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SECTION

EDITOR

A mathematical method anterior stop pin setting W. James

Wright,

Jr.,

University of Tennessee, The

geometry

ical

formulae

the

thickness

ments that

are the

formulae

articulator mounted

when casts

malocclusion stop further

pin

can

to allow

the

study.

therefore

predict

the

parallelism

of the

interocclusal

brought

is identified and

reliably

centric

into

occlusion.

as a source in the

cast

Testing

geometry

and

anterior

relation

maxillary

record

are

discussed.

and

relation The

record

vertical

of variability

compensates centric

articulator;

relation indicated

stop

pin

opening

mandibular

bows

is removed

and

component in the

setting

for

Measure-

mountings

of anterior

maxillary

of the the

of dental of the

it is defined

anterior as a need

for

(J PROSTHET DENT 1989;61:362-67.)

A

362

that

interocclusal

amount

of mathemac-

pin

30 diagnostic

rticulators have long been used by the dental profession to simulate maxillomandibular posture in static position and functional and parafunctional movements.1-3 Recent, more accurate description of mandibular movements have shown the importance of semiadjustable arcon principle articulators to fixed and removable prosthodontic rehabilitation.4-6 Studies by Weinberg,7 Hart et al.,s and Gordon et al.” illustrated occlusal errors that result from errors in hinge axis location. TsaolO addressed early mandibular movement as it relates to the occlusal plane, dentoskeletal malocclusion, and how bhe geometry of mandibular movement is affected. In recent studies, Hobo,“, I2 Curtis,13 and WachtelL4 investigated the horizontal component of condylar movement and described the limiting factors of semiadjustable articulators in their ability to simulate condylar movement. Gordon et al.9 stated that centric relation (CR) should be recorded at the planned vertical dimension of occlusion to minimize errors resulting from incorrect hinge axis location. Zuckerman15 recognized the importance of minimal vertical separation of the maxillary and mandibular teeth in interocclusal CR records, noting that errors increased as records increased in thickness. He also stated that occlusal error decreases as the distance between true hinge axis (THA) and the occlusal element increases. This finding implied that the greatest er-

*Assistant Professor, Fixed Prosthodontics.

development

stop

and

on

of the

Tenn.

Memphis,

articulator

of the

centric

formulae.

near

are

setting

mounted

in these

or’ Dentistry,

arcon

the

interocclusal

facebow used

necessary

College

of a semiadjustable to calculate

of the

record

D.D.S.*

Memphis,

of the

for calculating the compensatin of a semiadjustable arcon articulator

Department

of Prosthodont.ics,

Division

of

ror would compared

be found nearer the axis or :r: & with the incisor region.

moiai

region3

PROBLEM For mounting the mandibular cast with an in;er:)cclusai CR record, articulator manufacturers recommend compen~ sation for the opening effect of the interocc!usal CR record by estimating or by trial and error. E&nation can produce inconsistent results and trial and error methods can be time-consuming and frustrating. Both met,hods hare the potential for inaccuracy. This study developed and t,ested mathematical formuiae to calculate the effect of the vertical openicp of the interocciusal CR record on the geometry of the articulatt;r. Once calculated, the opening effect would be compensated for by adjusting the anterior stop pin +a allow for the ef7ect of a specific record.

METHOD The Hanau model 166-l (Teledyne Hanau, Buifa10, N-Y.> semiadjustable arcon articuiat,or was used in this stud:,:. This articulator exhibits the following constant measurements: 1. The distance from the condylar axis to i.he posterior aspect of the anterior stop pin along the siipermr surface of the maxillary bow is 110 mm. 2. The condylar axis is parallel (in the horizuntal plane) and 21.5 mm inferior to the superior surface of t,he maxillary bow at the mounting place. 3. The condylar axis is in the same horizontal plane as th.e

COMPENSATING

ANTERIOR

S I’OP

PIN

SETTING

Millimeters

1

2

3

4

5

6

7

8

9

10

11

12

Degrees Fig. 1. Graph of differences in tangent with constant radius of 110 mm.

