The analysis of the change in residual wavefront aberrations after laser refractive surgery is important for the development of visual correction technology. Based on the ablation profile for wavefront-guided refractive surgery including optical zone and transition zone, the effect of pupil size on residual wavefront aberrations was studied. The research revealed that the optical zone to pupil ratio had a significant influence on the residual wavefront aberrations. The residual spherical aberration and coma were obviously larger than other individual Zernike higher-order terms when pupil diameter was larger than the optical zone size, and they increased rapidly as the pupil size increased. In addition, when the ablation zone diameter was kept constant, the residual higher-order aberrations increased rapidly as the blend coefficient increased for a 6mm or 7mm pupil. Furthermore, the residual higher-order aberrations with treatment decentration were distinctly larger than those without decentration. In the achievement of the best postoperative visual performance, the design of ablation profile played a crucial role in decrease of the residual wavefront aberrations after refractive surgery, especially optical zone size and the ablation pattern of transition zone.
© 2013 OSA
It is important to further understand residual wavefront aberrations in optimizing the algorithms and decreasing postoperative aberrations for a customized correction. Customized corneal refractive surgery can be based on corneal topography or whole-eye wavefront aberrations for correcting refractive errors. Topography-guided refractive surgeries aim to treat the corneal irregularities in elevation in addition to the defocus and astigmatism [1–3]. Alternatively, wavefront-guided surgeries aim to address the total ocular wavefront aberrations in addition to the refractive errors [4,5]. This paper concerns the effect of pupil size on residual optical aberrations after a wavefront-guided corneal ablation.
The great impact of pupil size on wavefront aberration RMS (root mean square) has been known for a long time . Wang et al. indicated that Zernike polynomial coefficients increased with the increase of pupil size, and yet the different increase of values was induced for an equal increase of pupil size . Furthermore, several investi gators described the influence of pupil dilation on higher order aberrations after refractive surgery. Buhren et al. found that the changes in higher order aberration RMS and primary spherical aberration were significantly correlated with the optical zone diameter to pupil diameter ratio .
The pupil center shift is inevitable in corneal refractive surgery. The treatment decentration in refractive surgery has been observed in several studies and the results revealed that the centration errors had an important influence on the residual aberrations . Wang et al. reported that mean pupil centroid shift was 0.29mm during wavefront-guided corneal ablation and the centration error induced 4.9 times, 2.8 times, and 8.7 times higher values of total, low-order, and higher-order aberrations, respectively . Porter et al. showed that the postoperative aberrations were typically larger than those theoretically induced due to a pupil center offset of the treatment . Therefore, the treatment decentration should be considered by Monte Carlo simulation in our study. Theoretically, the translation and rotation of ablation profile could be simulated by wavefront transformation . Guirao et al. theoretically investigated the impact of translation and rotation on individual Zernike terms by ocular wavefront transformation .
Transition zone, a ring-shaped or ellipsoid area around the intended optical zone, is included in modern laser algorithms for refractive surgery . It connects the optical zone to the untreated cornea. With the application of transition zone, the curvature is continuous at the boundary between optical zone and transition zone and at the boundary between transition zone and the unaltered cornea. In fact, the exact size, shape and profile of the transition zone have profound impact on the residual aberrations. Several researches proved that the aspheric transition zone was safe and predictable [15,16]. The use of transition zone during LASIK resulted in a low incidence of postoperative glare and halos [17,18]. Also, a transition zone could be used to photorefractive keratectomy for high myopia . Arbelaez et al. reported that a multidynamic aspheric transition zone was included in a laser system in order to minimize the amount of induced aberrations . So the effect of transition zone on the postoperative residual aberrations would be considered in this theoretical analysis.
Until now the influence of the pupil size on residual aberrations with consideration of treatment decentration and transition zone for a wavefront-guided surgery has not been explicitly evaluated yet. In this research, based on the wavefront-guided corneal ablation profile including transition zone and optical zone, we evaluated the relationships among the amount of predicted residual aberrations, transition zone and pupil size. The influence of oblique incidence on the residual aberrations was also taken into account. Since transition zone, oblique incidence of laser and pupil size were all taken into account, our simulation analysis would be closer to the actual corneal ablation.
