November 2011
Volume 52, Issue 12
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Clinical and Epidemiologic Research  |   November 2011
Quantitative Evaluation of Changes in Eyeball Shape in Emmetropization and Myopic Changes Based on Elliptic Fourier Descriptors
Author Affiliations & Notes
  • Kotaro Ishii
    From the Department of Ophthalmology, Institute of Clinical Medicine, University of Tsukuba, Ibaraki, Japan; and
  • Hiroyoshi Iwata
    the Laboratory of Biometry and Bioinformatics, Department of Agricultural and Environmental Biology, Graduate School of Agricultural and Life Sciences, The University of Tokyo, Tokyo, Japan.
  • Tetsuro Oshika
    From the Department of Ophthalmology, Institute of Clinical Medicine, University of Tsukuba, Ibaraki, Japan; and
  • Corresponding author: Kotaro Ishii, Department of Ophthalmology, Institute of Clinical Medicine, University Hospital of Tsukuba, 2-1-1 Amakubo, Tsukuba, Ibaraki 305-8576, Japan; ishii_k@md.tsukuba.ac.jp
Investigative Ophthalmology & Visual Science November 2011, Vol.52, 8585-8591. doi:10.1167/iovs.11-7221
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      Kotaro Ishii, Hiroyoshi Iwata, Tetsuro Oshika; Quantitative Evaluation of Changes in Eyeball Shape in Emmetropization and Myopic Changes Based on Elliptic Fourier Descriptors. Invest. Ophthalmol. Vis. Sci. 2011;52(12):8585-8591. doi: 10.1167/iovs.11-7221.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose.: To evaluate changes in eyeball shape in emmetropization and myopic changes using magnetic resonance imaging (MRI) and elliptic Fourier descriptors (EFDs).

Methods.: The subjects were 105 patients (age range, 1 month–19 years) who underwent head MRI. The refractive error was determined in 30 patients, and eyeball shape was expressed numerically by principal components analysis of standardized EFDs.

Results.: In the first principal component (PC1; the oblate-to-prolate change), the proportion of variance/total variance in the development of the eyeball shape was 76%. In all subjects, PC1 showed a significant correlation with age (Pearson r = −0.314; P = 0.001), axial length (AL, r = −0.378; P < 0.001), width (r = −0.200, P = 0.0401), oblateness (r = 0.657, P < 0.001), and spherical equivalent refraction (SER, r = 0.438; P = 0.0146; n = 30). In the group containing patients aged 1 month to 6 years (n = 49), PC1 showed a significant correlation with age (r = −0.366; P = 0.0093). In the group containing patients aged 7 to 19 years (n = 56), PC1 showed a significant correlation with SER (r = 0.640; P = 0.0063).

Conclusions.: The main deformation pattern in the development of the eyeball shape from oblate to prolate was clarified by quantitative analysis based on EFDs. The results showed clear differences between age groups with regard to changes in the shape of the eyeball, the correlation between these changes, and refractive status changes.

