purpose. To evaluate a linear model that relates the glaucomatous loss in retinal nerve fiber (RNFL) thickness, measured with optical coherence tomography (OCT), to the loss in sensitivity, measured with standard automated perimetry (SAP).

methods. Fifteen patients with asymmetrical glaucoma, whose better eye was normal or near normal (mean deviations better than −3 dB) on SAP, were tested. SITA 24-2 standard and OCT RNFL thickness measures were made on three to five different occasions and the mean values were obtained. For each eye, the mean SAP loss was calculated for an upper and lower arcuate field region by averaging the loss in relative sensitivity on a linear scale. The average RNFL thickness for corresponding arcuate sectors of the lower and upper optic disc was obtained for each eye. A linear model was fitted to the plots of RNFL thickness versus SAP loss. According to the linear model, the RNFL thickness *R* = *s* _{o} *T* + *b*, where *T* is the SAP sensitivity loss relative to age-matched normal eyes (linear scale), (*s* _{o} + *b*) is the RNFL thickness in the healthy/normal state (*T* = 1), and *b* is the residual RNFL thickness measured when all sensitivity and all axons are lost.

results. The model provided a reasonable fit to the data with best fitting values of (*s* _{o}; *b*) of (upper field: 80.6 μm; 50.5 μm) and (lower field: 67.4 μm; 50.5 μm) and (upper field: 78.8 μm; 54.9 μm; *r* = 0.82) and (lower field: 59.2 μm; 61.5 μm; *r* = 0.70) for two different methods of best fit.

conclusions. A linear model that relates RNFL thickness to losses in SAP sensitivity describes the results for arcuate regions of glaucomatous visual fields. The linear model provides a framework for assessing the relative efficacy of structural and functional tests throughout the course of the disease.

^{ 1 }and in more accurately determining the stage of the disease in an individual eye.

^{ 2 }and a recent article by Bowd et al.

^{ 3 }) However, the theoretical treatment of these data has been relatively limited.

^{ 2 }Recently, Hood

^{ 4 }proposed a simple model that predicts the relationship between RNFL thickness and loss of SAP sensitivity. This model, based on a model

^{ 5 }proposed to explain the relationship between multifocal visual evoked amplitudes and SAP losses, assumes that the local loss in RNFL thickness is linearly related to the loss in SAP sensitivity, when SAP sensitivity loss is expressed on a linear scale. However, a complete loss in sensitivity does not result in an RNFL thickness of zero; rather, it is associated with a finite RNFL thickness. In particular, there is some minimum value beyond which the thickness cannot be reduced because glial cells or other factors do not contribute to the number of axons. According to this model, the portion of the RNFL thickness due to retinal ganglion cell (RGC) axons is linearly related to the proportion of RGC axons (“response”) remaining.

^{ 4 }reviewed the data from published studies

^{ 6 }

^{ 7 }

^{ 8 }

^{ 9 }

^{ 10 }that used optical coherence tomography (OCT) to measure RNFL thickness and concluded that these data were consistent with the proposed model. However, there are problems with evaluating this model with the published data. First, the results from both OCT and SAP may vary from day-to-day, due to measurement variability. Existing studies typically present the cross-sectional results for a single OCT and SAP test. Second, the best way to evaluate the model is to compare local sensitivity to local RNFL loss. Most of the studies in the literature present the SAP data averaged over a large region (e.g., hemifield or full-field of 24-2 SAP). Finally, in existing studies, the SAPs value for a region of the visual field are usually expressed as an arithmetic average of dB units for each location within the region considered, but, as detailed in the Methods section, according to the model, these values should be antilogged before averaging and then logged again after averaging.

^{ 1 }

^{ 5 }

^{ 11 }

^{ 12 }The data have been presented in abstract form (Hood DC, et al.

*IOVS*2007;48:ARVO E-Abstract 490).

*n*= 5; normal tension glaucoma,

*n*= 7; pigmentary glaucoma,

*n*= 2; pseudoexfoliative glaucoma,

*n*= 1) were participants in a larger prospective study and were chosen on the basis of interocular asymmetry of glaucoma. In particular, the least-affected eye was normal or had mild damage (better than a mean deviation of −3 dB). These patients were judged to be stable by visual field testing and disc appearance over the period during which each was studied. The patient characteristics are summarized in Table 1 . Procedures adhered to the tenets of the Declaration of Helsinki, and the protocol was approved by the committee of the Institutional Board of Research Associates of Columbia University and the University of Iowa Institutional Review Board (IRB) for Human Use.

