HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Figures Only Full Text (PDF) Electronic Appendix Services Similar articles in this journal Similar articles in ISI Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (7) Citing Articles via Google Scholar Google Scholar Articles by DeLong, R. Articles by Douglas, W.H. Search for Related Content PubMed PubMed Citation Articles by DeLong, R. Articles by Douglas, W.H.

Helical Axis Errors Affect Computer-generated Occlusal Contacts

¹ Minnesota Dental Research Center for Biomaterials and Biomechanics, Department of Oral Science, University of Minnesota School of Dentistry, Moos Health Science Tower, 515 Delaware Street, SE, Minneapolis, MN 55455; and
² Division of Biostatistics and Oral Health Clinical Research Center, Department of Preventive Sciences, University of Minnesota;

*corresponding author, delon002{at}tc.umn.edu

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
A helical axis describing mandibular motion can be calculated from two distinct positions of the mandible; however, as these positions come closer together, calculation errors increase. This study investigated the effects of errors in the calculated helical axis on simulated mandibular motion by the measurement of changes in occlusal contacts. A standard helical axis was calculated from a simulated lateral movement. A series of digital interocclusal records from centric to a 5° mandibular rotation about the standard helical axis was created. Digital dental cast models were aligned to the interocclusal records. Helical axis parameters and occlusal contacts calculated with the use of the aligned digital models were compared with those of the standard. Helical axes calculated from mandibular positions separated by 1.5° to 5.0° yielded equivalent occlusal contacts. Qualitatively, contacts for helical axes calculated from jaw rotations of 0.7° or larger were nearly identical to those of the standard.

KEY WORDS: helical axis • occlusal contacts • jaw motion • 3D scanning • interocclusal record

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
Clinical observations show that the potential for damage to the temporomandibular joint is high if proper chewing function is not maintained (Okeson, 1998). This is supported by computer simulations showing large reaction forces in the temporomandibular joints from bite forces (Smith et al., 1986; Koolstra et al., 1988; Koolstra and van Eijden, 1992). While we can measure tooth contacts, we cannot determine their effects on the temporomandibular joint, because we cannot quantitatively measure jaw function when the teeth are in contact or generating forces.

Mandibular motion can be described as a rotation about and a translation along an axis, called the helical axis, located in space (Gallo et al., 1997, 2000). The helical axis characterizes joint function without requiring the location of anatomical positions in the joints (Panjabi et al., 1981; Hart et al., 1991). This is important when the precise location of anatomical landmarks is difficult (Panjabi and White, 1971; Spoor, 1984; Zwijnenburg et al., 1996; Peck et al., 1997). The position and orientation of the helical axis are constrained by the anatomical restrictions of the joint; thus, the helical axis provides insight into the relationship between joint movement and joint anatomy (Soudan et al., 1979; de Lange et al., 1990a; Fioretti et al., 1990). This relationship has shown promise for the distinguishing of healthy from pathologic conditions (Panjabi et al., 1982; Woltring et al., 1994; Gallo et al., 1997).

The helical axis describing the motion between two jaw positions is calculated from target points defining the positions (Spoor and Veldpaus, 1980). Unfortunately, small jaw movements, such as those encountered while the teeth are in contact, can result in large errors in calculation of the helical axis' orientation and position (Spoor, 1984; Woltring et al., 1985; de Lange et al., 1990b). Therefore, helical axes are generally calculated from jaw positions separated by at least a 1° rotation about the axis. This threshold was set arbitrarily (Gallo et al., 1997, 2000). A more appropriate determination of the minimum jaw separation would evaluate the effects of helical axis errors on occlusal contacts, which are measurable.

The purpose of this study was to investigate the effects of errors in the helical axis orientation and position vectors on inferred occlusal contacts. Helical axis parameters and occlusal contacts were calculated from jaw motion simulations by means of clinical casts mounted in a dental articulator and by computer models.

	MATERIALS & METHODS

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
Maximum intercuspation and a lateral movement of dental casts mounted in an articulator were recorded from interocclusal records. The helical axis for the lateral movement was calculated from the two positions, then used to produce a set of simulated eccentric interocclusal records. Computer models of the casts were aligned to the simulated eccentric interocclusal records. New helical axes and occlusal contacts were calculated and compared with those obtained from the articulator.

Creation of Computer Models
Maxillary and mandibular dental casts were mounted in a semi-adjustable dental articulator. Interocclusal records, made with the use of an experimental vinyl polysiloxane registration paste with optimal digitizing properties (3M, St. Paul, MN, USA), recorded the casts at maximum intercuspal position (ICP) and an eccentric position (EP) of 2.3° rotation about the right condyle of the articulator. This rotation was equivalent to a 3-mm displacement of the lower incisal midline from ICP.

