Corneal changes induced by laser ablation: study

Journal of Modern Optics Vol. 53, No. 11, 20 July 2006, 1605–1618

Corneal changes induced by laser ablation: study of the visual-quality evolution by a customized eye model D. ORTIZ*y, R. G. ANERAz, J. M. SAIZy, J. R. JIMEŃEZz, F. MORENOy, L. JIMEŃEZ DEL BARCOz and F. GONZA´LEZy yDepartamento de Fı´ sica Aplicada, Facultad de Ciencias, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain zDepartamento de O´ptica, Edificio Mecenas, Facultad de Ciencias, Universidad de Granada, Avda. Fuentenueva s/n, 18071 Granada, Spain (Received 31 October 2005; in final form 16 January 2006) This study focuses on the changes induced in both the asphericity and homogeneity of the cornea for a group of myopic eyes undergoing LASIK surgery. Eyes were characterized by a Kooijman-based customized eye model in which changes were introduced in the form of Gaussian-distributed refractive-index variations of given correlation length for the inhomogeneities and in the form of an expression, based on the modified Munnerlyn’s paraxial formula, for the post-LASIK asphericity. Visual quality was evaluated in terms of the Modulation Transfer Function and the Point-Spread Function. The results show that, on average, the evolution of visual acuity is consistent with the change in corneal asphericity, while the evolution of contrast sensitivity requires a loss in corneal homogeneity in order to be explained. By including both effects in the model, the overall model performance in predicting visual quality is improved.

1. Introduction Geometrical models have consistently been used in optical design for all sorts of imaging devices and optical instruments. However, their role in the quantitative evaluation of visual quality has not been equally important. The main reason for this is that the eye, as an optical instrument, has been modelled only as an ‘average’ system, with well-defined regions of average thicknesses separated by well-known interfaces of averaged radii [1–3]. This statistical eye is useful for illustrating the effect on retinal image of the correction by spectacles or contact lenses; by corneal surgery and anterior chamber phakic intraocular lens [4]; and for studying the optical performance of eyes of different vertebrate species [5]. They are also used for modelling the average visual performance of the eye [6, 7] but not for individual diagnosis or predictions about a particular eye. With the introduction of modern ocular measurements and technical improvements (corneal topography, corneal *Corresponding author. Email: [email protected]

Journal of Modern Optics ISSN 0950–0340 print/ISSN 1362–3044 online 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/09500340600579136

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pachymetry, reliable biometry), calculations made on such bases have become valuable in making objective assessments for individuals. Wavefront evaluation techniques [8–10] applied to measurement of the PointSpread Function (PSF) and Modulation Transfer Function (MTF) of the eye system have improved the general knowledge of visual performance. These two functions are closely linked to parameters that characterize subjective visual quality, such as visual acuity (VA) and the characteristic values of the contrast-sensitivity function (CS). A relatively simple method to reproduce individual eye behaviour (the goal of customized procedures) is to introduce the biometric clinical data of an eye into an appropriate optical model and to calculate the above-mentioned eye-performance functions (PSF and MTF) by means of a ray-tracing process. It is possible to predict the evolution of the image quality when the eye is altered in some way, provided that the changes can be implemented in the model. A noteworthy example of change, though not the only one, is refractive surgery. Refractive-surgery techniques, and in particular laser in situ keratomileusis (LASIK), are commonly used to correct refractive errors of the human eye by changing the corneal refractive power [11]. Although the main objective of this procedure is to reach emmetropia, thereby avoiding the use of external refraction correction, some studies have shown that the procedure has other consequences on visual quality [12–15]. Moreover, several authors [16–18] have evaluated the corneal-tissue response after LASIK surgery, finding morphologic changes in the stroma, such as the presence of microfolds and particles at the corneal interface. Also, it has been suggested that high-order aberrations induced by LASIK surgery in the wavefront may be an effect of the micro-structures produced in the tissue. These inhomogeneities can be introduced into the eye model through the Pupil Function. It is difficult to predict, or even to assess, how the tissue is affected at a microscopic scale for each case, but it is possible to introduce into a customized eye model a parameter that accounts for the changes induced in the wavefront. In previous work, Ortiz et al. [19] showed that local refractive-index variations, characterized by its RMS (Root Mean Square) roughness and correlation length, induce changes in the wave aberration that may explain contrast-sensitivity evolution after LASIK. Another well-known effect of LASIK surgery is the important change in the corneal asphericity value [12, 20–21]. In recent work, Jimeńez et al. deduced an analytical expression to determine the change in corneal asphericity after LASIK surgery [22]. Their calculation is based on the algorithm given by Munnerlyn’s paraxial formula and can be modified in order to account for the effects that reflection losses and non-normal incidence on the anterior cornea have on laser ablation [23–26]. In this way, the change in asphericity can be easily implemented in a personalized model by direct application of the equations deduced by these authors. The main objective of the present work is to assess the relative importance of these two effects induced by refractive surgery (i.e. loss of homogeneity and change in corneal asphericity value), by comparing their respective influence on the change in the calculated PSF and MTF before and after LASIK surgery, as compared with the evolution observed in VA and CS for the same patients.

