purpose. Retinal nerve fiber (RNFL) thickness and visual field loss data from patients with glaucoma were analyzed in the context of a model, to better understand individual variation in structure versus function.

methods. Optical coherence tomography (OCT) RNFL thickness and standard automated perimetry (SAP) visual field loss were measured in the arcuate regions of one eye of 140 patients with glaucoma and 82 normal control subjects. An estimate of within-individual (measurement) error was obtained by repeat measures made on different days within a short period in 34 patients and 22 control subjects. A linear model, previously shown to describe the general characteristics of the structure–function data, was extended to predict the variability in the data.

results. For normal control subjects, between-individual error (individual differences) accounted for 87% and 71% of the total variance in OCT and SAP measures, respectively. SAP within-individual error increased and then decreased with increased SAP loss, whereas OCT error remained constant. The linear model with variability (LMV) described much of the variability in the data. However, 12.5% of the patients’ points fell outside the 95% boundary. An examination of these points revealed factors that can contribute to the overall variability in the data. These factors include epiretinal membranes, edema, individual variation in field-to-disc mapping, and the location of blood vessels and degree to which they are included by the RNFL algorithm.

conclusions. The model and the partitioning of within- versus between-individual variability helped elucidate the factors contributing to the considerable variability in the structure-versus-function data.

^{ 1 }or postmortem counts of retinal ganglion cells (RGCs),

^{ 2 }but the focus has shifted to quantifying retinal nerve fiber layer (RNFL) thickness and optic rim parameters measured with automated, noninvasive techniques, such as optical coherence tomography (OCT), confocal scanning laser ophthalmoscopy (SLO), and scanning laser polarimetry (SLP).

^{ 3 }

^{ 4 }

^{ 5 }

^{ 6 }

^{ 7 }

^{ 8 }for the relevant literature.) Recently, Hood, et al.

^{ 6 }

^{ 9 }

^{ 10 }

^{ 11 }showed that a simple linear model (SLM) described the general characteristics of these data. In its simplest form,

^{ 9 }

^{ 10 }this model assumes that RNFL thickness consists of two parts: RGC axons and everything else (e.g., glial cells, blood vessels). It further assumes that the loss in the thickness of the axon portion is proportional to local field sensitivity loss, when field loss is expressed on a linear scale. That is, a 3-dB loss (i.e., one half of normal sensitivity or a 50% luminance increase in threshold) is associated, on average, with a loss of one half of the axon portion of the RNFL thickness, whereas the nonaxon portion is assumed to remain constant with RGC loss. A variety of evidence is consistent with a linear model, at least in humans.

^{ 4 }

^{ 6 }

^{ 12 }

^{ 13 }

^{ 14 }On the other hand, Harwerth et al.

^{ 15 }

^{ 16 }

^{ 17 }

^{ 18 }have argued for nonlinear relationships between structure and function in humans

^{ 15 }

^{ 16 }and monkeys.

^{ 17 }

^{ 18 }

^{ 3 },

^{ 5 }

^{ 6 }

^{ 7 }

^{ 8 },

^{ 15 },

^{ 16 }). Hood and Kardon

^{ 6 }attempted to model some of the scatter in the data by modifying the SLM to predict the 95% confidence limits of RNFL measures. However, this model predicted only the variability in RNFL measurements; it ignored the variability in the SAP measurements.

^{ 6 }

^{ 10 }

^{ 11 }we were interested in the sensitivity loss in arcuate regions falling within the 24-2 SAP test field, and so we used the definitions of these regions provided by Garway-Health et al.

^{ 19 }The associated locations on the 24-2 SAP field are shown in Figures 2A and 2B . The decibel values for each location of the total deviation field within the arcuate region were converted to a linear scale (e.g., 0 dB converted to 1.0 and −30 dB to 0.001) before they were averaged within each sector.

^{ 6 }

^{ 20 }

^{ 21 }The decibel value of these sector averages is plotted in Figures 2 and 3 .

