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1 From the Glaucoma Unit, Moorfields Eye Hospital, London, United Kingdom; the 2 Glaucoma Division, Jules Stein Eye Institute, UCLA, Los Angeles, California; and the 3 Department of Visual Science, Institute of Ophthalmology, University College London, United Kingdom.
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Abstract |
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METHODS. Normative human data for ganglion cell density within 30° of retinal eccentricity were taken from the literature and corrected for lateral displacement of ganglion cells from the fovea to estimate ganglion cell receptive field density (g). The number of ganglion cell receptive fields within a Goldmann size III target (G) was calculated from target area (A) and receptive field density (g) [G = A (g)]. Normative data for DLS in the central 30° (Humphrey 30-2) were taken from the literature. The coefficient summation (k) was measured empirically at each Humphrey 30-2 test point in 8 normal subjects. The relationship between DLS and G was investigated by plotting DLS as decibels (dB) against G and DLS as 1/L (1/Lamberts) against Gk. The physiological relationship was extrapolated to glaucomatous ganglion cell loss by calculating hypothetical cell losses for 3 and 6 dB sensitivity defects at each test point.
RESULTS. Spatial summation increased with eccentricity. The relationship between DLS (dB) and G was curvilinear. The relationship between DLS (1/L) and Gk was linear (r2 = 0.73). The extrapolation to glaucomatous ganglion cell loss indicated that a proportionally greater loss of ganglion cells is required in the central compared with peripheral visual field for equal losses in dB sensitivity.
CONCLUSIONS. The number of underlying ganglion cells, adjusted for local spatial summation, is better reflected by the DLS scale of 1/L than by dB. If spatial summation is unchanged in glaucoma, this scale more accurately reflects the amount of neuronal damage.
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Introduction |
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Differential light sensitivity (DLS) is the ratio of background luminance (Lb) to target luminance (L) at threshold (i.e., DLS = Lb/L). The process underlying the loss in DLS in glaucoma is retinal ganglion cell loss.10 11 In most clinical practices, perimetry is the only method available to objectively quantify the amount of optic nerve damage. The relationship between ganglion cell numbers and DLS is, therefore, of fundamental importance in assessment of the severity of the neuropathy and of disease progression.
Direct measurement of ganglion cell numbers during life is not yet possible, and surrogates, such as the area of the neuroretinal rim at the optic nerve head, are used. The neuroretinal rim area is a measurement of a cross section, of variable obliquity, of the retinal nerve fibers, supporting glia and blood vessels in the optic nerve head, and as such is proportional to the numbers of retinal nerve fibers.12 Studies relating structural to functional measures in glaucoma frequently demonstrate a nonlinear relationship between neuroretinal rim area and dB DLS.13 14 This observation is supported by a recent study of experimental glaucoma in rhesus monkeys in which ganglion cell loss, measured histologically, was correlated with DLS, measured by behavioral perimetry. The relationship between percentage ganglion cell loss and dB DLS loss was shown to be curvilinear.15 A postmortem study of human glaucomatous eyes compared the number of remaining ganglion cells to dB sensitivity loss in three eyes.16 The results suggested that more ganglion cells have to be lost in the central field than in the peripheral field for equivalent sensitivity loss. These observations suggest that the relationship between ganglion cell numbers and DLS may not be linear.
It is thought that the DLS, for different visual field test spot sizes, is related to the receptive field size of the ganglion cells.17 Within a small retinal area ("Ricco’s area" or "critical area"), the visual system summates brightness information (spatial summation) so that DLS (the reciprocal of light intensity for threshold detection) is linearly related to the area of the stimulus17 18 19 20 : A ~ 1/L or A = C/L, where L = luminance, 1/L = DLS, A = stimulus area, and C = constant.
DLS (1/L) should, therefore, be linearly related to the number of ganglion cell receptive fields in the illuminated area (with a curvilinear relationship between dB DLS [which is 10 * log (1/L)] and ganglion cell receptive field numbers). When the retinal area illuminated is larger than the critical area, spatial summation is incomplete and is governed by the relationship18 21 22 23 : Ak ~ 1/L or Ak = C/L, where k = coefficient of summation.