Peripheral visual input plays an important role in emmetropization, and the quality of the peripheral visual input may affect postnatal refractive eye development in patients with myopia. However, changes in the shape of the eyeball during normal and pathologic postnatal eye development are still poorly understood. 1 3 A previous report described that when the angle on the retinal side of the extrafoveal area changes by 1°, the contrast of the retinal image changes. 4 Therefore, a quantitative evaluation of the shape of the eyeball is necessary to understand eye growth during infancy. 
Until now, few studies have been conducted on the shape of the eyeball itself. 1,5 7 Deller et al. 5 reported that the dimensions of most emmetropic eyes are similar, but that the rate of increase of length is approximately twice that of height and width as myopia increases. Atchison et al. 6 suggested that there are considerable individual variations in adult eyes. In general, myopic eyes are elongated relative to emmetropic eyes, with greater growth in length than height and even lesser growth in width. Moreover, some researchers determined that the variations are quantitative by using the Q-value of the shape of the surface of the adult retina. 8 Their study results showed that in both emmetropia and myopia, the retinal shape is oblate and that this oblateness is decreased according to the strength of the myopia. 
The current method of evaluating the shape of the eyeball uses the ratio of the length, radius of curvature, angle of the tangent on the retinal surface, and Q-value. 1,4 8 However, limitations in the evaluation of partial features and problems with the inaccuracy and lack of reproducibility are persistent with the use of these evaluation methods. In addition, the Q-value is not suitable for the evaluation of complex shapes, such as the shape of the entire eyeball. Therefore, it is necessary to introduce a method that can mathematically achieve a quantitative evaluation of an outline of the shape of the eyeball. 
Elliptic Fourier descriptors (EFDs), initially proposed by Kuhl and Giardina 9 in 1982, can numerically express any shape with a closed two-dimensional contour. There have been reports showing that measurements based on EFDs are helpful in the quantitation of plant and animal organ shapes. 10 17 Despite its potential, however, there have been relatively few reports on the use of EFDs in shape analysis within the field of clinical medicine. In our study, we used eyeball images taken by magnetic resonance imaging (MRI) to conduct a quantitative analysis of eyeball shape development based on EFDs. 
Materials and Methods
Informed consent was obtained from all patients participating in the present study or their guardians. The approval of the hospital's ethics committee was also obtained. The study was conducted in accordance with the guidelines of the Declaration of Helsinki. The study sample refers to 266 cases (ranging in age from 1 month to 19 years) in which magnetic resonance images (MRIs) of the head were obtained at regular physical checkups during a 6-month period from April to October 2008 at Tsukuba University Hospital. The exclusion criteria for analysis were intracranial space-occupying lesions, cerebral hemorrhage, cerebral contusions, hydrocephalus, congenital abnormalities, chromosomal abnormalities, metabolic diseases, and degenerative diseases, and the criteria were specified as the cases of those patients for whom the MRI results constituted “no abnormality.” As a result, 105 cases were included in the analysis, 30 of which were evaluated for refractive error. The refractive error was measured with an autorefractometer in subjects under cycloplegia with atropine sulfate or cyclopentolate hydrochloride. In subjects younger than 6 years, 0.5% atropine sulfate was applied with eyedrops. In addition, the MRIs and refractive error of seven subjects without ocular or intracranial diseases were measured for MRI validation. 
MRIs were obtained with a 1.5-Tesla superconducting magnet (Gyroscan Intera and Gyroscan Power Trak 1000; Philips Medical Systems, Best, The Netherlands) with a phased-array head coil. Horizontal T1- and T2-weighted images were obtained for all patients. T2-weighted images were obtained using the fast spin-echo method, with a pixel bandwidth of 100 to 150 kH, repetition time (TR) of 2500 to 2948 ms, echo time (TE) of 90 to 130 ms, section thickness of 1.2 to 3 mm, field of view of 70 × 70- to 95 × 95-mm, image frequency of 63.9, pixel spacing of 0.429 to 0.625 pixel/mm, flip angle of 90°, and an acquisition matrix of 256 × 256 to 320 × 320. T1-weighted images were obtained with a spin-echo sequence with a TR of 480 to 574 ms, TE of 14 to 18 ms, section thickness of 1.2 to 3 mm, image frequency of 63.9, pixel spacing of 0.313 to 0.429 pixel/mm, and flip angle of 90°. 
All patients were imaged while supine, and those aged 6 years or younger were sedated. All ocular dimension measurements were made from the horizontal images at approximately 16 magnifications on a computer monitor with a standard resolution of 512 × 512 pixels. The distances were measured with a line caliper in a software program, and the distance between the two points was converted from pixels to millimeters by using pixel-spacing data. Axial length (AL) and width of the right eye were measured from the horizontal MRIs. As the T2-weighted images were used to apply EFDs, the shaft length of the inside of the eyeball was measured in the AL measurements. AL was recorded as the distance between the posterior cornea and the approximate location of the fovea along the line that bisected the eye in the horizontal plane. The eye width was measured between the retinal surfaces on either side across the horizontal image at the point that visually appeared the widest. 
MRI Validation
As our examinations were part of a retrospective study and the MRIs of the infants in that study were taken with the subjects under sedation, it was not possible to fix the visual line on these images. Therefore, a technique using three-dimensional (3D) MRIs of the eyeball was used to obtain the horizontal MRIs of the eyeball in all analyses at an equal height. We used the unprocessed horizontal MRIs from approximately 7 to 15 unprocessed slices to make the 3D MRI of the eyeball (Figs. 1A, 1B). The x-, y-, and z-axes of these 3D images have been adjusted respectively (ZedView software program; Lexi Corp., Tokyo, Japan), as shown in Figure 1
Figure 1.
 
(A) In the unprocessed MRI, it was confirmed that the bilateral lenses, optic nerves, medial and lateral rectus muscles, and optic chiasm were depicted clearly in the horizontal images. (B) A 3D restructuring of the head MRIs from the horizontal MRI of a 3-mm slice. (C) To adjust the working reference plane, the optic chiasm of the unprocessed MRI was set to the intersection graph. The y-axis was set in parallel with both eyeballs. In addition, the x-axis was set orthogonal to the y-axis. The z-axis was vertically set as the working reference plane, orthogonal to the x- and y-axes. The x-axis was rotated to create a shape that ensured that the crystalline lenses of both eyes appeared the same. Then the y-axis was rotated so that the anterior chamber is at its greatest depth. Finally, the z-axis was moved up and down so that the crystalline lens might attain its maximum size; (D) A restructured horizontal MRI of patient 4 (Table 1, 73 years, female).
Figure 1.
 