^{ 19 }(see Figs. 2A 2B ) were averaged and are plotted in Figures 2 and 3 .

*increase*with age. For the superior and inferior arcuate regions, the positive slopes were 0.11 and 0.18 μm/y. Further, less than 1.5% of the variance in RNFL thickness was accounted for by age. The average thickness for the total RNFL profiles versus age had a slope of −0.10 μm/y. Although this value is low compared with most values in the literature, it is within the range of values obtained in recent studies.

^{ 22 }

^{ 23 }

^{ 24 }

^{ 25 }We do not know why the effects of age were minor in our study, but we speculate that it could be related to the quality of our scans. As mentioned, we routinely performed multiple scans at each visit and chose the scans of highest quality for analysis. Cheung et al.

^{ 26 }recently reported that RNFL thickness decreases with decreased signal strength. Perhaps part of the effect of age is due to lower signal strength in older populations, secondary to, for example, smaller pupils and/or media opacities (some the normal eyes in this study were also pseudophakic). The signal strength in our study for the 41 control subjects younger than 54.2 years was 9.1, close to the value (8.8) in the 41 control subjects older than 54.2 years. In any case, a correction for age would have relatively little effect on the variability seen in our study, even if we had used the values in the literature. For example, Budenz et al.

^{ 24 }in a study of 328 control subjects found a value of 0.2 μm/y. If we had used this value and normalized the OCT thickness relative to 50 years of age, as in 24-2 SAP, then the age-corrected points would change; but the change would be relatively small, between −6 and +6 μm for 20 to 80 years of age.

^{ 6 }

^{ 9 }

^{ 10 }has three assumptions:

- The RNFL thickness,
*r*, measured with the OCT technique for any individual, is made up of two components. One component,*s*, is the thickness due to the RGC axons, and the other is the residual or base level,*b*, thickness. That is,\[r{=}s{+}b\]The residual*b*includes glial cells, blood vessels, and anything that the proprietary algorithm includes in the determination of RNFL thickness, besides axons. - As SAP field sensitivity decreases, the value of
*s*, the thickness due to the axons, decreases, but the residual level*b*does not change. - The relationship between the loss in axons thickness,
*s*, is linearly related to the loss in sensitivity on a linear, not a decibel, scale.

*t*is relative sensitivity and is equal to 10

^{0.1d }, where

*d*is the individual’s total deviation in decibels from the mean age-matched machine norms;

*s*

_{o}is the value of

*s*in equation 1when sensitivity is normal (

*t*= 1.0); and

*b*is the residual thickness. Note that an individual’s RNFL thickness is

*s*

_{o}+

*b*when the field is normal (

*t*≥ 1.0); that is, RNFL thickness does not depend on

*t*for normal control subjects.

*R*,

*S*,

*B*, and

*T*are the average or median values for the group. Note that the average RNFL thickness is

*S*

_{o}+

*B*when the field sensitivity, on average, is normal (

*T*≥ 1.0); that is, it does not depend on

*T*.

*B*and

*S*

_{o}+

*B*indicated by the dashed lines. Fitting the model requires estimating

*S*

_{o}and

*B*.

*B*, the asymptotic value when all sensitivity and all axons are lost, can be estimated as the median of the patients’ data for field losses greater than −15 dB (where the RNFL becomes asymptotic).

*S*

_{o}, the thickness of the axon portion in a healthy RNFL, can be obtained by using the mean of the control RNFL thicknesses as the estimate of

*S*

_{o}+

*B*.