A Comet 100 optical digitizing system (Steinbichler Optical Technologies, Neubeuern, Bavaria, Germany) was used for scanning the casts and interocclusal records. The Comet 100 has an accuracy of 0.040 mm and precision of 0.130 mm in the X and Y directions and 0.005 mm in the Z direction (parallel to the line of sight of the Comet). Multiple views were required for complete 3D digitized images to be captured. Each view was filtered by means of FilterAC (in-house software developed in the Minnesota Dental Research Center for Biomaterials and Biomechanics under NIH/NIDCR grant DE RO1 12225: The Virtual Dental Patient [VDP]) and aligned by means of PolyWorks^TM (InnovMetric Software, Quebec, Canada). Filtered and aligned files were merged into a single data file (referred to as a model) with Stratus (in-house software; see above). Point resolution within the models averaged 0.130 mm in the X, Y, and Z directions. The Appendix gives a more detailed description (www.dentalresearch.org). All models were rendered as 3D surfaces by means of the VDP software (in-house software; see above).

Calculation of Helical Axis Parameters from Computer Models
Movement about a helical axis is defined by the orientation unit vector, E, and the position vector, P, the rotation about the axis, , and the translation along the axis, t (Fig. 1A). The parameters are calculated from two positions of the mandible, each position being defined by a set of target points (Spoor and Veldpaus, 1980). Data points in the mandibular cast computer model served as targets. The ICP and EP interocclusal record models defined the two mandibular positions. Within the VDP software, the maxillary side of the ICP interocclusal model was aligned to the maxillary cast model by minimizing distances between common anatomical areas. Similarly, the mandibular model was aligned with the mandibular side of the ICP model. The result was a transformation matrix, M_ICP, that aligned the mandibular and maxillary models in ICP. The process was repeated for the EP interocclusal record model, which defined an eccentric transformation matrix, M_EP. Helical axis parameters were calculated from the transformation matrix, M_ICPEP = M_EP^-1 M_ICP, which moved the mandible from ICP to the eccentric position (Spoor and Veldpaus, 1980). Model alignments were replicated 5 times with different target points. The helical axis parameters calculated from the replicate alignments were averaged to define a set of "standard" helical axis parameters.

View larger version (32K): [in this window] [in a new window]   Figure 1. Experimental vs. standard helical axis parameters. (A) Diagram of the experimental and standard helical parameters. Experimental helical axis parameters are calculated from the digital image of the mandible at two locations, A and B, defined by the centric and eccentric interocclusal record images (Spoor and Veldpaus, 1980). The unit vector E defines the orientation of the helical axis. The angle between the experimental and standard helical axis unit vectors, E, was calculated with the use of a second unit vector, ÉExp that points in the same direction as EExp and originates from PStd. The vector P, which points from the origin to a point on a line defined by E, defines the position of the helical axis. Differences in the experimental and standard position vectors, P, were calculated indirectly as the minimum distance, d, between the two lines defined by the helical axes (Zwillinger, 1996), because the 3D distance between PExp and PStd is not necessarily the minimum distance between the lines. Equations for the difference calculations are shown in (A). Mean differences between experimental and standard helical axis parameters are shown in (B) through (E) as a function of jaw movement expressed as a rotation about the helical axis (n = 5; error bars representing the standard deviations are smaller than the data symbols). Qualitatively, the angle (B) and the distance (C) between the experimental and standard helical axes are similar to the helical axis error model predictions represented by the "best fit" dashed line (Spoor, 1984; Woltring et al., 1985; de Lange et al., 1990b). The differences between the experimental and standard rotations about the helical axis (D) and translations along the helical axis (E) are nearly constant, as predicted by the helical axis error model, and are smaller than the 0.040 mm accuracy of the Comet 100 scanner.

The orientation vector error, E, was calculated as the angle between the experimental and standard unit vectors. The position vector error, P, was calculated indirectly as the minimum distance, d, between the two lines defined by the experimental and standard helical axes (Fig. 1A). Rotation, , and translation, t, errors were the differences between the experimental and standard rotations, , and translation, t, values, respectively.

Calculation of Occlusal Contacts
In the VDP software, a contact between two opposing models, Mx and Mn, is a set of points S. Set S contains at least one point from Mx and one point from Mn, and the distance between the points is less than a specified distance (Tolerance). Also, all the points from Mx (Mn) that are in S must be within a specified distance (Range) from at least one other point from Mx (Mn) that is in S, or be the only point from Mx (Mn) that is in S. Tolerance and range values were set at 0.050 mm and 0.65 mm, respectively. Three parameters described an occlusal contact: Area, Centroid, and Normal. Area is the 3D surface area of the contact. Centroid is the center of mass of the contact. Normal is a unit vector at the centroid pointing out at a right angle from the contact (Fig. 2A). The Appendix gives a more detailed description (www.dentalresearch.org).

