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Arrangement in the Jaws of the Roots of the Teeth
original
articles
. . . a detailed study in the finest classical tradition which may have far-reaching clinical implications, especially in orthodontics and endodontics . . .
ARRANGEMENT IN THE JAW S OF THE ROOTS OF THE TEETH W. T. Dempster ,* Sc.D.; W . J. Adams,f D.D.S., and R. A. Duddles,% D.D.S., Ann Arbor, Mich.
This is a qualitative and quantitative study of the arrangement of the roots of the teeth in 11 skulls with typical dentitions. The patterns of the roots are shown pictorially. Additional data show the relations of the center points of the aditus of the roots to the alveolar arch. The direction of root slanting and the amount of slanting were measured and treated statistically. In the light of these data, conventional descriptions of “ ideal” dental arches do not correspond with actual dentitions. This was so for dental arch patterns on a plane surface ( the Bonwill pattern) , for patterns on a cylindrical surface (the curve of Spee) and for patterns having a spherical surface of occlusion (the curve of Monson). These curvatures should be restudied.
This report examines the arrangement, inclination and angulation o f the roots of erupted teeth in the adult. This aspect of the anatomy o f the permanent denti tion has never been covered systematically in the literature on the jaws and teeth. One might assume that information about the anatomy and mechanics of the denti tion would be documented solidly since it bears on such fields o f clinical dentistry as periodontics, orthodontics, oral sur
gery, prosthodontics and dental roentgen ography. Yet, for a hundred years, when ever authors discussed the form and ge ometry o f the dental arches, they really were concerned only with the exposed crowns o f the teeth. Our work, in con trast, deals mostly with the hidden parts of the dentition. The whole complement of about 50 roots o f the dentition has not been studied as a whole, and our present information
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about roots is a congeries from extracted teeth or from dental roentgenograms and histologic sections. A study of the arrange ment of teeth cannot be made on living subjects, nor on occasional biopsy speci mens. It is only through a study of skulls with unusually complete and well-formed dentitions that such information can be acquired. W e have been able to work with such material. Q U A L IT A T IV E A P P R O A C H F O R T H E R O O TS
Figure 1 helps define our problem. This set o f teeth represents the complete denti tion o f one skull. The preparation of our specimen involved the following stages: making rubber-base impressions o f the crowns o f the teeth for both the upper and lower arches, extracting the teeth, setting them into the rubber mold and stabilizing the roots with acrylic resin. Thus, both the crowns and the roots of the teeth had the identical relationship that they had while in the intact skull. Th e dentition in Figure 1 shows that the roots o f the teeth from the cementoenamel junction to the apex are more or less elongate cones. The bicuspid roots are most nearly perpendicular to the plane of occlusion. The lower incisor and molar roots are directed obliquely back ward; in addition, the incisor roots are more or less medially directed, and the molar roots are tilted outward. The roots of the maxillary teeth mesial to the second bicuspid are directed backward and in ward, whereas the roots o f the maxillary molars are more vertical than the roots of the opposed lower molars. T h e palatal roots o f the upper molars tilt inward more than the buccal roots do. In this generalized description of the dentition, the roots have been treated as if they were cones; that is, right circular cones, with a straight central axis, ignor ing curvatures and surface irregularities. This probably can be justified from a me chanical standpoint for man since the line o f action of a force vector through a root;
that is, the direct thrust into an alveolus, seems to be a straight line. (It would not be so for a monkey canine or a rodent incisor tooth.) A straight line between the center o f the alveolar aditus and the apex of the socket is a simple approxima tion to such a direct line of thrust. Figure 2 shows other relations of the central axes o f the roots in the same skull. T o prepare this specimen, the apexes of the roots were cut transversely by grinding to locate pulp canals at the centers of the roots; a straight drill hole along each pulp canal toward the center o f the wid est part of the root served to define the central axes of the roots. After the roots were drilled through lengthwise (from apex to the occlusal surface o f the crow n), the teeth were reinserted into their alveoli. The skull, after soaking in glycerin, could be drilled easily. Each tooth and root was then drilled from its occlusal surface to the apex using a long drill until the drill emerged somewhere on the surface o f the skull at a point in line with the tooth axis. Orthodontic wires were then placed in each drill hole. Both front and side view cephalograms and comparable natural size photographs were made o f the specimens; these showed root angulations relative to the anatomy o f the skull so that composite drawings o f the skull could be made. Figure 2 shows wires through the roots of the left side of the jaws, and the afore mentioned root directions (Fig. 1) are even more clearly depicted, especially those of the mandibular roots. T h e second bicuspid and the mesiobuccal roots of the maxillary molars are directed well for ward toward the forehead region; the first bicuspid and the palatal (lingual) roots o f the molars point farther back, and the cuspids and incisors direct their thrust vectors still farther toward the vertex. Figure 3 shows the anterior aspect of the same skull. T h e bicuspid roots are about vertical, and they aim through the medial part o f the orbit. The buccal roots o f the first and second molars point through the central part of the orbit. All
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Fig. I • C o m p le te dentition with teeth and roots having same positional relations as they did in alveolar processes. A : Left lateral aspect. B: O b liq u e aspect. C : A n te rio r aspect. D: Posterior aspect
of the remaining upper roots— third m o lars, incisors and cuspids— tilt inward toward the region between the orbits; that is, ethmoid or juxta-ethmoid part of the frontal bone. (I f the reader wishes to make a more detailed study o f the 50 plus wires, he may use a straight edge, and he should carefully correlate the cod ing system o f the free ends of the wires with the teeth concerned.) The latter root axes, if projected upward, may cross the midline anywhere between the nasal cav ity and the region well above the scalp. The general arrangement of Figures 2 and 3 was corroborated by another skull with wires in all the roots. In addition, tracings were made o f split skull cephalograms o f the group o f 11 basic skulls with sockets coated with white lead so that axes could be determined. The axis of each root was then extended by drawing a straight line until it emerged from the skull. In several separate mandibles, lengths o f straight wire were cemented axially into the empty sockets with a
metal-plastic mixture (Sculp-metal) that hardened and fixed the wires. When the wires were twisted loose, the bone beyond the putty could be drilled so that the wire axis could be pushed on through the bone. These confirmed that the skull used for Figures 2 and 3 was representative of the material used in the study. Q U A N T IT A T IV E A P P R O A C H E S TO ALVEOLAR A RCH
Materials * Our study of the pattern of root arrangement was less a study of roots per se than o f the alveoli for the roots which, after careful extraction of the teeth, represent a negative mold o f the root forms. Thirteen adult male skulls obtained from commercial suppliers (from India) formed our study sample; nine had a complete dentition o f 32 teeth, and four, lacking third molars, had 28 teeth. The skulls were selected from many others for well-formed individual teeth, minimal wear and minimal absorption of
+ Di st obuccal r o o t s □ Pal at al r o o t s LOWER MOLARS ▲ Mesi al r oot s ■ Di stal r o o t s Fig. 2 • Lateral view o f skull with orthodontic wires in holes drilled axially through roots o f teeth and on through to exterior, showing variable orientation o f different axes. C o d e fo r wires relates to different roots
Fig. 3 • Relative orientation of axes of roots of teeth as seen in anterior view o f sam e skull shown in Figure 2; same co d in g of roots of teeth has been used as in Figure 2. The general arrangem ent shown in Figures 2 and 3 was corroborated by another skull with wires in all the roots
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Fig. 4 • Catenom eter built on vernier caliper shows standard 74 mm. catenary curve superim posed on a dental arch. Left: Two points (p and q ) are localized b y m easuring perpendicular distance to curve an d sagittal distance between point and com m on horizontal reference line. R ight: Best possible fit of catenary curve in relation to mean center points of all of roots of I I m andibles
alveolar bone. (A comparable group of skulls with caries-free teeth representing our present-day American population probably could not be obtained.) Tw o skulls were used for the qualitative data of Figures 1, 2 and 3; 11 skulls were sub jected to quantitative analysis. Methods • For general reference through out the study, standardized full-scale pho tographs and cephalograms weye made both with the teeth in occlusion and after the extraction of the teeth. For clarity and to avoid overlapping images of the right and left sockets, additional side view cephalograms were made after a sagittal sawcut through the mandible and palate; a film placed vertically in the sawcut then permitted us to take a roentgenogram of
each side separately with the skull in a cephalostat. Stone models of the dental arches and alveolar processes of each skull were made both before and after extrac tion o f the teeth. This gave us a perma nent record of both the teeth and the sockets. W ood’s metal casts of the root patterns were also made from a duplicate socket model of plaster of paris. For all o f these models o f sockets, we defined the plane of the alveolar margins (or common alveolar plane) in terms of three points: (1) the midpoint of the interradicular septum between the central incisors, (2) the midpoint o f the interradicular septum between the first and second right molars and (3) the com parable intermolar point o f the left. Each stone model was set into a rectangular
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block o f plaster o f paris with its alveolar plane below but parallel to the smooth upper surface o f the plaster; this surface later formed a reference plane for the measurement of root angles (Fig. 7, A, B ) . A life-size photograph of the arch in the plane o f the alveolar margins also was made showing alveoli and septums. Th e curve which was used as a refer ence in this study was a “ catenary” curve — a type o f curvature produced by a chain of many links suspended by its ends but otherwise allowed to hang freely. The suspension system and chain for defining such curves was called a “ catenometer.” MacConaill and Scher1 and later Scott2 felt that the catenary was the curvature which best fit the dental arch. No attempt was made in the present study to verify these findings. Figure 4 shows our adapta tion o f Scott’s catenometer using a stand ard 200 mm. length o f fine link jeweler’ s chain soldered to the endpoints (for in side measurements) of an ordinary metric vernier caliper. The catenary, which most nearly coin cided with the three points just men tioned for the plane o f the alveolar mar gin, was a catenary curve o f 74 mm. width between suspension points. W e arbitrarily selected this catenary (Fig. 4) as a refer ence standard for all subsequent use on both the upper and lower arches. (W e might have used equally well any other curve of reference— ellipse, parabola and so forth— since it was merely a line from which to measure.) The catenary was used since a convenient size could be ob tained easily just by suitably separating the suspension points of the chain. M E A N P O S IT IO N S OF R O O T S IN A L V E O L A R A R C H
T w o 10 X enlarged outlines of this 74 mm. catenary curve were drawn to repre sent the lower and upper jaws, respec tively. On these curves, the relative posi tion o f the center point of each alveolar aditus in all the upper and all the lower jaws was plotted. The photographic nega
tives o f the alveolar arches provided the raw data for locating the various center points. For each center point, we chose to define its locus in terms o f a point on the curve where a perpendicular from the center point intersected a line tangent to the curvature. (Tangents and perpen diculars may be drawn in fairly accurately on large scale curves using simple mechanical-mathematical methods.) We measured for each center point: (1) its perpendicular distance to a tangent to the catenary, and (2 ) the sagittal distance between this intersection point (that is, of the perpendicular and the curve) and a horizontal tangent to the lowest point of the curve (Fig. 4, left). This sagittal and perpendicular distance served to lo cate the center o f each alveolus mathe matically in relation to the catenary curve. From these measurements an average measurement for each alveolus could be determined. As shown (Fig. 5, 6 ), the centers of the different roots form a cluster o f points. W e sought to find an average point or center o f gravity o f the cluster for each root, then some measure o f the relative dispersion. The center of gravity of a cluster was found as follows: An arbitrary x and y coordinate was drawn in, and the measured distances from each point to coordinate x were summated and divided by the number o f points; this provided a mean center distance from the x co ordinate. Similarly, %dy = distance of the n mean center locus from the y coordinate; dv was the distance between each point and the y coordinate; n was the number of points. The mean distances were de termined separately for each root for both sides o f the 11 jaws, taken together. Thus, we obtained mean points for each root for both the lower (Fig. 4) and the upper jaws. As is shown graphically for the 11 skulls (Fig. 4, right), a best fit catenary is cer tainly not a perfect description of the dental arch curvature. Nevertheless, the catenary— fitted to the incisor and molar
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la n d m a r k s w e u se d — is p r o b a b ly as g o o d a n a p p r o x im a t io n t o fittin g th e a lv e o la r ce n te rs
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V A R IA B IL IT Y OF D IF F E R E N T A L V E O L A R A D IT U S
O u r p lo t s s h o w e d m a n y clu sters o f p o in ts e a c h n o r m a lly re p re se n tin g 22 ro o ts . T h e a re a o f t h e clu s te r w a s sm a ll w h e n th e d e v ia t io n
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bility from jaw to jaw was large. T o pro vide a simple measure o f the relative vari ability or dispersion o f the individual center points relative to the weighted mean point of the cluster for the different roots o f our graphs (Figs. 5, 6 ), the aver age deviation (A.D .) was computed. New perpendicular x and y coordinates were drawn through each mean center on our large plots; one axis was arbitrarily drawn parallel to the catenary and the other was perpendicular. Next, the distances be tween the x coordinate and each point,
Fig. 5 • C e n te r points of roots o f teeth of 22 m andibular sides are shown as clusters o f points about alternate sockets on left and right sides of standard catenary curve. Relative sizes and proportions of small ellipses with crossed m esiodistal and buccolingual axes indicate variability in position of several roots as measured b y average deviation (A.D .)