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The calculation is made by means of a wavefront estimate, derived from a customized ray-tracing process through the complete eye, which includes the inhomogeneities and the change in corneal asphericity. The work is organized as follows: section 2 describes some relevant theoretical elements and a description of the eye model developed to characterize individual eyes from the clinical data corresponding to each of the 55 eyes used in our postsurgical evolution analysis, 37 for the VA case and 18 for the CS case. Section 3 presents the main results, while section 4 discusses key points and establishes the main conclusions.

2. Materials and methods 2.1 Eye model Every eye included in this study is modelled by means of a complete theoretical eye, based on the Kooijman model [27], in which the most important parameters of the individual eye, the central values of each radius and thickness, have been implemented. In this way, the parameters introduced into the model can be classified into three groups: (i) Ocular data taken directly from the patients: axial length, corneal thickness, anterior-chamber thickness, lens thickness, corneal power and anterior corneal radius. (ii) Data from the Kooijman model [27]: refractive index of the cornea, lens and aqueous humour, pre-surgical shape of the aspheric surfaces and the ratio between the two lens radii (or aspect ratio: 1.7). (iii) Data deduced from all previous measurements together with the refractive error of the eye. These are: posterior corneal radius (determined from the corneal power and the anterior corneal radius measured by topography), and the lens radii (calculated from the overall refractive defect and the aspect ratio of the theoretical lens). As only the refractive indices, the conic constant, and the aspect ratio of the lens radii are taken from the Kooijman theoretical model, our calculation has a high degree of personalization. It is worth noting that the results found using this model are not severely altered when the aspect ratio between the two lens radii is changed to other similar values.

2.2 Calculation of visual-quality functions For each optically characterized eye, a ray-tracing process was performed from an object point (5 m) to the retina. In the entrance pupil, a square sampling pattern with a step size of less than 2% of the pupil size was chosen for the ray tracing (for the results presented here the pupil diameter was 3.5 mm, the step size 0.0665 mm

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and the total number of rays reaching the entrance pupil, 2171). The wavefront aberration and the PSF are determined in the exit pupil. The wavefront inside the eye is calculated from the optical path along each ray [28]. The difference between this wavefront and the spherical wavefront corresponding to an aberration-free system in the exit pupil provides the wave aberration W(, ), where and denote the pupil coordinates. The PSF is obtained by calculating the square modulus of the Fourier transform of the Pupil Function [29], defined as 2 ð1Þ Pð, Þ ¼ Tð, Þ exp i Wð, Þ where T(, ), the pupil transmittance, may account for the Stiles–Crawford effect [30] and , the wavelength, is 586.7 nm. The Modulation Transfer Function (MTF) is calculated from the modulus of the Fourier transform of the PSF. The double of the standard deviation of the PSF distribution is calculated to estimate the point-image size, this value being denoted by d. Image formation is the optical part of the vision process. While the functions PSF and MTF provide information on the optical quality of the eye as an instrument, the VA and CS evaluate the overall performance of the visual system. According to the nature and definition of these functions, a connection is commonly established between them. Because of their importance for our work, we give some details about these relationships throughout this section. To establish the relationship between the VA and the size of the PSF, we consider two object points separated by a distance y. Their paraxial images will be at a distance y0 ¼ y, with being the magnification; the possibility of distinguishing the two points will increase when increases and also when the size of the blur circle d decreases. This has been mentioned before in a general way [31, 32], stating generally that point-eye aberrations influence the resolution capability of the eye, and may be expressed approximately as a direct dependence among VA, and d. For a given light distribution at the image point, of diameter d, it follows that the shortest distance required between two points in the retina in order to be resolved, y0 , should be a function of d. In a preliminary attempt to model this dependence, we assume that y0 is proportional to the size of the blur circle, d, imposing some standard retinal and environmental conditions: y0 ¼ d

