Analytical Model of Scanning Laser Polarimetry for Retinal Nerve Fiber Layer Assessment

Abstract

purpose. To develop a quantitative understanding of scanning laser polarimetry (SLP) for retinal nerve fiber layer (RNFL) assessment in glaucoma diagnosis and management.

methods. The Mueller calculus was used to model the polarization optics of SLP. A birefringent retinal structure (RNFL or macula) was represented as a circularly symmetric linear retarder with a radial slow axis. The birefringent cornea and a corneal compensator within the SLP instrument were represented as fixed linear retarders. The model provided images of the radial retarder that were compared with retardance images obtained by SLP of the macula in eight normal subjects. Theoretical and experimental images were quantified with circular profiles around the center of the radial retarder or macula. Experimental retardance profiles were varied by tilting the subject’s head to rotate the corneal axis. The SLP model was fit to the experimental profiles by nonlinear least-squares curve fitting.

results. The combined retarder formed by the cornea and corneal compensator induced bow-tie patterns in images of the radial retarder. Macular SLP images exhibited similar patterns. Retardance profiles could be characterized by three parameters: modulation, mean, and axis. The SLP model fit the experimental profiles very well (r ² = 0.8–0.9).

conclusions. The SLP model provided a quantitative framework within which to interpret SLP studies. Modulation-based parameters were generally more sensitive to retinal birefringence than mean-based parameters. Corneal birefringence is an important source of variance in SLP, especially for mean-based parameters. The theory developed for this study may guide improvements in clinical SLP.

The diagnosis and management of glaucoma require sensitive methods for detecting and measuring damage to retinal ganglion cells. The potential value of retinal nerve fiber layer (RNFL) assessment in glaucoma has long been known, ¹ and recognizable loss of the RNFL often precedes measurable loss of vision. ^{1 2 3 4 5 6} Scanning laser polarimetry (SLP), a technology that incorporates ellipsometry into a confocal scanning laser ophthalmoscope, ^{7 8} detects the birefringence of the peripapillary RNFL. SLP is attractive, because it provides rapid assessment, a visual image of the RNFL, and the potential for reproducible, objective measures of RNFL loss. 

The RNFL exhibits substantial linear birefringence with the slow axis parallel to the direction of nerve fiber bundles (approximately radial around the optic disc) ^{8 9} and retardance that correlates well with RNFL thickness. ¹⁰ This birefringence probably results from form birefringence of closely spaced cylindrical structures. ¹¹ The normal RNFL thickness in humans at the location of typical SLP scans may range from 130 to 250μ m. ¹² Two current estimates of RNFL birefringence are 0.1 nm/μm in fixed macaque retina (calculated from Fig. 5 of Reference ¹⁰ ) and 0.23 nm/μm in ex vivo rat retina. ¹³ Estimating with these two values results in double-pass RNFL retardances of 25 to 50 nm and 60 to 120 nm, respectively. Because fixation decreases the RNFL birefringence, ¹³ the actual values in normal human subjects are likely to be closer to the higher estimate. 

A potential confounding variable for SLP is corneal birefringence. Commercial SLP instruments have incorporated a corneal compensator (CC) intended to neutralize it, ^{7 14 15} but no analysis of the cornea-SLP system has been published. Although a general model describes the cornea as a curved biaxial crystal, ^{16 17} for perpendicularly incident light the cornea is well-represented as a linear retarder. ^{16 18} The mode of the distribution of corneal slow axes falls between 10° and 20° nasally downward (ND), ^{18 19} and double-pass retardance ranges from near 0 to 250 nm. ^{18 20} Early concepts for a CC used a variable retarder, ^{14 15} but the current instrument uses a fixed 60-nm linear retarder with its fast axis oriented 15° ND (information courtesy of Qienyuan Zhou, Laser Diagnostic Technologies, Inc., San Diego, CA). Although the CC fast axis lies at the mode of the corneal slow axis distribution and is thus oriented to oppose the birefringence of many corneas, the CC is not aligned with the axes of many other corneas. Furthermore, CC retardance (120 nm in double-pass) is greater than approximately 80% of corneal retardance. ¹⁸ Thus, in spite of corneal compensation, corneal birefringence can contribute significant population variance to SLP measurements. For example, several SLP output parameters are strongly correlated with the slow axis of corneal birefringence. ¹⁹ SLP measurements, however, are reproducible, ^{21 22 23 24 25 26} at least in part because within individuals the corneal axis has good long-term stability. ²⁷  

Additional evidence that corneal birefringence complicates SLP measurements comes from SLP images of the macula. Henle’s fiber layer, long parallel photoreceptor axons extending radially from the fovea, imparts substantial radial birefringence to the macula ^{28 29 30} with an estimated double-pass retardance of approximately 30 nm (measured at 1.25° and 2.9° from the fovea at a wavelength of 568 nm). ²⁹ (This birefringence is distinct from the dichroism at short wavelengths associated with Haidinger’s brush. ^{17 28} ) Henle’s fiber layer is more circularly symmetric than the RNFL, yet SLP images of the macula show distinct“ bow-tie” or “double-hump” patterns that do not correspond to the underlying anatomy. ¹⁹  

This study was designed to gain a quantitative understanding of the role of corneal birefringence in SLP measurements, first by analyzing the interaction of corneal compensation and the cornea over a wide range of corneal birefringence, next by incorporating corneal compensation into a theoretical model of SLP of a radial retarder, and finally by experimentally testing the model with SLP measurements of the birefringent macula in normal subjects. The model produced bow-tie patterns similar to those in macular images. The theoretical and experimental patterns were quantified with circular profiles around their centers. The variation of theoretical profiles with corneal axis was then compared with the variation of experimental profiles obtained while tilting the subject’s head to rotate the corneal axis. This study found that the SLP model fit the experimental profiles very well and that it could explain many results from other, previously published studies of SLP. 

