October 2012
Volume 53, Issue 11
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Glaucoma  |   October 2012
An Anatomically Customizable Computational Model Relating the Visual Field to the Optic Nerve Head in Individual Eyes
Author Affiliations & Notes
  • Jonathan Denniss
    From the Departments of Optometry and Vision Sciences and
    Computing and Information Systems, The University of Melbourne, Melbourne, Victoria, Australia.
  • Allison M. McKendrick
    From the Departments of Optometry and Vision Sciences and
  • Andrew Turpin
    From the Departments of Optometry and Vision Sciences and
    Computing and Information Systems, The University of Melbourne, Melbourne, Victoria, Australia.
  • Corresponding author: Andrew Turpin, Computing and Information Systems, The University of Melbourne, Melbourne, Victoria 3010, Australia; aturpin@unimelb.edu.au
Investigative Ophthalmology & Visual Science October 2012, Vol.53, 6981-6990. doi:10.1167/iovs.12-9657
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      Jonathan Denniss, Allison M. McKendrick, Andrew Turpin; An Anatomically Customizable Computational Model Relating the Visual Field to the Optic Nerve Head in Individual Eyes. Invest. Ophthalmol. Vis. Sci. 2012;53(11):6981-6990. doi: 10.1167/iovs.12-9657.

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

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Abstract

Purpose.: To present a computational model mapping visual field (VF) locations to optic nerve head (ONH) sectors accounting for individual ocular anatomy, and to describe the effects of anatomical variability on maps produced.

Methods.: A previous model that related retinal locations to ONH sectors was adapted to model eyes with varying axial length, ONH position and ONH dimensions. Maps (n = 11,550) relating VF locations (24-2 pattern, n = 52 non–blind-spot locations) to 1° ONH sectors were generated for a range of clinically plausible anatomical parameters. Infrequently mapped ONH sectors (5%) were discarded for all locations. The influence of anatomical variables on the maps was explored by multiple linear regression.

Results.: Across all anatomical variants, for individual VF locations (24-2), total number of mapped 1° ONH sectors ranged from 12 to 90. Forty-one locations varied more than 30°. In five nasal-step locations, mapped ONH sectors were bimodally distributed, mapping to vertically opposite ONH sectors depending on vertical ONH position. Mapped ONH sectors were significantly influenced (P < 0.0002) by axial length, ONH position, and ONH dimensions for 39, 52, and 30 VF locations, respectively. On average across all VF locations, vertical ONH position explained the most variance in mapped ONH sector, followed by horizontal ONH position, axial length, and ONH dimensions.

Conclusions.: Relations between ONH sectors and many VF locations are strongly anatomy-dependent. Our model may be used to produce customized maps from VF locations to the ONH in individual eyes where some simple biometric parameters are known.

