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Location of the mandibular center of autorotation in maxillary impaction surgery
Location of the mandibular center of autorotationin maxillary impaction surgery E. Dianne Rekow, DDS, PhD," T. Michael Speidel, DDS, MSD, b and Richard A. Koenig, DDS, MSD c Balthnore, Md., Minneapolis, Mimz., and Lima, Peru Controversy exists about the location of the center of autorotation of the mandible after maxillary impaction surgery. This investigation focuses on the problems associated with locating that center of autorotation and identifies factors that can increase the probability of accurately identifying its location for predicting surgical outcomes. The reliability of the Rouleaux technique for calculating the centers of rotation is established and is shown to be acceptable, as long as the landmarks used for determining the center are properly selected, and the magnitude of the rotation required is sufficient. The location of the centers of autorotation of the mandibles after maxillary impaction surgery for 46 patients was used to investigate the errors associated with landmark selection and amounts of rotation. Although there is much variation in its location, the center does not lie within the body of the condyle but instead lies away from the condyle. Guidelines for maximizing the reliability of predicting surgical outcomes on the basis of autorotation of the mandible after maxillary impaction surgery are given. (AM J OFITHOD DENTOFACORTHOP 1993;103:530-6.)
A f t e r maxillaryimpaction surgery, the mandible autorotates to a new position anterior and superior to its original position. The magnitude of the autorotation is an integral part of the treatment planning process. Unfortunately, there is not much agreement concerning the point about which this autorotation occurs. The final position of the mandible, however, is completely determined by that point and the amount of rotation about it. The objective of this study is three fold: (1) to establish the reliability of the principle technique used for locating the center of rotation, (2) to establish the location of the center of rotation in a number of actual surgical impaction cases, and (3) to establish guidelines for selecting landmarks for reliably predicting surgical outcomes. REVIEW OF THE LITERATURE
Several studies have discussed the location of the center of rotation about which autorotation of the mandible occurs after maxillary impaction surgery. Yet there is little agreement about where that center lies. Shendal et al.t and Radney and Jacobs-" argue that it was located at some point at the top of the head of the condyle. Fish and Epker ~ also use the head of the condyle for
'Associate Professor, University of Maryland, Baltimore College of Dental Surgery Dental School. Department of Orthodontics. bDivision Chairman, University of Minnesota. School of Dentistry, Department of Diagnostic and Surgical Sciences, Division of Orthodontics. 'Private orthodontic practice, Lima, Peru. Copyright 9 1993 by the American Association of Orthodontists. 0389-5406/93151.00 ~- 0.10 8/1135237
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predicting the center of rotation of autorotation. ValnotP argued that the rotation occurred near a hinge axis angle formed by the intersection of the Frankfort horizontal plane and a line joining the condyle center and pogonion, placing it very near the head of the condyle. Sperry et al. 5 suggested that a more appropriate center of rotation lies in the mastoid process. Studies by Nevakari, 6 Grant, 7 and HalP support this conclusion. Brewka 9 found that in more than 50% of the cases he studied the center is located below and behind the cephalometric center of the condyle. Lepera t~ argues that it lies within the neck of the mandible. One of the major motivations for beginning this study was this disagreement. How could so many investigators come to so many different conclusions? The center of rotation is a simply a point around which a body rotates to move from one position to another. The most common method for locating the center of rotation is that described by Rouleaux." This method is the one most commonly used in studies that use centers of rotation that have been reported in the orthodontic literature. With this method, any two landmarks on a body (the mandible, for instance) are selected (points A and B in Fig. 1). The location of these two landmarks is identified in an initial position and then again after the rotation has occurred (points A and A' and points B and B'). A line is drawn between the initial and rotated positions (A to A' and B to B'). The perpendicular bisector to each of those lines is constructed. The intersection of the two bisectors is defined as the center of rotation.
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Fig. 2. Location el arbitrary reference points and lines.
Fig. 1. Location of center of autorotation.
With this Rouleaux method, the center of rotation can be found either graphically or computationally. When it is determined computationally, inaccuracies because of human error are eliminated. But some sources of error in locating the center o f rotation still remain. A n y error in constructing or calculating the perpendicular bisectors can result in remarkable 9 changes in the location o f the center o f rotation. 6'~2 Factors that can influence and contribute to the errors include (1) errors in coordinate measurement (errors in locating the same landmark in its original and rotated position), '3 (2) errors in placement of the position of a landmark on the postrotation x-ray film from either misidentification o f the landmark or changes in radiographic image as a result o f head variation, ~4 (3) location of the landmark selected relative to the location of the center o f rotation, and (4) the amount of rotation. Panjabi and his colleagues t''6 were perhaps among the first to address the problem o f the impact o f errors on the location o f the center o f autorotation. The results o f that work provided a basis for this investigation, which has three specific objectives. The first objective is to establish the reliability of the Rouleaux technique for locating the center o f autorotation for the mandible after maxillary impaction surgery. The second is to identify the location o f the actual center o f autorotation on the basis o f the results b f 46 maxillary impaction
surgical cases. The third is to seek guidance for selecting landmarks that minimize errors in predicting surgical outcomes.
