To deliver a custom correction, it is important to understand the types and magnitudes of wavefront aberrations induced by treatment decentration. Custom corneal ablations to correct refractive errors can be based on whole-eye wavefront aberrations and corneal topography. Topography-guided ablations aim to treat irregularities in corneal elevation in addition to the refractive errors of defocus and astigmatism.1 Alternatively, wavefront-guided ablations aim to address the wavefront aberrations in the entire eye in addition to the refractive errors.2 Based on the clinical refraction and whole-eye wavefront aberrations, the ablation profile of a wavefront-guided treatment in an ablation zone can be obtained.3
Pupil-center shift and eye cyclotorsion are inevitable in corneal refractive surgery. Patients are supine in refractive surgery, yet they are seated at the preoperative examination. A low to moderate amount of eye cyclotorsion has been found in the transition from the seated to supine position.4 In addition, treatment decentration in refractive surgery has been observed in several studies, and the results indicate that centration errors have an important influence on induced aberrations. Wang and Koch5 found that the mean pupil centroid shift was 0.29 mm during wavefront-guided corneal ablation and that the centration error induced 4.9 times, 2.8 times, and 8.7 times higher values of total aberration, lower-order aberrations (LOAs), and higher-order aberrations (HOAs), respectively, than cyclotorsion misalignment. Furthermore, centration error decreased the polychromatic modulation transfer function values by 31% to 66%. Porter et al.6 found that the mean magnitude of the shift between the natural pupil-center and dilated pupil-center locations was 0.29 mm and that the postoperative aberrations were typically larger than those theoretically induced due to a pupil-center offset of the treatment. Other studies report that the mean decentration was 0.26 mm after laser-assisted subepithelial keratectomy7 and 0.23 mm after active eye tracker–assisted myopic photorefractive keratectomy (PRK).8 Clinical data by Mrochen et al.9 showed that the postoperative aberrations caused by subclinical treatment decentration were significantly larger than the preoperative aberrations in photorefractive laser surgery.
Modern laser algorithms in refractive surgery include a ring-shaped or ellipsoid area around the optical zone,10 namely the transition zone. It connects the optical zone to the untreated cornea. When the transition zone is applied, the curvature is continuous at the boundary between the optical zone and the transition zone and at the boundary between the transition zone and the unaltered cornea. The transition zone is important because an abrupt change in corneal curvature at the edge of the optical zone may induce excessive epithelial and stromal tissue healing after surgery. In addition, the transition zone has a significant impact on postoperative HOAs. Arbelaez et al.11 found that the main HOA effect postoperatively (coma and spherical aberration) originated from decentration and edge effects, which represent the strong local curvature change from optical zone to transition zone and from transition zone to the untreated cornea. Hori-Komai et al.12 found that the change in root-mean-square (RMS) values for HOAs varied with the ablation profiles with an aspheric transition zone. Thus, the theoretical effect of the transition zone on wavefront aberrations deserves further study.
In theory, the actual wavefront aberration after ablation can be calculated by the translation and rotation transformation of the ideal corrected aberrations.13 Guirao et al.14 also theoretically evaluated the impact of translation and rotation on individual Zernike terms by ocular wavefront transformation. Yi et al.15 evaluated the visual outcomes of wavefront-only corneal ablation by computer simulation.
In addition, the oblique incidence of the laser plays an important role in the impact on the ablation depth of the cornea.16 Thus, for a custom wavefront-guided correction, the incidence angle of the laser beam at any arbitrary point on the cornea can be determined by the mathematic model of the anterior corneal surface for myopic astigmatism.17
The ablation profiles proposed by Manns et al.18 are used to calculate the ablation depth of the cornea in the optical zone for wavefront-guided correction. Based on these ablation profiles, what individual Zernike terms should be induced from treatment decentration and the transition zone and what are the relationships between the amount of aberrations, the transition zone, and the degree of decentration? These questions deserve further study.
In this study, the relationship between induced aberrations and the transition zone and treatment decentration was studied. The influence of the oblique incidence on the induced aberrations was also assessed. This was based on the ablation profile of the ablation zone including the transition zone and optical zone for the custom laser refractive surgery.
Patients and methods
This study comprised refractive surgery candidates for myopia correction. All patients were free of ocular disease. All candidates provided informed consent after they received an explanation of the nature and possible consequences of the research.
