Structure and Function in Glaucoma: The Relationship between a Functional Visual Field Map and an Anatomic Retinal Map

Abstract

purpose. To examine the relationship between an anatomic map relating the retinal nerve fiber layer (RNFL) distribution to the optic nerve head and a functional map derived from the interpoint correlation of raw sensitivities in visual field (VF) testing.

methods. Previously, interpoint correlations were generated for all possible pairs of VF test points in a dataset of 98,821 Humphrey VF test results taken from the Moorfields Eye Hospital archive. The relationship between these correlations and the physical distance between the VF test point pairs was evaluated by Pearson’s correlation coefficient and multiple regression analysis. The distance between the pairs of VF test points was calculated in two ways. First, the anatomic map was used to estimate the angular distance at the optic nerve head (ONH), between the RNFL bundles corresponding to the VF test points in each pair (ONHd). Second, the retinal distance between pairs of test points was calculated from the Humphrey VF template (RETd). A best-fit model for predicting functional correlation (FC) from ONHd and RETd was constructed and used to formulate a filter incorporating the anatomic-functional correlation data.

results. All scatterplots showed a negative association between interpoint retinal sensitivity correlation values and distance between points: ONHd (R ² = 0.60) and RETd (R ² = 0.33). The raw sensitivity correlation values could be predicted from a multiple regression model using ONHd, RETd, and a combined interaction of ONHd and RETd (R ² = 0.75, P < 0.00001). The construction of a new filter was based on the equation FC = 0.9325 − (0.0029 · ONHd) − (0.0077 · RETd) + (0.0001 · ONHd · RETd).

conclusions. A good level of association was observed between the strength of correlation between points in the VF and the relative location of those test points in the peripheral retina and in corresponding RNFL bundles at the ONH. These results help to validate the relationship between structure and function and may be of use in the further refinement of physiologically derived VF filters to reduce measurement noise.

Glaucoma is a progressive optic neuropathy in which structural changes to the optic nerve head (ONH) and peripapillary retina are associated with characteristic visual field (VF) defects. ¹ These functional deficits correspond to the anatomic distribution of the retinal nerve fiber layer (RNFL). Glaucomatous field loss, if allowed to progress, will result in functionally significant visual loss and ultimately affect the subject’s quality of life. In routine clinical practice, visual function is frequently assessed with static automated perimetry. The detection of progression using this technique is confounded by intertest and intratest variability inherent within the testing process. Threshold sensitivity may be influenced by factors including patient response fluctuation, patient experience, and patient fatigue. ^{2 3 4 5} Variability has also been shown to increase in areas of glaucomatous deficit in which there is established reduced sensitivity. ^{2 6 7} For progression to be detected reliably, the VF change due to glaucoma must be distinguished from this measurement “noise.” 

One approach to reducing measurement variability is the post hoc application of a spatial filter. Spatial filtering is a technique adapted from digital imaging processing whereby the measured sensitivity of a particular test point is adjusted based on the sensitivity of other points in the vicinity. A novel spatial filter has been designed with the intention that it closely mimics the physiological relationship between VF test points. ⁸ This filter predicts the sensitivity of each test point based on the sensitivities at other locations in the field, weighted according to the predictive power of each of those locations. Predictive power was determined by examining the correlations and covariances between sensitivities among all pairs of test locations, within a large database of 98,821 predominantly glaucomatous VFs. The derivation of this filter has generated an array of interpoint absolute correlations for the entire field. These values effectively constitute a mathematically derived “functional map” based on physiological data. The relationship between points demonstrated by the filter should therefore closely mirror the structural pattern of the nerve fiber layer, although it may be influenced by idiosyncrasies of the Humphrey Visual Field Analyzer (HFA; Carl Zeiss Meditec, Inc., Dublin, CA) testing algorithm used. 

In this study, we examined the relationship between this functional map and an anatomic map relating the RNFL distribution to the ONH. ⁹ The purpose of the study was to identify whether the magnitude of functional correlation between points is related to the relative proximity of the points at the ONH and in the retinal periphery. This information may give insight to the anatomic organization of the ONH, the glaucomatous disease process, and enable the refinement of filters applied to the VF series to reduce measurement noise. 