MATERIALS AND METHODS Records were obtained for 46 patients who had undergone LeFort ! osteotomies with superior repositioning of the maxilla. The subjects had been treated at the University of Minnesota School of Dentistry. All were nongrowing patients. None had mandibular surgery (with the possible exception of an occasional sliding genioplasty). For all subjects, clear preoperative, postoperative, and final treatment cephalometric radiographs with the teeth in occlusion wire available. The radiographs were traced according to standard procedures, and the outline of the anterior cranial base, the maxilla (including first molar and central incisor), and the mandible (including internal border of the symphysis and mandibular canal) were recorded. An arbitrary 10 mm reference line was drawn in the middle of the ramus, approximately perpendicular to the occlusal plane and passing approximately through the mandibular foramen (see Fig. 2). Another arbitrary reference line was placed in the body of the mandible, parallel to but below the occlusal plane, passing approximately through the most mesial portion of the mandibular first molar and the apex of the cingulum of the mandibular central incisor. Three templates of anatomic structures from the preoperative radiographs were made. The first included the anterior cranial fossa and outline of sella turcica. The second was for the maxilla. The third was for the mandible and included the two arbitrary reference lines. The templates were used to
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ensure maximum congruity between sequential tracings for each patient. The mandibular reference lines were established on the tracing of the presurgical ccphalometric radiograph. They were transferred to the mandible on subsequent radiographs by superimposing the mandibular template onto the mandible in those films. Digitizing of the end points was done with the template in place. When the tracings were completed, a number of landmarks were digitized. The digitizing was accomplished with a Summagraphic I-D digitizing tablet (Summagraphics Corp., Fairfield, Conn.) with resolution of 0.1 mm and accuracy of 0.05 mm. The x and y coordinates of the landmarks were automatically recorded in a DEC 11/44 computer (Digital Equipment Corp., Maynerd, Mass.). The landmarks digitized were as follows: 1. Nasion, the most anterior midline point on the frontonasal suture. 2. Sella, the center of the radi~raphic outline of sella turcica as determined by inspection. 3. Condyle point, a constructed point determined from a perpendicular bisector from the sella-nasion line to the most superior portion of the condyle. 4. The four end points of the arbitrary reference lines, the upper and lower ends of the ramal reference line defined by points 1 and 2 and the posterior and anterior ends of the mandibular body defined by points 3 and 4. All coordinates were measured relative to an x,y coordinate system with its origin at nasion. The x-axis was defined as the sclla-nasion line. To determine the accuracy of the Rouleaux method for establishing the location of the center of rotation (objective 1), surgical impaction were simulated. One subject was selected at random, and tracings were made from the preoperative radiographs as described previously. An arbitrary center of rotation was selected (in the head of the condyle). A template of the mandible was placed over the preoperative tracing. A pin was placed through the template at the arbitrary center of rotation. The template was rotated through 2~, simulating autorotation after maxillary impaction. Landmarks defined previously were digitized. The process was repeated 10 times on 10 different days. From the digitized landmarks, the center of rotation was calculated for each trial according to the Rouleaux method. The location of the center of rotation was calculated by computer pr~rams as follows: (1) Equations for lines were computed, joining the arbitrary reference points in their prerotation and postrotation positions. (2) The midpoint of each of these lines was calculated. (3) The equations for the lines representing the perpendicular bisectors were determined. (4) The intersection of those bisectors (defined as the center of rotation) was calculated and reported as x,y coordinates. Different combinations of the arbitrary reference points were used to calculate the position of the center of rotation. The location of the center of rotation was determined by using points I and 2, points 1 and 3, points I and 4, points 2 and 3, points 2 and 4, and points 3 and 4. The distance between the actual center of rotation (the pin) and the calculated center of rotation
American Journal of Orthodontics and Dentofacial Orthopedics June 1993
was calculated for each combination of sets of arbitrary reference points. This same process was repeated for 6~ 10~ and 20" of rotation. To locate the center of mandibular autorotation after actual surgery and to investigate the effect of errors in its location as a function of (1) landmark selection and (2) magnitude of rotation (the second objective), each of the 46 sets of radi~raphs were digitized as described previously. The centers of autorotation were calculated according to the Rouleaux method for each combination of arbitrary reference points. The distance between the actual center of rotation and the condyle (a digitized point) were calculated. In addition, after the centers of rotation were determined, the amount of rotation of the mandible resulting from the surgical maxillary impaction was calculated. RESULTS When the center of autorotation is calculated on the basis of simulated, precisely located landmarks, errors result. In theory, the calculated location of the simulated center of rotation should fall exactly at the location of the arbitrary center of rotation (defined by the pin). In fact, there are rather dramatic errors. For 2 ~ of rotation, the m i n i m u m of the mean error in locating the center of rotation occurred with points 2 and 3. In this case, the calculated center of rotation was 17.91 m m from the mean actual center. Standard deviation for all 10 trials was I 1.32 mm, the range was from 6.36 to 47.03 mm. Ths maximum value was recorded for points 3 and 4 with a mean error of 329.53 ram. For this combination of points, the standard deviation was 438.36 mm, and the range for the l0 trials was from 29.71 m m to 1 3 3 6 . 8 8 mm. For 6 ~ of rotation the m i n i m u m mean error occurred with points 1 and 3 (mean error = 2.75 rnm). The standard deviation for this combination of points was 1.23 mm, and the range of values for the I0 trials was from 0.91 to 4.21 mm. The maximum was obtained with points l and 2 where the mean error was 96.68 mm, well below the 329 m m found with only 2 ~ rotation. The standard deviation for points l and 2 was 65.54 mm, and the range for the I0 trials was from 12.01 to 210.19 mm versus 29.17 mm to 1336.88 mm for 2 ~ of rotation. For 10~ of rotation the m i n i m u m error occurred with points 2 and 4 (mean error -- 2.15 ram). The standard deviation was 1.40 mm; the range from 0.85 to 5.39 mm. Points 1 and 4 were nearly as good (mean error = 2.22 mm, standard deviation 1.08 mm, range from 0.81 to 3.87 mm). The maximum mean error occurred when the center of rotation was calculated by using the combination of points 1 and 2, giving a mean error of 17.74 mm. The
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Fig. 3. Calculated locations of center of autorotation versus arbitrary actual center of autorotation for simulated impactions. Arbitrary center indicated by *, For 20 ~ rotation with reference points 1 and 4.