^{ 6 }modified the linear model to take into consideration individual variations in RNFL thickness. In particular, they added the assumption that:

- Individuals with healthy vision differ in RNFL thickness. They have different
*s*_{o}+*b*values. Individuals with healthy vision also have different*b*values, which are approximated as ⅓(*s*_{o}+*b*) for the arcuate regions. In particular, based on data from a group of patients with severe unilateral anterior ischemic optic neuropathy (AION; correlating*b*from the affected eye with*s*_{o}+*b*from the fellow unaffected eye),^{ 11 }it was assumed that

*S*

_{o}+

*B*. By definition, this is the mean value for a group of age-matched control subjects. The predicted curve, shown as a bold black line in Figure 1B , for the mean RNFL thickness (filled square) can be determined from the mean of the control measures (filled square). This theoretical curve does not depend on the patients’ data. In other words, it is derived without any degrees of freedom.

*s*

_{o}+

*b*value and asymptote at one third that value, as shown by the top dashed curve in Figure 1B . Thus, an individual’s curve is determined by the RNFL thickness

*s*

_{o}+

*b*when it is healthy (

*d*= 0 dB and

*t*= 1.0). Therefore, the model predicts that there should be variability in the data, even if there is no measurement error, based on the starting RNFL thickness in a healthy eye, which varies among normal subjects. Hood and Kardon

^{ 6 }obtained an approximation of this range by plotting two dashed curves (Fig. 1B) , encompassing the 95% range in normal OCT values.

*x*-axis in the figures cited later). To extend the logic of the approach just described to variability on both axes requires calculating a 95% boundary that takes both axes into consideration.

^{ 27 }). If we assume that the two variables are normally distributed and independent, this ellipse will be the 95% confidence boundary. To a first approximation, these assumptions are supported by the data. Neither variable fails a test for normality (Kolmogorov-Smirnov test) and, although there is a weak positive correlation between RNFL thickness and SAP loss in normal eyes, it is very small (

*r*= 0.1) and not significant; further, the slope is very shallow.

*d*, which was taken as the “true” loss (no measurement error) in SAP field sensitivity and was a continuous variable from 0 dB (normal sensitivity for any individual) to −30 dB. In particular, let

*Y*

_{ ij }(

*d*) is the OCT RNFL thickness (micrometers) for the

*i*th individual on the

*j*th test day in disease state

*d*; μ

_{ y }(

*d*) is the mean RNFL thickness over a large (infinite) number of individuals and large number of test days, all with the same

*d*value (i.e., μ

_{ y }is the population means without individual variability or measurement error);

*e*

_{ ybi }(

*d*) is the between-individual error (individual differences) for the

*i*th individual and depends on

*d*; and

*e*

_{ ywij }is the within-individual error (measurement error) for the

*i*th individual on

*j*th test day. Similarly, for the SAP field loss (in dB)

*X*

_{ ij }(

*d*) is the SAP field loss for the

*i*th individual on the

*j*th test day; μ

_{ x }(

*d*) is the mean SAP field loss over many individuals and many test days;

*e*

_{ xbi }is the between-individual error for the

*i*th individual; and

*e*

_{ xwij }(

*d*) is the within-individual error for the

*i*th individual on

*j*th test day and depends on

*d*.

*e*

_{ ybi }(

*d*),

*e*

_{ xbi }

_{,}

*e*

_{ ywij }, and

*e*

_{ xwij }(

*d*) are independent, normal distributions with means of 0 and standard deviations of ς

_{ yb }(

*d*), ς

_{ xb }, ς

_{ yw }, and ς

_{ xw }(

*d*) (assumption A). In addition, we made two other assumptions. First, we assumed that random variables

*e*

_{ xbi }(assumption Ba) and

*e*

_{ ywij }(assumption Bb) do not vary with the level of

*d*. Figure 3Aprovides support for assumption Bb, whereas

*e*

_{ xbi }should depend on the sample of patients, not

*d*. Second, we assumed that

*e*

_{ ybi }(

*d*) (assumption Ca) and

*e*

_{ xwij }(

*d*) (assumption Cb) do vary with the level of

*d*. In particular, for assumption Cb we assumed that

*e*

_{ xwij }(

*d*) has a mean of 0 and a standard deviation of ς

_{ xw }(

*d*) and is estimated from the repeat-measure data in Figure 3C(solid green curve), as explained later. For assumption Ca, we assumed that

*e*

_{ ybi }(

*d*) has a mean of 0 and a standard deviation equal to:

*Y*

_{ yb }(without measurement error) for a given

*d*will be decreased by the same factor, for any given

*d*, the value of ς

_{ yb }for this value of

*d*will be decreased by the same factor. This factor is derived from equations 2 2and 4 4 .

