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From the LSU Eye Center, Louisiana State University Health Sciences Center, School of Medicine, New Orleans, Louisiana.
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Abstract |
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METHODS. Corneal topography maps were obtained from 32 keratoconus cases, 27 postoperative penetrating keratoplasty cases, and 29 postoperative conductive keratoplasty cases (88 total). The best spectacle-corrected visual acuity (BSCVA) for each case ranged from -0.2 to 1.3 logarithm of the minimum angle of resolution (logMAR) units (20/12.5–20/400). Topography was analyzed to determine wavefront error and the elevation fit error for a 4-mm optical zone. The 4th and 10th expansion series were analyzed with the 0th-order (piston) and 1st order (tip and tilt) removed. Linear regression analysis was performed. The difference in root mean square (RMS) error between the 4th- and 10th-order analyses was assessed for both wavefront and elevation fit error.
RESULTS. The correlation of BSCVA to wavefront error for 4th-order terms was moderately strong and significant (R2 = 0.581; P < 0.001). The 10th-order correlation for wavefront error had a similar result (R2 = 0.565; P < 0.001), but the regression was not significantly different from the 4th-order result. The correlation of BSCVA to the elevation fit error was strong and significant for the 4th order (R2 = 0.658; P < 0.001). The 10th-order data had a similar result (R2 = 0.509; P < 0.001), and there was no significant difference between the two regressions. Only 72% of the cases showed a shift toward increased wavefront error with the 10th-order series, whereas 18% lost wavefront error. All cases showed a shift toward improved elevation fit with the 10th-order expansion.
CONCLUSIONS. The wavefront error correlation to acuity was moderately strong, but the corneal elevation fit error also strongly correlated with visual acuity, indicating that Zernike polynomials do not fully characterize the surface shape features that influence vision and that exist in postsurgical or pathologic eyes. In addition, the change in wavefront error when using a larger expansion series was found to increase or diminish somewhat unpredictably. The authors conclude that Zernike polynomials fail to model all the information that influences visual acuity, which may confound clinical diagnosis and treatment.
Wavefront error data have the form of a complex, three-dimensional surface that can be mathematically decomposed into a canonical set of terms that describe individual aberration components such as spherical aberration and coma. These terms are also used for guiding laser ablation during customized refractive surgery. Currently, the preferred decomposition method uses the Zernike polynomial, which represents total wavefront error as a series of terms that describe surface shape components with respect to angular and radially arranged basis functions of different frequencies and orders.12 13 14 Each Zernike term has a coefficient with a magnitude and sign that indicate the relative strength and direction of the aberration contributed by that term. The wavefront error is often expressed as the sum of the root mean square (RMS) error to avoid sign discrepancies for certain terms, particularly when combining left and right eyes into a single cohort.15
The Zernike decomposition process is a reverse-fitting routine. That is, given a complete set of individual aberration components, the original surface shape can be theoretically reconstructed. When the process is applied to describing the optics of the eye, however, there are unresolved questions as to how many terms are sufficient to describe a given surface, and whether the accuracy of the decomposition process (i.e., Zernike fitting) is adequate12 16 17 (Smolek MK, et al. IOVS 2002;43:ARVO E-Abstract 3943).
The Zernike fitting process is not limited to analysis of wavefront error surfaces, but can be applied to other ocular surfaces as well, including the front surface elevation of the cornea. Given the significance of the shape of the front surface of the cornea to the refraction of the eye and the ability to correct refractive errors by laser ablation of the front surface of the cornea, detailed wavefront error analysis of corneal topography data is clinically useful and important. In addition, corneal topography is a major ocular component that defines the total ocular wavefront error, and so one can argue that the relevance and accuracy of Zernike polynomial analysis in general can be studied with corneal data alone. In this type of analysis, the corneal surface data are used to establish a mean reference wavefront through a best-fit curve to the elevation data and also to provide the raw elevation information used to calculate the wavefront error.
It has been recognized that the corneal first surface generally provides the bulk of the ocular aberrations in the postsurgical or pathologic eye. The corneal front surface in the normal eye contributes approximately half the total aberrations of the eye, but the number contributed is age dependent, and the contributions increase substantially with surgery and disease.18 Several thousand topographic data points are necessary for adequate detection of corneal surface irregularities that can decrease vision; however, the number of data points measured with wavefront-sensing instruments varies from the low hundreds to several thousand.