In this study, 112 eyes of 56 potential refractive surgery candidates for correction of myopia were enrolled. Patients with connective tissue disease, amblyopia, cataract, retinal disease, keratoconus, and previous ocular surgery were excluded. The age of the patients ranged from 18 to 34 years (mean, 24.3 ± 4.8). The mean preoperative spherical equivalent was −5.39 ± 1.06D (from −4 to −7.75D) in right eyes and −5.28 ± 0.92D (from −4 to −7.25D) in left eyes, with D representing diopter. The distribution of the mean spherical power is shown in Fig. 1(a) . Additionally, the mean preoperative astigmatic power was −0.95 ± 0.82D (from plano to −4.75D) in right eyes and −0.96 ± 0.68D (from plano to −2.75D) in left eyes. Figure 1(b) shows the astigmatic power as a scatter plot of the orthogonal components J0 and J45. After a complete ophthalmic examination and an explanation of the nature and possible consequences of the research, the written informed consent was obtained from all patients. The wavefront aberrations were measured using a Shack–Hartmann aberrometer  (WaveScan wavefront system, VISX, Inc., Santa Clara, CA) in the natural scotopic condition. All measurements were repeated at least three times for each eye, and the 3 best-matching measurements were used in this study. The wavefront aberrations for a 6-mm diameter pupil in all eyes were obtained by scaled transformation of Zernike aberrations. The contact lens wearers were excluded from this study.
2.2 Ablation profile for customized laser refractive surgery
According to the phase-conjugate principle, the ablation depth in optical zone is given directly at any arbitrary point by the wavefront information.
Here, the parameter n depicts the refractive index of cornea in visible light, and the value is 1.376 in this study. Also, (x,y) represents an arbitrary point in the optical zone on cornea. In addition, the wavefront aberrations are expressed as a Zernike polynomial expansion.
Here, the parameters p and q are the radial integer index and the meridional index, respectively.
In this section, an ablation profile for transition zone should be structured. R represents the radius of optical zone. If the width of transition zone is Rρ, the inside radius of transition zone is R, and then the outer radius is R(1 + ρ). Here, ρ depicts the blend coefficient of transition zone. The ablation profile for transition zone can be denoted as follows :
Here, Da(x,y) represents as a blend function. The function value is one at the boundary between optical zone and transition zone, but the value changes to be zero at the boundary between transition zone and untreated periphery.
Also, Db(x,y) indicates the extended ablation depth in transition zone, which is extended from the boundary value of optical zone.
In laser refractive surgery process, the laser beam is moved vertically parallel to the centration axis of the cornea ablation, and it is at oblique incidence (incident angle α or α’) in the peripheral area of pupil. So this will cause the reflection loss of laser energy and the change of the illuminated area on anterior corneal surface. Consequently, the effect of laser oblique incidence in laser-ablation profile can be evaluated by an adjustment factor (κ) with decentration for our study population , and it is denoted as follows:
Here, (x, y), α and R depict the parameters of the actual laser-ablation process. Also, (x’, y’), α’ and R’ are used in the laser-ablation profile.
2.3 Monte Carlo simulating treatment decentration
In our research, we simulated treatment decentration using Monte Carlo methods. The amount of treatment decentration was randomly selected from a probability distribution across a population, which was derived from a recent clinical treatment decentration study . The mean transverse translation in the Monte Carlo analysis was 0.27 ± 0.11mm (range 0.06 to 0.50mm) (Fig. 2 ). Additionally, the mean transverse translation along the horizontal meridian and vertical meridian was + 0.24 ± 0.12 mm (range + 0.06 to + 0.49 mm) and −0.06 ± 0.12 mm (range −0.23 to + 0.17 mm), respectively, in left eyes. Also, the mean translation along the horizontal and vertical meridian was −0.16 ± 0.17mm (range −0.49 to + 0.14 mm) and −0.05 ± 0.14 mm (range −0.29 to + 0.23 mm), respectively, in right eyes.