The value of k is between 0 and 1 and changes with retinal eccentricity, being closer to 0 near the fovea and closer to 1 peripherally.18 21 23 The Goldmann size III target, used in conventional automated perimetry, is larger than the critical area throughout the central 40°,18 so that there is incomplete spatial summation at all visual field test points. To relate the DLS to ganglion cell receptive field numbers, when tested with the size III target, a correction for incomplete spatial summation is needed.
The aim of this study is to evaluate the relationship between ganglion cell receptive field numbers and DLS for white-on-white perimetry with the Goldmann size III target and to test the hypothesis that the target area (A) in the equation Ak = C/L can be represented by the number of underlying ganglion cell receptive fields (G) to give Gk = C/L.
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Methods |
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Ganglion Cell Receptive Field Numbers
This was derived from the ganglion cell density at each test
point, after a correction for lateral displacement of ganglion cells
from the fovea, and the area of a Goldmann size III target at each test
point.
The ganglion cell density was calculated from normative data taken from the literature. Curcio and Allen24 reported a topographic ganglion cell density map of the human retina derived from dense sampling of 6 young (mean age, 34 years) human retinas of 5 individuals. The article includes profiles of the ganglion cell density along the vertical and horizontal meridia, in graphical format. A topographic map was reconstructed from these profiles assuming a linear change in density between meridia along lines of equal eccentricity.
Ganglion cell bodies are displaced laterally from their receptive fields in the central 2 to 3 mm of the retina as a result of the elongated cone fibers of Henle and oblique connection of bipolar cells.24 25 The average lateral displacement at a series of eccentricities was calculated by Curcio and Allen,24 assuming a central ganglion cell–to–cone ratio of 3 to 1 that gradually declines with eccentricity. This assumption is consistent with other reports.26 27
To make a correction for lateral displacement, the number of ganglion cells in concentric annuli around the fovea was calculated from the raw ganglion cell density data. The receptive field positions were determined from the average lateral displacement values given by Curcio and Allen, and the ganglion cell receptive field density calculated.
Target stimulus area (A) was calculated with a conversion factor (q) to transform the target size from degrees of visual angle to millimeters on the retina. At the fovea, q = 0.286 mm/°.28 This conversion factor changes with eccentricity,29 so that qP < q0, where qP = conversion factor in the periphery and q0 = conversion factor at the fovea. Target size in the periphery was calculated according to the following relationship30 : qP = q0 - 0.000014U2, where U = the angle of retinal eccentricity, in degrees.
Conventional white-on-white automated perimetry is performed using the Goldmann size III target, the diameter of which subtends 0.431° of visual angle. At the fovea, this represents 0.123 mm (0.431° * 0.286 mm/°) on the retina, and an area of 0.012 mm2.
The ganglion cell receptive field count for each visual field test point was calculated in the following way: G = A * g, where G = ganglion cell receptive field count, A = stimulus area, and g = ganglion cell receptive field density.
Differential Light Sensitivity
Normative data for the DLS at each Humphrey 30-2 test point were
taken from the literature.31
Heijl et al.31 gives pointwise dB DLS for the average 50-year-old and pointwise age slopes. The age slopes were used to derive average pointwise sensitivity values for a 34-year-old, for equivalence with the ganglion cell data.
The units of DLS are conventionally given in decibels. In the Humphrey perimeter, the dB is 10 times the log of the reciprocal of stimulus intensity as measured in Lamberts [10,000 apostilbs; i.e., dB = 10 * log (1/L)].
Spatial Summation
The relationship between an increase in threshold DLS with an
increase in test spot size is given by the following
equation18
21
22
23
: Ak =
C/L, where L = luminance,
1/L = DLS, A = stimulus area,
C = constant, and k = coefficient of
summation.
Alternatively, this can be written as follows: k * log A = log C - log L or log L = log C - k * log A, the slope of the line of a plot of log L against log A will, therefore, give k.