View larger version (46K): [in this window] [in a new window]   Figure 2. Experimental vs. standard contact parameters. (A) Diagram of the differences between the experimental and standard contact parameters. Contacts are defined by three parameters: Area, Centroid, and Normal. Area is the 3D surface area of the contact. We calculated differences in areas, A, by subtracting the experimental from the standard contact areas. The location of the contact is defined by the centroid, C, which is the center-of-mass of the points in the contact. The contact location error, C, was the 3D distance between the experimental and standard centroids. The normal, N, defines the orientation of the contact. It is a unit vector that originates at the contact centroid, points away from the contact, and is perpendicular to a plane that is tangent to the surface at the contact centroid. The angle between the experimental and standard contact Normals, N, was calculated by means of a second unit vector, N'Exp that points in the same direction as NExp and originates from CStd. Because it is possible to have non-identical contacts with identical centroids, areas, and normals, a fourth parameter, Overlap, was defined. Overlap was the average of the overlap of the two contact areas, O, divided by the two areas, expressed as a percentage. Equations for the differences in parameters and the overlap are shown in (A). See the Appendix for a more detailed description of the calculation of the contact parameters (www.dentalresearch.org). Mean differences between the experimental and standard contact parameters and the area overlap are shown in (B) through (E) as a function of the jaw movement used to calculate the helical axis parameters. The means represent the average over all contacts and 5 repeated measures (n = 5 times the number of contacts). The total number of contacts for the standard helical axis was 7. At 0.1°, 0.2°, 0.3°, and 0.4° jaw rotations, 1, 2, 6, and 8 contacts were averaged, respectively. The 7 "correct" contacts were averaged for the remaining jaw rotations. Error bars represent the standard deviations. All contacts were calculated for a mandibular rotation of 2.3° from maximum intercuspation based on the appropriate helical axis set. Contact parameter differences show a dependence on jaw position similar to that of the helical axis parameters (dashed line). Differences in the contact parameters with the use of helical axes calculated from jaw movements of 1.5° and larger were not significantly different (p > 0.05).

Statistical Analysis
One-way ANOVAs compared the standard and experimental contact parameters for the different jaw positions. Where significant differences existed (p < 0.05), post hoc pairwise comparisons were done by the Tukey-Kramer test. Correlation coefficients were calculated from scatterplots of the helical axis orientation and position errors, E and P, with the contact parameter errors, C, N, A, and Overlap.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
Differences between the experimental and standard helical axis parameters are displayed graphically as a function of the jaw movement defined by rotation about the standard helical axis (Figs. 1B-1E). The angles and distances between the experimental and standard helical axes behaved similarly, decreasing rapidly for very small jaw movements, then approaching zero asymptotically with larger jaw movements. Both angle and distances are well-represented by the equation y = m/x, where m is a constant and x is the rotation about the helical axis in degrees. Rotation, , and translation, t, errors were smaller than the measurement error.

Differences between the experimental and standard contact parameters are shown as a function of jaw movement (Figs. 2B-2E). Contact parameters show the same trends as the helical axis orientation and position parameters. Differences between the experimental and standard contact centroids, normals, and areas decreased and the area overlap increased with increasing angles. The total number of contacts for the standard helical axis was 7. At 0.1°, 0.2°, 0.3°, and 0.4° jaw rotations, 1, 2, 6, and 8 contacts were identified, respectively. The 7 "correct" contacts were found for the remaining jaw rotations. Statistically, the contact parameters for rotations of 1.5° or larger were not significantly different, p > 0.05. Qualitatively, differences in the contact parameters were not detectable for rotations of 0.7° or larger (Fig. 3).

View larger version (85K): [in this window] [in a new window]   Figure 3. Effects of helical axis parameters on contact shape. Occlusal contacts are shown for a 2.3° rotation of the mandible from maximum intercuspation based on the standard helical axis parameters and helical axis parameters calculated from 0.5°, 0.7°, 1.0° and 1.5° of jaw rotation. We calculated contacts by identifying the points on the aligned computer models of the maxillary and mandibular casts that were within 0.050 mm of each other. Contacts are shown as black regions on the occlusal surfaces of the teeth, and are identified by the black ovals on the full-arch standard image. Seven contacts were identified for the standard helical axis alignment of the casts. Visually, there is little difference between the contacts for the helical axis parameters calculated from 0.7° and larger jaw rotations.

