Palatal Surface Area of Maxillary Plaster Casts A Comparison Between Two-Dimensional and Three-Dimensional Measurements

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Tron A. Darvann, M.Sc., Ph.D., Nuno V. Hermann, D.D.S., Ph.D., Bjarne K. Ersbøll, Ph.D., Sven Kreiborg, D.D.S., Dr. Odont., Ph.D., Samuel Berkowitz, D.D.S., M.S., F.I.C.D.

Objective: To investigate the relationship between corresponding two-di-mensional and three-ditwo-di-mensional measurements on maxillary plaster casts taken from photographs and three-dimensional surface scans, respectively.

Materials and Methods: Corresponding two-dimensional and three-dimen-sional measurements of selected linear distances, curve lengths, and (surface) areas were carried out on maxillary plaster casts from individuals with unilat-eral or bilatunilat-eral cleft lip and palate. The relationship between two-dimensional and three-dimensional measurements was investigated using linear regres-sion.

Results and Conclusions:Error sources in the measurement of three-dimen-sional palatal segment surface area from a two-dimenthree-dimen-sional photograph were identified as photographic distortion (2.7%), interobserver error (3.3%), vari-ability in the orientation of the plaster cast (3.2%), and natural shape variation (4.6%). The total error of determining the cleft area/palate surface area ratio was 15%. In population studies, the effect of using two-dimensional measure-ments is a decrease of discriminating power. In well-calibrated setups, a two-dimensional measurement of the cleft area/palate surface area ratio may be converted to a three-dimensional measurement by use of a multiplication fac-tor of 0.75.

KEY WORDS: cleft lip and palate, dental casts, 3D measurement, timing of pal-atal closure, 2D measurement

Many current three-dimensional (3D) recording devices are able to capture the surface morphology of maxillary plaster casts with high spatial resolution and accuracy, using stereo-photogrammetry or laser or CT scans. These technologies pro-vide digital copies of the surface that are stored easily and are subsequently retrieved for measurement at the desired level of detail. Measurements carried out later on the virtual casts may

Dr. Darvann is Research Engineer, 3D Laboratory, University of Copenha-gen, Copenhagen University Hospital, CopenhaCopenha-gen, Denmark, and the Tech-nical University of Denmark, Lyngby, Denmark. Dr. Hermann is Associate Professor, Department of Pediatric Dentistry and Clinical Genetics, School of Dentistry, Faculty of Health Sciences, University of Copenhagen, Copenhagen, Denmark. Dr. Ersbøll is Associate Professor, Informatics and Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark. Dr. Kreiborg is Professor and Chairman, Department of Pediatric Dentistry and Clinical Ge-netics, School of Dentistry, Faculty of Health Sciences, University of hagen and Department of Clinical Genetics, The Juliane Marie Center, Copen-hagen University Hospital, CopenCopen-hagen, Denmark. Dr. Berkowitz is Research Director, South Florida Cleft Palate Clinic, Clinical Professor of Pediatrics and Surgery, University of Miami School of Medicine, and Consultant to Plastic Surgery Department, Miami Children’s Hospital, all in Miami, Florida.

Submitted December 2005; Accepted August 2006.

Address correspondence to: Dr. Tron A. Darvann, 3D Lab, School of Den-tistry, University of Copenhagen, Nørre Alle´ 20, DK 2200 Copenhagen N, Denmark. E-mail tdarvann@lab3d.odont.ku.dk.

DOI: 10.1597/05-213.1

be truly 3D (e.g., 3D distances, 3D curve lengths, and 3D surface areas) and are seen to be replacing previous two-di-mensional (2D) measurements (e.g., photographs or photocop-ies). Widespread use of new 3D technologies is, however, still lagging due to several disadvantages. One is the high cost of scanning hardware, but even more important drawbacks are their frequently cumbersome use and the lack of efficient 3D analysis software, standardization, and easy calibration. These obstacles, however, will be overcome eventually (e.g., Baum-rind et al., 2003). Meanwhile, one can ask if there are situa-tions where 2D measurements would be sufficient and would provide a trade-off between accuracy on the one hand and simplicity, cost, and time on the other. The situation is, in many respects, analogous to the use of 2D cephalometric variables measured on x-rays that are used as valid representations of their 3D counterparts as measured, for instance, in a CT scan. Although the accuracy and precision of the various 2D or 3D methods and instruments employed for plaster-cast mea-surements usually are determined and reported in the literature, there is a lack of discussion of the differences between 2D and 3D measurements and the consequences of selecting one of them in a particular situation. The purpose of the present article is to provide a comparison of the use of 2D versus 3D measurements for some typical applications.