Methods

Matrix Methods

The interaction of the optical components in the SLP model was determined by means of the Mueller calculus, ³¹ in which the polarization properties of a component X are completely specified by a 4 × 4 matrix, the Mueller matrix M _X. The properties of multiple components in series are then given by the Mueller matrix formed by successively multiplying the Mueller matrices of all components in the order in which a light beam encounters them. Similar calculations have been used previously in the eye, to determine the interaction of corneal and macular birefringence in retinal birefringence scanning. ³² All calculations were performed on computer (Matlab; The MathWorks, Inc., Natick, MA) using well-established Mueller matrices. ³¹ The Mueller matrix of a linear retarder with fast axis (ρ) and retardance (δ) is  

\[\mathbf{M}{=}\left(\begin{array}{cccc}1&0&0&0\\0&C_{4}\ \mathrm{sin}^{2}({\delta}/2){+}\mathrm{cos}^{2}({\delta}/2)&S_{4}\ \mathrm{sin}^{2}({\delta}/2)&\mathrm{-}S_{2}\ \mathrm{sin}({\delta})\\0&S_{4}\ \mathrm{sin}^{2}({\delta}/2)&\mathrm{-}C_{4}\ \mathrm{sin}^{2}({\delta}/2){+}\mathrm{cos}^{2}({\delta}/2)&C_{2}\ \mathrm{sin}({\delta})\\0&S_{2}\ \mathrm{sin}({\delta})&\mathrm{-}C_{2}\ \mathrm{sin}({\delta})&\mathrm{cos}({\delta})\end{array}\right)\]

where C ₂ is cos(2ρ), C ₄ is cos(4ρ), S ₂ is sin(2ρ), and S ₄ is sin(4ρ). 

In this article, retardance is expressed as a difference in optical path length (units of δ are in nanometers). It is understood that before a value for δ was used in equation 1 , it was converted to a phase difference at 780 nm. Subscripts on ρ and δ identify the optical component to which they refer. It was frequently necessary, both for convenience and for consistency with the existing literature, to describe the slow axis of a retarder. These cases are denoted with a superscript s; thus, for example, ρ_C ^s denotes the slow axis of the cornea, and a ρ_C ofρ _C ^s + 90° was used in equation 1 . All axis orientations are relative to the horizontal nasal meridian, as viewed by an observer facing the eye, with ND taken as negative. 

Model of SLP

The analysis of SLP used the model in Figure 1 . In SLP, images of the ocular fundus are formed by scanning the beam of a near-infrared laser (780 nm) in a raster pattern. ⁷ The ellipsometer (E) measures at each image point the total retardance in the optical path, by detecting the ellipticity induced in a linearly polarized input beam. ⁷ In the model, the measuring beam passed through three linear retarders: the corneal compensator (CC), the cornea (C), and a uniform radial retarder (R) that represented birefringent regions in the retina (e.g., peripapillary RNFL or macula). The slow axis of R was oriented radially, and distance around R was measured from the horizontal nasal meridian by angle β. At each point, therefore, the fast axis of R was ρ_R =β + 90°. Radial variation in retardance was not analyzed. The measuring beam was reflected at a deeper layer and traveled back through the three retarders to the ellipsometer. Reflection from the ocular fundus exhibits a high degree of polarization preservation, ⁹ and the reflector in the model (polarization-preserving reflector [PPR]) was assumed to preserve completely the polarization state of the incident beam, except for a 180° phase change due to the reversal in direction. This phase change reversed the sign of the azimuth and the handedness of the reflected polarization. ³³ Each optical component in the model experienced a double pass of the measuring beam. A component that had axis ρ in the incident beam had axis −ρ in the reflected beam and the relative orientation of the component axis to the polarization azimuth was maintained. Because the ellipsometer was insensitive to handedness, a double pass with reflection was equivalent to a single pass through the components of the model followed by the components in reverse order. Thus, an image of R was described by a profile around its center derived from  

\[\mathbf{M}({\beta}){=}\mathbf{M}_{\mathrm{CC}}\mathbf{M}_{\mathrm{C}}\mathbf{M}_{\mathrm{R}}({\beta})\mathbf{M}_{\mathrm{R}}({\beta})\mathbf{M}_{\mathrm{C}}\mathbf{M}_{\mathrm{CC}}\]

Letting m _ij be the elements of M(β), the total retardance in the image was, from equation 1 ,  

\[{\delta}_{\mathrm{T}}({\beta}){=}\ \frac{{\lambda}}{360{^\circ}}\ \mathrm{cos}^{-1}(m_{44})\]

where λ is 780 nm. 

