*DSc

^{†} PhD

School of Optometry and Vision Sciences, Cardiff University, Cardiff, Wales, United Kingdom.

Received October 1, 2007; accepted June 10, 2008.

It is of interest to researchers and clinicians, to have reliable estimates of D, the ganglion cell (GC) receptive field density per solid degree at stimulus locations used in static threshold perimetry , and to have an algorithm relating this to sensitivity. By this means a quantitative estimate of neural damage may be made from such data in glaucoma or other conditions involving GC loss. However, there has been some uncertainty as to the best estimates of D, how these should change with increasing age, and as to models of the effect of GC loss on sensitivity.

Recently an improved model of the distribution of D in the visual field was described.^{1} This provided new estimates of GC displacements from their receptors, which were greater than previously reported and in combination with earlier data,^{2} predicted D at eccentricities (0 > 30°) along the principal meridians in the visual field. If a suitable age correction factor could be determined, D could be estimated for subjects of any age, at any point within the central visual field. Another relatively recent possibility is that contrary to previous reports^{3,4} GC loss in glaucoma might be reflected in a proportionate change in sensitivity^{5–8} which is evident if this is expressed on a linear rather than dB scale.^{5} For the purpose of this report the sensitivity is expressed as S, the inverse of threshold luminance in Lamberts using the Goldman size III stimulus, so that S = 10^{(T/10)} , where T is the threshold in decibels. Given these recent developments, we propose some provisional methods of quantifying GC loss from perimetric data and consider their application and some implications of the theory on which they are based.

THEORY AND METHODS
The equations from a newly designed model of receptive field distribution^{1} were applied to determine D at the required eccentric angle for each stimulus point in the 24-2 test pattern, along each adjacent principal meridian. A process of interpolation along the iso-eccentric arc was then used to obtain an estimate for the same eccentricity along the required intermediate meridian. Linear interpolation is not quite suited to this purpose because it produces an unnaturally sharp transition in estimated densities when crossing the major axes. An interpolation based on a cosine function of twice the polar angle taken from the horizontal, for each stimulus location, was used to avoid this artifact. The D values obtained at the stimulus locations for the Humphrey 24-2 threshold test pattern for the right eye were determined in this way.

The data on which the model was based^{2} relate to a mean age of approximately 34 years, and many reports suggest that in the absence of any recognisable pathology the number of GCs declines progressively with increasing age.^{6,9–11} Some reports suggest that age related GC loss is unevenly distributed in the visual field. One study^{9} reported a relative loss (25%) of GCs in the central and nasal retina which was statistically significant between the mean ages of 34 and 76 years. More peripherally, the low densities and wide individual variations prevented the data achieving significance, indeed some peripheral areas showed an apparent increase in density. By contrast, others^{10} found slightly lower losses centrally and higher ones in the periphery. These conflicting and uncertain findings reduce confidence in the idea of a characteristic retinal distribution of age related GC loss. Furthermore, the average GC loss described by Curcio and Drucker^{9} in the central retina over the age range 34 to 76 years, is about 6500 per year. This is higher than the loss of optic nerve axons, generally reported in the literature at 0.36% to 0.62% per year^{6} but lower than the 7205 loss per year reported for an older age range of 55 to 95 years in another recent study.^{11} It is concluded that although the concept of age related local loss distribution in the visual field is important and further research is required, there is at present insufficient evidence to depart from the simple concept of uniform loss throughout the field and a rate of 6500 GCs per year from the age of 34, which equates to a percentage loss of 0.59% increasing to 0.79% at 76 years. This is achieved if D at 34 years is modified by a factor of |P[1 − [(age − 34)/169]|P] to obtain an estimate for any age above 34 years. These age corrected values of D may be compared with corresponding normative values of S calculated from the data of Heijl et al.^{12} using pointwise age correction. However, if the age related loss is assumed to be uniform over the whole field as outlined above, the effective number of GCs for each stimulus domain of 6 × 6° may be expressed as a percentage and will then be applicable to all age groups.

RESULTS
The estimates of D at age 34 years, for each stimulus location in the 24-2 test program for the right eye calculated from the original studies^{1,2} in the manner described above, are displayed in Table 1 . To relate this more appropriately to clinical application, age corrected estimates of D and S at 60 years are given in Table 2 . From this it is clear that D is not proportional to S across the field and their relationship is further revealed by plotting the 52 data points in Fig. 1 . The data are fitted with a fourth order polynomial function.