The wavefront aberrations in all eyes were measured under dim illumination using a Hartmann-Shack aberrometer (Wavescan wavefront system, Visx, Inc.) under natural accommodation. The wavefront aberration diameter was 6.0 mm in all cases. All measurements were repeated at least 3 times in each eye, and the 3 best-matching measurements were used in this study. The wavefront aberration was expressed as a 6th-order Zernike polynomial expansion. Contact lens wearers were excluded from the study.
Ablation Profile in the Optical Zone for Custom Laser Refractive Surgery
When the ablation depth of corneal tissue (D) is given, multiplication by n−1 results in optical path difference (OPD).
where n represents the refractive index of the cornea in visible light; the value is 1.376 in this study, and the corrected wavefront aberrations are the negative value of OPD according to the phase conjugate principle.
where (x,y) depicts an arbitrary point in the ablation zone on the cornea. The wavefront aberrations are expressed as a Zernike polynomial expansion.
The ablation depth is given directly at any arbitrary point by the wavefront information according to equations 1 to 3.
Effect of Oblique Incidence on the Effective Ablation Depth
The effect of oblique incidence on the loss of ablation efficiency is composed of reflection losses and geometric changes in the illuminated area. In addition, refractive surgery lasers use scanning approaches by angular projection of the beam. If the distance from the scanner mirror to the ablation plane is large, the angle of incidence of the beam onto a flat surface perpendicular to the axis of the laser is near zero.16 Thus, the laser beam can be considered as moving vertically parallel to the center axis of the human eye in refractive surgery. Subsequently, the incidence angle (Θ) of the laser beam at any arbitrary point Symbol on the cornea was determined by the model of anterior corneal surface. After that, an adjustment factor of the ablation depth of corneal tissue at any arbitrary point was deduced from the changes in the illuminated area and reflection loss of laser energy on the anterior corneal surface as the change in the incidence angle. In this study, 3 methods were used to evaluate the effect of the laser’s oblique incidence on the ablation depth in the study population, each with a unique interpretation. Method 1 takes into account the effect of oblique incidence on the anterior cornea in the laser ablation profile. In addition, the effect in the actual laser ablation process is included.17 Following is the method 1 equation:
where Symbol represents the adjustment factor of the ablation depth of cornea, α and R represent the parameters of the actual laser ablation process, and Symbol and Symbol are used in the laser ablation profile.
Method 2 takes into account the effect of oblique incidence in the actual laser ablation process, but without this effect in the laser ablation profile. Following is the method 2 equation:
Here, the parameters are as mentioned above.
Method 3 is without the effect of oblique incidence, not only in the actual laser ablation process but also in the laser ablation profile as performed in previous studies. Following is the method 3 equation:
where Symbol represents the adjustment factor of the ablation depth of the cornea.
Ablation Profile in the Transition Zone for Custom Laser Refractive Surgery
When the diameter of the optical zone is assumed as 6.0 mm, the ablation profile of the cornea in the optical zone can be indicated as a general-purpose function in refractive surgery as follows:
Here, R depicts the radius of optical zone.
The coordinate system used to represent the ablation profile shall be the standard ophthalmic coordinate system in accordance with International Organization for Standardization (ISO) 24157,19 in which the x-axis is local horizontal with its positive sense to the temporal side in the left eye or the nasal side in the right eye, the y-axis is local vertical with its positive sense superior with respect to the eye, and the z-axis is the line of sight of the eye with its positive sense in the direction from the inside to the outside of the eye. The coordinate system origin lies at the center of the exit pupil of the eye.
If the width of the transition zone is Rβ, the inside radius of the transition zone is R and the outer radius is R(1 + β). Here, β represents the coefficient of the blend zone. An ablation profile for the transition zone can be constructed as follows20:
where Symbol represents a blend function. The function value is 1 at the boundary between the optical zone and the transition zone; however, the value changes to zero at the boundary between the transition zone and the unaltered periphery. Symbol indicates the extended ablation depth in the transition zone, which is extended from the boundary value of the optical zone.
