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.

^{ 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 }

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^{ 11 }

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^{ 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 }

^{ 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 }

^{ 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.

*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.

^{ 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.

^{ 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).

^{ 1 }

^{ 11 }

^{ 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.

*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.

^{ 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.

^{ 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 }

*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).

*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.

^{ 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.

^{ 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.

^{ 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.

^{ 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.

^{ 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.

**Figure 1.**

**Figure 1.**

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.**

**Figure 2.**

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 (mm^{2}) | 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 |

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.**

**Figure 3.**

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