**Purpose.**:
To compare the effects of laser profile asphericity on the induction of wavefront aberrations, susceptibility to decentration, and depth of focus in a polymethylmethacrylate (PMMA) model.

**Methods.**:
Four PMMA lenses received an excimer laser ablation of −6 D with a 6-mm optical zone and different amounts of primary spherical aberration (Z_{4} ^{0}): 0, −0.346, −1.038, and −2.076 μm. The curvature of each lens was measured by using surface profilometry, and wavefront changes were computed from curvature differences. Changes in optical quality were compared by treatment simulation of 13 real myopic eyes. The influence of pupil diameter, ablation decentration, and defocus on retinal image quality was measured by using the optical transfer function–based visual Strehl ratio (VSOTF).

**Results.**:
Aspheric ablation profiles induced significantly less primary but higher secondary spherical aberration (Z_{6} ^{0}) than did the standard profile; however, Z_{4} ^{0} compensation was incomplete. Simulated treatments with aspheric profiles resulted in significantly better retinal image quality and higher decentration tolerance than did the standard profile. Optical depth of focus was not affected with a 3-mm pupil, whereas with a 6-mm pupil, there was a small but statistically significant decrease in depth of focus.

**Conclusions.**:
Aspheric laser profiles showed theoretical optical benefits over standard ablation profiles for the treatment of myopia, including terms of decentration tolerance. However, there remained profound induction and thus, undercorrection of Z_{4} ^{0}, due to loss of laser ablation efficiency in the lens periphery.

^{ 1–4 }Both spherical and comalike aberrations can significantly decrease optical quality. The induction of SA could be explained by a change in the cornea from a prolate to a more oblate shape.

^{ 5 }This shape change has been attributed to biomechanical effects after cutting of the stromal lamellae

^{ 6,7 }and to a decrease in laser efficacy at the corneal periphery,

^{ 8 }although a recent PMMA study favored the latter hypothesis.

^{ 9 }The induction of SA was found to correlate with higher attempted spherical equivalent and a small fractional clearance (ratio of the diameter of the optical zone to the pupil diameter).

^{ 3,10,11 }Coma induction was also linked to a small fractional clearance, which could be explained by the fact that a decentered SA translates into coma.

^{ 12 }Therefore, the induction of SA plays a pivotal role for the change in the wavefront error (WFE) after corneal refractive excimer surgery.

^{ 13–15 }Thus, the benefits of AAP should be a higher optical quality for larger pupils and a higher tolerance of decentration of the optical zone because of lower coma induction. On the other hand, SA can enhance the depth of focus,

^{ 16 }suggesting a lower depth of focus for treatments with AAP.

_{4}

^{0}) to precompensate for the induction of Z

_{4}

^{0}. Based on previous studies from our laboratory,

^{ 17 }an induction of 0.346 μm Z

_{4}

^{0}was expected over a pupil diameter of 6 mm for a −6-D treatment with an optical zone of 6 mm. The aspheric profiles contained the one-, three- and sixfold negative amount of expected Z

_{4}

^{0}(Table 1).

_{2}

^{0}, Z

_{4}

^{0}, and Z

_{6}

^{0}to the height difference data. The pupil diameter for the Zernike fit was 9 mm, and an air/cornea refractive index difference of 0.376 was assumed.

^{ 18 }Second-order aberration Zernike coefficients (lower order aberrations [LOAs]) were converted into dioptric power vectors (

*M*,

*J*

_{0}, and

*J*

_{45}), where

*M*, is the spherical equivalent,

*J*

_{0}is the 0°/90° and

*J*

_{45}is the 45°/135° astigmatic component. HOAs were broken down into coma root mean square (the RMS value of all coma terms

*C*

_{n}^{±1}), SA RMS (the RMS value of all coefficients

*C*

_{n}^{0}), and the RMS of the residual, noncoma, nonspherical aberrations (the RMS value of all remaining HOAs

*C*

_{n}^{≥ ±2}).

^{ 19 }The VSOTF for the best-corrected state (BCVSOTF) was calculated with commercial software (Visual Optics Laboratory [VOL]-Pro 7.14; Sarver and Associates, Carbondale, IL). This metric is obtained by modification of LOA coefficients to maximize the VSOTF, simulating the process of subjective refraction. Thus, the BCVSOTF reflects the influence of HOA on optical quality. WFEs and BCVSOTF values were reconstructed for pupil diameters over a range from 2.5 to 8 mm in 0.1-mm steps.

