The most common assessment of visual function in glaucoma utilizes perimetric measurements of visual sensitivity. The clinical version of perimetry, SAP, employs threshold measures for small, white-light test stimuli superimposed on a white background, with test locations spaced throughout the central 48 or 60 degrees of the visual field (Anderson, 1987
; Johnson, 1996
; Heijl, and Patella, 2002
). The validity of this pointwise measure of visual function to gauge glaucomatous neuropathy requires a systematic relationship between the neural loss in a region of the retina and the resulting loss of sensitivity in the corresponding location in the visual field. Although, in the most elementary sense, there must be a causal relationship between RGC densities and the functions of the visual pathways in which the RGCs are involved, the quantitative structure-function linking relationship is neither theoretically nor empirically straightforward. The difficulty in defining the relationship was illustrated clearly by the initial study of post-mortem retinas of glaucoma patients that compared RGC densities to static perimetry thresholds (Quigley, et al., 1989
). The point-by-point relationship between the logarithmic loss of visual sensitivity, with respect to mean normal values, expressed in decibels (dB) and the percent of RGC loss, with respect to mean normal values, showed that, on average, statistically significant abnormalities of visual sensitivity required neural losses of 20 – 50 %, depending on the retinal eccentricity, and for any given level of neural loss there was a very large range of visual defects. A later study with a larger number of glaucoma patients (Kerrigan-Baumrind, et al., 2000
) also reported a low pointwise correlation between visual sensitivity and RGC losses, but the relationship was improved if global measures of sensitivity, such as average sensitivity loss or mean deviation (MD), were used to assess vision loss.
The variability in the relationship is smaller in studies of monkeys with laser-induced, experimental glaucoma (Harwerth, 2009
). Monkeys that are trained to perform behavioural perimetry provide very reliable SAP data (Harwerth, et al., 1993a
) and the timing of collection and processing of retinal tissue can be controlled (Harwerth, et al., 1999
). The first report on experimental glaucoma (Harwerth, et al., 1999
) assumed the log-linear relationship between visual sensitivity and RGC density that had been applied in the studies of retinas from human patients (Quigley, et al., 1989
; Kerrigan-Baumrind, et al., 2000
). The animal work confirmed the results of the studies of human tissue that the structure–function relationship was systematic, but it was not linear in the log-linear coordinate space (Harwerth, et al., 1999
). The general relationship suggested that, when the RGC losses were less than about 50%, there were small reductions in visual sensitivity, but the functional losses were not proportional to the structural losses. With greater degrees of neuropathy, the structure-function relation was more systematic, although with considerable variability. Thus, these results suggested other factors need to be included in the translation of ganglion cell losses to visual field sensitivity, specifically, 1) more appropriate measurement scales for sensitivity and neural losses, and 2) retinal eccentricity as an independent parameter (Harwerth, et al., 2004
). First, the quantification of visual loss on a decibel (dB) scale and the neural loss as a percentage, though intuitive, does not provide an accurate predictive relationship. As alternatives, linear and logarithmic transformations have been advocated, but a log-log coordinate system has a stronger theoretical basis for predicting the relationship between neural mechanisms for visual perception, i.e., the statistical description of probability summation for the detection of a stimulus that is imaged on a retinal area with multiple detectors (Pirenne, 1943
; Nachmias, 1981
; Robson, and Graham, 1981
; Tolhurst, et al., 1983
; Meese and Williams, 2000
). The fundamental principle of probability summation is that an observer will detect a stimulus whenever at least one, or a small ensemble, of the potential detectors in the population actually detects the stimulus (Pirenne, 1943
). By this principle, visual thresholds are not determined by linearly summed responses of all of the detectors in the total population, but rather by nonlinear pooling among neural detectors.