2.4 Residual wavefront aberrations caused by treatment decentration
When treatment decentration occurs, the actual corrected aberrations can be computed by coordinate transformation. If a part of transition zone moves into the pupil, the actual corrected aberrations should include two parts. One is the part that moves into the pupil, which is calculated from the transition zone, and the other is the left part, which is calculated from the optical zone. Then, the corresponding Zernike coefficients can be obtained by the wavefront surface fitting. For simplicity, ablation profile decentration consisting of lateral displacements and rotations are explicitly considered in this research. The coordinate transformation formula is given as follows:
Here β represents the rotation angle, and Δx, Δy convey the lateral displacement in x- and y-axis, respectively.
With the coordinate transformation formula, the ablation depth in whole ablation zone can be obtained from the ablation profile for optical and transition zone. Additionally, the ablation depth is multiplied by an adjustment factor (κ) of the ablation depth of cornea, and the effective depth can be obtained. Finally, the depth may be converted into the actual corrected aberrations:
We can obtain the actual corrected Zernike aberrations by the surface fitting.
On the other hand, the constriction of pupil may range from 8mm in diameter in scotopic condition to 2 mm in photopic condition because of the change of illumination or accommodation. In addition, it has been proved that the pupil size plays an important role in the RMS value of the measured optical aberrations. Laser refractive surgery may increase the postoperative higher-order aberrations and decrease the visual performance, especially in night vision. Therefore, the quantitative relationship between the residual optical aberrations and the pupil size deserves further study.
Based on the ablation depth for the whole ablation zone, we can obtain the actual corrected aberrations in a given pupil diameter. When the ablation zone covers the pupil, the predicted residual optical aberrations can be calculated as the differences between the preoperative whole-eye aberrations (Wp(x, y)) and actual corrected aberrations. However, when the ablation zone cannot cover the pupil, the predicted residual optical aberrations should include two parts. One is in the overlap region of the ablation zone and pupil, which can be computed as the differences between the preoperative aberrations and actual corrected aberrations. The other is in the non-overlap region, which can be indicated as the preoperative aberrations. Then the Zernike coefficients of the predicted residual aberrations in pupil zone can be achieved by surface fitting.
Finally, the predicted residual aberrations in customized refractive surgery with treatment decentration are calculated as follows:
Here, Cpq is the Zernike coefficient for vision correction with treatment decentration, and apq is the coefficient of preoperative aberration.
3.1 Population statistics of the wavefront aberration
Figure 3 shows the average of the signed Zernike coefficients in a myopic population of 112 eyes (56 subjects) including mean value and standard deviation. The means of almost all Zernike coefficients are nearly zero. For example, vertical trefoil (C3−3) and spherical aberration (C40) have mean values of −0.050μm and + 0.052μm, respectively, in left eyes and −0.021μm and + 0.059μm, respectively, in right eyes. The standard deviation of vertical coma (C3−1) is maximal being 0.139μm in left eyes and 0.152μm in right eyes. The one clear exception is that the coefficient for spherical aberration is systematically biased toward positive values, as may be seen in the statistical summaries presented in Fig. 3. Note that distributions of aberrations for left and right eyes have similar means and variances.
3.2 Effect of the pupil size on the residual wavefront aberrations
Based on the ablation profile for wavefront-guided refractive correction, the exact ablation depth for the whole ablation zone including transition zone could be calculated. Monte Carlo simulation was performed for the transverse translation of the ablation centre. Then the actual ablation depth was computed by coordinate transformation. Furthermore, the laser beam is considered to be moved vertically parallel to the reference axis of laser ablation. According to the mathematical model of anterior corneal surface, the incidence angle (θ) of laser beam at any arbitrary point (x,y) on cornea could be calculated. After that, an adjustment factor (κ) was deduced from the change of the illuminated area and reflection loss of laser energy as the change of incidence angle. The ablation depth was multiplied by adjustment factor and then the effective depth could be obtained. Then, it was converted into the corrected aberrations. The residual aberrations were represented by the differences between the original and corrected aberrations. Finally, the residual Zernike aberration coefficients were achieved by the wavefront fitting. In this study, the relationship between the residual aberrations and pupil diameter was studied. Here, the original and residual aberrations are all up to sixth order and the diameter of optical zone is 6mm. In addition, the blend coefficient is 0.35, and then the diameter of ablation zone is up to 8.1mm. The results are shown in Fig. 4 . The panel A corresponds to the left eyes (OS) and the panel B corresponds to the right eyes (OD). The residual aberration coefficients include the lower-, higher-, 2nd, 3rd, 4th, 5th and 6th order aberrations.