standard deviation with this combination was 20.32 mm; the range for the 10 trials from 0.95 to 62.87 mm. When the simulated rotation was 20 ~ the minimum mean error occurred with points 1 and 4, with a value of just 0.88 mm. The standard deviation was 0.72 mm; the range from 0.25 to 2.53 mm. Fig. 3 shows the calculated locations of centers of autorotation versus the arbitrary actual center of rotation for 20" rotation, based on points 1 and 4. The star indicates the actual center of rotation (the pin). Maximum errors for 20 ~ of rotation occurred with points 1 and 2, but the value of the mean error had dropped to 8.26 m m (less than half of the maximum for 10 degrees of rotation). This maximum is well below the best result determined from a 2" rotation (where the minimum mean error was 17.91 mm). It is apparent that the error decreases with increasing amount of opening rotation. Another relationship, however, bears notice. If lines are drawn from the center of rotation to each of the four arbitrary reference points, the angle between the lines to points 1 and 2 is 2". The angle between the lines to points 3 and 4 is 8". However, the angle between lines to all the other combinations of points is substantially higher (31 ~ between points I and 3, 39 ~ between points I and 4, 32 ~ for points 2 and J
3, and 40 ~ for points 2 and 4). The mean errors for the simulated rotation are substantially lower for all of the combinations of points that are separated by large angles than for points that have a smaller angle between them. For instance, at 20 ~ of rotation, the mean error is 8.26 m m for points 1 and 2 (separated by 2~ and 3.83 mm for points 3 and 4 (separated by 8"). The points with larger angles between them yielded smaller mean errors (1.11 mm for points 1 and' 3, 0.88 mm for points 1 and 4, 1.62 mm for points 2 and 3, and 1.50 m m for points 2 and 4). The distance between the calculated location of the center of rotation for the 46 surgery patients and the condyle point (a digitized landmark) was calculated for different combinations of reference points. The distances from the c-6nter of rotation to the condyle ranged from 0.48 m m (patient 39, points 1 and 4) to 926.47 mm (patient 38, points 3 and 4). The mean values range from a m i n i m u m of 49.00 m m (points 1 and 3) to a maximum of 107.61 mm (points 3 and 4). The standard deviation ranges from 59.15 mm (points 1 and 3) to 168.85 mm (points 3 and 4). This emphasizes the importance of the relative positions of the p o i n t s - - t h o s e that have a small angle between them (such as points 3 and 4) give greater errors. Those with large angles
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Fig. 4. Locations of mandibular centers of autorotation for 46 maxillary impaction patients.
(such as points 1 and 4 and points 1 and 3) give smaller errors. Fig. 4 shows the distribution of location of the center of autorotation for points 1 and 3 for all 46 patients. The degree of autorotation of the mandible for each of the 46 patients was calculated. The average impaction was 4 ~ but individual values ranged from a - 1~ (a slight opening rather than closing as would result from a successful impaction) to 9 ~ Since, as seen in the simulated rotation portion of this study, an increase in rotation reduces the error, the patients with rotations of greater than 6 ~ were investigated as a separate subset. When this subset of patients is considered, the minimum difference between the calculated center of rotation and the condyle is 3.15 mm (patient 22, points 1 and 3). The maximum is 54.70 mm (patient 22, points 1 and 2) versus a maximum of 926.47 for all patients. The mean error ranges from a minimum of 23.64 mm (points 1 and 3) to a maximum of 34.24 mm (points 3 and 4). Equally importantly, the range in standard deviations drops dramatically, ranging from 12.19 mm (points 2 and 3) to 18.36 mm (points 1 and 2) versus 59.15 mm to 168.85 mm for all patients. These results are presented graphically in Fig. 5.
DISCUSSION Substantial errors can be found in the calculated position of the center of mandibular autorotation after maxillary impaction surgery. In the simulated study to establish the reliability of the technique, the calculated
center of rotation should have been identical to the arbitrary (the pin) center of rotation every time. In fact there is a rather dramatic variation. Care must be taken in selecting the landmarks from which the center of rotation is to be estimated or predicted. Marker angle is determined by the angle between lines drawn from the expected center of autorotation to the selected landmarks. (For example, one marker angle in this investigation is the angle between the line from the center of autorotation to point 1 and the line from the center of autorotation to point 2.) As the marker angle increases, the contribution to error dramatically decreases. Results presented by Panjabi ~6 and verified by this investigation show that the marker angle should be as close to 90 ~ as possible (see Fig. 6) but are acceptable as it nears 30 ~. In this study, by using points that are separated by more than 30 ~ (such as points 1 and 3, 1 and 4, or 2 and 4), an improvement of 10 times was obtained when compared with the error found by using calculations based on points that are separated by less than 10~ (such as points 1 and 2 or 3 and 4). As the angle between the points increases, the precision in locating the center of autorotation improves. Another contribution to the error in locating the center of autorotation is the magnitude of rotation. As predicted by Panjabi ~6and verified in this investigation, as the angle of rotation increases, the contribution to the error in locating the center of rotation falls exponentially. In the simulated study, maximum mean error
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Fig. 5. Locations of mandibular centers of autorotation for patients with 6 ~ or more of autorotation.
