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Optics Express

Optics Express

  • Vol. 16, Iss. 16 — Aug. 4, 2008
  • pp: 11808–11821
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Impact of ablation efficiency reduction on post-surgery corneal asphericity: simulation of the laser refractive surgery with a flying spot laser beam

Young Kwon, Myoung Choi, and Steven Bott  »View Author Affiliations


Optics Express, Vol. 16, Issue 16, pp. 11808-11821 (2008)
http://dx.doi.org/10.1364/OE.16.011808


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Abstract

We developed a rigorous simulation model to evaluate ablation algorithms and surgery outcomes in laser refractive surgery. The model (CASIM: Corneal Ablation SIMulator) simulates an entire surgical process,which includes calculating an ablation profile from measured wavefront errors, generating a shot pattern for a flying spot laser beam, simulation of the shot-by-shot ablation process based on a measured or modeled beam profile, and healing of the cornea after surgery. Using simulated post-surgery corneal shapes for various ablation parameters and beam fluences,we calculated angular dependence of ablation efficiency and the amount of increase in corneal asphericity. Without considering the effect of corneal healing, our result shows the following; 1) ablation efficiency reduction in the periphery depends on the peak fluence of the laser beam, 2) corneal asphericity increases even in the surgery using an ablation profile based on the exact Munnerlyn formula, contrary to previous reports, and 3) post-surgery corneal asphericity increases by a smaller amount in high fluence small Gaussian beam surgery than in low fluence truncated Gaussian beam.Our model can provide improved ablation profiles that compensate for the change of corneal asphericity and induction of spherical aberration in a flying spot laser system, resulting in better surgery outcomes in laser refractive surgeries.

© 2008 Optical Society of America

1. Introduction

Laser refractive surgery is an ophthalmic technique used to reshape the anterior corneal surface for the correction of refractive errors. In this technique, a laser beam is applied to the corneal surface for the ablation of tissue [1

1. R. R. Krueger and S. Trokel, “Quantitation of Corneal Ablation by Ultraviolet Laser Light,” Arch.Ophthalmol. 103, 1741–1742 (1985). [CrossRef] [PubMed]

]. Using a flying laser beam, better control over laser energy delivery at each corneal position is possible and therefore a greater demand to reproduce details accurately is put on the laser systems [2

2. F. Manns, J.-H. Shen, P. Söderberg, T. Matsui, and J.-M. Parel, “Development of an algorithm for corneal reshaping with a scanning laser beam,” Appl. Opt. 34, 4600–4608 (1995). [CrossRef] [PubMed]

]. To implement the wavefront based customized ablation, a surgical laser system must be capable of reproducing the details of complex wavefront-driven ablations while reducing the incidence of high-order aberrations after surgery [3

3. M. Mrochen, M. Kaemmerer, and T. Seiler, “Wavefront-guided Laser in situ Keratomileusis: Early Results in Three Eyes,” J. Refract. Surg. 16, 116–121 (2000). [PubMed]

, 4

4. E. Moreno-Barriuso, J. M. Lloves, S. Marcos, R. Navarro, L. Llorente, and S Barbero, “Ocular Aberrations before and after Myopic Corneal Refractive Surgery: LASIK-induced changes measured with Laser Ray Tracing,” IOVS 42, 1396–1403 (2001).

]. It is well known that a successful surgery depends on the correct design of an ablation profile, precise delivery of laser energy to the corneal position, and reliable understanding of the corneal tissue response.

When conventional refractive surgery is used to correct defocus and astigmatism, an increase of spherical aberration after surgery has been observed clinically. The surgery has a tendency to induce a positive spherical aberration after myopic treatment and amount of induction shows a strong correlation with the attempted dioptric correction [5

5. S. Marcos, S. Barbero, L. Llorente, and J. Merayo-Lloves, “Optical response to LASIK Surgery for Myopia from Total and Corneal Aberration Measurements,” IOVS 42, 3349–3356 (2001).

]. The induction of positive spherical aberration and associated increase of corneal asphericity are not yet fully understood.

A large number of factors can influence the laser ablation process and outcome. Among them, laser energy delivery technique [6

6. C. B. O'Donnell, J. Kemner, and Francis E. O'Donnell Jr., “Ablation smoothness as a function of excimer laser delivery system,” J. Cataract. Refract. Surg. 22, 682–685 (1996). [PubMed]

, 7

7. B. Muller, T. Boeck, and C. Hartmann, “Effect of excimer laser beam delivery and beam shaping on corneal sphericity in photorefractive keratectomy,” J. Cataract. Refract. Surg. 30, 464–470 (2004). [CrossRef] [PubMed]

], ablation decentration and registration [8

8. M. Mrochen, M. Kaemmerer, P. Mierdel, and T. Seiler, “Increased higher-order optical aberrations after laser refractive surgery: A problem of subclinical decentration,” J. Cataract. Refract. Surg. 27, 362–369 (2001). [CrossRef] [PubMed]

, 9

9. M. Mrochen, R. R. Krueger, M. Bueeler, and T. Seiler, “Aberration-sensing and Wavefront-guided Laser in situ Keratomileusis: Management of Decentered Ablation,” J. Refract. Surg. 18, 418–429 (2002). [PubMed]

], eye tracking [10

10. N. M. Taylor, R. H. Eikelboom, P. P. v. Sarloos, and P. G. Reid, “Determining the accuracy of an Eye Tracking System for Laser Refractive Surgery,” J. Refract. Surg. 16, S643–S646 (2000). [PubMed]

, 11

11. M. Bueeler, M. Mrochen, and T. Seiler, “Effect of spot size, ablation depth, and eye-tracker latency on the optical outcome of corneal laser surgery with a scanning spot laser,“ in Ophthalmic Technologies XIII (SPIE, 2003), pp.150–160.

], flap [12

12. D. Zadok, C. Carrillo, F. Missiroli, S. Litwak, N. Robledo, and A. S. Chayet, “The Effect of Corneal Flap on Optical Aberrations,” Am. J. Ophthalmol. 138, 190–193 (2004). [CrossRef] [PubMed]

], physical characteristics of ablation [13-19

13. M. Mrochen and T. Seiler, “Influence of Corneal Curvature on Calculation of Ablation Patterns used in Photorefractive Laser Surgery,” J. Refract. Surg. 17, S584–S587 (2001). [PubMed]

], wound-healing and biomechanics of the cornea [20-23

20. C. Roberts, “Biomechanics of the Cornea and Wavefront guided Laser Refractive Surgery,” J. Refract. Surg. 18, S589–S592 (2002). [PubMed]

] have been explored to predict or explain the clinically observed discrepancy between intended and actual outcomes. The quantification of influence of these factors is important for providing the optimal outcome with wavefront-based customized refractive surgeries.