*d*can be generated by using radii of 2.45 times the standard of

*X*and

*Y*, where these standard deviations equal [ς

_{yb}(

*d*)

^{2}+ ς

_{ yw }

^{2}]

^{1/2}and [ς

_{ xb }

^{2}+ ς

_{ xw }(

*d*)

^{2}]

^{1/2}. The ellipse is centered on the mean (black curve) at [μ

_{ x }(

*d*), μ

_{ y }(

*d*)], where

*d*ranging from 0 (normal) to −30 dB for the upper field. The smooth green curves in Figure 1Dare the envelopes of a large number of these ellipses and represent the 95% boundaries for all individuals. All parameters needed to derive these ellipses from the model were estimated from the data—that is, no parameter fitting was involved. For the upper field, μ

_{ y }(0) = 142.6 μm; ς

_{ yb }(0) = 16.4 μm; ς

_{ yw }= 6.4 μm; μ

_{ x }(0) = 0.33; ς

_{ xb }= 0.90 dB; ς

_{ xw }(0) = 0.57 dB; and ς

_{ yb }(

*d*) is given by equation 6and ς

_{ xw }(

*d*) by the solid green curve through the data in Figure 3C . For the lower field, μ

_{ y }(0) = 125.8 μm; ς

_{ yb }(0) = 16.4 μm; ς

_{ yw }= 6.4 μm; μ

_{ x }(0) = 0.00 dB; ς

_{ xb }= 0.90 dB; ς

_{ xw }(0) = 0.57 dB; and ς

_{ yb }(

*d*) is given by equation 6and ς

_{xw}(

*d*) by the solid green curve through the data in Figure 3C .

^{ 19 }map. Figures 2A and 2Bshow the RNFL disc sectors and associated SAP field regions for the upper visual field/inferior disc sector and the lower visual field/superior disc sector, respectively. In Figures 2C 2D 2E 2F , the OCT RNFL thickness for the appropriate disc sector is plotted against the visual field loss (decibels) for the corresponding visual field region. The data for the 82 control subjects are shown in Figures 2C and 2D , where each open symbol represents the data for one individual. The bold lines indicate the 95% CI (i.e., ±2 SD) for RNFL thickness (vertical) and field loss (horizontal), with the intersection of these lines indicating the mean of the control subjects’ data. These mean values are shown in Figures 2E and 2Fas the filled squares. See the caption to Figure 2for the CIs and the means.

^{ 6 }

^{ 9 }

^{ 10 }the patients’ RNFL thicknesses decreased with visual field loss, approaching an asymptotic value for field losses more extreme than approximately −10 dB.

^{ 28 }

^{ 29 }and is approximately the same for control subjects and patients—that is, 6.4 μm (Fig. 3A , dashed line), which supports assumption Bb. The same data are shown in Figure 3B , plotted against mean SAP field loss. As might be expected, the RNFL within-individual variability (repeat-measurement error) of normal eyes also appeared independent of the level of field loss.

^{ 30 }

^{ 31 }

^{ 32 }). This same pattern can be seen in the data for the arcuate regions in Figure 3C . In Figure 3C , the standard deviation for an individual’s five SAP examinations is plotted versus the mean of these values for the upper (red) and lower (blue) regions of the field. Note that the standard deviation increases from approximately 0.6 (the mean of the control values shown as the large green square and the black dashed line) to a peak in the range of −13 to −20 dB before decreasing again for extreme field losses. The open squares represent the grouped data obtained as in Figure 3A . These data support assumption Cb, and the solid green curve that approximates the grouped data was used for deriving the ellipses in Figures 1C and 1D , as described in the Methods section. (Note that the dashed green curve was drawn through the higher values. As mentioned earlier in the Discussion section, the LMV’s predictions were also derived with this curve.)