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neglecting + or — signs, were measured in millimeters, summated and divided by the number of measurements; similarly, the average deviation was determined rel ative to the y coordinate. An ellipse with its semi-axes the same as the two perpen dicular A.D. was drawn about the mean center. (The more frequently used stand ard deviation, a, relative to a mean in cludes 68.2 per cent of the deviates above or below the mean in a bell-shaped type o f population; whereas the A.D. includes 57.5 per cent.) In Figures 5 and 6 an ellipse encircles the points included by the A.D. in two perpendicular directions. Since the axes were parallel to and per pendicular to the catenary, the semi-axes
represent an average deviation that re lates to the conventional directions of the arch, that is mesial, distal, buccal and lingual. The size and spread of the clusters of points shown in an alternating sequence (Fig. 5) for the right and left sides of the mandibular arch represent the relative stability to the mean center position of root position. The crossing point o f the axis within the ellipse is the locus of the mean center of the root. All of the roots show more variability in the mesiodistal than in the buccolingual direction. The central incisor, for instance, in 22 sockets showed a limited positional variation in the labiolingual direction, but this tooth
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A
On each side o f the arch the 15 roots of the maxillary dentition show a degree o f variability (Fig. 6 ). Only the central incisor and the buccal root o f the first bicuspid are comparable to the anterior mandibular teeth in having tight clusters o f centers, that is, small ellipses. The cus pid and the second bicuspid roots are notably more diffuse relative to the catenary, and all of the molar roots are much more variable in their positions than any o f the mandibular roots. The lingual roots o f the three molars and first bicuspid and the buccal roots o f the third molar fall inside the standard 74 mm. catenary curve; whereas most of the roots from the lateral incisor to the buccal roots of the first molar are outside. R O O T IN C L IN A T IO N
Fig. 7 • A : Inclination angle of root is measured on stone m odel of maxillary alveoli set in plate o f plaster as shown. W ire placed axially in upper first m olar root socket is m easured b y transparent protractor with its axis of sym m etry parallel to midsagit+al plane of palate. B: M easurem ent of amount of angulation of wire relative to "p lan e of the alveolar m argins." U sin g sliding depth g a u g e of vernier caliper (b) and fixed height (a), angle (a ) of right triangle com pleted by wire m ay be determ ined trigonom etrically
had three times that range in the mesiodistal direction. In contrast the deviation from the mean in both directions is more nearly the same for the cuspid root. The first, the third and finally, the second molar roots are increasingly variable in their locations, in that order; in each in stance, both the mesial and the distal roots have about the same degree of variability. The two bicuspids, which show compar able amounts o f positional variability, are less variable than the molar roots but more than the incisor-cuspid roots. Char acteristically, 4 roots (lateral incisor to second bicuspid) lie on the buccal side of the standard catenary; whereas, the sec ond and third molars are slightly lingual.
Since the central axes o f the roots slant more or less as they intersect the plane of the alveolar margins, it was convenient to measure the degree o f tilt relative to a perpendicular to this plane. T w o meas urements were found to be necessary for each root: (1) inclination was regarded as the direction toward which the coronal extremity o f the root tilted; that is, anr terior, toward the midline, laterally and so forth, and (2) angulation was taken as the number of degrees that the root axis slanted beyond the vertical. Both inclina tion and angulation were measured on the stone replicas o f the alveolar sockets. These, for ease o f manipulation, were set into plaster blocks with a flat-surfaced plane parallel to the three reference points located beyond the apexes o f the sockets (Fig. 7, B ) . Straight lengths of orthodon tic wire were carefully set into the alve olus for each root to represent its longi tudinal axis; the rods were set into the model so that they could be pushed beyond the plane of the alveolar margin, be pushed on through in the opposite direction, be removed or reinserted. T o measure rod inclination, that is, the root inclination, the free end o f the rod was pushed through a central hole in a trans
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parent protractor scale, and this was clamped to the model just over the alve olar plane (Fig. 7, A ) . The 0-180 degrees coordinate was made parallel to the midsagittal plane o f the cast with the zero anterior. The direction o f maximum slant of the axis was determined and read di rectly in degrees reading from zero either way on the graph. Therefore, our con vention in reading inclination for either arch as seen from the occlusal view was as follows: A direct anterior slant was read as 0 degree, a slant in the posterior direction was 180 degrees, slants from + 1 degree to +179 degrees implied inclina
tion away from the midline and — 1 degree to —179 degrees referred to in clination toward the midline. A N G U L A T IO N F R O M T H E V E R TIC AL
Angulation was measured trigonometri cally from the apically located parallel plane using a simple device involving a vernier caliper mounted horizontally on a base with a fixed 25 mm. height (Fig. 7, B ) . A length o f the rod was treated as the hypotenuse ( h) of a right triangle having a vertical side (a) with a fixed
Fig. 8 • C ate n ary curves in relation to sockets of m andibular and maxillary arches. W ith in each socket, arrow -tipped lines indicate direction of inclination of coronal extremity of roots; angle between shaft and barb is average deviation o f all inclination measurements. Twenty-six small grap hs— I for each root socket— indicate num ber of degrees o f angulation of root axes from vertical; dashed lines on each side give average deviation of angulation relative to mean
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T a b le • Inclination and angulation o f roots o f teeth in 11 skulls
+ A .D .*
M ean angulation from v e rtica l (degrees)
± A .D .
N o . (n) o f roots m e asu red f
103 89 72 61 28 0 45 -2 8 -2 19 8 3 6 16 2
42 40 39 44 66 78 30 35 46 36 20 22 14 6 5
12.3 11.3 11.6 10.7 9.3 9.6 13.0 13.0 10.4 8.6 10.4 9.6 20.6 29.0 29.0
3.9 4.6 5.5 3.2 2.8 3.9 4.9 4.6 3.9 3.6 3.4 3.2 3.2 3.4 4.1
11 7 7 20 16 16 22 22 22 22 12 12 22 22 22
-5 3 -5 1 -5 1 -5 3 -5 8 -5 8 -3 4 14 13 17 2
15 16 19 18 36 23 54 35 27 10 15
32.6 38.5 25.4 28.0 12.8 16.8 10.1 9.8 14.6 21.2 19.3
4.5 8.0 4.6 3.4 5.6 4.0 3.4 3.9 6.4 6.4 7.5
10 9 22 22 22 22 22 22 22 22 22
M ean inclination (degrees)
M a x illa ry roots Third molar lingual Third m olar d isto buccal Third m olar m esiobuccal Second m o lar lingual Second m o lar d istobuccal Second m o lar m esiobuccal First m olar lingual First m olar distobuccal First m olar m esiobuccal Second bicuspid First bicuspid lingual First bicuspid buccal Cuspid Lateral in ciso r C e n tra l inciso r M andib ular roots Third m olar mesial Third m olar distal Second m olar mesial Second m o lar distal First m olar mesial First m olar distal Second bicuspid First bicuspid Cuspid la t e ra l inciso r C e n tra l inciso r
Root
*A.D., average deviation. -[Where the number (n) is less than 22, fused roots, which have not been included in the Table, or missing third molars will account for discrepancy.
