purpose. Differential light sensitivity (DLS) in white-on-white perimetry is
used as a measure of ganglion cell function to estimate the amount of
neuronal damage in glaucoma. The physiological relationship between DLS
and ganglion cell numbers is poorly understood. Within small retinal
areas, brightness information is summated, so that *A* ∗ *L* = *C*, or *A* = *C*/*L*, where *A* is target area, *L* is threshold luminance, and *C* is a constant. In
larger illuminated areas, as with a Goldmann size III target in
perimetry, summation is incomplete, so that *A* ^{ k } = *C/L*, where *k* is
the coefficient of summation, and 0 < *k* < 1.
This study tests the hypothesis that the target area (*A*) can
be represented by the number of underlying ganglion cells
(*G*) to give *G* ^{ k } = *C/L*.

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 *G* ^{ k }. 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 *G* ^{ k } was linear
(*r* ^{2} = 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.

^{ 1 }

^{ 2 }Inferences about the pathogenesis of the condition are made from the observed pattern of visual field defects.

^{ 3 }

^{ 4 }

^{ 5 }

^{ 6 }Disease progression may be estimated from analyses of serial field tests, either by clinical impression or by more quantitative means.

^{ 7 }

^{ 8 }

^{ 9 }

*L*

_{b}) to target luminance (

*L*) at threshold (i.e., DLS =

*L*

_{b}/

*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.

^{ 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.

^{ 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 stimulus

^{ 17 }

^{ 18 }

^{ 19 }

^{ 20 }:

*A*∼ 1/

*L*or

*A*=

*C*/

*L*, where

*L*= luminance, 1/

*L*= DLS,

*A*= stimulus area, and

*C*= constant.

*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 relationship

^{ 18 }

^{ 21 }

^{ 22 }

^{ 23 }:

*A*

^{ k }∼ 1/

*L*or

*A*

^{ k }=

*C*/

*L*, where

*k*= coefficient of summation.

*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.

*A*) in the equation

*A*

^{ k }=

*C*/

*L*can be represented by the number of underlying ganglion cell receptive fields (

*G*) to give

*G*

^{ k }=

*C*/

*L*.

*G*

^{ k }=

*C*/

*L*, the following were established for each Humphrey 30-2 visual field test point for a Goldmann size III target: ganglion cell receptive field numbers (

*G*), average DLS (1/

*L*), and coefficient of summation (

*k*). From these, a value for

*C*was calculated.

^{ 24 }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.

^{ 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 }

*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

*q*

_{P}<

*q*

_{0}, where

*q*

_{P}= conversion factor in the periphery and

*q*

_{0}= conversion factor at the fovea. Target size in the periphery was calculated according to the following relationship

^{ 30 }:

*q*

_{P}=

*q*

_{0}− 0.000014

*U*

^{2}, where

*U*= the angle of retinal eccentricity, in degrees.

^{2}.

*G*=

*A*∗

*g*, where

*G*= ganglion cell receptive field count,

*A*= stimulus area, and

*g*= ganglion cell receptive field density.

^{ 31 }

^{ 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.

*L*)].

^{ 18 }

^{ 21 }

^{ 22 }

^{ 23 }:

*A*

^{ k }=

*C*/

*L*, where

*L*= luminance, 1/

*L*= DLS,

*A*= stimulus area,

*C*= constant, and

*k*= coefficient of summation.

*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*.

*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.

*k*for each test point was calculated. These were compared with available data in the literature.

*A*

^{ k }=

*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

*G*

^{ k }=

*C*/

*L*.

*G*

^{ k }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.

*G*). DLS in 1/

*L*(1/Lambert) was plotted against the effective ganglion cell receptive field numbers per size III target (

*G*

^{ k }).

*L*) and effective ganglion cell receptive field numbers (

*G*

^{ k }), 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.

*P*< 5% level for each point.

^{2}at 0.05 mm from the foveal center.

*R*

^{2}(±SD) for the quadratic regression fits (log

*L*against log

*A*), for all subjects and test points, was 0.96 (±0.05), with a mean

*P*= 0.04.

*k*was 0.09.

*G*) is shown in Figure 4 .