*n*= 10 patients for five determinations;

*n*= 4 patients for four determinations;

*n*= 1 patient for three determinations). The average time between the first and last test was 19.3 ± 4.8 months. As mentioned earlier, the SAP fields and disc appearance of each patient were stable over the testing period. For the arcuate field regions studied (described later), the change in mean deviation between the first and last test averaged −0.45 ± 2.39 dB (lower field) and −1.06 ± 2.60 dB (upper field), where a negative number is an improvement in sensitivity.

^{ 12 }(Fig. 1) . For visual field threshold data, the decibel levels in each location of the total deviation field were converted to a linear scale (e.g., 0 dB converted to 1.0 and −30 dB to 0.001) before averaging the data within each sector.

^{ 1 }

^{ 5 }

^{ 11 }To understand the rationale for averaging the values on a linear scale rather than on the log (decibel) scale, one must consider a region of the retina in which the left half is normal (0 dB total deviation and a normal complement of RGCs) and in which the right half has no RGCs remaining and a maximum loss in sensitivity (a total deviation of ∼ −30 dB). If we take the average of the decibel levels, we get −15 dB, which is 1/30 on a linear scale. In contrast, if we take the average on the linear scale, we get 0.5 (1.001/2), which is −3 dB on the decibel scale. The model predicts that the OCT RNFL thickness due to RGC axons for the entire region should be one half the normal thickness.

^{ 2 }

^{ 4 }In particular, RNFL thickness

*D*is the total SAP deviation from normal sensitivity in decibels (the 0.1 in the exponent term converts from decibels to log units);

*s*

_{o}is the RNFL thickness attributable to the RGC fibers in the healthy normal state; and

*b*, the base level, is the residual response

*R*(RNFL thickness) obtainable when a patient has lost all sensitivity to light and all RGC axons. Note that when the total deviation,

*D*, is 0 dB and the region on average is normal, then

*R*equals (

*s*

_{o}+

*b*). When the total deviation is very abnormal (e.g.,

*D*is −30 dB), then

*R*is approximately equal to

*b*. So, the predicted curve goes from (

*s*

_{o}+

*b*) to

*b*as the field goes from normal sensitivity (0 dB field loss in Figs. 2A 2B ) to a total loss in light sensitivity (∼ −30 dB). To derive a predicted curve from equation 1 , we estimated the starting (

*s*

_{o}+

*b*) and ending (

*b*) points of the curve.

*s*

_{o}+

*b*) equal to the median of the arcuate RNFL thicknesses for the better eye. (

*s*

_{o}+

*b*) was calculated separately for the superior (131.1 μm) and inferior (117.9 μm) fields.

*b*was set equal to the median (50.5 μm) of the RFNL thickness for field loss greater than −10 dB for both superior and inferior field regions. (For field losses worse than −10 dB, the RNFL thickness reaches an asymptotic value.)

*T*, the relative sensitivity, equals 10

^{0.1×D}.

*T*equals 1.0 when there is no loss (0 dB difference from normal) and approaches 0 for large losses in sensitivity. For example, total deviations (

*D*) of −3 dB and −30 dB, yield

*T*of 0.5 and 0.001, respectively. For the solid lines, the

*s*

_{o}and

*b*were the same as in Figures 2A and 2B . For the dashed straight lines, the best-fitting linear regression was obtained by allowing both

*s*

_{o}and

*b*to vary. The parameters, expressed as

*s*

_{o}+

*b*;

*b*, of best fit were 133.7 μm; 54.9 μm (superior field) and 120.7 μm; 61.5 μm (inferior field), with

*r*= 0.82 and 0.70, respectively. The parameters for the two methods were similar with those of the best-fitting line, tending to be slightly larger. When the data from only the last test date were analyzed, the results were very similar. The best-fitting parameters (

*s*

_{o}+

*b*;

*b*) were similar (123.6 μm; 57.5 μm, superior field; and 116.7 μm; 57.8 μm, inferior field) as were the correlations (

*r*= 0.81 and 0.69, respectively).