Experimental Methods

Eight subjects (four women, four men, ages 25–58), who on clinical examination had normal anterior segments and maculas, were studied after providing written informed consent. The research followed the tenets of the Declaration of Helsinki and was approved by the Institutional Review Board of the University of Miami. One eye was tested in each subject. 

Macular birefringence was imaged with a commercially available SLP nerve fiber analyzer (GDx; Laser Diagnostic Technologies, Inc., San Diego, CA). The GDx imaged the fundus with a 256 × 256-pixel raster scan that covered an area of 15° × 15° in visual angle. To obtain macular images, the instrument was aligned on the cornea and pupil as usual, and the subject was instructed to fixate the center of the red rectangle formed by the 780-nm scanning laser. To achieve stable but tilted head position, the usual head-and-chin rest was replaced with one that pivoted in the plane of the subject’s face and gave support to the side of the subject’s head. 

For each scan, the GDx provided two images of the fundus, a reflectance image (intended as a means to judge scan quality) that showed retinal blood vessels at high contrast and a retardance image that provided the data from which perifoveal profiles were extracted. Reflectance images were cropped from the GDx screen display. Retardance images were written to a file by a software routine provided with the instrument. Subsequent image and data processing was performed on computer (Matlab software; The MathWorks, Natick, MA). 

The amount of head tilt was difficult to control and, because of compensatory torsional eye movements, ³⁴ did not necessarily correspond to the rotation of the corneal axis. Fortunately, the reflectance images provided a means to determine the rotation of the eye. Using one image as a reference, the rotation and translation of all other images were determined by registration with triple invariant image descriptors, ³⁵ followed by manual adjustment. Images were registered to within 1 pixel of translation and 0.5° of rotation. The average rotation of images obtained with the usual instrument headrest was used to define the vertical head position. The corneal rotation was assumed to be equal to the fundus rotation and was added to the corneal axis at vertical to provide a value of ρ_C for the particular image. 

For each retardance image, a perifoveal profile was extracted from an annular region with an inner radius of 1.8° and an outer radius of 2.2° (1.3 mm in diameter in an emmetropic eye), a region that corresponded approximately to the peak of macular birefringence, as observed in GDx images. The center of the annulus was determined from several retardance images with pronounced macular bow-tie patterns by manually placing a cursor in the middle of the saddle-shaped center of the bow tie. Contour lines superimposed on the image aided this placement. When corrected for translation and rotation and superimposed on the reference image, the selected points clustered close to the region in the reflectance image judged to be the anatomic fovea. The centroid of the cluster was used as the annulus center in all images. At each of 128 angular steps around the annulus, the pixel values across its width were averaged to produce a profile value. 

In each subject, the SLP model was fit to the perifoveal profiles by nonlinear least-squares curve fitting (routine LSQCURVEFIT of the Matlab Optimization Toolbox, ver. 2.0; MathWorks)—that is, the set of measured profiles was fit by a set of calculated profiles that minimized the square of the difference between the two sets. Standard errors of the parameters were calculated from the Jacobian of the curve fit. ³⁶  

To provide a comparison to values obtained by fitting the SLP model, in six subjects corneal birefringence was also measured with a corneal polarimeter described elsewhere. ¹⁸ Briefly, the polarimeter provided a view of the fourth Purkinje image of a 585-nm light through crossed polarizers and a variable retarder. The Purkinje image was extinguished by adjusting the fast axis and retardance of the retarder to match the slow axis and retardance of the cornea, and the corneal parameters were read from calibrated dials. The other two subjects were not available for corneal polarimetry, but their corneal axes, measured in an earlier study, were included. ¹⁹  

Results

Analysis of Corneal Compensation

It was useful to consider the combined retarder (CR) formed by a double pass through CC and C, namely,  

\[\mathbf{M}_{\mathrm{CR}}{=}\mathbf{M}_{\mathrm{CC}}\mathbf{M}_{\mathrm{C}}\mathbf{M}_{\mathrm{C}}\mathbf{M}_{\mathrm{CC}}\]

The slow axis (ρ_CR ^s) and retardance (δ_CR) of CR were used to characterize the effects of corneal compensation. (Although shown in Fig. 1 as 1/2CR, the system formed by a single pass through CC and C is not strictly a retarder, but is equivalent to a retarder followed by a rotator. ³⁷ The rotations cancel in the symmetric chain of linear retarders in equation 4 , and M _CR is the Mueller matrix of a linear retarder.) M _CC was calculated from equation 1 , whereδ _CC is 60 nm and ρ_CC is− 15°. 

The behavior of CR as a function of ρ_C ^s, calculated directly from equation 4 for realistic values ofδ _C, ^{16 18} is shown in Figure 2 . For all corneas, δ_CR declined to a minimum where the slow axis of C and the fast axis of CC were parallel (ρ_C ^s = ρ_CC = −15°; Fig. 2A ). The minimum value was δ_CR = 2|δ_CC − δ_C|. Forδ _C = δ_CC = 60 nm, this minimum was zero—that is, CC exactly canceled C, but forδ _C ≠ δ_CC the minimum could reach values comparable to the retardances of the macula or RNFL. The behavior of ρ_CR ^s was more complex (Fig. 2B) , Nearly any ρ_CR ^s could be produced by some combination of ρ_C ^s and δ_C, although in actual corneas some values of ρ_CR ^s were more common than others (see the Discussion section). 