distance of 25 mm. and a horizontal side ( b) to be measured. Side ( b) was meas ured quite accurately by extending the depth gauge at the end of the caliper until it touched the rod (Fig. 7, B) and reading the vernier. The angle “ a” which referred to the slant angle from the vertical could be found as follows: Tan a = (b) / (a) or ( b) /25. O ne axial rod at a time (while the others did not project to the reference plane) was measured, first for inclination angle and then for angulation. The table lists the inclination and an gulation for the various roots of the two jaws and shows the degree of variability found in the roots. Figure 8 shows the same information graphically. In the cen ter o f each root alveolus, the main arrow like lines show the mean inclination direc
tion relative to the sagittal plane; the tail of the arrow indicates the direction of the apex of the root; whereas the tip is the point o f intersection with the alveolar plane. The angle between the barb on each side and the main vector line repre sents the A.D., right and left, o f the inclination angle for each root. The small rectangular plot nearby shows for each root the mean angle of deviation from the vertical; the dotted lines on either side indicate the A.D. of the angulation angles measured. From Figure 8 and the table, it can be seen that the mandibular roots of the cen tral incisors through the first bicuspid are directed lingually; whereas the apical extremities of the second bicuspid and molar roots are inclined buccally. In the
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upper arcade all of the roots except the distobuccal roots o f the first molar tend to tilt palatally. Variability in direction of inclination, low for the anterior teeth, tends to increase from tooth to tooth through the second molar. Figure 8 clearly shows that the incisors and the third molar roots are most notably trans verse to the arch; whereas most of the other roots tilt at an angle. The maxillary incisors and the man dibular third and second molars have the greatest angulation (table, Fig. 8 ). The first and second bicuspids of each arch have the most perpendicular roots. The maxillary molar roots rarely tilt more than 15 degrees from the vertical. The greatest deviation from the mean in the mandibu lar dentition shows in the distal root of the third molar and in the anterior teeth from cuspid to central incisor. The least variable angulations are found in the lower bicuspids and lower first and second molars; most o f the roots of the maxillary dentition show a comparable low degree o f angular variation. Figures 9 and 10 pictorialize the angu lar arrangement o f the longitudinal axes of the roots as they lie in the alveolar bone o f both arcades. Each figure presents a composite that represents the mean obliquity o f the right and left roots from 11 jaws. The axes are shown as projec tions beyond the occlusal surfaces of the teeth so that they stick out like pegs or stakes and emphasize the overall angular pattern that characterizes the root system. The two views in both Figures 9 and 10 represent orthographic projections o f the mean directions of root tilting as seen from the side and front views, aspects just 90 degrees from one another. Thus, when the same root is viewed alternately in the two projections, its true orientation may be visualized. The side view in Fig ure 9 shows the maxillary pattern of the roots extending in an oblique and pos terior direction into the alveolar bone; the bicuspid axes are the least oblique; the roots of the anterior teeth are the
most oblique, and the molar roots are intermediate. From the front view, the central incisor and bicuspid roots are the most vertical; the lateral incisor and cus pid roots are directed medially. The buc cal roots of the molars are vertical or tilt slightly outward; whereas the palatal roots tilt inward. As a mean pattern with symmetrical right and left roots, the ar rangement, although similar to the single specimen of Figures 2 and 3, is much more regular. An even more striking pattern shows up for the mandibular roots (Fig. 10). As seen from the side, the incisors slant pos teriorly to a pronounced degree; the cus pids slant less; the second and third m o lars slant even less, and the bicuspid and first molar roots are the most nearly ver tical. From the anterior aspect, the incisor and cuspid roots point inward; the bicus pids are fairly vertical, and the molar roots in a mesiodistal sequence increas ingly slant outward. As mentioned, the axes o f the roots in Figures 9 and 10, as well as the crowns, project like pegs beyond the alveolar bone. The projecting parts, since they suggest the angulation of the buried roots, which the crowns do not, will readily per mit one to visualize the type of force application (on the exposed part of the tooth) required to produce a given type of root displacement. The latter may be either a minor displacement well within physiological limits; it may be a persistent force resulting in socket adaptation and gross tooth movement, as in orthodontics or periodontics, or it may be a destructive application of force from either oral sur gery or accidental trauma, which results in luxation. The tooth may be moved up or down along its axis in line with the angulation of the axis, as described here, without torque being applied. It may be rotated about this axis, twisting or tearing the periodontal fibers. It may be tilted much in the way that a peg in the ground or a cantilever beam may be moved by a trans
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Fig. 9 • Composite illustration of maxilla and upper teeth as seen from anterior and lateral aspects showing average arrangement of teeth. Amount of slant shown by roots represents mean of combined right and left teeth of 11 ¡aws. Longitudinal axes of roots of teeth have been extended beyond crowns and are shown in orthographic projection. C oding of roots same as for Figure 2
verse or oblique force; a resisting torque will be developed in the hidden part of the tooth and its socket. Finally, if a suit able torque (most easily visualized as per pendicular to the long axis) is applied to the crown of a tooth, the root will exert a steady pressure against a side of its alve olus. This force system may be visualized as comparable to that which a canoeist applies to a paddle; namely, above the blade and at the handle, in order to drive
the blade through the water and develop a forward thrust for his canoe. If the pro jecting rods o f Figures 9 and 10 are viewed as if one could apply forces to them, all of the various possible tooth displacements may be visualized. The dif ferent arrangements o f the different roots, as shown, make it clear, even when no other factor except axis obliquity is con sidered, that the mechanical problems of the teeth are quantitatively different.
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Fig. 10 • Complete orthographic projection of mandibular teeth as seen from anterior and lateral aspects showing arrangement of teeth relative to alveolar bone. Degree of obliquity of roots of right and left teeth of 11 ¡aws has been averaged. Root axes have been extended beyond crowns to emphasize angular pattern
D IS C U S S IO N
Although the arrangement of the roots of the teeth apparently has not been stud ied before, the dental and anatomical lit erature has had much to say about the dental arches. These comments have been based almost wholly on data relating to the crowns of teeth, to plaster models of dentitions or to experiences with articu lators and the designing o f dentures. The
generalizations that have followed have often been based on old data from few specimens and rather casual observations without measurements. Few people have distinguished between the alveolar arches and the dental arches proper. On the basis of simple qualitative observation, anthropologists3,4 have de scribed the general shape of the palatal arch using such terms as paraboloid, U-shaped, ellipsoid, rotund and horse
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shoe-shaped. From simple length to width ratios, narrow, midrange and broad pal ates have been distinguished— with an appropriate latinized terminology. W il liams5 took several measurements from plaster models to work out more discrimi nating ratios relating to the dental arches. Some of the differences in the descrip tions of arch form over the years have depended as much upon the landmarks selected as upon the dentitions dealt with. G . V . Black6 fitted a semiellipse to the curvatures of both the upper and lower dental arches. This semiellipse coincided with the incisor cutting edges of the upper arch, skirted the palatal side of the cus pids and then ran along the buccal cusps of the bicuspid and molar teeth. W hen the same sized semiellipse was fitted to the lower dental arch, it coincided with the cutting edges of the incisors, and it lay along the buccal surfaces of the bicuspid and molar teeth. Black7 was aware of variability in the form and placement of teeth. H e recognized that the cuspids might be more or less prominent and that the more posterior teeth might be either a straight line or a curve. As early as 1887, Davenport8 noted that a regular arch curvature could be identified along the line of occlusal con tact from the lingual cusps of the upper molars and the opposed sulci of the lower molars forward to the incisor contact points. Angle,9 who made much o f the functional significance of this line, re ferred to it as being paraboloid shaped. Later attempts to define arch relations included those of Franke10 and Goldstein and Stanton.11 T h e later authors used a special denture pantograph12 to trace the gingival margin of a plaster model. The midpoint of the area of each tooth as seen in the pantograph tracing was called the centroid of the tooth section, and these were taken as points which defined arch curvature. Using landmarks similar to those of Davenport,8 MacConaill and Scher1 in dicated that the curvature of the “ com mon occlusal line” was ideally described
by a catenary curve. Scott2 concurred; both reports recommended a standard 200 m m . catenometer which could be varied in its width to fit different arches. Hayashi13 defined a still different “curve of dental arch,” arbitrarily, using as land marks for both arches the incisal edges of the incisors, the tips of the cuspids, the tips o f the buccal cusps of the bicuspids and the deepest points of the notch be tween the buccal cusps of the molar teeth. This curve was projected to an occlusal plane which simultaneously touched the medial comers of the central incisors and the extreme points of the distal buccal cusps of the second molars. Hayashi’s curvature was laid out on rectangular co ordinate paper, and the curve was de fined mathematically by an empirical exponential formula of the type y = bx7c. It is noteworthy that all of the curve describers were apparently seeking a basic mean pattern or “ ideal.” Apart from Black’ s recognition6 of the ways in which teeth m ay vary in relation to a basic pat tern, only MacConaill and Scher1 actu ally measured the relative variability of the positions of the teeth. (Although these authors have dealt with the variability of occlusal points, and we have been con cerned with the variability of the positions of the root aditus, the two classes of data are so different that logically they can be neither contrasted nor harmonized.) Hrdlicka14 stated that harmony in oc clusion is “common in all primitive peo ple, but is frequently disturbed in m odem civilized man.” Years before, in 1904, Cryer15 vigorously dissented from the view that harmonious arch patterns had any real functional meaning. H e main tained that each dentition presented its own individual occlusal problems. T h e teeth were regarded as functionally re lated to the bony framework of the facial bones, and each specific occlusion, whether typical or atypical, was consid ered the outcome of such inborn or en vironmental factors as: available bone space, anthropological or racial character istics, orderly or disturbed growth, tooth
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Fig. 11 • In each sketch, plastic cube with central rod was cemented to occlusal cusps of each molar tooth. Rods (double lines) for mandibular teeth or their virtual extensions for maxillary teeth show points of intersection (black dots) at different levels
crowding and so forth. T h e view that bones and teeth were interrelated func tionally was amplified16 later by students of facial bone structure who believed that the directional properties, that is, “ split line” pattern or “grain,” of facial bones were related to the effective transmission of masticatory stresses. In Figures 5 and 6, which relate to the group of 11 intact dentitions, there are shown clusters of points and ellipses that suggest the relative variability of the cen ter points of the aditus of each root as projected to the common alveolar plane. It might be assumed that the central in cisor and the first and second molar roots would be the least variable since these were closest to the landmarks for super imposing the reference curves. But the variability was especially high for the sec ond molar. It was higher for all molars than for the other teeth. Since the teeth show different amounts of variation and do not align to form any regular curve, the idea of an ideal curve cannot be re garded as more than a convenient, rough approximation for descriptive purposes or as a suitable pattern for the placement of artificial teeth on dentures. T h e commonness of crowded, irregular and twisted teeth and occlusal dishar
monies has long been recognized.17'20 Hunter17 even correlated tooth irregular ity with unusual mechanical pressures. W hen in 1899 Angle9 introduced his classification of anomalous occlusions and the term “malocclusion,” he went a step further and made it clear that the forces of cuspal interdigitation were the essen tials for keeping the teeth in alignment. This outlook argued for adapting mech anisms rather than for “ideal” patterns which ignore variance. Th e occlusal surfaces of the teeth of the upper and lower dental arches have at times been regarded as a continuous interdental occlusal surface or “ plane.” As seen laterally, this surface was recog nized as a sagittal curvature,7, 21 concave upward. Spee21 interpreted it as a circular curve which could be laid out with a compass on a profile photograph of a skull; when the curve was continued backward on Spee’ s drawings of adult skulls, it passed between the head of the mandible and the articular eminence of the temporal bone. T h e center of curva ture was at the lacrimal bone. Langer22 and Luce23 had already demonstrated that the mandibular head followed a curved pathway (concave upward) when it moved sagittally over the articular emi
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nence of the temporal bone. Spee cor related their observations with his, and the “ curve o f Spee” came to be regarded as a compensating “ curve” for effec tive protrusive movements of the jaw. Nagao,24 however, showed that if this were so in humans, it certainly was not a general feature of primate (or other mamm al) jaws and teeth. Some years later, Monson25 and Villain26 argued that the idea of a sagittal occlusal curvature should be expanded into a concept of a three-dimensional spherical curvature in volving both the right and left bicuspid and molar cusps and the right and left mandibular condyles. A sphere 8 inches in diameter suggested by Monson (20 cm. by Villain) was regarded as a basic di mension that could be used by dentists concerned with the design of complete dentures. T h e Monson curvature included an earlier denture-arch design presented by Bonwill27 in 1887 along with an articu lator. Gysi28 and later Boyle29 introduced arch designs that were somewhat differ ent. Both Monson25 and Boyle29’ 30 were concerned that the upper and lower den tal arches came into occlusal contact along a spherical interface; however, Boyle specifically omitted the mandibular condyles as totally unrelated to the spheri cal occlusal pattern. From the small number of specimens studied and his crude way of measuring contours, Monson25 believed that perpen diculars from the occlusal surfaces o f the molar and bicuspid teeth converged to a common point. These radiuses of the spherical surface of contact, he inferred, illustrated “ lines of force.” Even further, and totally without evidenced, he assumed that “ lines from the center [of the sphere] obviously must pass through the center of each tooth.” For Boyle29,30 the concave and convex occlusal surfaces formed a mill with a radius of curvature of 11.1 c m .; the masticatory forces were imagined as merging at a fixed point above and forward of the crista galli of the ethmoid bone— at a so-called “dental center.”
