the effect of projection errors on cephalometric length measurements

European Journal of Orthodontics 8 (1986) 141-148 O "86 European Orthodontic Society

The effect of projection errors on cephalometric lengthmeasurements

J. Ahlqvist*, S. Eliasson** and U. Welander*•Department of Oral Radiology, University of Umea and "Department of Oral Radiology, KarolinskaInstitutet, Stockholm

SUMMARY The magnitude of projection errors, calculated theoretically on length measurementsin cephalometry was studied. Rotation of the object by up to 5° from the proper position,resulted in errors in length measurements that were usually less than one percent. Rotations ofmore than 5° may increase the error but imply misalignments of the head that should be evidentwhen positioning the patient. The advantage of extremely long focus film distances seemsdoubtful.

Introduction

The accuracy of cephalometric measurementshas generated great interest. Many studies havebeen published on errors associated with land-mark identification, errors arising from the regis-tration of landmarks, and errors due tomeasurement procedures (for a review seeMcWilliam, 1983).

Errors due to the projection of the three-dimensional object on to a two-dimensional filmhave been studied less extensively. Similarly,there has been little analysis of errors arisingfrom misalignment between the different com-ponents of the cephalographic equipment ormisalignment of the patient in the cephalo-graphic system. Although these errors may beof significance they are usually considered to beof less importance than other errors (Bjork,1974; Hatton and Grainger, 1958; Miller etal., 1966; Savara et al., 1966; Carlsson, 1967;Baumrind and Frantz, 1971; Mitgard et al.,1974; Houston et al., 1986).

In a study by van Aken (1963) projectionerrors were found to be small but might be ofsignificance in cephalographs of asymmetricalskulls or in the case of anatomical landmarksthat do not lie in the mid-sagittal plane. Berg-ersen (1980) studied magnification and distortionin cephalometric radiography and found discrep-ancies between distances measured on the filmand true distances in the object. Based on theseresults compensation tables for correction oflinear measurements were constructed.

The basic mathematics for calculating projec-tion errors in cephalometry have been describedpreviously (Eliasson et al., 1982). The authorspresented expressions that may be used to calcu-late the position of any landmark on the filmfor any misalignment between the components ofthe cephalographic system including the object.

The cephalographic projection is a so-calledcentral projection which implies that the beamdiverges from a point. With respect to thegeometry of the cephalographic projection, thefocal spot may be considered a point source.Ideally the cephalographic projection is ortho-gonal, i.e. the film plane is perpendicular to thecentral ray of the beam. In a straight lateralprojection, such as a cephalographic profile,the sagittal plane of the patient should beperpendicular to the central ray of the beam andparallel to the film plane.

In practice, however, the alignment of thecephalographic system may differ from thetheoretical ideal. The relations between the dif-ferent components of the cephalographic systemare affected by a number of factors (Fig. 2):

—the focal spot, the cephalostat, and the filmmay be linearly displaced in relation to eachother;

—the cephalostat and the film may be rotatedwith respect to each other;

—the patient may be linearly displaced and/orrotated in relation to the cephalographicsystem.

An analysis of the influence of these factors

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on the cephalometric image was presented byAhlqvist et al., 1983. The factors causing imagedistortion and the principal distortion effect werestudied.

The purpose of the present work was to studythe magnitude of projection errors on lengthmeasurements in cephalometry. It was assumedthat the alignment of the cephalographic equip-ment was accurate. Thus, the study was limitedto the effects of incorrect patients positioningon linear measurements.

Method

Errors in linear measurements due to misalign-ment of the patient was studied by performingmathematical calculations. By this approach allother errors are eliminated, such as those arisingfrom landmark identification, landmark regis-tration, and measurement procedures.

The projected distance on the film planebetween differently placed points in model ob-jects was calculated. The mathematics employedhave been described in a previous work (Eliassonet al., 1982).

A series of computer models of distances ofdifferent lengths and inclinations were con-structed. Three principally different situationswere modelled:—the distance between cephalometric land-

marks representing structures in the sagittalplane, such as the distance between sella andnasion;

—the distance between one landmark represent-ing a structure in the sagittal plane and a pairof landmarks representing bilaterally sym-metrical structures, such as the distancebetween pogonion and articulare;

—the distance between two pairs of landmarksrepresenting bilaterally symmetrical struc-tures, such as the distance between articulareand gonion.In the case of bilaterally symmetrical struc-

tures, the mean position of the two landmarkson the film plane was calculated and used todefine one endpoint in a calculated distance.

In order to perform calculations for a realisticsituation a patient was modelled. Three-dimen-sional coordinates for anatomical structuresused to define cephalometric landmarks werefound by determining the mean position of thestructures on ten dry skulls.