TABLE 1: Estimated ganglion cell receptive field densities per solid degree, on Humphrey 24-2 stimulus points for right eye age 34 yrs

TABLE 2: Estimated ganglion cell receptive field densities per solid degree (D) and perimetric (1/Lambert) sensitivities (S) on Humphrey 24-2 stimulus points for right eye at 60 yrs

FIGURE 1.:
Ganglion cell receptive field density per solid degree (D) at 60 years for 52 points in the Humphrey 24-2 field test, plotted against corresponding normative values of 1/Lambert sensitivity (S).The data are fitted by a fourth order polynomial function.

Since, the initially calculated values in Table 1 were densities per solid degree, and each stimulus samples a domain of 6 × 6°, the total number of GC receptive fields within the 24-2 test area is estimated at 645,480. Taking the total number of GCs at 34 years, for one eye^{2} as 1.1 million, the test pattern is found to sample 58% of all GCs in the eye. The percentage of the total number within the 24-2 test area which is sampled by each stimulus domain is displayed in Table 3 . Accepting the principle already outlined, of approximately uniform age related loss throughout the field in the absence of pathology, Table 3 is applicable to all age groups. Furthermore, if multiplied by 0.58 the percentages in Table 3 become percentages of GCs in the field as a whole.

TABLE 3: Estimated percentage of ganglion cell receptive fields within the Humphrey 24-2 test field which are sampled in each stimulus domain for the right eye

DISCUSSION
Ophthalmological, experimental, and theoretical studies have reported diverse relationships between perimetric sensitivity and GC loss^{3–8,11} Histological studies,^{3,4,11} suggest a complicated relationship and that considerable damage must occur before sensitivity is significantly affected. By contrast, in other studies using normative data and theoretical modeling,^{5,8} correlates of GC density, such as temporal neuro retinal rim area,^{6} retinal nerve fiber layer thickness,^{7} and pattern electroretinogram (PERG) amplitude,^{6} suggest that under some circumstances a linear relationship with more sensitivity to smaller GC losses may be demonstrated.

When the new estimates of D for the 24-2 test locations are plotted against values of S derived from an early study^{12} they appear to be non-linearly related (Table 2 and Fig. 1 ). The non-linearity may be partly attributable to the extent of spatial summation areas which are smaller than the stimulus and decreases progressively towards the fovea. The theory of this has been clearly outlined by Garway-Heath et al.^{5,6} The standard Goldman III stimulus subtends 0.1459 solid degrees and the number (N) of GCs stimulated is therefore 0.1459 D. However, it may be argued (that due to the application of Pieron Law), the number that contribute to sensation and thus determine S, is N^{k} , where k is the summation index. We found that atypically high values of k compared with those reported^{5} would be required in order to make N^{k} directly proportional to S, producing a straight line, so we conclude that the relationship varies across the field, not only due to spatial summation, but also due to local variations in other properties of GCs. However, at any fixed point in the field, it remained possible that a linear relationship between D and S might apply.

The Simplest Linear Model
Early studies on perimetry established the approximate validity of Ricco's Law, according to which, threshold luminance is inversely proportional to stimulus area. Since sensitivity S is the inverse of threshold luminance, we should expect that S would be directly proportional to stimulus area and hence to the number of GCs stimulated at that location. There is also persuasive evidence that on average a linear or near linear relationship between D and S may be demonstrated.^{5–8} For the simplest model, it is therefore assumed that D = SC locally, where C is an appropriate constant at each test location. The important feature of this simplifying model is not the value of C, but that D is assumed to be directly proportional to S. It follows that, at a given stimulus point, if there is any amount of surviving visual function S, compared to the age matched norm, it may be envisaged in terms of equivalent percentage of GC survival. The reduction in sensitivity, at each point, compared to a calculated age matched normative value is conveniently provided in the Humphrey single field analysis in the form of the signed (±) total deviation (TD) in decibels and the percentage equivalent GC survival is therefore 100 × (10^{(TD/10)} ). This rule in conjunction with the above data provides a simple means of modeling aspects of neural damage.

The linear relationship with sensitivity gives rise to greater estimates of GC losses for a given loss of sensitivity than do histological studies of GC or nerve fiber death. This is probably due to the fact that estimates from S values reflect dysfunction in addition to GC loss (death) and should be considered as equivalent GC loss. For example, histological reports^{3,11} indicated that a 5 dB sensitivity loss was associated with a 20 to 25% loss of GCs whereas according to the present model, 100 × (10^{(−5/10)} ) predicts 32% survival and therefore 68% loss and shows that even 1 dB overall depression corresponds to a 21% GC loss. Similarly, the earlier studies suggested a 40 to 45% loss for a 10 dB depression but the above linear algorithm would predict that with a 10 dB depression there would be a 90% equivalent GC or functional loss. However, although these dramatic increases with a given sensitivity loss predicted by the simple linear model are higher than previously expected, they are consistent with clinical studies which show markedly reduced PERG amplitudes in suspected or early glaucoma ^{13–16} where threshold losses in dBs are minimal.