Methods

Description and Application of the Anatomic Map

A map based on the anatomic structure of the ONH and RNFL has been described by our group. ⁹ This map was derived from 69 RNFL photographs from 69 patients with normal-tension glaucoma in whom the course of discrete RNFL defects or prominent nerve fiber bundles could easily be traced. Using this method, the ONH location, in degrees, corresponding to each test point was calculated, with the temporal meridian designated 0° and measured in a clockwise fashion for the left eye and counterclockwise for the right eye. In the present study, the angular distance between all possible pairings of test points were calculated using the map. Where the angular distances exceeded 180°, a correction was incorporated so that the shortest angular distance between points was calculated, by subtracting the distance from 360°. Angular distance between test points at the ONH was designated ONHd (Fig. 1) . 

The distance between test points within the peripheral retina was calculated using the Humphrey Visual Field Analyzer 24-2 template (Carl Zeiss Meditec, Inc.). Within the template, there is a 6° separation between horizontally or vertically adjacent points and 8.5° between diagonally adjacent points (using the Pythagorean theorem, h ² = a ² + b ² ). In this study the retinal distance between all possible pairs of test points was calculated and designated RETd (Fig. 1) . This was calculated for the global VF and for pairings within the upper and lower hemispheres. 

Description and Application of the Functional Map

The derivation of the novel spatial filter has been described in detail elsewhere. ⁸ Briefly, the filter was derived from VFs from the Moorfields Eye Hospital database. This contains 98,821 VFs, taken from 14,675 individual patients with suspected glaucoma. The tests performed were all standard white-on-white, full-threshold tests; only complete 24-2 tests were used in the generation of the filter, although all levels of test reliability were included. The relationship between individual test points was elucidated by examining the covariances and correlations between the sensitivities of all test-point pairings throughout the database. These interpoint correlations (52 × 51 = 2652 in total) were used in this study as a gauge of “functional relationship” between points, and the basis of the comparison with the physical distances obtained from the anatomic map. 

Comparison of the Anatomic Map and the Functional Map

Interpoint correlations were compared to RETd and ONHd using Pearson correlation coefficient and multiple regression analyses. Comparisons were performed for the entire template, for the upper and lower hemispheres and between hemispheres. Linear models for predicting VF correlations (FCs) from ONHd and RETd were assessed. The application of the prediction model to derive a “physiological” filter was illustrated for one VF test point. First, FCs predicted for that test point were generated; R ² was generated by squaring the predicted FCs. An R ² cutoff of 0.7 was selected to identify which test point pairings should be included in the filter for that test point. All statistical analyses were performed using S (AT & T Bell Laboratories, Murray Hill, NJ). This study adhered to the tenets of the Declaration of Helsinki. 

Results

A strong negative association was observed between the VF FCs and the angular distance at the ONH (R ² = 0.60). That is, points corresponding to locations closer together at the ONH tended to show better correlation than those farther apart (Fig. 2) . A weaker negative association was observed between the FC and the retinal distance (R ² = 0.33; Fig. 3 ). These results, for the entire VF template, are summarized in Table 1 . 

To elucidate whether the two distances are independent predictors of functional interpoint correlation, “best-fit” models incorporating both distances were investigated. A multiple regression model incorporating ONHd, RETd, and the product of the two (ONHd · RETd) was identified as having the highest R ² when plotted against functional interpoint correlation (R ² = 0.75, Fig. 4 ). The interaction coefficient of the regression model is very low (0.0001), though still significant (P < 0.00001) due to the large number of points involved. However, when the interaction between the two variables is removed from the model, the linear relationship between the raw VF values and the combination of ONHd and RETd is still present. The results of the regression models are summarized in Table 2 . 

Figure 5illustrates a plot of interpoint correlation against angular distance for both the upper and lower hemispheres. The association between functional correlations and anatomic distance was much better when restricted to one hemisphere at a time (Fig. 6) . 

