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Recording the dental cast in three dimensions
Recording the dental cast in three dimensions S. Richmond, B.D.S., D. Orth., R.C.S., M.Sc.D., F.D.SJ3.C.S (Edinburgh)* Heath Park, Cardiff,
Wales
A method is described to record the three-dimensional relationship of the teeth within the dental arches. (AM J ORTHOD DENTOFAC ORTHOP 1987;92:199-206.)
T
here have been many techniques described to quantify the characteristics of a malocclusion. It has been only recently with the advent of new technology that the dental cast can be readily measured and recorded in three dimensions. This facility allows a realistic reconstruction of the dental cast, which would prove useful in research. It is with the research aspect in mind that a quantitative analysis has been devised. MATERIALS AND METHODS
It is important to design an analysis that has features a clinician would use. The variables used are described and defined below. 1. Overjet and overbite Ovejet and overbite are measurements the clinician uses frequently during, before, and after treatment. The usual technique is to use a ruler or vernier calipers to the labial aspect of the lower incisors to the incisal edge of the upper incisors. With this technique, two different landmarks are used. In the proposed analysis, similar landmarks are chosen on the upper and lower central incisors using the mesial incisal edges. Overbite
Overbite is the distance of the upper central incisors from the lower incisors measured in relation to a perpendicular to the occlusal plane using the same points as those for overjet. This measurement would be difficult to make on dental casts using a ruler or vernier calipers. 2. Center-line discrepancy
The center line is an important assessment because it shows the symmetry of the arches. A shift in the center line would suggest differing molar positions or discrepancy between the tooth diameters of the upper and lower teeth. The center line is constructed using the midpoints between the canines and molars *Department of Orthodontics, University Dental Hospital Turner Dental School, Manchester, England.
of Manchester,
and
on the lower model because the clinician usually assesses the lower model first and fits the upper model to the lower. This facility enables the operator to determine whether the center lines of the incisors are coincident or how far they deviate from each other and from the constructed center line. Although the construction of the center line may not coincide with the midline of the face, a useful indication of discrepancy in center lines can be determined. In the case of mandibular asymmetry, a gross reading in center-line discrepancy may be noted that would warrant further investigation with radiographs to assess the skeletal symmetry. 3. Mesiodistal widths
Tooth size plays an important part in malocclusions, particularly in space requirements. In this analysis the three dimensional measurement is determined across the anatomic contact points. Results are determined for the anterior (3-3) and overall (6-6) mesiodistal widths. In the event of unerupted teeth, the mesiodistal width of its antimere can be used. In the case of both teeth being unerupted, an average mesiodistal width for that particular tooth can be substituted (Table I). 4. Three- and two-dimensional intercanine and incisal angles
The traditional method of assessing intercanine and incisal angles has been by the use of the lateral skull radiograph. A significant difference has been shown between two-dimensional and three-dimensional interincisor assessment on the lateral skull radiograph and dental cast, respectively. ’ This would suggest a difference between the crown and crown/root angulation measuring techniques. However, the authors also noted that the results for the dental cast assessment of the crown angulation were more consistent and reproducible than the assessments for the crown/root angulations on the lateral skull radiograph. 199
Am. J. Orthod.
200 Richmond
Dentofac. Orthop. September 1981
Table I. Average mesiodistal widths used when the individual teeth are unempted or missing Tooth tl0.
Central incisor Lateral incisor Canine First premolar Second premolar First permanent molar
1 2 3 4 5 6
Wper (mm)
Lower
8.79 6.83 1.74 6.68 6.59 10.38
5.5 5.98 6.77 6.84 7.145 10.5
(mm)
cusp tip of the lower molar (Fig. 3). This is comparable to the functional occlusal plane.’ The plane is constructed to the best fit of these six points. However, only three points are required to form a plane; therefore, a full complement of teeth is unnecessary. 6. Intercanine and intermolar distances
I
It is important during treatment to monitor intercanine and intermolar distances. In certain cases it will be necessary to expand the upper or lower intercanine distance if the arch is excessively narrow and crowded. The distance measured is between the occlusal tips of the canines and the points of the buccal groove of the molars where the buccal surface meets the occlusal surface. 7. Parabolic curve length
Fig. 1. lnterincisal and canine crown angulations. The central axis of the crown is constructed using points A and B defined by the mesioincisal edge and the corresponding point on the distal surface. C-D relate to the points that bisect the crown at the gingival margin when the dental cast is viewed occlusally.