Induced Wavefront Aberrations from Treatment Decentration
When the center of the ablation profile in refractive surgery is inconsistent with the center of the corrected wavefront aberrations in eyes, it means that the treatment is decentered. Figure 1 shows that X′Y′ is displaced and rotated with respect to XY. The circle, which is centered at O, the origin of XY, represents the ideal corrected aberrations, whereas the circle, which is centered at O′, origin of X′Y′, conveys the actual corrected aberrations. Therefore, the actual corrected aberrations should include 2 parts. One is the overlapping zone of 2 circles (III), and the other is a part of the transition zone (II). Then, the induced Zernike coefficients can be obtained by the wavefront surface fitting. For simplicity, only linear conformal mappings between reference frames are explicitly considered in this study; namely, ablation profile decentration consisting of lateral displacements and rotations.
A coordinate transformation formula obtained from Figure 1 is as follows:
where α represents the rotation angle and Symbol and Symbol convey the lateral displacement in the x-axis and the y-axis, respectively.
With the coordinate transformation formula, the ablation depth in the overlapping zone (III) in Figure 1 can be obtained. In addition, the ablation depth of part of the transition zone (II) can be calculated from the ablation profile for the transition zone. Thus, the ablation profile in the whole pupil zone can be obtained. Finally, the ablation profile is multiplied by an adjustment factor Symbol of the ablation depth of the cornea, after which the depth can be converted into the actual corrected aberrations as follows:
The actual corrected Zernike coefficients can be obtained by the surface fitting:
Finally, the induced wavefront aberrations in custom refractive surgery from treatment decentration are calculated as follows:
where Symbol is the Zernike coefficient for vision correction with treatment decentration and Symbol is the coefficient of original aberration.
Evaluating Induced Wavefront Aberrations
Based on the ablation profile for custom correction, the theoretical ablation depth for the whole ablation zone, including the transition zone, was calculated. The transverse translation of the ablation center in refractive surgery was simulated by coordinate transformation. Then, the ablation depth was converted into the actual corrected aberrations, and they were multiplied by the adjustment factors. The induced aberrations were represented by the differences between the original and the actual corrected aberrations. In addition, the angle mismatch in refractive surgery was simulated by coordinate rotating transformation. Furthermore, the induced aberrations from the combination of transverse translation and angle mismatch were studied. The combination follows the Guirao at al.14 approach as follows: decentration first, then rotation of the decentered area. The induced aberrations were obtained as in the above description.
Without the effect of the transition zone in laser ablation algorithms, the transverse translation of the ablation zone was simulated by ocular wavefront transformation. In particular, the unknown part of the wavefront that moved into the pupil could be extrapolated from the original wavefront. In addition, the effect of oblique incidence was not taken into account.
Finally, for wavefront-guided custom laser refractive surgery, because the optical path length is inconsistent at a different point on the cornea in the optical zone, the transition zone is effectively used to bring the ablation depth at the periphery of ablation zone to zero. In fact, when the translation maintains a constant, a different location of the center of the ablation zone may induce different amounts of wavefront aberrations for wavefront-guided custom correction. For conventional refractive surgery, previous results17 showed that the induced aberration RMS from the treatment decentration was maximum when the position vector of translation was perpendicular to the astigmatism axis. In particular, the induced astigmatism obviously changed with the angle difference between the position vector of the translation and the astigmatism axis.17 Because the astigmatism axis varies with the individual eye, only the relationship between the induced aberrations and the amount of decentration in the y-axis were analyzed to explain the effect of the position vector of translation on induced aberrations.
The study comprised 117 eyes of 77 refractive surgery candidates. The mean age was 24.5 years ± 4.8 (SD) (range 18 to 34 years). The range of spherical equivalent (SE) errors was 0.0 to −11.0 diopters (D). Figure 2, A, shows the distribution of the mean SE. The range of astigmatic refractive errors was from plano to −4.25 D, which is shown as a scatterplot of the orthogonal components J0 and J45 in Figure 2, B.
As stated above, the decentered wavefront aberrations can be obtained according to Figure 1; when decentration occurs, the part of the wavefront that moves outside the pupil is calculated from the transition zone. For example, Figure 3 shows the original wavefront map of 1 eye 6.0 mm in diameter and the induced wavefront aberrations with oblique incidence (method 1) and without oblique incidence (method 3) for decentration of 0.2 mm, 0.4 mm, and 0.6 mm.