ð2Þ

where is a constant to be determined. Taking into account that y0 ¼ y and dividing by the distance to the object, L, we get: u ¼ d=L, where u ¼ y/L is the angle subtended by the object (small). The visual acuity is numerically defined [32] as VA ¼ 1/u with u expressed in minutes. Thus we get: VA ¼

L : 10800 y0

ð3Þ


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Introducing equation (2) and the definition of the power of the ocular system (P ¼ (L)1) we have VA ¼

L 1 ¼ 10800 d 10800 P d

ð4Þ

where d and L are expressed in metres and P in diopters. Equation (3) contains only information already known in past theory [32] while equation (4) includes our resolution criterion. It is noteworthy that in the last expression, VA depends, for a given eye accommodation and under our assumptions, only on intrinsic ocular parameters. A useful way of expressing equation (4) for our purposes is VA ¼ a ð=dÞ

ð5Þ

where a ¼ L/10800 is a linear coefficient to be determined. Clearly, some assumptions have been made in order to formulate this expression, and only the clinical data may confirm this dependence of VA on the image-point size. In the case of the CS, other authors [32, 33] have used the relation between this function and the MTF to study the influence of ocular aberrations on visual performance. The relationship between them is determined by analysing the definition of each function [32]: (i) CS is defined for a given spatial frequency as the inverse of the minimum contrast necessary for a sinusoidal pattern of the frequency to be recognized by the eye; (ii) the MTF is defined as the transmission factor of the optical system for every spatial frequency. To connect these functions, we consider that, for a spatial frequency f, the minimum contrast that can be recognized is c. The contrast sensitivity is given by CS( f ) ¼ 1/c. The contrast c will be attenuated by the MTF of the visual system and should be equal to the minimum threshold of contrast in the retinal image to be detected, denoted by c0 . All these relationships can be summarized in the following expressions: c MTFð f Þ ¼ c0 ½CSð f Þ1 MTFð f Þ ¼ c0 MTFð f Þ ¼ c0 CSð f Þ:

ð6Þ

The conclusion drawn from the equations (6) is that the MTF and the CS are proportional for every spatial frequency [32]. This is not as useful as it may appear. The shapes of CS and MTF differ, in particular for small frequencies, evidencing the dependence of c0 on the spatial frequency f [33]. In summary, we shall take into account that the PSF and the MTF are closely connected with parameters that characterize subjective visual quality, such as VA and the characteristic values of CS. Thus, the individual eye behaviour will be assessed by introducing the biometric clinical data into the eye model and calculating the above-mentioned eye-performance functions (PSF and MTF), the values of which (or evolution values, more properly) will be compared to those of the VA and CS. In this way, by using the pre- and post-surgical clinical data, it is possible to propose plausible explanations for the evolution of image quality when the eye undergoes LASIK surgery.

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2.3 Model implementation of LASIK effects 2.3.1 Inhomogeneities induced by laser ablation. Changes that laser ablation induces in the properties of corneal stroma [19] (as local inhomogeneities) are introduced into the model as an uncertain variation in the corneal refractive index distributed according to a Gaussian distribution of mean value 0 and standard deviation n. Thus, a correlation length, L, is introduced by convolution of the refractive-index distribution with a Gaussian function of mean value 0 and width L, and the resulting distribution is rescaled to recover n. The correlation length may depend, for instance, on the local dimensions of the ablation, the characteristic volume of cell regeneration or other effects, such as the variable thickness of the tear film [34, 35]. Refractive-index inhomogeneities alter pupil transmittance, now given by ð4 n ðn þ nÞ 2 0 T ð; Þ ¼ Tð; Þ ð2 ðn þ nÞ and, more importantly, cause a phase change in the Pupil Function: ’ð, Þ ¼