Introduction
Imaging devices are increasingly available for measuring optic nerve head (ONH) and retinal nerve fiber layer (RNFL) parameters. These devices offer alternative, complementary information to functional tests such as perimetry and electrophysiology for the diagnosis and management of optic neuropathies. Commonly, such as in glaucoma, no single test presents a complete picture of a patient's disease state and as such, it is desirable to compare and combine information from different tests. 
In the case of glaucoma, combining imaging with perimetry is clinically desirable for at least two reasons. First, it is desirable to combine results from structural and functional tests post hoc to improve sensitivity and specificity of both disease detection and progression measures. Second, imaging data may be useful in developing patient-specific targeted functional tests, for example by customizing the locations tested in a perimetric procedure. 
In order for these objectives to be achieved, it is necessary to know the topographic relationship between regions of the ONH and retinal (and, therefore, visual field) locations. Several such schemes have been published as a result of different techniques being applied to populations of patients in order to relate ONH and visual field locations. Garway-Heath et al. 1 traced visible nerve fiber bundles from the ONH to points on a 24-2 visual field pattern overlaid on fundus photographs to produce the map which is perhaps most commonly used at present. Jansonius et al. 2 took a similar approach in tracing visible nerve fiber bundles in fundus photographs, and then fitting a mathematical model to the results thus allowing for generalization to other areas of the central retina. Gardiner et al. 3 investigated statistical correlations between visual field sensitivity (24-2 pattern) and neuroretinal rim area in 36 sectors in patients with or at risk of glaucoma. This method could be easily generalized to other visual field patterns, but is hampered somewhat by the noisy relationship between current structural and functional measures in glaucoma. Our group has previously published a scheme based on the methods of Gardiner et al., but using a computational model of axon growth to restrict the locations deemed to be related to those that are physiologically plausible. 4  
One barrier to combining imaging information with functional tests is the considerable noise common to relationships between all tests. 58 This noise is likely to occur from multiple sources, one of which is between-subject variations in how retinal locations relate to the ONH. 9 As all previously published schemes are based on population data, they share the common pitfall that they do not take into account between-subject differences in ocular anatomy, nor are they customizable to individual eyes. The facility to produce custom maps for individual eyes, relating any pattern of functional test locations to the ONH, would be extremely useful for combining or comparing tests for individual patient management. 
In this article, we describe a new computational model of axon paths in the RNFL that produces maps relating any central retinal location to the ONH. The model takes into account individual anatomical parameters, and can be customized to individual eyes given some simple biometric data. We explore the potential variation between maps according to individual anatomy, and compare this to the variability found in previous studies using the visible nerve fiber bundle tracing method. Our computational modeling approach holds the advantage that the model can produce maps for any given combination of anatomical parameters, including those that occur infrequently and are therefore unlikely to appear in small population studies. 
Methods
Computational Model of Axon Paths in the Retina
How retinal locations, and therefore points in the visual field, are related to sectors of the ONH is determined by the endpoints of retinal ganglion cell (RGC) axons connecting them. Our previously published model assumed the retina to be a spherical surface, and axon paths in the RNFL to be efficient; that is, they take the shortest available path around the spherical surface from cell body to ONH. 4 The ONH was divided into sectors, and the retina was divided into a grid of elements. The number of RGCs in each grid element was derived from empirical data. 10 The choice of path from an element to the ONH was governed by two things: First, the number of axons which could enter the ONH in each sector was determined by a representative normal RNFL thickness profile derived from studies using optical coherence tomography (OCT). If an axon's shortest path to the ONH led to an ONH sector that was already full of axons, it was redirected nasally until it reached a sector with available space. Second, the order in which axons “grew” from different areas of the retina was set such that the temporal sectors of the ONH were quickly filled, causing the remaining axons to be redirected away from these sectors. This had the effect of preventing fiber paths from crossing the fovea. 
The model reported herein is based on the previous model and makes the same basic assumptions of a spherical retinal surface and axons taking the shortest available path to the ONH. However, there are several differences: 
  1.  
    The number of axons that can enter an ONH sector is determined by the area of neuroretinal rim in that sector (this allows for the simulation of different ONH configurations and further customization of the model to individual eyes);
  2.  
    The papillomacular bundle is treated differently (see later);
  3.  
    Elements grow their axons in order of increasing distance from the ONH 11 ; and
  4.  
    The retina is divided into a square (rather than hexagonal) grid of elements for easier computation.
As visual field locations are conventionally described in Cartesian coordinates centered on the fovea, this is the coordinate system we use to describe the model. Given that visual field locations are inverted relative to their corresponding retinal locations, we must choose one system to describe the model. We use “retinal” coordinates as it simplifies the geometry. The model is described for a right eye, so in this article, positive latitude refers to superior retina, and longitude increases in the nasal direction. To convert to conventional visual field coordinates, one simply inverts the sign of the latitude. At the ONH, sectors are labeled in degrees from the temporal side (so for a right eye, 9 o'clock is 0, 12 o'clock is 90, 3 o'clock is 180, and 6 o'clock is 270). 
As data in the literature on RGC density across the retina is given for flat-mounted retinas, 10,12 the number of RGCs per grid element in the model was determined using stereographic projection of the grid onto a flat plane. The position of the center of each grid element on a flat plane was calculated, and the number of RGCs per mm2 derived by linear interpolation of Figure 5 in Curcio and Allen. 10 This density figure was then multiplied by the area of the stereographic projection of each element onto a flat plane to give the total number of RGCs in each grid element. 
Table 1 gives the algorithmic details of the model. To summarize, the ONH is divided into 1° sectors and the central retina is divided into a grid of elements. RGCs in each grid element (in order of increasing distance from the ONH 11 ) attempt to send their axons to the closest ONH sector. If this sector is already “full” of axons, as determined by the neuroretinal rim area in the sector divided by the average axonal diameter, then an alternative sector must be found. 
Table 1. 
 
Algorithmic Details for the Computational Model of Axon Growth
Table 1. 
 
Algorithmic Details for the Computational Model of Axon Growth
Input L, a list of grid squares sorted by increasing distance from the ONH;
Ag , the sector containing the angle from grid square g to the ONH;
Cg , the number of RGCs in grid square g; and
Ni , a count of the number of axons allowed into sector i of the ONH.
1. Set papDone ← false and S ← empty list.
2. For each grid square g∈L, in order of increasing distance from ONH do
3.  If distance of g from ONH is more than that of the fovea, then
4.   Set papDone ← true.
5.   Set NJ ← 0, where J is the set of foveal sectors.
6.  Set T ← empty list. // possible sectors containing Cg 's axons.
7.  While Cg > 0 do
8.   If Cg NAg // need more than sector Ag for these axons
9.    Record that NAg axons went into sector Ag in list T.
10.    Set Cg Cg NAg and set NAg ← 0.
11.    Set Ag ← next sector for g.
12.   Else
13.    Record that Cg axons went into sector Ag in list T.
14.    Set NAg NAg − Cg and set Cg ← 0.
15.  Choose the sector from T which has the largest axon count and assign it to Sg .
Output S, a list of sectors for all grid locations.
Steps 3 through 5 in Table 1 require explanation. The sectors that fall into the “shadow” of the fovea projected onto the ONH along the shortest path between the two (as shown in Fig. 1a) are defined as papillomacular sectors. If the total number of RGCs in the macular region is smaller than the total number of axons allowed into the papillomacular sectors, then once grid elements in the macular region have been processed, there will still be space in the papillomacular sectors for further axons. If this is allowed to be the case, then other elements beyond the fovea can send their axons to the papillomacular sectors, crossing the fovea. Since this is not observed in real eyes, we explicitly set the available space in papillomacular sectors to be zero once axon growth from macular elements is complete. 
Figure 1. 
 