(A) In the unprocessed MRI, it was confirmed that the bilateral lenses, optic nerves, medial and lateral rectus muscles, and optic chiasm were depicted clearly in the horizontal images. (B) A 3D restructuring of the head MRIs from the horizontal MRI of a 3-mm slice. (C) To adjust the working reference plane, the optic chiasm of the unprocessed MRI was set to the intersection graph. The y-axis was set in parallel with both eyeballs. In addition, the x-axis was set orthogonal to the y-axis. The z-axis was vertically set as the working reference plane, orthogonal to the x- and y-axes. The x-axis was rotated to create a shape that ensured that the crystalline lenses of both eyes appeared the same. Then the y-axis was rotated so that the anterior chamber is at its greatest depth. Finally, the z-axis was moved up and down so that the crystalline lens might attain its maximum size; (D) A restructured horizontal MRI of patient 4 (Table 1, 73 years, female).
The initial experiments confirmed that the bilateral lenses, optic nerves, medial and lateral rectus muscles, and optic chiasm were depicted clearly in the unprocessed horizontal MRIs. In addition, a slice was made in the working reference plane of the rotation. The center of the rotation axis used the optic chiasm to facilitate identification of the position of the eyeball and grasp of the rotation state. The y-axis was set parallel to both eyeballs, as shown in Figure 1C. In addition, the x-axis was set orthogonal to the y-axis. The z-axis was vertically set as the working reference plane, orthogonal to the x- and y-axes. To create a shape that ensured that the crystalline lenses of both eyes appeared the same, the x-axis was rotated. Then the y-axis was rotated so that the anterior chamber is at its greatest depth. Finally, the z-axis was moved up and down so that the crystalline lens could attain its maximum size. In this study, the MRIs from the 1.2- to 3-mm slices were used. 
In the evaluation and examination, the measurement of AL by A-scan ultrasonography (AL-3000; TOMEY, Nagoya, Japan) was compared in seven subjects (14 eyes) with that based on the MRI. The MRI of the 3-mm slice was used for evaluation and examination to ensure an error margin for the region. Because the AL based on the MRI did not contain the corneal thickness, AL based on A-scan ultrasonography was a value from which the corneal thickness had been subtracted. The results of the MRI validation are shown in Table 1. AL determined by A-scan ultrasonography showed a significant correlation with that generated by the unprocessed MR (Pearson r = 0.956; P < 0.001) and 3D restructured (r = 0.966; P < 0.001) images. A Bland-Altman plot, indicating the relationship between AL generated by the unprocessed MR and 3D restructured images with that based on A-scan ultrasonography was made, as shown in Figure 2. Because the horizontal image of the unprocessed MRI did not usually cut an eyeball at a major axis, AL determined from the unprocessed MRI was shorter than the actual length (Table 1). In addition, there was a significant difference between the error margin of AL determined by a combination of A-scan ultrasonography and the 3D-restructured image and the error margin of AL determined by a combination of A-scan ultrasonography and the unprocessed MRI (paired t-test, P = 0.0191). In this study, therefore, we used the T2-weighted horizontal MRIs that used the 3D restructuring technique to estimate the geometry of the eyeball. 
Table 1.
 
A Comparison of Axial Lengths Determined by MRI and A-scan Ultrasonography
Table 1.
 
A Comparison of Axial Lengths Determined by MRI and A-scan Ultrasonography
Participant Age, Sex Eye Mean SER (D) A-scan US AL (mm) MRI-AL (mm) 3D-AL (mm) (A-scan US AL) Minus (MRI-AL) (A-scan US AL) Minus (3D-AL)
56, female Right −0.75 23.53 23.31 23.5 0.22 0.03
Left −0.5 23.39 23.08 23.5 0.31 −0.11
73, male Right −1.25 25.31 24.41 24.8 0.9 0.51
Left −0.75 25.21 24.8 25 0.41 0.21
63, female Right 0.5 22.9 22.86 22.9 0.04 0
Left 0.75 22.83 22.64 23.1 0.19 −0.27
73, female Right −2.5 23.84 23.2 23.7 0.64 0.14
Left −1.75 23.57 22.9 23.6 0.67 −0.03
72, male Right −1.5 23.39 23.18 23.3 0.21 0.09
Left −1.75 23.57 23.43 23.1 0.14 0.47
30, male Right −2.25 25.31 25.48 25.1 −0.17 0.21
Left −2.00 25.21 24.96 25.6 0.25 −0.39
62, female Right 0.75 22.95 22.87 22.9 0.08 0.05
Left 0.25 23.12 22.65 22.9 0.47 0.22
Mean ± SD 23.87 ± 0.96 23.56 ± 0.94 23.79 ± 0.93 0.31 ± 0.28 0.08 ± 0.25
Figure 2.
 
Bland-Altman plots indicating the relationship between AL generated by the unprocessed MR and 3D restructured images and that based on A-scan ultrasonography. A-scan US, A-scan ultrasonography. (A) The mean value of the difference between the A-scan-US AL and MRI AL was 0.31. The average difference ±1.96 SD was from −0.24 to 0.87. (B) The mean value of the difference between the A-scan-US AL and 3D AL was 0.081. The average difference ±1.96 SD was from −0.40 to 0.57.
Figure 2.
 