at 2 ~ of rotation was 329.53 mm. It fell to 96.68 mm at 6 ~ of rotation, 17.74 mm at 10~ of rotation, and 8.26 mm at 20 ~ of rotation. In clinical applications, there is usually little freedom in manipulating the amount of rotation that will 9 be required. Certainly the physiologic and esthetic constraints will determine that value. Consequently, since amount of rotation cannot be maximized for optimal mathematical results, for any predictions using the Rouleaux technique for locating the center of autorotation, special care must be taken to ensure that landmarks are separated by as great a distance as possible. This study suggests that substantial errors can result in the location of the center of autorotation as calculated with the Rouleaux method (the method most commonly reported in the orthodontic literature). The location of the landmarks relative to each other and to the expected center of autorotation can significantly affect the location of the actual center of rotation. The influence on landmark position and amount of rotation provide some insight into the reason for the disagreement about the actual location of the center of autorotation of the mandible after maxillary impaction surgery. Depending on the landmarks selected, the vari-
ation in the range of locations for the 46 cases investigated is remarkable. When the findings from the simulated study are considered and maximum separation between landmarks is used for the calculations, the accuracy of the location improves. For maximally separated landmarks, and for patients with greater than 6 ~ of rotation, it appears that the "true" center of autorotation is still variable. In no case (even with the best possible combination of points and amrunts of rotation) did the center lie on the condyle. These findings provide some important guidelines for future predictions of mandibular position after maxillary impaction surgery. First, it is critical to remember that even with the best of all possible combinations of data from previous treatments, the actual location of the center of rotation will probably not lie in the condyle itself, but will be some unpredictable distance away from the condyle. Secondly, after selecting some arbitrary point to make the estimates and predictions, to minimize additional error contribution, it is critical that the landmarks selected for the analysis be separated by as great a distance as possible. In summary, three conclusions can be drawn from this investigation: (1) The Rouleaux method is reliable
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105 120 135 150 Marker Angle (degrees) 60
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Fig. 6. Errors as a function of marker angle (from Panjabi'5).
for calculating the location o f the center o f rotation, but only if landmarks are properly selected and rotation angles are greater than at least 6 ~. (2) There is great variation in the location o f the center o f autorotation o f the mandible after actual maxillary !mpaction surgery. (3) If this technique is to be used for treatment planning to determine the position o f t h e mandible after maxillary impaction surgery, it can only serve as a broad guideline and must not be considered to predict absolute locations. Authors" note: Tables providing values for (I) the distance
between the center of autorotation and the calculated center of rotation for the simulated study (for 2 ~ 6~ 10~ and 20 ~ of rotation); (2) the distance between the center of autorotation and the condyle, determined by using different combinations of reference points for all surgery patients and for those with 6 ~ or more of autorotation; and (3) the degree of rotation for each surgery patient are avilable, on request, directly from the author.
REFERENCES 1. Shendel SA, Eisenfeld JM, Bell WM, Epker BN. Superior repositioning of the maxilla: stability and soft tissue osseous relations. AM J ORTItOD DENTOFACOR'I-HOP 1976;70:663-74. 2. Radney L, Jacobs J. Soft-tissue changes associated with surgical total maxillary intrusion. AM J Own~ot) DF~"rOF,~,C ORTnOP 1981;80:191-212. 3. Fish LC, Epker BN. Surgical-orthodontic cephalometfic prediction tracings. J Clin Orthod 1980;14:36-52. 4. Valinoti JR. The hinge axis angle. J Clinc Orthod 1977;9: 551-9. 5. Sperry TP, Steinberg M J, Gans BJ. Mandibular movement during autorotation as a result of maxillary impaction surgery. Ar~1 J ORI"HODDENTOFACORTHOP 1982;81:! 16-22. 6. Navakari K. An analysis of the mandibular movement for rest to occlusal position. Acta Odontol Scand 1956;14:Suppl. 19. 7. GrantPG. Biomechanicalsignificanceoftheinstantaneouscenter of rotation: the human temporomandibular joint. J Biomech 1973;6:109-13. 8. Hall RE. An analysis of the work and ideas of investigators and 9authors of relations and movement of the mandible. J Am Dent Assoc 1929;16:1642-93. 9. Brewka RE. Pant~raphie evaluation of cephalometric hinge axis. AM J ORTIIOD DEN'rOFACORTHOP 1980;79:1-19. tO. Lepera F. Determination of the hinge axis clutches on condyle position. J. Prosthet Dent 1958;8:260-5. I I. Rouleaux F. The kinematics of machinery: outline of a theory of machines (translated by A.B.W. Kennedy). Dover, 1875. 12. Hultgren BW. A method for analyzing some effects of orthodontic mechanotherapy upon the directions and proportions of facial growth from two samples of Angle Class I. Division I 9 malocclusions utilizing computer graphic routines. [MS thesis], University of Minnesota, 1977. 13. Dimnet J, Carret JP, Gonon G, Fisher LP. A technique for joint center analysis using a stored pmgeram calculator. J Biomech 1974;10:659-73. 14. Soudan K, Van Audekercke R. Methods, difficulties, and inaccuracies in the study of human joint kinematics and pathokinematics by the instant axis concept. J Biomech 1979;12:27-33. 15. Panjabi MM. Centers and angles of rotation of body joints: a study of errors and optimization. J Biomech 1979;12:91 !-20. 16. Panjabi MM. Errors in kinematic parameters of a planar joint: guidelines for optimal experimental design. J Biomech 1982;15:537--44.