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TABLE 1 Overview of the Material

Designation Sample 1 Sample 2

Origin South Florida Cleft Palate Clinic Copenhagen University Hospital

n (casts) 122 41

n (individuals) 16 41

Age range 7 d to 8 y 50 to 100 d

Diagnosis 8 UCLP, 8 BCLP 41 UCLP

Treatment conservative (Berkowitz) none (primary anomaly)

2D acquisition photography⫹manual 2D tracing (Observer 2) CT-scanning⫹manual 3D tracing (Observer 2) 3D acquisition electromechanical digitizer (Observer 1) CT-scanningmanual 3D tracing (Observer 2)

Data stored 2D and 3D areas 3D surface meshes

very 3D in nature (e.g., relating to the length of the alveolar arch, the inclination of the palatal shelves, or the surface area of palatal segments) (Berkowitz, 1990, 1996). Measurement of such quantities would, strictly speaking, only be valid if car-ried out in 3D. Using a 2D measurement corresponds to pro-jecting the 3D points of the plaster-cast surface onto some 2D plane. The 2D configuration of projected 3D points in the projection plane (focal plane of the 2D recording device) de-pends on the orientation of the plaster cast. When using a photographic camera, the base or ‘‘support back-plate’’ of the plaster cast created during grinding determines the orientation as it rests on a table. In the case of a photocopier or flatbed scanner, the orientation of the cast is determined by the palatal shape as the cast rests on the glass plate of the capturing de-vice. The amount of error introduced by carrying out the mea-surements in a 2D projection plane depends on the type of measurement and the orientation of the plaster cast. It is as-sumed here that perspective and distortion effects are negli-gible during 2D capture of the object (an appropriate assump-tion in well-controlled setups; see below). On the one hand, if the length of a straight line (or equivalently, the distance be-tween two points or the position of a point) or the area of a flat surface (plane) is to be measured, then the resulting 2D value varies in general as the cosine of the angle␪ between the line/plane and the projection plane. If, on the other hand, the length of a nonstraight curve (e.g., a curve along the al-veolar crest) or the area of a nonflat surface (e.g., palatal sur-face) is to be measured, then the measured 2D value will de-pend on the shape of the object (anatomy) in addition to the angle␪. The 2D value always will be smaller than the corre-sponding 3D value, and, in general, the taller and deeper (more nonflat) the anatomy is, the larger the difference between the 2D and 3D values. In general, a measured 2D area (or curve length) may correspond to many different 3D surface areas (or 3D curve lengths). An estimate of this variation or ‘‘error bar’’ on the 2D measurement due to natural shape variability may be obtained by measuring corresponding 2D and 3D values in a representative population. Similarly, an estimate of a varia-tion or error-bar on the 2D measurement due to variavaria-tion in

the angle␪may be obtained in an experiment where 3D

sur-faces are rotated within a typical range of ␪ values and cor-responding 2D and 3D values are recorded. A typical range of

␪values may be determined by estimating the orientation of

the back-plate (base) of the plaster cast in a standardized

co-ordinate system defined by some reference points (e.g., tuber points and points at the palatal rugae). These errors of 2D measurement (due to natural shape variability and orientation, respectively) may be viewed alternatively as errors of esti-mating the 3D value from a 2D measurement. There are sev-eral types of measurements that exemplify these estimation errors: (1) distance between tuber points, (2) alveolar arch length, (3) palatal segment surface area, and (4) age at which the ratio between segment and cleft areas reaches a given val-ue. Examples of applications of the surface area measurement are the investigation of palatal tissue deficiency in some cleft types (Lo et al., 2003) and the monitoring of cleft area relative to palatal area over time for individualizing timing of cleft treatment, as proposed by Berkowitz (1996). The usefulness of these types of measurements are not being debated here. They are included solely in order to discuss the amount of errors involved in obtaining 3D values from 2D measurements. If one accepts that a particular cleft area/palate area ratio may be used to indicate timing of surgical closure of the pal-atal cleft space (as suggested by Berkowitz et al., 2005), a simple formula to convert from 2D to 3D measurements could assist surgeons to determine objectively when best to surgi-cally close the palatal cleft space.

