June 2003
Volume 44, Issue 6
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Glaucoma  |   June 2003
Fourier Analysis of Scanning Laser Polarimetry Measurements with Variable Corneal Compensation in Glaucoma
Author Affiliations
  • Felipe A. Medeiros
    From the Hamilton Glaucoma Center and Diagnostic Imaging Laboratory, Department of Ophthalmology, University of California, San Diego, California.
  • Linda M. Zangwill
    From the Hamilton Glaucoma Center and Diagnostic Imaging Laboratory, Department of Ophthalmology, University of California, San Diego, California.
  • Christopher Bowd
    From the Hamilton Glaucoma Center and Diagnostic Imaging Laboratory, Department of Ophthalmology, University of California, San Diego, California.
  • Antje S. Bernd
    From the Hamilton Glaucoma Center and Diagnostic Imaging Laboratory, Department of Ophthalmology, University of California, San Diego, California.
  • Robert N. Weinreb
    From the Hamilton Glaucoma Center and Diagnostic Imaging Laboratory, Department of Ophthalmology, University of California, San Diego, California.
Investigative Ophthalmology & Visual Science June 2003, Vol.44, 2606-2612. doi:10.1167/iovs.02-0814
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      Felipe A. Medeiros, Linda M. Zangwill, Christopher Bowd, Antje S. Bernd, Robert N. Weinreb; Fourier Analysis of Scanning Laser Polarimetry Measurements with Variable Corneal Compensation in Glaucoma. Invest. Ophthalmol. Vis. Sci. 2003;44(6):2606-2612. doi: 10.1167/iovs.02-0814.

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

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Abstract

purpose. To apply Fourier analysis to the retinal nerve fiber layer (RNFL) thickness measurements obtained with scanning laser polarimetry (SLP), by using variable corneal compensation, and to evaluate the ability of this method to discriminate glaucomatous from normal eyes.

methods. The study included one eye each of 55 patients with glaucoma and 52 healthy subjects. RNFL thickness measurements were obtained with a modified commercial scanning laser polarimeter (GDx Nerve Fiber Analyzer; Laser Diagnostic Technologies, Inc., San Diego, CA) so that corneal birefringence could be corrected on a subject-specific variable basis. The shape of the RNFL thickness double-hump pattern was analyzed by Fourier analysis of polarimetry data. Fourier coefficients and GDx parameters were compared between the two groups. A linear discriminant function was developed to identify and combine the most useful Fourier coefficients to separate the two groups. Receiver operating characteristic (ROC) curves were obtained for each measurement, and sensitivity values (at fixed specificities) were calculated.

results. The Fourier-based linear discriminant function (LDF Fourier) resulted in a sensitivity of 84% for a specificity set at 92%. For similar specificity, the GDx software–provided parameters had sensitivities ranging from 24% to 69%. The area under ROC curve for the LDF Fourier was 0.949, significantly larger than the ROC curve area for the single best GDx software–provided parameter, superior average (0.870).

conclusions. The combination of Fourier RNFL thickness measures in an LDF, obtained using SLP with variable corneal compensation, improved the ability to discriminate glaucomatous from healthy eyes, compared with the GDx software–provided parameters.