The best-fit model for predicting FC was therefore  

\[\mathrm{FC}{=}0.9325{-}(0.0029{\cdot}\mathrm{ONHd}){-}(0.0077{\cdot}\mathrm{RETd}){+}(0.0001{\cdot}\mathrm{ONHd}{\cdot}\mathrm{RETd})\]

 

A new filter was derived for test point 49 by using this predictive model; its derivation is illustrated in Figure 7 . The ONHd (Fig. 7a)and RETd (Fig. 7b)for point 49 were calculated. Using the regression equation, we predicted FCs for each test point in relation to point 49. The FC results were squared to generate R ² (Fig. 7c) . The test points included in the filter were those with R ² ≥ 0.7 (highlighted in bold in Fig. 7c ). Finally, the weightings by which each test point’s sensitivity influences the sensitivity of point 49 were calculated by dividing each R ² value by the sum of all R ² values included in the filter (Fig. 7d) . 

Discussion

Spatial filtering is an attractive proposition, as it may reduce measurement noise without the need for additional VF testing. This is particularly useful when analyzing long series of VF data acquired over several years. The first method of spatial filtering tested, Gaussian filtering, was shown to reduce test-retest variability and to reduce measurement noise. ^{10 11} However, the use of the Gaussian filter is limited by the simple nature of its construction; it was originally designed for use in digital image processing. It is based on a 3 × 3 test-point grid whereby the sensitivity of the central test point is adjusted according to the relative weights of the other points in the grid. The same filter is applied for each point in the field (in a methodical, not physiological, fashion) and the weightings of the other points in the grid remain fixed throughout the field. It is clear that measurement error should be attenuated by this approach, but its non-physiological nature results in blurring (or erasing) of established field defects. This was demonstrated by Spry et al., ¹² who examined the effect of Gaussian filtering on detection of glaucomatous progression using point-wise linear regression analysis compared with raw threshold sensitivity data. Although Gaussian filtering was able to improve specificity by reducing variability of non-progressing locations, sensitivity was reduced as it also attenuated useful signal. This was particularly a problem for small defects, indicating that Gaussian filtering would result in difficulty in the detection of early perimetric change. 

A more useful approach to constructing a spatial filter would be to exploit the functional or anatomic relationship between test points. Initially, the point by point spatial dependence was determined by multiple regression analysis of sensitivity values for each test point in a dataset of 440 Humphrey VFs. ¹³ A similar investigation reports the relationship between sensitivities of test points using the 32 program of Octopus 1-2-3 (Interzeag, Schlieren-Zurich, Switzerland). ¹⁴ In this study, linear regression analysis among each of the locations and the rest of the points in the field was performed. The methodology used in the construction of our filter was similar, although the mathematical relationship between sensitivities was assessed using covariances and correlations and the number of fields assessed was much larger. In particular, all available VF data were used in the construction of the filter, so as to be truly representative of a glaucoma clinic population. It may therefore not be suitable for use in normal subjects or subjects with nonglaucomatous—for example, neurological—field defects. With simulated progressing VF data, the novel filter was found to improve both specificity and sensitivity. ⁸ When used on a 50-patient sample of longitudinal field data, the filter has been shown to reduce variability, and it does not reduce detection of loss by total deviation maps (Artes PH et al., IOVS 2005;46:ARVO E-Abstract 3732). This method represents a clear improvement in the performance of the Gaussian filter, although the effect of the filter has yet to be fully assessed on prospective clinical data. An additional observation from this study is that the filter improved the “pattern” of progression compared with unfiltered VFs, so that it more closely resembled the defect appearance expected in glaucoma. This result would be expected if the physiological relationships exploited in the construction of the filter are valid. 

An encouraging level of agreement between the magnitude of functional correlation between points and the relative location of the points at the ONH and the retinal periphery was observed in this study. There was a negative association between functional correlation and both ONHd and RETd. Using a multiple regression model with the product of ONHd and RETd, we were able to predict interpoint correlations. It should be noted that the model continues to predict FC well with the interaction term removed, which may indicate that the dependent association between ONHd/RETd and interpoint sensitivity correlation may not be large. However, although the coefficient for the interaction term is small (0.0001), it cannot be dismissed completely, as an increase in R ² from 0.6 to 0.75 was observed when the interaction was included. The interaction term is intended to account for the nonlinearity observed in the models shown in Figures 2 and 3 . The minimal impact of the interaction term on the predictive model may suggest that the nonlinearity observed has a negligible influence. The regression equation used to construct the example filter was therefore derived from a predictive model that included the interaction between ONHd and RETd. In the construction of the filter, only predicted FCs with ONHd correlations >0.84 (R ² ≥ 0.7) were included, and the ONHd/FC relation is clearly linear over this range (Fig. 5) . However, as should be expected, the relationship between ONHd and the VF correlations appears to be wholly valid (and linear) within the same hemisphere but not between hemispheres (Fig. 6) . 