The results in Fig. 4 indicate that the pupil size plays a very critical role in the residual aberration. When the pupil diameter is larger than optical zone diameter (6mm in diameter), the residual aberrations will increase rapidly as the pupil diameter increases from 6mm to 7mm. In addition, the residual defocus and spherical aberration are obviously larger than other individual Zernike terms and they sharply increase as the pupil size increases. The sign of spherical aberration is positive, which is the same as the clinical results after refractive surgery. Additionally, the postoperative coma increases theoretically as the pupil diameter increases. Furthermore, the amounts of the residual fifth or sixth order aberrations are markedly lower than the second or third order aberrations, and then the main residual individual Zernike terms are secondary spherical aberration and secondary coma. By comparing panel A with panel B, it can be found that the residual aberrations in left eyes are slightly lower than those in right eyes. This result can be accounted by the differences in clinical treatment decentration and refractive errors between the left and right eyes.
3.3 Influence of the blend coefficient on the residual wavefront aberrations
Similarly, based on the ablation profile, the treatment translation was simulated by coordinate transformation. In addition, the effect of oblique incidence of laser was taken into account. In this section, the diameter of ablation zone is still 8.1mm. Consequently, the optical zone size decreased as the increase of blend coefficient. For example, when the diameter of optical zone was 6.48mm, 6.12mm, 5.76mm, 5.40mm, 5.04mm and 4.68mm, the corresponding blend coefficients was 0.250, 0.324, 0.406, 0.500, 0.607 and 0.731, respectively. The effective ablation depth could be converted into the corrected aberrations. The residual Zernike coefficients were obtained by surface fitting. Figure 5 shows the relationship between the residual aberrations and the diameter of optical zone. Here, the blend coefficient is from 0.250 to 0.731.
It can be seen in Fig. 5 that the residual higher-order aberrations will distinctly decrease with the increase of the optical zone diameter for a 5-mm or 6-mm pupil, especially spherical aberration. Additionally, when optical zone is relatively small, the residual spherical-like aberration RMS for a 7-mm pupil is significantly larger than that of other order aberrations, while the residual coma-like aberration RMS takes second place. Besides, the correlation between magnitude of the residual coma-like aberrations and optical zone diameter is significantly lower than that of the spherical-like aberrations. Furthermore, the residual coma-like aberrations for a 7-mm pupil are at a maximum when the optical zone diameter is about 5.5mm. However, the optical zone diameter has little influence on other order aberrations. For example, the residual sixth order aberration RMS changes little with the decrease of optical zone diameter. By comparing panel A with panel B in Fig. 5, it can be found that the residual aberrations in left eyes show a slight difference from those in right eyes.
4.1 Comparison with previous studies
In clinical practices, the entrance pupil center (EPC) is usually used as the ablation center. However, EPC changes with accommodation and the change of illumination . In this study, ablation decentration is defined as the offset of the pupil center in wavefront aberration measurement from the ablation center in refractive surgery. Therefore, subclinical ablation decentration is inevitable in refractive surgery. Wang et al. found that mean pupil centroid shift was 0.29mm during wavefront-guided refractive surgery . Porter et al. indicated that mean magnitude of pupil center shift was also 0.29mm in wavefront-guided laser refractive surgery . Lee reported that the mean decentration was 0.23mm in active eye-tracker–assisted myopic photorefractive keratectomy (PRK) using the VISX STAR S4 laser with ActiveTrak (Abbott Medical Optics [AMO]) . In addition, Cakmak et al. demonstrated that the mean decentration was 0.26mm after LASEK . Therefore, a Monte Carlo simulation was used to simulate the clinical treatment decentration in our analysis. It was noticed that the treatment decentration in our analysis existed binocular symmetry between left and right eyes as shown in Fig. 2. However, the optical performance of human eye is dynamic. Febbraro indicated that static cyclotorsion and dynamic cyclotorsion occurred during LASIK . In our study, one limitation is that cyclotorsional errors during refractive surgery are not taken into account.