^{ 2 }map in Figure 1 . To get a measure of whether this map was optimal for the eyes in this study, the following analyses were performed. First, the width of the Garway-Heath et al. arcuate disc sector was held constant (40°) and rotated in 5° steps around the optic disc. For each sector location, a correlation coefficient

*r*between the SAP field loss and the RNFL for that location was determined with our linear model. Figure 3shows the

*r*-value on the

*y*-axis as a function of the position of the center of the 40° sector for correlations with the superior arcuate SAP field values and inferior arcuate SAP field values. The best correlations are close to the center of the sectors in the map by Garway-Heath et al. For the superior arcuate field region, the best correlations are centered approximately 5° nasal from the center of the inferior arcuate disc sector in Garway-Heath et al. The agreement with the map of Garway-Heath et al. is excellent.

^{ 4 }This model has been shown to describe the relationship between local multifocal VEP amplitude and local SAP field loss.

^{ 5 }While the fit of our data to the model was reasonably good, there are still outlier points and some scatter of points about the predicted curve. Before discussing the clinical implications of these findings, we will consider the possible sources of the scatter of the data points.

*s*

_{o}

*T*, which represents the thickness of the retinal axons associated with a given relative sensitivity,

*T*, and the other,

*b*, which is the residual RNFL thickness measured when all the axons are lost. This residual portion includes glial cells and perhaps limitations imposed by the algorithm that determines the RNFL layer. In individuals with normal visual sensitivity (

*T*= 1.0), the RNFL thickness is the sum of

*s*

_{o}+

*b*, where

*s*

_{o}is the thickness of the axon portion in the normal healthy eye. There is a wide range of values for (

*s*

_{o}+

*b*) as defined by the 95% confidence interval for normal RNFL thickness as shown by the green region in Figure 4C . Assuming for the moment that

*b*is the same in different eyes, this confidence interval provides a range of normal (

*s*

_{o}+

*b*) values. (For the purposes of this example, the effects of age on

*s*

_{o}, and thus on the confidence interval, were not taken into consideration.) The upper and lower boundaries of the confidence interval each provide the parameter (

*s*

_{o}+

*b*) for a theoretical curve. Figure 5shows the data from Figure 2with the predicted curves associated with the upper and lower limits of (

*s*

_{o}+

*b*) estimated from the green region in Figure 4E . In particular, the upper curve describes the predicted course of glaucomatous progression in a patient who started with a relatively large

*s*

_{o}, whereas the lower curve shows the predicted curve in a patient with an

*s*

_{o}that was relatively small when normal. Regardless of the initial RNFL thickness, all curves have the same common shape, meaning that there is a loss in SAP sensitivity that is proportional to RNFL thickness attributable to RGC axons. Note that the linear model, combined with the normal confidence interval, predicts that most of our data points should fall between these curves.

*b*is the same in different individuals, whereas in fact there may be a variation in

*b*due to a variety of factors. In particular, individuals may have different amounts of glial contributions to the RNFL thickness in the normal eye, and this contribution may increase or decrease with progressive losses in RGC axons due to disease. Third, as detailed earlier, there may be interindividual variations in the map between a particular region of the SAP field and the corresponding sector of the optic disc.

^{ 12 }If we know the mapping for an individual patient, we suspect that a substantial portion of the scatter could be reduced. Finally, there may be sources of SAP sensitivity loss that are not associated with RNFL loss. For example, sensitivity loss due to disease occurring before the RGCs (e.g., media opacities, photoreceptor disease) would result in data points falling above the predicted curves. In isolated glaucoma, there may be cases in which SAP sensitivity loss precedes a loss of axons. In such cases, the associated data points also should fall above the predicted curve. In fact, the model provides a framework for identifying these so-called sick as opposed to dead RGCs.

^{ 4 }That is, if local RNFL thickness is reduced by one half, then the number of RGC axons is reduced by one half. If we assume that the local number of axons is proportional to RNFL thickness, then the model predicts that the loss in the relative number of local RGCs is linearly related to local sensitivity loss (in linear terms). On the one hand, this assumption is in apparent contradiction with the conclusions in the monkey

^{ 13 }

^{ 14 }

^{ 15 }and human

^{ 15 }

^{ 16 }

^{ 17 }studies in which RGCs were counted in postmortem tissue. On the other hand, it is in agreement with theoretical treatments of the human data that argue for a linear relationship.

^{ 1 }

^{ 2 }

^{ 5 }

^{ 18 }

^{ 19 }For example, the best articulated model of the human data by Swanson et al.

^{ 19 }predicts that the relationship is linear for eccentricities beyond 5° to 10°. There are several differences among the methodologies in these studies,

^{ 2 }

^{ 19 }and further work is needed before we can account for the differences in conclusions.