Scanning laser polarimetry (SLP) quantitatively and objectively measures parapapillary retinal nerve fiber layer (RNFL) thickness. It is based on the principle that polarized light passing through a birefringent medium undergoes a measurable phase shift, known as retardation, that is directly proportional to the thickness of the medium. 1 2 3 4 The measure of RNFL thickness is based on the linear relationship between the retardation of reflected light and histologically measured RNFL tissue thickness. 1 Differences in retardation between normal and glaucomatous eyes, as well as good correlation between retardation measurements and visual field loss, have been described. 5 6 7 8 9 10 11 12 13  
In normal subjects, the distribution of RNFL thickness around the optic disc has a characteristic double-hump pattern, due to a thicker RNFL in the superior and inferior regions, compared with temporal and nasal regions. 4 14 RNFL loss in glaucoma leads to a change in the appearance of this pattern, either by reducing its amplitude or changing its shape. However, as RNFL thickness is known to vary widely among healthy eyes, the usefulness of absolute thickness values to separate glaucomatous from normal eyes is limited. 6 7 8 9 10 11 Therefore, it is possible that an analysis of the global shape of the distribution of RNFL thickness around the optic disc may be more effective in detecting RNFL loss in glaucoma than an assessment of particular thicknesses. This analysis can be accomplished by using the mathematical procedure, Fourier analysis. 15 16  
Using Fourier analysis, a complex waveform, such as the double-hump curve, can be resolved into a set of sinusoidal related components that yield the original pattern, when added point by point. Fourier analysis provides the amplitudes and phases of the sinusoidal components that can then be used to study the original waveform quantitatively. Essock et al. 16 applied Fourier analysis to the RNFL measurements of scanning laser polarimetry (SLP) and found an improved detection of glaucoma. However, a source of error in these studies was most likely introduced by the erroneous compensation of anterior segment birefringence. 17 18 19 20  
The RNFL is not the only birefringent structure in the eye. The cornea and Henle fiber layer of the macula, and to a lesser extent the lens, also are birefringent. 21 22 To address anterior segment birefringence, the first commercial SLP had a fixed corneal compensator. The compensator was calibrated based on the assumption that all individuals had a slow axis of corneal birefringence 15° nasally downward with a magnitude of retardance of 60 nm. However, there is a wide variation, both in the axis and in the magnitude of corneal birefringence in normal and glaucomatous eyes. 23 24 25 An improvement of the SLP technology consisting of the variable compensation of anterior segment birefringence has been recently described. 26 Weinreb et al. 27 have shown that SLP with variable corneal compensation results in improvement of the sensitivity and specificity of several parameters to discriminate between healthy and glaucomatous eyes. 
In the present study, Fourier analysis was applied to the RNFL thickness measurements obtained with SLP by using variable corneal compensation, and the ability of this method to discriminate between glaucomatous and normal eyes was evaluated. 
Subjects and Methods
Fifty-two healthy subjects and 55 patients with glaucoma who met entry criteria were enrolled in this study. All patients were evaluated at the Hamilton Glaucoma Center (University of California, San Diego, CA), and informed consent was obtained from all participants. The University of California San Diego Human Subjects Committee approved all protocols and the methods described adhered to the tenets of the Declaration of Helsinki. Mean age ± SD of normal subjects and patients with glaucoma was 63.0 ± 9.5 years (range, 49–86) and 68.4 ± 9.4 years (range, 50–89), respectively (P = 0.004; Student’s t-test). Normal subjects were chosen so that the age range was similar to that of the patients with glaucoma. Among the healthy subjects, 33 (63%) were women and 19 (37%) were men. Among the patients with glaucoma, 29 (53%) were women and 26 (47%) were men. Ninety individuals were white, five were Hispanic, five were of African American descent, four were Asian American, and three were self-reported as unknown. One randomly chosen eye per subject was included in the study. 
Each subject underwent a comprehensive ophthalmic examination, including review of medical history, best corrected visual acuity, slit-lamp biomicroscopy, intraocular pressure (IOP) measurement with Goldmann applanation tonometry, gonioscopy, dilated fundoscopic examination with a 78-D lens, stereoscopic optic disc photography, and automated perimetry with 24-2 full-threshold standard automated perimetry (SAP) or Swedish interactive threshold algorithm (SITA; Humphrey Field Analyzer; Zeiss-Humphrey Systems, Dublin, CA). To be included, each subject had to have best corrected visual acuity of 20/40 or better. Refractive error ranged from −9.00 to +4.75 D. Eyes with coexisting retinal disease, uveitis, or nonglaucomatous optic neuropathy were excluded from the investigation. 
Normal control eyes had an IOP of 22 mm Hg or less, with no history of increased IOP, and a normal visual field result. Normal visual field was defined as a mean deviation (MD) and pattern standard deviation (PSD) within 95% confidence limits, and a Glaucoma Hemifield Test result (GHT; Zeiss-Humphrey Systems) within normal limits. Normal control eyes also had a healthy appearance of the optic disc and RNFL (no diffuse or focal rim thinning, cupping, optic disc hemorrhage, or RNFL defects), as evaluated by optic disc photographs. 
Eyes were classified as glaucomatous if they had repeatable (two consecutive) abnormal visual field test results, defined as a PSD outside the 95% normal confidence limits, or a GHT result outside 99% normal confidence limits, regardless of the appearance of the optic disc. Because we were evaluating SLP-measured RNFL thickness, we chose the best indicator of glaucoma that is not dependent of RNFL or optic disc appearance, to avoid bias in the evaluation of the technique. 28 Average MD of the glaucomatous eyes on the visual field test nearest the imaging date was −5.91 dB (range, −20.92 to +0.26 dB). According to the Hodapp-Parrish-Anderson 29 grading scale of severity of visual field defects, 21 (38%) patients were classified as having early visual field defects, 30 (55%) as having moderate defects, and 4 (7%) as having severe defects. 
SLP Measurements
In this study, all patients were imaged using a prototype version of the scanning laser polarimeter with variable corneal polarization compensation. This prototype consisted of a modification of the previously commercially available scanning laser polarimeter (GDx Nerve Fiber Analyzer, Laser Diagnostic Technologies, Inc., San Diego, CA) so that the original fixed corneal compensator was replaced with a variable custom corneal compensator (VCC) as described by Zhou and Weinreb. 26 This method of variable corneal polarization compensation has been incorporated into the new commercially available version of the scanning laser polarimeter (GDx VCC, software version 5.0.1; Laser Diagnostic Technologies, Inc.) Briefly, the VCC scanning laser polarimeter consists of a set of four linear retarders in the path of the measurement laser beam. The first two adjustable retarders are optical lenses that have equal retardance and form a variable corneal and lens compensator. The third retarder is composed of the cornea and lens, and the fourth retarder is the retinal birefringent structure (RNFL or macular Henle fibers). 