T h e conjecture of Boyle and Monson that the roots of the teeth converge to a center is disproved by the data shown in Figures 2, 3, 9 and 10. W e are not aware that anyone has so far made a statistical or mathematical study of the “ curve of Spee.” Likewise, it appears that no analytical study has been made of the spatial curvatures of the oc clusal faces of actual dentitions. Monson, Villain and Boyle merely fitted mandibles or plaster models and dentures to repre sentative spherical surfaces to determine what size of sphere gave the best fit. D if ferences in the way dentitions fit the sphere and the variability of the occlusal surfaces of individual teeth were in no way considered. For the 11 skulls of this study, clear plastic cubes were cemented to the oc clusal cusps of each of the three (or two) molars; perpendiculars from the midre gion of each cube appeared to converge when seen in profile view photographs of either the mandibular or the maxillary regions (Fig. 1 1 ). Each side of the jaw was studied separately. W here there were three molars, the converging lines as seen in profile, however, never met exactly at a single point; that is, they never repre sented the same radius of curvature for the arc between the first and second mo lars and between the second and third molars. T h e perpendiculars typically crossed in sequence at different heights rather than at a common point. Thus, instead of forming the arc of a circle, the occlusal surfaces of the molars showed a slight spiral; the toe of the curve, more over, was about as often directed mesially as it was distally (Fig. 1 1 ). T h e intersect ing perpendiculars from adjacent molars (and therefore, the mean radius for the circular arc tangent to the two teeth) cross one another in the facial region of the profile photographs between the lev els of the palate and the superciliary ridges. In anteroposterior photographs taken at a 90 degree angle to the occlusal plane (Fig. 1 1 ), the directions of perpendicu
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lars from the first, second and third m o lars could be shown simultaneously on the two sides of the dental arch. These pho tographs typically showed that perpen diculars from the third molars ( right and left) crossed at or near the midline in the facial region; perpendiculars from the second molars crossed symmetrically or asymmetrically at a higher level, and those from the first molars were usually vertical or they converged or diverged only a few degrees. Anatomically, the three levels for the crossing points fell within the nasal cavity, in the anterior region of the cranial cavity above or in the region well above the head. None of our 11 upper or lower dentitions showed any semblance of converging toward a common point (as postulated by Monson, Villain and B oyle). Clearly, both the sag ittal curvature (of Spee) and the trans verse curvatures of the molar occlusal surfaces should be restudied from a large statistical sample of models or dentitions. Both suitable numbers of specimens and some standard mathematical method for defining curvatures; namely, evoluteinvolute curve method and so forth, should be used. T h e 11 skulls in this study clearly show that the longitudinal axes of the roots do not converge toward a com mon center. They also affirm that the occlusal surfaces of the molar teeth can not be fully congruent with the surface of any sized sphere.
A co n trib u tio n of the d e p artm e n t o f anato m y, the U n iv e rsity o f M ic h ig a n , Ann A rb o r , M ic h . This research has been sup po rted by N S F G ra n ts no. G-13260 and GB-356. *Pro fesso r o f anato m y, xA rb o r, M ic h .
U n ive rsity o f M ic h ig a n , A nn
TC trn ical in stru cto r in d en tistry, U n ive rsity of M ic h ig a n , A nn A rb o r , M ich . ^Present a d d re ss, G ra n d B lan c, M ich . 1. M a c C o n a ill, M . A ., and Sch e r, E . A . Id e a ! form of the human d en tal a rc a d e , with some p ro sth etic a p p li ca tio n . D . R eco rd ¿9:285 N o v. ¡949. 2 . S c o tt, J . H . Sh ape o f the d en tal a rc h e s. J . D . R es. 36:996 D e c. 1957. 3. M a rtin , R . Lehrbuch d e r A n th ro p o lo g ie . J e n a , G . F isc h e r, 1914. 4 . H rd lic k a , A . A n th ro p o m e try . In stitu te of A n a to m y, 1920.
P h ila d e lp h ia , W is ta r
5. W illia m s , P. N . D eterm ining the shap e of the no r m al a rc h . D. C osm os 59:695 J u ly 1917.
6. B la ck , G . V. D e scrip tive an a to m y o f the human te eth , e d . 3. P h ila d e lp h ia , W ilm in g to n D ental M a n u fa c tu rin g C o ., 1894. 7 . B la ck , G . V . D e scrip tive anato m y o f the human te eth . P h ila d e lp h ia , W ilm in g to n D ental M anufacturing C o ., 1890. 8. D aven po rt, I. B. The sig n ific a n c e o f the natural form and a rra n g e m en t o f the d en ta l arches o f m an, w ith a co n sid eratio n o f the chang es w hich o c c u r as a result of th e ir a rtific ia l d era n g em en t by fillin g , o r by the e xtractio n of te eth . D . C osm os 29:413 J u ly 1887. 9 . A n g le , E . H . C la ss ific a tio n o f m a lo cclu sio n . D . C o s mos 41:248 M arch 1899. 10. Fra n k e, G . O b er W achstum und V e rb ild u n g e n des K ie fe rs und d e r N asenscheidew and a u f G ru n d V e rg le ic h en d er Kiefer-M essungen und e xp e rim e n te lle r U ntersuch ungen ü b er Knochenw achstum . Z tsch r. f . La ryn g . 10:187 J u ly 1921. 11. G o ld s te in , M . S ., and Stanto n, F . L . C han g e s în dim ensions and form o f the d en ta l arches w ith a g e . In te rn a t. J . O rth o d o n t. 21:357 A p ril 1935. 12. Stanto n, F. L . ; Fish, G . D ., and A shley-M on tag ue, M . F . D escrip tion o f th ree instrum ents fo r use in o rth o d o n tic and c e p h a lo m e tric in v e stig a tio n s. J . D . Res. 11:885 O c t. 1931. 13. H a y a s h î, T . A m a th e m atica l analysis of the curve o f the d en tal a rc h . Tokyo M . & D. U n iv . Bui. 3:175 N o . 2 1956. 14. H rd lic k a , A . N o rm al v a ria tio n of teeth and jaw s and ortho d o nty. A m . S o c . O rth o d ists. T r. 33:68 M ay 1935. 15. C ry e r, M . H . T y p ic a l and a ty p ic a l occlusion o f the teeth in re latio n to the co rrectio n o f irre g u la ritie s . D. C osm os 46:713 S e p t. 1904. 16. Bennînghoff, A . S p a ltlin ie n am Knochen, eine M ethode zu r Erm ittlu n g d e r A rc h ite k tu r p la tte m Knochen. A n a t. G e s . V e rh . 34:189 1925. 17. H u n ter, J . The natu ral history o f the human te e th ; e xp la in in g th e ir stru ctu re , use, fo rm a tio n , grow th and d ise ase s, e d . 2 . London, Jo hnso n , 1778. 18. Fox, J . The natu ral history of the te e th . London, Thom as C o x , 1803. 19. K n e isel, F . C . Der S c h liefsta n d d e r Z ähne dessen Ursachen und A b h ü lfe nach e in e r neuen, sichern und schm erzlusen H e ilm e th o d e . (W ith co n cu rre n t French v e r sio n .) B e rlin , E . S . M ittle r, 1836. 20. S te rn fe ld , A . U e b e r Bissarten und Bissano m alien. M unich, K no rr and H irth , 1888. 21. Sp e e, F . G . Die V e rschieb ung sbahn des U n te r kie fe rs am S c h ä d e l. A rc h . f . A n a t. u. Physiol. A n a t. A b t., 285-294, 1890. 22. La n g e r, K . Das K ie fe rg e le n k des M enschen. K . A k a d . d . W is s . W ie n . M a th .-N a tu rw iss C I. SItzungsber, 39:457 J a n .- F e b ., I860. 23. Lu c e , C . E . The m ovem ents o f the Boston M . & Su rg . J . 121:8 J u ly 1889.
low er ¡aw .
24. N a g a o , M . C o m p a ra tiv e stud ies on th e curve of Sp ee in m am m als, w ith a discussion of its re la tio n to the form of the fossa m a n d ib u la ris. J . D. Res. 1:159 Ju n e 1919. 25. M onson, G . S. O cclu sio n as a p p lie d to crown and b rid g e w o rk . J . N a t. D. A . 7:399 M ay 1920. 26. V illa in , G . P rin cip es g énéraux a p p liq u é s aux d if fé re n te s prothèses, v o l. I of Prothèse, M a rtîn ie r, P ., and V illa in , G . Paris, B a lllle re , 1922. 27. B o nw ill, W . G . A . The g e o m e tric a l and m echan ical laws o f the a rtic u la tio n o f human teeth— the a n a to m ica l a rtic u la to r. In L itc h , W . F ., e d . The A m e ric a n system of d en tistry in trea tises by va rio u s authors, v o l. 2. O p e ra tiv e and p ro sth etic d en tistry, P h ila d e lp h ia , Lea Brothers, 1886-1887, p . 486-498. 28. G y s i, A lfr e d . D ie g eo m etrische Konstruktion eines m en schlich en, ob ern, b le ib e n d en , no rm alen G e b isses m ittle re r G rö s s e . S chw . V ie rte lia h rs c h r. f . Z a h n h e ilk . 5:1 M arch 1895. 29. Bo yle, H . H . Design London, Sta p le s Press, 1952.
of
the
na tu ra l
d e n titio n .
30. B o yle, h l. H . T heo ry and tre a tm e n t o f fra ctu re s of the |aw s, in p ea ce and w a r. St. Louis, C . V . M osby C o ., 1940.
articles
. . . a detailed study in the finest classical tradition which may have far-reaching clinical implications, especially in orthodontics and endodontics . . .
ARRANGEMENT IN THE JAW S OF THE ROOTS OF THE TEETH W. T. Dempster ,* Sc.D.; W . J. Adams,f D.D.S., and R. A. Duddles,% D.D.S., Ann Arbor, Mich.