The central axis of the crown is constructed by the midpoints of the incisal edges (A and B) and gingival margin (C and D). The midpoints M and N define the central axis (Fig. 1). The analysis determines only the axial inclinations of the crowns of the teeth and does not relate to the crown root angle. However, the orthodontist is mainly concerned with the crowns of the teeth unless the crown root angle is acute and torquing will adversely position the root apex. Two-dimensional angles can be determined by projecting the central axis onto the anteroposterior plane (Fig. 2). 5. Occlusal plane construction
The occlusal plane is constructed using the buccal cusp tips of the lower premolars and the mesiobuccal
Several investigators have concluded that there is no consistency in nature and the shape of the dental arch varies so widely among individuals that geometric comparisons are impossible.3z4 Hawley’ proposed a geometric method for predetermining the dental arches for orthodontic purposes using the ideas of BonwilL6 although Hawley’ and Chuck7 advised against strict use of this. The ellipse’-” and catenary curve” have also been suggested. The polynominal equation of the fourth degree was proposed by Lu.‘~ Hechtert3 found that the paraboia had a high “goodness of fit” to the middle curve of teeth when compared with other generated curves. The parabola has also been suggested by other RichmondI concluded that it investigators. 4*9,10~‘4~15 was doubtful whether the ideal arch form ever exists, but in order to treat a malocclusion, it is necessary to align the teeth and to know the severity of crowding to a given arch form. The parabolic curve was chosen in this analysis because several authors have shown it to be the best fit; it was also selected for its ease in matbematic application. The parabolic curve is constructed to the best fit
Volume 92 Number 3
CONSTRUCTION
Recording
OF
e-ve&
2D ANGLES
dental
cast
in three
dimensions
201
OCCLUSAL PLANE CONSTRUCTION
etve
Fig. 2. Construction of two-dimensional angles. Positive angles are produced when the teeth are proclined relative to the perpendicular raised to the occlusal plane.
Fig. 3. Occlusal plane points. In three dimensions, three points are required to fit a plane. The occlusal plane is established using the buccal cusp tips of the lower premolars and the mesiobuccal cusp of the molars. The plane is constructed as the best fit using the six points.
of the mesiodistal anatomic contact points ending distal to the first molar (Fig. 4). The curve is constructed to each individual case rather than fitting the case to a preselected arch form. The length of the parabolic curve is noted. 8. Discrepancy measurements
A discrepancy measurement is a useful indication of the crowding or spacing within the dental arches that will determine an extraction or nonextraction approach. Two types of discrepancy measurements are used. a. Arch discrepancy This discrepancy measurement provides a guideline for how far the anatomic contact points are from the corresponding contact point on the adjacent teeth in relation to the occlusal plane (Fig. 5). Therefore, this measurement attempts to quantify the discrepancy between the contact points in two dimensions. The distance a-b when projected onto the occlusal plane c-d produces the two-dimensional discrepancy. These distances represent the movement required to align the dental arches to form a flat occlusal plane. b. Parabolic curve discrepancy The length of the parabolic curve and the overall mesiodistal widths are known; therefore, by subtracting one from the other, the discrepancy can be determined, noting whether the arch is spaced or crowded.
Fig. 4. Parabolic curve length. A parabola was constructed to the best fit of the mesiodistal anatomic contact points ending distal to the first molar (plotted from Fig. 7).
202
Am. J. Orthod.
Richmond
Dentofac.
Orthop. September 1987
Instrument
The points were recorded by means of the Reflex Metrograph,* which is capable of measuring objects in three dimensions (Fig. 6). This instrument basically consists of a semireflecting mirror; therefore, an object standing in front of the mirror has its image at an equal distance behind the mirror. Working on this principle, a moving light source connected to a three-dimensional (X-Y-Z) slide system behind the mirror can be used to record points corresponding to the image of the object. Movement of this slide system operates rotary encoders, which translate movement into electrical impulses that can be transmitted to a computer. With the aid of a computer program, this information can be accepted and measurements performed. One of the qualities of the Reflex Metrograph is that the 0.3-mm diameter light source can move into interproximal spaces and contact areas where a pair of dividers could not. The accuracy of the Reflex Metrograph has been tested by RichmondI and the error found to be less than 0.27 mm (<0.3%) with an angular error of less than 0.76%. This compares favorably with other authors.‘7-‘9 One hundred ten points were recorded. It takes an experienced operator less than 20 minutes to record 110 points. To illustrate the analysis, a Class II, Qivision 2 malocclusion is shown in Fig. 7. The following analysis shows a deep overbite of 6 mm and an overjet of 3 mm. The center lines are almost coincident, the mesial edges all being to the right of the constructed center line by approximately 1.5 mm. The distances for the mesiodistal widths are shown.
Fig. 5. Discrepancy measurement. The distance a-b is the three-dimensional distance between the anatomic contact points. When the points a-b are projected onto the occlusal plane o-o, the distance is now c-d. This “correction factor” provides a more reliable discrepancy reading and is related to the need for flattening the occlusal plane.
Sample
To test the accuracy and reproducibility of the analysis, 12 sets of dental casts were randomly selected from 150 patients, first checking to validate the occlusal relationships. It was essential that the dental casts be stable when positioned on their heels-together and independently. Procedure
Eight crossed lines were marked around the land area for each in the upper and lower dental casts (the fiducial points). The intersection of the two lines was used to orientate the dental casts so that when they were in occlusion, the spatial arrangement was noted; therefore, when the casts were moved apart to digitize the occlusal surfaces, all the points could be reorientated to the occluded position by re-recording the fiducial points.
CLASS II, DIVISION 2 MALOCCLUSION Left Right Overjet Overbite Center-line discrepancy Mesial edge of upper center line Mesial edge of upper center line Mesial edge of lower center line Mesial edge of right
3.05 mm 5.73 mm
left central right
central
left central central
*H. F. Ross, Ross Instruments
incisor,
3.35 mm 5.78 mm
incisor, incisor, incisor,
1.5 1 mm right
1.65 mm right of 1.42 mm right of
1.19 mm right
Ltd., Salisbury,
of
Wiltshire,
of center
England.
line
Recording
Volume 92 Number 3
Fig. 6. Reflex
Fig. 7. A through
D, Dental
casts
of Class
dental cast in three dimensions
Metrograph.