The general relationship between sensory events and neural substrates derived from probability summation is an exponential function of the number of detectors and the probability of detection for each of the available mechanisms (Robson, and Graham, 1981
; Tolhurst, et al., 1983
). To obtain a linear quantitative model of the structure–function relationship for clinical perimetry, the exponential function for visual sensitivity versus retinal ganglion cell density becomes linear via logarithmic transforms on both variables (Robson, and Graham, 1981
). The logarithmic transforms are an expression of visual sensitivity in dB, from the threshold value at a given test location and RGC density in dB, from 10-times the logarithm of the histological density of RGCs at the corresponding retinal location. It is obvious that the use of probability summation to model the relationship cannot be exact because it would require a homogeneous population of neural detectors. Homogeneity is unlikely for retinal mechanisms involved in image processing of the white-light perimetry stimulus, especially with glaucomatous neuropathy, but a linear neural-sensitivity relationship with logarithmic coordinates has been shown for clinical SAP (Harwerth, et al., 2004
The other source of variability in the psycho-physiological links between SAP and RGC density is retinal eccentricity. An eccentricity parameter was not considered in the initial studies of experimental glaucoma, although both visual sensitivity (Heijl, et al., 1987
; Harwerth, et al., 1999
) and RGC density (Rolls and Cowey, 1970
; Drasdo, 1989
; Curcio and Allen, 1990
; Wassle, et al., 1990
) vary as a function of eccentricity. For example, for the untreated, control eyes of rhesus monkeys (Harwerth, et al., 1999
) the perimetric sensitivity is highest near the fovea and falls by fivefold in the mid-periphery and inter-subject variability of sensitivity also varies, being lowest in the central field with systematic increases across the peripheral visual field (see ). Similarly, in normal monkey eyes the RGC density is dependent on retinal eccentricity with the highest concentrations of cells in the macular area and a 10-fold reduction in cell density in the mid-periphery.
Data from control and laser-treated eyes of monkeys that were used to derive an empirical structure-function relationship for glaucoma and the dynamic range of measurement for clinical perimetry.
In order to derive an empirical model of the neural-sensitivity relationship, the data for behavioral SAP measurements and histological RGC densities as a function of retinal eccentricity were analyzed by linear regression in log-log coordinate space, with ganglion cell density as the independent variable (Harwerth, et al, 2002
). An example of the regression analysis of the sensitivity and neural data for a retinal eccentricity of 21.2 deg, i.e., a perimetric test field location of 15 × 15 deg, is presented in . The data represent the control (open symbols) and experimental (filled symbols) eyes from 16 monkeys with laser-induced glaucoma. The parameters of the linear function are presented in the inset along with the coefficient of determination (R2
) showing that the linear function accounts for 79% of the variance. An analysis of data for 3 additional retinal eccentricities confirmed that the goodness-of-fit was similar at other eccentricities, but the parameters of the linear functions varied systematically with the retinal eccentricity. The histological and SAP data at the four eccentricities analyzed are presented in in both linear units (cells/mm2
for cell counts and apostilb (asb) for threshold intensity), and in logarithmic (dB) units. Examples of the eccentricity-dependent functions are illustrated in and with the slope and intercept values for the linear functions in log-log coordinates at four eccentricities are presented in . An additional regression analysis of the slope and intercept data showed that they are each also linear functions of eccentricity (deg) and, accordingly, provide a method to determine the slope and intercept parameters for the structure-function relationship at any given eccentricity. These linear regressions define the relationship between visual sensitivity and RGC density, but for clinical application the computation must be the opposite way, i.e., the quantification of RGC densities from SAP data. However, because the R2
values are high the functions presented in were used to derive RGC densities from SAP measures and the non-linear, structure-function model for SAP was based on three functions. One function to derive the slope (dB sensitivity/dB RGC density) and another to derive the y-intercept (dB sensitivity) of the linear functions in log-log coordinates. The third function is the relationship between RGC density (in dB units) as a function of SAP sensitivity based on the slope and intercept parameters derived from the eccentricity-dependent functions. The specific functions developed for experimental glaucoma in monkeys are:
Figure 1 A structure-function relationship between SAP visual sensitivity and histologically defined RGC density in monkeys with experimental glaucoma. (A) An example of the relationship for SAP visual sensitivity as a function of RGC density when both parameters (more ...)
Where m and b are the slope and y-intercept, respectively, for the function at retinal eccentricity ec, s is the SAP visual sensitivity, in dB units, and gl is the RGC density (somas/mm2), in dB units.