To illustrate some quantitative implications of these ideas we estimate the extent of loss of GCs with local and diffuse field defects. If we consider the effect of a 10 dB superior nasal step defect (TD = −10) covering three points indicated in Table 3 , the proportion of GC survival in this area would be 10^{(−10/10)} = 0.1 or a loss of 0.9 times the percentages in Table 3 which is 0.9(0.44 + 0.71 + 0.59) = 1.56% of the GCs in the 24-2 field. Multiplying by 0.58 gives 0.9% which is the percentage of the GCs in the whole retina. Although these defects could be a highly significant sign of primary open angle glaucoma ^{17} the amount of GC loss from isolated focal defects is minute in relation to the reported histological GC losses of >20% when primary open angle glaucoma can first be detected by automated perimetry .^{3} This might indicate that much neural damage may be diffuse and undetected. Other evidence including that based on earlier data on GC distribution and on PERG also supports this view.^{13–15} The PERG recorded from the central retina is mainly generated by GC activity and is considered to be proportional to the number of GCs.^{18} It is therefore attenuated in glaucoma , but loss of 0.9% of the GCs in the above example with the significant field defect alone would clearly be far too small to be detected by a change in PERG amplitude.

Conversely, according to this linear model in the case of a diffuse defect with a uniform depression of 2 dB, though this could be within normal limits of perimetry it is seen to amount to 10^{(−2/10)} = 0.63, i.e., 63% survival or 37% loss over the retina as a whole, according to the linear model. Here the factor 0.58 is not applicable because the uniform depression is assumed to apply to the whole field. This should result in a significant reduction in PERG amplitude. Although the local defects in peripheral vision have been recognized for over a century, the acceptance of the idea that diffuse defects might occur has been slow to develop, because conventional wisdom asserts that central vision is relatively unaffected in early glaucoma . The effect of even the simplest linear model is of interest in this respect since it helps to explain why such a diffuse loss would not be very noticeable. However, some reports suggest that the linear model gives misleading predictions in the central field, and that a non-linear model is needed which reveals further insight into the nature of central defects.

Non-Linear Models
Histological studies^{3} have suggested that more GCs must be lost in central vision to produce a given reduction in S, than is the case in the arcuate region where an approximately linear relationship of D to S applies. This suggests that a non-linear model is necessary to model changes in central vision and this concept has been supported in some important attempts to model GC loss and dysfunction.^{5,6,8}

In two significant reports^{5,6} Garway-Heath et al., analyzed the relationship between sensitivity S, and GC numbers, taking into account the local spatial summation properties. They presented persuasive evidence that there is an approximately linear, though not quite proportionate relationship between D and S, but with increased GC losses for the foveal region.^{5} Another advanced model to consider the role of dysfunction in addition to GC loss and other factors which are thought to determine perimetric sensitivity has been described by Swanson et al.^{8} This model used hypothetical sampling mosaics and spatial filters based on the properties of cortical neurones, and gave some further support to the concept of a linear relationship between sensitivity and cell density for data that would relate to the arcuate region. These studies provide more detailed theoretical insight into the detection of perimetric stimuli. However, despite the ingenuity of these analytical approaches they have not provided a general formula, or a simple procedure for estimating D from S.

Given the availability of the new data on GC receptive field density D and S in Table 1 and the data of an earlier study,^{12} age correction for D was achieved by the equation given above, and for S using data and point-wise age gradients of Heijl et al.^{12} enabling us to estimate the data at age 60 years (Table 2 ). We could then investigate the direct relationship between D and S. The plotted values of D vs. S, shown in Fig. 1 are clearly non-linearly related but may be fitted with a fourth order polynomial function. However, for values of S < 800 corresponding to 29 dB, there is a noticeable tendency to linearity, despite the known wide normal variability^{2,12} of D and S (Fig. 2 ). The regression equation was D = −0.9569 + 0.208S. In view of the minute intercept, on the scale of the vertical axis, the line closely approximated the functional relationship of direct proportionality D = 0.2065S, where the line passes through the origin. If this relationship, established from physiological data is approximately valid for pathologically reduced receptive field densities then this would cover many reduced sensitivities encountered in glaucoma and the conditions of the simple linear model would apply. Over the limited range, D would be reduced in exactly the same proportion as S is reduced as in the simple model. We could, therefore, estimate the surviving proportion of GCs at a given point in the field as 10^{(TD/10)} and therefore the loss proportion as, 1–10^{(TD/10)} and the percentage loss as

FIGURE 2.:
The data for S < 850, revealing a trend towards linearity for low to moderate values of S.