II, Division
2 malocclusion.
(For
analysis,
see text.)
203
204
Am. J. Orthod.
Richmond
Dentofac.
Orthop. September 1987
Table II. Intra- and interobserver errors Intraobserver
Overjet Overbite Mesial contact points of central incisors to center line Upper mesiodistal widths Lower mesiodistal widths Three-dimensional angles, incisors Three-dimensional angles, canines Upper two-dimensional angles Lower two-dimensional angles Overall mesiodistal widths Anterior mesiodistal widths Overall discrepancy Anterior discrepancy Intercanine distance Intermolar distance Length of fitted parabola Length of discrepancy of fitted parabola OBSERVER ERRORS:
Intraobserver-significant
error
Interobserver
error
n
Mean
SD
RMS
Mean
SD
RMS
24 24 48
0.04 0.13 -0.12
0.32 0.64 0.49
0.31 0.63 0.49
0.17 0.16 -0.17
0.49 0.82 0.63
0.49 0.85 0.64
44 144 24 19 67 67 24 24 24 24 20 21 24 24
0.05 0.15 0.24 0.44 -0.18 0.69 1.06 0.27 0.24 0.11 0.12 0.03 0.44 -0.71
0.26 0.27 2.12 5.82 3.54 3.05 1.37 0.97 1.92 1.43 0.52 0.40 0.93 1.57
0.27 0.32 2.04 5.5 3.48 3.08 1.70 1.01 1.87 1.39 0.52 0.38 1.01 1.72
0.34 0.25 0.89 - 2.48 1.65 - 1.55 3.45 0.66 0.1 -0.77 -0.08 0.01 1.36 -2.01
0.35 0.34 2.34 5.7 5.24 3.79 1.49 1.18 2.74 2.06 0.56 0.45 1.27 1.55
0.48* 0.43 2.7 5.27 5.37* 4.03 3.74* 1.32 3.32* 2.11* 0.54 0.43 1.93* 2.55
difference
P > 0.5;
interobserver-significant
difference
P > 0.5
except
‘I*”
when
0.5 > P > 0.1.
Mesiodistal Left Left Left Left Left Left Right Right Right Right Right Right
widths
Upper
first permanent molar second premolar first premolar canine lateral incisor central incisor central incisor lateral incisor canine first premolar second premolar permanent molar
(mm)
Lower
10.09 6.48 6.23 7.66 6.6 8.8 8.39 6.27 7.39 5.89 6.6 10.27
(mm)
10.13 6.55 6.33 6.34 6.39 5.52 5.83 5.61 6.61 6.8 6.28 10.52
CANINE AND INCISOR ANGULATIONS Three-dimensional angle (“) Left Left Left Right Right Right
canine lateral central central lateral canine
incisor incisor incisor incisor
133.70 148.61 167.42 167.45 129.60 158.80
Two-dimensional angle (“) Lower Upper 1.29 22.74 -1.68 23.63 - 12.73 22.82 - 14.89 25.80 13.48 28.71 -7.09 12.83 Upper
Overall mesiodistal widths (6-6) Anterior mesiodistal widths (3-3) Intercanine distance Intermolar distance Length of fitted parabola
(mm) 90.67 45.11 35.34 48.58 91.50
Lower
(mm)
82.91 36.30 27.36 44.16 85.22
Discrepancy measurements 1. Arch discrepancy Anterior (3-3) Overall (6-6) 2. Parabola length minus Mesiodistal widths
10.22 14.24
3.84 13.83
1.13
2.31
The canine and incisal angulations are measured in three and in two dimensions. The results suggest that the upper central incisors are retreclined to a perpendicular by 12” to 14”. This is supported by the three-dimensional angle of 167” indicating a high interincisal angle. Looking at the lateral incisors, it is noted that the upper right lateral incisor is proclined 14” more than the upper left lateral incisor. Therefore, by looking at the results, these would indicate a Class II, Division 2 incisor relationship. Looking at the arch discrepancy measurement, the upper anterior discrepancy is 10.22 mm and overall 14.24 mm. This would suggest that most of the discrepancy is in the incisor region caused mainly by the retroclination of the upper central incisors. In the lower arch, there appears to be minimal crowding in the anterior region (3.84 mm), but the overall reading is 13.83 mm. This is mainly due to the rotation of the lower left second pre-
Recording
Volume 92 Number 3
molar and insufficient space for the lower right second premolar. The rotations are indicated on the plot of the dental arches (Fig. 4). The parabolic-curve discrepancy shows minimal crowding in the upper arch; the parabolic curve suggests retroclination of the upper central incisors and a proclination of the upper right labial incisor. In the lower arch, minimal crowding is also noted; the parabolic curve suggests retroclination of the lower incisors. RESULTS
The root mean squares (RMS) and standard deviations (SD) are shown for the recorded measurements in Table II. The inter- and intraobserver errors were acceptable and no significant differences were found, the P value being greater than 0.5 for intraobserver and generally greater than 0.1 for interobserver readings. The results would indicate that the measurements undertaken were consistent and reproducible. The intercanine and incisal angulation readings have been grouped together. RichmondI has previously shown that the accuracy of the incisal angulations is generally better than that of the canine angulations. This is due to the distance between the midpoints m-n, which is generally less in canines than incisors, and because the morphology of the canines is slightly unfavorable since the defined points are not always readily identified. The interobserver error was not significant, which suggests that a technician could be trained to use the analysis to produce a series of readings upon which to build a data base that would be useful in research. CONCLUSIONS