As glaucomatous damage is believed to manifest at the ONH, it seems logical that VF locations that correspond to similar regions of the ONH should be well correlated; damage to that area of the ONH affects all such points. In this study, retinal proximity was also found to be a predictor of the strength of correlation between two points. This observation has implications for both disease process and anatomic organization, although with the caveat that it is unknown whether the finding is real or spurious. If the observation is “real,” it may support the hypothesis that RNFL bundles from similar peripheral eccentricities are closely located at the optic nerve head. Experimental studies in different species of the macaque monkey have generally suggested that a degree of retinotopic organization exists with respect to the eccentricity of axonal origin, although they tend to differ in terms of exact detail. ^{15 16 17 18} To date, there has been little by way of clinical observation to support this hypothesis, ¹⁹ although the results of this study may support such a finding in the context of a disease model where glaucomatous damage occurs at the ONH. An alternative explanation applies to a model in which damage occurs primarily in the retina. In this situation, damage may propagate from dysfunctional or dying retinal ganglion cells locally within the retina. ONHd has a much higher coefficient of determination than RETd, suggesting that the glaucomatous process more likely occurs at the ONH, although the ONH and retinal models are not mutually exclusive. The FC/RETd relationship, however, may in part be spurious, resulting from measurement error. The error may be systematic, perhaps related to inaccuracies in the anatomic map. Random error may relate to interindividual variation of ONH position in relation to the fovea, ⁹ or to fixation losses occurring during visual field testing. 

The comparison between interpoint functional correlation and the anatomic map is dependent on the assumption that the relationships described by the anatomic map are valid. Alternative maps have been described that were developed with similar techniques. ^{20 21} The map used in the present study has already been used in structure-function studies in glaucoma. ^{22 23 24} The adoption of an alternative map, such as that developed by Junemann et al. ²² has been based on the simplicity of use, as opposed to any perceived greater integrity compared to the map used in our study. ²⁵ Recently, a map has been described that was developed using both static automated perimetry and Heidelberg Retina Tomograph (HRT) data. ²⁶ This newer map therefore differs from the map used in our study, in that it incorporated both structural and functional information in its development, although the result is similar to the map used in the present study. 

Structural data have been incorporated into the construction of a spatial filter, based on the multiple regression predictive model that incorporates the angular distance between test points at the optic nerve head, the angular distance between test points in the retinal periphery, and the interaction between the two distances. In the example used in this study, which is for a single test point, the filter has a similar distribution of test-point associations compared with those generated using the “physiological” filter of Gardiner et al. ⁸ Both filters follow an “arcuate” pattern, in keeping with what might be expected given the distribution of the retinal nerve fiber layer. The newly developed “structural” filter does include fewer points, however, that have more similar weightings relative to each other, compared with the physiological filter. This may be explained by the fact that all the points, bar one, included in the structural filter are directly adjacent to the point of interest and as such may be expected to have a similar relation, according to a linear model. The method of constructing the physiological filter downplayed points that strongly covaried with, but had lower predictive value than, other predicting points. This method was not used in the new structural filter. By not downplaying strongly associated points, measurement noise reduction may be improved through increased signal averaging; however, whether this confers an advantage in the detection of signal should be tested and will be the subject of further work. The use of the predictive equation developed in this study enables the construction of filters that may be customized on a point-wise basis. A “bespoke” spatial filter is particularly useful if one wants to exclude test points from a longitudinal series if they are consistently depressed by a mechanism other than glaucoma—for example, by lid artifact or chorioretinal scarring. Likewise, the physiological filter is limited as it is designed for use with the 24-2 program of the HFA. A point-wise customizable filter may be adopted in the context of different Humphrey programs (such as 10-2) and may also be used in alternative proprietary perimeters. 

