Quantification of Retinal Nerve Fiber and Retinal Artery Trajectories Using Second-Order Polynomial Equation and Its Association With Axial Length

Abstract

Purpose.: To determine whether a second-degree polynomial equation can fit the retinal nerve fiber (RNF) and retinal artery (RA) trajectories in the posterior pole of eyes and whether the RNF and RA trajectories are correlated with the axial length of the eye.

Methods.: This was a prospective observational cross-sectional study of 109 right eyes of 109 healthy participants. All participants underwent axial length measurements, optical coherence tomography (OCT) to determine the peripapillary retinal nerve fiber layer (RNFL) thickness, and red-free fundus photography. The supratemporal and infratemporal peaks of the RNFL thickness were determined in the OCT RNFL circle scan images. The trajectories of the RNF passing through the peaks of the RNFL thickness were plotted in the red-free fundus photographs and were fitted to a second-degree polynomial equation (ax ²/100 + bx + c) by ImageJ. The coefficient a represented the steepness of the trajectories. Intraclass correlation coefficient was used to measure the reliability between the raters. The relationships between the RNF or RA trajectories and the axial length were investigated using linear regression analyses.

Results.: The mean axial length was 25.5 ± 1.4 mm, and the mean RNF trajectory and the mean RA trajectory, a, were 0.472 ± 0.123 and 0.442 ± 0.109, respectively. The intrarater and interrater correlation coefficients of the RNF trajectories were 0.954 and 0.881, respectively. The RNF and RA trajectories were significantly and positively correlated with the axial length (R = 0.28, 0.33, P < 0.01).

Conclusions.: A longer axial length is associated with narrower RNF and RA trajectories. (www.umin.ac.jp/ctr number, UMIN000006040.)

Introduction

Primary open-angle glaucoma (POAG) is a chronic, progressive optic neuropathy characterized by the death of the retinal ganglion cells and their axons; that is, glaucomatous optic neuropathy is associated with deaths of these retinal nerve fiber (RNF) bundles. ^1,2 Thus, an assessment of the RNF defects in eyes suspected of having glaucoma is important before functional defects develop. 

Individuals with myopia have a higher risk of glaucoma than nonmyopic individuals, and thus myopia is a risk factor for POAG. ^3,4 This is important because the prevalence of myopia is increasing worldwide. ^5–7 In the early stage of glaucoma in myopic eyes, the retinal nerve fiber layer (RNFL) defects and corresponding visual field defects are more likely to be detected in the paracentral area. ^8–10 The paracentral RNFL defects and corresponding paracentral visual field defects in myopic glaucoma patients can lead to severe visual impairments. 

The paracentral RNFL defects should be possible to detect in optical coherence tomographic (OCT) images of RNFL circle scans. In normal eyes, there are two peaks in the RNFL thickness profile, the supratemporal peak and the infratemporal peak. ^11–16 Earlier studies demonstrated that these two peaks shifted temporally as the axial length increased. ^17–19 This shift may explain the paracentral RNFL defects in eyes at the early stage of myopic glaucoma. However, the shift in the RNFL peaks in myopic eyes has been investigated only in circumferential scans. 

There have been several studies that simulated the distribution of RNFL in glaucomatous eyes, but these simulations are generally too complicated and not easy to use in clinical settings. ^20,21 We have investigated whether a mathematical model of the RNFL trajectories can help in understanding how the RNFL defects develop in myopic eyes. 

To develop the mathematical model, we examined red-free fundus photographs and noted that the thick RNF bundles can be seen in the supratemporal and infratemporal margins of the optic disc. When the fundus photographs of the right eye are rotated clockwise 90°, the trajectories of the thick RNF bundles resemble a second-degree polynomial curve (Fig. 1). 

Figure 1

 

Red-free fundus photographs and retinal nerve fiber trajectories (yellow curves in fundus photographs). The photographs are rotated 90° clockwise. (A, C) A low-myopic eye; (B, D) a highly myopic eye.