This is a qualitative and quantitative study of the arrangement of the roots of the teeth in 11 skulls with typical dentitions. The patterns of the roots are shown pictorially. Additional data show the relations of the center points of the aditus of the roots to the alveolar arch. The direction of root slanting and the amount of slanting were measured and treated statistically. In the light of these data, conventional descriptions of “ ideal” dental arches do not correspond with actual dentitions. This was so for dental arch patterns on a plane surface ( the Bonwill pattern) , for patterns on a cylindrical surface (the curve of Spee) and for patterns having a spherical surface of occlusion (the curve of Monson). These curvatures should be restudied.
This report examines the arrangement, inclination and angulation o f the roots of erupted teeth in the adult. This aspect of the anatomy o f the permanent denti tion has never been covered systematically in the literature on the jaws and teeth. One might assume that information about the anatomy and mechanics of the denti tion would be documented solidly since it bears on such fields o f clinical dentistry as periodontics, orthodontics, oral sur
gery, prosthodontics and dental roentgen ography. Yet, for a hundred years, when ever authors discussed the form and ge ometry o f the dental arches, they really were concerned only with the exposed crowns o f the teeth. Our work, in con trast, deals mostly with the hidden parts of the dentition. The whole complement of about 50 roots o f the dentition has not been studied as a whole, and our present information
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about roots is a congeries from extracted teeth or from dental roentgenograms and histologic sections. A study of the arrange ment of teeth cannot be made on living subjects, nor on occasional biopsy speci mens. It is only through a study of skulls with unusually complete and well-formed dentitions that such information can be acquired. W e have been able to work with such material. Q U A L IT A T IV E A P P R O A C H F O R T H E R O O TS
Figure 1 helps define our problem. This set o f teeth represents the complete denti tion o f one skull. The preparation of our specimen involved the following stages: making rubber-base impressions o f the crowns o f the teeth for both the upper and lower arches, extracting the teeth, setting them into the rubber mold and stabilizing the roots with acrylic resin. Thus, both the crowns and the roots of the teeth had the identical relationship that they had while in the intact skull. Th e dentition in Figure 1 shows that the roots o f the teeth from the cementoenamel junction to the apex are more or less elongate cones. The bicuspid roots are most nearly perpendicular to the plane of occlusion. The lower incisor and molar roots are directed obliquely back ward; in addition, the incisor roots are more or less medially directed, and the molar roots are tilted outward. The roots of the maxillary teeth mesial to the second bicuspid are directed backward and in ward, whereas the roots o f the maxillary molars are more vertical than the roots of the opposed lower molars. T h e palatal roots o f the upper molars tilt inward more than the buccal roots do. In this generalized description of the dentition, the roots have been treated as if they were cones; that is, right circular cones, with a straight central axis, ignor ing curvatures and surface irregularities. This probably can be justified from a me chanical standpoint for man since the line o f action of a force vector through a root;
that is, the direct thrust into an alveolus, seems to be a straight line. (It would not be so for a monkey canine or a rodent incisor tooth.) A straight line between the center o f the alveolar aditus and the apex of the socket is a simple approxima tion to such a direct line of thrust. Figure 2 shows other relations of the central axes o f the roots in the same skull. T o prepare this specimen, the apexes of the roots were cut transversely by grinding to locate pulp canals at the centers of the roots; a straight drill hole along each pulp canal toward the center o f the wid est part of the root served to define the central axes of the roots. After the roots were drilled through lengthwise (from apex to the occlusal surface o f the crow n), the teeth were reinserted into their alveoli. The skull, after soaking in glycerin, could be drilled easily. Each tooth and root was then drilled from its occlusal surface to the apex using a long drill until the drill emerged somewhere on the surface o f the skull at a point in line with the tooth axis. Orthodontic wires were then placed in each drill hole. Both front and side view cephalograms and comparable natural size photographs were made o f the specimens; these showed root angulations relative to the anatomy o f the skull so that composite drawings o f the skull could be made. Figure 2 shows wires through the roots of the left side of the jaws, and the afore mentioned root directions (Fig. 1) are even more clearly depicted, especially those of the mandibular roots. T h e second bicuspid and the mesiobuccal roots of the maxillary molars are directed well for ward toward the forehead region; the first bicuspid and the palatal (lingual) roots o f the molars point farther back, and the cuspids and incisors direct their thrust vectors still farther toward the vertex. Figure 3 shows the anterior aspect of the same skull. T h e bicuspid roots are about vertical, and they aim through the medial part o f the orbit. The buccal roots o f the first and second molars point through the central part of the orbit. All
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Fig. I • C o m p le te dentition with teeth and roots having same positional relations as they did in alveolar processes. A : Left lateral aspect. B: O b liq u e aspect. C : A n te rio r aspect. D: Posterior aspect
of the remaining upper roots— third m o lars, incisors and cuspids— tilt inward toward the region between the orbits; that is, ethmoid or juxta-ethmoid part of the frontal bone. (I f the reader wishes to make a more detailed study o f the 50 plus wires, he may use a straight edge, and he should carefully correlate the cod ing system o f the free ends of the wires with the teeth concerned.) The latter root axes, if projected upward, may cross the midline anywhere between the nasal cav ity and the region well above the scalp. The general arrangement of Figures 2 and 3 was corroborated by another skull with wires in all the roots. In addition, tracings were made o f split skull cephalograms o f the group o f 11 basic skulls with sockets coated with white lead so that axes could be determined. The axis of each root was then extended by drawing a straight line until it emerged from the skull. In several separate mandibles, lengths o f straight wire were cemented axially into the empty sockets with a
metal-plastic mixture (Sculp-metal) that hardened and fixed the wires. When the wires were twisted loose, the bone beyond the putty could be drilled so that the wire axis could be pushed on through the bone. These confirmed that the skull used for Figures 2 and 3 was representative of the material used in the study. Q U A N T IT A T IV E A P P R O A C H E S TO ALVEOLAR A RCH
Materials * Our study of the pattern of root arrangement was less a study of roots per se than o f the alveoli for the roots which, after careful extraction of the teeth, represent a negative mold o f the root forms. Thirteen adult male skulls obtained from commercial suppliers (from India) formed our study sample; nine had a complete dentition o f 32 teeth, and four, lacking third molars, had 28 teeth. The skulls were selected from many others for well-formed individual teeth, minimal wear and minimal absorption of
+ Di st obuccal r o o t s □ Pal at al r o o t s LOWER MOLARS ▲ Mesi al r oot s ■ Di stal r o o t s Fig. 2 • Lateral view o f skull with orthodontic wires in holes drilled axially through roots o f teeth and on through to exterior, showing variable orientation o f different axes. C o d e fo r wires relates to different roots
Fig. 3 • Relative orientation of axes of roots of teeth as seen in anterior view o f sam e skull shown in Figure 2; same co d in g of roots of teeth has been used as in Figure 2. The general arrangem ent shown in Figures 2 and 3 was corroborated by another skull with wires in all the roots
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Fig. 4 • Catenom eter built on vernier caliper shows standard 74 mm. catenary curve superim posed on a dental arch. Left: Two points (p and q ) are localized b y m easuring perpendicular distance to curve an d sagittal distance between point and com m on horizontal reference line. R ight: Best possible fit of catenary curve in relation to mean center points of all of roots of I I m andibles
alveolar bone. (A comparable group of skulls with caries-free teeth representing our present-day American population probably could not be obtained.) Tw o skulls were used for the qualitative data of Figures 1, 2 and 3; 11 skulls were sub jected to quantitative analysis. Methods • For general reference through out the study, standardized full-scale pho tographs and cephalograms weye made both with the teeth in occlusion and after the extraction of the teeth. For clarity and to avoid overlapping images of the right and left sockets, additional side view cephalograms were made after a sagittal sawcut through the mandible and palate; a film placed vertically in the sawcut then permitted us to take a roentgenogram of
each side separately with the skull in a cephalostat. Stone models of the dental arches and alveolar processes of each skull were made both before and after extrac tion o f the teeth. This gave us a perma nent record of both the teeth and the sockets. W ood’s metal casts of the root patterns were also made from a duplicate socket model of plaster of paris. For all o f these models o f sockets, we defined the plane of the alveolar margins (or common alveolar plane) in terms of three points: (1) the midpoint of the interradicular septum between the central incisors, (2) the midpoint o f the interradicular septum between the first and second right molars and (3) the com parable intermolar point o f the left. Each stone model was set into a rectangular
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block o f plaster o f paris with its alveolar plane below but parallel to the smooth upper surface o f the plaster; this surface later formed a reference plane for the measurement of root angles (Fig. 7, A, B ) . A life-size photograph of the arch in the plane o f the alveolar margins also was made showing alveoli and septums. Th e curve which was used as a refer ence in this study was a “ catenary” curve — a type o f curvature produced by a chain of many links suspended by its ends but otherwise allowed to hang freely. The suspension system and chain for defining such curves was called a “ catenometer.” MacConaill and Scher1 and later Scott2 felt that the catenary was the curvature which best fit the dental arch. No attempt was made in the present study to verify these findings. Figure 4 shows our adapta tion o f Scott’s catenometer using a stand ard 200 mm. length o f fine link jeweler’ s chain soldered to the endpoints (for in side measurements) of an ordinary metric vernier caliper. The catenary, which most nearly coin cided with the three points just men tioned for the plane o f the alveolar mar gin, was a catenary curve o f 74 mm. width between suspension points. W e arbitrarily selected this catenary (Fig. 4) as a refer ence standard for all subsequent use on both the upper and lower arches. (W e might have used equally well any other curve of reference— ellipse, parabola and so forth— since it was merely a line from which to measure.) The catenary was used since a convenient size could be ob tained easily just by suitably separating the suspension points of the chain. M E A N P O S IT IO N S OF R O O T S IN A L V E O L A R A R C H
T w o 10 X enlarged outlines of this 74 mm. catenary curve were drawn to repre sent the lower and upper jaws, respec tively. On these curves, the relative posi tion o f the center point of each alveolar aditus in all the upper and all the lower jaws was plotted. The photographic nega
tives o f the alveolar arches provided the raw data for locating the various center points. For each center point, we chose to define its locus in terms o f a point on the curve where a perpendicular from the center point intersected a line tangent to the curvature. (Tangents and perpen diculars may be drawn in fairly accurately on large scale curves using simple mechanical-mathematical methods.) We measured for each center point: (1) its perpendicular distance to a tangent to the catenary, and (2 ) the sagittal distance between this intersection point (that is, of the perpendicular and the curve) and a horizontal tangent to the lowest point of the curve (Fig. 4, left). This sagittal and perpendicular distance served to lo cate the center o f each alveolus mathe matically in relation to the catenary curve. From these measurements an average measurement for each alveolus could be determined. As shown (Fig. 5, 6 ), the centers of the different roots form a cluster o f points. W e sought to find an average point or center o f gravity o f the cluster for each root, then some measure o f the relative dispersion. The center of gravity of a cluster was found as follows: An arbitrary x and y coordinate was drawn in, and the measured distances from each point to coordinate x were summated and divided by the number o f points; this provided a mean center distance from the x co ordinate. Similarly, %dy = distance of the n mean center locus from the y coordinate; dv was the distance between each point and the y coordinate; n was the number of points. The mean distances were de termined separately for each root for both sides o f the 11 jaws, taken together. Thus, we obtained mean points for each root for both the lower (Fig. 4) and the upper jaws. As is shown graphically for the 11 skulls (Fig. 4, right), a best fit catenary is cer tainly not a perfect description of the dental arch curvature. Nevertheless, the catenary— fitted to the incisor and molar
26/786 • T H E JO U R N A L O F T H E A M E R IC A N DEN TAL A S S O C IA T IO N
la n d m a r k s w e u se d — is p r o b a b ly as g o o d a n a p p r o x im a t io n t o fittin g th e a lv e o la r ce n te rs
as
any
r e g u la r
ty p e
of
cu rve
c o u ld b e .