Schematic showing how papillomacular sectors were defined (a) and the direction of stepwise search for next available ONH sector (b). (a) Papillomacular sectors are shaded. (b) The solid line passing through the ONH (actually part of a great circle on the assumed sphere through the centers of the ONH and fovea) is defined as the midline. The arrow for each shaded sector indicates which direction the search for the next available sector will go, assuming the shaded sectors are full.
Figure 1. 
 
Schematic showing how papillomacular sectors were defined (a) and the direction of stepwise search for next available ONH sector (b). (a) Papillomacular sectors are shaded. (b) The solid line passing through the ONH (actually part of a great circle on the assumed sphere through the centers of the ONH and fovea) is defined as the midline. The arrow for each shaded sector indicates which direction the search for the next available sector will go, assuming the shaded sectors are full.
Steps 8 through 14 in Table 1 also require further explanation. If there are more axons from an element g than available space in the sector representing the shortest path from the element to the ONH Ag , then as many axons as possible enter sector Ag but then the rest must enter the next available sector. The next available sector is found by stepwise search away from the fovea and midline as shown in Figure 1b. The midline is defined as part of the great circle passing through the centers of the fovea and ONH, as shown in Figure 1b. 
Anatomical Parameters and Production of Maps
Maps were produced from the ONH to each of the 52 non–blind spot locations in the 24-2 visual field pattern using the computational model. Maps varied according to the input of eight anatomical parameters to the model, as described in Table 2. Note that while the model can accept different input values for CUPh, CUPv, and FD, they were kept constant in this study to keep the number of variables manageable (i.e., five parameters varied). The inputs for each parameter were taken from population data in the literature to represent clinically plausible values. 1,10,1315  
Table 2. 
 
Input Parameters to the Computational Model of Axon Paths and the Values Used in this Study
Table 2. 
 
Input Parameters to the Computational Model of Axon Paths and the Values Used in this Study
Parameter Description Range
AL Axial length of eye (used as diameter of sphere) [20, 21, 22 … 30] mm
ONHx Longitude of the center of the ONH [13, 14, 15 … 18] degrees
ONHy Latitude of the center of the ONH [−1, 0, 1 … 5] degrees
DH Horizontal ONH diameter [1.2, 1.4, 1.6 … 2.4] mm
DV Vertical ONH diameter [1.4, 1.6, 1.8 … 2.6] mm
CUPh Horizontal cup diameter 0.83 mm
CUPv Vertical cup diameter 0.77 mm
FD Foveal diameter 0.4 mm
A map was produced for every possible combination of the parameter values shown in Table 2, with the constraint that (DV − 0.6) < DH < (DV + 0.4). 
This constraint was applied to prevent unlikely ONH shapes from being modeled. In this way 11,550 unique maps were produced, each representing a different combination of the above anatomical parameters. For each map we recorded the 1° ONH sector mapped from each of the 52 non–blind spot locations in the 24-2 visual field test pattern. 
Statistical Analysis
Consideration was first given to removing anatomically unlikely parameter combinations. For example, an eye with very short axial length is probably not likely to have a very eccentrically located ONH. Rather than attempt to predict which parameter combinations would produce erroneous configurations in advance, we opted for removing these post hoc, simply discarding the least-frequently mapped 5% of ONH sectors for each visual field location. 
The influence of anatomical variables on the mapping between individual visual field locations and the ONH was explored by multiple linear regression. As the ONH sectors form a circular domain (that is sectors 1 and 360 are adjacent despite their large numerical difference) ranges of mapped ONH sectors for each visual field location were examined for discontinuities around the 1/360 border. In all cases where such a discontinuity existed, we subtracted 360 from all sectors >180 in order to make the range continuous for the regression. For example, a visual field location mapping to sectors from 1 to 10 and 350 to 360 would have been transformed to a continuous range from −10 to 10 prior to performing the regression. Care was taken when applying this transform that it did not induce any new discontinuities 180° around from the corrected discontinuity. Separate linear models (n = 52) were fitted to data from each non-blind spot visual field location and standardized regression coefficients were calculated for all parameters in each model. This technique computes the change in mapped ONH sector per standard deviation change in each anatomical variable. It is a unit-independent measure of the effect of each anatomical variable on the mapping for each visual field location. Linear models had the form 
where c is a constant and α1–5 are the regression coefficients for each parameter as defined in Table 2. A measure of the relative effect of each parameter on each location was also calculated as the standardized regression coefficient multiplied by the variation in mapped sector at the same location. This allows for comparison of the effect of changes in an anatomical parameter at different locations. 
The distribution of ONH sectors mapped to each visual field point was compared with the ranges given by Garway-Heath et al. 1 and Jansonius et al. 2 Neither of these represents an ideal reference standard for our model for two reasons: First, each study used methods of aligning the photographs which would substantially reduce the effect of between-subject variation in anatomy. Second, the sample sizes of each study were small relative to the potential variation in the population. We simulate 11,550 equally weighted parameter combinations (in effect 11,550 unique eyes), whereas the population studies analyze 552 or 691 eyes each whose parameters are probabilistically distributed and therefore may share common characteristics. As such, we expect some overlap between the results of these studies and our model, but we do not expect perfect concordance. However, in the absence of an ideal reference we compared our model results to these maps as a way of checking whether our results are within the approximate range expected. 
Both the implementation of the computational model and all subsequent statistical analyses were carried out in the open-source environment, R (http://www.r-project.org/, in the public domain). Statistical significance was defined as P < 0.0002 to account for multiple (260) comparisons. 
Results
Effect of Anatomy on Maps
Figure 2 shows the unique 1° ONH sectors that mapped to each visual field location. For individual locations across the range of anatomical variants, the total number of unique mapped 1° ONH sectors ranged from 12 to 90. Note that as can be seen in Figure 2, the sectors are not necessarily continuous, nor does this inform us of the frequency distribution within the range (see “Comparison with Existing Schemes” later). 
Figure 2. 
 