To determine the eye-specific corneal polarization axis (CPA) and magnitude (CPM), the compensating retarders were adjusted to 0 nm, and the macula was imaged. The retardation at a locus of points along a circle centered on the fovea was calculated. This measurement circle was 80 pixels in diameter, which corresponds to 1.41 mm in an emmetropic eye. The resultant retardation profile represented the additive effects of cornea, lens, and macular Henle fiber birefringence. The axis and magnitude of anterior segment birefringence were determined from the measured macular retardation profile. The macula was imaged three times in each subject, and the average CPMs and CPAs from the three macular scans that resulted in adequate compensation were recorded. The compensating retarders were then adjusted to minimize the effects of anterior segment birefringence. Once the VCC was correctly adjusted, the total retardation measured with the scanning laser polarimeter was the birefringence of the RNFL or that of the Henle fiber layer. As a result, an SLP image of the macula should show a radially symmetric uniform pattern—that is, a flat profile. To confirm the effectiveness of the VCC, the macula was imaged again with a subject-specific CPM and CPA, to assure that the macular retardation profile was flat when these values were used. Next, three corneal birefringence-compensated parapapillary SLP images from each eye were obtained with the appropriate eye-specific VCC CPM and CPA. The three images were automatically combined by GDx software version 3.0 to create a composite mean image that was used for analysis of RNFL thickness. 
The disc margin was established by an experienced operator who used an ellipse to outline the inner margin of the peripapillary scleral ring, based on the information from the SLP reflectance image. A 10-pixel-wide elliptical band was automatically positioned concentric with the disc margin outline and at 1.7 disc diameters from the center of the optic disc. This elliptical band was divided into 32 sectors (11.25° each), and the average retardation was recorded in each of the 32 sectors. These measurements were used in the Fourier analysis, to be described later. The GDx software estimates RNFL thickness (in micrometers) by correcting retardation measurements (in degrees) with a factor determined in a previous histologic comparison in monkey eyes. 1  
The GDx software also calculates summary parameters based on quadrants, which are defined as temporal (335°–24° unit circle), superior (25°–144°), nasal (145°–214°), and inferior (215°–334°). The GDx software–provided parameters investigated in this study were symmetry (superior quadrant thickness/inferior quadrant thickness), superior ratio (superior quadrant thickness/temporal quadrant thickness), inferior ratio, superior/nasal ratio, maximum modulation (thickest quadrant/thinnest quadrant)/thinnest quadrant, average thickness, superior maximum (average of the 1500 thickest points in the superior quadrant), inferior maximum, ellipse average, ellipse modulation, superior average, inferior average, and superior integral. The latter five parameters are measured in relation to the 10-pixel-wide elliptical band. Each of these parameters have been described in detail elsewhere. 11  
Fourier Analysis
Fourier analysis is a mathematical procedure in which a complex waveform pattern (such as the double-hump pattern of the RNFL) is resolved into a series of harmonically related sinusoids of specific frequencies, amplitudes, and phases. 30 These sinusoids, when summed point by point, reproduce the original pattern. The fast Fourier transform (FFT) was used to determine the coefficients (i.e., the amplitude and phase of each sinusoid). Mathematical software (Mathematica ver. 4; Wolfram Research, Inc., Champaign, IL) was used to perform the FFT algorithm on the data from SLP. Details from the FFT procedure have been published elsewhere. 31 32 33 In brief, when Fourier analysis is performed on a set of discrete data that describe a waveform, the number of sine waves generated equals half the number of data points. The lowest frequency component, also called the fundamental (F1), corresponds to a sine wave pattern with one cycle. The other components are sine waves with frequencies that are integer multiples of the fundamental. For example, the second harmonic has a frequency that is two times that of the fundamental. 
The pattern of thickness measurements from SLP in the 32 sectors around the optic disc was analyzed to obtain the Fourier coefficients. Because the Fourier analysis was performed on 32 data points, 16 sine waves were obtained. The amplitude of each sine wave was the peak-to-trough distance, and the phase value described where the waveform begins in relation to some reference. The amplitude of each component indicated its relative contribution to the shape of the composite curve fitting the RNFL thickness measurements. Also produced by Fourier analysis was the component DC, which represented the overall level of all amplitude measures and corresponded to the mean RNFL thickness. The use of more than 32 sectors (data points) resulted in higher-frequency components with very low amplitudes that did not contribute significantly to the discrimination of glaucomatous from normal eyes, even when combined with other lower-frequency components. 
Statistical Analysis
Student’s t-tests were used to evaluate differences in GDx software–provided parameters and Fourier coefficients between glaucomatous and healthy eyes. The Hotelling T-squared generalized means test was used to compare the set of means of Fourier measures between the two groups. This test controls the inflation of type I error that arises when making a series of t-tests to compare group means of several dependent variables. Results of Bonferroni adjustments based on the number of comparisons were also provided. 
The Fourier components (DC component and all amplitude and phase values) from all subjects were entered into a stepwise discriminant analysis to develop a classification function (linear discriminant function; LDF) designed to identify and combine the best measures to differentiate glaucomatous from normal eyes. The relative importance of each variable in the discriminant analysis was assessed by using the ratio of within-group variance to total-group variance (Wilks λ). The absolute phase coefficients were used for this procedure to emphasize deviations in shape, regardless of direction. 
Receiver operating characteristic (ROC) curves were used to describe the ability to differentiate glaucomatous from normal eyes of each GDx software–provided parameter and also of the LDF obtained from Fourier measures. The ROC curve shows the tradeoff between sensitivity and 1 − specificity. An area under the ROC curve of 1.0 represents perfect discrimination whereas an area of 0.5 represents chance discrimination. The method of DeLong et al. 34 was used to compare areas under the ROC curve. Minimum specificity cutoffs of 80% and 90% were used for comparing the sensitivity of the GDx parameters and the Fourier-based LDF by the McNemar’s test for paired proportions. 
The use of the same population to identify and test a discriminant function can lead to a significant bias or optimism with respect to the predictive ability of the model when applied to an independent data set. To provide an internal validation estimate, we used a computer-intensive bootstrap approach. 35 36 The bootstrap procedure provides stable and nearly unbiased estimates of performance, and is said to be the most efficient validation method available, comparing favorably with standard cross-validation techniques. 35 36 37 38 The superiority of bootstrap estimates over traditional methods seems to be particularly evident for error rate estimation in two-class discrimination problems when samples sizes are relatively small. 39 Bootstrapping is a resampling method that allows one to make inferences about the population from which the sample originated by drawing B samples (B = 1000 in the present study), with replacement from the original data set, of the same size as the original data set. In the 0.632 bootstrap method, a model is built for each bootstrap sample and evaluated only in those subjects not sampled. The prediction errors are then averaged over all bootstrap samples (test performance). Because the evaluation is based on an independent data set, this method can be seen as a direct extension or as a smoothed version of cross-validation. Because the data set is sampled with replacement, on average 63.2% of the subjects are included at least once in a bootstrap sample, giving the method its name. 35 The estimated performance is a weighted combination of the apparent performance (resubstitution error estimate on the full data set) and test performance. The 0.632+ bootstrap is an extension of the 0.632 method applying a regularizing coefficient based on the amount of overfitting and has the advantage of performing well even when there is severe overfitting. 36 In our study, the area under the ROC curve was used as a measure of predictive performance of the LDF, and the 0.632+ bootstrap method was applied to estimate the optimism or bias of this measure. The application of the 0.632+ method in a similar situation has been reported elsewhere. 37 Bootstrap methods were also used to calculate bootstrap bias-corrected and accelerated (BCa) 95% confidence intervals for the area under ROC curves. 40 41  