In this retrospective study, we examined the adequacy of the Zernike polynomial function to model wavefront error and corneal surface shape by using corneal topography data from surgical and pathologic eyes. We correlated visual acuity to both the Zernike fit data of wavefront error and to the Zernike fit for the corneal shape (expressed as the result of the residual shape data not represented by the Zernike fit). We also examined the wavefront error and elevation fit error difference between the 4th- and 10th-order expansion series to determine whether the Zernike fitting process performs in a predictable manner when higher order terms are included in the analysis.
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Methods |
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Image analysis software for computer tomography (CTView 4.07; Sarver and Associates, Merritt Island, FL) was used for retrospective analysis of the corneal topography for two parameters: the corneal wavefront error and the corneal elevation surface fit error, based on a Zernike polynomial fitting procedure. With CTView analysis, a zone of the corneal surface is used to define the "pupil" across which the wave aberration is determined. The number of orders of Zernike terms desired to fit the surface is user selectable, and the user also has control over the inclusion and extraction of specific aberrations during the analysis by using an individual Zernike term-masking option. The wavefront error is presented both in terms of the signed coefficient values and as RMS values.
To determine the positive and negative values of the Zernike terms, a reference surface sphere is needed. This reference sphere is determined from the average surface curvature of the corneal topography. In the present study, a 4-mm diameter zone was used to establish the reference wavefront, as well as to analyze the wavefront error and surface elevation fit error. A 4-mm pupil was chosen, because it corresponds closely to the average adult pupil diameter during photopic viewing conditions and under the typical clinical conditions in which visual acuity is assessed. The analysis was performed with a 4th-order Zernike polynomial expansion and then repeated using a 10th-order expansion. For the wavefront error, the 0th-order piston term and the 1st-order tip and tilt terms were removed, using the CTView Zernike term-masking option, leaving only 2nd-order and higher terms to define the aberrations that generate blur. The wavefront error and elevation fit error obtained from CTView analysis of each map were recorded as the sum of the RMS error and expressed in micrometers.
The BSCVA of each case was plotted as a function of the wavefront error and the elevation fit error for each of the two expansion series. BSCVA was obtained under ordinary photopic viewing conditions, which would have resulted in a pupil diameter of approximately 4 mm on average. Because pupil size can fluctuate slightly during the course of acuity testing, assigning a constant pupil diameter to the BSCVA measure is not meaningful, but using an average diameter is acceptable. Linear regression analysis was performed to determine the strength (R2) of each correlation from the Pearson product moment correlation coefficient (R). If it had been possible to determine the patient’s precise pupil diameter for the BSCVA measurement, the correlations we obtained would have been more significant, but the correlation coefficient would not necessarily have changed. A t-test on regressions was performed to assess whether the correlations were significant among the different test conditions.
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Discussion |
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The correlations between BSCVA and the corneal surface elevation fit error (Figs. 2 4) showed that visual acuity is strongly correlated to surface features that were not contained in the Zernike representation of the surface. If the Zernike representation were an exact fit, then there would be zero RMS error and no correlation to visual acuity by the residual error. The magnitude of the Zernike fit error tends to increase as the cornea becomes more irregular, because such surfaces are more difficult to fit by a Zernike polynomial. However, adding more Zernike terms to the fit by increasing the expansion order should tend to improve the fit, even with aberrant surfaces. When we increased the Zernike expansion order from 4 to 10, it reduced the elevation fitting error as expected, but the reduction was not significant in terms of a diminished correlation with BSCVA. Therefore, even the 10th-order expansion contained information that was not fully captured by the Zernike fitting process. It remains unknown how many more orders of terms are needed to eliminate effectively the residual elevation fit error and to make the correlation to BSCVA insignificant.
When looking at the wavefront error representation of the cornea, we find that there is also a moderately strong correlation with BSCVA (Figs. 1 3) . Clearly, visually relevant information is contained in the Zernike representation of wavefront error. Increasing the Zernike expansion order from 4 to 10 had no significant effect on strengthening the correlation to BSCVA (i.e., increasing the R2 value).