As been well known, pupil diameter exerts a significant influence on the wavefront aberration RMS by scaling Zernike expansion coefficients to different pupil sizes [28,29]. Oshika et al. found that the total aberration RMS was up to 2.5μm in post–laser in situ keratomileusis eyes for a 7-mm pupil . Additionally, aberrations of all types increase significantly with increasing pupil diameter in myopic eyes . In general, the pupil size and optical zone size are usually inconsistent in clinical practices. Our results are in agreement with the result from Buhren et al. that the optical zone to pupil ratio has a significant impact on higher-order aberrations induction after wavefront-guided refractive surgery . In addition, Vinciguerra has found that larger optical zones have fewer postoperative spherical aberrations . Therefore, the optical zone should be matched with the pupil diameter in the design of refractive surgery.
In fact, an abrupt change in corneal curvature at the edge of optical zone may induce excessive epithelial and stoma tissue healing after surgery. The corneal curvature can be continuous at the boundary between the inner zone and the transition zone and at the boundary between the transition zone and the unaltered periphery by creating a transition zone. Additionally, the haze may be confined to the wound edge, and the central cornea may remain relatively clear and provide good vision. Corneal optical aberrations after photorefractive keratectomy with a larger ablation zone and a transition zone were less pronounced than those associated with no transition zone . Therefore, the transition zone must be designed in a wavefront-guided refractive surgery. Our results have shown that the ablation profile with transition zone may account for a main portion of the postoperative increases in higher-order aberrations, especially spherical and coma aberrations. In particular, the power of transition zone is less than that of optical zone, which would lead to the remain of some refractive errors . Furthermore, because of stromal and epithelial healing, the shape of the cornea changes in the postoperative period by creating a transition zone that may increase spherical aberrations . In fact, our results are also in agreement with the conclusion drawn by Arbelaez et al. that the main higher order aberration effects postoperatively (coma and spherical aberration) originated from decentration and “edge” effects, the strong local curvature change from optical zone to transition zone and from transition zone to the untreated cornea . Also, our results are sufficient to support the conclusion that changes in HOA root mean square and spherical aberration are significantly correlated with the optical zone to pupil ratio .
The postoperative increases in corneal asphericity and induction of spherical aberration can be partly explained by the ablation efficiency reduction due to laser angle of incidence [35–38]. Also, Jimenez et al. argued that the consideration of the angular dependence of laser-ablation rates was important in efforts to improve the ablation algorithms used in refractive surgery . Our previous results revealed that the effect of laser oblique incidence exerted an important influence on the residual wavefront aberrations . Consequently, the ablation efficiency reduction was taken into account in this study.
Porter et al. indicated that the postoperative aberrations were typically larger than those theoretically induced due to a pupil center offset of the treatment . These results might be different from those of clinical observations because the transition zone is ignored in the analysis of theoretically induced aberrations. Our results demonstrate that the transition zone has a critical influence on the residual aberrations. In addition, our results are sufficient to support the conclusion that the postoperative coma-like and spherical-like aberrations tend to increase [34, 40]. Besides, ablation area irregularity may influence optical and functional outcomes of the refractive surgery and can be improved in regularity of the ablated surface by final smoothing [31, 41].
In this paper, the transition zone, effect of laser oblique incidence and pupil size were all taken into account in simulation analysis, which may be closer to clinical practices.
4.2 Without effect of treatment decentration
As described above, the ablation depth in ablation zone was computed and no translation of ablation centre occurred. The effect of oblique incidence was taken into account. In the same way, we obtained the residual Zernike coefficients by the surface fitting. In this section, we investigated the relationship between the residual aberrations without translation and pupil diameter. The diameter of optical zone is 6mm. In addition, the blend coefficient is 0.35, and then the diameter of ablation zone is up to 8.1mm. The results are shown in Fig. 6 .
