## 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]. 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]. 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, 4]. 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]. 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, 7], ablation decentration and registration [8, 9], eye tracking [10, 11], flap [12], physical characteristics of ablation [13-19], wound-healing and biomechanics of the cornea [20-23] 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] 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]. 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] 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] shows that the Munnerlyn profile should not increase the asphericity of corneas with typical preoperative asphericities,while Jimenez et al. [27] 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] 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]. 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]:

$$=0,\mathrm{if}\phantom{\rule{.2em}{0ex}}{F}_{r}\le {F}_{\mathit{TH}},$$

where *d* is the ablation depth, *α* is the absorption coefficient in the material at the laser wavelength,*F _{r}* is the fluence at position

*r*, and

*F*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

_{TH}*α*=2.9 μm

^{-1}and

*F*=40 ~ 60 mJ/cm

_{TH}^{2}.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].

Many studies have shown that the efficiency of the laser changes across the cornea [13, 15], 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 *R _{c}* is (Fig. 1)

The aspheric corneal surface has a sagittal depth *z* given by

where Q is the asphericity of the meridian section of the corneal surface. The fluence
delivered at the same location, *F _{r}*, is approximated by the following equation:

The reason that Eq. (4) holds true is as follows: the intensity of a laser beam varies with cos^{2}Θ 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]:

where *D _{R}* and

*D*are given by:

_{I}$${D}_{I}={\varpi}^{2}{\mu}_{0}{\epsilon}_{I},$$

where *ω* is angular frequency of the laser, *μ*
_{0} is permeability of vacuum, *ε _{R}* and

*ε*are the real and imaginary parts of permittivity of corneal tissue at the laser wavelength (

_{I}*ε*=

_{C}*ε*

_{R}-

*jε*),respectively and can be calculated from complex refractive index of cornea,

_{I}*n*=1.52-

_{C}*j*0.04. It is well known that reflectance also varies with incidence angle. So, the final expression of the ablation depth Eq. (1) becomes:

where *R _{REFL}* is reflectance at the beam position on the cornea and

*α*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.

_{Z}#### 2.2 Ablation profiles and shot pattern generation

The number of diopters to correct is related to the radius, by:

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, *z _{M}*(

*r*), is given by [24]:

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]:

An Alcon Laboratories’ proprietary algorithm [33] 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:

The volume of uniform height (VPS^{*}number of shots) is equivalent to the volume of the ablation profile. The laser beam shots are applied to the corneal tissue in a spatially distributed pattern spread over an area equivalent to the surface area of the OZ diameter. To generate the non-uniform shot pattern of the desired ablation profile, we stretch the uniform shot density pattern as a function of azimuthal angle and radial distance from a reference position on the ablation profile to produce the shot density necessary to achieve the desired ablation depth at every point within the ablation zone. The sequence of these shots is chosen to satisfy the following conditions; 1) no adjacent shots fall in the neighboring area of the previous shot and 2) the sequence is chosen to provide progressive correction over time. Figure 2(a) shows an example of shot patterns generated for the correction of -3D eye with OZ diameter of 6mm.The total number of shots required to correct certain diopters depends on many parameters.For example, with a peak fluence of 540 mJ/cm^{2} and a Gaussian-shaped beam of 0.4 mm radius (corresponding of*VPS*=418∙10^{-6}
*mm*
^{3}), we would need 5,923 shots to correct -12 D myopic eyes with a 6 mm optical zone based on the Munnerlyn formula. For a low fluence laser, peak fluence of 120 mJ/cm^{2}, truncated Gaussian at 50% fluence level of 1.0 mm radius (Fig. 2), and a corresponding of*VPS*=383∙10^{-6}
*mm*
^{3}, we would need 6,457 shots to correct same -12 D myopic eye.

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, 34, 35]. 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, 28].