Reprhlt requests to: Dr. E. Dianne Rekow The University of Maryland Baltimore College of Dental Surgery Dental School 666 W. Baltimore S t . Baltimore, MD 21201-1586
A f t e r maxillaryimpaction surgery, the mandible autorotates to a new position anterior and superior to its original position. The magnitude of the autorotation is an integral part of the treatment planning process. Unfortunately, there is not much agreement concerning the point about which this autorotation occurs. The final position of the mandible, however, is completely determined by that point and the amount of rotation about it. The objective of this study is three fold: (1) to establish the reliability of the principle technique used for locating the center of rotation, (2) to establish the location of the center of rotation in a number of actual surgical impaction cases, and (3) to establish guidelines for selecting landmarks for reliably predicting surgical outcomes. REVIEW OF THE LITERATURE
Several studies have discussed the location of the center of rotation about which autorotation of the mandible occurs after maxillary impaction surgery. Yet there is little agreement about where that center lies. Shendal et al.t and Radney and Jacobs-" argue that it was located at some point at the top of the head of the condyle. Fish and Epker ~ also use the head of the condyle for
'Associate Professor, University of Maryland, Baltimore College of Dental Surgery Dental School. Department of Orthodontics. bDivision Chairman, University of Minnesota. School of Dentistry, Department of Diagnostic and Surgical Sciences, Division of Orthodontics. 'Private orthodontic practice, Lima, Peru. Copyright 9 1993 by the American Association of Orthodontists. 0389-5406/93151.00 ~- 0.10 8/1135237
530
predicting the center of rotation of autorotation. ValnotP argued that the rotation occurred near a hinge axis angle formed by the intersection of the Frankfort horizontal plane and a line joining the condyle center and pogonion, placing it very near the head of the condyle. Sperry et al. 5 suggested that a more appropriate center of rotation lies in the mastoid process. Studies by Nevakari, 6 Grant, 7 and HalP support this conclusion. Brewka 9 found that in more than 50% of the cases he studied the center is located below and behind the cephalometric center of the condyle. Lepera t~ argues that it lies within the neck of the mandible. One of the major motivations for beginning this study was this disagreement. How could so many investigators come to so many different conclusions? The center of rotation is a simply a point around which a body rotates to move from one position to another. The most common method for locating the center of rotation is that described by Rouleaux." This method is the one most commonly used in studies that use centers of rotation that have been reported in the orthodontic literature. With this method, any two landmarks on a body (the mandible, for instance) are selected (points A and B in Fig. 1). The location of these two landmarks is identified in an initial position and then again after the rotation has occurred (points A and A' and points B and B'). A line is drawn between the initial and rotated positions (A to A' and B to B'). The perpendicular bisector to each of those lines is constructed. The intersection of the two bisectors is defined as the center of rotation.
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'1
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Fig. 2. Location el arbitrary reference points and lines.
Fig. 1. Location of center of autorotation.
With this Rouleaux method, the center of rotation can be found either graphically or computationally. When it is determined computationally, inaccuracies because of human error are eliminated. But some sources of error in locating the center o f rotation still remain. A n y error in constructing or calculating the perpendicular bisectors can result in remarkable 9 changes in the location o f the center o f rotation. 6'~2 Factors that can influence and contribute to the errors include (1) errors in coordinate measurement (errors in locating the same landmark in its original and rotated position), '3 (2) errors in placement of the position of a landmark on the postrotation x-ray film from either misidentification o f the landmark or changes in radiographic image as a result o f head variation, ~4 (3) location of the landmark selected relative to the location of the center o f rotation, and (4) the amount of rotation. Panjabi and his colleagues t''6 were perhaps among the first to address the problem o f the impact o f errors on the location o f the center o f autorotation. The results o f that work provided a basis for this investigation, which has three specific objectives. The first objective is to establish the reliability of the Rouleaux technique for locating the center o f autorotation for the mandible after maxillary impaction surgery. The second is to identify the location o f the actual center o f autorotation on the basis o f the results b f 46 maxillary impaction
surgical cases. The third is to seek guidance for selecting landmarks that minimize errors in predicting surgical outcomes.