When no decentration occurred, the correction vision system showed an optimum effect and the induced wavefront aberration was zero. But with the increase in lateral displacement, decentration played a more signficant role in the corrected effect. In addition, the induced wavefront aberrations with oblique incidence in the laser ablation profile were not markedly different from those without the oblique-incidence effect.
Population Statistics of the Wavefront Aberration
Figure 4 shows the average of the signed Zernike coefficients in the 117 eyes, including the mean value and standard deviation (SD). The mean values of almost all Zernike coefficients were approximately zero. However, the mean Zernike coefficients of 5 terms, C(3,−3), C(3,1), C(4,0), C(4,2), and C(6,2), were significantly different from zero at the 5% level. Vertical trefoil C(3,−3) and spherical aberration C(4,0) had mean values of −0.039 μm and +0.055 μm, respectively. The SD of vertical coma C(3,−1) was maximum at 0.161 μm, and the SD of spherical aberration C(4,0) was apparently larger than other 4th-order Zernike terms (Figure 4).
Induced Aberrations from Treatment Decentration
Figure 5 shows the relationship between the induced aberrations and the amount of decentration in the x-axis. In this section, the adjustment factors in Method 1 were used. The results in Figure 5, A, indicate that the induced coma and defocus were larger than other individual Zernike terms, and they sharply increased with increasing translation of the center of ablation. In addition, postoperative spherical aberration increased theoretically as the translation increased, and its sign was positive, which is the same as the clinical results after refractive surgery. Figure 5, B, shows that the amounts of induced 5th- or 6th-order were markedly lower than 2nd- or 3rd-order aberrations. Thus, the main induced individual Zernike terms from treatment decentration were secondary coma and secondary spherical aberration.
Figure 6 shows the relationship between the induced aberrations and the mismatch angle from rotation of the ablation zone when the transverse decentration in the x-axis is 0.5 mm. The induced HOAs were only slightly related to the mismatch angle when the amount of decentration in the x-axis was 0.5 mm. This suggests that the mismatch angle from the subclinical eye cyclotorsion may account for a very small portion of the induced aberrations and yet the increase in postoperative aberrations may be mainly attributed to transverse decentration. Induced astigmatism increased slowly with the increase in the mismatch angle, as did the induced LOAs (Figure 6, A).
Figure 7 shows the induced aberrations from the combination of transverse translation and angle mismatch. For simplicity, only 4 main Zernike aberration terms are shown. In fact, all Zernike terms are induced from treatment decentration. Figure 7 shows that the centration error induced significantly greater HOAs than cyclotorsional misalignment. The induced astigmatism was closely related to the rotation of the ablation zone (Figure 7, A). In addition, the astigmatism increased with an increase in translation. The postoperative coma, spherical aberration, and secondary coma increased with the increase in centration error (Figure 7, B to D). However, they remained almost constant with the change in mismatch angle. This result implies that they are hardly induced from the rotation of ablation zone.
Without Effect of Oblique Incidence as in Previous Studies
Figure 8 shows the relationship between the induced aberrations without the effect of oblique incidence and decentration in the x-axis. In this section, the adjustment factors in Method 3 were used. Analyzed were the induced wavefront aberrations from decentration with consideration of oblique incidence in the laser ablation profile (method 1) and without consideration of the effect of oblique incidence (method 3). In both cases, the most significant induced individual Zernike terms were coma and spherical aberration. However, the induced HOAs with oblique incidence in laser ablation profile, especially induced coma, were slightly larger than that without oblique incidence when the decentrations were the same in both cases. This can be attributed to the transition zone.
With the Effect of Oblique Incidence in Actual Laser Ablation Process
Figure 9 shows the relationship between the induced aberrations with the effect of oblique incidence in the actual laser ablation process and decentration in the x-axis. In this section, the adjustment factors in method 2 were used. The RMS values of the residual LOAs and HOAs were distinctly different from zero, even with no subclinical decentration.
Without Effect of Transition Zone in Laser Ablation Algorithms
Figure 10 shows the induced aberrations without the effect of the transition zone in laser ablation algorithms from the decentration in the x-axis. In this section, the transverse translation of the ablation zone was simulated by ocular wavefront translation transformation. The residual HOA RMS was less than 0.4 μm, and the signficant induced higher-order term was coma. However, the induced spherical aberration RMS was very small. In addition, when the original aberrations were up to the 6th order, the coefficients of the 6th-order individual Zernike terms in the induced aberrations without transition zone became zeros. That is, these coefficients were only up to 5th order.