The associations identified in our study help to validate the structure-function relationship in glaucoma and give insight into the anatomic organization of the ONH and glaucomatous disease process. The incorporation of structural data may be of benefit in the development of more refined spatial filters to reduce measurement noise in VF testing. 

 Supported by a Friends of Moorfields Research Fellowship and an unrestricted grant from Heidelberg Engineering.

 Submitted for publication December 29, 2005; revised May 17 and August 23, 2006; accepted October 11, 2006.

 Disclosure: N.G. Strouthidis, Heidelberg Engineering (F); V. Vinciotti, None; A.J. Tucker, None; S.K. Gardiner, None; D.P. Crabb, None; D.F. Garway-Heath, Carl Zeiss Meditec (C)

 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Method used to calculate interpoint RETd and ONHd. The Humphrey 24-2 template was superimposed on a retinal nerve fiber layer (RNFL) photograph. The boundaries of a prominent RNFL defect have been demarcated. The distance between the limits of the defect both at the optic nerve head and in the retina have also been marked.

Figure 2.

 

Scatterplot demonstrating the relationship between the interpoint FC and ONHd. A linear trend is observed from a 0° to 90° angular distance from the test point of interest. R ² = 0.6 assumes a linear relationship between the FC and ONHd.

Figure 3.

 

Scatterplot demonstrating a weak linear relationship between the interpoint FC and RETd (R ² = 0.33).

Table 1.

 

Multivariate Correlation Coefficients

	ONHd	RETd
ONHd	1.00	0.34
RETd	0.34	1.00
FC	0.60	0.33

 Matrix demonstrating R ² for the interactions between the anatomic and functional relationships. ONHd, angular distance (deg) between visual field test points at the optic nerve head; RETd, angular distance (deg) between visual field test points at the retinal periphery; FC, interpoint functional correlation of visual field threshold sensitivity values.

Figure 4.

 

Scatterplot demonstrating a linear relationship between the interpoint FCs, and a multiple regression model incorporating ONHd, RETd, and the product of the two (ONHd · RETd).

Table 2.

 

Multiple Regression Coefficients

	Intercept	ONHd	RETd	ONHd*RETd
Coefficient	0.9325	−0.0029	−0.0077	0.0001
Probability	<0.00001	<0.00001	<0.00001	<0.00001

 Shown are multiple regression coefficients and probabilities for the interactions between FC and ONHd, RETd and the combined product of the two, Abbreviations are defined in Table 1 .

Figure 5.

 

Scatterplot demonstrating the relationship between interpoint FC and distance at the ONHd for each hemisphere in isolation. A linear relationship between FC and ONHd is observed for test points in the upper and in the lower hemispheres.

Figure 6.

 

Scatterplot comparing the relationship between the interpoint FC and the ONHd within the same hemispheres and between different hemispheres. A linear association between test points within the same hemisphere is observed from 0° to 90° angular ONHd.

Figure 7.

 

The derivation of a new filter for test point 49 based on the regression equation shown in the Methods section. (a) The angular distances (in degrees) between each test point and point 49 at the optic nerve head (ONHd) are illustrated. X, point 49; ONHd is 0 at this point. (b) The angular distances between each test point and point 49 in the retinal periphery (RETd) are illustrated. X, point 49; RETd is 0 at this point. (c) R ² values generated by squaring the FC predicted by the regression equation. Values achieving an R ² cutoff of 0.7 are highlighted in bold. These test points are included in the filter for point 49. (d) The “weightings” for each test point in the filter based on the predictive model for point 49 (bold) are shown. The weightings are estimated by dividing each point’s R ² by the sum of R ²s included in the filter. When applying the filter, the sensitivity for each test point is multiplied by its relative weighting; the “weighted” sensitivities are summed for each point included in the filter, to generate the “filtered” sensitivity for point 49. (e) The weightings for each test point associated with point 49, based on the physiologically derived filter described by Gardiner et al. ⁹