Freer, Grew, and Little” reported that a difference exists between the subjective assessment and diagnosis of orthodontists. It would be desirable to make consistent assessments of orthodontic cases; by using the above-described analysis, a consistent approach to diagnosis and treatment planning is possible. All the landmarks were recorded directly on the dental casts. Using the points recorded, planes can be constructed to assess positions of teeth in the three planes of space. Therefore, planes can be directly produced on the dental casts; this offers an advantage over radiographically derived planes as the latter may be subjected to rotational errors and magnification problems. It is important to remember that the analysis is a
dental cast in three dimensions
205
recent development and has not as yet been calibrated. This would best be achieved by mensurating ideal occlusions to attain standard values. Although the full analysis is described, it is possible to measure individual variables alone without digitizing the 110 points. It is also possible to construct other planes and angles if so desired. In conclusion, this analysis can illustrate and quantify the characteristics of a given malocclusion; it is both consistent and reproducible. The analysis may prove useful to the orthodontist in the clinical setting or for use as a research tool. The analysis can describe the dental cast in the three planes of space and could be useful as a record of the dental cast to alleviate the problems of storage. I wish to thank Dr. P. J. Scott for his contribution to the preparation of the software for three-dimensional cast measurement to attain prepared Mrs. L.
and Mr. M. L. Jones for measuring the dental casts the interobserver error. The illustrative work was by the Cardiff Audio-Visual Aids Department and Lowe, who typed the manuscript.
REFERENCES Richmond S, Jones ML. A comparison of two and three dimensional incisor angles. Br J Orthod 1985;12:90-6. Jacobson A. The “Wits” appraisal of jaw disharmony. AM J ORTHOD1975;67:125-38. Hellman M. Dimensions versus form in teeth and their bearing on the morphology of the dental arch. Int J Orthod 1919; 5:615-51. 4. Wheeler RC. A textbook of dental anatomy and physiology. 2nd ed. Philadelphia: WB Saunders Co, 1950:196-215, 352-406. 5. Hawley CA. Determination of the normal arch and its application to orthodontics. Dent Cosmos 1905;47:541-57. 6. Bonwill WGA. Geometrical and mechanical laws of articulation. Trans Odont Sot Penn 1884-1885:119-33. 7. Chuck GC. Labial arch form. Angle Orthod 1934;4:312-27. 8. Black GV. Descriptive anatomy of the human teeth. 5th ed. Philadelphia: S.S. White Mfg. Co., 1902. 9. Izard G. New method for the determination of the normal arch by the function of the face. Int J Orthod 1927;13:582-95. 10. Sicher H. Oral anatomy. 2nd ed. St. Louis: The C. V. Mosby Company, 1952:262-3. 11. MacConnaill MA, Scher EA. The ideal form of the human dental arcade with some prosthetic application. Dent Ret 1949;69: 285-302. 12. Lu KH. Analysis of dental arch symmetry [Abstract]. J Dent Res 1964;43:780. 13. Hechter FJ. Symmetry and dental arch form of orthodontically treated patients. J Can Dent Assoc 1978;44: 173-84. 14. Broomwell IN. Anatomy and histology of the mouth and teeth. 2nd ed. Philadelphia: P. Blakistone Sons & Co, l902:99. 1.5. Currier JH. A computerized geometric analysis of human dental arch form. AM J ORTHOD 1969;56:164-79. 16. Richmond S. The feasibility of categorising orthodontic treatment difficulty-The use of three dimensional plotting [M.Sc.D. thesis]. Cardiff, Wales: University of Wales, 3984.
206 Richmond
Am. J. Orthod. Dcntojhc.
Orthop
September
17. Butcher GW, Stephens CD. The reflex optical plotter. A preliminary report. Br Dent J 1981;151(9):304-5. 18. Scott PJ. The reflex plotters: Measurement without photographs. Photogrammetric Ret 1981;10:435-46. 19. Takada K, Lowe AA, De Cou R. Operational performance of the Reflex Metrograph and its applicability to the three-dimensional analysis of dental casts. AM J ORTHOD1983;83: 195-9. 20. Freer FJ, Grew JM, Little RM. Agreement among the subjective severity assessment of 10 orthodontists. Angle Orthod 1973; 43:185-90.