^{2}) of the within-individual (or repeat measure) 6.4 or 41.0 μm

^{2}. The total variance for the single examinations of the 82 control subjects is 17.6 or 309.8 μm

^{2}for the control subjects. Thus, the between-individual RNFL variance is 268.8 μm

^{2}(309.8–41.0) or approximately 87% of the total variance. (Note that based on a similar analysis for the SAP field results, the between-individual threshold variance for normal eyes is 0.81 dΒ

^{2}or approximately 71% of the total variance.)

*P*= 0.06, Fisher’s exact test, two-tailed), it is of interest because the presence of an ERM certainly adds to the RNFL thickness measured.

*n*= 13) of the 14 upper extremes had two major temporal BVs within the arcuate analysis window, whereas only one had one BV and none had no BVs within the arcuate sector analysis window. On the other hand, of the 14 lower extremes, only 2 had two BVs, whereas 12 had either one (

*n*= 8) or none (

*n*= 4) within the analysis window. In addition, for the two extremes with two BVs, the algorithm did not include the BVs. The difference between the number of eyes with BVs within the arcuates for the upper versus lower extremes was statistically significant (

*P*= 0.0001; two-tailed Fisher’s exact test using a 2 × 2 table with the 0-BV and 1-BV cells in Table 1combined).

^{ 33 }However, a BV contributes to the RNFL thickness measured only if it is incorporated within the RNFL by the algorithm. Figure 6Aprovides an example in which the signals from the two major inferior BVs are incorporated within the RNFL thickness determined by the algorithm (Fig. 6A , middle, white lines). The BVs are seen on the fundus photograph (bottom left) lying within the arcuate analysis window shown as the black radial lines in relation to the peripapillary scan line (green). The locations of the signals from these vessels are confirmed by the shadows they cast (arrows) in the expanded scan (bottom right). When the RNFL becomes thin and the signals from the BVs are not surrounded by significant signals from axons, the full extent of the BVs is not included within the boundaries of the algorithm. This point is illustrated in Figure 6B . The signal from a BV is indicated by the red arrow in the middle panel. The lower insets show a magnified version of this scan (right bottom inset) and a higher resolution version (left bottom inset). The BV signal was largely excluded from the RNLF thickness, as determined by the algorithm (white lines).

^{ 6 }

^{ 9 }

^{ 10 }

^{ 11 }showed that an SLM relating structure to function describes the general shape and central tendency of the data when local RNFL thickness is plotted against local visual field loss. This model was confirmed in the present study (Figs. 2E 2F , black curves). However, the data from individuals show a large degree of scatter around the mean theoretical curve, as is true of similar plots in the literature (see, for example, Refs.

^{ 3 },

^{ 5 }

^{ 6 }

^{ 7 }

^{ 8 },

^{ 15 },

^{ 16 }). Our concern in the current study was with the variability in the data. To better understand this variability, we obtained estimates of within- and between-individual variability and tested a model that incorporated the information. In particular, to obtain a prediction of the variability to be expected, a model proposed by Hood and Kardon

^{ 6 }was modified and extended by making explicit assumptions about the within- and between-individual variability in both OCT and SAP measurements.

^{ 23 }for a review). These results are similar to those of two recent studies of patients with glaucoma, which reported good OCT repeatability that was independent of mean RNFL level.

^{ 27 }

^{ 28 }In any case, within-individual variability accounted for only 13% of the variance in the control OCT values. It was not a major factor in the scatter seen in the control data, as shown by comparing the vertical dimensions of the blue and green ellipses in Figure 7 . With disease progression, the relative contributions of within- and between-individual OCT variability approached similar values.