Effect of Position Vector of Translation on the Induced Aberrations
Figure 11 shows the relationship between the induced aberrations and the amount of decentration in the y-axis. The residual LOA RMS was up to 2.0 μm when the transverse decentration in the y-axis was 1.0 mm. In addition, the induced coma was larger than the induced spherical aberration.
According to the ISO,19 the line of sight serves as the reference axis in reporting aberrations in the human eyes. Therefore, in clinical practice, the entrance pupil center is usually used as the ablation center; however, it changes due to accommodation or a change in illumination.21 As a consequence, the pupil center in wavefront aberration measurement is usually inconsistent with the ablation center. Therefore, subclinical ablation decentration frequently occurs in laser refractive surgery. In addition, the refractive outcome of cylinder correction depends on the accuracy of the axis treatment.22 The significant magnitude of dynamic cyclotorsion has been detected by several authors.23,24 In fact, the dynamic movements of human eyes occur continually throughout the refractive surgery procedure; thus, although laser in situ keratomileusis (LASIK) with active cyclotorsion error correction increases the accuracy of cylinder correction,24,25 these movements are unpredictable.23 Therefore, no dynamic cyclotorsion of human eyes was considered for analysis in this study. Furthermore, the sensitivity of the position in custom correction is dependent on the individual Zernike terms and the amount of misalignment. For this reason, the sensitivity of the position of each eye may be determined by the particular composition of the aberration pattern in terms of Zernike polynomials and the position vector of translation.17 Clinically, the use of active eye tracking may improve the optical and visual outcomes after photorefractive laser surgery.26 In clinical application, laser treatment decentration can be diagnosed using tangential topography corneal maps.27
Comparison of Figure 5 and Figure 9 indicates that the RMS of the wavefront aberrations induced by decentration with the effect of oblique incidence in the laser ablation profile is lower than that in the actual laser ablation process for slight subclinical decentration (<0.3 mm). The reason is that the actual ablation center deviates from the ideal correction center, and this misalignment leads to the possibility that the actual ablation depth in the transition zone with decentration is lower than the ideal ablation depth. In particular, slightly larger spherical aberration and coma are induced from decentration in the actual laser ablation process. Previous studies have shown that the effect of oblique incidence across the cornea must be taken into consideration in ablation profile calculations.16 For instance, efficiency reduction is a primary source of the change in asphericity observed clinically. Our results show that this effect is not negligible, especially for slight subclinical decentration. However, to achieve accurate correction of the eye, corneal ablation algorithms should include analytical information on deviations from Lambert-Beer law.28
Comparison of Figure 5 and Figure 10 shows that the induced HOAs without a transition zone are distinctly lower than those with it. In addition, when the transition zone is not taken into account in the theoretical analysis, the induced spherical aberration RMS is very small. Therefore, induced wavefront aberrations may be underestimated with this method. The exact size, shape, and profile of the transition zone have a profound effect on residual aberrations. Several studies have shown that the aspheric transition zone is safe and predictable.12 In addition, the use of the transition zone during LASIK results in a low incidence of postoperative glare and halos.29 Elliptical transition zones may improve the optics and biologic tolerance of excimer laser treatments10 and reinforce the orientation of the axis of cylinder. Also, a transition zone can be used in PRK for high myopia.30 Furthermore, the use of larger ablation diameters in LASIK for hyperopia and hyperopia with mixed astigmatism may produce accurate results, early refractive stability, and good visual performance.31 Corneal optical aberrations after PRK with a larger ablation zone and transition zone are less pronounced than those with no transition zone.32 In our study, a limitation is that only 1 transition zone ablation profile was used.
Comparison of Figure 5 and Figure 11 shows that the features of the induced aberration structure are the same even though the position vectors of the translation are different. However, the induced LOA RMS with decentration in the y-axis has a 28% increase over that with the same decentration in the x-axis. In particular, the coefficients of induced defocus in both cases are markedly different. Furthermore, the induced HOA RMS with decentration in the y-axis has a 14% increase over that with the same decentration in the x-axis. Similarly, the coefficients of induced coma in both cases are different. The reason may be because the J45 astigmatism components are lower than the J0 components and the induced aberrations are closely correlated with the astigmatism axis.17 However, induced spherical aberration is only slightly related to the astigmatism axis. This may be because spherical aberration is an axial aberration and is defined for objects in the axis.