Bland-Altman plots indicating the relationship between AL generated by the unprocessed MR and 3D restructured images and that based on A-scan ultrasonography. A-scan US, A-scan ultrasonography. (A) The mean value of the difference between the A-scan-US AL and MRI AL was 0.31. The average difference ±1.96 SD was from −0.24 to 0.87. (B) The mean value of the difference between the A-scan-US AL and 3D AL was 0.081. The average difference ±1.96 SD was from −0.40 to 0.57.
Elliptic Fourier Descriptors
To obtain a sharply defined contour of the eyeball, the T2-weighted MRIs were used for elliptic Fourier descriptors (EFDs; Fig. 3A). In all cases, images of the right eyes were used for image processing and contour recording. In the derivation of EFDs, the x- and y-coordinate information of the contour of eyeball was taken in the form of a closed curve, as is illustrated in Figure 3B. The shape was mathematically delineated with the Fourier series expansion of the closed curve. The obtained Fourier coefficients (an , bn , cn , and dn ) were further standardized to be invariant under a change of size and direction of a contour and starting point of contour trace. In this study, standardization was done on the basis of the two landmark points that were used in axial length measurements. In standardizing EFDs, the corneal apex of the right eye was placed on the right, the temporal side on the top, and the nasal side at the bottom (Fig. 3C). 
Figure 3.
 
(A) In reference to the T2-weighted MRIs that were obtained of the right eye. (B) The closed curve on the graph refers to the seam of the contour that was image processed from the eyeball images derived via MRI. The x and y coordinates of the contour are delineated, respectively, as periodic functions of distance t with a period of T. (C) The location of point E* on the graph, which is a reference point for standardizing the elliptic Fourier descriptors obtained in (B) was identified manually. With this standardization, the size and direction of all contours were aligned, respectively, to the size and direction of the corneal apex on the right eyeball. (D) A PCA was performed based on the variance–covariance matrix of the coefficients.
Figure 3.
 
(A) In reference to the T2-weighted MRIs that were obtained of the right eye. (B) The closed curve on the graph refers to the seam of the contour that was image processed from the eyeball images derived via MRI. The x and y coordinates of the contour are delineated, respectively, as periodic functions of distance t with a period of T. (C) The location of point E* on the graph, which is a reference point for standardizing the elliptic Fourier descriptors obtained in (B) was identified manually. With this standardization, the size and direction of all contours were aligned, respectively, to the size and direction of the corneal apex on the right eyeball. (D) A PCA was performed based on the variance–covariance matrix of the coefficients.
With regard to the harmonic number n, the larger the maximum value of n becomes, the better the descriptive power of the shape. In this analysis, the value of n was set at 20 (n = 20). When n = 20, the standardized EFDs comprise 80 standardized Fourier coefficients. Since the number of coefficients was large, it was not easy to analyze the variation of each coefficient and understand the results of the analysis. To summarize the information contained in the EFDs, we conducted a principal component analysis (PCA) of EFDs. The PCA reduced the dimension of the original data (i.e., 80) to a much lower dimension. The PCA was performed based on the variance–covariance matrix of the coefficients, and the component scores of the first several components were used as the measurements of the eyeball shape (Fig. 3D). The shape analysis method that uses standardized EFDs evaluates the shape provided from an eyeball image (such as Fig. 3A) by several shape principal components (s 1, s 2, s 3, …), as shown in Figure 3D. 
Evaluation of Oblateness
Oblateness indicates the shape of a spheroid body compared with that of a sphere. Oblateness f is defined as f = 1 − AL/a, where a is the equatorial diameter. For an oblate spheroid for which a > AL, f > 0, and for a prolate spheroid for which a < AL, f < 0. For a sphere, f = 0. 
Statistical Analyses
The measurements of the eye dimensions of AL and width were recorded in millimeters and are expressed as the mean ± standard deviation. The Pearson correlation coefficients, determined using a bivariate correlation analysis, were used to compare factors such as AL, width, age, principal components of standardized EFDs, and oblateness. P ≤ 0.05 was significant (StatView ver. 5.0; SAS, Cary, NC). 
Results
There were 105 patients (105 eyes) enrolled in the study; they consisted of 55 males and 50 females, ranging in age from 1 month to 19 years (8.47 ± 6.64 years). The AL (Pearson r = 0.794; P < 0.001) and width (r = 0.754; P < 0.001) of the right eyes showed a significant correlation with age. AL and age were closely approximated via a logarithmic approximation. AL (Pearson r = −0.443; P < 0.001) and age (r = −0.356; P = 0.002) showed a significant correlation with oblateness (Figs. 4A, 4B). The average of the spherical equivalent refraction (SER) was −0.68 ± 1.19 D (n = 30). The SER showed a significant correlation with AL (r = −0.695, P < 0.001), width (r = −0.547; P = 0.0014), oblateness (r = 0.423; P = 0.0189), and age (r = −0.682; P < 0.001). 
Figure 4.
 
Scatterplots indicating the relationship between oblateness, age, axial length, and PC1 and -2 (A, B). Age (Pearson r = −0.356; P = 0.002) and axial length (r = −0.443; P < 0.001) showed a significant correlation with oblateness. (C) PC1(r = 0.657; P < 0.001) and (D) PC2 (r = −0.289; P = 0.0027) showed a significant correlation with oblateness. PC1 and -2 almost intersected oblateness in the origin. Therefore, the eyeball shape of the mean value of PC1 and -2 was approximately a sphere.
Figure 4.
 