MATERIALS AND METHODS Records were obtained for 46 patients who had undergone LeFort ! osteotomies with superior repositioning of the maxilla. The subjects had been treated at the University of Minnesota School of Dentistry. All were nongrowing patients. None had mandibular surgery (with the possible exception of an occasional sliding genioplasty). For all subjects, clear preoperative, postoperative, and final treatment cephalometric radiographs with the teeth in occlusion wire available. The radiographs were traced according to standard procedures, and the outline of the anterior cranial base, the maxilla (including first molar and central incisor), and the mandible (including internal border of the symphysis and mandibular canal) were recorded. An arbitrary 10 mm reference line was drawn in the middle of the ramus, approximately perpendicular to the occlusal plane and passing approximately through the mandibular foramen (see Fig. 2). Another arbitrary reference line was placed in the body of the mandible, parallel to but below the occlusal plane, passing approximately through the most mesial portion of the mandibular first molar and the apex of the cingulum of the mandibular central incisor. Three templates of anatomic structures from the preoperative radiographs were made. The first included the anterior cranial fossa and outline of sella turcica. The second was for the maxilla. The third was for the mandible and included the two arbitrary reference lines. The templates were used to
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ensure maximum congruity between sequential tracings for each patient. The mandibular reference lines were established on the tracing of the presurgical ccphalometric radiograph. They were transferred to the mandible on subsequent radiographs by superimposing the mandibular template onto the mandible in those films. Digitizing of the end points was done with the template in place. When the tracings were completed, a number of landmarks were digitized. The digitizing was accomplished with a Summagraphic I-D digitizing tablet (Summagraphics Corp., Fairfield, Conn.) with resolution of 0.1 mm and accuracy of 0.05 mm. The x and y coordinates of the landmarks were automatically recorded in a DEC 11/44 computer (Digital Equipment Corp., Maynerd, Mass.). The landmarks digitized were as follows: 1. Nasion, the most anterior midline point on the frontonasal suture. 2. Sella, the center of the radi~raphic outline of sella turcica as determined by inspection. 3. Condyle point, a constructed point determined from a perpendicular bisector from the sella-nasion line to the most superior portion of the condyle. 4. The four end points of the arbitrary reference lines, the upper and lower ends of the ramal reference line defined by points 1 and 2 and the posterior and anterior ends of the mandibular body defined by points 3 and 4. All coordinates were measured relative to an x,y coordinate system with its origin at nasion. The x-axis was defined as the sclla-nasion line. To determine the accuracy of the Rouleaux method for establishing the location of the center of rotation (objective 1), surgical impaction were simulated. One subject was selected at random, and tracings were made from the preoperative radiographs as described previously. An arbitrary center of rotation was selected (in the head of the condyle). A template of the mandible was placed over the preoperative tracing. A pin was placed through the template at the arbitrary center of rotation. The template was rotated through 2~, simulating autorotation after maxillary impaction. Landmarks defined previously were digitized. The process was repeated 10 times on 10 different days. From the digitized landmarks, the center of rotation was calculated for each trial according to the Rouleaux method. The location of the center of rotation was calculated by computer pr~rams as follows: (1) Equations for lines were computed, joining the arbitrary reference points in their prerotation and postrotation positions. (2) The midpoint of each of these lines was calculated. (3) The equations for the lines representing the perpendicular bisectors were determined. (4) The intersection of those bisectors (defined as the center of rotation) was calculated and reported as x,y coordinates. Different combinations of the arbitrary reference points were used to calculate the position of the center of rotation. The location of the center of rotation was determined by using points I and 2, points 1 and 3, points I and 4, points 2 and 3, points 2 and 4, and points 3 and 4. The distance between the actual center of rotation (the pin) and the calculated center of rotation
American Journal of Orthodontics and Dentofacial Orthopedics June 1993
was calculated for each combination of sets of arbitrary reference points. This same process was repeated for 6~ 10~ and 20" of rotation. To locate the center of mandibular autorotation after actual surgery and to investigate the effect of errors in its location as a function of (1) landmark selection and (2) magnitude of rotation (the second objective), each of the 46 sets of radi~raphs were digitized as described previously. The centers of autorotation were calculated according to the Rouleaux method for each combination of arbitrary reference points. The distance between the actual center of rotation and the condyle (a digitized point) were calculated. In addition, after the centers of rotation were determined, the amount of rotation of the mandible resulting from the surgical maxillary impaction was calculated. RESULTS When the center of autorotation is calculated on the basis of simulated, precisely located landmarks, errors result. In theory, the calculated location of the simulated center of rotation should fall exactly at the location of the arbitrary center of rotation (defined by the pin). In fact, there are rather dramatic errors. For 2 ~ of rotation, the m i n i m u m of the mean error in locating the center of rotation occurred with points 2 and 3. In this case, the calculated center of rotation was 17.91 m m from the mean actual center. Standard deviation for all 10 trials was I 1.32 mm, the range was from 6.36 to 47.03 mm. Ths maximum value was recorded for points 3 and 4 with a mean error of 329.53 ram. For this combination of points, the standard deviation was 438.36 mm, and the range for the l0 trials was from 29.71 m m to 1 3 3 6 . 8 8 mm. For 6 ~ of rotation the m i n i m u m mean error occurred with points 1 and 3 (mean error = 2.75 rnm). The standard deviation for this combination of points was 1.23 mm, and the range of values for the I0 trials was from 0.91 to 4.21 mm. The maximum was obtained with points l and 2 where the mean error was 96.68 mm, well below the 329 m m found with only 2 ~ rotation. The standard deviation for points l and 2 was 65.54 mm, and the range for the I0 trials was from 12.01 to 210.19 mm versus 29.17 mm to 1336.88 mm for 2 ~ of rotation. For 10~ of rotation the m i n i m u m error occurred with points 2 and 4 (mean error -- 2.15 ram). The standard deviation was 1.40 mm; the range from 0.85 to 5.39 mm. Points 1 and 4 were nearly as good (mean error = 2.22 mm, standard deviation 1.08 mm, range from 0.81 to 3.87 mm). The maximum mean error occurred when the center of rotation was calculated by using the combination of points 1 and 2, giving a mean error of 17.74 mm. The
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Fig. 3. Calculated locations of center of autorotation versus arbitrary actual center of autorotation for simulated impactions. Arbitrary center indicated by *, For 20 ~ rotation with reference points 1 and 4.