^{ 12 }Briefly, a custom algorithm (MATLAB; The MathWorks) was used to calculate decentered WFE differences from PMMA data for the size of a 6-mm subaperture along Cartesian decentrations Δ

*x*and Δ

*y.*Δ

*x*and Δ

*y*were changed in steps of 100 μm, covering the entire 9-mm area of the PMMA WFE difference. This method resulted in a maximum decentration range of 3000 μm (−1500 to 1500 μm) over a circular region. Zernike polynomials for the second to sixth order were fitted to the data of each decentered wavefront, resulting in 709 WFEs: 1 centered and 708 decentered per eye. All postoperative WFEs were calculated by adding the centered or decentered WFE difference to each of the preoperative human WFEs. WFEs and corresponding VSOTF metrics were calculated over pupil diameters from 3 to 6 mm (0.5-mm steps).

^{ 19 }Vectors (

**) between the centered position (**

*r**x*,

*y*) and each outmost coordinate below the criterion (threshold coordinates

*x*′,

*y*′) were calculated. The mean value (

*r̄*) reflects the average maximum permissible decentration in micrometers that allows one to remain below the threshold criterion and equals the radius of a circle around the centered position. Tolerance values (

*r̄*) were calculated based on the entire set of 709 data points for pupil diameters between 3 and 6 mm at 0.5-mm steps.

*n*) and the aspheric profiles were compared by paired, two-tailed Student's

*t*-test. If data were not normally distributed according to the Kolmogorov-Smirnov-Lilliefors test, the Wilcoxon-Mann-Whitney U test was used instead. If multiple comparisons were made,

*P*values were adjusted with the Bonferroni method (BiAS 8.2 software; Epsilon Verlag, Hochheim, Germany).

*n*, 6-mm pupil) to 5.84 D (0

*n*, 6-mm pupil). There was induction and profound undercorrection of positive primary SA (Z

_{4}

^{0}) (Table 1, Fig. 2). When calculated over a 6-mm pupil diameter, the 0

*n*profile (standard profile) induced 0.4 μm of Z

_{4}

^{0}. Both the 1

*n*and 3

*n*profiles induced smaller but still significant amounts of positive Z

_{4}

^{0}. Overcorrection (induction of negative SA) was observed only with the

*6n*profile. Numerical interpolation using a second-order polynomial fit (

*y*= −0.0596

*x*

^{2}+ 0.1376

*x*+ 0.3913,

*R*

^{2}= 0.98) to the data revealed that a precompensation factor of 4.79

*n*(−1.656 μm of Z

_{4}

^{0}) resulted in a treatment without induction of primary SA (Fig. 2). Figures 3A–C show the change in sphere and primary and secondary SA as a function of pupil diameter. Whereas the 0

*n*, 1

*n*, and 3

*n*ablations had similar curve characteristics, with a steep increase at pupil diameters beyond 6 mm, the 6

*n*profile showed induction of negative Z

_{4}

^{0}at pupil diameters <7 mm and a steep Z

_{4}

^{0}increase beyond 7.5-mm diameters (Fig. 3B). A higher precompensation factor resulted in higher induction of secondary SA (Z

_{6}

^{0}; Table 2, Fig. 3C).

Profile | ΔM (D) | ΔSA RMS (μm) | ΔZ_{4} ^{0} (μm) | ΔZ_{6} ^{0} (μm) | ||||
---|---|---|---|---|---|---|---|---|

3 mm | 6 mm | 3 mm | 6 mm | 3 mm | 6 mm | 3 mm | 6 mm | |

0n | 6.65 | 5.84 | 0.018 | 0.401 | 0.018 | 0.400 | 0.000 | 0.026 |

1n | 5.88 | 5.26 | 0.012 | 0.323 | 0.012 | 0.321 | 0.000 | 0.028 |

3n | 5.49 | 5.20 | 0.001 | 0.197 | 0.000 | 0.193 | 0.001 | 0.042 |

6n | 4.35 | 4.95 | 0.031 | 0.171 | −0.031 | −0.153 | 0.001 | 0.077 |

*n*) and treatment with an aspheric profile (Table 3, Fig. 3D). Treatment simulations with the 3

*n*profile showed the least decrease of VSOTF throughout a large range of pupil diameters with a mean difference of 0.22 ± 0.06 log units (minimum, 0.14; maximum, 0.33 log units) compared with the standard profile over a 6-mm pupil diameter (Table 3).

Profile | ΔLog BCVSOTF | |
---|---|---|

3-mm PD | 6-mm PD | |

0n | −0.03 ± 0.02 (−0.06 to −0.01) | −0.29 ± 0.07 (−0.36 to −0.14) |

1n | −0.02 ± 0.01 (−0.04 to 0) | −0.21 ± 0.07* (−0.29 to −0.04) |

3n | 0 ± 0† | −0.08 ± 0.05† (−0.14 to 0.03) |

6n | −0.02 ± 0.03 (−0.07 to 0.03) | −0.21 ± 0.02‡ (−0.35 to 0.20) |

*n*profile induced larger amounts of coma when decentered up to 1000 μm (Fig. 4A). However, this pattern was reversed when coma induction was examined over a 6-mm pupil diameter (Fig. 4B).