MATERIAL

The material used in the present investigation originated from two distinct sources (Table 1). Sample 1 was used for obtaining a relationship between corresponding 2D and 3D measurements of surface area, as well as for estimation of a total error of obtaining 3D surface areas from 2D measure-ments. Furthermore, it was used for the study of the temporal change of the ratio between cleft area and palatal area. The sample consisted of a total of 122 plaster casts from 16 indi-viduals, 8 of whom had unilateral complete cleft lip and palate (UCLP) and 8 of whom had bilateral complete cleft lip and palate (BCLP). All were treated conservatively (Berkowitz, 2003) without the use of presurgical appliances and were rep-resented by longitudinally studied cases of excellent treatment outcome from the South Florida Cleft Palate Clinic in Florida. Ages ranged from 7 days to 8 years.

Sample 2 was used in order to investigate some of the errors contributing to the total error of estimating 3D values from 2D measurements. In particular, errors due to orientation and

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FIGURE 1 3D drawings of UCLP and BCLP palatal surfaces created by use of an electromechanical digitizer. Anatomical landmarks used in the text are indicated.

FIGURE 2 Photographs of plaster casts and corresponding electromechanical digitizer surfaces for two example individuals from Sample 1: one UCLP individual at 8 different ages (top) and one BCLP individual at 10 different ages (bottom).

natural shape variability were investigated using this sample, because 3D coordinates of the points on the surface represen-tations from Sample 1 were not available in digital format (only total surface area was available, as well as 2D plots of the surfaces). Sample 2 represented a young and age-wise nar-rowly distributed sample of untreated infants with UCLP and consisted of 41 plaster casts from individuals between 50 and 100 days of age from the total Danish nonsyndromic cleft pop-ulation born between 1976 and 1981 (Jensen et al., 1988).

METHODS 2D Versus 3D Measurement

Corresponding 2D and 3D measurements were obtained and a linear regression of the 3D values on the 2D values was carried out. A relation between 2D and 3D measurements was given by the regression line and was used in order to obtain a 3D value from a 2D measurement.

Measurements and Definitions

Measured quantities are shown in Figure 1 and were defined as follows. (1) Anterior cleft width represented by the linear

distance between points AC and AC⬘(Sample 2, UCLP only).

(2) Palatal segment arch length represented by the length of

the curve from P to AC and from AC⬘to P⬘measured along

the alveolar crest (Sample 2, UCLP only). (3) Palatal surface areas were as defined by Berkowitz (1990, 1996). In UCLP, the total palatal surface area (P) was the sum of the lateral segment areas measured to the top of the alveolar crest. It was defined similarly in BCLP, but the area of the premaxilla was included. The cleft area (C) was defined as the area of the smallest possible 3D surface bridging the cleft space (Fig. 1). (4) The ratio of cleft area to total palatal surface area (C/P) was computed as a function of age for each subject.

Measurements on Sample 1

Figure 2 shows photographs of plaster casts from two of the individuals included in Sample 1. The figure also shows cor-responding computer-generated drawings of 3D surfaces cre-ated using an electromechanical hand-held digitizer previously described by Berkowitz (Berkowitz, 1990). Corresponding 2D areas were computed by tracing outlines of the segments and cleft on acetate paper superimposed on standardized cast pho-tos (scale 1:1, because this was seen to be a sufficient mag-nification for the current purpose), followed by scanning of the tracings in a flatbed image scanner. Finally, area computation in the resulting images was carried out automatically by use of a custom software program. The 3D and 2D tracings were completed by different observers. The 2D observer, however, did use the same pencil markings on the casts along the seg-ment outline (visible in most of the photographs) as did the

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FIGURE 3 Surface reconstruction from CT scan of one of the Sample 2 plaster casts. Alveolar arch segments are shown hatched and cleft space is shown in black. Letters P, P, and i indicate points used for orientation standardization (see the text).

3D observer, for visual guidance. For measurements on Sam-ple 1, the 3D cleft area (C) was determined as the area of a flat (2D) approximation to the cleft surface.

Measurements on Sample 2

Figure 3 shows an example of a computer-generated plaster cast surface from Sample 2 created by CT scanning and iso-intensity thresholding (Hermann et al., 1999; Darvann et al., 2001). Outlines of the alveolar crest and cleft space were de-lineated manually in 3D on the computer-generated surface using custom software developed using the Visualization Toolkit (VTK; Kitware Inc., Clifton Park, NY) (Schroeder et al., 1998). The surface was cut accordingly, yielding the pal-atal surfaces as well as a Delaunay triangulation (Schroeder et al., 1998) of the cleft space, such that the corresponding sur-face areas (P and C) could be computed.