View this table: [in this window] [in a new window]   Table. Correlation Parameters ( = 0.2° to 5.0°; n = 85)

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
Helical axis and contact parameters were calculated from computer models of dental casts aligned in maximum intercuspation and in eccentric positions. How well the position and orientation of the helical axis were known depended on the separation of the two locations of the mandibular model as measured by a rotation about the helical axis, . Mean differences between experimental and standard values for the position and orientation of the helical axis had an inverse relationship to similar to the error model predictions for their standard deviations (Woltring et al., 1985; de Lange et al., 1990b) (Figs. 1B-1E). Mean differences between experimental and standard contact parameters showed a similar trend (Figs. 2B-2E); however, more variations occurred at the smaller angles, because fewer occlusal contacts were identified at these angles. Also, the contact parameter errors did not show the asymptotic behavior of the helical axis parameter errors. Instead, the contact parameter errors increased slightly for jaw rotations greater than 3°.

Contact parameters computed from cast models aligned based on the helical axes calculated from mandibular separations of 1.5° to 5° were not significantly different. Clinically, this means that normal lateral interocclusal records made with the upper and lower buccal cusps aligned (approximately 2° of jaw rotation) provide sufficient movement for accurate helical axis parameters to be produced. Smaller rotations may be acceptable if the clinical significance of the contact parameters is considered. Helical axis parameters calculated from a jaw rotation of 0.7° produced mean errors in the contact centroid, normal, and area of 0.08 mm, 1.4°, and 0.26 mm², respectively. Although these are about twice the corresponding values for the 1.5° rotation, they are still small differences that may have little impact on the clinical assessment of jaw function. Visually, there were few detectable differences between contacts produced from casts aligned by the standard and experimental helical axes calculated from jaw rotations of 0.7° or larger (Fig. 3). These results support a 0.7° step limitation.

Correlations between the helical axis parameter and contact parameter errors indicated that errors in helical axis position have less impact on contact parameters than do errors in its orientation. This is beneficial, because the helical axis error model (Woltring et al., 1985; de Lange et al., 1990b) implies that the position is the least-well-defined helical axis parameter. The location of the contact centroid was most sensitive to helical axis orientation and position errors. The contact normal and area overlap parameters were least sensitive (Table). Contact parameters are important for the definition of occlusal stress distributions, which may have a direct impact on the temporomandibular joints.

This study showed that accurate helical axis parameters can be calculated with the use of digital images of dental casts and interocclusal records; however, the errors reported here are strictly mathematical errors related to the rigid geometry of the experimental setup. Clinically, non-rigid properties of the functioning mandible, which are not recorded in full-arch replica casts, could significantly affect the results (Korioth and Hannam, 1994). Interocclusal records, however, do capture the functional changes. With minor alterations in procedure, the helical axis parameters can be calculated directly from interocclusal records. ICP and eccentric interocclusal records are aligned with the use of the maxillary occlusal imprints. Following this, the mandibular occlusal imprints are aligned, directly yielding the transformation matrix, M_ICPEP, from which helical axis parameters are calculated. Functional differences in the ICP and eccentric records will affect the alignments, and thus the calculation accuracy; however, the magnitude should be less than that found with the use of rigid casts. Clinical effects of non-rigid tissues and biologic variability on helical axis parameters and inferred contacts remain to be measured.

	ACKNOWLEDGMENTS

 
This study was supported in part by USPHS Research Grant RO1 DE-12225-05 from the National Institute of Dental and Craniofacial Research, National Institutes of Health, Bethesda, MD 20892, and by the Minnesota Dental Research Center for Biomaterials and Biomechanics.

	FOOTNOTES

 
A supplemental appendix to this article is published electronically only at http://www.dentalresearch.org.

Received November 7, 2001; Last revision February 11, 2002; Accepted February 13, 2002

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
de Lange A, Huiskes R, Kauer JMG (1990a). Effects of data smoothing on the reconstruction of helical axis parameters in human joint kinematics. J Biomech Eng 112:107–113.[Medline]

de Lange A, Huiskes R, Kauer JMG (1990b). Measurement errors in Roentgenstereophotogrammetric joint-motion analysis. J Biomech 23:259–269.[Medline]

Fioretti S, Jetto L, Leo T (1990). Reliable in vivo estimation of the instantaneous helical axis in human segmental movements. IEEE Trans Biomed Eng 37:398–409.[Medline]