*n*). Generally, decentration tolerance decreased at larger pupil diameters (Fig. 5, Table 4) and BCVSOTF showed the smallest decentration tolerance of all parameters tested (0

*n*: 581 ± 124 μm, 6

*n*: 967 ± 228 μm;

*P*< 0.01).

Profile | Spherical Equivalent (Threshold −0.5 D) | Sphere (Threshold −0.5 D) | Cylinder Magnitude (Threshold −0.5 D) | BCVSOTF (Threshold −0.2 log units) | ||||
---|---|---|---|---|---|---|---|---|

3 mm | 6 mm | 3 mm | 6 mm | 3 mm | 6 mm | 3 mm | 6 mm | |

0n | 1020 ± 0 | 829 ± 0 (821–830) | 1281 ± 90 (1183–1405) | 1043 ± 66 (969–1161) | 1193 ± 65 (1071–1293) | 1007 ± 89 (853–1122) | 936 ± 88 (814–1070) | 581 ± 124 (266–941) |

1n | 1171 ± 0‡ | 894 ± 0* (894–895) | 1378 ± 70 (1300–1455) | 1121 ± 67 (1056–1247) | 1298 ± 62† (1208 to 1367) | 1073 ± 90 (914 to 1190) | 869 ± 62* (1017–1454) | 668 ± 189 (414–1009) |

3n | 1455 ± 0* | 985 ± 0* | 1451 ± 4* (1444 to 1455) | 1222 ± 74* (1147–1353) | 1455 ± 0* | 1146 ± 86‡ (994 to 1246) | 1416 ± 15* (1387–1439) | 799 ± 174 (495–1147) |

6n | 1455 ± 0* | 1294 ± 0* | 1455 ± 0* | 1421 ± 35* (1347–1455) | 1329 ± 44* (1251–1372) | 1340 ± 45* (1247–1384) | 1161 ± 122* (975–1455) | 967 ± 228‡ (588–1323) |

*n*profile, defocus curves had a similar appearance without multifocality (Figs. 6A, 6B). At a 3-mm pupil diameter, depth of focus of all profiles was comparable. Over a 6-mm pupil diameter, the 3

*n*and 6

*n*aspheric profiles had a significantly lower depth of focus (Table 5), with the greatest difference occurring between 0

*n*and 3

*n*(mean difference 0.24 ± 0.07 D, minimum 0.17 D, maximum 0.42 D).

Profile | 3-mm PD | 6-mm PD |
---|---|---|

0n | 0.52 ± 0.04 (0.47–0.59) | 0.67 ± 0.14 (0.31–0.99) |

1n | 0.50 ± 0.04 (0.46–0.56) | 0.58 ± 0.13 (0.44–0.86) |

3n | 0.49 ± 0.03 (0.45–0.54) | 0.43 ± 0.11† (0.24–0.74) |

6n | 0.51 ± 0.02 (0.48–0.56) | 0.52 ± 0.23‡ (0.32–1.11) |

^{ 20,21 }and biological effects, such as biomechanics and wound-healing reactions.

^{ 22–24 }The simulation experiment allowed direct comparison of retinal image quality of the different profiles in the same subjects. Three major findings emerged from the study: (1) in a PMMA model, the myopic ablation induced positive primary SA, overriding the SA correction of aspheric profiles and consecutively causing undercorrection of SA; (2) a perfectly centered ablation with an aspheric profile has optical benefits over treatment with a standard profile without significant compromise of depth of focus; (3) aspheric ablation profiles were less susceptible to decentration-induced image quality deterioration.

^{ 6,7,23,25 }and secondarily to the variability of laser energy across the cornea.

^{ 8 }This popular biomechanics hypothesis has been questioned recently in a study that applied a PMMA model.

^{ 9 }A well-centered myopic ablation for −6 D with a 6-mm optical zone and a standard profile induced an amount of positive primary SA on PMMA comparable to that observed after LASIK in human eyes.

^{ 3,11 }The results of the present study and comparison of SA induction on PMMA and feline eyes using our cat PRK model (Bühren J, et al.

*IOVS*2008 49:ARVO E-Abstract 2922) strongly suggest that the main source of SAs in laser refractive surgery are of physical (i.e., laser fluence loss toward the corneal periphery) rather than biological (i.e., biomechanical effects and wound healing) origins. This hypothesis is supported by the present finding that the correction of SA with an aspheric profile is insufficient if the expected amount obtained from a standard correction is used (1

*n*approach). Interpolation showed that a 4.79

*n*-fold amount of the expected Z

_{4}

^{0}induction would be necessary to obtain an ablation free of Z

_{4}

^{0}induction. Significant Z

_{4}

^{0}undercorrection with induction overriding the correction, as found on PMMA (Fig. 2), has also been observed in human eyes after wavefront-guided LASIK.