Reprint requests to: Dr. Stephen Richmond Department of Orthodontics Dental Hospital of Manchester and Turner Dental School Higher Cambridge St. Manchester MI5 6FH England
BOUND VOLUMES AVAILABLE TO SUBSCRIBERS
Bound volumes of the AMERICAN JOURNAL OF ORTHODDN~CS AND DENTOFACIAL ORTHOPEDICS are available to subscribers (only) for the 1987 issues from the Publisher, at a cost of S40.00 ($50.00 international) for Vol. 91 (January-June) and Vol. 92 (July-December). Shipping charges are included. Each bound volume contains a subject and author index and all advertising is removed. Copies arc shipped within 60 days after publication of the last issue in the volume. The binding is durable buckram with the journal name, volume number, and year stamped in gold on the spine. Payment must accompany all orders. Contact The C. V. Mosby Company, Circulation Department, 11830 Westline Industrial Drive, St. Louis, Missouri 63146, USA; phone (800) 3254177, ext. 351. Subscriptions must be in force to qualify. Bound volumes are not available in place of a regular Journal subscription.
1987
Wales
A method is described to record the three-dimensional relationship of the teeth within the dental arches. (AM J ORTHOD DENTOFAC ORTHOP 1987;92:199-206.)
T
here have been many techniques described to quantify the characteristics of a malocclusion. It has been only recently with the advent of new technology that the dental cast can be readily measured and recorded in three dimensions. This facility allows a realistic reconstruction of the dental cast, which would prove useful in research. It is with the research aspect in mind that a quantitative analysis has been devised. MATERIALS AND METHODS
It is important to design an analysis that has features a clinician would use. The variables used are described and defined below. 1. Overjet and overbite Ovejet and overbite are measurements the clinician uses frequently during, before, and after treatment. The usual technique is to use a ruler or vernier calipers to the labial aspect of the lower incisors to the incisal edge of the upper incisors. With this technique, two different landmarks are used. In the proposed analysis, similar landmarks are chosen on the upper and lower central incisors using the mesial incisal edges. Overbite
Overbite is the distance of the upper central incisors from the lower incisors measured in relation to a perpendicular to the occlusal plane using the same points as those for overjet. This measurement would be difficult to make on dental casts using a ruler or vernier calipers. 2. Center-line discrepancy
The center line is an important assessment because it shows the symmetry of the arches. A shift in the center line would suggest differing molar positions or discrepancy between the tooth diameters of the upper and lower teeth. The center line is constructed using the midpoints between the canines and molars *Department of Orthodontics, University Dental Hospital Turner Dental School, Manchester, England.
of Manchester,
and
on the lower model because the clinician usually assesses the lower model first and fits the upper model to the lower. This facility enables the operator to determine whether the center lines of the incisors are coincident or how far they deviate from each other and from the constructed center line. Although the construction of the center line may not coincide with the midline of the face, a useful indication of discrepancy in center lines can be determined. In the case of mandibular asymmetry, a gross reading in center-line discrepancy may be noted that would warrant further investigation with radiographs to assess the skeletal symmetry. 3. Mesiodistal widths
Tooth size plays an important part in malocclusions, particularly in space requirements. In this analysis the three dimensional measurement is determined across the anatomic contact points. Results are determined for the anterior (3-3) and overall (6-6) mesiodistal widths. In the event of unerupted teeth, the mesiodistal width of its antimere can be used. In the case of both teeth being unerupted, an average mesiodistal width for that particular tooth can be substituted (Table I). 4. Three- and two-dimensional intercanine and incisal angles
The traditional method of assessing intercanine and incisal angles has been by the use of the lateral skull radiograph. A significant difference has been shown between two-dimensional and three-dimensional interincisor assessment on the lateral skull radiograph and dental cast, respectively. ’ This would suggest a difference between the crown and crown/root angulation measuring techniques. However, the authors also noted that the results for the dental cast assessment of the crown angulation were more consistent and reproducible than the assessments for the crown/root angulations on the lateral skull radiograph. 199
Am. J. Orthod.
200 Richmond
Dentofac. Orthop. September 1981
Table I. Average mesiodistal widths used when the individual teeth are unempted or missing Tooth tl0.
Central incisor Lateral incisor Canine First premolar Second premolar First permanent molar
1 2 3 4 5 6
Wper (mm)
Lower
8.79 6.83 1.74 6.68 6.59 10.38
5.5 5.98 6.77 6.84 7.145 10.5
(mm)
cusp tip of the lower molar (Fig. 3). This is comparable to the functional occlusal plane.’ The plane is constructed to the best fit of these six points. However, only three points are required to form a plane; therefore, a full complement of teeth is unnecessary. 6. Intercanine and intermolar distances
I
It is important during treatment to monitor intercanine and intermolar distances. In certain cases it will be necessary to expand the upper or lower intercanine distance if the arch is excessively narrow and crowded. The distance measured is between the occlusal tips of the canines and the points of the buccal groove of the molars where the buccal surface meets the occlusal surface. 7. Parabolic curve length
Fig. 1. lnterincisal and canine crown angulations. The central axis of the crown is constructed using points A and B defined by the mesioincisal edge and the corresponding point on the distal surface. C-D relate to the points that bisect the crown at the gingival margin when the dental cast is viewed occlusally.
