purpose. To describe and evaluate a new statistical technique for detecting topographic changes in the optic disc and peripapillary retina measured with confocal scanning laser tomography.

methods. The 256 × 256-pixel array of topographic height values obtained with each image from the Heidelberg Retina Tomograph (Heidelberg Engineering, Heidelberg, Germany) was divided into an array of 64 × 64 superpixels, where each superpixel contained 16 (i.e., 4 × 4) pixels. An analysis of variance technique was developed to analyze each superpixel with three baseline and three follow-up images. The performance of the technique was tested with and without adjustment for spatial correlation of topographic values using computer simulations and with real data from a normal control subject and a patient with progressive glaucomatous disc change.

results. Computer simulation with fixed population means and variance, and varying spatial correlation showed a monotonically increasing number of superpixels with significant test results (false positives), with 20% false-positives when the spatial correlation was 0.8 (the approximate median value in real patient data). The number of false-positive results was similar (17%) in serial images of a normal subject. When corrected for spatial correlation, the number of false-positives was independent of the level of spatial correlation and remained at the expected value of less than 5% in both simulations and real data. Although the number of significant test results in the patient with progressive glaucoma decreased after correction for spatial correlation, the change was readily apparent. Statistical power to detect mean differences in topographic values ranging from 0.5 to 4.0 SDs in computer simulation showed low power for changes of 1 SD or less, but increased dramatically with larger changes.

conclusions. This technique has a high level of sensitivity to detect changes in the optic disc while maintaining a high level of specificity.

^{ 1 }Quantitative clinical assessment of the optic disc for the follow-up of glaucoma is frequently performed with cup-to-disc ratio estimates.

^{ 2 }With stereo or monoscopic photography and planimetry, it is also possible to compute the neuroretinal rim area, whereas stereophotogrammetry

^{ 3 }

^{ 4 }and stereovideographic techniques

^{ 5 }allow volumetric estimation of the optic cup.

^{ 6 }

^{ 7 }but briefly, confocal sections of the optic disc, where the focal plane of the laser and the detector plane are optically conjugate, are obtained. The focal plane of the laser is changed incrementally to obtain the sections. The optical setup ensures that the information contained in a given image section is derived largely from the focal plane of the laser. After the confocal sections are aligned and processed, topographic heights of discrete locations in the scanned area are estimated.

^{ 8 }

^{ 9 }

^{ 10 }The most important advantages of the technique are ease of operation, rapid image acquisition and processing times, and, unlike conventional photography, the ability to obtain images with natural pupils in most patients.

^{ 11 }

^{ 10 }

^{ 12 }Because these anatomic features are unique to each eye, local variability estimates should be made for each eye to gauge whether corresponding local topographic differences between two sets of images separated by time are statistically significant.

*z*-axis perpendicular to the optical axis. The image acquisition time is 1.6 seconds.

*z*-axis. For a given pixel, the area under the reflectivity profile is the sum reflectivity, and the position of maximum reflectivity along the

*z*-axis is assumed to be the topographic height value. After the calculations have been made for all pixels, reflectivity and topographic images of the scanned area are determined. Therefore, the result of each scan is a topographic image representing the topographic height of each pixel from the focal plane of the eye.

^{ 9 }) from which a mean topographic and reflectivity image are computed after horizontal, vertical, and rotational alignments are made with a correlation procedure using the reflectivity images. Alignment for depth and tilt are made using the topographic images.

^{ 10 }In this case, there are 48 measurements (3 × 4 × 4) per superpixel. Because the size of a superpixel in a 10° × 10°-scan is approximately 47 × 47μ m, variability estimates based on the 48 measurements will be influenced by the topography of the imaged structure in the superpixel. Therefore, although the pooling has minimal effect on a superpixel situated in an area with flat topography, in an area with steep contours such as the optic cup edge, the variability estimates are increased. To remove the topographic component in variability estimates, we subtracted the respective topographic measurement in each pixel from its respective mean across the three images to determine the adjusted value. After this process, we computed three corrected images from which estimates of test–retest variability were made. These estimates are expressed as variability maps and have been described in detail elsewhere.

^{ 10 }

*I*baseline and follow-up images and that the topographic measurements from a superpixel of 4 × 4 pixels is a vector of length 16 indexed by

*l*, an analysis of variance model for each superpixel in vector form is:

*h*

_{ tli }represents the topographic height value at time of examination

*t*, at location in image array

*l*, in image

*i*(

*t*= 1,2;

*l*= 1,. . ., 16;

*i*= 1, . . .,

*I*)

*.*μ represents the overall mean topographic value;

*T*

_{ t }the time effect, which allows for differences between the baseline and follow-up examinations;

*L*

_{ l }, the location effect, which allows for differences among pixels;

*TL*

_{ tl }, the time by location effect, which allows the time effect to differ by location;

*I*(

*T*)

_{ it }, the image within time effect, which allows for variability in the images at baseline and at follow-up; and

*e*

_{ tli }, the error effect assumed to be independently normally distributed with a mean of zero and variance ς

_{ e }

^{2}.