Scatterplots indicating the relationship between oblateness, age, axial length, and PC1 and -2 (A, B). Age (Pearson r = −0.356; P = 0.002) and axial length (r = −0.443; P < 0.001) showed a significant correlation with oblateness. (C) PC1(r = 0.657; P < 0.001) and (D) PC2 (r = −0.289; P = 0.0027) showed a significant correlation with oblateness. PC1 and -2 almost intersected oblateness in the origin. Therefore, the eyeball shape of the mean value of PC1 and -2 was approximately a sphere.
A PCA was performed based on the variance–covariance matrix of the coefficients (Fig. 3D). Therefore, each principal component was independent of the shape without the correlation. In the PCA of standardized EFDs, the proportion of the variance/total variance of the first principal component (PC1) was 76.0%. The proportions of PC2 and PC3 were 7.7% and 4.1%, respectively. Figure 5 shows the changes in shape variations within the value range of −2 to 2 SD for the three principal components. The solid line in Figure 5 indicates the average value, and the numerical value is set at 0 in the PCA. The dotted line represents −2 SD, and the dashed line represents 2 SD. Regarding the components following PC4, their ratio to the eyeball shape was small, and the change was minute. As a result, it was found that PC1 showed a significant correlation with age (Pearson r = −0.314; P = 0.001) as well as with AL (r = −0.378; P < 0.001), width (r = −0.200; P = 0.0401), and oblateness (r = 0.657; P < 0.001; Fig. 4C). PC1 also was significantly correlated with SER (r = 0.438, P = 0.0146; n = 30). PC2 showed a significant correlation with oblateness (Pearson r = −0.289, P = 0.0027), but not AL (Fig. 4D). PC1 and -2 almost intersected oblateness in the origin (Figs. 4C, 4D). Therefore, the eyeball shape of the mean value of PC1 and -2 was approximately spherical. 
Figure 5.
 
Eyeball images where the PCA by standardized EFDs was visualized. Shape variations were accounted for by the first three principal components. Contours drawn in solid lines are the average shape, and the numerical value is set to 0 in the PCA. Contours drawn in dotted and dashed lines correspond to shapes having the component scores of −2 and 2 SD, respectively.
Figure 5.
 
Eyeball images where the PCA by standardized EFDs was visualized. Shape variations were accounted for by the first three principal components. Contours drawn in solid lines are the average shape, and the numerical value is set to 0 in the PCA. Contours drawn in dotted and dashed lines correspond to shapes having the component scores of −2 and 2 SD, respectively.
Emmetropization is usually completed by about age 6 years. For those subjects in the group aged 1 month to 6 years (n = 49), AL (Pearson r = 0.733; P < 0.001), width (r = 0.681; P < 0.001), and oblateness (r = −0.309; P = 0.0301) showed a significant correlation with age. PC1 showed a significant correlation with AL (r = −0.421; P = 0.0024), oblateness (r = −0.715; P < 0.001), and age (r = −0.366; P = 0.0093), whereas PC2 only showed a significant correlation with oblateness (r = −0.355; P = 0.0118). The average SER of the younger subjects was 0.09 ± 0.75 D (age, 1 month–6 years; n = 14). For those subjects aged 7 to 19 years (n = 56), AL (r = 0.459; P = 0.003) and width (r = 0.312; P = 0.0187) showed a significant correlation with age. This result seems to reflect the trend of increasing myopia with increasing age within this subject population. PC1 showed a significant correlation with oblateness (r = −0.524; P < 0.001). The average SER of the young adult subjects was −1.36 ± 1.11 D (age, 7–19 years; n = 16). PC1 showed a significant correlation with SER (r = 0.640; P = 0.0063), as did oblateness (r = 0.534, P = 0.0317). The correlations between PC1 and -2 in the two age groups and AL, width, oblateness, age, and SER are summarized in Table 2. Figure 6 shows the scatterplots comparing PC1 with age and SER in the two age groups. 
Table 2.
 
Correlation Coefficients between PCA of Eyeball Shape and Geometric Data, Age, and Refraction in the Two Age Groups
Table 2.
 
Correlation Coefficients between PCA of Eyeball Shape and Geometric Data, Age, and Refraction in the Two Age Groups
Age Group Principal Component AL Width Oblateness Age SER (n = 30)
1 mo–6 y (n = 49) PC1 −0.421* −0.205 0.715† −0.366* 0.015
PC2 −0.134 −0.266 −0.355‡ −0.108 −0.251
7–19 y (n = 51) PC1 −0.200 0.227 0.524† −0.173 0.640*
PC2 0.232 0.097 −0.161 −0.147 0.329
Figure 6.
 
Scatterplots indicating the relationship of PC1 with age and SER in the two age groups. (A, C) In those subjects aged 1 month to 6 years, PC1 showed a significant negative correlation with age (Pearson r = −0.366; P = 0.0093), but not with SER. (B, D) In those subjects aged 7 to 19 years, PC1 showed a significant correlation with SER (r = 0.640; P = 0.0063), but not with age.
Figure 6.
 