Corresponding 2D areas (p and c; note the lowercase letters, as opposed to the capital letters of the 3D counterparts) were computed from 2D projections of the 3D surfaces after adopt-ing a standard view angle. The standard view angle was de-fined in terms of a line of sight perpendicular to the plane defined by the tuber points (P and P⬘in Fig. 1) and a point i at the intersection between the alveolar crest and a plane pass-ing through the midpoint of P-P⬘and being perpendicular to that line (Darvann et al., 2001). Similarly, the P-P⬘distance and the length of the alveolar crest were computed as 2D pro-jections of the corresponding 3D quantities.

Error Sources and Their Corresponding Standard Error Total Error of a 3D Value Estimated From 2D

Measurement

The ability of obtaining a 3D value from a measured 2D value was determined as the linear Pearson correlation coef-ficientRand the amount of scatter around the regression line

as represented by 95% confidence and prediction bands (Rice, 1996). The total error of the 3D value calculated from a 2D measurement was obtained by regression analysis using Sam-ple 1.

3D Measurement Error

The error of 3D measurement comprised an instrument error (due to miscalibration, distortion, or error due to limited spatial resolution of the capturing device) and an error due to the landmarking or tracing process, because the observer has a limited ability to recognize and to pinpoint the structures on the surface. It is assumed here that the instrument error is negligible; hence:s3D艐slandmarking,3D⫽sd/兹2, wheresdis the

standard deviation of the differences between duplicate mea-surements and it is divided by the square root of 2, because the variance of a single measurement is half that of a differ-ence between two measurements. A coefficient of variation (relative error) for the 3D measurement thus becomes CV3D⫽ slandmarking,3D/v3D, where v3D is the actual measured 3D value. Inter- and Intraobserver Error

Inter- and intraobserver errors of 3D surface measurement of palatal area P were determined by tracing by two different observers/duplicate tracing, respectively, of 41 casts from Sample 2.

2D Measurement Error

The error of 2D measurement contained contributions from instrument error (due to miscalibration, distortion, or error due to limited spatial resolution of the capturing device) and from error of the landmarking or tracing:

2 2

CV2D⫽兹sdistortion,2D⫹slandmarking,2D/v .3D Error Due to Distortion of 2D Photographs

An error due to distortion of 2D photographs of the plaster casts was estimated by photographing (by use of a 4-megapixel Olympus Camedia E10 camera; Olympus America Inc., Center Valley, PA) a number of objects (LEGO DUPLO plastic toy bricks; Lego Corp., Billund, Denmark) of comparable size as the plaster casts, but with known geometrical dimensions. Camera parameters were varied within realistic limits and cor-responding distortion was recorded. The main distortion pres-ent was radial off-axis distortion leading to objects being viewed approximately as if rotated slightly away from the ob-server. An effective corresponding rotation angle was estimat-ed visually by comparing images of the DUPLO bricks with artificially rotated surface representations of a brick.

Inter- and Intraobserver Error

Inter- and intraobserver errors of 2D area measurement of palatal area p were determined by tracing by two different

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FIGURE 4 Regression analysis for alveolar segment area measurement carried out on plaster casts from Sample 1. The 3D segment area measured by use of an electromechanical digitizer is plotted against the correspond-ing 2D area measured on photographs. The linear relationship between 2D and 3D measurements is expressed by the solid regression line and a correlation coefficientRas indicated; 95% confidence and prediction bands are shown by short- and long-dashed curves, respectively.

FIGURE 5 Regression analysis for alveolar segment area measurement carried out on plaster casts from Sample 2. Regression line (solid) and confidence (short dash) and prediction (long dash) bands are shown. The dotted line indicates the regression line for Sample 1 shown in Figure 4.

observers/duplicate tracing (1 month apart) of 2D outlines on 30 casts.

Error Sources in Calculating 3D Values From 2D Measurement

The error of calculating 3D values from 2D measurements contains errors due to 2D measurement and, in addition, in-cludes errors due to plaster cast orientation and shape vari-ability. Thus,

CV3D, calculated from 2D

2 2 2 2

⫽兹sdistortion,2D⫹slandmarking,2D ⫹sorientation⫹sshape/v .3D

The total variability within a group of subjects becomes the sum of the error variance and the natural variance.