SLP of a Radial Retarder

The retardance pattern in SLP images of the radial retarder in Figure 1 was determined from δ_T(β) in equation 3 . Some results are displayed in Figure 3 as retardance images and profiles. The patterns can be understood as an interaction between R and CR. As ρ_R varied withβ , δ_CR alternately added and subtracted with the double-pass radial retardance (2δ_R) to produce the bow-tie pattern seen in the images. The bright arms of the bow tie (i.e., the peaks of the profile) occurred whereρ _R and ρ_CR were aligned. The profile modulation depended on the relative values ofδ _R and δ_CR. The peak-to-peak amplitude (A _p-p) of the modulation was  

\[A_{\mathrm{p-p}}{=}{\vert}{\delta}_{\mathrm{CR}}{+}2{\delta}_{\mathrm{R}}{\vert}-{\vert}{\delta}_{\mathrm{CR}}-2{\delta}_{\mathrm{R}}{\vert}\]

In theory, profile modulation could be governed either by the retarder R undergoing measurement, by the combined retarder CR formed by CC and C, or by a combination of both. When R had no birefringence (2δ_R = 0) there was no modulation of the profile; it was constant at a value equal toδ _CR (flat dashed lines in Fig. 3 ). 

To compactly display the results in a wide range of corneas, each retardance profile was characterized by three measures: peak-to-peak modulation (A _p-p), mean level (average around which the profile swings), and orientation (i.e., bright axis of the bow tie). Figure 4 shows the variation with ρ_C ^s of these measures for five values of δ_C and three values ofδ _R. Modulation (Fig. 4 , top row) varied nearly linearly with δ_R except nearρ _C ^s = −15°. Near ρ_C ^s =ρ _CC = −15°, especially for corneas with 2δ_C ≈ 2δ_CC = 120 nm, the modulation exhibited a dip toward 0. On the slope of this dip, the modulation did not change with δ_R, except at very low values (e.g., Fig. 3B and the corresponding points labeled b in Fig. 4 ). The mean retardance (Fig. 4 , middle row) increased with the difference between ρ_C ^s andρ _CC, resulting in a minimum atρ _C ^s = −15°. The height of the minimum increased with the difference between 2δ_C and 2δ_CC. In the region of the minimum, the mean varied with δ_R, but away from the minimum the mean was much less sensitive than the modulation to changes inδ _R. The variation with ρ_C ^s of the bow-tie bright axis (Fig. 4 , bottom row) was the same as that ofρ _CR ^s in Figure 2B , because maximum total retardance occurred when the slow axes of R and CR were aligned. The variation of bow-tie axis with ρ_C ^s did not depend onδ _R. The bright axis of the bow tie was approximately vertical over a wide range of corneal parameters, but for 2δ_C > 120 nm and ρ_C ^s >− 15° the bow-tie axis lay closer to horizontal. 

Experimental Test of the SLP Model

To test the theoretical concepts developed in the foregoing section, we reproduced experimentally the situation depicted in Figures 3 and 4 and then attempted to fit the data with the model expressed in equations 2 and 3 3 . Henle’s fiber layer in the macula of the eye provided a circularly symmetric birefringent structure corresponding to R in Figure 1 . It was assumed that corneal retardance (δ_C) remained constant; the corneal axis (ρ_C ^s) was varied by tilting the subject’s head. 

Three examples of macular retardance images and perifoveal profiles are shown in Figure 5 . Pronounced bow-tie patterns, similar to those in Figure 3 but with retardance that varied with distance from the center, were apparent in this subject for head tilts to either side of vertical (images A and C). Near vertical, there was only weak modulation of the macular retardance (image B). The bow-tie patterns were quantitatively evaluated by means of perifoveal profiles (left column) obtained from the annular region of maximum modulation. Each profile was approximated by a simpler, smooth profile synthesized from terms 0, 2, and 4 of its Fourier series (i.e., the mean and first two even harmonics for a total of five independent coefficients per profile). To aid the comparison of data to theory, the three parameters used in Figure 4 to characterize the theoretical profiles—A _p-p, mean level, and bow-tie orientation—were derived from the synthesized profile. In each subject the data fitted by the model consisted of 19 to 45 perifoveal profiles acquired with various amounts of fundus tilt. 

To fit the corneal compensation model to the head tilt data, two additional properties of the GDx were taken into account. First, the output of the GDx was in arbitrary units, which we termed GDx units (GU), and we added as a parameter the scale factor s required to convert GU to retardance in nanometers. Second, the retardance values in GDx images never approached zero (as they can in theory). We hypothesized a “noise floor,” a minimum value determined by instrumental factors and by random variation in the raw images from which retardance was calculated. This concept was introduced into the model by means of the arbitrary transfer function shown in Figure 6A , which had the form (δ_T ⁿ + F ⁿ )^1/n , where F was the noise floor and n controlled the sharpness of the transition between F and the line of equality. A value of n = 3 gave a good balance between an unrealistic clipping of the troughs of some fitted profiles and a rapid approach to equality. Thus, the SLP model fitted to the data was  

\[G({\beta}){=}s({\delta}_{\mathrm{T}}({\beta})^{3}{+}F^{3})^{1/3}\]

where G(β) is a set of profiles expressed in GU that depend on five parameters: s, F, and the three ocular properties that control δ_T(β) through equations 2 and 3 3 : macular retardance (δ_R), corneal retardance (δ_C), and corneal axis at vertical head position. 

