**Purpose.**:
To introduce a method to optimize structural retinal nerve fiber layer (RNFL) models based on glaucomatous visual field data and to show how such an optimized model can be used to reduce noise in visual fields while probably preserving clinically important features.

**Methods.**:
Correlation coefficients between age-adjusted deviation values of pairs of visual field test locations were calculated from 103 visual fields of eyes with moderate glaucomatous damage. Distances between those test locations were defined for various parameters of a mathematical RNFL model. Then, the correspondence between the structural and functional data was defined by the spread, or variance, of the correlation coefficients for all distances. The model parameters that minimized this spread constituted the optimized model. To reduce noise in visual fields, the optimized model was used to smooth visual field data according to the RNFL's structure. The resulting fields were compared with visual fields that were smoothed based on the regular testing grid.

**Results.**:
The optimal parameters for the RNFL model reduced the variance of the correlation coefficients by 78% and were well within the range of parameters previously determined from fundus photographs. Smoothing the visual fields based on the optimized RNFL model strongly reduced noise while keeping important features.

**Conclusions.**:
Mathematic RNFL models can be optimized based on visual field data, resulting in a strong structure–function relationship. Taking the RNFL's shape, as defined by such an optimized model, into account when smoothing visual fields results in better noise reduction while preserving important details.

^{ 1 }By determining local sensitivity values at various test locations (commonly called retinal sensitivity), it produces an estimate of the visual field. However, the reproducibility of such visual field sensitivity measurements is rather poor, due to high levels of noise caused by, for example, eye movement, blinking, or imprecisions of the measuring device, and most importantly the variability of the patient's response.

^{ 2–5 }A number of different factors have been shown to affect visual field measurements.

^{ 6 }While noise may be reduced by using summary parameters, such as the mean defect (MD) or visual field index (VFI), these parameters hide important details of the visual field, such as its spatial structure. Noise may also be reduced by exploiting spatial and/or temporal relationships in the measurements.

^{ 2 }These methods aim to average or smooth points that are highly correlated, while points that are uncorrelated are excluded in this average.

^{ 2 }The presence of wedge-shaped retinal nerve fiber layer (RNFL) defects in many glaucomatous eyes

^{ 7 }underlines this. In images of these wedge-shaped defects, there is a strong correlation between the points within the wedge. However, points within the defect are largely uncorrelated with points outside the defect, regardless of their Euclidean distance. The same applies to locations within the visual field: in the case of an arcuate scotoma, locations within the scotoma are highly correlated, while neighboring points at the edge of the scotoma are largely uncorrelated. The Euclidean distance, which is for instance used in Gaussian filters, is not well suited to capture this characteristic of the correlation between test locations. Therefore, we would like to find a different measure that defines distance in a way that is more coherent with the correlations found in visual fields and what is known about the RNFL.

^{ 8 }and is therefore of limited use.

^{ 9 }Several authors developed improved filters that incorporate the RNFL structure. Crabb et al.

^{ 9 }built a filter based on point-wise multiple regression of each test location on all other test locations, where the regression coefficients function as weights for the estimation of the sensitivity value at that first location. Gardiner et al.

^{ 2 }pursued a similar approach by applying regression with some constraints on the covariances to derive weights. Betz-Stablein et al.

^{ 10 }used weights that were based on adjacency in the regular grid of test locations as well as physical adjacency to incorporate the spatial structure of glaucomatous damage into their model.

^{ 11,12 }This is a very time-consuming approach and its accuracy may depend on the operator.

^{ 13 }

^{ 11,12,14 }However, the complexity of these models may be very different. For example, an axial model is very simple but quickly becomes inaccurate with increasing distance from the optic nerve head (ONH), while a model with many parameters may be able to largely capture normal morphologic variation between different eyes. With such an RNFL model, the distance between two test locations may be defined in a different way than the conventional Euclidean distance. For example, the distance may be measured along or across fibers.

*x*and

*y*coordinates, is defined by

*α*and

*β*, representing the angle of that point measured at the ONH and the fovea, respectively.

*A*and

*B*are two shape parameters that may be adjusted to adapt the model to the actual nerve fiber bundles. A more detailed description and interpretation of

*α*,

*β*, and the parameters

*A*and

*B*can be found elsewhere.

^{11}In Equation 1, the value of

*Ψ*selects a specific nerve fiber bundle. The angle

*α*

^{*}at which this nerve fiber bundle meets the ONH can be derived from Equation 1 by setting

*y*to 0 and

*β*to

^{ 11 }optimized the shape parameters

*A*and

*B*on observed nerve fiber bundle patterns in fundus photographs.

^{ 15 }

^{ 16 }For the present study, all visual fields measured in the years 1998 through 2011 with an MD between −6 dB and −12 dB were selected. If a patient had more than one field with an MD in that range, the field from the mean time point was included. In all, we used 103 visual fields from 103 patients, which is a subset of the data used by Bryan et al.