#### 3. Results and discussion

### 3.1 Ablation efficiency reduction at oblique incidence

In the previous section, we explained the key features of CASIM (Corneal Ablation SIMulator), developed for a rigorous simulation of laser surgery process. The change in corneal asphericity produced by refractive surgery can be explained by ablation profile design,ablation efficiency reduction in the periphery of the ablation zone and healing of the cornea after surgery. CASIM allows the modeling of both factors; however, in this paper, the corneal healing is neglected. The effects of corneal healing and remodeling are addressed in [38].Recently proposed models for explaining the ablation efficiency reduction have been limited to computing the ablation profile without considering the progressive nature of surgery, which is an important aspect with flying spot lasers. During the ablation process, the laser beam gradually etches the corneal surface and changes the surface profile. Therefore, incidence angles of final laser shots on the cornea are expected to be different from the initial ones.Using a realistic shot pattern and beam profile, we can examine the reduction of ablation efficiency more accurately.

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, 34]. 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.

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]. We found the same reduction with truncated Gaussian beams with a VPS of 832×10^{-6}mm^{3}, (F_{PK}=540 mJ/cm^{2}). This pattern has been generated assuming the ablation threshold is 40mJ/cm^{2}. However, if we use a VPS of 383×10^{-6}mm^{3} shot pattern (F_{PK}=120 mJ/cm^{2}), based on the ablation threshold of 60mJ/cm^{2}, the reduction is larger as shown in the graph and this confirms that efficiency reduction largely depends on the ratio of *F _{PK}*/

*F*[13]. Marcos et al., showed that if they use 60 mJ/cm

_{TH}^{2}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], 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], 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/cm^{2} 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] is closer to our results with a high fluence Gaussian beam. When using truncated Gaussian beams, the Jimenez calculation provides smaller estimated asphericities.

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

In the previous section, the impact of: 1) the type of formula on which ablation profiles is based, 2) the laser beam fluence profile and 3) pre-surgery corneal asphericity, on the pre-to-post surgery changes in asphericity was explored, without considering post-surgery corneal healing process. Only a few clinical studies have reported on pre- and post-surgery corneal asphericity changes as a function of the amount of attempted corrections [14, 23, 28, 37]. However, one such study by Marcos et al. included appropriate data for our analysis here.

Figure 5, below, includes digitized data taken from Fig. 2(a),in Marcos et al. [28] and Fig.8 in Dorronsoro et al. [19]. 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/cm^{2} 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].

In Fig. 5, below, the data indicated by the star symbols is taken from Fig. 2(a) in reference
[28]. Also graphed with diamond symbols are the Dorronsoro et al., [19] predictions for post-surgery corneal asphericity calculated with threshold fluence of 60 mJ/cm^{2} after applying an experimentally derived correction factor. In Fig. 8 in [19], Dorronsoro et al., showed two more predictions for post-surgery corneal asphericity, one calculated with lower threshold fluence of 40 mJ/cm^{2} and another one using an efficiency reduction factor proposed by Jimenez et al [15]. We did not reproduce the two predictions in Fig. 5, because one calculated with lower threshold fluence (40mJ/cm^{2}) did predict smaller change for post-surgery asphericity than one with higher threshold fluence (60 mJ/cm^{2}). Also, as discussed in [19, 28],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]. It should be noted that the correction factor proposed by Jimenez et al. was obtained for an ablation profile based on the parabolic approximation.

As explained in previous sections, the CASIM calculations are based on the exact Munnerlyn formula, realistic beam fluence profiles and shot patterns. Figure 5 plots CASIM results obtained using the truncated Gaussian beam at two fluence threshold levels for the ablation of corneal tissue. Although there are still noticeable discrepancies between the CASIM results and clinical data, the CASIM-predicted post-surgery asphericity is closer to the clinical data than the data plotted with diamond symbols. The results show that the ablation efficiency reduction, which is incorporated into CASIM’s excimer shot-by-shot simulation of the ablation, is the main cause of the large change in corneal asphericity. Since the ablation efficiency reduction depends on the fluence of laser beam and corneal ablation fluence threshold, both built into the CASIM model, CASIM produces results closer to clinical data. It should be noted that the impact of corneal remodeling or wound healing were not modeled in this paper and the differences between clinical data and our simulation results may be explainable by individual post-surgery healing process.

The Spearman correlation coefficient for simulated asphericity with the high threshold
truncated Gaussian beam and clinical data is R^{2}=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.

#### 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]. 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, 28].

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

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