To obtain estimates of spatial summation at each visual field test
point, DLS was measured in 8 normal subjects with 5 target sizes
(Goldmann I–V). Perimetry was performed with the Humphrey type II 750
perimeter and the 30-2 and "macula" full threshold programs, after
informed consent was obtained. The present study conformed to the
tenets of the Declaration of Helsinki. Each subject performed one test
at each test spot size in a randomized order, in one eye, chosen at
random. All tests were performed within 1 week for each subject, with
no more than 2 tests performed in one session. Subjects adapted to the
perimeter illumination for 10 minutes before each session, and a near
addition, calculated according to the subject’s age, was used.
Restriction criteria for the subjects were as follows: corrected visual
acuity 
20/20, ametropia < 5 D sphere and < 2 D
cylinder, no known ocular disease, and visual field test reliability
indices of fixation losses (excluding the size V target) < 15%,
and false-positive and false-negative responses < 15%.
Log L was plotted against log A for each subject. Visual inspection suggested a curvilinear relationship at all eccentricities. A quadratic regression line was fitted to each plot, and the slope of the tangent to the curve at the point corresponding to the size III stimulus was used to determine k for this stimulus size.
The mean subject k for each test point was calculated. These were compared with available data in the literature.
Comparison of DLS and Ganglion Cell Numbers
In the relationship Ak =
C/L, A refers to the test spot area.
At any point on the retina, there will be a certain number of ganglion
cell receptive fields (G = A * g,
above) within the stimulus area. Substitution of G for
A gives Gk =
C/L.
Gk may be defined as the "effective ganglion cell receptive field number," which is the number of ganglion cell receptive fields that would have the same sensational effect in the case of complete spatial summation (where k = 1). In the central 30° of the visual field, 0 < k < 1.
Two comparisons between DLS and ganglion cell numbers were made. DLS in decibels was plotted against the ganglion cell receptive field numbers per size III target (G). DLS in 1/L (1/Lambert) was plotted against the effective ganglion cell receptive field numbers per size III target (Gk).
Extrapolation to Glaucomatous Ganglion Cell Loss
The relationship (equation of the regression line) between DLS
(1/L) and effective ganglion cell receptive field numbers
(Gk), derived from the second comparison,
was used to calculate theoretical ganglion cell losses for 3- and 6-dB
sensitivity losses at each visual field test point. A 3-dB loss
represents a doubling of the test spot intensity, and a 6-dB loss a
quadrupling.
In addition, theoretical ganglion cell losses were calculated for a dB sensitivity loss at the P < 5% level for each point.
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Results |
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A typical curve for the plot of log test spot intensity against log test spot area, for the point at 9°, 9° in the superonasal field in one subject, is shown in Figure 2 .
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The mean coefficients of summation are shown plotted against eccentricity, together with curves derived from data taken from the literature, in Figure 3 . The mean pointwise SD for estimations of k was 0.09.
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Discussion |
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Additional supportive evidence comes from the comparison of the detection thresholds in acuity (high pass resolution) perimetry and conventional (DLS) perimetry. Acuity perimetry measures spatial resolution, which is, theoretically, related to the density of intact sensory units.32 The spatial threshold units are the logarithm of spatial extent, in the same way that dB DLS threshold units are the logarithm of test spot intensity. Spatial thresholds and dB DLS thresholds have been found to be linearly related.33 34 One would, therefore, expect spatial extent and test spot intensity, and their reciprocals, sensory unit density and DLS (1/Lambert), to be linearly related. Bartz Schmidt and Weber33 compared pointwise sensitivity in the two tests, and calculated that a 6-dB DLS loss equated with a 50% loss of sensory units (and 12 dB with a 75% loss). This is consistent with the model proposed in this article only for test points at approximately 27° eccentricity, with a 6-dB loss representing, on average, greater ganglion cell loss (mean 71%). Differences at other points may in part be explained by location-specific differences in spatial summation, which were not taken into account in Bartz Schmidt’s calculations.