Error Due to Natural Variability of Palatal Shape

Error due to natural variability of palatal shape was esti-mated using the casts of Sample 2. Assuming that orienting the casts according to the standard reference system (defined by the points P, P⬘, and i as explained above) brings them into perfect general anatomical alignment, the remaining variability of a 2D area in a sample is due purely to natural shape vari-ability. (This assumption is believed to be good as long as the age difference in the sample is small [Darvann et al., 2001], which was the case for Sample 2. If large age differences exist, then different anterior reference point[s] than the i point [de-fined above] should be used, for example, exploiting the pal-atal rugae if they can be located.)

Error Due to Plaster Cast Orientation

An approximate upper limit to the error due to the variation in orientation was estimated by visual assessment of the ap-pearance of a plaster cast while its back-plate was gradually tilted. A rotation angle larger than 20⬚ was considered to be unlikely. Actual angles of inclination between the back-plate and a plane representing a standard view angle (see above) were measured after fitting a plane through five landmarks placed on the back-plate.

Error of C/P Ratio Measurement

A coefficient of variation of the C/P ratio was estimated as a combination of errors from separate measurements of C and P, assuming equal contributions from the two (because delin-eation of the cleft and of the segments are similar processes); resulting in CVC/P⫽CVP⫻兹2. In this way, a problematic

determination of a CV as C approaches 0 (as for the older subjects) was avoided.

RESULTS 2D Versus 3D Measurement

Figure 4 shows corresponding 3D and 2D values of segment area P measured for all the 122 plaster casts of Sample 1. The

regression line shown is given by P3D ⫽1.10 ⫻P2D ⫹205.

Figure 5 shows a similar plot for P of Sample 2, with P3D⫽

1.19⫻P2D⫹83.6. A similar analysis for alveolar arch length

for the UCLP subjects of Sample 2 resulted in l3D ⫽1.15⫻

l2D⫺0.451. For the distance between alveolar endpoints from

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FIGURE 6 Coefficient of variability for the orientation error as a func-tion of maximum rotafunc-tion angle. Each CV was determined by regression analysis as explained in the text. Error bars correspond to 1 SD of the variation in CV values due to variation in surface area across the sample.

FIGURE 8 Cleft/palate ratio as a function of age in a single subject. Error intervals correspond to total error (1 SD) of the 3D and 2D mea-surements, respectively. Horizontal lines indicate particular C/P ratio val-ues; C/P0.1 is indicated by the long-dashed horizontal line; C/P0.133 is a corresponding value derived for the 2D measurement (see the text).

FIGURE 7 Cleft/palate ratio as a function of age in a single subject. Error intervals correspond to total error (1 SD) of the 3D and 2D mea-surements, respectively. Horizontal lines indicate particular C/P ratio val-ues; C/P0.1 is indicated by the long-dashed horizontal line; C/P0.133 is a corresponding value derived for the 2D measurement (see the text). Error Sources and Their Corresponding Standard Errors Total Error of Calculating 3D Values From 2D

Measurement

Using Sample 1, an estimate of the total error of obtaining a 3D segment area from a 2D measurement was obtained by regressing the 3D values on the 2D values for all 122 plaster casts, as shown in Figure 4. The 95% prediction interval is shown as the long-dashed curve in Figure 4, leading to a mean

CV (1 SD 艐 one half of the 95% prediction interval) of

CV3D,calculated from 2D⫽10.4%, ranging from 6.1% (for the

small-est plaster casts) to 20.5% (for the largsmall-est plaster casts). Note that the prediction interval is used instead of the confidence

interval (short-dashed curve in Fig. 4), because 3D values are to be ‘‘predicted’’ for individual subjects. For example, a

fu-ture measurement of a 2D segment area of 700 mm2 would

imply a best guess for the 3D surface area of 977 mm2 , but

with a 5% chance of this value being lower than 771 mm2or

higher than 1183 mm2. Note that Figure 4 was created using

a particular sample (Sample 1) and that its valid use is restrict-ed to subjects from a similar population. Note also that Figure 4 was created for a particular definition of the segment areas (Fig. 1).

3D Measurement Error:Inter- and intraobserver error

The inter- and intraobserver error of 3D segment surface area measurement was estimated to be 3.0% (range: 2.5% to 4.2%) and 3.3% (range: 2.7% to 4.3%), respectively.

2D Measurement Error:Error due to distortion of 2D photographs

Photographic distortion was measured by photographing a scene consisting of known objects (see ‘‘Methods’’ section above) and estimating the distortion as camera parameters (zoom, object distance, and F ratio) were varied within realistic limits. Off-axis distortion increased with distance from the camera axis and decreased with object distance, whereas image spatial resolution decreased with distance. Maximum distortion to be expected under realistic conditions and a field-of-view of 28 cm was estimated. An upper limit to the effective rota-tion due to distorrota-tion under realistic condirota-tions was estimated to be about 12.5⬚, corresponding to a CV of 2.7% (range: 2.1% to 3.4%).