The main purpose of this paper is to explore details of surgery models incorporating most of the factors mentioned above. Most of the studies on computational modeling of refractive surgeries assume a standard ablation profile based on the exact Munnerlyn formula [24

24. C. R. Munnerlyn, S. J. Koons, and J. Marshall, “Photorefractive keratectomy: A technique for laser refractive surgery,” J. Cataract. Refract. Surg. 14, 46–52 (1988). [PubMed]

] or parabolic approximation to the Munnerlyn formula for conventional refractive surgery. For wavefront based customized ablation, an ablation profile must be calculated from wavefront data to correct defocus and astigmatism, as well as higher order aberrations. With a flying spot laser, the resulting ablation profile must be deconvolved into a series of shot positions, often requiring more than 10,000 shots for the surgery [25

25. R. W. Frey, J. H. Burkhalter, and G. P. Gray, “Laser Sculpting System,” (2001), US Patent #6,261,220.

]. Alcon’s proprietary shot pattern algorithm is used for this deconvolution. For a comprehensive simulation model, the contribution of the progressive nature of ablation to the final surgery outcome is included, by using Alcon Laboratories’ proprietary shot sequence algorithm and we show how the laser fluence distribution and beam size can substantially alter the resulting corneal shape.

Although the Alcon shot pattern and shot sequence algorithms [33

33. R. W. Frey, J. H. Burkhalter, and G. P. Gray, “Laser Sculpting Method and System,” US Patent 5,849,006 (1998).

] are used in the model,the results generated by the model would be extremely similar if other, generic algorithms were used in place of Alcon’s proprietary algorithms, provided that the generic algorithms were: 1> accurate in reproducing the ablation profile shape with the shot pattern, 2> removed the minimum possible tissue consistent with reproducing the ablation profile, and 3> distributed the shots in a sequence which removes the tissue smoothly and progressively across the whole cornea over the course of the ablation. Therefore the results reported here are believed to be general and not restricted to the proprietary shot pattern or shot sequence algorithms used in this work.

Next, we use our model to explain the causes of the changes in corneal asphericity produced by refractive surgery. Analysis by Gatinel et al. [26

26. D. Gatinel, T. Hoang-Xuan, and D. T. Azar, “Determination of Corneal Asphericity after Myopia Surgery with the Excimer Laser: A Mathematical Model,” IOVS 42, 1736–1742 (2001).

] shows that the Munnerlyn profile should not increase the asphericity of corneas with typical preoperative asphericities,while Jimenez et al. [27

27. J. R. Jiménez, R. G. Anera, and L. J. d. Barco, “Equation for Corneal Asphericity After Corneal Refractive Surgery,” J. Refract. Surg. 19, 65–69 (2003). [PubMed]

] explain that a profile with parabolic approximation must increase corneal asphericity, which is consistent with clinical findings. Marcos et al. report that the increase in corneal asphericity is not due to an inappropriate design of ablation profiles [28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

] and wound healing and biomechanics of the cornea may play an important role. Our new results will be important both in the optimization of laser system parameters for refractive surgeries and pre-adjusting an ablation profile for better surgery outcomes.

2. Method

2.1 Interaction of the laser beam at oblique incidence

The interaction of 193 nm excimer laser radiation and corneal tissue is a complex process,involving both ultraviolet photochemistry and rapid thermal decomposition [29

29. A. Vogel and V. Venugopalan, “Mechanisms of Pulsed Laser Ablation of Biological Tissues,” Chem. Rev. 103, 577–644 (2003). [CrossRef] [PubMed]

]. With the flying spot laser system, the corneal ablation behavior is mainly governed by the relationship between the per-pulse tissue ablation depth and the fluence (energy per illuminated area) of the incident laser radiation. Over a broad fluence range at normal incidence, organic material typically exhibits an ablation behavior described by Lambert-Beer’s law [30

30. T. F. Deutsch and M. W. Geis, “Self-developing UV photoresist using excimer laser exposure,” J. Appl.Phys. 54 (12), December 1983 54, 7201–7204 (1983). [CrossRef]

]:

d=1αlnFrFTH,ifFr>FTH,
=0,ifFrFTH,
(1)

where d is the ablation depth, α is the absorption coefficient in the material at the laser wavelength,Fr is the fluence at position r, and FTH is the ablation threshold fluence. Establishing precise values for the cornea is challenging, but based on the collected ablation data from many published studies, typical values are α=2.9 μm-1 and FTH=40 ~ 60 mJ/cm2.It may be noted that our choice of absorption coefficient can only be correct in average-sense,as recent dynamic ablation model development has shown that ablation rate can be more accurately represented by a dynamic model with varying absorption coefficient and local water content in the ablated tissue during the time course of ablation pulse [31

31. B. Fisher and D. Hahn, “Development and Numnerical Solution of a Mechanistic Model for Corneal Tissue Ablation with the 193-nm Argon Fluoride Excimer Laser,” J. Opt. Soc. Am. A 24, 265–277 (2007). [CrossRef]

].

Fig. 1. Schematic representation of a laser beam at oblique incidence on the surface of the cornea. (A) Laser beam with fluence F incident on the corneal at angle Θ. The radius of curvature of the cornea is RC. (B) Laser beam incidents and splits into a propagated/absorbed beam inside the cornea and a reflected beam. The reflectance and absorption vary with angle. (C) Progressive nature of surgery is shown. The radius of curvature of the cornea varies from RI to RF in myopic surgery as ablation progresses. Each laser shot is expected to remove the tissue of thickness d. Shown also is a complex permittivity of cornea. The thickness d measured along z varies over incidence angle for a given fluence of laser beam.

Many studies have shown that the efficiency of the laser changes across the cornea [13

13. M. Mrochen and T. Seiler, “Influence of Corneal Curvature on Calculation of Ablation Patterns used in Photorefractive Laser Surgery,” J. Refract. Surg. 17, S584–S587 (2001). [PubMed]

, 15

15. J. R. Jimenez, R. G. Anera, L. J. d. Barco, and E. Hita, “Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea,” Appl. Phys. Lett. 81, 1521–1523 (2002). [CrossRef]

], primarily because of the enlargement of the laser spot as it moves away from the corneal apex and because of differences in reflected/absorbed energy as a function of the angle of incidence. The angle Θ of the laser beam at a distance y from the apex of the cornea with radius of curvature Rc is (Fig. 1)

sinΘ=yRC.
(2)

The aspheric corneal surface has a sagittal depth z given by

z=y2RC1+1(1+Q)y2RC2,
(3)

where Q is the asphericity of the meridian section of the corneal surface. The fluence delivered at the same location, Fr, is approximated by the following equation:

Fr=FcosΘ.
(4)

The reason that Eq. (4) holds true is as follows: the intensity of a laser beam varies with cos2Θ while the illuminated area under the laser beam is enlarged by 1/cosΘ, with the resulting fluence varying ‘approximately’ with cosΘ. The relationship holds exactly with a spherical shape, but also holds ‘approximately’ for aspheric shapes when a large beam is employed. In the real simulation, we calculate the local incidence angle for each shot to minimize the error in the estimation of local fluence.