Statistical analyses were performed on computer (S-Plus 2000 software; MathSoft, Inc., Seattle, WA). 
Results
The mean RNFL thickness estimates for the 32 sectors around the optic disc in normal and glaucomatous eyes are represented in Figure 1 . The general double-hump pattern is evident in both groups, but the glaucomatous mean RNFL profile is clearly blunted in relation to the mean profile of normal subjects. 
Fourier analysis of the thickness measurements in the 32 sectors for each patient resulted in 16 components. The mean amplitudes and phases of the Fourier components in glaucomatous and normal subjects are shown in Table 1 . The Hotelling T-squared test, when applied to the set of Fourier measures, showed a statistically significant difference between the two groups (P < 0.001). Several Fourier measures showed statistically significant differences between glaucomatous and healthy eyes at P < 0.05 (Table 1) . After Bonferroni correction, significant differences were observed for the amplitudes of the F2, F5, and F14 components and also for the DC component and phase of the second component. The stepwise discriminant analysis applied to the Fourier measures resulted in the following LDF (LDF Fourier): LDF = −7.504 + (0.170 × F2 amplitude) + (0.744 × F2 phase) + (0.481 × F5 amplitude) + (1.280 × F14 amplitude) + (0.057 × DC). 
This discriminant function had an area under the ROC curve of 0.949 (bias-corrected 95% confidence interval: 0.895–0.978). The bootstrap estimate of bias under the ROC curve was 0.016. With a cutoff of −0.186 as normal, the LDF correctly labeled 48 of 52 healthy eyes (specificity 92%) and 46 of the 55 glaucomatous eyes (sensitivity 84%). 
The most important variable in the LDF (in terms of discriminative power) was the amplitude of the second Fourier component. Figure 2 shows the graph resulting from inverse Fourier transform using only the mean values from the F2 component and the overall mean RNFL thickness (DC component) of glaucomatous and normal eyes. The F2 component represents a sine wave with a frequency that is two times that of the fundamental (sine wave with one cycle). Thus, the F2 component has two cycles, or humps. The curve profile resulting from the F2 component roughly corresponds to the double-hump pattern of RNFL thickness shown in Figure 1
Table 2 shows the measures for the GDx software–provided parameters in normal and glaucomatous eyes. A Hotelling T-squared test of the set of GDx parameters showed a statistically significant difference between the two groups (P < 0.001). After Bonferroni correction, significant differences were observed for all evaluated parameters except symmetry. Table 3 shows ROC curve areas for all parameters and also for the LDF Fourier. ROC curve areas for the GDx parameters ranged from 0.612 to 0.870. The GDx software–provided parameter with largest area under the ROC curve was the superior average. The ROC curve area for the LDF Fourier was significantly larger than the area for the superior average (P = 0.01; Fig. 3 ). Our study had a power ranging from 100% to 78% to detect statistically significant differences (at α = 0.05) between an area under the ROC curve of 0.95 and areas under the ROC curve ranging from 0.61 to 0.87. Table 3 also shows the sensitivities at fixed specificities (≥80% and ≥90%) for the GDx parameters and LDF Fourier. The LDF Fourier had significantly higher sensitivities (P < 0.05, McNemar test) than did the single best GDx software–provided parameter (superior average) at both specificities. 
Discussion
The present study shows that the Fourier analysis of the RNFL thickness measurements obtained with SLP using variable corneal compensation improves the discrimination of glaucomatous from normal eyes, compared with the parameters provided by the GDx software. The area under the ROC curve for the LDF combining Fourier measures (LDF Fourier) was greater than the ROC area of any of the GDx software–provided parameters investigated. In this study, we also calculated sensitivities at high (≥90%) and moderate (≥80%) specificities. Depending on the patient population and diagnostic criteria, each of these conditions may be more desirable. At a fixed specificity of at least 90%, the LDF Fourier had a sensitivity of 84% to detect glaucoma, whereas the GDx parameters had sensitivities ranging from 24% to 69%. At a fixed specificity of at least 80%, the sensitivities of GDx parameters ranged from 40% to 82%, whereas the LDF Fourier had sensitivity of 93%. 
Previous reports have also shown that GDx software–provided parameters have a limited ability to discriminate between normal and glaucomatous eyes. 6 7 8 9 10 11 The lower diagnostic ability of the GDx parameters may be related to the wide variability of absolute RNFL thickness measurements in healthy subjects. The RNFL thickness has been found to vary widely in the normal population. 4 14 This may limit the identification of glaucomatous eyes with loss of nerve fibers, but with absolute RNFL thickness still within normal limits. Rather than emphasizing absolute thickness itself, the Fourier analysis of RNFL thickness measurements provides a global measure that takes into account the whole shape of thickness distribution around the optic disc, emphasizing relative differences between local areas. In addition, by comparing relative differences in the shape of RNFL thickness distribution curve, the Fourier method may better identify patients with glaucoma with localized RNFL defects compared with the GDx parameters, which are usually calculated based on RNFL thicknesses averaged over a large region. 
Although different methods of analyzing thickness measurements from SLP may result in an improved detection of eyes with glaucomatous RNFL loss compared with the GDx parameters, a persistent source of error may be related to the erroneous compensation for anterior segment birefringence. 17 18 19 20 The erroneous compensation for anterior segment birefringence produces a wider range of retardation measurements in normal eyes, which may complicate the identification of abnormalities. The effects of the axis and magnitude of corneal birefringence on RNFL retardation measurements have been described, 18 23 24 25 and algorithms designed to correct for this have been reported. 19 20 26 Garway-Heath et al. 20 proposed a correction of RNFL retardation measurements obtained using the GDx with fixed corneal compensation using perifoveal or peripapillary temporal retardation values. This method resulted in a narrower normal range of retardation measurements and improvement in the discrimination between normal and glaucomatous eyes. In another approach, Greenfield et al. 19 showed that the incorporation of CPMs improved the discriminatory ability of some GDx parameters. We used SLP data obtained using variable corneal compensation according to the method described by Zhou and Weinreb. 