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FIGURE 9 Cleft/palate ratio as a function of time for each of the 16 individuals of Sample 1. Results of 3D measurements are shown, together with curves derived from 2D measurements (solid line)1 SD (short-dashed line). Long-dashed horizontal line is drawn at a C/P ratio of 1/10.

2D Measurement Error:Inter- and intraobserver error

The inter- and intraobserver errors of 2D segment surface area measurement was estimated to be 1.9% (range: 1.5% to 3.1%) and 3.5% (range: 2.5% to 4.2%), respectively.

Error Sources in Calculating 3D Values From 2D Measurement:Error due to natural variability of palatal shape

Corresponding 2D and 3D segment areas for the 41 UCLP subjects of Sample 2 who were 2 months old are plotted in Figure 5. Again, long-dashed and short-dashed lines represent 95% prediction and confidence bands, respectively. Adopting half of the 95% prediction interval for the mean of the obser-vations as the error, the corresponding CV can be expressed as CVnatural variability⫽4.6% (range: 3.6% to 6.0%). Note that the

plaster cast surface orientation was standardized as explained in the ‘‘Methods’’ section prior to 2D area computation. Con-sequently, orientation error was removed, at least partially.

For alveolar arch lengths of Sample 2, the coefficient of variation was 2.1% (range: 1.9% to 3.0%), whereas for the distance between left and right anterior alveolar endpoints, it was 7.6% (range: 4.3% to 27%).

Error Sources in Calculating 3D Values From 2D Measurement:Error due to plaster cast orientation

As described above, the error due to shape variability alone was estimated using the corresponding 2D and 3D segment areas of the plaster casts when they were rotated to a standard orientation (Fig. 5). If there had been no shape variability, Figure 5 would have shown a perfect correlation with all points lying perfectly on the regression line. Consequently, in order to estimate the error purely due to variation in orienta-tion, 2D values for the standard orientation were corrected to fall on the regression line before all 41 plaster casts were ro-tated artificially by known amounts (drawn from a Gaussian distribution) and corresponding 2D segment areas were re-corded. Similar experiments were carried out for maximum rotation angles of 5⬚, 10⬚, and 20⬚, and corresponding CVs were computed as shown in Figure 6. Actual rotation angles in the sample were computed as the angle between the plaster cast back-plate and a plane representing the standard orienta-tion. No rotation angles were larger than 15⬚. It seemed rea-sonable to assume that the maximum orientation error expected to occur under realistic conditions would be 20⬚, corresponding to a coefficient of variation of 3.2% (range: 2.5% to 4.0%).

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FIGURE 10 Standard relative error of five different error sources con-tributing to the total error of a measurement of 3D palatal segment area using 2D photographs. P: Variability due to photographic distortion. I(2D): Intra-observer variability of measurement of 2D palatal segment area. B(2D): Inter- (or between-) observer variability of measurement of 2D palatal segment area. N: Error of 3D segment area when calculated from 2D measurement, due to natural shape variability (this error also is shown in Fig. 12.) O: Error of 3D segment area when calculated from 2D measurement, due to variation in plaster cast orientation during 2D measurement. Vertical bars indicate the range between minimum and max-imum values in the sample.

FIGURE 12 Standard relative error of 3D quantities when calculated from 2D measurement, due to natural shape variability. N: Palatal segment area (also shown in Fig. 10).l(UCLP): Alveolar arch length in a population of 2-month-old UCLP infants.l(UICL): Alveolar arch length in a popula-tion of 2-month-old unilateral incomplete cleft lip infants.d: Distance be-tween alveolar endpoints in a population of 2-month-old UCLP infants. Vertical bars indicate the range between minimum and maximum values in the sample.

FIGURE 11 Standard relative error. P: 3D palatal segment area deter-mined from 2D measurements. C/P: 3D cleft palate area ratio deterdeter-mined from 2D measurements. I(3D): Intraobserver error of 3D palatal segment area measurement. B(3D): Intra- (or between-) observer error of 3D pal-atal segment area measurement. Vertical bars indicate the range between minimum and maximum values in the sample.

FIGURE 13 Plot of the number of additional subjects needed in order for 3D alveolar segment area estimated from 2D measurement to match the discriminating power of a direct 3D measurement, as a function of difference between group means. Differences between group means are assumed tested by Student’sttest with equal n and variance in the two groups.