Figures 6B and 7 show the result of fitting the SLP model to perifoveal profiles from three subjects. The fitted parameters of the model for all eight subjects are displayed in Table 1 . Values for noise floor (sF) and the SD of the residuals (SD) are reported in GU to allow comparison with the output display of the GDx instrument. As seen in Figure 6B , in six examples (one high-modulation and one low-modulation profile from each subject) most model profiles followed closely the actual profiles, although exceptions occurred, usually near the modulation minimum (e.g., Fig. 6Ba ). The fit across all head positions is shown in Figure 7 , where the measures used to characterize both actual and fitted profiles are plotted in the same format as Figure 4 . The solid curves in Figure 7 were calculated from theoretical profiles, i.e., profiles generated from the parameter values in Table 1 inserted into the SLP model (equations 2 , 3 , and 6 ). The circles show measures of the smooth Fourier description of each actual profile (as exemplified in Fig. 5 ); these measures were independent of the model parameters. Overall, the SLP model fit the data quite well, explaining 80% to 90% of the profile variance (r ² = 0.81–0.91 in Table 1 ; P < 0.0001). 

For Table 1 the SLP model was applied to the profiles from each subject individually to produce eight separate tests of the model. This also produced eight estimates of the scale factor s and the noise floor F. In fact, these two parameters were properties of the instrument and should have been the same for each subject. For this reason, the entire set of 250 profiles was reanalyzed in a single curve-fitting procedure with 26 free parameters (3 × 8 subject parameters + 2 instrument parameters). The subject values obtained were similar to those in Table 1 . The instrument values were s = 0.58 ± 0.01 GU/nm and sF = 28 ± 0.6 GU with SD = 3.6 GU and r ² = 0.86. 

Discussion

In this study, we analyzed SLP, including the role of corneal compensation, and developed a model for SLP of a radial retarder. The SLP model was then tested with experiments in which the radial birefringence of the macula was imaged with the GDx instrument while the corneal axis was varied by tilting the subject’s head. The range of corneal parameter values used to test the model (Table 1) , although not as wide as in Figure 4 , included a majority of the normal population. The range of corneal retardance (2δ_C = 21–138 nm, measured with the corneal polarimeter) spanned more than 85% of normal values, including one unusually low value. ¹⁸ Unfortunately, no subjects with corneas in the upper 10% of the normal retardance range were available to test the model. The range of corneal axes (ρ_C ^s =− 8° to −40°) included more than 75% of normal eyes. ¹⁸ The range of radial retardance was limited (2δ_R = 42–53 nm, as provided by the fit to macular retardance) and could not be measured independently. The corneal axes determined by the model were statistically identical with those determined by polarimetry (P < 0.002). In six subjects, the corneal retardance determined by the model did not correspond well to that measured with the corneal polarimeter. This is consistent with a report that corneal retardance measured with a modified GDx, although well-correlated with retardance measured with the corneal polarimeter, is not the same. ²⁰ Wavelength (780 nm vs. 585 nm) and the area of cornea measured are both possible reasons for the difference, but require further investigation. It was encouraging that the SLP model fit the experimental profiles very well and elicited confidence that the model provided a framework for understanding clinical SLP. 

An ideal imaging polarimeter should portray accurately the spatial distribution of specimen retardance. For example, the ideal image of R in Figure 1 should have uniform brightness and a profile that is flat and equal to the double-pass retardance. The model showed, however, that the birefringent C and the CC combined to produce bow-tie patterns in the images and double-hump patterns in the retardance profiles. Clearly, a clinical SLP retardance pattern does not necessarily (or even usually) match the retinal retardance pattern. Nevertheless, the model showed that SLP images encode clinically useful information about retinal retardance. 

The SLP model assumes that the retinal structure imaged is a uniform radial retarder (Fig. 1R) . This assumption fits macular anatomy very well, but it does not apply quite as well to the RNFL. Two properties of the peripapillary RNFL clearly differ from R. First,ρ _R ^s ≠ β for the RNFL. Nerve fiber bundles are only approximately radial as they emanate from the optic nerve head. They are crowded toward the temporal direction and spread apart nasally. ^{38 39 40} When introduced into the model, this should distort the symmetry of a vertical bow tie by broadening the temporal trough, narrowing the nasal trough, and moving the peaks to the nasal side of vertical. Exactly these features are present in actual RNFL profiles obtained clinically. ^{41 42} Second, the RNFL does not have uniform thickness; hence, δ_R is not a constant. The peripapillary RNFL is thinner temporally and nasally and thicker superiorly and inferiorly. ¹² This RNFL anatomy elevates the retinal surface into a double-hump pattern that is apparent in scanning laser tomography ⁴³ and optical coherence tomography (OCT), ⁴⁴ and most observers expect to see a double hump in SLP profiles of the RNFL. Although these differences are important for a detailed interpretation of RNFL retardance patterns, to develop a global picture of the information provided by SLP, it is useful to ignore the differences and maintain the concept of the RNFL as a uniform radial retarder. 