The central axis of the crown is constructed by the midpoints of the incisal edges (A and B) and gingival margin (C and D). The midpoints M and N define the central axis (Fig. 1). The analysis determines only the axial inclinations of the crowns of the teeth and does not relate to the crown root angle. However, the orthodontist is mainly concerned with the crowns of the teeth unless the crown root angle is acute and torquing will adversely position the root apex. Two-dimensional angles can be determined by projecting the central axis onto the anteroposterior plane (Fig. 2). 5. Occlusal plane construction
The occlusal plane is constructed using the buccal cusp tips of the lower premolars and the mesiobuccal
Several investigators have concluded that there is no consistency in nature and the shape of the dental arch varies so widely among individuals that geometric comparisons are impossible.3z4 Hawley’ proposed a geometric method for predetermining the dental arches for orthodontic purposes using the ideas of BonwilL6 although Hawley’ and Chuck7 advised against strict use of this. The ellipse’-” and catenary curve” have also been suggested. The polynominal equation of the fourth degree was proposed by Lu.‘~ Hechtert3 found that the paraboia had a high “goodness of fit” to the middle curve of teeth when compared with other generated curves. The parabola has also been suggested by other RichmondI concluded that it investigators. 4*9,10~‘4~15 was doubtful whether the ideal arch form ever exists, but in order to treat a malocclusion, it is necessary to align the teeth and to know the severity of crowding to a given arch form. The parabolic curve was chosen in this analysis because several authors have shown it to be the best fit; it was also selected for its ease in matbematic application. The parabolic curve is constructed to the best fit
Volume 92 Number 3
CONSTRUCTION
Recording
OF
e-ve&
2D ANGLES
dental
cast
in three
dimensions
201
OCCLUSAL PLANE CONSTRUCTION
etve
Fig. 2. Construction of two-dimensional angles. Positive angles are produced when the teeth are proclined relative to the perpendicular raised to the occlusal plane.
Fig. 3. Occlusal plane points. In three dimensions, three points are required to fit a plane. The occlusal plane is established using the buccal cusp tips of the lower premolars and the mesiobuccal cusp of the molars. The plane is constructed as the best fit using the six points.
of the mesiodistal anatomic contact points ending distal to the first molar (Fig. 4). The curve is constructed to each individual case rather than fitting the case to a preselected arch form. The length of the parabolic curve is noted. 8. Discrepancy measurements
A discrepancy measurement is a useful indication of the crowding or spacing within the dental arches that will determine an extraction or nonextraction approach. Two types of discrepancy measurements are used. a. Arch discrepancy This discrepancy measurement provides a guideline for how far the anatomic contact points are from the corresponding contact point on the adjacent teeth in relation to the occlusal plane (Fig. 5). Therefore, this measurement attempts to quantify the discrepancy between the contact points in two dimensions. The distance a-b when projected onto the occlusal plane c-d produces the two-dimensional discrepancy. These distances represent the movement required to align the dental arches to form a flat occlusal plane. b. Parabolic curve discrepancy The length of the parabolic curve and the overall mesiodistal widths are known; therefore, by subtracting one from the other, the discrepancy can be determined, noting whether the arch is spaced or crowded.
Fig. 4. Parabolic curve length. A parabola was constructed to the best fit of the mesiodistal anatomic contact points ending distal to the first molar (plotted from Fig. 7).
202
Am. J. Orthod.
Richmond
Dentofac.
Orthop. September 1987
Instrument
The points were recorded by means of the Reflex Metrograph,* which is capable of measuring objects in three dimensions (Fig. 6). This instrument basically consists of a semireflecting mirror; therefore, an object standing in front of the mirror has its image at an equal distance behind the mirror. Working on this principle, a moving light source connected to a three-dimensional (X-Y-Z) slide system behind the mirror can be used to record points corresponding to the image of the object. Movement of this slide system operates rotary encoders, which translate movement into electrical impulses that can be transmitted to a computer. With the aid of a computer program, this information can be accepted and measurements performed. One of the qualities of the Reflex Metrograph is that the 0.3-mm diameter light source can move into interproximal spaces and contact areas where a pair of dividers could not. The accuracy of the Reflex Metrograph has been tested by RichmondI and the error found to be less than 0.27 mm (<0.3%) with an angular error of less than 0.76%. This compares favorably with other authors.‘7-‘9 One hundred ten points were recorded. It takes an experienced operator less than 20 minutes to record 110 points. To illustrate the analysis, a Class II, Qivision 2 malocclusion is shown in Fig. 7. The following analysis shows a deep overbite of 6 mm and an overjet of 3 mm. The center lines are almost coincident, the mesial edges all being to the right of the constructed center line by approximately 1.5 mm. The distances for the mesiodistal widths are shown.
Fig. 5. Discrepancy measurement. The distance a-b is the three-dimensional distance between the anatomic contact points. When the points a-b are projected onto the occlusal plane o-o, the distance is now c-d. This “correction factor” provides a more reliable discrepancy reading and is related to the need for flattening the occlusal plane.
Sample
To test the accuracy and reproducibility of the analysis, 12 sets of dental casts were randomly selected from 150 patients, first checking to validate the occlusal relationships. It was essential that the dental casts be stable when positioned on their heels-together and independently. Procedure
Eight crossed lines were marked around the land area for each in the upper and lower dental casts (the fiducial points). The intersection of the two lines was used to orientate the dental casts so that when they were in occlusion, the spatial arrangement was noted; therefore, when the casts were moved apart to digitize the occlusal surfaces, all the points could be reorientated to the occluded position by re-recording the fiducial points.