J. AHLQVIST, S. ELIASSON AND U. WELANDER

Z

Figure 1 Directions of possible misalignments of thepatient.

The length on the film of the distance betweenthe landmarks in all modelled situations wascalculated when the model objects were rotatedaround their _y-axes (Fig. 1). For a real patientthis would mean tilting the head. Rotationaround the z-axis, i.e. rotation of a patient'shead, will result in analogous effects and wasnot treated separately. Since rotation around thex-axis alone will only affect the position of thelandmarks on the film and not the distancebetween them, rotation around the x-axis wasnot considered. Rotations between —10° and+ 10° were considered. In the case of themodelled patient the result of simultaneous ro-tation around the y- and z-axes was calculated.Here, only rotations within ± 5° were consideredsince careful positioning of the patient shouldeliminate more severe malpositioning in clinicalpractice. A focus to object distance of 1400 mmand a focus to film distance of 1550 mm wereused.

The computer programs were constructed toallow for translation of the model objects alongall three axes of the cephalostat. Calculationswere performed when rotation was combinedwith a maximum of ± 10 mm translation alongthe three axes.

The magnitude of measurement errors wasstudied by means of diagrams in which therelative length of distances between modelledlandmarks were plotted as functions of rotationand/or translation of the model objects.

Results

In one investigated case one object point co-incided with the origin of the coordinate systemof the object. Another point was given different

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PROJECTION ERRORS IN CEPHALOMETRY 143

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Figure 2 Relative length on the film plane of objectdistances plotted as functions of rotation around the >>-axis.(A) A distance of 50 mm with one endpoint at the origin.The four functions represent different inclinations to the y-axis of the distance as indicated in the diagram, (B) Adistance of 100 mm with one endpoint at the origin, (c)Same as (B) with the addition of 10 mm translation alongall three axes.

positions in the y-z-plane, corresponding tothe sagittal plane of a real patient. Differentpositions of the latter point were chosen so thata line connecting the two object points hadconstant length but formed different angles tothe y-axis, the rotational axis. The relativeprojected length of such lines, 50 mm long, whenrotated around the y-axis is illustrated in Fig.2a. When a line connecting the object points

forms a small angle to the rotational axis, theerror of its projected length on to the filmplane is insignificant. The error increases withincreased inclination. Still, the error is less than1 per cent between —10° and 5° rotation whenthe inclination of the line is at 90° to therotational axis. If the length of line is increasedto 100 mm there is an insignificant change ofthe error for distances forming small anglesto the horizontal plane (Fig. 2b). For largerinclinations the error decreases with negativerotation. With positive rotation, the error in-creases with increased inclination of the distance.The magnitude of this change is trivial (compareFigs 2a and b). When rotation was combinedwith translations up to ±10 mm the additionaleffect on the error was found to be negligiblefor practical purposes (Fig. 2c).

It should be noted that when the head isrotated so that the forehead approaches the film,the projected length of a line such as S-N isreduced, while when the rotation is in the op-posite direction, the projected length is increased(Fig. 3).

Results of calculations performed on themodelled patient are exemplified for the follow-ing distances: sella-nasion (Fig. 4), nasion-sub-spinale (Fig. 5), and articulare-pogonion (Fig.6). Here all combinations of simultaneous ro-tation from — 5° to + 5° around the y- and z-axes are plotted.

Sella-nasion forms a small angle to the y-axis.When the patient is rotated around the y-axisthe length of this distance decreases as the

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144 J. AHLQVIST, S. ELIASSON AND U. WELANDER

Figure 3 The projected length of an object will vary with its inclination toward the focal spot or toward the film.

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Figure 4 Relative projected length, L, of the distance sella-nasion as a function of combined rotations around the y- and z-ajtes, respectively. The variation of the relative length of the distance is represented by a curved plane which is illustratedby a three-dimensional diagram shown in three aspects: in a perspective plot (a) and in plots that demonstrate the effect ofrotation around the z-axis (b) and the >>-axis (c).

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Figure 5 Relative projected length, L, of the distance nasion-subspinale as a function of combined rotations around the y-and z-axes, respectively. Three-dimensional diagram shown in three aspects.

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Figure 6 Relative projected length, L, of the distance articulare-pogonion as a function of combined rotation around they- and z-axes, respectively. Three-dimensional diagram shown in three aspects.