Porter et al.6 found that postoperative aberrations were typically larger than those theoretically induced due to a pupil-center offset of the treatment. In addition, the mean ratio of the theoretically induced higher-order RMS to the actual higher-order RMS measured 6 months postoperatively was 0.26. In that study, the optical zone was surrounded by a 0.875 mm fixed transition zone. However, the transition zone was not taken into account in the analysis of theoretically induced aberrations. In our study, the induced aberrations without considering the transition zone were lower than when it was considered; this implies that the transition zone exerts an important influence on the residual aberrations in refractive surgery. Therefore, this difference may be partially attributed to no consideration of the transition zone.
If the transition zone is considered, decentration of the ablation profile during surgery may account for a main portion of the postoperative increases in HOA, especially spherical and coma aberrations. It is well known that subclinical ablation decentration is a major factor in increased coma-like aberrations after corneal laser surgery.33 Our results support the conclusion that postoperative coma-like aberrations tend to increase.9 However, fitting Zernike polynomials to wavefront data relative to a shifted reference axis may introduce reference axis–dependent aberrations such as coma. In addition, our results are in agreement with the conclusion drawn from Wang and Koch5 that centration error induces significantly greater HOAs than cyclotorsional misalignment in induced wavefront aberrations. Furthermore, our results show that induced spherical aberration is closely related to treatment decentration. This result is consistent with previous findings.9 Nevertheless, some studies have found that aspheric ablation profiles can lead to a significant reduction in induced spherical aberration.34 These results cannot be extrapolated to hyperopic patients. In fact, the change in refractive power shows a much steeper change in power in the transition zone for hyperopic correction than for myopic correction. Therefore, the effect of the transition zone in hyperopic correction deserves further research.
On the other hand, pupil size plays an important role in the influence on induced postoperative aberrations. In this study, the pupil size and optical zone size were assumed to be 6.0 mm; however, they are usually inconsistent in clinical practices. Thus, the impact of pupil size should be considered when the clinical wavefront data are analyzed.
Clinical residual aberrations are correlated with several factors. First, wavefront aberrations in human eyes are dynamic and clinical aberrations are different for each measurement. Second, the corneal flap is meticulously created and repositioned in laser refractive surgery. Third, ablation area irregularity may influence the optical and functional outcomes in refractive surgery and can be improved by increasing the regularity of the ablated surface by final smoothing.35 Finally, laser surgical systems include many systematic errors, such as errors related to registration, fitting, and tracking. To achieve better postoperative visual performance, treatment decentration should be minimized. For wavefront-guided custom correction, an optimum transition zone should be designed to decrease the induced HOAs in refractive surgery.
In conclusion, based on the ablation profile for wavefront-guided custom correction, the influence of treatment decentration and the transition zone on induced wavefront aberrations was studied. Theoretical results indicated that induced coma and defocus were larger than other individual Zernike terms when the transition zone was taken into account, and they increased rapidly with the increase in the lateral translation of the center of ablation. Theoretically, the induced spherical aberration also increased rapidly with the translation increase, and its sign was positive. In addition, induced wavefront aberrations when oblique incidence was considered in the laser ablation profile were lower than those in the actual laser ablation process for slight subclinical decentration. Furthermore, the induced aberrations were not closely related to the subclinical unmatched angle of ablation profile from eye cyclotorsion. Without considering the transition zone, the HOAs induced by decentration were significantly lower than those when the transition zone was considered; this indicates that the transition zone may account for a main portion of the postoperative increases in HOAs. Therefore, the induced HOAs from decentration may be decreased by the design of the transition zone.
What Was Known
- It is well known that HOAs distinctly increase after laser refractive surgery and the increase was closely related to the treatment decentration in many clinical studies.
What This Paper Adds
- A quantitative approach can be used to analyze the increase in postoperative optical aberrations from treatment decentration. In addition, the influences of the transition zone and laser oblique incidence are taken into account in analysis.
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