Another factor that affects the absorption of the laser beam in the cornea is the incidence angle dependence of the absorption coefficient. For the beam absorbed in the cornea, the transmitted field inside the cornea would experience an angle-dependent absorption [32

32. S. J. Orfanidis, Electromagnetic Waves & Antennas (2004), http://www.ece.rutgers.edu/~orfanidi/ewa/.

]:

αZ=[DR2+DI2DR2]12,
(5)

where DR and DI are given by:

DR=ϖ2μ0(εRsin2Θ),
DI=ϖ2μ0εI,
(6)

where ω is angular frequency of the laser, μ 0 is permeability of vacuum, εR and εI are the real and imaginary parts of permittivity of corneal tissue at the laser wavelength (εC=ε R-I),respectively and can be calculated from complex refractive index of cornea, nC=1.52-j0.04. It is well known that reflectance also varies with incidence angle. So, the final expression of the ablation depth Eq. (1) becomes:

d=12αZln[FFTHcosΘ(1RREFL)],
(7)

where RREFL is reflectance at the beam position on the cornea and αZ is the angle-dependent absorption coefficient. We note that the factor of two in Eq. (1) is required to take into account laser beam intensity instead of field amplitude. With the use of Eq. (7), we can calculate the individual ablated depth profile associated with each shot on the cornea.

2.2 Ablation profiles and shot pattern generation

D=0.376(1R'1R),
(8)

where R’ and R are the radii of the curvature of the post-surgery and pre-surgery cornea,respectively. We assume that the refractive index of the cornea is 1.376. Then, the Munnerlyn formula for the ablation profile, zM(r), is given by [24

24. C. R. Munnerlyn, S. J. Koons, and J. Marshall, “Photorefractive keratectomy: A technique for laser refractive surgery,” J. Cataract. Refract. Surg. 14, 46–52 (1988). [PubMed]

]:

ZM(r)=R2r2R'2r2+R'2OZ24R2OZ24,
(9)

where OZ is the ablation optical zone diameter and r is the radial distance from the apex of the cornea. The parabolic approximation of the Munnerlyn formula, zP(r), is given by [16

16. R. G. Anera, J. R. Jiménez, L. J. d. Barco, and E. Hita, “Changes in corneal asphericity after laser refractive surgery, including reflection losses and nonnormal incidence upon the anterior cornea,” Opt. Lett. 28, 417–419 (2003). [CrossRef] [PubMed]

]:

ZP(r)=4Dr23DOZ23.
(10)

An Alcon Laboratories’ proprietary algorithm [33

33. R. W. Frey, J. H. Burkhalter, and G. P. Gray, “Laser Sculpting Method and System,” US Patent 5,849,006 (1998).

] for beam shot pattern generation is summarized here. After calculation of the ablating profile in accordance with the treatment of a specified eye condition, a plurality of laser beam shots of uniform energy and fluence distribution are first selected to form a shot pattern of uniform shot density and therefore depth. Each shot removes a known amount of tissue volume, called volume per shot (VPS) and is found by integrating the depth profile from Eq. (7) over the entire ablated area:

VPS=0d2πrdr.
(11)

Fig. 2. Laser beam shot pattern and profiles on the surface of the cornea used in the simulation: (a) Shot pattern for the correction of -3D myopic eye based on Munnerlyn formula. Number of shots in the pattern is 1,417. (b) Gaussian beam with 0.4 mm radius and truncated Gaussian beam with 1 mm radius. For the sake of comparison, we used the same peak fluence of 120mJ/cm2 for the plot.

Table 1 lists the types of laser beam and shot patterns used in the simulations as explained below. The use of a small size Gaussian beam is typical in flying spot laser systems. We include the case of a truncated Gaussian beam to allow a comparison with the experimental data reported by Marcos et al. [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

, 34

34. C. Dorronsoro, J. Merayo-Lloves, and S. Marcos, “An Experimental Correction Factor of Radial Laser Efficiency Losses in Corneal Refractive Surgery,” IOVS 47 E-Abstract 3611 (2006).

, 35

35. S. Marcos, “Spherical Aberration: Biomechanics or Physical Laser Effects?,“ presented in Wavefront Congress 2006 Meeting (Nassau, Bahamas. January 06, 2006).

]. For truncated Gaussian laser beams, we picked two VPS values corresponding to different ablation threshold values, since the ablation threshold influences the ablation process significantly with a low-fluence laser beam. If two ablation threshold values were used with the exact same shot pattern, the result would be an under- or overcorrection with one of the threshold values.. Therefore the shot pattern algorithm adjusts the shot patterns corresponding to the two ablation thresholds to target the same intended correction, so that the impact of different thresholds on the surgery outcome could be accurately studied. For myopic eyes, we picked two patterns; the Munnerlyn and the parabolic approximation. For the Munnerlyn profile, the post-surgery radius of curvature R=7.8 mm was used. We will use these laser shot patterns to calculate the asphericity changes in post-surgery eyes and compare our results with reported data in the literature [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

, 28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

].

Table 1. Surgeries and shot patterns used for the simulations. For each sphere correction, we include the number of shots for each of the profiles calculated by the Munnerlyn formula (M) and the parabolic approximation formula (P).In all patterns, a 6 mm optical zone (OZ) size is used without transition zone

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3. Results and discussion

3.1 Ablation efficiency reduction at oblique incidence

Figure 3 summarizes the simulation results on the ablation efficiency reduction as a function of beam profile. We used a Gaussian beam of radius 0.4mm and a truncated Gaussian beam of radius 1mm, with the pre-surgery radius of corneal curvature, R, equal to 8 mm for direct comparison to Marcos’ empirically-determined correction factor [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

, 34

34. C. Dorronsoro, J. Merayo-Lloves, and S. Marcos, “An Experimental Correction Factor of Radial Laser Efficiency Losses in Corneal Refractive Surgery,” IOVS 47 E-Abstract 3611 (2006).