*L*against effective ganglion cell receptive field numbers (

*G*

^{ k }) is shown in Figure 5 . The equation of the regression line (

*r*

^{2}= 0.73,

*P*< 0.000) is 1/

*L*= −392 + (208 ∗

*G*

^{ k }).

^{ 2 }Despite this, the relationship between such indices and the amount of structural damage has been unclear. It is not known whether there is a linear correspondence, so that a MD of −6 dB represents twice as much damage as a MD of −3 dB. The results of this study indicate that this is not the case. The dB scale is logarithmic, and there is a curvilinear relationship between dB DLS and underlying ganglion cell numbers (Fig. 4) . When DLS is appropriately scaled, as the reciprocal of test spot intensity, there is a linear relationship with underlying ganglion cell numbers adjusted for spatial summation (Fig. 5) . The curvilinear relationship between dB DLS and structural measures is supported by studies comparing visual field MD and neuroretinal rim area

^{ 13 }

^{ 14 }and by recent primate work comparing visual field MD and ganglion cell numbers counted histologically.

^{ 15 }

^{ 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 Weber

^{ 33 }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.

^{ 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 }

^{ 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.

*G*

^{ k }) − 392; from this (DLS + 392)/208 =

*G*

^{ k }, where

*k*= location-specific coefficient of summation; and[ (DLS + 392)/208]

^{1/k }=

*G*.

*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.

*G*

^{ k }=

*C*/

*L*. If true, the relationship between

*G*

^{ k }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) .

^{ 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.

^{ 22 }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.

^{ 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.

^{ 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.

^{ 24 }and the wide interindividual range of psychophysical measurements.

^{ 31 }

^{ 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.

*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.

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

*.*1992;70:679–686. [PubMed]

*.*1994;101:1445–1455. [CrossRef] [PubMed]

*.*1981;99:137–143. [CrossRef] [PubMed]

*Am J Ophthalmol.*1987;104:98]. Am J Ophthalmol.

*.*1987;103:512–517. [CrossRef] [PubMed]

*.*1984;62:658–664. [PubMed]

*.*1998;126:763–771. [CrossRef] [PubMed]

*.*1997;81:1037–1042. [CrossRef] [PubMed]

*.*1997;104:808–815. [CrossRef] [PubMed]

*Ophthalmology.*1998;105:7]. Ophthalmology

*.*1997;104:1228–1236. [CrossRef] [PubMed]

*.*1982;100:135–146. [CrossRef] [PubMed]

*.*1995;36:200–205. [PubMed]

*.*1998;116:493–497. [CrossRef] [PubMed]

*.*1985;103:203–204. [CrossRef] [PubMed]

*.*1997;124:488–497. [CrossRef] [PubMed]

*.*1999;40:2242–2250. [PubMed]

*.*1989;107:453–464. [CrossRef] [PubMed]

*.*1970;207:611–622. [CrossRef] [PubMed]

*.*1975;15:1381–1384. [CrossRef] [PubMed]

*.*1977;17:867–873. [CrossRef] [PubMed]

*.*1961;1:121–138. [CrossRef]

*.*1971;6:311–319. [PubMed]

*.*1992;557–562. Kugler Publications Kyoto, Japan.

*.*1990;300:5–25. [CrossRef] [PubMed]

*.*1988;269:479–505. [CrossRef] [PubMed]

*.*1990;30:1897–1911. [CrossRef] [PubMed]

*.*1994;232:432–437. [CrossRef] [PubMed]

*.*1998;82:643–649. [CrossRef] [PubMed]

*.*1988;72:228–230. [CrossRef] [PubMed]

*.*1994;232:361–367. [CrossRef] [PubMed]

*.*1987;105:1544–1549. [CrossRef] [PubMed]

*.*1988;12:169–174. [CrossRef] [PubMed]

*.*1993;17:171–178. [CrossRef] [PubMed]

*.*1993;28:306–311. [PubMed]

*.*1961;142:469–486. [CrossRef]

*.*1986;18:301–308. [CrossRef] [PubMed]

*.*1997;38:2134–2143. [PubMed]

*.*1988;313–324. Kugler & Ghedini Publications Amsterdam.

*.*1974;91:463–468. [CrossRef] [PubMed]

*.*1992;33:611–617. [PubMed]