Thus, the purpose of this study was to determine whether the RNF trajectories can be fit by a second-degree polynomial equation. Because the positions of the supratemporal and infratemporal peaks of the RNFL thickness appeared to be correlated with the positions of the major supratemporal and infratemporal retinal arteries (RAs), we also determined whether the trajectories of the RAs can be fit by a second-degree polynomial equation. ^17,22,23 Since myopia is a risk factor for glaucoma and myopic eyes tend to have longer axial lengths, we also determined whether there was a significant correlation between the RNF and RA trajectories and the axial lengths of the eyes. 

Methods

All of the procedures used conformed to the tenets of the Declaration of Helsinki. Written informed consent was obtained from all of the subjects after an explanation of the procedures to be used. The study was approved by the Ethics Committee of Kagoshima University Hospital, and it was registered with the University Hospital Medical Network (UMIN) clinical trials registry. The registration title was “Morphological analysis of the optic disc and the retinal nerve fiber in myopic eyes,” and the registration number was UMIN000006040. A detailed protocol is available at https://upload.umin.ac.jp/cgi-open-bin/ctr/ctr.cgi?function=brows&action=brows&type=summary&recptno=R000007154&language=J. The results presented in this paper are part of the overall study. ¹⁷  

Subjects

This was a cross-sectional, prospective observational study. We studied 133 eyes of 133 volunteers who were enrolled between November 1, 2010 and February 29, 2012. Volunteers with no known eye diseases as determined by examining their medical charts were studied, and only the data from the right eyes were analyzed. The eligibility criteria were age ≥ 20 years but <40 years; eyes normal by slit-lamp biomicroscopy, ophthalmoscopy, and OCT; best-corrected visual acuity (BCVA) ≤ 0.1 logarithm of the minimum angle of resolution (logMAR) units; and intraocular pressure (IOP) ≤ 21 mm Hg. The exclusion criteria were eyes with known ocular diseases such as glaucoma, staphyloma, and optic disc anomalies; volunteers with systemic diseases such as hypertension and diabetes; presence of visual field defects; and history of refractive or intraocular surgery. None of the eyes were excluded because of poor OCT image quality caused by poor fixation. 

Measurement of Axial Length and Refractive Error

All of the eyes had a standard ocular examination including slit-lamp biomicroscopy of the anterior segment, ophthalmoscopy of the ocular fundus, IOP measurements with a pneumotonometer (CT-80; Topcon, Tokyo, Japan), and axial length measurements with the AL-2000 ultrasound instrument (Tomey, Nagoya Japan). The refractive error (spherical equivalent) was measured with the Topcon KR8800 autorefractometer/keratometer. 

Determination of Position of Supratemporal and Infratemporal RNFL Thickness Peaks and Calculation of Retinal Nerve Fiber and Retinal Artery Trajectories

The RNFL thickness was measured with the Spectralis SD-OCT (Heidelberg Engineering, Inc., Heidelberg, Germany) using the RNFL circle scan, and the temporal-superior-nasal-inferior-temporal (TSNIT) thickness curves were used to identify the positions of the supratemporal and infratemporal peaks of the RNFL thickness. The fundus images and OCT images were taken at the same time. Then, the scan circle was laid over the fundus image as a green circle. Using these images, the position of the supratemporal or infratemporal peak RNFL thicknesses in the fundus images was determined with the Spectralis OCT software (Fig. 2). 

Figure 2

 

Identification of the supratemporal peak of the retinal nerve fiber layer thickness position in the temporal-superior-nasal-inferior-temporal (TSNIT) profile of the Spectralis OCT RNFL thickness output (black triangle). The corresponding position in the fundus image is shown by the white arrow. The infratemporal peak RNFL position was assessed by the same method by the Spectralis embedded software.