]. Figure 3 shows efficiency reduction in the 12 D and 6 D correction patterns using profiles based on the Munnerlyn formula. The profile based on the parabolic approximation formula, not shown,provided a similar result, confirming that the ablation profile itself is not the source of ablation efficiency reduction.

Fig. 3. Ablation efficiency reduction with Gaussian and truncated Gaussian beams. (a) 12 D correction. (b) 6 D correction. Gaussian beam with 0.4 mm size and Truncated Gaussian beam with 2 mm size are used for the simulation. The reduction is calculated from pre-healed corneal shape. For the comparison with results in the literature, we used R=8mm for the pre-surgery radius of curvature.

We observe that the efficiency reduction is slightly smaller in the 12 D correction than in the 6 D correction for all three beam types. This is expected since the post-surgery surface is flatter with higher corrections, resulting in smaller reduction in the periphery. For the 3D case (not shown here), the difference in efficiency reduction 2 mm from the apex is less than 1% for both Gaussian beam and truncated Gaussian beam with a low threshold fluence shot pattern, but over 1% with a truncated Gaussian beam with a high threshold fluence shot pattern. This indicates that one scheme for the aberration compensation may be difficult to cover a wide range of myopic corrections using a single correction profile. Also we observe that the efficiency reduction for small Gaussian beams is smaller than the reduction for wide truncated Gaussian beams, indicating that induced change in the corneal asphericity and spherical aberration are smaller in laser surgery with a small Gaussian beam.

Marcos et al. showed that the ablation efficiency reduction on the cornea 2 mm from the apex with a 12D correction is 0.04 [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

]. We found the same reduction with truncated Gaussian beams with a VPS of 832×10-6mm3, (FPK=540 mJ/cm2). This pattern has been generated assuming the ablation threshold is 40mJ/cm2. However, if we use a VPS of 383×10-6mm3 shot pattern (FPK=120 mJ/cm2), based on the ablation threshold of 60mJ/cm2, the reduction is larger as shown in the graph and this confirms that efficiency reduction largely depends on the ratio of FPK/FTH[13

13. M. Mrochen and T. Seiler, “Influence of Corneal Curvature on Calculation of Ablation Patterns used in Photorefractive Laser Surgery,” J. Refract. Surg. 17, S584–S587 (2001). [PubMed]

]. Marcos et al., showed that if they use 60 mJ/cm2 for the calculation of post-surgery corneal asphericity, the asphericity calculated by using the experimental correction factor came closer to the clinically observed value. Our results support their findings.

3.2 Impact of ablation efficiency reduction on post-surgery corneal asphericity

The impact of ablation efficiency reduction on the post-surgery corneal asphericity over different beam shapes and ablation profiles for the corrections of 3 D to 12 D is shown in Fig.4. As noted by Gatinel et al. [26

26. D. Gatinel, T. Hoang-Xuan, and D. T. Azar, “Determination of Corneal Asphericity after Myopia Surgery with the Excimer Laser: A Mathematical Model,” IOVS 42, 1736–1742 (2001).

], based on theoretical modeling of post-surgery corneal shape without considering ablation efficiency reduction, the asphericity increases slowly with the amount of correction for corneas with positive pre-surgery asphericity and decreases slowly for those with negative pre-surgery asphericity. The post-surgery asphericity does not change as a function of the amount of attempted corrections for zero asphericity. Figure 4(a) shows all of the above findings in the case of ablation profiles based on the Munnerlyn formula. In contrast, Jimenez et al. [36

36. J. R. Jimenez, R. G. Anera, J. A. D?az, and F. Perez-Ocon, “Corneal asphericity after refractive surgery when the Munnerlyn formula is applied,” J. Opt. Soc. Am. A 21, 98–103 (2004). [CrossRef]

], based on a mathematical model for post-surgery corneal asphericity without considering ablation efficiency reduction, showed that in the case of a profile based on the parabolic approximation of the Munnerlyn formula, asphericity increased with the amount of correction, regardless of initial asphericity. We found the same result as Jimenez et al. as summarized in Figs. 4(c) and 4(d), where Fig. 4(c) summarizes the post-surgery corneal asphericities using profiles based on parabolic approximations without efficiency reduction and Fig. 4(d) with efficiency reduction.

Many clinical studies show that the post-surgery corneal asphericity increases for medium and high myopia corrections even when profiles based on the Munnerlyn formula are used.For some lasers, errors due to the use of the parabolic approximation have been suspected to be the source of the increase in asphericity. However, we find that when we take into account the ablation efficiency reduction, the post-surgery corneal asphericity increases substantially even using the ablation profile based on the exact Munnerlyn formula. Figure 4(b) shows that even using profiles based on the exact Munnerlyn formula, the post-surgery asphericity increases with the amount of attempted corrections. Thus, we conclude that the efficiency reduction is a primary source of the change in asphericity observed clinically; the type of formula upon which ablation profile is based plays only a secondary role.

Figure 4 shows modeled results of post-surgery corneal asphericity with and without ablation efficiency included for various combinations of: 1> pre-surgery corneal asphericity, 2> excimer beam type (Gaussian and two truncated Gaussians with ablation threshold fluences of 40 and 60mJ/cm2 respectively, 3> ablation profile (exact Munnerlyn or parabolic approximation) for various magnitudes of attempted correction. When efficiency reduction is ignored, Figs. 4(a) and 4(c) show a very small difference in post-surgery asphericity between the three beam types. However, with efficiency reduction included, Figs. 4(b) and 4(d) clearly show different asphericities among three laser beam types. The largest increase of asphericity is observed with the low fluence truncated Gaussian beam at high threshold fluence and the smallest increase is with the high fluence Gaussian beam for a given correction and a given pre-surgery asphericity. Thus we conclude that the impact of ablation efficiency reduction on the change in asphericity is less significant with a high fluence Gaussian beam than a low fluence truncated Gaussian beam.

It is apparent from Fig. 4 that post-surgery corneal asphericity increases regardless of which ablation profile (exact Munnerlyn or parabolic approximation) is used, for all values of pre-surgery asphericity when ablation efficiency reduction is included in the model. However,when ablation profiles are based on the exact Munnerlyn formula, the pre-to-post asphericity change is much smaller with high fluence Gaussian beams than with truncated Gaussian beams. High fluence Gaussian beams also induce a smaller change in asphericity with the Munnerlyn profile than one based on the parabolic approximation, regardless of attempted correction. It is also observed that with ablation profiles based on the parabolic approximation, the post-surgery asphericity calculated by Jimenez et al. [36

36. J. R. Jimenez, R. G. Anera, J. A. D?az, and F. Perez-Ocon, “Corneal asphericity after refractive surgery when the Munnerlyn formula is applied,” J. Opt. Soc. Am. A 21, 98–103 (2004). [CrossRef]

] is closer to our results with a high fluence Gaussian beam. When using truncated Gaussian beams, the Jimenez calculation provides smaller estimated asphericities.