Scatterplots indicating the relationship of PC1 with age and SER in the two age groups. (A, C) In those subjects aged 1 month to 6 years, PC1 showed a significant negative correlation with age (Pearson r = −0.366; P = 0.0093), but not with SER. (B, D) In those subjects aged 7 to 19 years, PC1 showed a significant correlation with SER (r = 0.640; P = 0.0063), but not with age.
Discussion
To quantitatively evaluate the patterns of development of the shape of the eyeball, we performed PCA using standardized EFDs from the MRIs of 105 right eyes. The significance of PC1 and -2 in such development, which can be understood by visualizing the aforementioned PCA, is shown in Figure 5. Such visualization helps us to understand the morphologic meaning of each principal component axis, which is not easy to do in a multivariate analysis. 17 In addition, visualization may give us an idea for a novel shape characteristic that has not been evaluated so far. 
PC1 is the width expansion and contraction. The oblateness showed a significant correlation with AL and PC1 (Figs. 4B, 4C). In addition, PC1 showed a significant correlation with age. Therefore, the deformation pattern of PC1 in the development of the shape of the eyeball was thought to be a change from oblate to prolate (i.e., a gradual change in the eyeball to a spherical shape from an oblate one and to a prolate shape from a sphere). This result shows the extension of AL to be more dominant than the extension of the width in the development of the eyeball. While the findings associated with myopic changes were consistent with those of many previous reports, 1,5 7 few studies have addressed the eyeball transformation pattern in emmetropization, most likely because most evaluations of the eyeball shape of emmetropia have been conducted during later childhood. 1,7  
PC2 is the posterior pole elongation (Fig. 5). It should be noted that the oblate-to-prolate change cannot be evaluated accurately in terms of oblateness alone. However, oblateness was correlated with both PC1 and -2 according to this examination (Fig. 4). Therefore, since oblateness is an evaluation made using the ratio, the true oblate-to-prolate change and posterior pole elongation cannot be distinguished. That the difference of the shape variation cannot be evaluated by the ratio, but can be distinguished, is one of the advantages of using EFDs in this research. 
A major advantage of shape analysis based on the principal components of EFDs is that it requires no prior knowledge about morphologic variations of analysis objects. As suggested by PC2 estimated in this study, we were able to discover a novel shape characteristic and measure it via the shape analysis without any prior knowledge. On the other hand, in analyses based on conventional shape characteristics, we should identify characteristics that are appropriate for evaluating analysis objects by referring to previous studies before taking measurements (e.g., the ratio of the length and the radius of curvature). 1,5 7 If the established characteristics are not appropriate for our analysis objects, the shape variations of those objects will not be adequately evaluated. It should be noted that EFDs are not necessarily summarized by PCA alone. Other multivariate analysis methods may also be useful to extract significant information for our study objectives from EFDs. To evaluate local shape characteristics, such as the shape of the corneal or retinal surfaces, more precisely, a different approach may be necessary. One example of such an evaluation method is measuring the angle of the tangent on the retinal surface and the Q-value. 4,8  
The main result of this study is that there are clear differences between age groups with regard to changes in the shape of the eyeball, the correlation between these changes, and changes in refractive status. At 6 years of age, when emmetropization is generally complete, PC1 showed a significant negative correlation with age, but not with SER (Figs. 6A, 6C). Previous studies have generally assumed that visual signals processed in the fovea dominate the emmetropization process and are the genesis of common refractive errors in children. 18 From our results, it was not possible to explain changes in the shape of the eyeball during the emmetropization period by using the refraction value alone. Therefore, the pattern of development of the shape of the eyeball during emmetropization was thought to be a fixed form change from the oblate shape to the sphere (i.e., oblateness is 0 at approximately 6 years of age; Fig. 4A). During the period from age 7 to 19 years, PC1 showed a significant correlation with SER, but not with age (Figs. 6B, 6D). To explain the fact that the oblate-to-prolate changes did not correlate with age, we proposed that the main eyeball shape transformation after age 7 is global expansion. In a past report, the shape of the eyeball in adult eyes was almost the same as that described for emmetropia. 5 In contrast, another past study indicated that myopic eyes in children are typically prolate. 1 Our results showed that the development of myopia was due to the oblate-to-prolate change that occurred during the period from 7 to 19 years of age (Fig. 6D). 
Recent studies have suggested that the mouse eye grows in two phases—that is, a period of rapid growth that lasts until postnatal day (P)40 to P60 and a period of very slow eye expansion that continues up to P300. 19,20 When the evaluation of the crystalline lens and corneal radius of curvature were added, Tkatchenko et al. 21 suggested that the mouse eye grows in three phases. In the human eye, AL has a growth pattern approximated in the logarithmic function, with a rapid growth phase up to 6 years of age followed by a slow growth phase from 6 years onward. 22,23 Our research results suggest that there is a difference in the development of the shape of the eyeball between these two phases of growth. In addition, the change in the eyeball shape did not essentially correlate with age in the young adult subjects. On the other hand, the change from oblate to prolate was shown in the myopic subjects during the period from 7 to 19 years of age. Therefore, it was suggested that two pattern changes in the eyeball shape exist in the young adult subjects. 
There were some limitations in this study. One constraint was the possibility that the sample data were biased because the subjects were not prospectively selected randomly. Another potential limitation is that, since only the horizontal images were examined, the overall height of the eyeball was not known. It is necessary to examine the MRI in a slice of 1 mm or smaller to restructure the sagittal image with the horizontal image. 24 Because this examination was part of a retrospective study, it was difficult to obtain other data. In addition, the cases that had an appreciable refractive error were limited. If it had been possible to examine patients by subdividing them into those with emmetropia and those with myopia, a more interesting result might have been obtained. 
In conclusion, the main deformation pattern in the development of the shape of the eyeball from oblate to prolate was clarified by quantitative analysis based on EFDs. Our findings suggest that there are differences between age groups with regard to changes in the shape of the eyeball, the correlation between these changes, and changes in refractive status. Previous studies were unable to evaluate complex shapes, such as that of the entire eyeball. This research clearly distinguished two or more deformation patterns of the eyeball shape that could not be distinguished by the aspect ratio (i.e., the oblate-to-prolate change, the posterior pole elongation, and the global expansion). We believe that our new technique based on EDFs, which allows quantitative evaluation of the shape, is effective for research on emmetropization and myopic changes and will ultimately serve as a useful tool in the field of ophthalmology. 
Footnotes
 Disclosure: K. Ishii, None; H. Iwata, None; T. Oshika, None
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Figure 1.
 