The model developed with SAP and RGC histology has implicit validity for quantifying the RGC density in an area of the retina from the corresponding SAP measurement, but it must be ascertained whether it is more accurate and precise than alternative models based on a linear translation of SAP data to the underlying RGC populations (Garway-Heath, et al., 2000
; Swanson, et al., 2004
; Hood and Kardon, 2007
). The most appropriate neural-sensitivity relationship is fundamental to the psycho-physiological linking proposition for glaucoma and the differences in nonlinear models (NLM) and linear models (LM) should be evaluated for a common set of data (Harwerth, et al., 2005
). Therefore, both algorithms were applied to data from monkeys with experimental glaucoma using SAP sensitivity measurements at a given test location and the histological counts of RGCs from the corresponding retinal area. The perimetry data were used to derive the predicted RGC densities and plotted as a function of the histologic counts. For the NLM () the predicted RGC densities were determined by functions 1
, above. The predicted RGC densities for the LM () were based on the simple assumption of a reciprocal relationship between visual sensitivity and RGC density and, therefore, the SAP sensitivity, in dB, was converted to an inverse linear light intensity and then scaled to equate the models for RGC densities for the control eyes at the most central SAP test location (i.e., 4.2 deg eccentricity). The specific methods are as follows:
Figure 2 A comparison of the relationships between the modeled and measured RGC densities. (A) Application of the NLM described by equations 1 – 3 and (B) application of the LM described by equations 4 and 5 for the translation between SAP visual sensitivity (more ...)
Where: s_loss is the reciprocal of the SAP stimulus intensity, s is the SAP sensitivity in dB units, k is a constant of proportionality (175,534) to scale the data for a linear relationship at a 4.2 deg eccentricity to the data for a logarithmic relationship, and gc is the predicted RGC density (dB).
The results of the nonlinear and linear model predictions of RGC densities with respect to measured RGC densities () were compared by several goodness-of-fit metrics that are presented as graph insets. First, the results can be visually compared to the perfect unity relationship, represented by the solid line, and by the coefficient of determination for the 1:1 relationship. The second metric, the accuracy of each model, is represented by the mean residual deviation (MRD) between the predicted and measured RGC densities, with the errors negative when the predicted values are greater than measured values, or the errors positive when the measured values are larger than the predicted values. Statistical indices for the precision of each model were based on three analytical methods; 1) the distributions of residual errors are presented as insets with the MRD ± standard deviation (SD) shown. The 95% limits of agreement, represented by the dashed lines on the main graphs, were determined by 1.96 times the SD of the error distribution., 2) the mean absolute deviation (MAD), which represents the average unsigned error with respect to the unity relationship, and 3) the root mean squared deviation (RMSD) as a measure of the variance with respect to the unity relationship that is influenced more by large deviations than small deviations, because large errors are especially important in evaluating the strength of the relationship.
In comparing the two models, it is interesting that their accuracies are very similar, with MRDs of less than 1 dB, but the distribution of residual errors is broader and, consequently, the 95% limits of agreement are larger for the LM compared to the NLM. The R2
, MAD, and RMSD, all indicate a greater degree of precision for the NLM than the LM. Therefore, for purposes of assessing population data the accuracies of the NLM and LM are similar, but for individual subjects the NLM provides a higher probability of an accurate prediction of RGC densities from SAP measurements (Harwerth, et al., 2005
Based on the analysis of the accuracy and precision, it is appropriate to apply the NLM to provide important descriptions of the neural mechanisms underlying perimetric visual field defects and the interpretation of stage of glaucoma from SAP measurements. For example, an interesting aspect of relating visual sensitivity to neural mechanisms is the sensitivity of the individual detection mechanisms. Because the structure-function relationship defines visual sensitivity as a function of the RGC density, the y-intercept represents the visual sensitivity (threshold) of a single detecting mechanism (Robson and Graham, 1981
), or in this case, a single mechanism per mm2
of retinal area. The y-intercept values, which also have been converted to light intensities in row 7 of , are 2.5 to 5.5 orders of magnitude brighter than the maximum intensity of the standard clinical instrument (104
asb) and, therefore, the dynamic range of measurement cannot assess the full range of RGC losses (Bengtsson, and Heijl, 2003
). On the other hand, the general characteristics of the neural detectors, i.e.