26 This method of anterior segment polarization compensation has been incorporated into the new commercially available scanning laser polarimeter (GDx VCC; Laser Diagnostic Technologies, Inc.) and is based on the determination of the magnitude and axis of anterior segment birefringence by polarimetry imaging of the Henle fiber layer. Individualized anterior segment compensation can be achieved with this method so that the measured retardation largely reflects the RNFL retardance. In a recent work, Weinreb et al. 27 showed that the diagnostic ability of several GDx parameters to classify eyes as glaucomatous or normal is improved considerably with SLP, using variable corneal compensation compared with SLP using fixed corneal compensation. This improvement was stronger for thickness parameters than for ratio–modulation parameters, probably because the latter may already compensate for some of the changes in retardation measurements caused by an inadequate corneal compensation in some patients. In our study, the areas under the ROC curve for the thickness parameters were generally greater than the ROC areas for the ratio/modulation parameters. The three summary parameters—superior average, ellipse average and inferior average—performed comparably. However, the area under the ROC curve for the best GDx parameter, superior average, was still significantly inferior to the area under the ROC curve for the LDF Fourier. 
The GDx parameters evaluated in our study were the ones shown in the standard GDx printout and previously identified as having the best performance to discriminate glaucomatous from normal eyes. Thus, it is not surprising that the means of all GDx parameters (except symmetry) showed significant differences between glaucomatous and normal eyes. In contrast, among all the Fourier components, just a few seem to be important for discriminating glaucomatous from normal eyes. This is also not a surprising result, because the RNFL thickness profile has certain characteristics (like its double-hump shape) which are better described by some Fourier components than by others. When the Fourier components are combined, as in the LDF developed in our study, they become a powerful tool for analysis of polarimetry data, showing higher sensitivity and specificity than GDx parameters. In the present study, we did not examine the diagnostic ability of the GDx parameter, the number, because this parameter has been developed as a best means to interpret data obtained from SLP, using fixed corneal compensation. This parameter is calculated with a neural network approach and has been reported to be the best of the GDx software–provided parameters in the discrimination of glaucomatous from normal eyes. 9 10 11 42 43 44 It is possible that a new neural network based parameter using SLP-VCC data will have better diagnostic precision than the GDx parameters reported in our study, and the comparison of this method to the Fourier analysis of RNFL measurements should be investigated. 
The application of Fourier analysis to the RNFL polarimetry data was initially reported by Essock et al. 16 Using SLP with fixed corneal compensation, they found a sensitivity and specificity of 96% and 90%, respectively, in the differentiation of glaucomatous from normal eyes. In our study, for a similar level of specificity, we found a lower level of sensitivity (84%). Different methods to evaluate the Fourier measurements and different population characteristics may be related to the different results. In the study by Essock et al., the average visual field MD of patients with glaucoma was −8.9 dB, considerably higher than the average MD of the patients included in our study (−5.9dB). Therefore, one of the reasons for the discrepancies in the results of the two studies may be related to different severity of glaucoma in the patient population. In another study using RNFL data obtained from SLP with fixed corneal compensation, Sinai et al. (Sinai MJ, Bowd C, Essock EA, Zangwill LM, Weinreb RN, ARVO Abstract 717, 2001) found an area under the ROC curve of 0.928 for discrimination between glaucoma and healthy eyes, using a Fourier-based LDF. Although this LDF significantly outperformed GDx parameters in that study, the overall unbiased assessment of its performance by split-half analysis resulted in a sensitivity of 73% with specificity of 73%. Fourier measures included in their LDF were the amplitudes of the 2nd, 12th, and 13th components, and also the phases of the 12th and 14th components. In the present study, we found an area under the ROC curve of 0.949 for the LDF, using RNFL data obtained from SLP using variable corneal compensation. The bootstrap estimate of bias of the area under the ROC curve was small and the application of a standard split-half analysis as an internal validation procedure for our LDF resulted in a sensitivity and specificity of 86% and 89%, respectively. This is an improvement over the previously reported Fourier-based LDF obtained with fixed corneal compensation data. Furthermore, the internal validation analysis of our LDF indicates that its robustness seems to be superior to the discriminant function developed using SLP with fixed corneal compensation data. The lower degree of robustness of the Fourier LDF developed with SLP FCC may be related to the wider variability of RNFL retardance measurements introduced by the erroneous compensation of anterior segment birefringence. 
In our LDF, the second Fourier component was the most important (in terms of discriminative power) in the LDF equation. Both the amplitude and phase of this component were entered in the LDF after stepwise discriminant analysis. This is an expected result, because the second component has two humps and hence contributes the most to the shape of the RNFL double-hump curve. The other components serve to shape the pattern provided by the second component, so that the composite curve matches the original RNFL pattern. It is likely that the incorporation of higher-frequency components in the LDF may also be involved in the identification of glaucomatous eyes with localized RNFL defects, although our study did not directly address this question. Also, the pattern of the RNFL thickness profile of normal and glaucomatous eyes shows an apparent indentation of the superior hump (Fig. 1) . This indentation may be related to the split of nerve fiber layer bundles, as described by Colen and Lemij. 45 Higher-frequency Fourier components may be also necessary to better describe this characteristic of the RNFL thickness profile. The mean overall RNFL thickness, or DC component, was also incorporated in our LDF obtained from SLP-VCC data, as opposed to the previously reported LDF obtained from SLP-FCC data. In that the absolute thickness parameters have improved diagnostic precision with the VCC compared with the FCC, 27 this is not a surprising finding. 
One limitation of our study was that we did not test our discriminant function on another cohort of subjects to estimate its sensitivity and specificity independently. However, the provided bootstrap estimate of bias under the area of the ROC curve was relatively small, resulting in a reduction of the area under the ROC curve of 1.7%. Therefore, we expect that our results of sensitivity and specificity may be overstated by approximately the same amount. A recent analysis of our LDF with an independent sample resulted in an area under the ROC curve of 0.941 with sensitivity of 80% and specificity of 90% to discriminate glaucomatous from normal eyes (Michael Sinai, Ph.D., Laser Diagnostic Technologies, unpublished data, September 2002). 
In conclusion, we showed that the combination of Fourier RNFL measures in an LDF obtained using SLP with variable corneal compensation improved the ability to differentiate between healthy eyes and eyes with glaucomatous visual field loss, compared with the GDx software–provided parameters. Further studies using larger samples and different populations are needed to develop and validate standard Fourier-based measures to be incorporated and tested in clinical practice. 
 