Compensation of Normal Corneas

The practical consequences of corneal compensation were explored by calculating the properties of CR for the untilted corneas of a population of normal subjects. Birefringence values, measured at 585 nm in each of the two corneas of 73 normal subjects, ¹⁸ were inserted into equation 4 for two situations, with corneal compensation as used in SLP and without corneal compensation (i.e., with the cornea alone). Although corneal retardance at 585 nm and at the SLP wavelength of 780 nm may not be the same, the two are well correlated, ²⁰ and the calculation provides useful insight. Recall that the bow-tie patterns in SLP images of R (Fig. 3) were induced by CR, with the bright arms of a bow tie lined up with the slow axis of CR and the mean and modulation determined byδ _CR and δ_R. The results of the calculation are plotted in Figure 8 , which shows δ_CR and ρ_CR ^s in polar coordinates. In polar coordinates, a line connecting a point to the origin has the same orientation as ρ_CR ^s and a length equal to δ_CR. Thus, the clusters of points in Figure 8 provide a sense of the population variability that CR introduces into SLP images of a radial retarder. 

From Figure 8A it is evident that if SLP were performed without corneal compensation, the bright arms of the bow tie pattern in most eyes would oppose the anatomic double hump of the RNFL. This could produce RNFL images that defy expectations and may suppress information from the important superior and inferior RNFL where the majority of nerve fiber bundles occur. The addition of CC rotated the distribution of bow-tie axes to near vertical (Fig. 8B) . Some bow-tie patterns could fall at any orientation, but those far from vertical usually had weak modulation (usually, δ_CR was low). Extreme cases, however, (e.g., the stars in Fig. 4 and in Fig. 8B ) could induce a strong horizontal bow tie that could overwhelm the anatomic pattern of the RNFL. ²⁰ The addition of CC also decreased the median of δ_CR in the 146 corneas from 86 (Fig. 8A) to 56 nm (Fig. 8B) . 

Relation of the SLP Model to Clinical Studies of the RNFL

The chief clinical application of SLP is the assessment of the peripapillary RNFL, a birefringent structure with retardance that varies directly with its thickness. ¹⁰ In part because commercial SLP instruments have used “thickness” as the name for the displayed output variable, papers in the literature frequently state or imply that the quantity measured is (or is proportional to) the actual anatomic thickness of the RNFL. The model and data show that this is a misconception. Except occasionally, when corneal birefringence is well compensated (ρ_C ^s ≈ρ _CC = −15°; 2δ_C ≈ 2δ_CC = 120 nm), the peripapillary retardance image produced by SLP cannot be the same as the spatial distribution of RNFL thickness and, indeed, a direct comparison of the two variables in a macaque showed only moderate correlation. ⁴⁵ (The measurements that originally demonstrated proportionality between retardance and thickness were made on fixed tissue without the cornea. ¹⁰ ) 

The model described herein suggests that the ability of SLP to measure loss of RNFL and thereby detect eyes with glaucoma resides mostly in the bow-tie pattern that CR induces in the radial retardance of the peripapillary RNFL. Figure 4 illustrates the characteristics of this pattern over a wide range of corneal and retinal birefringences. Because the distributions of ρ_C ^s andδ _C in the normal population and the CC designed into SLP instruments usually combine to produce a vertical bow tie (Fig. 8B) , only two parameters, modulation and mean, are required to characterize most patterns. The behavior of these parameters in Figure 4 is complex but, with important exceptions, generally the modulation is more sensitive to δ_R (the variable of interest) and the mean is more sensitive to ρ_C ^s. 

The predicted sensitivity to δ_R of profile modulation is borne out by a number of clinical studies that correlate SLP parameters to measures of visual field loss (mean deviation, corrected-pattern SD) that can serve as surrogates for decreases in thickness, or to the actual RNFL thickness as measured by OCT. Hoh et al. ⁴⁶ found that the modulation-based GDx parameters Maximum Modulation and Ellipse Modulation are better than the mean-based parameters Ellipse Average and Total Integral at discriminating between normal and glaucomatous eyes, are better correlated with visual field loss and are also better correlated with RNFL thickness obtained by OCT. Sinai et al. ⁴⁷ use sectors of SLP output to form two parameters, peak-to-trough amplitude (modulation-based) and average thickness (mean-based). The modulation-based parameter discriminates between the normal and glaucomatous condition and is significantly correlated with mean deviation; the mean-based parameter has neither attribute. Chen et al. ⁴⁸ form SLP measures from retardances in different quadrants: sums (mean-based), ratios to the nasal value, and modulation in the same sense as used herein. (Ratios are complex, containing modulation- and mean-based information in both the numerator and denominator.) Modulation is significantly correlated with visual field loss, ratios are correlated less but still significantly, and sums are not well correlated. Similarly, Weinreb et al. ⁴⁹ form ratios of inferior and superior retardance to temporal retardance that are correlated with visual field loss of the corresponding hemifields. 