The functional relationship D = 0.2065S can also be applied directly to the threshold data to provide a best estimate of D, from S between the limits of 0 to 800 since S = 10^{(T/10)} it follows that

When the data relating to higher values of S are considered, the non-linear relationship is apparent (Fig. 1 ) and is fitted with Eq. 3 . This relates to sensitivities between S = 600 and 1700, (26 dB to 32.3 dB) for the data of Table 2 . The estimates of D using Eqs. 2, and 3 coincide at S = 600 and are approximately similar between S = 600 to 800.

where S = 10^{(T/10)} .

To provide continuous coverage and avoid ambiguity, the linear rule is applied from S = 0 to S = 600, and the formulas jointly provide a bipartite model continuous coverage of the range from S = 0 to 1700 (32.3 dB). However, due to the wide variations in S and D in individuals, formulas (2) and (3) are best reserved for studying group average effects or simple modeling. To apply the equations to hypothetical problems it is necessary to produce a baseline model relating S and D precisely by Eqs. 2, and 3 . This is shown in Table 4 . The values of D were generated by substituting the normative values of S in the equations. Although it might be argued that the original data on GCs^{1,2} should ideally not be modified, this is necessary when modeling hypothetical losses and it should be noted that the S values were obtained from 140 subjects and were less variable than the GC data which were obtained from only six eyes of five individuals.^{2}

TABLE 4: Baseline data

The baseline model (Table 4 ) is designed for the age of 60 years, but it will permit many general effects to be simulated and since both S and D change in the same direction with age, residual error if it is applied to other age ranges will be small. The alternative is to generate baseline models for each decade. The model may be applied by changing S by a given amount. For example, for a −2 dB depression, S will be reduced by 37% or for −6 dB by 75%. If S < 600 Eq. 2 is used to estimate D, if >600 Eq. 3 is used. The reduced value of D is then compared with the value in the baseline model to note the percentage loss. For example if due to a 5 dB reduction in sensitivity S changes from S = 1585 (32 dB) to 501 (27 dB), by using formulas 3 and 2, respectively, the reduction in S of 68% is shown to change D from 1274 to 103, a reduction of 92%, showing the non-linear effect with higher and more centrally occurring sensitivities (Fig. 3 ).

FIGURE 3.:
According to the model comprising Eqs. 2/3 and Table 4, each stimulus point on the 24-2 array has a characteristic S/D response function. This may be calculated by reducing the S value in Table 4, by a given percentage, and calculating the value of D using the appropriate (Eq. 2 or 3). The straight line and the curve relate to the most and least eccentric points, respectively, in the upper nasal quadrant. These functions appear to be closely similar those of Swanson et al.8

This illustrates the principle first reported from histological studies^{3} and later supported by theoretical modeling,^{5,8} that large losses of GCs in the central vision are associated with only moderate losses of sensitivity. As mentioned in the simple linear model description, theoretical and non-histological studies give estimates of equivalent GC losses for a given loss of sensitivity that are greater than those reported in histological studies. The model now proposed predicts even higher “losses” but the “losses” quoted in non-histological studies are GC equivalent losses which may be due not only to GC deaths, but also to GC dysfunction^{8} perhaps associated with cell and fiber shrinkage which may occur before apoptosis.^{19} Again, although these dramatic increases with a given sensitivity loss predicted by the models are higher than previously expected, they are consistent with clinical studies which show markedly reduced PERG amplitudes in suspected or early glaucoma ^{13–17} where major perimetric sensitivity losses are absent.