standard deviation with this combination was 20.32 mm; the range for the 10 trials from 0.95 to 62.87 mm. When the simulated rotation was 20 ~ the minimum mean error occurred with points 1 and 4, with a value of just 0.88 mm. The standard deviation was 0.72 mm; the range from 0.25 to 2.53 mm. Fig. 3 shows the calculated locations of centers of autorotation versus the arbitrary actual center of rotation for 20" rotation, based on points 1 and 4. The star indicates the actual center of rotation (the pin). Maximum errors for 20 ~ of rotation occurred with points 1 and 2, but the value of the mean error had dropped to 8.26 m m (less than half of the maximum for 10 degrees of rotation). This maximum is well below the best result determined from a 2" rotation (where the minimum mean error was 17.91 mm). It is apparent that the error decreases with increasing amount of opening rotation. Another relationship, however, bears notice. If lines are drawn from the center of rotation to each of the four arbitrary reference points, the angle between the lines to points 1 and 2 is 2". The angle between the lines to points 3 and 4 is 8". However, the angle between lines to all the other combinations of points is substantially higher (31 ~ between points I and 3, 39 ~ between points I and 4, 32 ~ for points 2 and J
3, and 40 ~ for points 2 and 4). The mean errors for the simulated rotation are substantially lower for all of the combinations of points that are separated by large angles than for points that have a smaller angle between them. For instance, at 20 ~ of rotation, the mean error is 8.26 m m for points 1 and 2 (separated by 2~ and 3.83 mm for points 3 and 4 (separated by 8"). The points with larger angles between them yielded smaller mean errors (1.11 mm for points 1 and' 3, 0.88 mm for points 1 and 4, 1.62 mm for points 2 and 3, and 1.50 m m for points 2 and 4). The distance between the calculated location of the center of rotation for the 46 surgery patients and the condyle point (a digitized landmark) was calculated for different combinations of reference points. The distances from the c-6nter of rotation to the condyle ranged from 0.48 m m (patient 39, points 1 and 4) to 926.47 mm (patient 38, points 3 and 4). The mean values range from a m i n i m u m of 49.00 m m (points 1 and 3) to a maximum of 107.61 mm (points 3 and 4). The standard deviation ranges from 59.15 mm (points 1 and 3) to 168.85 mm (points 3 and 4). This emphasizes the importance of the relative positions of the p o i n t s - - t h o s e that have a small angle between them (such as points 3 and 4) give greater errors. Those with large angles
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(such as points 1 and 4 and points 1 and 3) give smaller errors. Fig. 4 shows the distribution of location of the center of autorotation for points 1 and 3 for all 46 patients. The degree of autorotation of the mandible for each of the 46 patients was calculated. The average impaction was 4 ~ but individual values ranged from a - 1~ (a slight opening rather than closing as would result from a successful impaction) to 9 ~ Since, as seen in the simulated rotation portion of this study, an increase in rotation reduces the error, the patients with rotations of greater than 6 ~ were investigated as a separate subset. When this subset of patients is considered, the minimum difference between the calculated center of rotation and the condyle is 3.15 mm (patient 22, points 1 and 3). The maximum is 54.70 mm (patient 22, points 1 and 2) versus a maximum of 926.47 for all patients. The mean error ranges from a minimum of 23.64 mm (points 1 and 3) to a maximum of 34.24 mm (points 3 and 4). Equally importantly, the range in standard deviations drops dramatically, ranging from 12.19 mm (points 2 and 3) to 18.36 mm (points 1 and 2) versus 59.15 mm to 168.85 mm for all patients. These results are presented graphically in Fig. 5.
DISCUSSION Substantial errors can be found in the calculated position of the center of mandibular autorotation after maxillary impaction surgery. In the simulated study to establish the reliability of the technique, the calculated
center of rotation should have been identical to the arbitrary (the pin) center of rotation every time. In fact there is a rather dramatic variation. Care must be taken in selecting the landmarks from which the center of rotation is to be estimated or predicted. Marker angle is determined by the angle between lines drawn from the expected center of autorotation to the selected landmarks. (For example, one marker angle in this investigation is the angle between the line from the center of autorotation to point 1 and the line from the center of autorotation to point 2.) As the marker angle increases, the contribution to error dramatically decreases. Results presented by Panjabi ~6 and verified by this investigation show that the marker angle should be as close to 90 ~ as possible (see Fig. 6) but are acceptable as it nears 30 ~. In this study, by using points that are separated by more than 30 ~ (such as points 1 and 3, 1 and 4, or 2 and 4), an improvement of 10 times was obtained when compared with the error found by using calculations based on points that are separated by less than 10~ (such as points 1 and 2 or 3 and 4). As the angle between the points increases, the precision in locating the center of autorotation improves. Another contribution to the error in locating the center of autorotation is the magnitude of rotation. As predicted by Panjabi ~6and verified in this investigation, as the angle of rotation increases, the contribution to the error in locating the center of rotation falls exponentially. In the simulated study, maximum mean error
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Fig. 5. Locations of mandibular centers of autorotation for patients with 6 ~ or more of autorotation.