One-degree ONH sectors mapped from each 24-2 visual field location. Circle size is proportional to the number of unique 1° ONH sectors mapped, which is also represented by the number in the circle. Darker shaded areas represent the mapped ONH sectors. Points shaded green mapped to within 30° across all parameters, while points shaded orange were more variable in their mapping. Points shaded red are the bimodal points referred to in the text. Note that in general, superior locations map to the inferior ONH and vice versa, this is due to the inversion of the visual field relative to the retina.
Figure 2. 
 
One-degree ONH sectors mapped from each 24-2 visual field location. Circle size is proportional to the number of unique 1° ONH sectors mapped, which is also represented by the number in the circle. Darker shaded areas represent the mapped ONH sectors. Points shaded green mapped to within 30° across all parameters, while points shaded orange were more variable in their mapping. Points shaded red are the bimodal points referred to in the text. Note that in general, superior locations map to the inferior ONH and vice versa, this is due to the inversion of the visual field relative to the retina.
Eleven of 52 locations always mapped to within 30° across the range of anatomical parameters modeled, these are shaded green in Figure 2. The variation in ONH sector mapped from these locations was thought unlikely to be clinically significant. All of these locations were in the temporal field. In the nasal field, there was more variability in mapping, and more inferiorly than superiorly (due to the tendency for the ONH to be above the horizontal midline of the retina rather than below). The distribution of mapped ONH sectors for five points in the nasal-step area (shaded red in Fig. 2) was bimodal, mapping to vertically opposite sides of the ONH depending on parameter combination. Points shaded orange in Figure 2 are those for which mapped ONH sectors varied 30° or more, but were unimodally distributed. 
Figure 3 shows the standardized regression coefficients (β = number of standard deviations change in mapped ONH sector per standard deviation change in anatomical variable) for each anatomical variable from multiple linear regression analysis, color-coded as in Figure 2. Note that this does not take the total variation in mapping into account (this is addressed later), it only reflects the influence of each parameter on the variation shown in Figure 2. We can see from Figure 3 that the ONH sector mapped from the above mentioned “bimodal locations” is determined largely by ONHy, the vertical position of the ONH. This can be explained with reference to Figure 1b, which shows the “midline” as part of a great circle on the sphere intersecting the centers of the ONH and the fovea. The position of a retinal location above or below this great circle determines whether the shortest path between it and the ONH is to a sector above or below the midline on the ONH. If this sector is “full” then the search for an alternative sector follows the direction of the arrow in Figure 1b. As the vertical ONH position (ONHy) increases (moves further above the horizontal), a certain point is reached at which the great circle moves past the retinal locations, making their shortest path be to the opposite side of the ONH. 
Figure 3. 
 
Standardized regression coefficients for effect of each anatomical variable on mapped ONH sector for each visual field location in a multiple linear regression. Bars at each location represent (left to right): AL, ONHx, ONHy, DH, DV and the y-axes are scaled from 0 to 1.5, as shown in the example at the top left. Bar plots at each location are color-coded as in Figure 2.
Figure 3. 
 
Standardized regression coefficients for effect of each anatomical variable on mapped ONH sector for each visual field location in a multiple linear regression. Bars at each location represent (left to right): AL, ONHx, ONHy, DH, DV and the y-axes are scaled from 0 to 1.5, as shown in the example at the top left. Bar plots at each location are color-coded as in Figure 2.
Other locations in the visual field are affected differently by anatomical variables, largely as would intuitively be expected, for example horizontal ONH position has a greater relative influence on locations immediately above and below the ONH. On average across all visual field locations, vertical ONH position explained the most variance in mapped ONH sectors (mean R 2 = 0.52, β = 0.88), followed by horizontal ONH position (mean R 2 = 0.20, β = 0.38), axial length (mean R 2 = 0.15, β = 0.31), horizontal ONH diameter (mean R 2 = 0.03, β = 0.25) and vertical ONH diameter (mean R 2 = 0.01, β = 0.10). 
A more complete picture of the effect of each parameter is given in Figure 4, which shows a relative measure of the effect size of each parameter on each visual field location significantly (P < 0.0002) affected by that parameter. The size of the circles in Figure 4 is proportional to β multiplied by the number of unique ONH sectors mapped at each location (as shown in Fig. 2). Locations not significantly affected by each parameter are marked with an “X.” Figure 4 allows us to compare the relative effects of each parameter on mapped ONH sector across all locations. From Figures 4 and 5, it is clear that in the majority of locations where mapped ONH sector varies more than 30°, vertical ONH position (ONHy) has the predominant effect on mapped ONH sector. 
Figure 4. 
 