V A R IA B IL IT Y OF D IF F E R E N T A L V E O L A R A D IT U S
O u r p lo t s s h o w e d m a n y clu sters o f p o in ts e a c h n o r m a lly re p re se n tin g 22 ro o ts . T h e a re a o f t h e clu s te r w a s sm a ll w h e n th e d e v ia t io n
w a s lo w , b u t th e clu s te r w a s
la rg e r a n d m o r e d iffu s e w h e r e th e v a r ia
bility from jaw to jaw was large. T o pro vide a simple measure o f the relative vari ability or dispersion o f the individual center points relative to the weighted mean point of the cluster for the different roots o f our graphs (Figs. 5, 6 ), the aver age deviation (A.D .) was computed. New perpendicular x and y coordinates were drawn through each mean center on our large plots; one axis was arbitrarily drawn parallel to the catenary and the other was perpendicular. Next, the distances be tween the x coordinate and each point,
Fig. 5 • C e n te r points of roots o f teeth of 22 m andibular sides are shown as clusters o f points about alternate sockets on left and right sides of standard catenary curve. Relative sizes and proportions of small ellipses with crossed m esiodistal and buccolingual axes indicate variability in position of several roots as measured b y average deviation (A.D .)
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neglecting + or — signs, were measured in millimeters, summated and divided by the number of measurements; similarly, the average deviation was determined rel ative to the y coordinate. An ellipse with its semi-axes the same as the two perpen dicular A.D. was drawn about the mean center. (The more frequently used stand ard deviation, a, relative to a mean in cludes 68.2 per cent of the deviates above or below the mean in a bell-shaped type o f population; whereas the A.D. includes 57.5 per cent.) In Figures 5 and 6 an ellipse encircles the points included by the A.D. in two perpendicular directions. Since the axes were parallel to and per pendicular to the catenary, the semi-axes
represent an average deviation that re lates to the conventional directions of the arch, that is mesial, distal, buccal and lingual. The size and spread of the clusters of points shown in an alternating sequence (Fig. 5) for the right and left sides of the mandibular arch represent the relative stability to the mean center position of root position. The crossing point o f the axis within the ellipse is the locus of the mean center of the root. All of the roots show more variability in the mesiodistal than in the buccolingual direction. The central incisor, for instance, in 22 sockets showed a limited positional variation in the labiolingual direction, but this tooth
# 8 /7 8 8 • T H E JO U R N A L O F T H E A M E R IC A N DENTAL A S S O C IA T IO N
A
On each side o f the arch the 15 roots of the maxillary dentition show a degree o f variability (Fig. 6 ). Only the central incisor and the buccal root o f the first bicuspid are comparable to the anterior mandibular teeth in having tight clusters o f centers, that is, small ellipses. The cus pid and the second bicuspid roots are notably more diffuse relative to the catenary, and all of the molar roots are much more variable in their positions than any o f the mandibular roots. The lingual roots o f the three molars and first bicuspid and the buccal roots o f the third molar fall inside the standard 74 mm. catenary curve; whereas most of the roots from the lateral incisor to the buccal roots of the first molar are outside. R O O T IN C L IN A T IO N
Fig. 7 • A : Inclination angle of root is measured on stone m odel of maxillary alveoli set in plate o f plaster as shown. W ire placed axially in upper first m olar root socket is m easured b y transparent protractor with its axis of sym m etry parallel to midsagit+al plane of palate. B: M easurem ent of amount of angulation of wire relative to "p lan e of the alveolar m argins." U sin g sliding depth g a u g e of vernier caliper (b) and fixed height (a), angle (a ) of right triangle com pleted by wire m ay be determ ined trigonom etrically
had three times that range in the mesiodistal direction. In contrast the deviation from the mean in both directions is more nearly the same for the cuspid root. The first, the third and finally, the second molar roots are increasingly variable in their locations, in that order; in each in stance, both the mesial and the distal roots have about the same degree of variability. The two bicuspids, which show compar able amounts o f positional variability, are less variable than the molar roots but more than the incisor-cuspid roots. Char acteristically, 4 roots (lateral incisor to second bicuspid) lie on the buccal side of the standard catenary; whereas, the sec ond and third molars are slightly lingual.
Since the central axes o f the roots slant more or less as they intersect the plane of the alveolar margins, it was convenient to measure the degree o f tilt relative to a perpendicular to this plane. T w o meas urements were found to be necessary for each root: (1) inclination was regarded as the direction toward which the coronal extremity o f the root tilted; that is, anr terior, toward the midline, laterally and so forth, and (2) angulation was taken as the number of degrees that the root axis slanted beyond the vertical. Both inclina tion and angulation were measured on the stone replicas o f the alveolar sockets. These, for ease o f manipulation, were set into plaster blocks with a flat-surfaced plane parallel to the three reference points located beyond the apexes o f the sockets (Fig. 7, B ) . Straight lengths of orthodon tic wire were carefully set into the alve olus for each root to represent its longi tudinal axis; the rods were set into the model so that they could be pushed beyond the plane of the alveolar margin, be pushed on through in the opposite direction, be removed or reinserted. T o measure rod inclination, that is, the root inclination, the free end o f the rod was pushed through a central hole in a trans
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parent protractor scale, and this was clamped to the model just over the alve olar plane (Fig. 7, A ) . The 0-180 degrees coordinate was made parallel to the midsagittal plane o f the cast with the zero anterior. The direction o f maximum slant of the axis was determined and read di rectly in degrees reading from zero either way on the graph. Therefore, our con vention in reading inclination for either arch as seen from the occlusal view was as follows: A direct anterior slant was read as 0 degree, a slant in the posterior direction was 180 degrees, slants from + 1 degree to +179 degrees implied inclina
tion away from the midline and — 1 degree to —179 degrees referred to in clination toward the midline. A N G U L A T IO N F R O M T H E V E R TIC AL
Angulation was measured trigonometri cally from the apically located parallel plane using a simple device involving a vernier caliper mounted horizontally on a base with a fixed 25 mm. height (Fig. 7, B ) . A length o f the rod was treated as the hypotenuse ( h) of a right triangle having a vertical side (a) with a fixed
Fig. 8 • C ate n ary curves in relation to sockets of m andibular and maxillary arches. W ith in each socket, arrow -tipped lines indicate direction of inclination of coronal extremity of roots; angle between shaft and barb is average deviation o f all inclination measurements. Twenty-six small grap hs— I for each root socket— indicate num ber of degrees o f angulation of root axes from vertical; dashed lines on each side give average deviation of angulation relative to mean
30/790 • T H E JO U R N A L O F TH E A M E R IC A N DEN TAL A S S O C IA T IO N
T a b le • Inclination and angulation o f roots o f teeth in 11 skulls
+ A .D .*
M ean angulation from v e rtica l (degrees)
± A .D .
N o . (n) o f roots m e asu red f
103 89 72 61 28 0 45 -2 8 -2 19 8 3 6 16 2
42 40 39 44 66 78 30 35 46 36 20 22 14 6 5
12.3 11.3 11.6 10.7 9.3 9.6 13.0 13.0 10.4 8.6 10.4 9.6 20.6 29.0 29.0
3.9 4.6 5.5 3.2 2.8 3.9 4.9 4.6 3.9 3.6 3.4 3.2 3.2 3.4 4.1
11 7 7 20 16 16 22 22 22 22 12 12 22 22 22
-5 3 -5 1 -5 1 -5 3 -5 8 -5 8 -3 4 14 13 17 2
15 16 19 18 36 23 54 35 27 10 15
32.6 38.5 25.4 28.0 12.8 16.8 10.1 9.8 14.6 21.2 19.3
4.5 8.0 4.6 3.4 5.6 4.0 3.4 3.9 6.4 6.4 7.5
10 9 22 22 22 22 22 22 22 22 22
M ean inclination (degrees)
M a x illa ry roots Third molar lingual Third m olar d isto buccal Third m olar m esiobuccal Second m o lar lingual Second m o lar d istobuccal Second m o lar m esiobuccal First m olar lingual First m olar distobuccal First m olar m esiobuccal Second bicuspid First bicuspid lingual First bicuspid buccal Cuspid Lateral in ciso r C e n tra l inciso r M andib ular roots Third m olar mesial Third m olar distal Second m olar mesial Second m o lar distal First m olar mesial First m olar distal Second bicuspid First bicuspid Cuspid la t e ra l inciso r C e n tra l inciso r
Root
*A.D., average deviation. -[Where the number (n) is less than 22, fused roots, which have not been included in the Table, or missing third molars will account for discrepancy.