Eq. 2 of the bipartite model predicts similar losses for the peripheral defects at the points shown in Table 3 as the simple linear model, since these are in the peripheral zone where linearity applies, but the effects with more central defects using Eq. 3 are different. Due to the non-linearity a 2 dB local reduction in sensitivity (TD = −2) which may appear relatively inconsequential in perimetry will be shown to have a profound effect in terms of equivalent GC loss. In the range where the linear model applies, applying Eq. 1 , would amount to an estimated loss 100(1 − 10^{(−2/10)} ) = 37%, but if the defect occurred on one of the four paracentral points of the 24-2 array, which is indicated in Table 4 , where S was 1514, (31.8 dB), then a 2 dB reduction would predict not 37%, but to a 78% equivalent loss of GCs as may be shown by substituting the reduced S which would be 955 in Eq. 3 , which estimates D at 219 instead of the 1007, hence a 78% reduction. Furthermore, as these points are those relating to the high GC densities which make a greater contribution to the PERG, the model would explain an enhanced sensitivity of the PERG to the diffuse loss of a −2 dB overall depression in early glaucoma . However, the equations also draw attention to the peculiarly irregular distribution of GC loss that would be required to give a uniform depression of S over the visual field (Table 5A ). This shows how uniform loss of sensitivity does not result from uniform loss of GCs.

TABLE 5: Calculated effects of nonlinearity on The R visual field 24–2 test point at 60yrs according to equations 2 and 3.

Since the model shows that the effect of uniform loss across the field would require very high losses of GCs centrally and losses proportional to the reduction in S at peripheral points it might be concluded that uniform diffuse defects are unlikely to occur. It is more logical to consider the consequence of a uniform loss of GCs of 37% which according to the equations would give the 2 dB depression outside about 15° but no significant loss (<0.5 dB) of sensitivity on the four central stimulus points of the 24-2 array (Table 5B ).

If the effect of a uniform loss of 37% of GCs in central vision as suggested by these equations, is not very significant in terms of perimetric sensitivity, it might be expected that this would be revealed by a loss of acuity. This can be simulated by a recent model^{1} relating to foveal visual acuity which is affected by both D and optical degradation. Taking the values from this model,^{1} visual resolution (R_{v} ) is affected by neural (R_{n} ) and optical (R_{o} ) resolution which are linked by the following equation R_{v} ^{2} = R_{n} ^{2} + R_{o} ^{2} . Since R_{n} ^{2} is proportional to D, it may be shown that a 37% reduction in GCs would only cause approximately 14% reduction in visual acuity. This corresponds to only half the increment between lines in the LogMAR chart and might not be considered significant in aging subjects. However, the central-most points of the 24-2 array sample is an area which is rich in GCs,^{1} and the loss of 37% could be reflected in reductions in neuroretinal rim area, optic nerve fiber layer thickness and PERG amplitude since about 80% of the P50-N95 potential is probably related to GC activity^{18} a 30% reduction might be expected. We have previously reported such a reduction in Ocular Hypertensives who had no significant field defects.^{13}

In summary, the data tables and equations provide the basis for interim methods of quantification, which may give improved insight into neural damage or dysfunction in glaucoma . As these are consensus models, they may not always give useful results when applied to individual cases due to the wide variation in perimetric sensitivity^{12} and GC density.^{2} The methods are probably most useful when applied to group averaged or follow up data, or for modeling results in hypothetical examples.

Surprisingly, these calculations may change our perception of the nature of some visual field defects. They reveal the minute GC losses for the eye (<1%) in early glaucoma , that would be estimated with significant defects in the arcuate region when the total losses are believed much higher,^{3} indicating that there must be more widespread damage not detected by standard perimetric tests. Application of the non-linear model suggests that this damage may not be associated with sub-clinical uniform diffuse loss of sensitivity, but rather with a uniform loss of GCs which has little effect on central sensitivity due to the non-linearity. This gives an additional explanation for the location of detectable neural damage in the arcuate region, additional that is to the location of nerve fiber bundles; and may caste some doubt on the conventional wisdom which asserts that central vision is relatively unaffected in early glaucoma . The central area may in fact be affected but the effect concealed by the non-linearity. This also explains the high sensitivity of the PERG in early glaucoma and ocular hypertension since the PERG arises from the region of highest GC density where perimetry is least sensitive, illustrating the complementary nature of these tests. These studies also reinforce the observations in other recent reports which emphasize the importance of linear scales in assessing progression of neural damage.^{5,8} Furthermore, if future research reveals that the non-linearity in the second model is excessive, the above deductions will still be applicable, though to a slightly reduced extent.

ACKNOWLEDGMENTS
The support of the Leverhulme Trust, and the Wellcome Trust is acknowledged for the work which led to the production of this report.

Rachel V. North

School of Optometry and Vision Sciences

Cardiff University

Maindy Road

Cardiff, Wales CF24 4LU

United Kingdom

e-mail: north@cardiff.ac.uk

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Keywords: retinal ganglion cells ; receptive fields ; perimetry ; glaucoma ; pattern electroretinogram

© 2008 American Academy of Optometry