Relative effects of (a) axial length, (b) horizontal ONH position, (c) vertical ONH position, (d) horizontal ONH diameter, (e) vertical ONH diameter on mapped ONH sectors at each visual field location. Relative effects of each anatomical parameter on mapped ONH sectors at each visual field location. Circle size is proportional to standardized regression coefficient scaled according to total variation (as in Fig. 2), larger circles mean a given change in the parameter has a larger effect on the ONH sector mapped from the visual field location. Locations where mapped ONH sector is not significantly affected (P < 0.0002) by the parameter are marked with an “X.” The plots are color-coded as in Figure 2.
Figure 4. 
 
Relative effects of (a) axial length, (b) horizontal ONH position, (c) vertical ONH position, (d) horizontal ONH diameter, (e) vertical ONH diameter on mapped ONH sectors at each visual field location. Relative effects of each anatomical parameter on mapped ONH sectors at each visual field location. Circle size is proportional to standardized regression coefficient scaled according to total variation (as in Fig. 2), larger circles mean a given change in the parameter has a larger effect on the ONH sector mapped from the visual field location. Locations where mapped ONH sector is not significantly affected (P < 0.0002) by the parameter are marked with an “X.” The plots are color-coded as in Figure 2.
Figure 5. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (superior arcuate areas). Each plot represents one of the four arcuate regions of the Garway-Heath et al. map, and each row within a plot represents one visual field location as labeled. The histograms on each row show the distribution of mapped ONH sectors from our model. The shaded grey area represents the sectors used in the Garway-Heath et al. map, the blue line represents the mean ±2*median standard deviation as reported by Garway-Heath et al. 1 The red line represents the mean and upper/lower limits of the predicted ONH sectors from the mathematical model proposed by Jansonius et al. 2 The green dots represent a single “typical eye” from our model (see “Comparison with Existing Schemes”).
Figure 5. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (superior arcuate areas). Each plot represents one of the four arcuate regions of the Garway-Heath et al. map, and each row within a plot represents one visual field location as labeled. The histograms on each row show the distribution of mapped ONH sectors from our model. The shaded grey area represents the sectors used in the Garway-Heath et al. map, the blue line represents the mean ±2*median standard deviation as reported by Garway-Heath et al. 1 The red line represents the mean and upper/lower limits of the predicted ONH sectors from the mathematical model proposed by Jansonius et al. 2 The green dots represent a single “typical eye” from our model (see “Comparison with Existing Schemes”).
Comparison with Existing Schemes
Figures 5 and 6 show the distribution of ONH sectors mapped from visual field points in the arcuate areas of the visual field, and how these correspond to those found in previous studies. 1,2 As discussed in the methods section, we do not expect our results to directly relate to those found by these previous studies due to methodological and study population differences; however, we use them as a reference for whether our results are plausible. The range plotted in Figures 5 and 6 for the study by Jansonius et al. (in red) was derived by calculating the mean and upper and lower limits at each location from their published mathematical model. 2 The range plotted for the study by Garway-Heath et al. 1 (in blue) represents their reported mean ONH sector ± 14.4° (two median standard deviations as reported). In Figures 5 and 6, the shaded grey area represents the area of the ONH the visual field locations map to in the final map produced by Garway-Heath et al. 1 (their Fig. 7), and this is also shaded on the diagrammatic representation of the ONH and 24-2 visual field locations in each plot. Green dots in Figures 5 and 6 represent the ONH sector mapped from each visual field location in our model with a single set of “typical” parameters (AL = 25 mm, ONHx = 15°, ONHy = 2°, DH = 1.6 mm, DV = 1.8 mm). Qualitative examination of these plots shows that in most locations there is reasonable overlap between our model and the two previous studies. The larger disparities shown are generally either in the locations far from or close to the ONH (i.e., the greatest concordance appears to be in the mid-peripheral arcuate areas). 
Figure 6. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (inferior arcuate areas). See Figure 5 for description.
Figure 6. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (inferior arcuate areas). See Figure 5 for description.
Discussion
This article presents a computational model that relates visual field locations to corresponding sectors of the ONH. The model is a new modification of an earlier model which was reported in a correlational study of ONH and visual field damage in glaucoma patients. 4 The updated model follows the same basic principles and assumptions as the previous model, but now takes some simple biometric information as input, allowing it to be customized to individual patients' ocular anatomy. Although we used the 24-2 visual field test pattern in this study, the model can easily be used to map any given location in the visual field to the ONH. 
According to our model, variations in ocular anatomy have little effect on how some visual field locations relate to the ONH (Fig. 2). Mapping from other locations is more strongly affected by variation in anatomy, and individual anatomical parameters contribute differently to this depending on location (Figs. 3, 4). Because of these variations due to anatomy, the use of generalized maps such as that published by Garway-Heath et al. 1 is a potential source of noise in studies aiming to relate structural and functional measures, as has been noted in a previous clinical study. 9  
The model presented herein allows for customized maps to be produced for individual patients, potentially reducing this noise. To produce such a map for a patient only requires some simple biometric information: axial length, ONH position relative to the fovea, and vertical and horizontal ONH diameters. The information on ONH position and diameters can be measured from fundus photographs, provided that head position is appropriately controlled. Alternatively, we have had some success in our lab in obtaining these measures from confocal scanning laser ophthalmoscope topography images (HRTII; Heidelberg Engineering GmbH, Heidelberg, Germany) using custom software and with fixation adequately controlled at a known location (the internal fixation light). Using this method it is also possible to obtain measures of ONH radius in 256 meridians, which could be incorporated into our model with some simple modifications, replacing the horizontal and vertical ONH diameters used in this study. This method could also easily be applied to other instruments such as optical coherence tomographers or scanning laser polarimeters. In situations where this biometric data is incomplete, there may still be some benefit in producing custom maps using the available information. For example, if axial length is not available, then since ONH position explained the most variance in mapping then it would still be worthwhile to account for this when attempting to combine or compare structural and functional information. 
To our knowledge, the “bimodal” distribution of mapped ONH sectors either side of the horizontal ONH midline for five nasal-step visual field locations has not been previously reported. Jansonius et al. reduced the effects of anatomical variation on their model by superimposing photographs using rotation and scaling before tracing nerve fiber bundles. 2 Jansonius et al. also fitted separate equations to the superior and inferior retina. 2 These factors would have precluded their study from finding this result. Our experience with tracing nerve fiber bundles in fundus photographs suggests another possible reason why studies using this method have not reported this result; in all but exceptional quality photographs, it is extremely difficult to reliably trace nerve fiber bundles out as far as these nasal-step locations. This is evident in Garway-Heath et al. where individual visual field locations could only be traced back to the ONH in median 22 (range 4–58) of a possible 69 photographs, with central and arcuate locations being more often traced than more peripheral locations. 1 Further to this, studies have typically used ophthalmically trained observers to carry out the tracing of nerve fiber bundles. Such observers may have an inherent preconception of a horizontal temporal raphe (rather than a sloping or curved raphe), which may lead to them being unlikely to trace fibers beyond the horizontal midline where there is any doubt. 
