Abstract
Objective: To present a comprehensive mathematical analysis of dental arch curvature in subjects with normal occlusion.
Materials and Methods: The materials studied were 40 sets of upper and lower plaster dental casts of subjects presenting with normal occlusion. The sample was equally divided into casts from male and female subjects with an age range from 18 to 25 years. Curve-fitting analyses was carried out and four main categories of functions were considered: the beta function, natural cubic splines, polynomial equations, and Hermite cubic splines.
Results: The polynomial function (fourth order) was found to be a reasonable analysis when the objective is to describe the general smooth curvature of the dental arch, while a Hermite cubic spline is more appropriate when it is desired to track arch irregularities, such as evaluating treatment changes.
Conclusions: Due to its advantage in providing a more naturally smooth curve, the fourth-order polynomial function may be used as a guide to fabricate customized arch wires, or even an entire fixed orthodontic appliance system.
INTRODUCTION
Extensive effort has been made to define the form of the human dental arch. Formerly, the shape of the arch was described in simple qualitative terms such as elliptic, parabolic, and U-shaped. Such descriptions were obviously inadequate to precisely define the dental arch. There was an increasing demand in orthodontics to quantify arch forms for both clinical and research purposes. Linear measurements such as arch width, depth, and circumference are methods used to describe arch shapes. However, they are insufficient in characterization of arch form because they provide an incomplete description of all arch traits.
BeGole1 suggested that the ideal means for describing the dental arch curvature should be a mathematical curve. This curve should have considerable flexibility so that it may be fitted to any size or shape of the dental arch, and must include asymmetries if present. Since the dental arch is an imaginary curve, the descriptive information is then a series of discrete points. Hence, the mathematically produced curve should be fitted to individual points.
This approach is referred to as curve fitting or interpolation. Curve fitting is a concept that utilizes a mathematically generated curve to approximate dental arch curvature. The curve produced by the proposed mathematical function is fitted against dental landmarks that are believed to reliably define the dental arch. In an attempt to find the best mathematical equation that would describe dental arch curvatures, several authors tested various mathematical functions (models) that would best fit that purpose. Some of these models are: conic sections,23 catenary curves,45 cubic spline curves,1 second- to eighth-degree polynomials,46 mixed models,6 and the beta function.7 Nonetheless, it is difficult to generalize their findings or draw conclusions for a number of reasons: differing objectives, dissimilar study samples with different selection criteria, and dissimilar methodology.
In the published literature, thus far, there exist no comprehensive mathematical analyses of dental arch configuration on normal occlusion subjects. Therefore, it is our aim in this work to attempt to present a comprehensive analysis of dental arch curvature of normal occlusion subjects.
MATERIALS AND METHODS
The materials used in this study were upper and lower plaster models of 40 Saudi subjects presenting with normal occlusion (age range 18–25 years). Models were equally distributed for both female and male subjects. Each model was checked to ensure that the occlusal plane and the base plane of the model were parallel, and that the heels were trimmed with the upper and lower models in good articulation. Exclusion criteria were as follows: previous orthodontic treatment; cast restorations; proximal, or extensive restorations involving cusp tip; obvious cuspal or incisal attrition; and tooth fracture, ectopically erupted teeth, deciduous teeth, congenitally missing, or extracted teeth (excluding third molars). The following criteria were adopted for normal occlusion68: bilateral Class I first molar and canine relations, overbite and overjet between 2–4 mm, no crowding or spacing of more than 2 mm, free from rotation of teeth, and absence of anterior or lateral crossbite.
Digitization
The digitization approach consists of sample imaging, landmark capturing, and numerical manipulation. For sample imaging, a medium-to-high resolution digital camera (Nikon digital camera model 995) was used to directly image the dental models. The digital camera was placed directly above the model (55 cm) for direct imaging (standard setting) (Figure 1). Prior to imaging, a black 0.5 mm pencil was used to mark points of interest on the upper and lower dental casts9 (Figure 2). The captured image was then stored in a computer file in pixels. The Nikon camera used in the current study is capable of capturing over 3.15 megapixels, but nonetheless, it is not possible to convert the image captured from pixels to mm directly because of the zooming effect and because the size of the object imaged is not necessarily preserved. Therefore, to associate the pixel count of each imaged model with the true length measurements (mm), each model was placed within a 10 × 15 cm box (Figure 2). The dimensions of this box were then used to convert the number of pixels captured to the required measurements (mm); the dimensions were also used for horizontal and vertical orientations.