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rotation angle increases from — 5° to + 5° (Fig.4a and c), i.e. when the forehead is tilted towardthe film. When rotated around the z-axis thelength of the distance increases as the rotationangle increases from — 5° to + 5° (Figs 4a andb), i.e. when the forehead is rotated away fromthe film. A distance such as sella-nasion whichforms a small angle to the y-axis, i.e. close tohorizontal, is most sensitive to rotation of thepatient's head and less sensitive to tilting of thehead. This is evident from the fact that the planerepresenting the varying projected length ofsella-nasion in Fig. 5 is more markedly curvedin the dimension representing rotation aroundthe z-axis, than in the dimension representingrotation around the y-axis.

Since the distance nasion-subspinale is essen-tially vertical this distance demonstrates theopposite variation during rotation and tilting ascompared to sella-nasion (Fig. 5).

The distance articulare-pogonion is more orless diagonal. As can be expected from the

above, the projected length of this distance ismost sensitive to combinations of rotation andtilting of the patient's head (Fig. 6). The mini-mum projected length occurs at simultaneousnegative rotations of —5° around both axes.The maximum projected length occurs at simul-taneous positive rotations of 5° around bothaxes. When negative rotation around one axisis combined with positive rotation around theother axis, there are only insignificant errorsin the projected length. All other calculateddistances followed these patterns depending onthe degree of inclination.

The effect of a varied focus to object distanceon errors of length measurements is limited.While relatively long focus to object distancesare favourable, extremely long distances donot change the magnitude of projection errorsmarkedly. This is exemplified in Fig. 7 wherethe relative projected length of the distancearticulare-pogonion is plotted for four differentfocci to object distances.

Figure 7 Relative projected length, L, of the distance articulare-pogonion as a function of rotations around the y- and z-axes. Four different focci to object distances were employed: a) 1400 mm, b) 2800 mm, c) 4200 mm, and d) 5600 mm. Theobject to film distance was held constant at 150 mm. It will be noted that the projection error demonstrates limited changeswhen the focus to object distance is longer than 2800 mm.

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Discussion

Errors caused by the cephalographic projectiontechnique represent only a part of the total errorof length measurements in cephalometry. Othererrors originate in the identification and regis-tration of landmarks and in the measurementtechnique. Since all these errors would influenceany experimental study, it was decided to investi-gate the importance of projection errors incephalometry by means of theoretical calcu-lations. As stated above the true magnitude ofprojection errors is hidden in any experimentutilizing measuring techniques, van Aken (1963)used a mathematical approach in a study of'geometrical errors in lateral skull x-ray projec-tions'. Although his study is excellent and ofgreat importance, it is limited to specific aspectsof projection errors.

In the present work the aim was to analyse theeffect of projection errors. Thus, the computerprograms constructed to perform the calcu-lations were written to allow for independentvariation of the following factors: translation inspace of the focal spot, the cephalostat, theobject, and the film; rotation of the cephalostat,the object, and the film plane around their axes;focus to object and focus to film distances. Thislist includes 22 variables. In fact, the number ofvariables is so great that a complete systematicanalysis of the influence of all of them, includingall possible combinations, is impossible for prac-tical reasons. In the present work, the investi-gation was limited to the effect of rotation andtranslation of the patients head and to variationsof the focus to film distance. These limitationsare justified by previous results indicating thatmisalignment between the different componentsof the cephalographic system is of minor impor-tance provided that the equipment is properlyadjusted (Ahlqvist et al., 1983). Furthermore,when several cephalographs are exposed usingthe same cephalostat, misalignment of the equip-ment will cause consistent systematic errors,while the positioning of the patient causes non-systematic and varying projection errors.

In general, projection errors in lengthmeasurements are minor in cephalometry. Infact, rotation of ±5° from the ideal positionusually results in errors that are less than onepercent. Such as error is usually insignificantand will in most instances be concealed by othererrors. If the rotation is increased the error

increases and may become significant even atrotations of a few degrees more than +5°.On the other hand, careful patient positioningshould eliminate such errors. Rotation or tiltingof the patient's head by more than 5° is discern-ible and should not arise in skilful clinical work.

Application of long focus to object distanceshas been suggested to minimize projection errors(Nawrath, 1961, Carlsson, 1967, van Aken,1963). Although it is true that short focus toobject distances result in greater projectionerrors than long focus to object distances, theresults of the present study indicate that there isa limited gain in using extremely long distances.

Address for correspondence

Dr J. AhlqvistDepartment of Oral RadiologyUniversity of UmeaS-901 87 UmeaSweden

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Mitgard J, Bjork G, Linder-Aronsson S 1974 Reproduci- Savara B S, Tray W E, Miller P A 1966 Analysis of errorsbility of cephalometric landmarks and errors of measure- in cephalometric measurements of three-dimensional dis-ments of cephalometric cranial distances. Angle tances on the human mandible. Archives of Oral BiologyOrthodontist 44: 56-62 11: 209-217

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the effect of projection errors on cephalometric length measurements

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