In common with previous studies, the absence of an ideal reference standard for maps linking visual field locations to the ONH makes it difficult to validate our model with certainty. Computational modeling approaches such as ours do, however, provide a valuable framework for empirical testing and development of future experimental hypotheses. The ability to produce a map for any given combination of anatomical parameters is unmatched by population studies, where infrequently occurring combinations are unlikely to occur in small samples and so are not represented by the maps produced. Another advantage to the computational modeling approach is the ease with which the model can be updated and improved as new empirical evidence becomes available. This plasticity will allow the model, through frequent updates, to keep pace with a quickly progressing evidence base. 
Comparison of our model's output with the mathematical model proposed by Jansonius et al. 2 and the map produced by Garway-Heath et al. 1 appears to show reasonable qualitative agreement for most locations, with the exception of the “bimodal” locations as discussed above. This is despite methodological and study population differences that mean that we can only expect approximate agreement between the two. Concordance between our model and the previous studies appears greatest in the midperipheral arcuate locations, with greater discordance occurring in the peripheral locations or those close to the ONH. Discordance in peripheral locations may be explained due to the nerve fiber tracing issues discussed above. One possible reason for discordance in locations close to the ONH is that when tracing nerve fiber bundles close to the ONH, it is extremely difficult to tell which fibers originate from the point of interest, and which originate further away and pass over the point of interest. According to both our model and the previous studies, axons originating further out enter the ONH more nasally, and so inadvertently tracing fibers that actually originate further out than the point of interest will result in a more nasal sector being recorded. This is consistent with the observed discordance between our model and the two previous studies. It should be noted, however, that comparing new models with previous models might be a poor means of validation as it holds the potential to perpetuate errors and hold back improvements. 
The finding in this study of “bimodal” points in the nasal step area may provide some explanation for why glaucoma patients exist who have “nasal step” visual field defects in the same hemifield as their apparent structural damage, and for why nasal step defects in clinical visual field tests often cross the horizontal midline. One study used statistical methods to investigate the relationship between locations in the 24-2 test pattern in the visual field database of a large glaucoma clinic in order to produce a spatial filter for reducing noise in test results. 16 An interesting finding, shown in their Figure 4, was a strong interrelation between the two most peripheral nasal locations, crossing the horizontal midline. The authors suggested this could be as a result of some fibers crossing the raphe slightly as observed in histological studies. 17 We propose that a more likely reason for these two points to be related in a large database is an anatomical link between them in some patients. This would be the result of an ONH sufficiently above or below the horizontal midline through the fovea for the shortest path from the visual field location to the ONH to be to the opposite side of the ONH to that conventionally expected. In that case our model would predict that the two locations would map to similar ONH sectors and so would both be affected by focal glaucomatous damage to those sectors. Further validation for our model can be taken from the finding by Garway-Heath et al. that ONH position explained almost half of the variance in mappings to more than half of visual field locations in their study. Our study is in agreement that ONH position is the major contributor to variability in mapping, and that this applies to varying degrees depending on location. 
One reference standard to which our model could ideally be compared would be a histological or autoradiographic study of human retinas, preserving their anatomical differences. In the absence of this, it may be possible in the future to obtain some further validation by using the model to investigate anatomical variation as a possible cause of apparent structure-function dissociation in the nasal step area of patients with glaucoma. 
A noteworthy difference between the present model and that reported previously 4 is how the space available for axons to enter each ONH sector is defined. The previous model used a representative normal RNFL profile from an OCT study, whereas in the present model, we base this on the neuroretinal rim area in the ONH sector. This makes the model more customizable, as it allows ONH shape and neuroretinal rim configuration (predisease) to influence the maps produced. In addition, due to the typically vertically elliptical shape of the ONH and horizontally elliptical shape of the neuroretinal rim in normals 13 the profile of neuroretinal rim area around the ONH is commonly of a “double-hump” shape, similar to that of the RNFL thickness profile. This means that on average, this method makes little difference to the maps produced, but eliminates sources of variability such as blood vessel location 18,19 and scan circle placement 20 in customizing the model to individuals. It also allows more detailed shape parameters of the ONH to be incorporated in future with a simple modification to the model, as described above. 
Our model assumes a spherical retinal surface, but human retinal surfaces may often be better approximated by slightly oblate or prolate ellipsoids. 21 If the shape of the modeled eye is significantly elliptical, for example in high myopia, then Ag in Table 1 will be affected as it is determined by the shortest path from RGC body to ONH around the modeled surface. These initial deviations may be partly removed by later stages of the model (steps 2–15 in Table 1), and so the effect on the final structure-function mapping is unclear. We have not examined the magnitude of the effect of using an elliptical retinal curvature because histological data for ganglion cell density across the retina is only currently available projected onto a spherical surface. 10 It is also conceivable that some eyes are not well approximated by any regular shape; these present a challenge for future generations of our model. 
The assumption underlying this model is that clinically measurable biometric information can be used to more accurately map visual field locations to the ONH. For some eyes, there will be factors not in our model that may influence such a mapping. For example, variations in ganglion cell number and distribution, other potential influences on RNFL bundle topography such as blood vessel positions, 18,19 and differences in growth cues during development may cause departures from our model in some eyes. This model concentrates only on factors that are clinically measurable, thus enabling further validation in future studies. 
In conclusion, the computational model presented in this article represents a step forward in usefully combining individual patients' structural and functional test data in the management of optic neuropathies such as glaucoma. By providing some simple biometric data as input to the model, individually customized maps relating any visual field location to the ONH can be derived. These maps may be useful in guiding the placement of test locations for customized visual field assessment, or in the post hoc combination or comparison of structural and functional measures. 
Acknowledgments
The authors thank Nomdo Jansonius (University of Groningen, The Netherlands) and Chota Matsumoto and Fumi Tanabe (Kinki University, Japan) for informative discussions. 
References
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Footnotes
 Supported by Australian Research Council Linkage Project LP100100250 (with Heidelberg Engineering GmbH, Germany); Australian Research Council Future Fellowship FT0990930 (AMM); Australian Research Council Future Fellowship FT0991326 (AT)
Footnotes
 Disclosure: J. Denniss, Heidelberg Engineering GmbH (F); A.M. McKendrick, Heidelberg Engineering GmbH (F); A. Turpin, Heidelberg Engineering GmbH (F)
Figure 1. 
 