Citation: The Angle Orthodontist 78, 2; 10.2319/121806-516.1
Citation: The Angle Orthodontist 78, 2; 10.2319/121806-516.1
By using the appropriate imaging software, landmark capturing was performed by viewing the captured landmarks (points) numerically. In other words, the (x, y) location coordinates of each pixel representing the landmarks of interest were highlighted and converted into a matrix of numerical data. These numerical data represent the location coordinates of each landmark with respect to a fixed reference frame (Figure 3).
Citation: The Angle Orthodontist 78, 2; 10.2319/121806-516.1
Eighteen points were captured on each arch to represent the anatomical dental arch. The selected points were: the midincisal points of the incisors, the canine cusp tip, the buccal cusp tip of the premolars, and the mesiobuccal and distobuccal cusp tips of the first and second molars. A single point was captured for each incisor in the middle of the incisal edge, rather than two points at the incisal angles, to minimize the effect of rotations on the curve-fitting procedure. To mark the anatomical y-axis, two points were captured on the midpalatal raphe16: the point immediately distal to the incisive papilla and the furthest point posteriorly on the midpalatal raphe. Increasing the distance between the two points minimizes the effect of any error in the capturing of these two landmarks on the y-axis orientation. To facilitate superimposition of the upper and lower arches, the two points at the corners of the heels of the trimmed models were digitized.
To accomplish the process of landmark capturing, a computer program (number of procedures called m-files) was written specifically for this project using the MATLAB environment. MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. The object of the m-files when called from within the MATLAB environment is to open the specified image file, allow the user to freely highlight the landmark or point of interest, and capture the coordinates associated with it. As shown in Figure 3, each image has five different landmark types: upper arch landmarks, lower arch landmarks, anatomical center landmarks of the upper arch, base landmarks for upper and lower arches, and scale points which represent the four corners of the scale box. For each image read, a numerical matrix was created and stored in a file using the sex, group number, and case number as part of the file name to ensure uniqueness of each data set and ease for later retrieval.
The last step of digitization was the numerical manipulation. Before any mathematical study can be performed, the data must be manipulated into the appropriate format to convert the numerical coordinates from pixel count to length measurements in millimeters.
Mathematical Functions
Following the model-digitizing step, each case data set was retrieved, and curve fitting was carried out by utilizing a number of mathematical functions described below.
Beta function
This function is an empirical formulation derived statistically by Braun et al.7 It is expressed as:
Here, W is the molar width in millimeters; it is the measured distance between the right and left second molar distobuccal cusp tips. The arch depth, D, is the average perpendicular distance from the central incisors to the molar cross-arch dimension in millimeters at the points at which W is measured.
Polynomial equations
Generally, polynomial equations are expressed as:
y = anxn + an−1xn−1 + … + a2x2 + a1x + a0where the ai (a0, a1, a2, …, an−1, an) are polynomial coefficients and n is referred to as the order or the degree of the polynomial. The odd-numbered coefficients a1 and a3 are interpretable in terms of left and right symmetry, and the even-numbered coefficients a2 and a4 describe arch shape in terms of taperedness and squareness, respectively. Polynomial approximations of second to twelfth order (n = 2, 3, …, 12) were tested using least square error (polynomial curve fitting).
Natural cubic splines
A cubic spline (referred to as natural cubic spline) was used. The general form of this cubic spline is:
y = a3x3 + a2x2 + a1x + a0For each segment of the arch bound by two consecutive points (called knots), a cubic polynomial equation was defined. Here, two sets of knots were tested: five-knot (4 cubic polynomial pieces) and seven-knot splines (6 cubic polynomial pieces). The five-knot splines utilize points from second molar to second molar (beginning and end of curve), and cusp tips of the canines (the greatest curvature anteriorly), while the seven-knot spline utilizes the same points mentioned above in addition to the right and left first molars.
Hermite cubic splines
For each curve segment, a cubic Hermite spline was derived. Each cubic Hermite spline is expressed as a linear combination (blend) of the four functions:
And the associated ith cubic Hermite spline (for the ith cubic segment) is given as;
y i = y1pi + y2pi+1 + y3ti + y4ti+1Here, pi and pi+1 are the location (position) of the two end points of the ith curve segment and the ti and ti+1 are the two end-point tangents (slopes).