distance of 25 mm. and a horizontal side ( b) to be measured. Side ( b) was meas ured quite accurately by extending the depth gauge at the end of the caliper until it touched the rod (Fig. 7, B) and reading the vernier. The angle “ a” which referred to the slant angle from the vertical could be found as follows: Tan a = (b) / (a) or ( b) /25. O ne axial rod at a time (while the others did not project to the reference plane) was measured, first for inclination angle and then for angulation. The table lists the inclination and an gulation for the various roots of the two jaws and shows the degree of variability found in the roots. Figure 8 shows the same information graphically. In the cen ter o f each root alveolus, the main arrow like lines show the mean inclination direc
tion relative to the sagittal plane; the tail of the arrow indicates the direction of the apex of the root; whereas the tip is the point o f intersection with the alveolar plane. The angle between the barb on each side and the main vector line repre sents the A.D., right and left, o f the inclination angle for each root. The small rectangular plot nearby shows for each root the mean angle of deviation from the vertical; the dotted lines on either side indicate the A.D. of the angulation angles measured. From Figure 8 and the table, it can be seen that the mandibular roots of the cen tral incisors through the first bicuspid are directed lingually; whereas the apical extremities of the second bicuspid and molar roots are inclined buccally. In the
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upper arcade all of the roots except the distobuccal roots o f the first molar tend to tilt palatally. Variability in direction of inclination, low for the anterior teeth, tends to increase from tooth to tooth through the second molar. Figure 8 clearly shows that the incisors and the third molar roots are most notably trans verse to the arch; whereas most of the other roots tilt at an angle. The maxillary incisors and the man dibular third and second molars have the greatest angulation (table, Fig. 8 ). The first and second bicuspids of each arch have the most perpendicular roots. The maxillary molar roots rarely tilt more than 15 degrees from the vertical. The greatest deviation from the mean in the mandibu lar dentition shows in the distal root of the third molar and in the anterior teeth from cuspid to central incisor. The least variable angulations are found in the lower bicuspids and lower first and second molars; most o f the roots of the maxillary dentition show a comparable low degree o f angular variation. Figures 9 and 10 pictorialize the angu lar arrangement o f the longitudinal axes of the roots as they lie in the alveolar bone o f both arcades. Each figure presents a composite that represents the mean obliquity o f the right and left roots from 11 jaws. The axes are shown as projec tions beyond the occlusal surfaces of the teeth so that they stick out like pegs or stakes and emphasize the overall angular pattern that characterizes the root system. The two views in both Figures 9 and 10 represent orthographic projections o f the mean directions of root tilting as seen from the side and front views, aspects just 90 degrees from one another. Thus, when the same root is viewed alternately in the two projections, its true orientation may be visualized. The side view in Fig ure 9 shows the maxillary pattern of the roots extending in an oblique and pos terior direction into the alveolar bone; the bicuspid axes are the least oblique; the roots of the anterior teeth are the
most oblique, and the molar roots are intermediate. From the front view, the central incisor and bicuspid roots are the most vertical; the lateral incisor and cus pid roots are directed medially. The buc cal roots of the molars are vertical or tilt slightly outward; whereas the palatal roots tilt inward. As a mean pattern with symmetrical right and left roots, the ar rangement, although similar to the single specimen of Figures 2 and 3, is much more regular. An even more striking pattern shows up for the mandibular roots (Fig. 10). As seen from the side, the incisors slant pos teriorly to a pronounced degree; the cus pids slant less; the second and third m o lars slant even less, and the bicuspid and first molar roots are the most nearly ver tical. From the anterior aspect, the incisor and cuspid roots point inward; the bicus pids are fairly vertical, and the molar roots in a mesiodistal sequence increas ingly slant outward. As mentioned, the axes o f the roots in Figures 9 and 10, as well as the crowns, project like pegs beyond the alveolar bone. The projecting parts, since they suggest the angulation of the buried roots, which the crowns do not, will readily per mit one to visualize the type of force application (on the exposed part of the tooth) required to produce a given type of root displacement. The latter may be either a minor displacement well within physiological limits; it may be a persistent force resulting in socket adaptation and gross tooth movement, as in orthodontics or periodontics, or it may be a destructive application of force from either oral sur gery or accidental trauma, which results in luxation. The tooth may be moved up or down along its axis in line with the angulation of the axis, as described here, without torque being applied. It may be rotated about this axis, twisting or tearing the periodontal fibers. It may be tilted much in the way that a peg in the ground or a cantilever beam may be moved by a trans
32 /792 • T H E J O U R N A L O F T H E A M E R IC A N D E N T A L A S S O C IA T IO N
Fig. 9 • Composite illustration of maxilla and upper teeth as seen from anterior and lateral aspects showing average arrangement of teeth. Amount of slant shown by roots represents mean of combined right and left teeth of 11 ¡aws. Longitudinal axes of roots of teeth have been extended beyond crowns and are shown in orthographic projection. C oding of roots same as for Figure 2
verse or oblique force; a resisting torque will be developed in the hidden part of the tooth and its socket. Finally, if a suit able torque (most easily visualized as per pendicular to the long axis) is applied to the crown of a tooth, the root will exert a steady pressure against a side of its alve olus. This force system may be visualized as comparable to that which a canoeist applies to a paddle; namely, above the blade and at the handle, in order to drive
the blade through the water and develop a forward thrust for his canoe. If the pro jecting rods o f Figures 9 and 10 are viewed as if one could apply forces to them, all of the various possible tooth displacements may be visualized. The dif ferent arrangements o f the different roots, as shown, make it clear, even when no other factor except axis obliquity is con sidered, that the mechanical problems of the teeth are quantitatively different.
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Fig. 10 • Complete orthographic projection of mandibular teeth as seen from anterior and lateral aspects showing arrangement of teeth relative to alveolar bone. Degree of obliquity of roots of right and left teeth of 11 ¡aws has been averaged. Root axes have been extended beyond crowns to emphasize angular pattern
D IS C U S S IO N
Although the arrangement of the roots of the teeth apparently has not been stud ied before, the dental and anatomical lit erature has had much to say about the dental arches. These comments have been based almost wholly on data relating to the crowns of teeth, to plaster models of dentitions or to experiences with articu lators and the designing o f dentures. The
generalizations that have followed have often been based on old data from few specimens and rather casual observations without measurements. Few people have distinguished between the alveolar arches and the dental arches proper. On the basis of simple qualitative observation, anthropologists3,4 have de scribed the general shape of the palatal arch using such terms as paraboloid, U-shaped, ellipsoid, rotund and horse
34/794 • T H E JO U R N A L O F TH E A M E R IC A N DENTAL A S S O C IA T IO N
shoe-shaped. From simple length to width ratios, narrow, midrange and broad pal ates have been distinguished— with an appropriate latinized terminology. W il liams5 took several measurements from plaster models to work out more discrimi nating ratios relating to the dental arches. Some of the differences in the descrip tions of arch form over the years have depended as much upon the landmarks selected as upon the dentitions dealt with. G . V . Black6 fitted a semiellipse to the curvatures of both the upper and lower dental arches. This semiellipse coincided with the incisor cutting edges of the upper arch, skirted the palatal side of the cus pids and then ran along the buccal cusps of the bicuspid and molar teeth. W hen the same sized semiellipse was fitted to the lower dental arch, it coincided with the cutting edges of the incisors, and it lay along the buccal surfaces of the bicuspid and molar teeth. Black7 was aware of variability in the form and placement of teeth. H e recognized that the cuspids might be more or less prominent and that the more posterior teeth might be either a straight line or a curve. As early as 1887, Davenport8 noted that a regular arch curvature could be identified along the line of occlusal con tact from the lingual cusps of the upper molars and the opposed sulci of the lower molars forward to the incisor contact points. Angle,9 who made much o f the functional significance of this line, re ferred to it as being paraboloid shaped. Later attempts to define arch relations included those of Franke10 and Goldstein and Stanton.11 T h e later authors used a special denture pantograph12 to trace the gingival margin of a plaster model. The midpoint of the area of each tooth as seen in the pantograph tracing was called the centroid of the tooth section, and these were taken as points which defined arch curvature. Using landmarks similar to those of Davenport,8 MacConaill and Scher1 in dicated that the curvature of the “ com mon occlusal line” was ideally described
by a catenary curve. Scott2 concurred; both reports recommended a standard 200 m m . catenometer which could be varied in its width to fit different arches. Hayashi13 defined a still different “curve of dental arch,” arbitrarily, using as land marks for both arches the incisal edges of the incisors, the tips of the cuspids, the tips o f the buccal cusps of the bicuspids and the deepest points of the notch be tween the buccal cusps of the molar teeth. This curve was projected to an occlusal plane which simultaneously touched the medial comers of the central incisors and the extreme points of the distal buccal cusps of the second molars. Hayashi’s curvature was laid out on rectangular co ordinate paper, and the curve was de fined mathematically by an empirical exponential formula of the type y = bx7c. It is noteworthy that all of the curve describers were apparently seeking a basic mean pattern or “ ideal.” Apart from Black’ s recognition6 of the ways in which teeth m ay vary in relation to a basic pat tern, only MacConaill and Scher1 actu ally measured the relative variability of the positions of the teeth. (Although these authors have dealt with the variability of occlusal points, and we have been con cerned with the variability of the positions of the root aditus, the two classes of data are so different that logically they can be neither contrasted nor harmonized.) Hrdlicka14 stated that harmony in oc clusion is “common in all primitive peo ple, but is frequently disturbed in m odem civilized man.” Years before, in 1904, Cryer15 vigorously dissented from the view that harmonious arch patterns had any real functional meaning. H e main tained that each dentition presented its own individual occlusal problems. T h e teeth were regarded as functionally re lated to the bony framework of the facial bones, and each specific occlusion, whether typical or atypical, was consid ered the outcome of such inborn or en vironmental factors as: available bone space, anthropological or racial character istics, orderly or disturbed growth, tooth
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Fig. 11 • In each sketch, plastic cube with central rod was cemented to occlusal cusps of each molar tooth. Rods (double lines) for mandibular teeth or their virtual extensions for maxillary teeth show points of intersection (black dots) at different levels
crowding and so forth. T h e view that bones and teeth were interrelated func tionally was amplified16 later by students of facial bone structure who believed that the directional properties, that is, “ split line” pattern or “grain,” of facial bones were related to the effective transmission of masticatory stresses. In Figures 5 and 6, which relate to the group of 11 intact dentitions, there are shown clusters of points and ellipses that suggest the relative variability of the cen ter points of the aditus of each root as projected to the common alveolar plane. It might be assumed that the central in cisor and the first and second molar roots would be the least variable since these were closest to the landmarks for super imposing the reference curves. But the variability was especially high for the sec ond molar. It was higher for all molars than for the other teeth. Since the teeth show different amounts of variation and do not align to form any regular curve, the idea of an ideal curve cannot be re garded as more than a convenient, rough approximation for descriptive purposes or as a suitable pattern for the placement of artificial teeth on dentures. T h e commonness of crowded, irregular and twisted teeth and occlusal dishar
monies has long been recognized.17'20 Hunter17 even correlated tooth irregular ity with unusual mechanical pressures. W hen in 1899 Angle9 introduced his classification of anomalous occlusions and the term “malocclusion,” he went a step further and made it clear that the forces of cuspal interdigitation were the essen tials for keeping the teeth in alignment. This outlook argued for adapting mech anisms rather than for “ideal” patterns which ignore variance. Th e occlusal surfaces of the teeth of the upper and lower dental arches have at times been regarded as a continuous interdental occlusal surface or “ plane.” As seen laterally, this surface was recog nized as a sagittal curvature,7, 21 concave upward. Spee21 interpreted it as a circular curve which could be laid out with a compass on a profile photograph of a skull; when the curve was continued backward on Spee’ s drawings of adult skulls, it passed between the head of the mandible and the articular eminence of the temporal bone. T h e center of curva ture was at the lacrimal bone. Langer22 and Luce23 had already demonstrated that the mandibular head followed a curved pathway (concave upward) when it moved sagittally over the articular emi
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nence of the temporal bone. Spee cor related their observations with his, and the “ curve o f Spee” came to be regarded as a compensating “ curve” for effec tive protrusive movements of the jaw. Nagao,24 however, showed that if this were so in humans, it certainly was not a general feature of primate (or other mamm al) jaws and teeth. Some years later, Monson25 and Villain26 argued that the idea of a sagittal occlusal curvature should be expanded into a concept of a three-dimensional spherical curvature in volving both the right and left bicuspid and molar cusps and the right and left mandibular condyles. A sphere 8 inches in diameter suggested by Monson (20 cm. by Villain) was regarded as a basic di mension that could be used by dentists concerned with the design of complete dentures. T h e Monson curvature included an earlier denture-arch design presented by Bonwill27 in 1887 along with an articu lator. Gysi28 and later Boyle29 introduced arch designs that were somewhat differ ent. Both Monson25 and Boyle29’ 30 were concerned that the upper and lower den tal arches came into occlusal contact along a spherical interface; however, Boyle specifically omitted the mandibular condyles as totally unrelated to the spheri cal occlusal pattern. From the small number of specimens studied and his crude way of measuring contours, Monson25 believed that perpen diculars from the occlusal surfaces o f the molar and bicuspid teeth converged to a common point. These radiuses of the spherical surface of contact, he inferred, illustrated “ lines of force.” Even further, and totally without evidenced, he assumed that “ lines from the center [of the sphere] obviously must pass through the center of each tooth.” For Boyle29,30 the concave and convex occlusal surfaces formed a mill with a radius of curvature of 11.1 c m .; the masticatory forces were imagined as merging at a fixed point above and forward of the crista galli of the ethmoid bone— at a so-called “dental center.”