^{ 17 }All data that was used in our analyses are freely available at http://orgids.com. Since our goal was to relate defects in the visual field (and not sensitivity values) to the RNFL model and the deviation from age adjusted normal threshold values represent the pattern of loss better than the raw threshold sensitivities, we used those threshold deviations for our analysis. The raw data were exported from the HFA and analyzed with the open source software R, version 3.0.0.

^{ 18 }The observations directly above and below the ONH were not used in our analysis, leaving 52 test locations per eye.

^{ 19 }In a B-spline approach a weighted sum of a set of piecewise defined polynomials is used to approximate and/or interpolate the data. A more detailed description as well as a graphic illustration may be found elsewhere.

^{ 19 }The second property was then evaluated by calculating the root mean square error (RMSE) of the fit of this function to the correlation data. Consequently, the objective function to optimize (i.e., minimize) is given by this RMSE.

*x*and

*y*coordinates, the distance between test locations is given by the Euclidean distance. Figure 1 shows a scatter plot of the Pearson correlation coefficients as a function of the Euclidean distance, for all pairs of visual field test locations, similar to what has been shown before.

^{ 20 }The distances are quantized by the spacing of the test grid. The red line indicates the monotonically decreasing penalized B-spline fit, which had a corresponding RMSE of 0.3. For the whole range of measured Euclidean distances, the variability of the correlation coefficients was rather large, as illustrated by the large spread of the data around the red line.

**Figure 1**

**Figure 1**

*A*= 0.8 and

*B*= 0.015 were selected; the course of the nerve fiber bundles for all 52 test locations are shown in the inset graph of Figure 2. The red line again indicates the result of the fit of the correlation curve. Compared with the Euclidean distance, the angular distance of this model resulted in a reduced variation of the data points around this line (RMSE = 0.23).

**Figure 2**

**Figure 2**

^{ 19,21 }In brief, this method represents the field by a relatively large number of B-splines, which are then smoothed up according to some smoothing parameters. The choice of these smoothing parameters determines the complexity of the smoothed visual field. One way to express complexity is by the effective dimension (ED), which can be calculated for one-dimensional smoothing by

^{ 21 }

*B*is the B-spline matrix (also called design matrix),

*λ*is the smoothing parameter, and

*D*is the difference matrix used to penalize differences in adjacent coefficients. Since

*B*and

*D*are predefined for all fields, the ED only depends on the smoothing parameter used. Related concepts are effective or equivalent degrees of freedom.

^{ 22 }More details, as well as the extension to EDs for 2D penalized B-splines, may be found elsewhere.

^{ 21 }

^{ 11 }

^{ 3 }the average noise level for visual fields in our data set was determined to be approximately 4 dB. Note that this should be considered only as a rough indication of what accuracy may be considered reasonable; the variability of single visual field points can be considerably worse.

*x-y*smoothing, anisotropic

*x-y*smoothing, and smoothing across and along nerve fibers as given by our optimized RNFL model. Here, isotropic means that the same amount of smoothing was done along the

*x*-axis (

*λ*

_{x}) and

*y*-axis (

*λ*

_{y}). For anisotropic smoothing, this may be different. Isotropic smoothing for the RNFL model was not considered meaningful; smoothing across (

*λ*

_{across}) or along (

*λ*

_{along}) fibers operates on very different scales (ONH angle versus length).

*A*was evaluated from 0.1 to 0.8 with a step size of 0.1, and

*B*was evaluated from 0 to 0.04 with a step size of 0.005. For each combination of shape parameters, the monotonic penalized B-spline model was fitted and the corresponding RMSE was computed. Figure 3 shows a contour plot of the RMSE, with the minimum error at

*A*= 0.4 and

*B*= 0.02. Figure 4 shows the correlation-distance plot and nerve fiber layout of this optimized model. Again, the red line indicates the monotone fitted line. The RMSE was calculated to be 0.14 for the optimized model.

**Figure 3**

**Figure 3**

**Figure 4**

**Figure 4**

*x-y*smoothing (

*λ*

_{x}and

*λ*

_{y}) and for smoothing along and across the nerve fibers as given by the optimized RNFL model (

*λ*

_{along}and

*λ*

_{across}). Different smoothing parameters resulted in a different complexity (expressed by ED) and representation error (expressed by RMSE). For

*x-y*smoothing, the results are presented in Figure 5A. The solid and dashed lines mark the sets of optimal smoothing parameters for anisotropic and isotropic smoothing, respectively. Points on those lines are optimal in the sense that one criterion (ED or RMSE) cannot be improved without adversely affecting the other one. The results for smoothing according to the optimized RNFL model are shown in Figure 5B. Again, the solid line marks the optimal smoothing parameters.