Figure 1.
 
Mean RNFL thickness distribution around the optic disc in healthy and glaucomatous eyes.
Figure 1.
 
Mean RNFL thickness distribution around the optic disc in healthy and glaucomatous eyes.
Table 1.
 
Mean Amplitudes and Phases of the Fourier Components in Healthy and Glaucomatous Eyes
Table 1.
 
Mean Amplitudes and Phases of the Fourier Components in Healthy and Glaucomatous Eyes
Fourier Component Amplitude Phase
Healthy Glaucomatous P Healthy Glaucomatous P
DC* 50.7 ± 6.24 40.2 ± 7.43 <.001
F1 5.39 ± 2.52 4.00 ± 2.20 .003 2.40 ± 0.89 2.01 ± 0.84 0.020
F2* , † 12.0 ± 3.05 6.85 ± 3.11 <.001 2.73 ± 0.30 2.47 ± 0.48 0.001
F3 3.60 ± 2.20 2.65 ± 1.35 .009 0.96 ± 0.69 1.00 ± 0.88 0.803
F4 1.66 ± 0.95 1.80 ± 0.97 .450 1.93 ± 0.77 1.92 ± 0.87 0.980
F5* 2.18 ± 0.99 1.44 ± 0.71 <.001 0.73 ± 0.54 1.05 ± 0.79 0.015
F6 1.29 ± 0.64 1.12 ± 0.54 .137 1.02 ± 0.85 1.23 ± 0.73 0.183
F7 1.01 ± 0.51 1.00 ± 0.51 .806 2.04 ± 0.89 1.77 ± 0.88 0.116
F8 0.83 ± 0.45 0.71 ± 0.34 .096 1.94 ± 0.86 1.73 ± 0.78 0.195
F9 0.69 ± 0.35 0.69 ± 0.33 .919 1.50 ± 0.98 1.53 ± 0.94 0.885
F10 0.67 ± 0.37 0.55 ± 0.32 .081 1.29 ± 0.98 1.31 ± 0.86 0.886
F11 0.55 ± 0.25 0.47 ± 0.23 .072 1.62 ± 0.96 1.63 ± 0.96 0.956
F12 0.45 ± 0.25 0.40 ± 0.20 .235 1.80 ± 0.96 1.40 ± 0.86 0.026
F13 0.42 ± 0.21 0.40 ± 0.24 .648 1.59 ± 0.87 1.71 ± 1.00 0.514
F14* 0.48 ± 0.27 0.32 ± 0.17 <.001 1.66 ± 0.91 1.92 ± 0.84 0.120
F15 0.37 ± 0.24 0.27 ± 0.14 .013 1.38 ± 0.79 1.24 ± 0.78 0.366
F16 0.36 ± 0.32 0.28 ± 0.21 .111 1.21 ± 1.54 1.49 ± 1.59 0.362
Figure 2.
 