CLASS II, DIVISION 2 MALOCCLUSION Left Right Overjet Overbite Center-line discrepancy Mesial edge of upper center line Mesial edge of upper center line Mesial edge of lower center line Mesial edge of right
3.05 mm 5.73 mm
left central right
central
left central central
*H. F. Ross, Ross Instruments
incisor,
3.35 mm 5.78 mm
incisor, incisor, incisor,
1.5 1 mm right
1.65 mm right of 1.42 mm right of
1.19 mm right
Ltd., Salisbury,
of
Wiltshire,
of center
England.
line
Recording
Volume 92 Number 3
Fig. 6. Reflex
Fig. 7. A through
D, Dental
casts
of Class
dental cast in three dimensions
Metrograph.
II, Division
2 malocclusion.
(For
analysis,
see text.)
203
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Am. J. Orthod.
Richmond
Dentofac.
Orthop. September 1987
Table II. Intra- and interobserver errors Intraobserver
Overjet Overbite Mesial contact points of central incisors to center line Upper mesiodistal widths Lower mesiodistal widths Three-dimensional angles, incisors Three-dimensional angles, canines Upper two-dimensional angles Lower two-dimensional angles Overall mesiodistal widths Anterior mesiodistal widths Overall discrepancy Anterior discrepancy Intercanine distance Intermolar distance Length of fitted parabola Length of discrepancy of fitted parabola OBSERVER ERRORS:
Intraobserver-significant
error
Interobserver
error
n
Mean
SD
RMS
Mean
SD
RMS
24 24 48
0.04 0.13 -0.12
0.32 0.64 0.49
0.31 0.63 0.49
0.17 0.16 -0.17
0.49 0.82 0.63
0.49 0.85 0.64
44 144 24 19 67 67 24 24 24 24 20 21 24 24
0.05 0.15 0.24 0.44 -0.18 0.69 1.06 0.27 0.24 0.11 0.12 0.03 0.44 -0.71
0.26 0.27 2.12 5.82 3.54 3.05 1.37 0.97 1.92 1.43 0.52 0.40 0.93 1.57
0.27 0.32 2.04 5.5 3.48 3.08 1.70 1.01 1.87 1.39 0.52 0.38 1.01 1.72
0.34 0.25 0.89 - 2.48 1.65 - 1.55 3.45 0.66 0.1 -0.77 -0.08 0.01 1.36 -2.01
0.35 0.34 2.34 5.7 5.24 3.79 1.49 1.18 2.74 2.06 0.56 0.45 1.27 1.55
0.48* 0.43 2.7 5.27 5.37* 4.03 3.74* 1.32 3.32* 2.11* 0.54 0.43 1.93* 2.55
difference
P > 0.5;
interobserver-significant
difference
P > 0.5
except
‘I*”
when
0.5 > P > 0.1.
Mesiodistal Left Left Left Left Left Left Right Right Right Right Right Right
widths
Upper
first permanent molar second premolar first premolar canine lateral incisor central incisor central incisor lateral incisor canine first premolar second premolar permanent molar
(mm)
Lower
10.09 6.48 6.23 7.66 6.6 8.8 8.39 6.27 7.39 5.89 6.6 10.27
(mm)
10.13 6.55 6.33 6.34 6.39 5.52 5.83 5.61 6.61 6.8 6.28 10.52
CANINE AND INCISOR ANGULATIONS Three-dimensional angle (“) Left Left Left Right Right Right
canine lateral central central lateral canine
incisor incisor incisor incisor
133.70 148.61 167.42 167.45 129.60 158.80
Two-dimensional angle (“) Lower Upper 1.29 22.74 -1.68 23.63 - 12.73 22.82 - 14.89 25.80 13.48 28.71 -7.09 12.83 Upper
Overall mesiodistal widths (6-6) Anterior mesiodistal widths (3-3) Intercanine distance Intermolar distance Length of fitted parabola
(mm) 90.67 45.11 35.34 48.58 91.50
Lower
(mm)
82.91 36.30 27.36 44.16 85.22
Discrepancy measurements 1. Arch discrepancy Anterior (3-3) Overall (6-6) 2. Parabola length minus Mesiodistal widths
10.22 14.24
3.84 13.83
1.13
2.31
The canine and incisal angulations are measured in three and in two dimensions. The results suggest that the upper central incisors are retreclined to a perpendicular by 12” to 14”. This is supported by the three-dimensional angle of 167” indicating a high interincisal angle. Looking at the lateral incisors, it is noted that the upper right lateral incisor is proclined 14” more than the upper left lateral incisor. Therefore, by looking at the results, these would indicate a Class II, Division 2 incisor relationship. Looking at the arch discrepancy measurement, the upper anterior discrepancy is 10.22 mm and overall 14.24 mm. This would suggest that most of the discrepancy is in the incisor region caused mainly by the retroclination of the upper central incisors. In the lower arch, there appears to be minimal crowding in the anterior region (3.84 mm), but the overall reading is 13.83 mm. This is mainly due to the rotation of the lower left second pre-
Recording
Volume 92 Number 3
molar and insufficient space for the lower right second premolar. The rotations are indicated on the plot of the dental arches (Fig. 4). The parabolic-curve discrepancy shows minimal crowding in the upper arch; the parabolic curve suggests retroclination of the upper central incisors and a proclination of the upper right labial incisor. In the lower arch, minimal crowding is also noted; the parabolic curve suggests retroclination of the lower incisors. RESULTS