For a detailed description of curve-fitting methods used in this study, refer to Burden and Faires,10 Cheney and Kincaid,11 and Fausett.12
To test the reliability in digitization and imaging, 18 randomly selected sets of models were imaged again, and the same process of landmark capturing and numerical manipulation was applied. Correlation coefficients and paired t-tests were computed for the xy data to assess the reliability in both tests.
RESULTS
Effort was made to consider all known possible sources of error throughout the imaging and digitizing procedures. Table 1 shows four experiments that were used to test the different types of errors that could be produced, with the mean error value and standard deviation for both coordinates (x, y) for each of the four tests.
Table 2 shows the correlation coefficients and the significance of correlation between the original (x, y) data and the redigitized and reimaged data. The results show that the correlation was significant between both at the .000 level.
Table 3 shows the results of the paired t-test for the same set of data, which shows no significant difference when the data were redigitized.
Figures 1 and 2 are an example of the dental arch of a male subject with normal occlusion (m4001). The arch curves produced by the use of the mathematical functions (beta functions, polynomial equations, natural cubic splines, and Hermite cubic splines) tested in the current study are presented in these figures. Figure 4 shows that the fourth-order polynomial function best approximates the data points as compared with the other functions, in addition to the irregular pattern the splines produce when they approximate the same set of points. Figure 5 displays second- to twelfth-degree polynomials and also reveals better approximation of the points with the fourth-order polynomial, as compared with the lower and higher degrees of polynomials.
Citation: The Angle Orthodontist 78, 2; 10.2319/121806-516.1
Citation: The Angle Orthodontist 78, 2; 10.2319/121806-516.1
DISCUSSION
In an attempt to present a comprehensive mathematical analysis of dental arch curvature, we tested most of the functions with possible predictive potential that was suggested in the literature. Four main categories of functions were considered: the beta function, as described by Braun et al7; polynomial equations, with variations according to the highest power of the x variable (second-order polynomial through twelfth-order polynomial)46; natural cubic splines1; and Hermite cubic splines. Conic sections were not tested in this study. One of the basic requirements for the ideal method for describing the dental arch mathematically is that it should be sufficiently flexible so that it may be fitted to any size or shape of dental arch.1 Conic sections, being second-order curves, have an inherent limitation of fitting the arch to specific shapes (ie, circles, ellipses, parabolas, and hyperbolas).
Mathematical Function Analysis
In this section each individual function is discussed with respect to its behavior and its error of fit.
Natural Cubic Spines
Figure 4 illustrates the possibility of the spline to misbehave. In this case both the 5 and 7 splines misbehaved anteriorly. The seven-knot spline showed severe curvature on one side while the opposite side showed a much flatter curvature. The five-knot spline was irregular on both sides around the middle knot. It also shows that while the error at the knots is eliminated as the spline is forced to pass through them, the error at the rest of the points between the knots is great. By increasing the number of knot points used, the error will logically be reduced. However, the behavior of the spline between knots will become more erratic.
Beta Function
The beta function has two major limitations:
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It is an empirical curve based on two parameters: arch depth and molar width; therefore, it does not take the rest of the dental landmarks into consideration. Braun et al7 reported a high correlation between the measured true arch width and molar depth and the mathematically estimated arch shape data as expressed by the beta function. Yet, high correlation between the measured and curve fit values alone, does not provide evidence on the accuracy of fit of the beta function to the entire dental arch. One can imagine an unlimited number of cases with different arch shapes that are of the same depth and width. For example, the arrangement of the teeth in an arch, and/or the different mesiodistal width of the teeth, would provide two different arch shapes with the same depth and width (the first could be pointed while the other is curved).
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It has no respect for asymmetry because it is inherently a symmetrical function.