2 z nð, Þ

ð7Þ

where n is the refractive index increment and z is the thickness of the region involved. Now, equation (1) should be expressed as 2 ð8Þ Pð, Þ ¼ T 0 ð, Þ exp i Wð, Þ þ ’ð, Þ : For our calculations, we propose a value of z ¼ 50 m, which was selected from the literature [16] and used in a previous study [19]. For its relevance to the present work, we summarize here the main result of that study. The MTF of each eye in a group of 18 eyes was calculated before and three months after surgery for different values of the refractive-index dispersion n and correlation length L. The evolution of the MTF was compared to the evolution of the measured CS for the measured spatial frequencies 3, 6, 12 and 18 cycles/deg. Mean values of n ¼ 0.007 0.003 and L ¼ 0.3 0.1 mm were calculated. It can be readily seen that high values of n and low values of L both contribute to the local roughness of the media, increasing the change in the wavefront and attenuating the MTF values for all frequencies.

2.3.2 Changes in corneal asphericity. Variations in the corneal asphericity value decisively influence eye aberrations and visual functions. Anera et al. [24] calculated the change in the corneal asphericity value (p-factor) after refractive surgery using Munnerlyn’s paraxial formula and considering an adjustment factor taking into account the reflection losses and the non-normal incidence when the laser beam


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is moved vertically parallel to the optical axis. The following equation is proposed in their work for the p-factor value after surgery: R3post aD 2 oz ð0:62 þ 1:333pÞ 1:333R2 ppost ¼ 3 p þ ð9Þ R R where p and ppost are, respectively, the p-factor before (for the Kooijman model p ¼ 0.75) and after surgery; R and Rpost are the anterior corneal radius before and after surgery; D is the number of diopters to correct; and oz is the ablation diameter. The parameter a is given by 1/ln(F0/Fth), where F0 is the incident exposure of the laser pulse, and Fth is the threshold exposure for the ablation. This parameter is a constant for the laser used and, on average, it ranges from 0.62 to 1.142. For this study, the mean value a ¼ 0.88 is used [24]. Although real corneal asphericity is even higher than predicted by equation (9) [24, 26], the level of prediction reaches 90%, this being a good approximation.

3. Results 3.1 Visual acuity A total of 37 myopic eyes of 23 patients (mean age 32, range 19 to 44 years old) were fully characterized, including topography, pachymetry, biometry and refraction. None had pathologies, underwent any surgical procedure or was under topical treatment. The mean myopia was 9.00 5.00 D and mean astigmatism was 1.25 0.95 D. All patients had myopia and spectacle-corrected visual acuity covering a wide range between 0.2 and 0.95. The best corrected visual acuity higher than 1 was not considered in the clinical evaluation, and consequently cases of BCVA ¼ 1 were not included in order to avoid ambiguities. The first result we present simply shows the dependence between the observed BCVA and the calculated /d, with being the magnification and d the PSF size. We analyzed the post-LASIK evolution in terms of these parameters. In figure 1, BCVA (best corrected visual acuity) is plotted against the ratio : d for the 37 eyes of our study. An approximate linear dependence can be found in the form: BCVA ¼ 5:8

þ 0:3: d

The correlation (R ¼ 0.7) supports the assumption of visual acuity understood as resolution ability, obviously increasing for small calculated image points. A value of 0.25 is obtained from the slope of the linear dependence. This means that the diameter of two blur images could be, on average, four times the shortest distance required between two points in the retina in order to be resolved. There is no doubt that this has to be understood as an average result and therefore it may change slightly depending on the eye model used but it is also worth noticing that it is the result of a fit over real VA, not a raw result of a model. Of course there

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Figure 1. Best corrected visual acuity (BCVA) versus the ratio between the magnification , and the PSF size d, for spectacle-corrected eyes. Continuous line represents the linear fit for 37 myopic eyes. (The colour version of this figure is included in the online version of the journal.)