^{ 29 }

^{ 30 }

^{ 31 }The relative impact of the SAP within-individual error can be seen in Figure 7 . For SAP sensitivity better than −6 dB, the horizontal width of the blue ellipses (only within-individual error) was smaller than that of the red ellipses (only between-individual error). The reverse was true for field sensitivity worse than or equal to −6 dB, where within-individual error predominated until extreme field losses were attained.

^{ 34 }

^{ 35 }

^{ 36 }

^{ 6 }

^{ 10 }

^{ 19 }Garway-Heath et al.

^{ 19 }estimated that the 95% CI for the location on the optic disc associated with a particular SAP field location spanned a range of almost 30°. Figures 6C 6D 6Eillustrate how mapping may contribute to variability. In Figure 6C , the filled red symbols shows the RNFL for two patients, P3 and P4, whose RNFL thickness profiles and sample scans are provided in Figures 6D and 6E , respectively. (Note that the peaks of the RNFL profiles in these two subjects appear displaced from those of the average normal subject shown in the overlaid green, yellow, and red boundaries.) The solid red vertical lines are the boundaries of the analysis window for the lower arcuate disc sector. The dashed vertical lines show these boundaries shifted so that the major temporal BVs now fall within these boundaries. The result is a considerable increase in the RNFL thickness (Fig. 6C , red arrows). Shifts of approximately 14° (10 on a 256-point scale) and 18° (13 points) were required—shifts well within the 95% CI of nearly 30° reported by Garway-Heath et al. These two patients were chosen because, although the original analysis window (solid black line on fundus photographs in Figs. 6D 6E ) did not include the major inferior temporal BVs, the shifted window did (dashed lines on fundus photographs).

^{ 6 }

^{ 10 }

^{ 11 }

^{ 37 }

^{ 38 }That is, the between-individual variation in this asymptotic part of the function is largely due to the number of BVs within the disc region of analysis, and the degree to which these BVs are incorporated into the RNFL thickness by the algorithm (see Fig. 6B ) (Kay KY et al., manuscript submitted).

^{ 33 }

^{ 39 }

^{ 40 }It should be possible to decrease OCT between-individual error if the BVs were treated in a consistent way, independent of the state of the RNFL.

^{ 16 }

^{ 18 }Both models assume that the RNFL is composed of axons and nonaxons (what we call the residual). We assume that the residual portion remains constant with disease progression.

^{ 6 }

^{ 10 }Harwerth et al.

^{ 16 }assumed that the residual, or at least the glial portion, increases with axon loss due to aging and/or glaucoma. Although it is not clear at this point which assumption will ultimately prove to be more accurate, it is clear that that the OCT data used to test these assumptions impose limitations. Both our approaches assume implicitly that the RNFL algorithm treats the nonaxon portion (residual) the same regardless of the thickness of the axon portion. We have found that this is not true of the RNFL algorithm of the Stratus OCT3 (Fig. 6and Ref.

^{ 40 }). Normal control subjects typically have the full extent of their BVs included, whereas patients with extreme field loss typically have very little of their BVs incorporated. It should be pointed out that the frequency domain OCT does not necessarily solve this problem, as the critical factor is the algorithm, not the machine. The better resolution available with the frequency domain OCT, however, allows for a clearer identification of the BVs and offers the possibility of developing a procedure to either explicitly include or exclude their contribution. In any case, an appropriate test of a structure–function model depends on a consistent treatment of signals from BVs or alternatively, analysis of retinal regions not containing large BVs.

^{ 16 }although this is unlikely to explain the extremes of relatively small field losses. In any case, further tests of the basic assumptions of the linear model will be possible as patients are observed over times long enough to see progression.

Outlier Location | Extremes (n) | Extremes with BVs in Arcuate (n) | Extremes with BVs within Algorithm (n) | ||||
---|---|---|---|---|---|---|---|

0 BV | 1 BV | 2 BV | |||||

Above | 14 | 0 | 1 | 13 | 18 | ||

Below | 14 | 4 | 8 | 2 | 0 |

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