at 2 ~ of rotation was 329.53 mm. It fell to 96.68 mm at 6 ~ of rotation, 17.74 mm at 10~ of rotation, and 8.26 mm at 20 ~ of rotation. In clinical applications, there is usually little freedom in manipulating the amount of rotation that will 9 be required. Certainly the physiologic and esthetic constraints will determine that value. Consequently, since amount of rotation cannot be maximized for optimal mathematical results, for any predictions using the Rouleaux technique for locating the center of autorotation, special care must be taken to ensure that landmarks are separated by as great a distance as possible. This study suggests that substantial errors can result in the location of the center of autorotation as calculated with the Rouleaux method (the method most commonly reported in the orthodontic literature). The location of the landmarks relative to each other and to the expected center of autorotation can significantly affect the location of the actual center of rotation. The influence on landmark position and amount of rotation provide some insight into the reason for the disagreement about the actual location of the center of autorotation of the mandible after maxillary impaction surgery. Depending on the landmarks selected, the vari-
ation in the range of locations for the 46 cases investigated is remarkable. When the findings from the simulated study are considered and maximum separation between landmarks is used for the calculations, the accuracy of the location improves. For maximally separated landmarks, and for patients with greater than 6 ~ of rotation, it appears that the "true" center of autorotation is still variable. In no case (even with the best possible combination of points and amrunts of rotation) did the center lie on the condyle. These findings provide some important guidelines for future predictions of mandibular position after maxillary impaction surgery. First, it is critical to remember that even with the best of all possible combinations of data from previous treatments, the actual location of the center of rotation will probably not lie in the condyle itself, but will be some unpredictable distance away from the condyle. Secondly, after selecting some arbitrary point to make the estimates and predictions, to minimize additional error contribution, it is critical that the landmarks selected for the analysis be separated by as great a distance as possible. In summary, three conclusions can be drawn from this investigation: (1) The Rouleaux method is reliable
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for calculating the location o f the center o f rotation, but only if landmarks are properly selected and rotation angles are greater than at least 6 ~. (2) There is great variation in the location o f the center o f autorotation o f the mandible after actual maxillary !mpaction surgery. (3) If this technique is to be used for treatment planning to determine the position o f t h e mandible after maxillary impaction surgery, it can only serve as a broad guideline and must not be considered to predict absolute locations. Authors" note: Tables providing values for (I) the distance
between the center of autorotation and the calculated center of rotation for the simulated study (for 2 ~ 6~ 10~ and 20 ~ of rotation); (2) the distance between the center of autorotation and the condyle, determined by using different combinations of reference points for all surgery patients and for those with 6 ~ or more of autorotation; and (3) the degree of rotation for each surgery patient are avilable, on request, directly from the author.
REFERENCES 1. Shendel SA, Eisenfeld JM, Bell WM, Epker BN. Superior repositioning of the maxilla: stability and soft tissue osseous relations. AM J ORTItOD DENTOFACOR'I-HOP 1976;70:663-74. 2. Radney L, Jacobs J. Soft-tissue changes associated with surgical total maxillary intrusion. AM J Own~ot) DF~"rOF,~,C ORTnOP 1981;80:191-212. 3. Fish LC, Epker BN. Surgical-orthodontic cephalometfic prediction tracings. J Clin Orthod 1980;14:36-52. 4. Valinoti JR. The hinge axis angle. J Clinc Orthod 1977;9: 551-9. 5. Sperry TP, Steinberg M J, Gans BJ. Mandibular movement during autorotation as a result of maxillary impaction surgery. Ar~1 J ORI"HODDENTOFACORTHOP 1982;81:! 16-22. 6. Navakari K. An analysis of the mandibular movement for rest to occlusal position. Acta Odontol Scand 1956;14:Suppl. 19. 7. GrantPG. Biomechanicalsignificanceoftheinstantaneouscenter of rotation: the human temporomandibular joint. J Biomech 1973;6:109-13. 8. Hall RE. An analysis of the work and ideas of investigators and 9authors of relations and movement of the mandible. J Am Dent Assoc 1929;16:1642-93. 9. Brewka RE. Pant~raphie evaluation of cephalometric hinge axis. AM J ORTIIOD DEN'rOFACORTHOP 1980;79:1-19. tO. Lepera F. Determination of the hinge axis clutches on condyle position. J. Prosthet Dent 1958;8:260-5. I I. Rouleaux F. The kinematics of machinery: outline of a theory of machines (translated by A.B.W. Kennedy). Dover, 1875. 12. Hultgren BW. A method for analyzing some effects of orthodontic mechanotherapy upon the directions and proportions of facial growth from two samples of Angle Class I. Division I 9 malocclusions utilizing computer graphic routines. [MS thesis], University of Minnesota, 1977. 13. Dimnet J, Carret JP, Gonon G, Fisher LP. A technique for joint center analysis using a stored pmgeram calculator. J Biomech 1974;10:659-73. 14. Soudan K, Van Audekercke R. Methods, difficulties, and inaccuracies in the study of human joint kinematics and pathokinematics by the instant axis concept. J Biomech 1979;12:27-33. 15. Panjabi MM. Centers and angles of rotation of body joints: a study of errors and optimization. J Biomech 1979;12:91 !-20. 16. Panjabi MM. Errors in kinematic parameters of a planar joint: guidelines for optimal experimental design. J Biomech 1982;15:537--44.
Reprhlt requests to: Dr. E. Dianne Rekow The University of Maryland Baltimore College of Dental Surgery Dental School 666 W. Baltimore S t . Baltimore, MD 21201-1586