t.-x(2)~ y ( 2 ) =
169
(Equation 16)
As the mandible is not deformed during movement, but performs a rotation (by definition), all the points measured should be transformed into the corresponding points after rotation. However, this will not
Condylar position in superior maxillary repositioning and its effect on the temporomandibular joint. J. Oral Maxillofac. Surg. 48 (1990) 690 Hohl, T. H. : The use of an anatomical articulator in segmental orthognathic surgery. Am. J. Orthod. 73 (1978) 428 [ngerslev, C. H., B. Solow." Sex differences in craniofacial morphology. Acta Odontol. Scand. 33 (1975) 85 Kahnberg, K. E., B. Sunzel, P. Astrand: Planning and control of vertical dimension in Le Fort I osteotomies. J. Cranio-Max.Fac. Surg. 18 (1990) 267 Marko, J. V. : Simple hinge and semi-adjustable articulators in orthognathic surgery. Am. J. Orthod. Dentofac. Orthoped. 90 (1986) 37 McCollum, B. B. : Fundamentals involved in prescribing restorative dental remedies. Dent. Items Interest 61 (1939) 522, 641,724, 852, 942 Nattestad, A., P. Vedtofte, E. Mosekilde : The significance of an erroneous recording of the mandibular centre of rotation in orthognathic surgery. J. Cranio-Max.-Fac. Surg. 19 (1991) 254 Nevakari, K. : A new 'triangular transfer' method for studying mandibular movements on the basis of cephalometric roentgenograms. Acta Odontol. Scand. 12 (1954) 293 Nevakari, K. : An analysis of the mandibular movement from rest to occlusal position. Acta Odontol. Scand. 14 (1956) suppl. 19 Polido, W. D., E. Ellis, D. P. Sinn: An assessment of predictability of maxillary surgery. J. Oral Maxillofac. Surg. 48 (1990) 697
170
Journal of Cranio-Maxillo-Facial Surgery
Pospisil, O. A.. Reliability and feasibility of prediction tracing in orthognathic surgery. J. Cranio-Max.-Fac. Surg. 15 (1987) 79 Rekow, E. D., F. W. Worms, A. G. Erdman, T. M. Speidel : Treatment induced errors in occlusion following orthognathic surgery. Am. J. Orthod. 88 (1985) 425 Schallhorn, R. G..' A study of the arbitrary centre and the kinematic centre of rotation for face-bow mountings. J. Prosthet. Dent. 7 (1957) 162 Solow, B,, A. Tallgren : Head posture and craniofacial morphology. Am. J. Phys. Anthrop. 44 (1976) 417 Sperry, T. P., M. J. Steinberg, B. J. Gans." Mandibular movement during autorotation as a result of maxillary impaction surgery. Am. J. Orthod. 81 (1982) 116 Stanehina, R., E. Ellis, W. J. Gallo, R. J. Fonseea: A comparison of two measures for repositioning the maxilla during orthognathic surgery. Int. J. Adult Orthod. Orthognat. Surg. 3 (1988) 149 Torii, K. : Analysis of rotation centers of various mandibular closures. J. Prosthet Dent. 61 (1989) 285
Turvey, T. A., D. J. Hall, L. C. Fish, B. N. Epker : Surgicalorthodontic treatment planning for simultaneous mobilisation of the maxilla and mandible in correction of dentofacial deformity. Oral Surg. 54 (1982) 491 Viteporn, S., B. Melsen, M. Bundgaard: Postsurgical change of mandibular position in patients following Le Fort I osteotomy. Int. J. Adult Orthod. Orthognat. Surg. 5 (1990) 91 Walker, P. M. : Discrepancies between arbitrary and true hinge axes. J. Prosthet. Dent. 43 (1980) 279 Winstanley, R. B. : Hinge axis location on the articulator. J. Prosthet. Dent. 42 (1979) 135
A. Nattestad, DDS, PhD, Department of Oral and Maxillofacial Surgery, Royal Dental College, Norre Alle 20, DK-2200 Copenhagen, Denmark. Paper received 31 May 1991 Accepted 28 October 1991