Color fundus photographs were taken with a color fundus camera (Topcon 3D OCT-1000 MARK II) with the RNFL 3.4-mm circle scan protocol. The RAs and the center of the optic discs were identified in the color fundus photographs. The OCT optical system detected the edge of the optic disc in the fundus image, and the scan circle was centered automatically on the optic disc just before taking the OCT and color fundus image. To exclude the effects of errors in the centering of the scan circle, one examiner (YK) checked that the center of the scan circle was located at the center of the optic disc. The center of the scan circle was used as the center of the disc. 

Red-free, 55° fundus photographs were taken with the Spectralis HRT (Heidelberg Engineering, Inc.). The red-free fundus photographs of the right eyes were rotated 90° clockwise. The trajectories of the RNF passing through the supratemporal and infratemporal peaks were plotted for at least 20 points on the red-free fundus photographs (Fig. 3A). The coordinates were determined automatically using the ImageJ program (ImageJ version 1.47; National Institutes of Health, Bethesda, MD, USA; http://imagej.nih.gov/ij/ [in the public domain]) (Fig. 3B). The x and y coordinates in the red-free fundus photograph were converted to a new set of data (x and y) with the center of the disc as the origin. Finally, the converted coordinate data were fitted to a second-degree polynomial (ax ²/100 + bx + c) equation with the curve-fitting program of ImageJ. The a, b, and c are constants calculated by the curve-fitting program of ImageJ. The curve-fitting program automatically determined the best-fit second-degree polynomial (ax ²/100 + bx + c) equation by the least squares method. Under these conditions, a larger a will make the curve steeper and narrower and will bring the arms of the curve closer to the fovea. Thus, the a constant was used as the degree of the RNF trajectory. The pixel coordinates and curve fittings of the RA trajectories were calculated using the same method as for the RNF trajectories. In eyes in which the RA branched, the plotting was made with the following rules. If a branch artery was smaller than the main artery, the main artery was used to determine the RA trajectory. If the branch artery was as large as the main artery, the plotting was not done after the branching point. Even so, the plotting was done with more than 20 points. Additionally, to eliminate the effect of ocular torsion of individual eyes, we also evaluated the images in which the fovea–disc axis was rotated vertically (model 2). 

Figure 3

 

Trajectories of the retinal nerve fibers passing through the supratemporal and infratemporal peaks of the retinal nerve fiber layer thickness are plotted in a red-free fundus photograph (A). The pixel coordinate data were determined automatically using ImageJ software and converted to a new set of data with the center of the optic disc as the origin. The converted coordinate data were fitted to a user-defined second-degree polynomial equation (ax ²/100 + bx + c) using the curve-fitting program of ImageJ (B).

Statistical Analyses

All statistical analyses were performed with the SPSS Statistics 19 for Windows (SPSS, Inc.; IBM, Somers, NY, USA). The intrarater or interrater correlation coefficients of the RNF trajectories were calculated using a two-way mixed-effects model for measurements of absolute agreement. The relationships between the RNF trajectories, the trajectories of RAs, and the axial length were determined by Spearman's correlation analyses. 

Results

One hundred thirty-three Japanese volunteers were screened for this study. Seven eyes were excluded due to ocular diseases or prior ocular surgery, three cases because of superior segmental optic hypoplasia, one case because of glaucoma, and three cases because of laser-assisted in situ keratomileusis. Five other eyes were excluded because of difficulty in identifying the position of the peak RNFL thickness in the RNFL thickness profiles. Twelve eyes were excluded because of difficulty in identifying the RNF trajectories in the red-free fundus photographs. In the end, the right eyes of 109 individuals (78 men and 31 women) were used for the analyses. 

The demographic information of the participants is presented in the Table. The mean ± standard deviation of the age was 25.6 ± 3.5 years, and the mean refractive error (spherical equivalent) was −4.76 ± 3.39 diopters (D). The mean axial length was 25.5 ± 1.4 mm. The refractive errors and axial lengths were significantly and negatively correlated (R = −0.83, P < 0.001; Fig. 4A). 