Schematic showing how papillomacular sectors were defined (a) and the direction of stepwise search for next available ONH sector (b). (a) Papillomacular sectors are shaded. (b) The solid line passing through the ONH (actually part of a great circle on the assumed sphere through the centers of the ONH and fovea) is defined as the midline. The arrow for each shaded sector indicates which direction the search for the next available sector will go, assuming the shaded sectors are full.
Figure 1. 
 
Schematic showing how papillomacular sectors were defined (a) and the direction of stepwise search for next available ONH sector (b). (a) Papillomacular sectors are shaded. (b) The solid line passing through the ONH (actually part of a great circle on the assumed sphere through the centers of the ONH and fovea) is defined as the midline. The arrow for each shaded sector indicates which direction the search for the next available sector will go, assuming the shaded sectors are full.
Figure 2. 
 
One-degree ONH sectors mapped from each 24-2 visual field location. Circle size is proportional to the number of unique 1° ONH sectors mapped, which is also represented by the number in the circle. Darker shaded areas represent the mapped ONH sectors. Points shaded green mapped to within 30° across all parameters, while points shaded orange were more variable in their mapping. Points shaded red are the bimodal points referred to in the text. Note that in general, superior locations map to the inferior ONH and vice versa, this is due to the inversion of the visual field relative to the retina.
Figure 2. 
 
One-degree ONH sectors mapped from each 24-2 visual field location. Circle size is proportional to the number of unique 1° ONH sectors mapped, which is also represented by the number in the circle. Darker shaded areas represent the mapped ONH sectors. Points shaded green mapped to within 30° across all parameters, while points shaded orange were more variable in their mapping. Points shaded red are the bimodal points referred to in the text. Note that in general, superior locations map to the inferior ONH and vice versa, this is due to the inversion of the visual field relative to the retina.
Figure 3. 
 
Standardized regression coefficients for effect of each anatomical variable on mapped ONH sector for each visual field location in a multiple linear regression. Bars at each location represent (left to right): AL, ONHx, ONHy, DH, DV and the y-axes are scaled from 0 to 1.5, as shown in the example at the top left. Bar plots at each location are color-coded as in Figure 2.
Figure 3. 
 