^{ 11 }With increasing amounts of attempted primary SA (Z

_{4}

^{0}) correction, increasing amounts of secondary SA (Z

_{6}

^{0}) were induced. Thus, a Z

_{4}

^{0}-neutral ablation is not SA neutral, although the Z

_{6}

^{0}-related wavefront distortion is quantitatively lower than that related to Z

_{4}

^{0}(Figs. 3B, 3C) and is probably caused by ablation efficiency changes across the cornea.

^{ 17,26 }It should be considered whether ablations on relatively flat PMMA spheres are less affected by peripheral fluence loss than similar ablations on steeper human corneas. In addition, a potential influence of preoperative toricity and asphericity on SA induction must be determined in future model studies. Another caveat is the assumption that the ablation rate on corneal tissue is as uniform as on PMMA. There may be variation of ablation rate with ablation depth leading to different amounts of SA induction in living tissue. However, since our results on PMMA are comparable to results obtained from human

^{ 3,11 }and feline

^{ 17 }eyes, we assume that the PMMA model used in the present study yields valid results, at least for the −6-D treatment.

_{4}

^{0}using aspheric ablation profiles led to an improved retinal image quality (BCVSOTF) compared with the standard treatment (0

*n*). Although the differences were marginal at smaller pupil diameters, they were statistically significant at a 6-mm pupil diameter, where the greatest BCVSOTF difference (ΔBCVSOTF) could be found between the 0

*n*and the 3

*n*profile. This difference (0.22 log units) roughly equals two high-contrast logMAR steps and therefore can be considered clinically relevant.

^{ 19 }Because of induction of negative Z

_{4}

^{0}(overcorrection) and the relative high Z

_{6}

^{0}induction, the 6

*n*profile contained steeper slopes and thus showed a behavior different from that of the 0

*n*, 1

*n*, and 3

*n*profiles. Compared with the 3

*n*profile, the 6

*n*profile decreased BCVSOTF more at smaller pupil diameters. This effect, which could also be explained by the Z

_{4}

^{0}overcorrection and higher induction of Z

_{6}

^{0}, was reduced for diameters larger than 6 mm (Fig. 3D). Because SA can enhance depth of focus

^{ 16,27 }and there is evidence that the reduction of SA by aspheric intraocular lenses (IOLs) can compromise depth of focus,

^{ 28 }it seemed necessary to investigate depth of focus of the different profiles. We calculated depth of focus by defocusing the LOA-optimized simulated postoperative WFE and used the metric VSOTF as the criterion of optical quality (optical depth of focus). The cutoff from the maximum VSOTF (in-focus) was set to −0.2 log VSOTF units. Depth of focus simulation showed that there were no differences between spherical and aspheric profiles for a 3-mm pupil diameter and only marginal differences for a 6-mm pupil diameter (Table 5, Fig. 6). This could be explained by the fact that the treatment left residual primary and secondary SA in postoperative eyes that still affected depth of focus. Like the curves of WFE and VSOTF as a function of pupil diameter, the defocus curves of the 6

*n*profile were of different shape with a sharp dip around −2.25 D (3-mm pupil diameter) and −1 D (6-mm pupil diameter), most likely due to induction of negative Z

_{4}

^{0}and positive Z

_{6}

^{0}. Even though they were statistically significant, we believe the differences in depth of focus to be negligible because they were not noticeable for the near-vision–important 3-mm pupil diameter and they were small over a 6-mm diameter.

^{ 3,10,11 }Second, although the BCVSOTF metric was the most predictive in a recent study of post-LASIK patients,

^{ 29 }some patients may not notice an enhanced retinal image quality or do not show functional improvements

^{ 15,30–32 }as reported psychophysically for aspheric IOLs.

^{ 33 }

^{ 12 }This hypothesis was confirmed by the present study. When decentered, aspheric profiles induced less coma than the standard profile (0

*n*, Fig. 4), leading to a higher decentration tolerance (Table 4, Fig. 5) with all pupil diameters. Most likely because of Z

_{4}

^{0}overcorrection and higher Z

_{6}

^{0}induction, decentration tolerance of BCVSOTF at smaller pupil diameters was lower for the 6

*n*than for the 3

*n*profile (Fig. 5D). For BCVSOTF obtained over a 6-mm pupil diameter, minimum values of decentration tolerance were below 500 μm for the 0

*n*, 1

*n*, and 3

*n*profiles. Given the frequency of random microdecentrations of (≤500 μm)

^{ 20,21 }in uneventful LASIK, some eyes from our collective were at risk of experiencing a decrease of >0.2 log BCVSOTF units, even if no obvious complication occurred. Our results suggest that because of lower coma induction, aspheric profiles are likely to cause less image quality decline in cases of microdecentration, especially at larger pupil diameters.

^{ 17,24,26 }