T h e conjecture of Boyle and Monson that the roots of the teeth converge to a center is disproved by the data shown in Figures 2, 3, 9 and 10. W e are not aware that anyone has so far made a statistical or mathematical study of the “ curve of Spee.” Likewise, it appears that no analytical study has been made of the spatial curvatures of the oc clusal faces of actual dentitions. Monson, Villain and Boyle merely fitted mandibles or plaster models and dentures to repre sentative spherical surfaces to determine what size of sphere gave the best fit. D if ferences in the way dentitions fit the sphere and the variability of the occlusal surfaces of individual teeth were in no way considered. For the 11 skulls of this study, clear plastic cubes were cemented to the oc clusal cusps of each of the three (or two) molars; perpendiculars from the midre gion of each cube appeared to converge when seen in profile view photographs of either the mandibular or the maxillary regions (Fig. 1 1 ). Each side of the jaw was studied separately. W here there were three molars, the converging lines as seen in profile, however, never met exactly at a single point; that is, they never repre sented the same radius of curvature for the arc between the first and second mo lars and between the second and third molars. T h e perpendiculars typically crossed in sequence at different heights rather than at a common point. Thus, instead of forming the arc of a circle, the occlusal surfaces of the molars showed a slight spiral; the toe of the curve, more over, was about as often directed mesially as it was distally (Fig. 1 1 ). T h e intersect ing perpendiculars from adjacent molars (and therefore, the mean radius for the circular arc tangent to the two teeth) cross one another in the facial region of the profile photographs between the lev els of the palate and the superciliary ridges. In anteroposterior photographs taken at a 90 degree angle to the occlusal plane (Fig. 1 1 ), the directions of perpendicu
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lars from the first, second and third m o lars could be shown simultaneously on the two sides of the dental arch. These pho tographs typically showed that perpen diculars from the third molars ( right and left) crossed at or near the midline in the facial region; perpendiculars from the second molars crossed symmetrically or asymmetrically at a higher level, and those from the first molars were usually vertical or they converged or diverged only a few degrees. Anatomically, the three levels for the crossing points fell within the nasal cavity, in the anterior region of the cranial cavity above or in the region well above the head. None of our 11 upper or lower dentitions showed any semblance of converging toward a common point (as postulated by Monson, Villain and B oyle). Clearly, both the sag ittal curvature (of Spee) and the trans verse curvatures of the molar occlusal surfaces should be restudied from a large statistical sample of models or dentitions. Both suitable numbers of specimens and some standard mathematical method for defining curvatures; namely, evoluteinvolute curve method and so forth, should be used. T h e 11 skulls in this study clearly show that the longitudinal axes of the roots do not converge toward a com mon center. They also affirm that the occlusal surfaces of the molar teeth can not be fully congruent with the surface of any sized sphere.
A co n trib u tio n of the d e p artm e n t o f anato m y, the U n iv e rsity o f M ic h ig a n , Ann A rb o r , M ic h . This research has been sup po rted by N S F G ra n ts no. G-13260 and GB-356. *Pro fesso r o f anato m y, xA rb o r, M ic h .
U n ive rsity o f M ic h ig a n , A nn
TC trn ical in stru cto r in d en tistry, U n ive rsity of M ic h ig a n , A nn A rb o r , M ich . ^Present a d d re ss, G ra n d B lan c, M ich . 1. M a c C o n a ill, M . A ., and Sch e r, E . A . Id e a ! form of the human d en tal a rc a d e , with some p ro sth etic a p p li ca tio n . D . R eco rd ¿9:285 N o v. ¡949. 2 . S c o tt, J . H . Sh ape o f the d en tal a rc h e s. J . D . R es. 36:996 D e c. 1957. 3. M a rtin , R . Lehrbuch d e r A n th ro p o lo g ie . J e n a , G . F isc h e r, 1914. 4 . H rd lic k a , A . A n th ro p o m e try . In stitu te of A n a to m y, 1920.
P h ila d e lp h ia , W is ta r
5. W illia m s , P. N . D eterm ining the shap e of the no r m al a rc h . D. C osm os 59:695 J u ly 1917.
6. B la ck , G . V. D e scrip tive an a to m y o f the human te eth , e d . 3. P h ila d e lp h ia , W ilm in g to n D ental M a n u fa c tu rin g C o ., 1894. 7 . B la ck , G . V . D e scrip tive anato m y o f the human te eth . P h ila d e lp h ia , W ilm in g to n D ental M anufacturing C o ., 1890. 8. D aven po rt, I. B. The sig n ific a n c e o f the natural form and a rra n g e m en t o f the d en ta l arches o f m an, w ith a co n sid eratio n o f the chang es w hich o c c u r as a result of th e ir a rtific ia l d era n g em en t by fillin g , o r by the e xtractio n of te eth . D . C osm os 29:413 J u ly 1887. 9 . A n g le , E . H . C la ss ific a tio n o f m a lo cclu sio n . D . C o s mos 41:248 M arch 1899. 10. Fra n k e, G . O b er W achstum und V e rb ild u n g e n des K ie fe rs und d e r N asenscheidew and a u f G ru n d V e rg le ic h en d er Kiefer-M essungen und e xp e rim e n te lle r U ntersuch ungen ü b er Knochenw achstum . Z tsch r. f . La ryn g . 10:187 J u ly 1921. 11. G o ld s te in , M . S ., and Stanto n, F . L . C han g e s în dim ensions and form o f the d en ta l arches w ith a g e . In te rn a t. J . O rth o d o n t. 21:357 A p ril 1935. 12. Stanto n, F. L . ; Fish, G . D ., and A shley-M on tag ue, M . F . D escrip tion o f th ree instrum ents fo r use in o rth o d o n tic and c e p h a lo m e tric in v e stig a tio n s. J . D . Res. 11:885 O c t. 1931. 13. H a y a s h î, T . A m a th e m atica l analysis of the curve o f the d en tal a rc h . Tokyo M . & D. U n iv . Bui. 3:175 N o . 2 1956. 14. H rd lic k a , A . N o rm al v a ria tio n of teeth and jaw s and ortho d o nty. A m . S o c . O rth o d ists. T r. 33:68 M ay 1935. 15. C ry e r, M . H . T y p ic a l and a ty p ic a l occlusion o f the teeth in re latio n to the co rrectio n o f irre g u la ritie s . D. C osm os 46:713 S e p t. 1904. 16. Bennînghoff, A . S p a ltlin ie n am Knochen, eine M ethode zu r Erm ittlu n g d e r A rc h ite k tu r p la tte m Knochen. A n a t. G e s . V e rh . 34:189 1925. 17. H u n ter, J . The natu ral history o f the human te e th ; e xp la in in g th e ir stru ctu re , use, fo rm a tio n , grow th and d ise ase s, e d . 2 . London, Jo hnso n , 1778. 18. Fox, J . The natu ral history of the te e th . London, Thom as C o x , 1803. 19. K n e isel, F . C . Der S c h liefsta n d d e r Z ähne dessen Ursachen und A b h ü lfe nach e in e r neuen, sichern und schm erzlusen H e ilm e th o d e . (W ith co n cu rre n t French v e r sio n .) B e rlin , E . S . M ittle r, 1836. 20. S te rn fe ld , A . U e b e r Bissarten und Bissano m alien. M unich, K no rr and H irth , 1888. 21. Sp e e, F . G . Die V e rschieb ung sbahn des U n te r kie fe rs am S c h ä d e l. A rc h . f . A n a t. u. Physiol. A n a t. A b t., 285-294, 1890. 22. La n g e r, K . Das K ie fe rg e le n k des M enschen. K . A k a d . d . W is s . W ie n . M a th .-N a tu rw iss C I. SItzungsber, 39:457 J a n .- F e b ., I860. 23. Lu c e , C . E . The m ovem ents o f the Boston M . & Su rg . J . 121:8 J u ly 1889.
low er ¡aw .
24. N a g a o , M . C o m p a ra tiv e stud ies on th e curve of Sp ee in m am m als, w ith a discussion of its re la tio n to the form of the fossa m a n d ib u la ris. J . D. Res. 1:159 Ju n e 1919. 25. M onson, G . S. O cclu sio n as a p p lie d to crown and b rid g e w o rk . J . N a t. D. A . 7:399 M ay 1920. 26. V illa in , G . P rin cip es g énéraux a p p liq u é s aux d if fé re n te s prothèses, v o l. I of Prothèse, M a rtîn ie r, P ., and V illa in , G . Paris, B a lllle re , 1922. 27. B o nw ill, W . G . A . The g e o m e tric a l and m echan ical laws o f the a rtic u la tio n o f human teeth— the a n a to m ica l a rtic u la to r. In L itc h , W . F ., e d . The A m e ric a n system of d en tistry in trea tises by va rio u s authors, v o l. 2. O p e ra tiv e and p ro sth etic d en tistry, P h ila d e lp h ia , Lea Brothers, 1886-1887, p . 486-498. 28. G y s i, A lfr e d . D ie g eo m etrische Konstruktion eines m en schlich en, ob ern, b le ib e n d en , no rm alen G e b isses m ittle re r G rö s s e . S chw . V ie rte lia h rs c h r. f . Z a h n h e ilk . 5:1 M arch 1895. 29. Bo yle, H . H . Design London, Sta p le s Press, 1952.
of
the
na tu ra l
d e n titio n .
30. B o yle, h l. H . T heo ry and tre a tm e n t o f fra ctu re s of the |aw s, in p ea ce and w a r. St. Louis, C . V . M osby C o ., 1940.