Fig. 4. Corneal asphericity calculated for different magnitudes of refractive correction with three initial asphericity values. Gaussian beam and truncated Gaussian beams are used (see the text). Ablation profiles in (a-b) are based on the Munnerlyn formula and (c-d) are based on a parabolic approximation. Initial radius of curvature R=7.8 and initial asphericity of cornea p=0.7, 1.0, and 1.3, respectively, are used for the simulation. (a) and (c) show results without ablation efficiency reduction. (b) and (d) show results with ablation efficiency reduction. The fitting zone size for asphericity is 4.5mm. The Q values shown in the left of each plot are pre-surgery Q values.

3.3 Post-surgery corneal asphericity: comparison of simulation and clinical data

Figure 5, below, includes digitized data taken from Fig. 2(a),in Marcos et al. [28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

] and Fig.8 in Dorronsoro et al. [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

]. The Marcos et al.’s data are clinical results which include attempted refractive correction and pre-surgery and post-surgery corneal asphericity values.The clinical data was obtained from surgeries conducted using scanning spot laser (Chiron Technolas 217-C; Bausch & Lomb). The Dorronsoro et al.’s data are calculated predictions of post-surgery corneal asphericity based on the attempted refractive correction and corneal asphericity data from the patients included in the Marcos et al. study. The Dorronsoro’s predictions are based on applying the exact Munnerlyn equation, adjusted by an empirical ablation efficiency correction factor based on PMMA (polymethylmethacrylate) ablations of flat and spherical PMMA. The empirically measured PMMA ablation efficiency correction factor is adjusted for corneal tissue with the assumption of an ablation fluence threshold of 60mJ/cm2 for corneal tissue.

Using attempted refraction and pre-surgery asphericities from Marcos et al., CASIM was used to calculate, using exact Munnerlyn ablation profiles, the post surgery corneal asphericities to compare to the clinical results of Marcos et al. A truncated Gaussian beam was used in the simulation to approximate the laser used by the Technolas. In all eyes, an initial corneal radius curvature of 7.8 mm was used for the CASIM calculations and the post surgery corneal asphericity was computed over the center 5.5mm, the midpoint of the optical zone values of 4.4mm ~ 7mm used in the Marcos et al. study. Marcos et al. calculated the clinical data by fitting a biconic surface to the corneal topography data within each individual optical zone of 4.4mm ~ 7mm [28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

].

In Fig. 5, below, the data indicated by the star symbols is taken from Fig. 2(a) in reference [28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

]. Also graphed with diamond symbols are the Dorronsoro et al., [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

] predictions for post-surgery corneal asphericity calculated with threshold fluence of 60 mJ/cm2 after applying an experimentally derived correction factor. In Fig. 8 in [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

], Dorronsoro et al., showed two more predictions for post-surgery corneal asphericity, one calculated with lower threshold fluence of 40 mJ/cm2 and another one using an efficiency reduction factor proposed by Jimenez et al [15

15. J. R. Jimenez, R. G. Anera, L. J. d. Barco, and E. Hita, “Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea,” Appl. Phys. Lett. 81, 1521–1523 (2002). [CrossRef]

]. We did not reproduce the two predictions in Fig. 5, because one calculated with lower threshold fluence (40mJ/cm2) did predict smaller change for post-surgery asphericity than one with higher threshold fluence (60 mJ/cm2). Also, as discussed in [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

, 28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

],there was a large discrepancy between the clinically observed post-surgery corneal asphericity and the calculated asphericity after applying the ablation correction factor proposed by Jimenez et al [27

27. J. R. Jiménez, R. G. Anera, and L. J. d. Barco, “Equation for Corneal Asphericity After Corneal Refractive Surgery,” J. Refract. Surg. 19, 65–69 (2003). [PubMed]

]. It should be noted that the correction factor proposed by Jimenez et al. was obtained for an ablation profile based on the parabolic approximation.

The Spearman correlation coefficient for simulated asphericity with the high threshold truncated Gaussian beam and clinical data is R2=0.96. The result shown here verifies that CASIM predictions of corneal asphericity using a rigorous model can be compared to clinical observations for real patients.

Fig. 5. Corneal asphericity from the simulations using truncated Gaussian beams, post-surgery without healing. All profiles are based on the Munnerlyn formula. The number of corrections and clinical data are obtained by digitizing Fig. 2(a) in reference [28] and Fig. 8 in reference [19]. Pre-surgery curvature of radius R=7.8 and clinically measured pre-surgery asphericity with individual corrections (obtained from 8 Fig. 8 in reference [19]) are used for the simulation.Also shown are the data points after Ka corrected by Marcos et al. and clinical data reported by Marcos et al.

4. Conclusions

A rigorous simulation model (CASIM: Corneal Ablation SIMulator) has been used to evaluate the impact of ablation parameters on the post-surgery corneal shape. The CASIM simulates the entire surgical process, based on generally accepted assumptions of the ablation physics. It incorporates an efficient, generalized shot pattern for a flying spot laser, calculates the shot-by-shot removal of corneal tissue by the excimer via the “blow off” model and allows selection of excimer beam characteristics and corneal tissue ablation parameters to assess the impact of these choices on the post-surgical corneal shape. In this paper, CASIM is used to calculate ablation efficiency reduction due the effects of corneal curvature and compare these simulations to experimental data, reported by Marcos et al., [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

]. The agreement between the CASIM simulations and the experimental data demonstrates that the ablation efficiency reduction phenomenon can be explained by the dependence of the volume of tissue ablated by an excimer pulse on the angle of incidence of the pulse onto the cornea.

Through exploration of the impact of pre-surgery corneal shape, laser beam fluence profiles and ablation profile (exact Munnerlyn or parabolic approximation) on the pre-to-post corneal asphericity changes, the CASIM modeling confirms that the ablation efficiency reduction in the periphery of cornea increases corneal asphericity even when the exact Munnerlyn ablation profile is used, regardless of pre-surgery corneal asphericity or laser beam type. The amount of change depends on the amount of refractive correction, type of laser beam and the type of ablation profile. The CASIM modeling also demonstrates that post-surgery corneal asphericity increases more with a low fluence truncated Gaussian beam than with a high fluence Gaussian beam. This finding explains why with many commercial laser platforms, believed to employ the exact Munnerlyn ablation profiles, large asphericity increases are reported in the post-surgery eyes [14

14. P. S. Hersh, K. Fry, and J. W. Blaker, “Spherical aberration after laser in situ keratomileusis and photorefractive keratectomy Clinical results and theoretical models of etiology,” J. Cataract. Refract. Surg. 29, 2096–2104 (2003). [CrossRef] [PubMed]

, 28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

].