The root mean squares (RMS) and standard deviations (SD) are shown for the recorded measurements in Table II. The inter- and intraobserver errors were acceptable and no significant differences were found, the P value being greater than 0.5 for intraobserver and generally greater than 0.1 for interobserver readings. The results would indicate that the measurements undertaken were consistent and reproducible. The intercanine and incisal angulation readings have been grouped together. RichmondI has previously shown that the accuracy of the incisal angulations is generally better than that of the canine angulations. This is due to the distance between the midpoints m-n, which is generally less in canines than incisors, and because the morphology of the canines is slightly unfavorable since the defined points are not always readily identified. The interobserver error was not significant, which suggests that a technician could be trained to use the analysis to produce a series of readings upon which to build a data base that would be useful in research. CONCLUSIONS
Freer, Grew, and Little” reported that a difference exists between the subjective assessment and diagnosis of orthodontists. It would be desirable to make consistent assessments of orthodontic cases; by using the above-described analysis, a consistent approach to diagnosis and treatment planning is possible. All the landmarks were recorded directly on the dental casts. Using the points recorded, planes can be constructed to assess positions of teeth in the three planes of space. Therefore, planes can be directly produced on the dental casts; this offers an advantage over radiographically derived planes as the latter may be subjected to rotational errors and magnification problems. It is important to remember that the analysis is a
dental cast in three dimensions
205
recent development and has not as yet been calibrated. This would best be achieved by mensurating ideal occlusions to attain standard values. Although the full analysis is described, it is possible to measure individual variables alone without digitizing the 110 points. It is also possible to construct other planes and angles if so desired. In conclusion, this analysis can illustrate and quantify the characteristics of a given malocclusion; it is both consistent and reproducible. The analysis may prove useful to the orthodontist in the clinical setting or for use as a research tool. The analysis can describe the dental cast in the three planes of space and could be useful as a record of the dental cast to alleviate the problems of storage. I wish to thank Dr. P. J. Scott for his contribution to the preparation of the software for three-dimensional cast measurement to attain prepared Mrs. L.
and Mr. M. L. Jones for measuring the dental casts the interobserver error. The illustrative work was by the Cardiff Audio-Visual Aids Department and Lowe, who typed the manuscript.
REFERENCES Richmond S, Jones ML. A comparison of two and three dimensional incisor angles. Br J Orthod 1985;12:90-6. Jacobson A. The “Wits” appraisal of jaw disharmony. AM J ORTHOD1975;67:125-38. Hellman M. Dimensions versus form in teeth and their bearing on the morphology of the dental arch. Int J Orthod 1919; 5:615-51. 4. Wheeler RC. A textbook of dental anatomy and physiology. 2nd ed. Philadelphia: WB Saunders Co, 1950:196-215, 352-406. 5. Hawley CA. Determination of the normal arch and its application to orthodontics. Dent Cosmos 1905;47:541-57. 6. Bonwill WGA. Geometrical and mechanical laws of articulation. Trans Odont Sot Penn 1884-1885:119-33. 7. Chuck GC. Labial arch form. Angle Orthod 1934;4:312-27. 8. Black GV. Descriptive anatomy of the human teeth. 5th ed. Philadelphia: S.S. White Mfg. Co., 1902. 9. Izard G. New method for the determination of the normal arch by the function of the face. Int J Orthod 1927;13:582-95. 10. Sicher H. Oral anatomy. 2nd ed. St. Louis: The C. V. Mosby Company, 1952:262-3. 11. MacConnaill MA, Scher EA. The ideal form of the human dental arcade with some prosthetic application. Dent Ret 1949;69: 285-302. 12. Lu KH. Analysis of dental arch symmetry [Abstract]. J Dent Res 1964;43:780. 13. Hechter FJ. Symmetry and dental arch form of orthodontically treated patients. J Can Dent Assoc 1978;44: 173-84. 14. Broomwell IN. Anatomy and histology of the mouth and teeth. 2nd ed. Philadelphia: P. Blakistone Sons & Co, l902:99. 1.5. Currier JH. A computerized geometric analysis of human dental arch form. AM J ORTHOD 1969;56:164-79. 16. Richmond S. The feasibility of categorising orthodontic treatment difficulty-The use of three dimensional plotting [M.Sc.D. thesis]. Cardiff, Wales: University of Wales, 3984.
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September
17. Butcher GW, Stephens CD. The reflex optical plotter. A preliminary report. Br Dent J 1981;151(9):304-5. 18. Scott PJ. The reflex plotters: Measurement without photographs. Photogrammetric Ret 1981;10:435-46. 19. Takada K, Lowe AA, De Cou R. Operational performance of the Reflex Metrograph and its applicability to the three-dimensional analysis of dental casts. AM J ORTHOD1983;83: 195-9. 20. Freer FJ, Grew JM, Little RM. Agreement among the subjective severity assessment of 10 orthodontists. Angle Orthod 1973; 43:185-90.
Reprint requests to: Dr. Stephen Richmond Department of Orthodontics Dental Hospital of Manchester and Turner Dental School Higher Cambridge St. Manchester MI5 6FH England
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1987