*MS*{NUM} =[

*SS*{T} +

*SS*{TL}]/16; the mean squared denominator, or

*MS*{DEN} =[

*SS*{I(T)} +

*SS*{e}]/υ; where

*SS*{T} is the sum of squares associated with time;

*SS*{TL} is the sum of squares associated with time by location interaction;

*SS*{I(T)} is the sum of squares associated with image within time;

*SS*{e} is the sum of squares associated with the residual or error; and degrees of freedom υ = 2(

*I*− 1)16.

*T*

_{ t },

*TL*

_{ tl }, and

*I*(

*T*)

_{ it }are assumed to be fixed effects, then the statistic in equation 2 has an

*F*distribution with 16 and υ degrees of freedom. (See Appendix A for a detailed description of the test statistic).

*I*(

*T*)

_{ it }is a random variable with a mean of zero and varianceς

_{ I(T)}

^{2}that are independent of

*e*

_{ tli }, then the model (equation 1) allows for spatial correlation between topographic values within a superpixel. The correlation between

*h*

_{ tli }and

*h*

_{ tmi }(two locations within the same superpixel) is

^{ 13 }appropriately reduces the degrees of freedom of the approximating

*F*distribution for the test statistic to

*MS*{T},

*MS*{TL}, and

*MS*{e}are the mean squared errors of the corresponding effects. The dependence of the degrees of freedom on the spatial correlation is measured by the intraclass correlation and is shown in Appendix B. In particular, the degrees of freedom are a monotonically decreasing function of ρ, and are 16 and υ for ρ = 0. The effect of the reduced degrees of freedom is to increase

*P*(reduce significance).

*P*< 0.05. When the corrected test procedure was used, however, the number of superpixels with

*P*< 0.05 was 4.0%. The distribution of the intraclass correlation coefficients within all superpixels in the imaged area was skewed toward higher values (Fig. 4) with a median value of 0.790.

^{ 14 }derived after a contour line defining the border of the optic disc is defined. The serial analysis can comprise regression analyses or a set of paired tests comparing baseline and follow-up images. An advantage of this approach is that a clinically familiar summary measure can be derived from the thousands of individual topographic values. Potential limitations are inadequate estimates of intraindividual variability, because one image yields only one estimate of the index, and the loss of spatial information not captured by summary measurements. Many of the limitations of summary indices for complex data sets are also present in the serial analysis of computed perimetry results,

^{ 15 }where change in visual field status has to be determined.

*F*statistic is adjusted to account for the spatial correlation between topographic values of pixels within a superpixel.

*h*

_{ tli }, the topographic height value at time

*t*, location

*l*, and image

*i*. In this appendix we give an explicit expression for, and justification for, the test statistic given in equation 2 .

*MS*{NUM} and

*MS*{DEN}

**.**

*MS*

**{**NUM

**}**is a scaled measure of the total squared deviation between the estimated mean heights at the two times, whereas

*MS*

**{**DEN

**}**is a scaled measure of the total squared deviation of individual heights around their means. Under the null hypothesis of no mean differences among the images at each time and of no mean difference between the two times—that is, no

*T*,

*TL*, or

*I*(

*T*) effects—the

*F*ratio is approximately equal to one. Large values of

*F*give evidence against the null hypothesis of no difference and suggest that the differences are real.

*t*, the mean heights are μ

_{ tl }= μ +

*T*

_{ t }+

*L*

_{ l }+

*TL*

_{ tl }. These means are estimated by the average heights at each location

*l*

*T*

_{ t },

*TL*

_{ l }, +

*I*{T}

_{ it }are assumed to be fixed effects, then each difference has variance 2ς

^{2}/

*I*, and

*SS*{NUM}/(2ς

^{2}/

*I*) has aχ

^{2}distribution with 16 degrees of freedom. The numerator of the test statistic is

*h*

_{ tli }− μ̂

_{ tl }estimates the error term

*e*

_{ tli }and their sum of squares

^{2}, where υ = 2(

*I*− 1)16, so an estimate ofς

^{2}is given by

*SS*{DEN}/ς

^{2}has a χ

^{2}distribution with υ degrees of freedom, independent of

*SS*{NUM}. It can be shown that

*SS*{I(T)} is the total variation among images within different follow-up times, and

*SS*{e} is the total residual variation.

*F*distribution with 16 and υ degrees of freedom. If

*I*(

*T*)

_{ it }is assumed to be a random effect, however, then the Satterthwaite approximation is used to approximate the distribution of the test statistic as an

*F*with

*f*

_{N}and

*f*

_{D}degrees of freedom, as in equation 3 .

*I*− 1)16. If there is no spatial dependence (ρ = 0), then the expected number of degrees of freedom are (16,υ), which agrees with the classic result. As ρ increases, the number of degrees of freedom in both the numerator and denominator decrease in a monotonic fashion. In the limit, as ρ approaches 1,

*E*[

*f*

_{ N }] approaches 1 and

*E*[

*f*

_{ D }] approaches 2(

*I*− 1)

**.**

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

**Figure 7.**

**Figure 7.**

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