Finally, the CASIM model, used with attempted refractive correction and pre-surgical asphericity from clinical patient data presented by Marcos et al. [28

28. S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

], more accurately predicts post-surgical corneal asphericity than the semi-empirical model which used plastic ablation measurements to estimate the ablation efficiency reduction at the periphery of the cornea (Dorronsoro et al. [19

19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

]). The remaining differences variance between CASIM predictions and post-surgical clinical measurements may be explainable by corneal remodeling or wound healing effects.

CASIM is believed to have advantages over other corneal ablation models because the model is built with minimal assumptions. The model has the flexibility to permit simulations of most experimental parameters, ablation profile shape, laser characteristics and “blow off” model parameters for ablation physics and to predict the impact of each of these variables on the post surgical corneal shape. Also built into the CASIM is a corneal healing model which is explained in [38

38. Y. Kwon and S. Bott, “Post-surgery asphericity and spherical aberration due to ablation efficiency reduction and corneal remodeling in refractive surgeries,” in prep. (2008).

]. CASIM may be used in the future to plan more efficient shot patterns to reduce the induction of spherical aberration by laser refractive surgery.

Acknowledgments

This effort was supported by Alcon.

References and links

1.

R. R. Krueger and S. Trokel, “Quantitation of Corneal Ablation by Ultraviolet Laser Light,” Arch.Ophthalmol. 103, 1741–1742 (1985). [CrossRef] [PubMed]

2.

F. Manns, J.-H. Shen, P. Söderberg, T. Matsui, and J.-M. Parel, “Development of an algorithm for corneal reshaping with a scanning laser beam,” Appl. Opt. 34, 4600–4608 (1995). [CrossRef] [PubMed]

3.

M. Mrochen, M. Kaemmerer, and T. Seiler, “Wavefront-guided Laser in situ Keratomileusis: Early Results in Three Eyes,” J. Refract. Surg. 16, 116–121 (2000). [PubMed]

4.

E. Moreno-Barriuso, J. M. Lloves, S. Marcos, R. Navarro, L. Llorente, and S Barbero, “Ocular Aberrations before and after Myopic Corneal Refractive Surgery: LASIK-induced changes measured with Laser Ray Tracing,” IOVS 42, 1396–1403 (2001).

5.

S. Marcos, S. Barbero, L. Llorente, and J. Merayo-Lloves, “Optical response to LASIK Surgery for Myopia from Total and Corneal Aberration Measurements,” IOVS 42, 3349–3356 (2001).

6.

C. B. O'Donnell, J. Kemner, and Francis E. O'Donnell Jr., “Ablation smoothness as a function of excimer laser delivery system,” J. Cataract. Refract. Surg. 22, 682–685 (1996). [PubMed]

7.

B. Muller, T. Boeck, and C. Hartmann, “Effect of excimer laser beam delivery and beam shaping on corneal sphericity in photorefractive keratectomy,” J. Cataract. Refract. Surg. 30, 464–470 (2004). [CrossRef] [PubMed]

8.

M. Mrochen, M. Kaemmerer, P. Mierdel, and T. Seiler, “Increased higher-order optical aberrations after laser refractive surgery: A problem of subclinical decentration,” J. Cataract. Refract. Surg. 27, 362–369 (2001). [CrossRef] [PubMed]

9.

M. Mrochen, R. R. Krueger, M. Bueeler, and T. Seiler, “Aberration-sensing and Wavefront-guided Laser in situ Keratomileusis: Management of Decentered Ablation,” J. Refract. Surg. 18, 418–429 (2002). [PubMed]

10.

N. M. Taylor, R. H. Eikelboom, P. P. v. Sarloos, and P. G. Reid, “Determining the accuracy of an Eye Tracking System for Laser Refractive Surgery,” J. Refract. Surg. 16, S643–S646 (2000). [PubMed]

11.

M. Bueeler, M. Mrochen, and T. Seiler, “Effect of spot size, ablation depth, and eye-tracker latency on the optical outcome of corneal laser surgery with a scanning spot laser,“ in Ophthalmic Technologies XIII (SPIE, 2003), pp.150–160.

12.

D. Zadok, C. Carrillo, F. Missiroli, S. Litwak, N. Robledo, and A. S. Chayet, “The Effect of Corneal Flap on Optical Aberrations,” Am. J. Ophthalmol. 138, 190–193 (2004). [CrossRef] [PubMed]

13.

M. Mrochen and T. Seiler, “Influence of Corneal Curvature on Calculation of Ablation Patterns used in Photorefractive Laser Surgery,” J. Refract. Surg. 17, S584–S587 (2001). [PubMed]

14.

P. S. Hersh, K. Fry, and J. W. Blaker, “Spherical aberration after laser in situ keratomileusis and photorefractive keratectomy Clinical results and theoretical models of etiology,” J. Cataract. Refract. Surg. 29, 2096–2104 (2003). [CrossRef] [PubMed]

15.

J. R. Jimenez, R. G. Anera, L. J. d. Barco, and E. Hita, “Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea,” Appl. Phys. Lett. 81, 1521–1523 (2002). [CrossRef]

16.

R. G. Anera, J. R. Jiménez, L. J. d. Barco, and E. Hita, “Changes in corneal asphericity after laser refractive surgery, including reflection losses and nonnormal incidence upon the anterior cornea,” Opt. Lett. 28, 417–419 (2003). [CrossRef] [PubMed]

17.

D. Cano, S. Barbero, and S. Marcos, “Comparison of real and computer-simulated outcomes of LASIK refractive surgery,” J. Opt. Soc. Am. A 21, 926–936 (2004). [CrossRef]

18.

J. R. Jiménez, F. Rodríguez-Marín, R. G. Anera, and L. J. d. Barco, “Deviations of Lambert-Beer's law affect corneal refractive parameters after refractive surgery,” Opt. Express 14, 5411–5417 (2006). [CrossRef] [PubMed]

19.

C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, “Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape,” Opt. Express 14, 6142–6156 (2006). [CrossRef] [PubMed]

20.

C. Roberts, “Biomechanics of the Cornea and Wavefront guided Laser Refractive Surgery,” J. Refract. Surg. 18, S589–S592 (2002). [PubMed]

21.

D. Huang, M. Tang, and R. Shekhar, “Mathematical Model of Corneal Surface Smoothing after Laser Refractive Surgery,” Am. J. Ophthal. 135, 267–278 (2003). [CrossRef] [PubMed]

22.