Error of C/P Ratio Measurement

A CV of the C/P is estimated from Figure 4, assuming sim-ilar error contributions from C and P, leading to CVC/P⫽10.4

⫻兹2 ⫽14.7% (range: 8.6% to 29%).

C/P Ratio as a Function of Time

Figures 7 and 8 show the decrease of the ratio (measured in 2D as well as in 3D) as a function of age in two example

subjects (a UCLP subject and a BCLP subject, respectively, corresponding to the plaster casts shown in the photographs of Fig. 2). The 1 SD error intervals shown in Figures 7 and 8 represent a combined error of determining cleft area and pal-atal segment area (CVC/P). For the 3D and 2D measurements,

these CVs (interobserver error) were 5.1% and 14.7% (as ex-plained above), respectively.

Using the regression line for the relationship between 2D and 3D area measurements (Fig. 5) it is possible to estimate 3D values from corresponding 2D measurements. This may be carried out by applying the appropriate conversion to each of

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FIGURE 14 Comparison of the discriminative power of direct and cal-culated (from 2D) 3D alveolar segment area measurements. The number of subjects (n) needed in each group in order to discriminate (at a 5% level) between two segment area mean values using Student’sttest is plot-ted (solid curves) against the difference between group means. Dashed curve shows the difference between the two solid curves: for example, ap-proximately 20 more subjects are needed using 2D than 3D measurement if a 6% difference between group means is to be detected.

FIGURE 15 Comparison of the discriminative power of direct and cal-culated (from 2D) 3D alveolar segment area measurements. Thepvalue from a Student’sttest (with population sizes in accordance with the study of Lo et al. [2003], as indicated) is plotted against the difference between group means that may be detected. Solid curve: 3D measurement. Dashed curve: 3D calculated from 2D measurement. Dash-dotted curve: 3D cal-culated from 2D measurement; but with increased sample sizes, as indi-cated.

the 2D measurements. Alternatively, the conversion may be carried out using a regression between 3D and 2D values of the average C/P values for each patient. This is the method adopted in Figure 9, where the development with time in terms of the 3D C/P ratio has been calculated from the 2D C/P ratio and is shown for all 16 subjects of Sample 1. Note that a conversion of 2D to 3D in this case corresponds to a simple multiplication factor of 0.75⫾0.1 (1 SD) to be applied to the 2D curve, moving it downward in the plots in Figure 9.

Al-ternatively, the conversion may be looked at as a 1/0.75 ⫽

0.133艐13% factor to be multiplied with a chosen C/P level

to be reached (e.g., with a chosen C/P level of 0.1 for a 2D measurement, this becomes 0.13 for a 3D measurement) (Figs. 7 and 8). The difference between the age when C/P reached 1/10 using 3D measurement and the corresponding age deter-mined indirectly by 2D measurement was calculated for the individuals of Sample 1. The median age difference was 24 days, with a maximum of 200 days. Figures 10 through 12 present summaries of the errors.

DISCUSSION ANDCONCLUSIONS

The results of this study provide insight into error sources that may be avoided if measurements are carried out in 3D as opposed to 2D. If 2D measurements are still to be carried out, or for interpretation of results based on 2D measurements in the literature, the present study may be of value. In general, it is concluded that 3D quantities may be derived from 2D mea-surements in well-controlled setups (e.g., in terms of orienta-tion of plaster casts). Note that when the conversion from 2D to 3D values is discussed, there is an implicit premise that it is desirable to obtain true 3D values. Under some circum-stances however, a 2D value could be used as a proxy for some

quantity, or differences between populations might be dis-closed using 2D measurements without necessarily having to relate them to any physical 3D quantity. Usually, however, it is the physical 3D quantity that is of interest, because it typi-cally relates better to the medical problem in question. In order to carry out the measurement with as little error as possible, a 3D measurement device should be used.