Based on the results for the corneal elevation fit error, we believe that the Zernike fitting process fails to capture fully all the important aberration characteristics of the wavefront error surface, particularly when the Zernike fitting order is small. Other explanations for a reduced correlation in wavefront error may be that the BSCVA measurements were made without an accurate spherocylinder correction, but this explanation seems highly unlikely. Other nonoptical factors affect BSCVA, but the patients in the study had no known neurologic or retinal deficits in vision—only optical deficits were reported. Furthermore, we assert that the results of the goodness-of-fit of the corneal elevation data confirm that the Zernike fitting method itself fails to capture fully the relevant aberration information—at least for two different series expansion orders that are often reported in the literature (4th and 10th). Unfortunately, we are unable to determine the corresponding goodness-of-fit for the wavefront error surface, because the wavefront error has no original counterpart surface for comparison; the wavefront error measurement is itself a product of the Zernike fitting routine. Fortunately, we had both the original topographic surface and the Zernike representation of that surface to perform the elevation fit error analysis and thus show the limitations of Zernike fitting to corneal surfaces.
When we performed a 4th-order minus 10th-order analysis of the topographic elevation fit error, we found that 100% of the cases exhibited less error with more terms (Fig. 6) . This result was expected, because adding more Zernike terms to a modeled surface should result in a better fit, and if the fit improves even slightly, then the residual fit error difference to the original topography should decrease. In contrast, unexpected results were found when we calculated the difference between the RMS errors for the 4th- and 10th-order expansions of the wavefront error. Figure 5 indicates that 18% of the cases had less total wavefront error, even after the addition of six more orders to the expansion series (an additional 51 terms). The expected outcome was seen in the remaining 72% of cases—namely, that wavefront error increased as more error terms were added with the 10th-order expansion. We believe that the apparent loss of wavefront error information is due to the phase nature of individual wavefront error terms that, when added together, can result in the total wavefront error being less than the anticipated amount based on the magnitude of individual error components. In other words, higher order Zernike terms can occasionally cancel out lower order Zernike terms and vice versa in surprising ways. The 18% of cases that did not follow the expected outcome were spread nearly equally across the three test categories, and therefore this effect was not isolated to the highly aberrant eyes, but involved mildly aberrant postsurgical eyes as well. This unpredictable aspect of wavefront error analysis can be a serious drawback to the effective clinical interpretation of wavefront error data and may explain the unpredictable visual outcomes that have been seen occasionally after customized ablation procedures. If the basis for this problem is indeed inherent to the Zernike fitting routine, then the Zernike method may be unreliable and unacceptable for clinical use.
The results of this study are relevant not only to visual acuity assessment, but also to refractive surgery outcomes when the laser ablation is guided by a Zernike algorithm. Such procedures tend to use only a 4th-order Zernike expansion to fit the wavefront error. This may be adequate, if not ideal, on normal corneas, but aberrant corneas, on average, such as those in surgical retreatment cases, cannot be fully represented by a 4th-order expansion (Smolek MK, et al. IOVS 2002;43:ARVO E-Abstract 3943). On the contrary, there are physical limitations on the current diameter of the laser beam that also play a major role in determining the resolution of correction that can be achieved, irrespective of the number of Zernike terms used for fitting.19
In conclusion, we have shown that the Zernike fitting method is inexact for the elevation fit, and it also does not fully capture important information about wavefront error aberrations that influence visual acuity. Increasing the number of terms used in the Zernike fitting process from 4 orders to 10 has an effect on the total error, but appears to be insufficient with respect to significantly altering the correlation to BSCVA. Accurately characterizing the wavefront error may require a great number of Zernike terms—far more than is practical to analyze in a clinical environment. Axisymmetric basis functions, like the Zernike basis function set, may be well suited for tasks such as describing commercial optics, but may be less than ideal for describing the complex, irregular, and varying wavefront errors and optical surface shapes of eyes. At the very least, vision scientists and clinicians must be fully aware that the Zernike fitting method can result in the loss of visually significant shape information in a generally unpredictable manner, particularly when the analysis involves a relatively low number of Zernike orders. Consequently, laser-treated eyes with wavefronts that cannot be accurately fit with the Zernike method tend to be at a greater risk for poor postoperative vision. As a corollary, wavefront sensors that do not achieve a high enough spatial sample density are unable to detect important imperfections in wavefront structure and therefore are of limited use for both diagnostics and treatment.
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Footnotes |
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Submitted for publication February 24, 2003; revised June 18, 2003; accepted July 20, 2003.
Disclosure: M.K. Smolek, Alcon (C), Nidek (C); S.D. Klyce, Nidek (C)
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.
Corresponding author: Michael K. Smolek, LSU Eye Center, 2020 Gravier Street, Suite B, New Orleans, LA 70112; msmole{at}lsuhsc.edu.
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