C. Roberts, “Biomechanical customization: The next generation of laser refractive surgery,” J. Cataract.Refract. Surg. 31, 2–5 (2005). [CrossRef] [PubMed]

23.

G. Yoon, S. MacRae, D. R. Williams, and I. G. Cox, “Causes of spherical aberration induced by laser refractive surgery,” J. Cataract. Refract. Surg. 31, 127–135 (2005). [CrossRef] [PubMed]

24.

C. R. Munnerlyn, S. J. Koons, and J. Marshall, “Photorefractive keratectomy: A technique for laser refractive surgery,” J. Cataract. Refract. Surg. 14, 46–52 (1988). [PubMed]

25.

R. W. Frey, J. H. Burkhalter, and G. P. Gray, “Laser Sculpting System,” (2001), US Patent #6,261,220.

26.

D. Gatinel, T. Hoang-Xuan, and D. T. Azar, “Determination of Corneal Asphericity after Myopia Surgery with the Excimer Laser: A Mathematical Model,” IOVS 42, 1736–1742 (2001).

27.

J. R. Jiménez, R. G. Anera, and L. J. d. Barco, “Equation for Corneal Asphericity After Corneal Refractive Surgery,” J. Refract. Surg. 19, 65–69 (2003). [PubMed]

28.

S. Marcos, D. Cano, and S. Barbero, “Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm,” J. Refract. Surg. 19, S592–S596 (2003). [PubMed]

29.

A. Vogel and V. Venugopalan, “Mechanisms of Pulsed Laser Ablation of Biological Tissues,” Chem. Rev. 103, 577–644 (2003). [CrossRef] [PubMed]

30.

T. F. Deutsch and M. W. Geis, “Self-developing UV photoresist using excimer laser exposure,” J. Appl.Phys. 54 (12), December 1983 54, 7201–7204 (1983). [CrossRef]

31.

B. Fisher and D. Hahn, “Development and Numnerical Solution of a Mechanistic Model for Corneal Tissue Ablation with the 193-nm Argon Fluoride Excimer Laser,” J. Opt. Soc. Am. A 24, 265–277 (2007). [CrossRef]

32.

S. J. Orfanidis, Electromagnetic Waves & Antennas (2004), http://www.ece.rutgers.edu/~orfanidi/ewa/.

33.

R. W. Frey, J. H. Burkhalter, and G. P. Gray, “Laser Sculpting Method and System,” US Patent 5,849,006 (1998).

34.

C. Dorronsoro, J. Merayo-Lloves, and S. Marcos, “An Experimental Correction Factor of Radial Laser Efficiency Losses in Corneal Refractive Surgery,” IOVS 47 E-Abstract 3611 (2006).

35.

S. Marcos, “Spherical Aberration: Biomechanics or Physical Laser Effects?,“ presented in Wavefront Congress 2006 Meeting (Nassau, Bahamas. January 06, 2006).

36.

J. R. Jimenez, R. G. Anera, J. A. D?az, and F. Perez-Ocon, “Corneal asphericity after refractive surgery when the Munnerlyn formula is applied,” J. Opt. Soc. Am. A 21, 98–103 (2004). [CrossRef]

37.

R. G. Anera, J. R. Jimenez, L. J. d. Barco, J. Bermudez, and E. Hita, “Changes in corneal asphericity after laser in situ keratomileusis,” J. Cataract. Refract. Surg. 29, 762–768 (2003). [CrossRef] [PubMed]

38.

Y. Kwon and S. Bott, “Post-surgery asphericity and spherical aberration due to ablation efficiency reduction and corneal remodeling in refractive surgeries,” in prep. (2008).

OCIS Codes
(170.1020) Medical optics and biotechnology : Ablation of tissue
(170.3890) Medical optics and biotechnology : Medical optics instrumentation
(170.4470) Medical optics and biotechnology : Ophthalmology
(220.1000) Optical design and fabrication : Aberration compensation
(330.4460) Vision, color, and visual optics : Ophthalmic optics and devices

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: April 28, 2008
Revised Manuscript: July 7, 2008
Manuscript Accepted: July 8, 2008
Published: July 23, 2008

Virtual Issues
Vol. 3, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Young Kwon, Myoung Choi, and Steven Bott, "Impact of ablation efficiency reduction on post-surgery corneal asphericity: simulation of the laser refractive surgery with a flying spot laser beam," Opt. Express 16, 11808-11821 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11808