Gallo LM, Airoldi GB, Airoldi RL, Palla S (1997). Description of mandibular finite helical axis pathways in asymptomatic subjects. J Dent Res 76:704–713.[Abstract/Free Full Text]

Gallo LM, Fushima K, Palla S (2000). Mandibular helical axis pathways during mastication. J Dent Res 79:1566–1572.[Abstract/Free Full Text]

Hart RA, Mote CD Jr, Skinner HB (1991). A finite helical axis as a landmark for kinematic reference of the knee. J Biomech Eng 113:215–222.[Medline]

Koolstra JH, van Eijden TM (1992). Application and validation of a three-dimensional mathematical model of the human masticatory system in vivo. J Biomech 25:175–187.[Medline]

Koolstra JH, van Eijden TM, Weijs WA, Naeije M (1988). A three-dimensional mathematical model of the human masticatory system predicting maximum possible bite forces. J Biomech 21:563–576.[Medline]

Korioth TW, Hannam AG (1994). Deformation of the human mandible during simulated tooth clenching. J Dent Res 73:56–66.[Abstract/Free Full Text]

Okeson JP (1998). Management of temporomandibular disorders and occlusion. 4th ed. St. Louis: Mosby-Year Book, Inc.

Panjabi MM, White AA III (1971). A mathematical approach for three-dimensional analysis of the mechanics of the spine. J Biomech 4:203–211.[Medline]

Panjabi MM, Krag MH, Goel VK (1981). A technique for measurement and description of three-dimensional six degree-of-freedom motion of a body joint with an application to the human spine. J Biomech 14:447–460.[Medline]

Panjabi MM, Goel VK, Walter SD, Schick S (1982). Errors in the center and angle of rotation of a joint: an experimental study. J Biomech Eng 104:232–237.[Medline]

Peck CC, Murray GM, Johnson CWL, Klineberg IJ (1997). The variability of condylar point pathways in open-close jaw movements. J Prosthet Dent 77:394–404.[Medline]

Smith DM, McLachlan KR, McCall WD Jr (1986). A numerical model of temporomandibular joint loading. J Dent Res 65:1046–1052.[Abstract/Free Full Text]

Spoor CW (1984). Explanation, verification and application of helical-axis error propagation formulas. Hum Movement Sci 3:95–117.

Spoor CW, Veldpaus PE (1980). Rigid body motion calculated from spatial co-ordinates of markers. J Biomech 13:391–393.[Medline]

Soudan K, van Audekercke R, Martens M (1979). Methods, difficulties and inaccuracies in the study of human joint kinematics and pathokinematics by the instant axis concept. Example: the knee joint. J Biomech 12:27–33.[Medline]

Woltring HJ, Huiskes R, de Lange A, Veldpaus FE (1985). Finite centroid and helical axis estimation from noisy landmark measurements in the study of human joint kinematics. J Biomech 18:379–389.[Medline]

Woltring HJ, Long K, Osterbauer PJ, Fuhr AW (1994). Instantaneous helical axis estimation from 3-D video data in neck kinematics for whiplash diagnostics. J Biomech 27:1415–1432.[Medline]

Zwijnenburg A, Megens CCEJ, Naeije M (1996). Influence of choice of reference point on the condylar movement paths during mandibular movements. J Oral Rehabil 23:832–837.[Medline]

Zwillinger D, editor (1996). CRC standard mathematics tables and formulas. 30th ed. Boca Raton, FL: CRC Press, LLC.

This article has been cited by other articles:

				K. Hayashi, J. Uechi, S.-P. Lee, and I. Mizoguchi Three-dimensional analysis of orthodontic tooth movement based on XYZ and finite helical axis systems Eur J Orthod, December 1, 2007; 29(6): 589 - 595. [Abstract] [Full Text] [PDF]

				S. B. PATEL, V. V. GORDAN, A. A. BARRETT, and C. SHEN The effect of surface finishing and storage solutions on the color stability of resin-based composites J Am Dent Assoc, May 1, 2004; 135(5): 587 - 594. [Abstract] [Full Text] [PDF]

				R. DeLong, M. Heinzen, J.S. Hodges, C.-C. Ko, and W.H. Douglas Accuracy of a System for Creating 3D Computer Models of Dental Arches J. Dent. Res., June 1, 2003; 82(6): 438 - 442. [Abstract] [Full Text] [PDF]

This Article Abstract Figures Only Full Text (PDF) Electronic Appendix Services Similar articles in this journal Similar articles in ISI Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (7) Citing Articles via Google Scholar Google Scholar Articles by DeLong, R. Articles by Douglas, W.H. Search for Related Content PubMed PubMed Citation Articles by DeLong, R. Articles by Douglas, W.H.