The SLP model explains two problematic cases presented by Hoh et al. ⁵⁰ In one case with absolute glaucoma, “SLP demonstrated moderate retardation” but the “characteristic‘ double-hump’ pattern … was absent”. ⁵⁰ When 2δ_R is 0 nm in Figure 3 (i.e., the RNFL is absent), the profile is flat (no modulation) with a value equal toδ _CR. If such a patient has a cornea that produces a substantial value for δ_CR, the mean-based parameters can have apparently normal values, but the modulation-based parameters should be small. In the second case, “SLP demonstrated an apparent 90° rotation of nerve fiber distribution,” a phenomenon that occurs in approximately 1% of patients. ⁵⁰ This patient probably possessed a cornea similar to that represented by the stars in Figures 4 and 8 , a highly birefringent cornea with its slow axis aligned nearρ _CC. With such a cornea, the resultant largeδ _CR with ρ_CR ^s ≈ 0° could overwhelm the vertical double-hump of RNFL anatomy to produce the observed horizontal pattern. 

The SLP model also explains most of the results of a study that is conceptually similar to the head-tilt experiment of Figures 5 6 and 7 , except that ρ_C ^s varied across a population rather than within an individual. Greenfield et al. ¹⁹ report SLP of the RNFL and macula from normal subjects in whom they had also measured ρ_C ^s. For both RNFL and macula the GDx parameters related to the mean—Average Thickness, Ellipse Average, Superior Average, Inferior Average, Superior Integral, and Total Integral—show a strong linear correlation with ρ_C ^s (r ² = 0.6–0.8) from +4° to −76°. (Note that Greenfield et al. express nasally downward angles as positive, the opposite to this study.) It is clear from their figures that the correlation is even stronger when the data are limited to axes more nasally downward than −15°. ^{19 51} The profile mean in Figure 4 (middle row) showed exactly this behavior for all values ofδ _C; the mean increased as the difference between ρ_C ^s and ρ_CC increased. Unlike the model and the results for subjects 1 and 3 in Figure 7 , the data of Greenfield et al., do not increase forρ _C ^s nasally upward from −15°, but there are only five subjects covering a 20° range. A coincidental combination of retardances and axes could have easily obscured the small increase expected. In contrast to parameters related to the mean, the correlation with ρ_C ^s of GDx parameters related to modulation—Inferior Ratio, Superior/Nasal, Maximum Modulation, and Ellipse Modulation—is weak in the RNFL (r ² = 0.13–0.24) and absent in the macula. ¹⁹ The difference between RNFL and macula follows directly from the modulation curves produced by the SLP model (Fig. 4 , top row). Except for a narrow range of corneal parameters, the modulation curve for a 2δ_R of 30 nm (similar to macula) did not vary with ρ_C. The curve for a 2δ_R of 90 nm (similar to RNFL), however, showed a broader range of variation with its minimum at −15°. 

Conclusions

The CC used in SLP followed by a birefringent cornea forms a CR in the light path of the measurement, and in most corneas the slow axis of CR is approximately vertical. An SLP image of a radial retarder exhibits a bow-tie pattern with its bright arms aligned with the slow axis of CR. In clinical SLP, the bow tie due to CR usually exaggerates the anatomic double-hump pattern of the RNFL. In general, therefore, the peripapillary SLP profile is not the same as the RNFL thickness profile. The SLP model explains why corneal properties are a significant source of variance for SLP measurements within a population, especially for SLP parameters based on the profile mean. 

The SLP model also reveals the strengths of SLP for assessing the RNFL. First, the model shows that SLP should be sensitive to RNFL birefringence, the variable it was designed to detect. The induction of a double-hump pattern requires the presence of RNFL birefringence, and loss of this birefringence produces a decline in SLP measurements, especially in parameters based on profile modulation. Second, SLP should be reproducible, because corneal birefringence is stable. ²⁷ If the RNFL is always imaged through the same corneal location, the effects modeled in the current study are the same for every measurement. Observed changes can then be attributed to change in δ_R. Reproducibility, of course, is key to detecting progression of RNFL damage. Finally, the model shows that SLP measurements depend on the cornea in a predictable way. It may be possible to correct for this dependence with auxiliary information obtained by measuring the cornea, ⁵² imaging the macula, ^{53 54} or both. ²⁰  

 Supported by National Institutes of Health Grant EY08684.

 Submitted for publication May 29, 2001; revised September 25, 2001; accepted October 8, 2001.

 Commercial relationships policy: R (RWK, DSG), C (X-RH).

 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked“ advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Model for SLP of a radial retarder. One-half of the combined retarder (CR) formed by a double pass through CC and C is indicated by 1/2CR. E, imaging ellipsometer; CC; corneal compensator, C, cornea, R, radial birefringent structure at the fundus of the eye, PPR, polarization-preserving reflector at the fundus of the eye.

Figure 2.

 

Characteristics of CR as a function of corneal slow axis (ρ_C ^s) for five values of double-pass corneal retardance (2δ_C). Negative axes are nasally downward (ND). The vertical dashed line (ρ_CC) locates the fixed axis of CC. (A) Retardance (δ_CR); (B) slow axis (ρ_CR ^s). H, horizontal; V, vertical.

Figure 3.