measuring point of the maxillary bow is to the anterior stop pin. 4. The anterior stop pin is marked in millimeters, not in degrees. One degree of rotation with a radius of 110 mm produces an arc 1.918 mm in length. The millimeter marks on the anterior stop pin are tangents to the arc created by the opening or closing of the maxillary bow of the articulator. 5. The machined reference marks on the stop pin extend 5 mm superiorly and inferiorly from the zero mark. Thus, they do not measure an angle of opening or closing of the maxillary bow that exceeds 2’36’18.9” (2.60525 degrees) from the zero mark. 6. Calculations of differences between tangents and chords of angles less than 12 degrees, with a constant radius of 110 mm, revealed that angles up to 7 degrees resulted in difference of tangents and chords that were 0.1 mm or less. Therefore, it was assumed that for any 0.5 mm visually estimated measurement of the anterior stop pin within the ? 2.605 degrees, the tangent and chord of the angle of opening could be considered equal (Fig. 1). These constants allowed the formation of a geometric problem that incorporated measured variables made from the arbitrary hinge axis facebow mounted maxillary cast and the interocclusal CR record. It was proposed that a point in the horizontal plane created by the intersection of the sagittal and frontal planes of the mesiolingual cusp tip of the maxillary second molar be projected onto the midsagittal plane of the articulator. From this projected intersection, a radius (X) would be calculated to the condylar axis. Using the radius X and the measured thickness of the interocclusal CR record, the angle of opening (0) caused by the thickness of the record would be calculated. Cnce known, the angle of opening would be converted to the amount of opening (mm) that would be necessary to compensate for the thickness of

THE

JOURNAL

OF PROSTHETIC

DENTISTRY

and chord length in millimeters

of various angles

the record at the anterior stop pin. After the stop pin was set to the amount of compensated opening, the mandibular cast would be mounted with the interocclusal CR record. Upon removal of the record, the measuring point of the maxillary bow would return to the zero mark of the anterior stop pin at the first contact of the casts. The measurements were obtained as follows (Fig. 2). 1. V measurement. With a Boley gauge (Buffalo Dental Manufacturing Co., Buffalo, N.Y.) as a T square, the perpendicular distance from the superior surface of the maxillary bow of the articulator to the mesiolingual cusp tip of the maxillary right second molar in the frontal plane was measured. 2. Y ccdculation. V -21.5 mm located the cusp tip in the frontal plane relative to the vertical placement of the condylar axis in the horizontal plane. 3. P measurement. The distance was measured from the posterior depth of the curve of the channel for the anterior stop pin in the maxillary bow, along the midsagittal plane on the superior surface of the maxillary bow to the intersection of the frontal plane of the mesiolingual cusp tip of the maxillary right second molar. 4. B calculation. The calculation 110 mm - P located the cusp tip to the condylar axis on the midsagittal plane and equalled variable B. 5. X calculation. The radius X (to the condylar axis) of the point projected by the mesiolingual cusp tip in the midsagittal plane could then be calculated by the Pythagorean theorem: a2 = b2 + c2, or in this instance. x = &lo-P)2

+ (V-21.5)2

6. W measurements. The thickness of material remaining in the interocclusal CR record in the indentation created by the maxillary second molar mesiolingual cusp tip and its opposing tooth was measured. The measurement must be at the depth of the impression made by the maxillary mesiolingual cusp tip and parallel to

Fig. 2. Schematic of articulator with geometric figure superimposed. V = measurement from superior surface of maxillary bow to mesiolingual cusp tip of posterior tooth used; P = measurement from posterior depth of curve of stop pin channel to vertical plane of mesiolingual cusp tip of posterior tooth used; Y = V - 21.5 mm; R = 110 mm - P; X = calculated solution to triangle with I3 and Y as other sides; W = measured thickness of wax interocclusal centric relation record in indentation of posterior tooth used; X = X’ (used with u’ to calculate angle of opening caused by interocclusal record). Angle of opening as converted into compensating stop pin setting in millimeters by describing angle of opening as tangent with radius of 110 mm.

the arc of opening, regardless of what portion of the opposing tooth made the impression in the interocclusal record. The W measurement represents the chord of the angle of opening at radius X. 7. 0 calculation. The angle of opening was calculated by the formula:

SinO=[(z)(2)] The angle of opening stop pin and converted

was then projected to the anterior to millimeters by the formula:

(Tan

O)(llO

mm)

A computer program (Basic 80, Microsoft, Rellevue, Wash.) was written to perform the calculations and a relational data base (Personal Pearl, Pearlsoft, Salem, Ore.) was designed to store data and allow variable sorts of data to produce specific reports. Data were obtained from measurements of arbitrary hinge axis (AHA) mounted maxillary casts and interocclusal CR records made by junior and senior dental students at the University of Tennessee, Memphis, College of Dentistry. The mountings were selected for use in the study before any impressions were made and the patients were examined to be certain they met with the study’s design. Measurements and the resulting calculations were recorded in the data base and

364

the anterior stop pin was se: at the calcuiated compensating opening before the mandibular cast was moonted. After the mandibular cast was mounted, laterai mterocclusal records were made and the instrument was programmed for lateral movement. At this point, the anterior stop pin was reset a! first contact of the casts in CR and the position was recorded. The condylar locks were then released, the stop pin was eievated, and the articulator was