*losses*in sensitivity (i.e.,

*D*≤ 0 dB or

*T*≤ 1.0); it makes no prediction about individuals who may have greater than normal sensitivity. We know that both SAP sensitivity and OCT RNFL thickness decrease with age. What remains to be determined is the quantitative relationship between these changes in the arcuate regions, as well as the nature of the changes, if any, within groups of normal individuals of the same age.

^{ 17 }reported that a loss of 25% to 35% of the RGCs was associated with a statistically significant SAP field loss. According to our model, a loss of 25% to 35% of the RGCs would result in a loss in local sensitivity between −1.2 and −4.6 dB, within the 5% confidence interval (∼ −5 dB for local SAP points). However, this assumption does not necessarily mean that the RNFL measure is more sensitive than SAP for detecting early damage. In practice, which test is more sensitive to early damage will depend on the relative variation of each normal control measure and the initial RNFL thickness (

*s*

_{o}+

*b*) when the eye is healthy.

*s*

_{o}values. According to the linear model, each patient would show a proportional −3-dB field loss and normal SAP test results. In contrast, a 50% RGC loss and the consequent 50% decrease in

*s*

_{o}in the patient with the smaller

*s*

_{o}would result in a much smaller RNFL thickness than that in the patient with the larger

*s*

_{o}. Thus, the patient with the smaller initial RNFL thickness may show an abnormal RNFL thickness, whereas the other is still in the normal range. Thus, a significant change in the OCT RNFL thickness may occur before a significant change in the SAP in the patient with the smaller initial RNFL thickness. Conversely, the patient with the thicker initial RNFL may show abnormalities on the SAP before the RNFL thickness drops below the 5th percentile of the normal range. In general, to determine which of two tests is the more sensitive, one must know both the function that relates the two tests and the variability in test scores in healthy individuals.

^{ 1 }

^{ 5 }

^{ 14 }

*b*, as the SAP loss approaches −10 to −15 dB. The value of

*b*, approximately 50 μm, provides an estimate for the lower bound of the OCT RNFL thickness. On average, the thickest part of the RNFL profile is <200 μm (Fig. 1 , bottom). Thus, the largest range typically encountered spans a factor of less than 4. According to the model, a −15-dB loss would reduce a 200-μm thickness to 54.7 μm {[150 μm × 0.03 (antilog of −15 dB)] + 50 μm}. This value is within the measurement error of the lower bound of 50 μm. Further, this is close to a best-case scenario; in practice, local SAP losses of more than −10 dB will not yield an RNFL thickness detectably different from the base level,

*b*. Thus, the RNFL thickness is of limited use for observing regions with extensive damage. In contrast, the SD of SAP under conditions of a −10-dB loss is very large.

^{ 20 }For both tests, progression is probably best observed by examining relatively healthier regions of the same eye.

Patient | Age | Gender | Mean Deviation | Interocular Visual Field Asymmetry (dB) | Test Sessions (n) | Duration of Testing (y) | ||
---|---|---|---|---|---|---|---|---|

OD (dB) | OS (dB) | |||||||

MA | 54 | Female | −0.93 | −7.04 | 6.11 | 5 | 1.7 | |

CB | 55 | Female | −2.51 | −7.37 | 4.86 | 3 | 0.6 | |

BD | 65 | Female | −5.13 | −2.38 | 2.75 | 5 | 1.1 | |

KRD | 66 | Female | −14.64 | −0.83 | 13.81 | 4 | 1.6 | |

KD | 61 | Female | −0.3 | −13.09 | 12.79 | 5 | 2.0 | |

DG | 65 | Female | −1.13 | −16.11 | 14.98 | 5 | 1.6 | |

VG | 53 | Female | −0.32 | −5.45 | 5.13 | 4 | 1.3 | |

JM | 67 | Female | −2.03 | −5.05 | 3.02 | 5 | 1.9 | |

CR | 71 | Female | −2.67 | −5.48 | 2.81 | 5 | 2.0 | |

CHR | 76 | Male | 0.23 | −3.86 | 4.09 | 5 | 1.9 | |

JR | 52 | Male | −7.07 | −2.65 | 4.42 | 4 | 1.5 | |

AS | 71 | Male | −2.75 | −0.77 | 1.98 | 4 | 1.5 | |

MS | 51 | Male | −0.57 | −11.44 | 10.87 | 5 | 2.1 | |

HS | 70 | Female | 0.19 | −4.56 | 4.75 | 5 | 1.9 | |

WS | 61 | Male | 0.42 | −6.57 | 6.99 | 5 | 1.5 |

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