 

Retardance images (right) and profiles (left) produced by the SLP model for three hypothetical corneas. A right-eye coordinate system was used for theoretical calculations. The corneas had a double-pass retardance (2δ_C) of 120 nm and slow axes that represented the range of normal: (A) horizontal (ρ_C ^s = 0°), (B) 20° ND (ρ_C ^s = −20°), and (C) 45° ND (ρ_C ^s = −45°). Profiles are shown for four values of double-pass retardance (2δ_R); the images are for a 2δ_R of 60 nm. Profiles show the variation of total retardance (δ_T) on a circle around the center of the image. Images of a uniform radial retarder displayed a bow-tie pattern that varied in orientation and magnitude with the parameters of C and R. The bow-tie axis was specified by the angle of maximum retardance (axis of highest brightness in the images).

Figure 4.

 

Retardance profiles were characterized by three measures: modulation, mean, and the orientation of the bright axis of the bow tie. The graphs show the variation of these measures as a function of corneal axis for five values of corneal retardance. The dots labeled a–c mark values from corresponding profiles in Figure 3A 3B 3C . Stars: an extreme value for corneal birefringence (see the Discussion section); vertical dashed lines: whereρ _C ^s is equal to ρ_CC; horizontal dashed lines: horizontal (H) and vertical (V) orientations of the bow-tie axis.

Figure 5.

 

Macular retardance images (right) and perifoveal profiles (left) from the left eye of subject 3 for three different head tilts. Image contrast has been enhanced to emphasize the retardance pattern. The profiles were obtained from the annular regions outlined on each image and each was characterized by a smooth approximation (superimposed dashed line). Instrument output values were expressed in arbitrary units (GU). Note that the coordinate system used to measure angle β in this left eye is mirror symmetric to that in Figure 3 . CW, clockwise; CCW, counterclockwise.

Figure 6.

 

Fit of the SLP model to head tilt data from three subjects. (A) Transfer function used to introduce a noise floor (F) into the retardance measurement. F is 25 in this example, but was allowed to vary among subjects. (B) Examples of perifoveal profiles (solid lines) with fitted profiles superimposed (dashed lines). All panels have identical scaling. Profiles e and f are the same perifoveal profiles as in Figure 5B and 5A , respectively, but in this figure the smooth profiles come from the set produced by the model.

Figure 7.

 

Three measures that characterize the actual perifoveal profiles (circles) and the fitted model (solid lines) are arranged in the same format as in Figure 4 , except that retardance scales are now in GU. Filled circles (a–f) correspond to profiles a–f in Figure 6B . Thin horizontal and vertical dashed lines: as in Figure 4 . Thick vertical dashed lines: fitted values forρ _C ^s with no head tilt. The calibration bars in the central row of panels represent a double-pass retardance of 50 nm calculated from the scale factor s for each subject.

Table 1.

 

Birefringence Properties of Ocular Structures for Eight Subjects

Subject	Perifoveal Profiles Fitted by SLP Model*															Corneal Polarimetry, †
		n	Cornea			Macula 2δR (nm)	s (GU/nm)	sF (GU)	SD (GU)	r 2	ρC s (deg)	2δC (nm)
				ρC s (deg)	2δC (nm)
1	19	−11 ± 0.7	95 ± 2.5	53 ± 3.9	0.64 ± 0.03	24 ± 1.5	3.3	0.88	−8	71
2	20	−35 ± 0.6	106 ± 1.5	44 ± 2.4	0.66 ± 0.02	30 ± 1.4	3.1	0.91	−40	94
3	34	−18 ± 0.2	148 ± 2.0	45 ± 1.7	0.57 ± 0.01	27 ± 1.0	2.7	0.91	−22	120
4	28	−26 ± 0.7	77 ± 2.4	45 ± 2.8	0.62 ± 0.02	25 ± 1.7	3.4	0.90	−22, ‡	—
5	26	−37 ± 2.1	119 ± 5.6	42 ± 3.4	0.51 ± 0.04	32 ± 2.4	3.0	0.87	−31, ‡	—
6	36	−14 ± 0.3	130 ± 1.9	46 ± 2.3	0.66 ± 0.02	26 ± 1.6	4.4	0.87	−18	98
7	42	−45 ± 2.3	83 ± 2.9	53 ± 3.0	0.46 ± 0.02	30 ± 1.6	2.8	0.87	−37	138
8	45	−34 ± 2.2	63 ± 2.6	51 ± 3.0	0.48 ± 0.02	34 ± 1.6	3.2	0.81	−36	21

 n, number of profiles in fitted set.

*  Parameters of fit ± SE.

^†  Ref. ¹⁸ , except

^‡  from ref. ¹⁹ ; repeatability is ±2.2° in axis, ±10% in retardance.

Figure 8.

 

Polar plots of δ_CR and ρ_CR ^s calculated for the corneas of 73 normal subjects. The calculation used values forρ _C ^s and 2δ_C measured at 585 nm in a previous study. Each cornea was plotted twice at ρ_CR ^s and ρ_CR ^s + 180° to visually emphasize the bow-tie pattern that CR induces in R. (A) Cornea alone (i.e.,δ _CC = 0). (B) Compensated cornea. Coordinate systems for the right and left eyes are mirror symmetric to show axis orientations as they would appear when facing the subject. The stars mark CR for a hypothetical extreme (rare, but possible) value for corneal birefringence (ρ_C ^s =− 8° and 2δ_C = 220 nm). NU, nasally upward; ND, nasally downward.

The authors thank William J. Feuer, MS, for statistical advice.