Specular-reflection topographers are commonly used to measure the shape of the cornea. These topographers use patterns reflected by the cornea to form topographic reconstructions of the corneal surface. The VU-Topographer (VU University Medical Center, Amsterdam, The Netherlands) is a corneal topographer of this sort, with the advantage of providing a one-to-one correspondence between source points and captured points, thus eliminating skew ray ambiguity. 26,27 Apart from the standard corneal height map, the VU Topographer also describes the shape of the cornea by means of a Zernike polynomial expansion.29 This model extends the spherical model of the cornea into one that has finer details. In the Zernike convention, each polynomial term represents a unique shape and is either rotationally symmetric (e.g. defocus) or not rotationally symmetric (e.g. astigmatism). The coefficient of each term specifies how strong a particular corneal shape feature exists. In corneal topography, the first three Zernike terms are usually neglected. The first term Z(0,0) is a piston term which is related to the distance from the apex of the cornea to the nodal point of the lens camera system. This distance is a calibrated instrument constant and can therefore be neglected. The second Z(1,-1) and third Z(1,1) terms describe the tilt of the corneal surface with respect to the optical axis of the lens camera system and are called vertical tilt and horizontal tilt, respectively.30

By standard convention, the monomial representation of the Zernike x-tilt term is defined as:30

The ztilt and the x variable form an angle ρtilt which can be determined directly from the coefficient Ctilt:

It can be shown that this angle ρtilt(Figure 1) is related to a surface orientation as measured in corneal topography. This is demonstrated by a series of measurements carried out on artificial surfaces that are translated horizontally. In figure 1, the x-z plane of a typical spherical surface is shown where the apex of the surface is translated along the x-axis with respect to the instrument axis. In this configuration the angle ρ can be theoretically calculated based on the value of that translation (Δx; displacement of the surface apex relative to the instrument axis), the surface radius (Rs) and the radius of the surface area over which Zernike polynomials are fitted (Rp), using geometrical modeling:

Figure 1.

Two-dimensional schematic model of the translation experiment. Δx describes the translation of the surface apex along the x-axis. Points C and D are the borders of the measured surface area, which has a radius of Rp. ρ is the angle formed by line segment CD¯ and the x-axis and η is the angle between the instrument axis and the surface normal.

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The Zernike tilt is often mistakenly linked with the angle η that the normal to the surface makes with the instrument axis (Figure 1). The angle η can be derived using the same spherical geometrical model of figure 1:

To show that the Zernike tilt is related to angle ρ and not to angle η, several measurements are done in which an artificial surface is translated in steps of 0.5 mm along the x-axis, with Δx ranging from -2 mm to 1.5 mm with respect to the instrument axis. These measurements are repeated for four types of artificial surfaces to investigate the dependence of the Zernike tilt on the surface features, including the orientation of the surface normals. The artificial surfaces include a pure sphere, a sphere + coma feature, a sphere + trefoil feature, and a sphere + quadrafoil feature. All surfaces have a base radius of 8.00 mm and a feature amplitude of 5 μm. The surfaces are made of PMMA material and were manufactured by SUMIPRO BV, Almelo, The Netherlands. Shape fitting is done following Zernike convention up to the 6th radial order, within a surface zone diameter of 6.0 mm. The experimental relation (2) and theoretical relations (3) and (4) are compared using chi-square statistics evaluated on the absolute value of the angles. The chi-square values will indicate how close the theoretical values are to the experimental values.

The angle λ can be determined in real eyes by applying equation (2) to corneal topography measurements. A similar model to the one proposed here was simulated in a previous investigation by Salmon and Thibos.11 In this simulation, the instrument axis of the videokeratography system is aligned to the line of sight and a linear relationship was observed between the angle λ and the Zernike tilt coefficient. In the present study, several off-axis fixation points are used where the instrument axis and the fixation axis are not co-axial (parallel). The measurements in this study are not done along the line of sight and, therefore, they can not be used for the determination of the corneal aberrations. However, they can be useful when the aim is to get the tilt orientation of the cornea. Although this is a different situation, compared to that of the Salmon/Thibos study,11 a similar linear relationship can be observed, this time between the angle ρ and the fixation angle θ. This linear relationship is a natural consequence of the geometrical properties of the cornea.

Subjects are measured with the VU Topographer (Figure 2) and are asked to look at eight fixation points at the stimulator's back-panel, for which the color-coded pattern, as seen by the subject, is shown in figure 3. The color-coded squares are used to define the eight fixation points and are each marked by the border between four squares. The fixation points are distributed in such a way that redundancy exists in terms of fixation angles (both in the x- and y-directions), in order to record repeated measurements of the same angles. Since the limbus is proven to be a more reliable landmark than, for instance, the pupil center,31 the measurements are centered on the subject's limbus.

Figure 2.

The VU Topographer, as seen from the subject's perspective: the head and chin rest for patient fixation and the topographer casing holding the color-coded stimulator pattern of squares.

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Figure 3.

The stimulator back-panel of the VU Topographer, as seen by the subject, with markings on the eight fixation points used for the Zernike-tilt method.

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The known fixation angle θ of the stimulator point is shown in figure 4: it is the angle between the instrument axis and the line connecting the center of the limbus plane with the fixation point. The model in figure 4 represents an eye with an angle λ equal to zero for a clear visualization of the geometrical model. In the case where the angle λ is not equal to zero, the angle θ in figure 4 can be considered to be the summation of the angle λ and the fixation angle θ. Angle ρtilt is calculated from the Ctilt output of the VU Topographer using equation (2). Assuming a spherical cornea with small angles θ, the relation between angle ρtilt and an angle θ is:

Figure 4.