The values obtained for spatial summation in this study are very similar to those of previous studies (Fig. 3) .18 21 22 23 35 36 The mean SD of pointwise estimates for the coefficient of summation, at 0.09, is very similar in magnitude to the findings of Gougnard.35
It can be appreciated from Figure 2 that the coefficient of summation at a particular location changes with the size of the stimulus target. This means that the value of the coefficient of summation itself changes over the range of target sizes used to measure it.22 This is in addition to variation according to retinal location (Fig. 3) . This suggests that it may be the number of ganglions cells stimulated that determines summation. The summation characteristics of a particular location also vary with the level of background luminance.17 The analysis in this article, therefore, applies to the specific experimental conditions in this study. That is Humphrey/Goldmann perimetry (background luminance 31.5 apostilbs) and the Goldmann size III target.
The equation that relates DLS (1/Lambert) to effective ganglion cell receptive field count can be used to predict the expected ganglion cell number, as a proportion of normal, at any point for various sensitivity losses.
The equation derived from the linear regression of DLS against "effective" ganglion cell numbers (Fig. 5) is DLS = (208 * Gk) - 392; from this (DLS + 392)/208 = Gk, where k = location-specific coefficient of summation; and [(DLS + 392)/208]1/k = G.
For a given field test point, substitution in the equation of the population average DLS (in 1/Lambert) for the term "DLS," and the location-specific spatial summation value for "k," yields the population average ganglion cell numbers at that point. Substitution of the measured DLS yields the actual number of ganglion cells at the test point.
The hypothesis tested in the study was that Gk = C/L. If true, the relationship between Gk and L should be linear and proportional, with the regression line passing through 0. In this study, a small offset was found. Possible explanations for this include an underestimation of the coefficient of summation at more peripheral points, an underestimation of DLS, or an overestimation of the number of ganglion cells, at these points. An underestimation of DLS might occur in the case of refractive blur in areas of incomplete summation. Refractive errors are corrected for foveal vision, and it is known that refraction in the periphery can deviate substantially from that at the fovea and that defocus affects detection acuity.37 The offset accounts for the predicted loss of <50% for a 3-dB loss seen at peripheral points (Fig. 6a) .
Extrapolation of the physiological relationship between DLS and ganglion cell numbers to the pathologic condition of glaucoma presupposes that spatial summation remains unchanged as ganglion cells are lost. There are limited data in the literature, but two studies suggest that spatial summation is either unchanged, or little changed, in glaucoma. Fellmann et al.38 studied locations with large sensitivity differences between size III and size V test spots in the visual field of 19 glaucomatous eyes. They found that the differences between the sensitivities to size III and V targets could be explained 73% of the time by recruitment of "undamaged units" (into the size V area) and normal spatial summation. "Pathologic summation" (greater than expected) explained the remaining 27%. Lower than expected (for normal spatial summation) sensitivity for smaller target sizes was found in fewer than half the 16 glaucomatous eyes studied by another investigator.39 The remainder had normal summation or increased variability of DLS measurements. Increasing spatial summation with ganglion cell loss would have the effect of overestimating the number of ganglion cells (if normal summation is assumed) in the model proposed in this article.
As the prevalence of glaucoma increases with age, it is important to know whether spatial summation changes with age. Dannheim and Drance22 studied summation in 35 normal subjects 20 to 79 years of age. Although lower retinal sensitivity was found in older subjects, the shape of the summation curves was preserved, indicating that spatial summation is unaffected by aging. Therefore, in applying this model to older age groups, no extra assumptions need to be made when relating actual sensitivity to the age-related average.
If a specific value of dB DLS loss is taken to indicate that a certain level of glaucomatous damage is present, the present study demonstrates that the sensitivity of the visual field test to detect damage varies with eccentricity. Figure 6a demonstrates the case for a 3-dB loss (a doubling of test spot intensity). At this level, around 70% of ganglion cells have to be lost at 4° eccentricity, but only 50% at 21°. The data of Quigley et al. lend support to this observation.16 Although a linear regression fit was made to the plot of dB sensitivity loss against percentage of normal ganglion cells, the analysis suggested that a greater proportion of ganglion cells had to be lost in the central, compared to more peripheral, field for equivalent dB sensitivity losses.