**Figure 5**

**Figure 5**

*x-y*smoothing, anisotropic smoothing produces significantly better results (Wilcoxon signed-rank test,

*P*< 0.0008 for each value of the ED ≤ 20) in terms of RMSE and ED. The best method, however, was smoothing based on the optimized RNFL model. Over the relevant range of ED (cf. estimated noise level) its optimal set lay below the two other optimal (

*P*< 0.025). That means that RNFL-based smoothing can produce a less complex representation of a visual field with the same error or a representation with a smaller error for the same complexity. It also implies that the RNFL-based smoothing method provided the least complex representation for reproducing a visual field up to the level of its reproducibility: ED is 14.2 for the RNFL-based model compared with ED equaling 16.0 for anisotropic

*x-y*smoothing, and ED equaling 18.0 for isotropic

*x-y*smoothing.

**Figure 6**

**Figure 6**

**Figure 7**

**Figure 7**

*A*= 0.4 and

*B*= 0.02) are well within the range of the shape parameters found by fitting the RNFL model to fundus photos.

^{ 11 }

^{ 2 }This concurs with our results: when smoothing based on the

*x-y*coordinates is applied to the visual field, either by isotropic smoothing (Fig. 7B) or anisotropic smoothing (Fig. 7C), the relatively small defect in the central visual field is blurred. Optimized RNFL models provide a great opportunity to incorporate the structure of the RNFL in the smoothing procedure. This is clearly illustrated in Figure 7D, which shows that this defect is preserved when smoothing based on the RNFL model. Due to the defect's location close to the fovea, such a defect should be considered important and it should therefore not be degraded by processing of the visual field.

*x-y*coordinates or coordinates based on the RNFL model). Smoothing of visual fields could be further improved by including weights in the fitting process. Such weights may, for example, be based on the variability of measurements for different sensitivity levels

^{ 3,23 }or on the density of ganglion cells over the measured area of the retina.

^{ 24,25 }It is important to note that our approach does not take into account that the strength of relation between points with the same distance is probably not uniform over the whole field, depending on the density of the ganglion cells.

^{ 2,9,10,20 }As mentioned above, and in contrast to other authors,

^{ 14,26 }we do not aim to produce an individualized RNFL map, but a general model that can provide a sufficient approximation to the RNFL structure, as shown in Figure 4. Our optimized RNFL model captures the variability in the individual RNFL structures in the RMSE, which contains the interindividual variability as well as the error made by approximating the real RNFL structure with the model. Our RNFL model differs from other models for instance in simplifying assumptions like symmetric hemispheres and that the ONH and fovea lie on a horizontal line. It has been shown previously that several ocular parameters may have significant influence on the layout of the RNFL.

^{ 27 }Although many of those parameters are currently not available for our study, it would be possible to include information on an individual eye's distance and angle between fovea and ONH into the model we used. Another factor to consider when comparing structural and functional measurements near the fovea is the displacement of ganglion cells.

^{ 28 }We estimated that the resulting additional error (∼1.7° for the four points nearest to the fovea) was insignificant in our analysis, but correction may be required when analyzing correlation of structure and function for an individual or when looking at denser grids.

^{ 29 }

^{ 15 }that relates visual field test locations to ONH angles based on fundus photos. Recently, it has been shown that tracing nerve fiber bundles in such photos is associated with poor reproducibility, suggesting that the resulting nerve fiber maps have a high variability (Denniss J, et al.

*IOVS*2013;54:ARVO E-Abstract 1883). An optimized mathematical RNFL model, as proposed in the present work, does establish the same kind of relationship, but for continuous rather than discretized ONH angles, and can, therefore, also be used in these and other filters. Furthermore, ONH angles derived from an optimized RNFL model, as done in this study, relate any location in the visual field to an angle at the ONH, without being restricted to a certain pattern of test locations.

^{ 30–32 }or by a Heidelberg Retina Tomograph.

^{ 33 }Our method is able to establish and optimize such a relationship without additional measurements, thereby showing a clear and strong structure–function relationship between the RNFL morphology and the visual field threshold deviation values.

^{ 34 }Temporal averaging of repeated visual fields is the most straightforward method to achieve this, but it requires too many resources for clinical use. However, it may be used in a study setup to produce a ground truth. Future studies are needed to quantify the effect of filtering visual fields based on the presented approach and its contribution to the assessment of glaucoma and glaucomatous progression.

**N.S. Erler**, Carl Zeiss Meditec (F), Heidelberg Engineering (F);

**S.R. Bryan**, None;

**P.H.C. Eilers**, None;

**E.M.E.H. Lesaffre**, None;

**H.G. Lemij**, Carl Zeiss Meditec (F), Heidelberg Engineering (F);

**K.A. Vermeer**, Carl Zeiss Meditec (F), Heidelberg Engineering (F)

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