Figure 4

 

Scatterplots of the refractive errors (spherical equivalent) (A), the retinal nerve fiber trajectories (B), and the retinal artery trajectories (C) as a function of the axial length. Scatterplot of the RA trajectories as a function of the RNF trajectories (D).

Table

 

Participant Data

	Mean ± SD	Range
Age, y	25.6 ± 3.5	22–38
Sex, M/F	78/31
Spherical equivalent, D	−4.76 ± 3.39	−14.25 to 0.50
Axial length, mm	25.5 ± 1.4	22.4 to 30.4
Retinal nerve fiber trajectory	0.472 ± 0.123	0.288 to 1.005
Retinal artery trajectory	0.442 ± 0.109	0.220 to 0.871

 SD, standard deviation.

Intrarater and Interrater Repeatability of RNF and RA Trajectories

The intrarater and interrater repeatabilities of the RNF trajectory measurements were investigated for the right eyes of the 109 participants. The intrarater correlation coefficient of the RNF trajectories was 0.954 (95% confidence interval [CI] 0.934–0.968, P < 0.001). The interrater correlation coefficient of the RNFL trajectories was 0.881 (95%CI 0.831–0.917, P < 0.001). The intrarater and the interrater repeatabilities of the RNF trajectories were excellent. 

An RA occasionally branches soon after it exits the optic disc. Among our eyes, there were nine eyes with an artery in which the branch was as large as the main artery at 2.5 disc diameters from the optic disc. In the other cases, a branch that was as large as the main artery occurred at 2.5 disc diameters or farther from the optic disc. Therefore, all of the RA trajectories could be fit to the approximation formula within a 2.5 disc diameter distance from the optic disc. The interrater correlation coefficient of the RA trajectories was 0.944 (95% CI 0.871–0.971, P < 0.001). The interrater repeatability of the RA trajectories was also excellent. Therefore, the mean values of the RNF or RA trajectories of the two raters were used for the analyses. The means and standard deviations of the RNF and RA trajectories were 0.472 ± 0.123 and 0.442 ± 0.109, respectively. 

Spearman's Correlation Coefficients Between RNF Trajectories, RA Trajectories, and Axial Length

The RNF trajectories were significantly and positively correlated with the axial length (R = 0.28, P = 0.003), and the RNF trajectories, expressed as the a constant of second degree polynomial, increased by 0.03 per millimeter increase of the axial length (Fig. 4B). The RA trajectories were significantly and positively correlated with the axial length (R = 0.33, P < 0.001), and the RA trajectories, expressed as the a constant of second degree polynomial, increased by 0.03 per millimeter increase of the axial length (Fig. 4C). The RNF trajectories were significantly correlated with the RA trajectories (R = 0.82, P < 0.001; Fig. 4D). 

Effect of Ocular Torsion

We performed the curve-fitting analysis on the fundus images rotated by 90° (model 1). To eliminate the effect of ocular torsion of the individual eyes, we also analyzed the images in which the fovea–disc axis was rotated vertically (model 2). The calculations showed that the means and standard deviations of the RNF and RA trajectories of model 2 were 0.478 ± 0.125 (range, 0.271–0.975) and 0.460 ± 0.113 (range, 0.245–0.899), respectively. The mean and standard deviation of the degree of rotation were 93.59 ± 2.65 (range, 90.0–105.0). The correlation coefficient between the original RNF trajectories and the RNF trajectories of model 2 was high (R = 0.98, P < 0.001). The correlation coefficient between the original RNF trajectories and the RA trajectories of model 2 was also high (R = 0.94, P < 0.001). The RNF trajectories of model 2 were significantly and positively correlated with the axial length (R = 0.32, P = 0.001). The RA trajectories of model 2 were significantly and positively correlated with the axial length (R = 0.29, P = 0.002). The RNF trajectories of model 2 were significantly correlated with the RA trajectories of model 2 (R = 0.87, P < 0.001). The results of the RNF and RA trajectories of model 2 were very similar to the original RNF and RA trajectories. Thus, it is possible that the width of trajectories of RNF or RA can be evaluated by either model because the approximation formula was used for the comparison (see Supplementary Fig. S1). 