is a significant variation between the values obtained for different real eyes, but it is very interesting to establish a preliminary resolution criterion for the vision of real eyes. The use of a two-point resolution criteria to model the behaviour of VA is a quite simplistic approach, considering that there are other optical metrics that are proving themselves very useful for assessing visual acuity (like the Strehl ratio), but for the wide range of VA contained in our study, the PSF size, combined with a resolution criteria, appears to us as a very efficient procedure. To study the evolution of the visual acuity after LASIK, in figure 2 we represent the visual acuity for the 37 eyes before (best corrected, black diamonds) and after surgery (without correction, blank squares) versus the values /d estimated for these eyes with the model. In figure 2(a), the only change made in the model was the variation in the central curvature and corneal thickness. The results show /d values to be much higher than those expected from the pre-surgical linear dependence. When inhomogeneities ( n ¼ 0.007 and L ¼ 0.3 mm) are introduced (figure 2(b)), the increase in /d is slightly smaller, as corresponds to a larger PSF. And, if only the change in corneal asphericity is introduced (figure 2(c)), the increase in /d causes a dependence between VA and /d that resembles the original. These results show that, with regard to VA, the dispersion induced in the image point by the local refraction variations is less important than the size increase induced by the post-surgical change of the corneal asphericity. Therefore, this effect should be taken into consideration in customized refractive procedures.

3.2 Contrast sensitivity In the case of the contrast sensitivity (CS), we consider a group of 18 myopic eyes of 13 patients (mean age 32, range 23 to 46 years old). The mean myopia (spherical equivalent) was 3.5 1.6 D. The contrast sensitivity was measured for every eye


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Figure 2. Evolution of the visual acuity after LASIK. In all cases pre- and post-surgical BCVA are represented on the ordinate axis, and pre-surgical values of /d, as calculated by means of the model, are plotted with symbol ^. (a) œ: post-surgical values of /d are calculated by the model; (b) m: post-surgical values of /d when the model includes the influence of the inhomogeneities characterized by n ¼ 0.5% and L ¼ 0.3 mm; (c) : post-surgical values of /d when the model includes the influence of the change in corneal asphericity (a ¼ 0.88). (The colour version of this figure is included in the online version of the journal.)

with spectacle-correction, using the CSV-1000E test before and 3 months after surgery. Figure 3(a) shows histograms of CS values for two spatial frequencies 6 (left column) and 18 cycles/deg (right column) before (first line) and after surgery (second line). The main feature of these histograms is that their distribution is quite similar before and after surgery. For every eye, optically characterized with the customized eye model in which the spectacle-correction of each case has been introduced, the MTF may be calculated before and after surgery. In the latter case, we first considered no additional effects, and secondly we introduced either a change in corneal asphericity or the effect of inhomogeneities. This MTF is used as a tool to explain CS evolution for

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Figure 3. Histograms of eye-performance evolution for the 18 eyes: (a) CS values (in linear units (Imax þ Imin)/(Imax Imin), from test CSV-1000E) for 6 cycles/deg (left) and 18 cycles/deg (right) before (first line) and after (second line) surgery. (b) Normalized MTF calculated values for 6 cycles/deg (left) and 18 cycles/deg (right), before surgery (first line), after surgery (second line), after surgery introducing the asphericity change predicted by equation (9) (third line) and after surgery with inhomogeneities (bottom line). Mean values n ¼ 0.5% and L ¼ 0.3 mm are selected. (The colour version of this figure is included in the online version of the journal.)