Inverse Fourier transform obtained from mean values of the second and DC Fourier components in healthy and glaucomatous eyes.
Figure 2.
 
Inverse Fourier transform obtained from mean values of the second and DC Fourier components in healthy and glaucomatous eyes.
Table 2.
 
GDx Software–Provided Parameters in Healthy and Glaucomatous Eyes
Table 2.
 
GDx Software–Provided Parameters in Healthy and Glaucomatous Eyes
Parameter Healthy Glaucoma P *
Symmetry 1.02 ± 0.15 0.97 ± 0.21 0.113
Superior ratio 2.79 ± 0.98 1.84 ± 0.60 <0.001
Inferior ratio 2.75 ± 0.96 1.93 ± 0.61 <0.001
Superior/nasal ratio 2.37 ± 0.60 1.84 ± 0.42 <0.001
Maximum modulation 2.11 ± 0.89 1.30 ± 0.59 <0.001
Ellipse modulation 3.84 ± 1.20 2.58 ± 0.85 <0.001
Average thickness (μm) 47.3 ± 6.08 40.3 ± 7.44 <0.001
Ellipse average (μm) 51.5 ± 6.54 40.7 ± 7.50 <0.001
Superior average (μm) 60.7 ± 8.33 45.3 ± 10.9 <0.001
Inferior average (μm) 59.1 ± 8.98 44.9 ± 10.1 <0.001
Superior integral (mm2) 0.167 ± 0.023 0.128 ± 0.032 <0.001
Superior maximum (μm) 68.7 ± 12.3 52.2 ± 13.1 <0.001
Inferior maximum (μm) 67.6 ± 11.3 55.0 ± 12.8 <0.001
Table 3.
 
Areas under the ROC Curves for Discriminating between Healthy and Glaucomatous Eyes
Table 3.
 
Areas under the ROC Curves for Discriminating between Healthy and Glaucomatous Eyes
Area under ROC Curve (95% CI)* Sensitivity/Specificity (Specificity ≥90%) Sensitivity/Specificity (Specificity ≥80%)
LDF Fourier 0.949 (0.895–0.978) 84/92 93/81
Superior average 0.870 (0.799–0.935) 69/90 82/81
Ellipse average 0.863 (0.782–0.923) 69/90 80/81
Inferior average 0.849 (0.757–0.912) 66/90 78/81
Superior maximum 0.831 (0.744–0.898) 55/90 72/81
Superior integral 0.829 (0.747–0.898) 66/90 67/81
Ellipse modulation 0.819 (0.721–0.889) 49/90 69/81
Superior ratio 0.808 (0.720–0.885) 35/90 62/81
Average thickness 0.791 (0.690–0.872) 53/90 64/88
Superior/nasal 0.783 (0.680–0.866) 49/90 60/81
Inferior maximum 0.782 (0.744–0.898) 53/90 64/81
Maximum modulation 0.781 (0.664–0.847) 36/90 56/81
Inferior ratio 0.771 (0.675–0.846) 40/90 49/81
Symmetry 0.612 (0.503–0.720) 24/90 40/81
Figure 3.
 
ROC curves for the LDF obtained from Fourier measures and for the single parameter with the largest ROC curve area (superior average).
Figure 3.
 
ROC curves for the LDF obtained from Fourier measures and for the single parameter with the largest ROC curve area (superior average).
The authors thank Michael Sinai, Ph.D., for valuable suggestions provided during a critical review of the manuscript. 
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Figure 1.
 
Mean RNFL thickness distribution around the optic disc in healthy and glaucomatous eyes.
Figure 1.
 
Mean RNFL thickness distribution around the optic disc in healthy and glaucomatous eyes.
Figure 2.
 
Inverse Fourier transform obtained from mean values of the second and DC Fourier components in healthy and glaucomatous eyes.
Figure 2.
 
Inverse Fourier transform obtained from mean values of the second and DC Fourier components in healthy and glaucomatous eyes.
Figure 3.
 
ROC curves for the LDF obtained from Fourier measures and for the single parameter with the largest ROC curve area (superior average).
Figure 3.
 
ROC curves for the LDF obtained from Fourier measures and for the single parameter with the largest ROC curve area (superior average).
Table 1.
 