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References

  1. R. R. Krueger and S. Trokel, "Quantitation of Corneal Ablation by Ultraviolet Laser Light," Arch. Ophthalmol. 103, 1741-1742 (1985). [CrossRef] [PubMed]
  2. F. Manns, J.-H. Shen, P. Söderberg, T. Matsui, and J.-M. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 34, 4600-4608 (1995). [CrossRef] [PubMed]
  3. M. Mrochen, M. Kaemmerer, and T. Seiler, "Wavefront-guided Laser in situ Keratomileusis: Early Results in Three Eyes," J. Refract. Surg. 16, 116-121 (2000). [PubMed]
  4. E. Moreno-Barriuso, J. M. Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, "Ocular Aberrations before and after Myopic Corneal Refractive Surgery: LASIK-induced changes measured with Laser Ray Tracing," IOVS 42, 1396-1403 (2001).
  5. S. Marcos, S. Barbero, L. Llorente, and J. Merayo-Lloves, "Optical response to LASIK Surgery for Myopia from Total and Corneal Aberration Measurements," IOVS 42, 3349-3356 (2001).
  6. C. B. O'Donnell, J. Kemner, and FrancisE. O'Donnell Jr., "Ablation smoothness as a function of excimer laser delivery system," J. Cataract. Refract. Surg. 22, 682-685 (1996). [PubMed]
  7. B. Muller, T. Boeck, and C. Hartmann, "Effect of excimer laser beam delivery and beam shaping on corneal sphericity in photorefractive keratectomy," J. Cataract. Refract. Surg. 30, 464-470 (2004). [CrossRef] [PubMed]
  8. M. Mrochen, M. Kaemmerer, P. Mierdel, and T. Seiler, "Increased higher-order optical aberrations after laser refractive surgery: A problem of subclinical decentration," J. Cataract. Refract. Surg. 27, 362-369 (2001). [CrossRef] [PubMed]
  9. M. Mrochen, R. R. Krueger, M. Bueeler, and T. Seiler, "Aberration-sensing and Wavefront-guided Laser in situ Keratomileusis: Management of Decentered Ablation," J. Refract. Surg. 18, 418-429 (2002). [PubMed]
  10. N. M. Taylor, R. H. Eikelboom, P. P. v. Sarloos, and P. G. Reid, "Determining the accuracy of an Eye Tracking System for Laser Refractive Surgery," J. Refract. Surg. 16, S643-S646 (2000). [PubMed]
  11. M. Bueeler, M. Mrochen, and T. Seiler, "Effect of spot size, ablation depth, and eye-tracker latency on the optical outcome of corneal laser surgery with a scanning spot laser," in Ophthalmic Technologies XIII (SPIE, 2003), pp. 150-160.
  12. D. Zadok, C. Carrillo, F. Missiroli, S. Litwak, N. Robledo, and A. S. Chayet, "The Effect of Corneal Flap on Optical Aberrations," Am. J. Ophthalmol. 138, 190-193 (2004). [CrossRef] [PubMed]
  13. M. Mrochen, and T. Seiler, "Influence of Corneal Curvature on Calculation of Ablation Patterns used in Photorefractive Laser Surgery," J. Refract. Surg. 17, S584-S587 (2001). [PubMed]
  14. P. S. Hersh, K. Fry, and J. W. Blaker, "Spherical aberration after laser in situ keratomileusis and photorefractive keratectomy Clinical results and theoretical models of etiology," J. Cataract. Refract. Surg. 29, 2096-2104 (2003). [CrossRef] [PubMed]
  15. J. R. Jimenez, R. G. Anera, L. J. d. Barco, and E. Hita, "Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea," Appl. Phys. Lett. 81, 1521-1523 (2002). [CrossRef]
  16. R. G. Anera, J. R. Jiménez, L. J. d. Barco, and E. Hita, "Changes in corneal asphericity after laser refractive surgery, including reflection losses and nonnormal incidence upon the anterior cornea," Opt. Lett. 28, 417-419 (2003). [CrossRef] [PubMed]
  17. D. Cano, S. Barbero, and S. Marcos, "Comparison of real and computer-simulated outcomes of LASIK refractive surgery," J. Opt. Soc. Am. A 21, 926-936 (2004). [CrossRef]
  18. J. R. Jiménez, F. Rodríguez-Marín, R. G. Anera, and L. J. -d. Barco, "Deviations of Lambert-Beer's law affect corneal refractive parameters after refractive surgery," Opt. Express 14, 5411-5417 (2006). [CrossRef] [PubMed]
  19. C. Dorronsoro, D. Cano, J. Merayo-Lloves, and S. Marcos, "Experiments on PMMA models to predict the impact of corneal refractive surgery on corneal shape," Opt. Express 14, 6142-6156 (2006). [CrossRef] [PubMed]
  20. C. Roberts, "Biomechanics of the Cornea and Wavefront guided Laser Refractive Surgery," J. Refract. Surg. 18, S589-S592 (2002). [PubMed]
  21. D. Huang, M. Tang, and R. Shekhar, "Mathematical Model of Corneal Surface Smoothing after Laser Refractive Surgery," Am. J. Ophthal. 135, 267-278 (2003). [CrossRef] [PubMed]
  22. C. Roberts, "Biomechanical customization: The next generation of laser refractive surgery," J. Cataract. Refract. Surg. 31, 2-5 (2005). [CrossRef] [PubMed]
  23. G. Yoon, S. MacRae, D. R. Williams, and I. G. Cox, "Causes of spherical aberration induced by laser refractive surgery," J. Cataract. Refract. Surg. 31, 127-135 (2005). [CrossRef] [PubMed]
  24. C. R. Munnerlyn, S. J. Koons, and J. Marshall, "Photorefractive keratectomy: A technique for laser refractive surgery," J. Cataract. Refract. Surg. 14, 46-52 (1988). [PubMed]
  25. R. W. Frey, J. H. Burkhalter, and G. P. Gray, "Laser Sculpting System," (2001), US Patent #6,261,220.
  26. D. Gatinel, T. Hoang-Xuan, and D. T. Azar, "Determination of Corneal Asphericity after Myopia Surgery with the Excimer Laser: A Mathematical Model," IOVS 42, 1736-1742 (2001).
  27. J. R. Jiménez, R. G. Anera, and L. J. d. Barco, "Equation for Corneal Asphericity After Corneal Refractive Surgery," J. Refract. Surg. 19, 65-69 (2003). [PubMed]
  28. S. Marcos, D. Cano, and S. Barbero, "Increase in Corneal Asphericity After Standard Laser in situ Keratomileusis for Myopia is not Inherent to the Munnerlyn Algorithm," J. Refract. Surg. 19, S592-S596 (2003). [PubMed]
  29. A. Vogel and V. Venugopalan, "Mechanisms of Pulsed Laser Ablation of Biological Tissues," Chem. Rev. 103, 577-644 (2003). [CrossRef] [PubMed]
  30. T. F. Deutsch and M. W. Geis, "Self-developing UV photoresist using excimer laser exposure," J. Appl. Phys. 54, 7201-7204 (1983). [CrossRef]
  31. B. Fisher and D. Hahn, "Development and Numnerical Solution of a Mechanistic Model for Corneal Tissue Ablation with the 193-nm Argon Fluoride Excimer Laser," J. Opt. Soc. Am. A 24, 265-277 (2007). [CrossRef]
  32. S. J. Orfanidis, Electromagnetic Waves & Antennas (2004), http://www.ece.rutgers.edu/~orfanidi/ewa/.
  33. R. W. Frey, J. H. Burkhalter and G. P. Gray, " Laser Sculpting Method and System," US Patent 5,849,006 (1998).
  34. C. Dorronsoro, J. Merayo-Lloves, and S. Marcos, "An Experimental Correction Factor of Radial Laser Efficiency Losses in Corneal Refractive Surgery," IOVS  47 E-Abstract 3611 (2006).
  35. S. Marcos, "Spherical Aberration: Biomechanics or Physical Laser Effects?," presented in Wavefront Congress 2006 Meeting (Nassau, Bahamas. January 06, 2006).
  36. J. R. Jimenez, R. G. Anera, J. A. Dıaz, and F. Perez-Ocon, "Corneal asphericity after refractive surgery when the Munnerlyn formula is applied," J. Opt. Soc. Am. A 21, 98-103 (2004). [CrossRef]
  37. R. G. Anera, J. R. Jimenez, L. J. d. Barco, J. Bermudez and E. Hita, "Changes in corneal asphericity after laser in situ keratomileusis," J. Cataract. Refract. Surg. 29, 762-768 (2003). [CrossRef] [PubMed]
  38. Y. Kwon and S. Bott, "Post-surgery asphericity and spherical aberration due to ablation efficiency reduction and corneal remodeling in refractive surgeries," in prep. (2008).

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