The error introduced by measuring a 3D quantity in 2D leads to a decrease in discriminating power when seeking to detect significant differences between populations. The added noise may be converted to an equivalent increase in the size of the populations needed for a 2D measurement to be as equally discriminating as a 3D measurement. This is illustrated in Figure 13 in the case of alveolar segment area measurement. The number of additional casts needed in order to obtain the same discriminating power is plotted, assuming use of Stu-dent’sttest with equal n and variances in the two populations under study. For instance, it is seen that approximately 20 ad-ditional casts are needed in the 2D case if a difference between

population means of 50 mm2 is to be detected. Figure 14

shows, similarly, the relation between the number of subjects and the difference between mean values that may be detected. As an example, Lo et al. (2003) compared the alveolar segment

area in a group of 30 UCLP subjects (mean value: 710⫾116

mm2) with a group of 23 isolated cleft palate subjects (mean

value: 826⫾89 mm2), representing a significant difference of

116 mm2. In Figure 15 the solid line shows the relation

be-tween pvalue and difference between group means for

pop-ulation sizes as described by Lo et al. (2003). From Figure 15 it is seen that despite an increased noise, the difference be-tween the groups would have been detected (at a 1% level) even by a 2D measurement (dashed line). It is also seen that in order for a 2D measurement to reach a discriminating power

(10)

similar to that of the 3D measurement, the number of subjects should be increased from 30 to 44 and 23 to 34 in the two groups, respectively (dash-dotted line).

In an attempt to estimate 3D quantities (e.g., segment sur-face area) from 2D measurements, it is important to keep in mind that the calibration curves (the regression curves in Figs. 4 and 5) have been obtained for a particular set of conditions. Calibration curves should be created for each population under study (e.g., a different curve for unilateral isolated cleft lip and UCLP), and possibly for different age ranges. Furthermore, separate calibration curves must be created separately for each quantity to be measured (e.g., alveolar arch length and palatal segment area), and the definition of these quantities must not differ between calibration and measurement. Such a discrep-ancy in the definition of a quantity may be exemplified by the different definitions of alveolar segment area used by Lo et al. (2003) and Berkowitz (1996).

With respect to the use of the C/P ratio for determination of timing for surgical closure of the palate (as suggested by Berkowitz et al., 2005), keeping the above limitations in mind, it may be concluded that 2D photocopy or scanned-image area measurements may be converted to 3D measurements by using the multiplication factor of 0.75.

REFERENCES

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Berkowitz S.Cleft Lip and Palate. With an Introduction to Other Craniofacial Anomalies. Perspectives in Management.Vol. 1. San Diego: Singular Pub-lishing Group; 1996.

Berkowitz S. A comparison of the effects of the Latham-Millard procedure with those of a conservative treatment approach for dental occlusion and facial aesthetics in unilateral and bilateral complete cleft lip and palate: part I. Dental occlusion.Plast Reconstr Surg.2003;113:1–18.

Berkowitz S. The complete unilateral cleft lip and palate: serial three-dimen-sional studies of excellent palate growth. In: Bardach J, Morris HL, eds.

Multidisciplinary Management of Cleft Lip and Palate. Philadephia: WB Saunders; 1990:456–473.

Berkowitz S, Duncan R, Evans C, Friede H, Kuijpers-Jagtman AM, Prahl-Andersen B, Rosenstein S. Timing of cleft palate closure should be based on the ratio of the area of the cleft to that of the palatal segments and not on age alone.Plast Reconstr Surg.2005;115:1483–1499.

Darvann TA, Hermann NV, Huebener DV, Nissen RJ, Kane AA, Schlesinger JK, Dalsgaard F, Marsh JL, Kreiborg S. The CT-scan method of 3D form description of the maxillary arch. Validation and an application. In: Trans-actions 9th International Congress on Cleft Palate and Related Craniofacial Anomalies.Go¨teborg: Erlanders Novum; 2001:223–333.

Hermann NV, Darvann TA, Huebener DV, Nissen RJ, Kane AA, Schlesinger JK, Kreiborg S, Marsh JL. A method for 3D shape description of the max-illary alveolar arch. Presented at the 56th Annual Meeting of the American Cleft Palate–Craniofacial Association; 1999; Scottsdale, Arizona. Jensen BL, Kreiborg S, Dahl E, Fogh-Andersen P. Cleft lip and palate in

Den-mark 1976–1981. Epidemiology, variability, and early somatic development.

Cleft Palate J.1988;25:1–12.

Lo L-J, Wong F-H, Chen Y-R, Lin W-Y, Wen-Ching E. Palatal surface mea-surement: comparisons among different cleft types.Ann Plast Surg.2003; 50:18–24.

Rice JA.Mathematical Statistics and Data Analysis.2nd ed. California: Dux-bury Press; 1995.

Schroeder W, Martin K, Lorensen B. The Visualization Toolkit. An Object-Oriented Approach to 3D Graphics.2nd ed. Upper Saddle River, NJ: Pren-tice Hall; 1998.

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