(A) In the unprocessed MRI, it was confirmed that the bilateral lenses, optic nerves, medial and lateral rectus muscles, and optic chiasm were depicted clearly in the horizontal images. (B) A 3D restructuring of the head MRIs from the horizontal MRI of a 3-mm slice. (C) To adjust the working reference plane, the optic chiasm of the unprocessed MRI was set to the intersection graph. The y-axis was set in parallel with both eyeballs. In addition, the x-axis was set orthogonal to the y-axis. The z-axis was vertically set as the working reference plane, orthogonal to the x- and y-axes. The x-axis was rotated to create a shape that ensured that the crystalline lenses of both eyes appeared the same. Then the y-axis was rotated so that the anterior chamber is at its greatest depth. Finally, the z-axis was moved up and down so that the crystalline lens might attain its maximum size; (D) A restructured horizontal MRI of patient 4 (Table 1, 73 years, female).
Figure 1.
 
(A) In the unprocessed MRI, it was confirmed that the bilateral lenses, optic nerves, medial and lateral rectus muscles, and optic chiasm were depicted clearly in the horizontal images. (B) A 3D restructuring of the head MRIs from the horizontal MRI of a 3-mm slice. (C) To adjust the working reference plane, the optic chiasm of the unprocessed MRI was set to the intersection graph. The y-axis was set in parallel with both eyeballs. In addition, the x-axis was set orthogonal to the y-axis. The z-axis was vertically set as the working reference plane, orthogonal to the x- and y-axes. The x-axis was rotated to create a shape that ensured that the crystalline lenses of both eyes appeared the same. Then the y-axis was rotated so that the anterior chamber is at its greatest depth. Finally, the z-axis was moved up and down so that the crystalline lens might attain its maximum size; (D) A restructured horizontal MRI of patient 4 (Table 1, 73 years, female).
Figure 2.
 
Bland-Altman plots indicating the relationship between AL generated by the unprocessed MR and 3D restructured images and that based on A-scan ultrasonography. A-scan US, A-scan ultrasonography. (A) The mean value of the difference between the A-scan-US AL and MRI AL was 0.31. The average difference ±1.96 SD was from −0.24 to 0.87. (B) The mean value of the difference between the A-scan-US AL and 3D AL was 0.081. The average difference ±1.96 SD was from −0.40 to 0.57.
Figure 2.
 
Bland-Altman plots indicating the relationship between AL generated by the unprocessed MR and 3D restructured images and that based on A-scan ultrasonography. A-scan US, A-scan ultrasonography. (A) The mean value of the difference between the A-scan-US AL and MRI AL was 0.31. The average difference ±1.96 SD was from −0.24 to 0.87. (B) The mean value of the difference between the A-scan-US AL and 3D AL was 0.081. The average difference ±1.96 SD was from −0.40 to 0.57.
Figure 3.
 
(A) In reference to the T2-weighted MRIs that were obtained of the right eye. (B) The closed curve on the graph refers to the seam of the contour that was image processed from the eyeball images derived via MRI. The x and y coordinates of the contour are delineated, respectively, as periodic functions of distance t with a period of T. (C) The location of point E* on the graph, which is a reference point for standardizing the elliptic Fourier descriptors obtained in (B) was identified manually. With this standardization, the size and direction of all contours were aligned, respectively, to the size and direction of the corneal apex on the right eyeball. (D) A PCA was performed based on the variance–covariance matrix of the coefficients.
Figure 3.
 
(A) In reference to the T2-weighted MRIs that were obtained of the right eye. (B) The closed curve on the graph refers to the seam of the contour that was image processed from the eyeball images derived via MRI. The x and y coordinates of the contour are delineated, respectively, as periodic functions of distance t with a period of T. (C) The location of point E* on the graph, which is a reference point for standardizing the elliptic Fourier descriptors obtained in (B) was identified manually. With this standardization, the size and direction of all contours were aligned, respectively, to the size and direction of the corneal apex on the right eyeball. (D) A PCA was performed based on the variance–covariance matrix of the coefficients.
Figure 4.
 