Standardized regression coefficients for effect of each anatomical variable on mapped ONH sector for each visual field location in a multiple linear regression. Bars at each location represent (left to right): AL, ONHx, ONHy, DH, DV and the y-axes are scaled from 0 to 1.5, as shown in the example at the top left. Bar plots at each location are color-coded as in Figure 2.
Figure 4. 
 
Relative effects of (a) axial length, (b) horizontal ONH position, (c) vertical ONH position, (d) horizontal ONH diameter, (e) vertical ONH diameter on mapped ONH sectors at each visual field location. Relative effects of each anatomical parameter on mapped ONH sectors at each visual field location. Circle size is proportional to standardized regression coefficient scaled according to total variation (as in Fig. 2), larger circles mean a given change in the parameter has a larger effect on the ONH sector mapped from the visual field location. Locations where mapped ONH sector is not significantly affected (P < 0.0002) by the parameter are marked with an “X.” The plots are color-coded as in Figure 2.
Figure 4. 
 
Relative effects of (a) axial length, (b) horizontal ONH position, (c) vertical ONH position, (d) horizontal ONH diameter, (e) vertical ONH diameter on mapped ONH sectors at each visual field location. Relative effects of each anatomical parameter on mapped ONH sectors at each visual field location. Circle size is proportional to standardized regression coefficient scaled according to total variation (as in Fig. 2), larger circles mean a given change in the parameter has a larger effect on the ONH sector mapped from the visual field location. Locations where mapped ONH sector is not significantly affected (P < 0.0002) by the parameter are marked with an “X.” The plots are color-coded as in Figure 2.
Figure 5. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (superior arcuate areas). Each plot represents one of the four arcuate regions of the Garway-Heath et al. map, and each row within a plot represents one visual field location as labeled. The histograms on each row show the distribution of mapped ONH sectors from our model. The shaded grey area represents the sectors used in the Garway-Heath et al. map, the blue line represents the mean ±2*median standard deviation as reported by Garway-Heath et al. 1 The red line represents the mean and upper/lower limits of the predicted ONH sectors from the mathematical model proposed by Jansonius et al. 2 The green dots represent a single “typical eye” from our model (see “Comparison with Existing Schemes”).
Figure 5. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (superior arcuate areas). Each plot represents one of the four arcuate regions of the Garway-Heath et al. map, and each row within a plot represents one visual field location as labeled. The histograms on each row show the distribution of mapped ONH sectors from our model. The shaded grey area represents the sectors used in the Garway-Heath et al. map, the blue line represents the mean ±2*median standard deviation as reported by Garway-Heath et al. 1 The red line represents the mean and upper/lower limits of the predicted ONH sectors from the mathematical model proposed by Jansonius et al. 2 The green dots represent a single “typical eye” from our model (see “Comparison with Existing Schemes”).
Figure 6. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (inferior arcuate areas). See Figure 5 for description.
Figure 6. 
 
Comparison of the distribution of mapped ONH sectors from our model to schemes published by Garway-Heath et al. 1 and Jansonius et al. 2 (inferior arcuate areas). See Figure 5 for description.
Table 1. 
 
Algorithmic Details for the Computational Model of Axon Growth
Table 1. 
 
Algorithmic Details for the Computational Model of Axon Growth
Input L, a list of grid squares sorted by increasing distance from the ONH;
Ag , the sector containing the angle from grid square g to the ONH;
Cg , the number of RGCs in grid square g; and
Ni , a count of the number of axons allowed into sector i of the ONH.
1. Set papDone ← false and S ← empty list.
2. For each grid square g∈L, in order of increasing distance from ONH do
3.  If distance of g from ONH is more than that of the fovea, then
4.   Set papDone ← true.
5.   Set NJ ← 0, where J is the set of foveal sectors.
6.  Set T ← empty list. // possible sectors containing Cg 's axons.
7.  While Cg > 0 do
8.   If Cg NAg // need more than sector Ag for these axons
9.    Record that NAg axons went into sector Ag in list T.
10.    Set Cg Cg NAg and set NAg ← 0.
11.    Set Ag ← next sector for g.
12.   Else
13.    Record that Cg axons went into sector Ag in list T.
14.    Set NAg NAg − Cg and set Cg ← 0.
15.  Choose the sector from T which has the largest axon count and assign it to Sg .
Output S, a list of sectors for all grid locations.
Table 2. 
 
Input Parameters to the Computational Model of Axon Paths and the Values Used in this Study
Table 2. 
 
Input Parameters to the Computational Model of Axon Paths and the Values Used in this Study
Parameter Description Range
AL Axial length of eye (used as diameter of sphere) [20, 21, 22 … 30] mm
ONHx Longitude of the center of the ONH [13, 14, 15 … 18] degrees
ONHy Latitude of the center of the ONH [−1, 0, 1 … 5] degrees
DH Horizontal ONH diameter [1.2, 1.4, 1.6 … 2.4] mm
DV Vertical ONH diameter [1.4, 1.6, 1.8 … 2.6] mm
CUPh Horizontal cup diameter 0.83 mm
CUPv Vertical cup diameter 0.77 mm
FD Foveal diameter 0.4 mm
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