Schematic diagram showing the geometrical model used to determine the angle λ. CC: centre of curvature of the cornea; A and B: the edges of the limbus; E: the midpoint on the line segment AB¯; H: the intercept of the line CC E¯ with the cornea; Point F: the intercept of the instrument axis with the cornea; Rp: the radius of the measured surface area, with points C and D being its borders on the cornea.

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where e is related to the distance from the limbal plane to the corneal apex. The dependence on e in finding angle λ can be eliminated by combining the measurements of all stimulator points. The angles ρtilt are plotted against the known fixation angles θ and a linear regression line is fitted to all measured points. Angle λ can be then directly determined from the θ-intercept of the fit. By applying this method to the ρtilt values obtained from both the Zernike x- and y-tilt coefficients, both components (in both x- and y-directions) of the angle λ can be determined. A typical example of the linear fitting of ρtilt in the x-direction is shown in figure 5. The complete derivation of relation (5) can be found in the Appendix.

Figure 5.

A typical plot of angle ρ in the x-direction, obtained by plotting measured Zernike x-tilt values against fixation angles θ. The linear relation between angle ρ and angle θ is shown. The error bars indicate the spread of angle ρ obtained from three repeated measurements for two fixation points with the same angle θ.

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The angle λ derived from the VU topographer is compared with the angle-κ values obtained from Orbscan II (Bausch & Lomb, Rochester, NY, USA). The Orbscan II was chosen because it has a readily available feature to measure angle κ, which has been previously validated with synoptophore procedures.6 The synoptophore validation study showed that despite of the Orbscan II giving significantly higher values of angle κ, there was a high correlation between the two instruments, which means that the differences are systematic.

The comparison of the Zernike tilt method is validated with 9 eyes (average age 36 years, ranging from 22 to 63 years; 8 males and 1 female; equivalent refractive error ranging from -1.25 to +4 diopters). The research followed the tenets of the Declaration of Helsinki. Written informed consent was obtained from the subjects after explanation of the nature and consequences of the study. The study was reviewed and approved by the Medical Ethics Committee of the VU University Medical Center, Amsterdam, The Netherlands.

The shape fitting of the VU Topographer is done following Zernike convention up to the 6th radial order, within a surface zone diameter of 6.0 mm.

Three repeated measurements for every subject and for every fixation point are carried out both with the Orbscan II as well as with the VU Topographer. In preparation for the statistical analysis, the resulting values from the Orbscan II are averaged, while for the VU Topographer for every fixation point the average ± standard deviation is used in the linear fit of relation (5). The Zernike-tilt method described above calculates the value of angle λ in one direction only: the x-direction for the Zernike x-tilt and the y-direction for the Zernike y-tilt. Since no anatomical dependency between the λx and λy is known in literature, they are treated as separate measurements in the statistical analysis. Consequently, the measured angle-κ values from the Orbscan II are decomposed in their x- and y-components using the given meridian angle:

Statistical analyses for instrument comparison are performed using a Pearson correlation test, a Bland-Altman plot,32,33 and a non-parametric Wilcoxon matched-pairs signed rank test. Possible dependency between λx and λy is investigated by means of a Pearson correlation test. The precision of the VU Topographer in measuring angle λ is quantified by means of the standard deviation for three trials, which is averaged over all subjects.

The effect of surface translation on angle ρ is shown in figure 6. The theoretically expected ρ is shown as a black line, the theoretically expected η is shown as a red line, and the experimentally obtained values for angle ρ are shown as colored dots. The chi-square statistic test values for both models for all surfaces as well as all together are presented in table 1. In general, the theoretical ρ model predicts the experimental ρ values more accurately than the theoretical η model, with overall chi-square values of 0.050 degrees and 0.990 degrees respectively. The chi-square value for the theoretical ρ model is below 0.025 degrees for all surface types, while it is greater than 0.185 degrees for the theoretical η model.

Figure 6.

The results of the translation experiments, where the experimental values obtained for angle ρ are shown as colored objects: red squares for the quadrafoil surface, green circles for the trefoil surface, yellow upright triangles for the coma surface, and blue inverted triangles for the spherical surface. The theoretical models are depicted as a black line for ρ and a red line for η.

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No significant correlation is found between λx and λy for the measurements carried out with the VU Topographer, as indicated by the Pearson correlation test (r=0.43, P>0.2). The correlation between the angle λ measured with the VU Topographer and the angle κ measured with the Orbscan II is shown in figure 7. There is a strong and significant correlation between the angles yielded by the VU Topographer and those obtained with the Orbscan II: r=0.95, P<0.001. A Bland-Altman plot of the comparison is shown in figure 8, reporting a mean difference of -0.52 degrees with a 95% limits of agreement of [-2.67,1.64]. A Wilcoxon matchedpairs signed rank test proves that the difference found is not statistically significant, with P<=0.1748. The precision in determining angle λ, as measured with the VU Topographer, is 0.6±0.3 degrees.

Figure 7.

Correlation plot of angle-λ measurements carried out with the VU Topographer vs. angle-κ measurements made with the Orbscan II. The error bars indicate the spread of the experimental values across three repeated measurements made with both instruments.

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Figure 8.

Bland-Altman plot comparing the VU-topographer with the Orbscan II, and indicating a difference of -0.52 degrees with a 95% limits of agreement of [-2.67,1.64].

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