Scatterplots indicating the relationship between oblateness, age, axial length, and PC1 and -2 (A, B). Age (Pearson r = −0.356; P = 0.002) and axial length (r = −0.443; P < 0.001) showed a significant correlation with oblateness. (C) PC1(r = 0.657; P < 0.001) and (D) PC2 (r = −0.289; P = 0.0027) showed a significant correlation with oblateness. PC1 and -2 almost intersected oblateness in the origin. Therefore, the eyeball shape of the mean value of PC1 and -2 was approximately a sphere.
Figure 4.
 
Scatterplots indicating the relationship between oblateness, age, axial length, and PC1 and -2 (A, B). Age (Pearson r = −0.356; P = 0.002) and axial length (r = −0.443; P < 0.001) showed a significant correlation with oblateness. (C) PC1(r = 0.657; P < 0.001) and (D) PC2 (r = −0.289; P = 0.0027) showed a significant correlation with oblateness. PC1 and -2 almost intersected oblateness in the origin. Therefore, the eyeball shape of the mean value of PC1 and -2 was approximately a sphere.
Figure 5.
 
Eyeball images where the PCA by standardized EFDs was visualized. Shape variations were accounted for by the first three principal components. Contours drawn in solid lines are the average shape, and the numerical value is set to 0 in the PCA. Contours drawn in dotted and dashed lines correspond to shapes having the component scores of −2 and 2 SD, respectively.
Figure 5.
 
Eyeball images where the PCA by standardized EFDs was visualized. Shape variations were accounted for by the first three principal components. Contours drawn in solid lines are the average shape, and the numerical value is set to 0 in the PCA. Contours drawn in dotted and dashed lines correspond to shapes having the component scores of −2 and 2 SD, respectively.
Figure 6.
 
Scatterplots indicating the relationship of PC1 with age and SER in the two age groups. (A, C) In those subjects aged 1 month to 6 years, PC1 showed a significant negative correlation with age (Pearson r = −0.366; P = 0.0093), but not with SER. (B, D) In those subjects aged 7 to 19 years, PC1 showed a significant correlation with SER (r = 0.640; P = 0.0063), but not with age.
Figure 6.
 
Scatterplots indicating the relationship of PC1 with age and SER in the two age groups. (A, C) In those subjects aged 1 month to 6 years, PC1 showed a significant negative correlation with age (Pearson r = −0.366; P = 0.0093), but not with SER. (B, D) In those subjects aged 7 to 19 years, PC1 showed a significant correlation with SER (r = 0.640; P = 0.0063), but not with age.
Table 1.
 
A Comparison of Axial Lengths Determined by MRI and A-scan Ultrasonography
Table 1.
 
A Comparison of Axial Lengths Determined by MRI and A-scan Ultrasonography
Participant Age, Sex Eye Mean SER (D) A-scan US AL (mm) MRI-AL (mm) 3D-AL (mm) (A-scan US AL) Minus (MRI-AL) (A-scan US AL) Minus (3D-AL)
56, female Right −0.75 23.53 23.31 23.5 0.22 0.03
Left −0.5 23.39 23.08 23.5 0.31 −0.11
73, male Right −1.25 25.31 24.41 24.8 0.9 0.51
Left −0.75 25.21 24.8 25 0.41 0.21
63, female Right 0.5 22.9 22.86 22.9 0.04 0
Left 0.75 22.83 22.64 23.1 0.19 −0.27
73, female Right −2.5 23.84 23.2 23.7 0.64 0.14
Left −1.75 23.57 22.9 23.6 0.67 −0.03
72, male Right −1.5 23.39 23.18 23.3 0.21 0.09
Left −1.75 23.57 23.43 23.1 0.14 0.47
30, male Right −2.25 25.31 25.48 25.1 −0.17 0.21
Left −2.00 25.21 24.96 25.6 0.25 −0.39
62, female Right 0.75 22.95 22.87 22.9 0.08 0.05
Left 0.25 23.12 22.65 22.9 0.47 0.22
Mean ± SD 23.87 ± 0.96 23.56 ± 0.94 23.79 ± 0.93 0.31 ± 0.28 0.08 ± 0.25
Table 2.
 
Correlation Coefficients between PCA of Eyeball Shape and Geometric Data, Age, and Refraction in the Two Age Groups
Table 2.
 
Correlation Coefficients between PCA of Eyeball Shape and Geometric Data, Age, and Refraction in the Two Age Groups
Age Group Principal Component AL Width Oblateness Age SER (n = 30)
1 mo–6 y (n = 49) PC1 −0.421* −0.205 0.715† −0.366* 0.015
PC2 −0.134 −0.266 −0.355‡ −0.108 −0.251
7–19 y (n = 51) PC1 −0.200 0.227 0.524† −0.173 0.640*
PC2 0.232 0.097 −0.161 −0.147 0.329
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