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the spatial frequencies observed. In the first line of figure 3(b), the histograms of pre-surgical MTF for the spatial frequencies 6 (left column) and 18 cycles/deg (right column) are represented, while the second line stands for post-surgery values of MTF when only corneal curvature and thickness changes are introduced. Clearly, the evolution of the MTF calculated in this way shows a substantial increase, and does not reproduce the CS evolution observed after surgery for the same eyes. In the third line the change in corneal asphericity is introduced, producing an overall decline in MTF values and therefore a shift of the distribution to the left, but this is not sufficient to reach the pre-surgical values, as in the CS distribution. In the bottom line of figure 3(b), only the inhomogeneities are introduced, with values n ¼ 0.007 and L ¼ 0.3 mm [19]. Now, the post-LASIK evolution of MTF has a similar appearance to the CS evolution. The change in asphericity, necessary to explain the evolution of visual acuity, does not appear sufficient to explain CS evolution after LASIK surgery. This result shows the importance of including any effect induced by laser ablation in a model used to calculate the optical performance of the eye after refractive surgery. To illustrate the calculation process, figure 4 shows the pre- and post-surgical MTF curves, calculated for a typical case. Black squares correspond to pre-surgical values, and blank circles to post-surgical ones, when no effect other than curvature and thickness changes, was considered. Blank triangles stand for the post-surgical calculation with inhomogeneities ( n ¼ 0.007 (0.5%) and L ¼ 0.5 mm, optimum values found for this eye) while stars ( ) stand for the values resulting from a change in asphericity. It is easy to see that when only the corneal curvature and thickness are changed in the model, the MTF calculated after surgery is clearly higher than the MTF before surgery for all the spatial frequencies, something not observed for the CS, showing that the model lacks the elements induced by the surgery. When the two main additional effects analysed in this work are implemented separately in the model, the reduction in low-frequency MTF values induced by the inhomogeneities is more important than the effect caused by the change in corneal asphericity. The change in corneal asphericity, on the other hand, was more determinant for explaining VA evolution, showing that it has a strong effect for high spatial-frequency values of the MTF (in figure 4, the curve of clearly decreases with spatial frequency).

4. Discussion In this work, we have presented a procedure supported by the improvement of clinical-data acquisition, which may be quite effective in modelling eye performance and is easy to implement in terms of clinical routine. Our results provide enough evidence on the ability of the model to take advantage of improvements in ocular measurement. In this personalized eye model, it is possible to implement the effects induced by LASIK surgery, which are required to reproduce the behaviour of the visual system.

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Figure 4. MTF for eye N.8, calculated before (g) and after surgery () with the plain model, compared with the MTF calculated introducing the change in corneal asphericity, with a ¼ 0.88 ( ), and MTF calculated introducing the inhomogeneities, with L ¼ 0.5 mm and n ¼ 0.5% (optimum values for this eye) (r).

Based on the connection between VA and PSF as well as between CS and MTF, we conclude that the analysis of the evolution slopes in the plot VA versus /d offers an appropriate framework for understanding and quantifying the differences between the expected and real results after LASIK. The results for post-surgical evolution reveal that LASIK surgery not only modifies the corneal curvature, but also may decrease the medium quality as an effect of either the flap cut or the laser ablation. In general, the new corneal radius corrects the refractive defect and low-order aberrations, but visual quality is affected by the regeneration of corneal tissue, a process in which the cornea loses homogeneity over the short term. LASIK surgery and other types of surgical procedures alter both low and high spatial frequencies in the MTF, and consequently VA and CS decrease. In the calculations the mean values reported in previous work [19] are used for the characteristic parameters of the inhomogeneities. This involves a degree of approximation in the procedure, but in this work the goal is to compare the effect that, on average, the change in corneal asphericity and the induced corneal-index inhomogeneities have on both visual acuity and contrast sensitivity. Recognizing the crucial role of PSF size in assessing visual quality, we should point out that a precise evaluation of individuals by means of a model like this would often require the knowledge of the PSF shape, particularly in cases of irregular topography. In addition, many other effects not considered are involved in refractive surgery, such as biomechanical effects, wound healing, optical role of the flap, type of laser, and technical procedures.


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It worth remarking that our customized model, which is reliable in predicting the individual PSF and MTF, could be used as a tool for predicting the effect on the visual quality exerted by any change in optical parameters. For instance, it may be applied to the design of many optical solutions, such as customized intraocular lenses and corneal refractive surgery (PRK, LASIK). This points a way to personal optimization of ablation in refractive surgery, in line with similar works [36, 37].