Polynomial Equations
In Figure 4, the fourth-order polynomial showed better approximation of the data points than both the splines and the beta function. The approximation is uniform around the entire data set producing a more natural curvature. Furthermore, the fourth-degree polynomial that included the odd-power terms tracks arch asymmetry. Figure 5 shows the curve fit of even-powered polynomials from second degree through twelfth degree. The choice of the degree of polynomial depends on the number of points selected. For example, in the current study where 18 points were used, lower-order polynomials produced a large amount of error, and therefore, may not be sufficiently descriptive of the dental arch. Figure 5 shows a second-order polynomial which produces pointed curves. It has a sharp curvature in the areas between the canines and as a result does not have the required flexibility to accommodate arches that are broad anteriorly. Depending on the number of points used for curve fitting, the resulting curves may reach a threshold beyond which the polynomial will start to behave erratically—this behavior being very obvious with the twelfth-order polynomial. Pepe4 suggested that the sixth-degree polynomial should be considered to describe arch form since it affords more reduction in error of fit over the fourth-degree polynomial. She tested polynomials on seven children with normal occlusion. In her study, the reduction in mean error was between 0.0048 mm2 and 1.9022 mm2 at the maximum.
For the normal arch, although the sixth-order polynomial may offer some reduction in error, both polynomials are almost equally descriptive, while for an irregular arch, the sixth degree will theoretically produce greater reduction in error. This reduction is undesirable as it is only an expression of tooth irregularity as opposed to arch form definition. That makes the fourth order universal for use in both cases, ie, regular and irregular arches. Therefore, when the purpose is to produce a natural smooth curvature of the arch, the fourth-order polynomial function is the recommended function to use.
Hermite Cubic Splines
In situations where it is intended to trace the irregularities of the arch (regardless of the general shape of the curve), such as in before and after comparative studies to evaluate treatment changes or relapse tendencies, it would be more appropriate to employ higher-degree polynomials or even spline curves to fit the dental arch more precisely. The natural cubic spline described by BeGole1 can serve this purpose. However, BeGole's original test of the natural cubic spline was on regular arches of subjects with normal occlusion. As discussed earlier, while the natural spline perfectly fits the knot points, it can become erratic between knots. This behavior tends to be more severe if the arch is irregular. Although it is intended to fit tooth displacements, it is advantageous to improve the behavior of the spline between points of interest.
On the basis of our findings, if a spline is to be used, we recommend the use of a Hermite spline instead due to its flexibility and smoothness around the knots. The blend of functions produces a “visually pleasing” interpolant of data. Such an interpolant may be more reasonable than a natural cubic spline if the data contain both steep and flat sections, ie, irregular dental arches.
Since prior publications put emphasis on error reduction, findings were mostly reported in terms of error. In this study we were interested in the overall behavior of functions when fitted to the dental arch. This issue was rarely addressed in the literature previously. Taking this into account, in addition to the introduction of the Hermite spline, we retested functions recommended in the literature to generate thorough data. The two advantages gained were: first, variables were controlled; and second, comprehensive data on different functions tested (on the same sample) were available so that intrastudy comparison became more significant than interstudy comparison.
CONCLUSIONS
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When defining the dental arch mathematically, the reduction in the amount of error should not be the only factor to consider. In fact, the more the error was reduced, the more irregular were the curvatures that were produced. Certain factors should be considered when selecting a specific mathematical function such as the objective of the study, the accuracy of the function, the number of dental landmarks used to represent the dental arch, and whether it is intended to reproduce arch asymmetry (or symmetry).
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The fourth-order polynomial was found to be a reasonable function to fit the dental arch when the objective is to describe the general smooth curvature of the arch.
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The Hermite cubic spline is more appropriate when it is desired to track arch irregularities. These splines can be useful with before and after comparative studies to evaluate treatment changes or tendencies to relapse.
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Due to its advantage in providing a more naturally smooth curve, the fourth-order polynomial function can be used to predict an individualized ideal arch for each particular patient. Therefore, it may be used as a guide to fabricate customized arch wires, or even an entire fixed orthodontic appliance system.
The imaging setting: digital camera mounted directly above the models which lie facing up on a flat platform
Digital image of each model set is placed within a 10 × 15 cm box with points of interest highlighted for clarity
Capturing of (x, y) location coordinates of pixels representing landmarks of interest
Curve fitting for male upper arch of subject (m4001)
Polynomial fitting for male upper arch of subject (m4001)
Contributor Notes
Corresponding author: Dr Eman Abdulrahman Alkofide, Division of Orthodontics, College of Dentistry, King Saud University, PO Box 60169, Riyadh, 11545, Saudi Arabia (ealkofide@hotmail.com)