, an increasing threshold with retinal eccentricity, are in agreement with retinal anatomy, i.e.,
a decrease in packing density of cone photoreceptors (Curcio, et al., 1990
; Jonas, et al., 1992
) and a lower efficiency of shorter outer segments (Rodieck, 1998
) with increasing eccentricity, which provides facial credibility to the form of structure-function relationship developed from the empirical data.
The single mechanism detection thresholds also suggest that the dynamic range of measurement varies with eccentricity (see ). At the high end of visual sensitivity, a certain amount of neural loss must occur in early glaucoma before a significant abnormality of visual sensitivity can be identified statistically. The minimum sensitivity loss at a given location in the visual field that is generally considered to be clinically significant is a loss greater than the lower 95% confidence interval (CI). The reductions in the normal RGC densities (row 1, ) caused by a decrease in normal SAP sensitivity by 2 SD units (row 4, ) are presented in row 8 of , with the RGC losses expressed as percent loss or dB loss in rows 9 and 10, respectively. The calculated losses of RGC density caused by reductions of 2 SD in visual sensitivity, which are also illustrated graphically in , demonstrate that the 2 SD-loss in the absolute number of RGCs is considerably larger for eccentricities near the fovea, but because the normal densities are higher, the proportion of loss either as a percent, or in dB units, is much smaller for locations near fixation than for more eccentric locations.
With advanced stages of glaucomatous neuropathy, the visual sensitivity becomes too low to obtain a measurement and SAP fails to accurately define the neural losses. As was illustrated by the single detector threshold, the failure point that is indicated by a measurement of zero dB sensitivity occurs with a non-zero population of RGCs. The calculated RGC densities at zero dB sensitivity at each of four eccentricities are presented in linear units in row 11, in dB units in row 12, and as a percent of the normal population in row 13 of . These data also are presented graphically in to illustrate that the RGC density that is insufficient to produce a visual sensation during the SAP measurement, is smallest near fixation (0.2% of the normal population) and increases systematically with eccentricity and is up to 5.6% of the normal density at an eccentricity of 24 deg.
The range of neural loss between the RGC density at the initial statistically significant loss (95% CI) and the density at which SAP fails as a measurement of RGC function provides the dynamic range of measurement (see row 14, and ). The eccentricity dependent function shows a variation in the dynamic range from about 26 dB near the center of the visual field to just over 9 dB at the 24 deg eccentricity. It should be noted that the data used for this example, although based on data from experimental glaucoma, can be considered representative of young-adult humans (i.e.
, 25 – 30 years of age), but the normal cell density decreases with age (Blanks, et al., 1996
; Harman, et al., 2000
; Kerrigan-Baumrind, et al., 2000
; and see section 5) and, thus, the dynamic range is also age-dependent. For example, the upper limit of the dynamic range of measurement for a normal 65 year-old patient would be about 1.2 dB lower than illustrated in (Harwerth, et al., 2008
Taken altogether, these data from an application of the structure-function model that was developed to relate visual sensitivity and neuronal populations are useful for explaining some of the typical characteristics of visual field defects caused by the neuropathy of primary open-angle glaucoma (Anderson, 1987
; Epstein, 1993
; Quigley, 1993
). For example, because of the probabilistic nature of visual sensitivity, it would be expected that perimetric defects would occur in the peripheral visual field before the central field. At locations in the retinal periphery the structure-function relationship is steeper and, thus, a given loss of neurons will cause a greater loss of visual sensitivity, compared to more central locations with shallow structure-function relationships. Conversely, vision in the central field will be preserved until late stages because of the shallow slope of the relationship and because the initial RGC density is high. It is also important for the clinical interpretation of SAP data that the loss of retinal neurons associated with a given loss of visual sensitivity is dependent on retinal eccentricity. As an example, a 3 dB loss in the central field represents a much greater loss of RGCs than the same loss of sensitivity in the peripheral field, and the central defect would be significant at the p <0.05 level, while the peripheral defect would not.