References [1] W.N. Charman, Optics of the eye in Handbook of Optics. Fundamentals, Techniques and Design, edited by M. Bass, et al., Vol. 1 (MacGraw-Hill, New York, 1995). [2] H.L. Liou and N.A. Brennan, J. Opt. Soc. Am. A 14 1684 (1997). [3] I. Escudero-Sanz and R. Navarro, J. Opt. Soc. Am. A 16 1881 (1999). [4] E.J. Sarver and R.A. Applegate, J. Refract. Surg. 16 S611 (2000). [5] G. Li, H. Zwick, B. Stuck and D.J. Lund, J. Biomed. Opt. 5 307 (2000). [6] J.B. Doshi, E.J. Sarver and R.A. Applegate, J. Refract. Surg. 17 414 (2001). [7] J.E. Greivenkamp, J. Schwiegerling, J.M. Miller, et al., Am. J. Ophthalmol. 120 227 (1995). [8] J. Liang and D.R. Williams, J. Opt. Soc. Am. A 14 2873 (1997). [9] L.N. Thibos, X. Hong, A. Bradley, et al., J. Opt. Soc. Am. A 19 2329 (2002). [10] O. Nestares, R. Navarro and B. Antona, J. Opt. Soc. Am. A 20 1371 (2003). [11] M.C. Knorz and T. Neuhann, Ophthalmology 107 2072 (2000). [12] E. Moreno-Barriuso, J.M. Lloves, S. Marcos, et al., Invest. Ophthalmol. Vis. Sci. 42 1396 (2001). [13] J.N. Fernańdez del Cotero, F. Moreno, D. Ortiz, et al., J. Refract. Surg. 17 305 (2001). [14] J.W. Chan, M.H. Edwards, G.C. Woo, et al., J. Cataract. Refract. Surg. 28 1774 (2002). [15] K. Nakamura, H. Visen-Miyajima, I. Toda, et al., J. Cataract. Refract Surg. 27 357 (2001). [16] P.J. Pisella, O. Auzerie, Y. Bokobza, et al., Ophthalmology 108 1744 (2001). [17] M. Vesaluoma, J. Pe´rez-Santonja, W.M. Petroll, et al., Invest. Ophthalmol. Vis. Sci. 41 369 (2000). [18] S. Patel, J. Alio´ and J. Pe´rez-Santonja, Invest. Ophthalmol. Vis. Sci. 45 3523 (2004). [19] D. Ortiz, J.M. Saiz and F. Gonza´lez, Opt. Lett. 29 739 (2004). [20] J.T. Holladay, D.R. Dudeja and J. Chang, J. Cataract Refract. Surg. 25 663 (1999). [21] R.G. Anera, J.R. Jimeńez, L. Jimeńez del Barco, et al., J. Cataract Refract. Surg. 29 762 (2003). [22] J.R. Jimeńez, R.G. Anera and L. Jimeńez del Barco, J. Refract. Surg. 19 65 (2003). [23] J.R. Jimeńez, R.G. Anera, L. Jimeńez del Barco, et al., Appl. Phys. Lett. 81 1521 (2002). [24] R.G. Anera, J.R. Jimeńez, L. Jimeńez del Barco, et al., Opt. Lett. 28 417 (2003). [25] M. Mrochen and T. Seiler, J. Refract. Surg. 17 S584 (2001). [26] D. Cano, S. Barbero and S. Marcos, J. Opt. Soc. Am. A. 21 926 (2004). [27] A.C. Kooijman, Opt. Soc. Am. 73 1544 (1983). [28] W.N. Charman, Optom. Vis. Sci. 68 574 (1991). [29] J.W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill International Editions, New York, 1996). [30] R.A. Applegate and V. Lakshminaraynan, J. Opt. Soc. Am. A 10 1611 (1993). [31] Lijun Zhu, D.U. Bartsch, W.R. Freeman, et al., Optom. Vis. Sci. 75 827 (1998).

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[32] J.M. Artigas, P. Capilla, A. Felipe, et al., O´ptica fisilo´gica: Psicofı´sica de la visioń (Interamericana de MacGraw-Hill, Madrid, 1995). [33] S. Marcos, J. Refract. Surg. 17 S596 (2001). [34] R. Monte´s-Mico´, J.L. Alio´, G. Munõz, et al., Invest. Ophtalmol. Vis. Sci. 45 1752 (2004). [35] R. Monte´s-Mico´, A. Ca´liz and J.L. Alio´, J. Refract. Surg. 20 243 (2004). [36] D. Ortiz, J.M. Saiz, F. Gonza´lez, et al., J. Refract. Surg. 18 S327 (2002). [37] J. Schwiegerling and R.W. Snyder, J. Opt. Soc. Am. A 15 2572 (1998).