Discussion

There were significant and positive correlations between the RNF and RA trajectories and the axial length. The coefficient of determinant of the curve fitting was >0.9 for all of the cases, which means that the second-degree polynomial equation was a good fit with the shape of the RNF and RA trajectories. 

We also tested a linear regression model: RNF trajectory = −0.26 + 0.03 × axial length and RA trajectory = −0.25 + 0.03 × axial length. Earlier, the RNF bundle trajectories were described by a polar coordinate system. ^20,21 Although these methods also described the RNF bundle trajectories well, their complexities prevented their general use for research and clinical evaluations. In comparison, our mathematical model is relatively simple and easy to use and requires only a fundus photograph and the easily accessible free public software, ImageJ, provided by the National Institutes of Health. 

We were able to obtain the following information using this method. It is known that visual field defects develop in the cecocentral field in the early stages of glaucoma in myopic eyes. ^9,10,24 In nonmyopic eyes, such changes usually occur in the intermediate to late stages of glaucoma. The present findings may explain the reason for the location of the defect. Eyes with longer axial lengths are usually myopic and have RNFL bundles running closer to the fovea than those with normal axial lengths, that is, nonmyopic eyes. Because the RNFL bundle defects in glaucomatous eyes generally begin in the supra- or infratemporal sectors (thicker RNFL bundles), the visual field damage would develop closer to the fovea from the earlier stages in myopic eyes than in nonmyopic eyes. Nonetheless, a direct comparison of RNFL defects and RNF trajectories will be still necessary to validate this interpretation. 

In the standard clinical setting, RNF photography is still important for studying myopic eyes. However, it is difficult to identify the RNF bundles and defects in myopic eyes because the RNFL is obscured by a low-pigmented fundus. In the Caucasian population, the overall low-pigmented eyes may further exacerbate this phenomenon. Even in more pigmented eyes such as Asian eyes, it is not easy to observe the RNFL in myopic eyes because of the lower pigmentation than in those without myopia. In this study, the distribution of RNFL fibers through the thickest peaks of RNFL coincided well with that of the temporal RA. Therefore, it is possible that the RA can be a good way to monitor the RNFL even when the RNFL location is not clearly observed. 

A recent study showed that the RNF bundle defects often appeared in the cecofoveal area in the early stage of glaucoma in myopic eyes; these are easily detected as visual field defects by Humphrey field analyzer (HFA) 10-2. ²⁵ If the RA is located closer to the fovea in myopic eyes, it would be advisable to examine patients with the HFA 10-2 so as not to overlook early glaucomatous damage. 

There was a tendency for the RNF bundles, which might indicate RNFL bundles, to run closer to the cecofoveal area; but the correlation coefficient 0.28 was not high, and there were some cases that differed. This would suggest that there was some discrepancy between the empiric data and the mathematical model. Thus, factor(s) other than the axial length may exist that determine the trajectories. One possibility would be the presence of a paradoxical eye, which is defined as one with a short axial length with myopic fundus changes, for example, conus, elliptic optic disc, and greater RNF trajectories (short paradoxical fundus, Fig. 5A). Or, there may be eyes with a longer axial length with no myopic fundus changes and lower RNF trajectories (long paradoxical fundus, Fig. 5B). An earlier study showed that there were large variations in the axial length at birth. ²⁶ Thus, a long axial length does not necessarily mean that the axial length will be longer after attainment of full growth. More specifically, even though two eyes have the same axial length in adulthood, if the axial length differed at birth, the degree of elongation must have been different between these eyes during the growth period. This may affect the trajectories of the RNF and the arcade arteries. The present model may help in determining the mechanism for the formation of these eyes and obtaining a correct diagnosis. Furthermore, if a modification of the present formula fits the empiric data better than the original formula, the modified factor might prove to be an important factor for determining the trajectories. 