Figure 6c illustrates the projected ganglion cell losses required to produce a test point sensitivity lower than two SD from the average, for selected points in a 30° field. By this measure, the sensitivity of the visual field test does not change much with eccentricity, only slightly greater losses are required centrally. This pattern, however, cannot be explained by interindividual variation in ganglion cell numbers, which is lower centrally and increases with eccentricity.24 Factors other than interindividual variation in ganglion cell numbers, such as the greater susceptibility of central thresholds to uncorrected refractive errors,21 40 may play a role.
If dB sensitivity loss means different things in terms of ganglion cell loss in different parts of the visual field, it follows that simply averaging the defect values for each test point, to provide a summary measure, will not necessarily reflect the true underlying ganglion cell loss. A more appropriate scale for DLS would overcome this problem.
The shape of the curve relating dB DLS to underlying anatomy has often been interpreted as demonstrating a built-in redundancy of neural units in the retina, so that a certain proportion of ganglion cells has to be lost before function is compromised. The present study suggests that this is an impression given by the logarithmic nature of the scale for DLS. In fact, there is a continuous, linear, relationship between DLS and underlying neural units. The difficulty in detecting early anatomic damage by tests of function may be a reflection of the wide interindividual range in the number of ganglion cells in the retina24 and the wide interindividual range of psychophysical measurements.31
Perimetric indices and pointwise DLS measurements have been used to quantify disease progression.7 8 9 The true pattern of progression in glaucoma is unknown and is likely to vary from patient to patient, and within a patient according to prevailing risk factors. The simplest model to describe progression, which might represent an average pattern, is a linear model. Comparison of Figures 6a and 6b shows that dB loss is not linear with respect to underlying ganglion cell numbers. At 21° eccentricity, a 3-dB sensitivity loss represents 50% ganglion cell loss. However, a further 3-dB loss represents only a further 20% ganglion cell loss. The assumption of linear progression of dB DLS would underestimate "true" (anatomic) progression when sensitivity values are near normal and overestimate progression when sensitivity loss is already advanced. There is a need to reevaluate models of progression in the context of the new DLS scale proposed in this article.
There are several implications for clinical practice of the nonlinear relationship between dB DLS and ganglion cell numbers, with small dB changes representing a large reduction in ganglion cell numbers when sensitivity is near normal and a small reduction in ganglion cell numbers when sensitivity is low. Appreciation of this relationship reinforces the widely held belief that structural changes at the optic disc and nerve fiber layer frequently occur earlier than recognizable changes in the visual field. It suggests a reappraisal of "staging" disease by perimetric indices. The loss of the first 5dB, often regarded as early disease, in fact represents moderately advanced disease. And the difference between a MD of -10 and -15 dB represents a lesser change. Detection of disease before these levels of functional loss is, therefore, of great importance, so that appropriate treatment can be instigated. Recognition of progression of functional loss in early disease requires an understanding of the nonlinear relationship between dB DLS and ganglion cell numbers, and of fluctuation of sensitivity measurements at near-normal levels. A linear model applied to dB DLS values is likely to be less sensitive in early disease. It may be possible to detect change earlier in the disease process with a linear (1/L) model. Recognition of the presence, and patterns, of field loss will be aided by the knowledge that similar dB losses in different parts of the field imply different levels of structural damage and that the "probability symbols" are a much better guide than the raw sensitivity values.
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Footnotes |
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Submitted for publication August 31, 1999; revised December 15, 1999; accepted January 5, 2000.
Commercial relationships policy: N.
Corresponding author: David F. Garway–Heath, Glaucoma Research Unit, Moorfields Eye Hospital, City Road, London EC1V 2PD, UK. ted.garway-heath{at}moorfields.nthames.nhs.uk
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