The results in Fig. 6 indicate that the residual aberrations will increase rapidly as the pupil diameter increases from 6mm to 7mm. However, the RMS of residual aberrations is zero when pupil diameter is smaller than optical zone diameter. In comparison with Fig. 4 and Fig. 6, the results reveal that the residual wavefront aberration RMS without decentration is lower than that with it. In both cases, the most important residual individual Zernike terms are spherical aberration. The coma in Fig. 6 is obviously smaller that in Fig. 4. This can be attributed to that the treatment decentration is taken into account in Fig. 4. By comparison, the residual aberrations in left eyes are substantially the same as that in right eyes. These results indicate that the treatment decentration may induce observably higher order aberrations.
4.3 Analysis of pupil size on residual wavefront aberration with transition zone
Clinical data and theoretical results revealed that the mismatch between pupil and optical zone may cause significant induced spherical aberration after refractive surgery. When treatment decentration occurred, the induced aberrations were distinctly larger than those without treatment decentration. For example, the induced aberration RMS was zero when the pupil diameter was smaller than optical zone diameter. However, the mean coefficient of the induced spherical aberration from the ablation profile was 0.50μm in left eyes and 0.54 μm in right eyes respectively, when the diameters of pupil and optical zone were 6mm. Additionally, when the ablation zone diameter was kept constant, the blend coefficient was determined by the optical zone size. In this case, the induced higher-order aberrations rapidly increased with the decrease of the optical zone diameter for a 6mm pupil. Therefore, the match between pupil and optical zone was the determining factor in the residual aberrations after refractive surgery. In clinical practice, a relatively large scotopic pupil for high myopia correction may lead to an increase in the risk of visual symptoms after surgery. Our results demonstrated that the ablation profile with transition zone may account for a significant portion of the postoperative aberrations increase in clinical results. In order to achieve better visual performance after surgery, some steps must be taken to decrease the induced high-order aberrations in refractive surgery. Treatment decentration should be minimized and an optimum transition zone should be designed for a wavefront-guided customized correction.
4.4 Influence of residual wavefront aberration on visual performance
It is well known that the wavefront aberrations have an important influence on retinal image quality. Even if the RMS value of aberrations maintains constant, the impact of a set of Zernike aberrations on retinal image quality varies greatly with aberration structure. In addition, the retinal image quality is closely related with visual performance. Therefore, the optical aberrations may degrade the eye's visual performance. The refractive surgery is known to induce an increase in optical aberrations that may cause night vision disturbances such as starbursts, halos, ghosting and glare. For instance, coma may be associated with monocular diplopia. Additionally, spherical aberration may cause a reduction in formation of halos and visual acuity, particularly when the pupil size is large. In addition, pupil diameter exerts greater influence on the relation between induced changes in ocular higher-order wavefront aberrations and changes in visual performance . However, Stiles-Crawford effect (SCE) can ameliorate the influence of induced aberration on visual performance and the neural visual system can be adapted or partly compensated to the eye's residual aberrations.
Based on the ablation profile with transition zone for wavefront-guided customized correction, the influence of pupil size on residual wavefront aberration was studied. Theoretical results revealed that the optical zone to pupil ratio exerted a significant influence on the residual wavefront aberrations. In addition, the residual spherical aberration and coma were obviously larger than other individual Zernike higher order terms when pupil size was larger than the optical zone diameter, and they increased rapidly with the increase of pupil diameter. Theoretically, the residual higher-order aberrations with treatment decentration were distinctly larger than those without treatment decentration. When the ablation zone diameter was kept constant, the blend coefficient was determined by the optical zone size and then the induced higher-order aberrations rapidly increased with the blend coefficient increase for a 6mm or 7mm pupil. Our results demonstrated that the transition zone may account for a main portion of increase in higher-order aberrations. Therefore, the residual higher-order aberrations may be decreased by the design of transition zone and optical zone.
This research is supported by the National Natural Science Foundation of China (No. 81170873).
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