Figure 5

 

Relationship between the axial length and retinal nerve fiber trajectories in atypical cases. (A) Shorter axial length (24.3 mm) but obvious myopic fundus changes (conus, tessellated fundus, oval optic disc) with wider retinal nerve fiber trajectories (0.490). (B) Longer axial length (27.7 mm) but few myopic fundus changes with narrower retinal nerve fiber trajectories (0.345).

It is well known that the “magnification effect” can affect the results of morphological analyses of OCT images. In this study, we used the trajectories instead of measurement of the distance from the fovea, because the trajectories are not strongly affected by the magnification effect. If the RNF and RA trajectories were closer to the fovea, the optic disc should also be closer to the fovea because the x-axis and y-axis change proportionately to the axial length. There is another reason why the correction method for the magnification effect was not suitable for this study. Bennett's formula is supposedly the most suitable method to adjust the deviated results from the magnification effect, and this formula contains the axial length variable. ^27,28 However, this was not applicable to the present data to determine the correlation between the fovea–RNF distance and the axial length because each variable contains the common variable of axial length. Thus, using the present analysis, the trajectories were found to be narrower with an elongation of the axial length. However, these results did not fully support the idea that the fovea–RNF distance is smaller in eyes with narrow trajectories than with wide trajectories because we could not measure the distance directly. This issue needs further investigation. 

There are several limitations to this study. First, this was not a population-based study. Epidemiological studies have shown that the Japanese population is one of the most myopic groups, ⁶ and the individuals studied were university students, who are known to be myopic. Thus, our results describe the characteristics of young myopic eyes and might not hold for older and nonmyopic populations. On the other hand, the reliability of the examination was very high because no pathological factors such as cataract or vitreal opacities were present in young healthy individuals, and understanding of the examination was high. In addition, the narrow range of age prevented interference of cohort effects and age effects. There are many factors that affect the trajectories of the RNF or RA, for example, the size and shape and torsion of the eye. ^29,30 None of these were considered in our analyses. Only the effect of the eye torsion could be minimized by use of the second-degree polynomial approach (model 2). There will always be limitations in mathematical models. The present model cannot consider the various and complex factors that could affect the structures of the posterior pole of the eye, especially in those with glaucoma. Study of glaucomatous eyes will be mandatory. In comparison to previous reports, in which the RNFL shift was depicted on the circle of optic disc, ^17–19 the present approach plotted the RNF trajectories on a much broader area. However, it still covered only the area from the optic disc to the foveal area. Therefore, we do not know about the area temporal to the fovea even from the present model. Finally, it is known that the shadow of an artery affects the identification of the RNFL peak, and this is a limitation of the present analysis of the OCT images. ^22,23 In B-scans of the Spectralis OCT, we could not draw an accurate borderline of the RNFL (Fig. 2) because the blood vessel shadow masked the borderline of the RNFL in most of the images. Thus, we adopted the method commonly used in earlier studies, using the RNFL thickness peak position in the OCT images ^17–19 for analysis of the trajectories. These limitations should be remembered in interpretation of the results. 

In summary, we were able to establish a mathematical model that can determine the RNF/RA trajectories in normal eyes. Our results showed that eyes with longer axial lengths had narrow RNF/RA trajectories. A shift in the trajectories of the RNF or the RA may lead to ceocentral visual field defects in myopic glaucomatous eyes; however, further studies are necessary to validate this interpretation. In any case, the trajectories of the RNF or the RA should be taken into account in assessment of myopic glaucomatous eyes and to analyze the findings. The present mathematical model may be useful for these purposes. 

Supplementary Materials

pdf S1 

Acknowledgments

Disclosure: T. Yamashita, None; T. Sakamoto, None; H. Terasaki, None; M. Tanaka, None; Y. Kii, None; K. Nakao, None