Mean Amplitudes and Phases of the Fourier Components in Healthy and Glaucomatous Eyes
Table 1.
 
Mean Amplitudes and Phases of the Fourier Components in Healthy and Glaucomatous Eyes
Fourier Component Amplitude Phase
Healthy Glaucomatous P Healthy Glaucomatous P
DC* 50.7 ± 6.24 40.2 ± 7.43 <.001
F1 5.39 ± 2.52 4.00 ± 2.20 .003 2.40 ± 0.89 2.01 ± 0.84 0.020
F2* , † 12.0 ± 3.05 6.85 ± 3.11 <.001 2.73 ± 0.30 2.47 ± 0.48 0.001
F3 3.60 ± 2.20 2.65 ± 1.35 .009 0.96 ± 0.69 1.00 ± 0.88 0.803
F4 1.66 ± 0.95 1.80 ± 0.97 .450 1.93 ± 0.77 1.92 ± 0.87 0.980
F5* 2.18 ± 0.99 1.44 ± 0.71 <.001 0.73 ± 0.54 1.05 ± 0.79 0.015
F6 1.29 ± 0.64 1.12 ± 0.54 .137 1.02 ± 0.85 1.23 ± 0.73 0.183
F7 1.01 ± 0.51 1.00 ± 0.51 .806 2.04 ± 0.89 1.77 ± 0.88 0.116
F8 0.83 ± 0.45 0.71 ± 0.34 .096 1.94 ± 0.86 1.73 ± 0.78 0.195
F9 0.69 ± 0.35 0.69 ± 0.33 .919 1.50 ± 0.98 1.53 ± 0.94 0.885
F10 0.67 ± 0.37 0.55 ± 0.32 .081 1.29 ± 0.98 1.31 ± 0.86 0.886
F11 0.55 ± 0.25 0.47 ± 0.23 .072 1.62 ± 0.96 1.63 ± 0.96 0.956
F12 0.45 ± 0.25 0.40 ± 0.20 .235 1.80 ± 0.96 1.40 ± 0.86 0.026
F13 0.42 ± 0.21 0.40 ± 0.24 .648 1.59 ± 0.87 1.71 ± 1.00 0.514
F14* 0.48 ± 0.27 0.32 ± 0.17 <.001 1.66 ± 0.91 1.92 ± 0.84 0.120
F15 0.37 ± 0.24 0.27 ± 0.14 .013 1.38 ± 0.79 1.24 ± 0.78 0.366
F16 0.36 ± 0.32 0.28 ± 0.21 .111 1.21 ± 1.54 1.49 ± 1.59 0.362
Table 2.
 
GDx Software–Provided Parameters in Healthy and Glaucomatous Eyes
Table 2.
 
GDx Software–Provided Parameters in Healthy and Glaucomatous Eyes
Parameter Healthy Glaucoma P *
Symmetry 1.02 ± 0.15 0.97 ± 0.21 0.113
Superior ratio 2.79 ± 0.98 1.84 ± 0.60 <0.001
Inferior ratio 2.75 ± 0.96 1.93 ± 0.61 <0.001
Superior/nasal ratio 2.37 ± 0.60 1.84 ± 0.42 <0.001
Maximum modulation 2.11 ± 0.89 1.30 ± 0.59 <0.001
Ellipse modulation 3.84 ± 1.20 2.58 ± 0.85 <0.001
Average thickness (μm) 47.3 ± 6.08 40.3 ± 7.44 <0.001
Ellipse average (μm) 51.5 ± 6.54 40.7 ± 7.50 <0.001
Superior average (μm) 60.7 ± 8.33 45.3 ± 10.9 <0.001
Inferior average (μm) 59.1 ± 8.98 44.9 ± 10.1 <0.001
Superior integral (mm2) 0.167 ± 0.023 0.128 ± 0.032 <0.001
Superior maximum (μm) 68.7 ± 12.3 52.2 ± 13.1 <0.001
Inferior maximum (μm) 67.6 ± 11.3 55.0 ± 12.8 <0.001
Table 3.
 
Areas under the ROC Curves for Discriminating between Healthy and Glaucomatous Eyes
Table 3.
 
Areas under the ROC Curves for Discriminating between Healthy and Glaucomatous Eyes
Area under ROC Curve (95% CI)* Sensitivity/Specificity (Specificity ≥90%) Sensitivity/Specificity (Specificity ≥80%)
LDF Fourier 0.949 (0.895–0.978) 84/92 93/81
Superior average 0.870 (0.799–0.935) 69/90 82/81
Ellipse average 0.863 (0.782–0.923) 69/90 80/81
Inferior average 0.849 (0.757–0.912) 66/90 78/81
Superior maximum 0.831 (0.744–0.898) 55/90 72/81
Superior integral 0.829 (0.747–0.898) 66/90 67/81
Ellipse modulation 0.819 (0.721–0.889) 49/90 69/81
Superior ratio 0.808 (0.720–0.885) 35/90 62/81
Average thickness 0.791 (0.690–0.872) 53/90 64/88
Superior/nasal 0.783 (0.680–0.866) 49/90 60/81
Inferior maximum 0.782 (0.744–0.898) 53/90 64/81
Maximum modulation 0.781 (0.664–0.847) 36/90 56/81
Inferior ratio 0.771 (0.675–0.846) 40/90 49/81
Symmetry 0.612 (0.503–0.720) 24/90 40/81
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