Recently, two papers “Dorronsoro et al., Experiment on PMMA models to predict the impact of corneal refractive surgery on corneal shape, Opt. Express 14, 6142 (2006)” and “JR Jiménez et al., Deviations of Lambert-Beer’s law affect corneal refractive parameters after refractive surgery, Opt. Express 14, 5411 (2006)” have been published on an important question in corneal refractive surgery: to explain the differences between post-surgical corneal elevation maps and those predicted after theoretical ablation. An analysis of Dorronsoro et al. data demonstrates that the failures in Lambert-Beer’s law could be in the origin of shape discrepancies. New models and experimental data on deviations of Lambert-Beer’s law might help to minimize post-surgical corneal discrepancies and thus optimise eye emmetropization.
© 2007 Optical Society of America
The outcome of wavefront-guided laser surgery is affected by the limited ability to transfer the aimed ablation profile onto the cornea [1-5]. Many variables such as, physical aspects of ablation, corneal biomechanics, hydration, influence the predictability of laser surgery. The work by Dorronsoro et al.  has experimentally analysed the real and expected ablation shape on spherical and flat surfaces of PMMA (polymethyl-methacrylate) finding postoperative shape discrepancies between measured shapes and those simulated by simple subtraction on the programmed ablation profile.
In a previous work, Jiménez et al.  proposed that deviations from Lambert-Beer’s law may partially explain deviations from the expected post-operative corneal shape. Lambert-Beer’s law [4,6-7] is given by dp =m ln(F/Fth), with dp being the ablation depth per pulse, F the incident exposure, Fth the threshold exposure, and m a constant. An analysis of data published by Dorronsoro et al.  could offer evidence supporting this hypothesis. These authors measured an experimental ablation efficiency factor on PMMA, obtained as the ratio between two ablation profiles, one from a spherical surface and the other from a flat surface. Ablations were made by programming the same correction in the laser (-12 D).
This ablation-efficiency factor was analysed (experimental data were provided by the authors) considering possible deviations of Lambert-Beer’s law. The ablation profile measured on PMMA with a flat surface is given by Ndp (0,y), where N indicates the number of pulses and dp(0, y) the depth ablation per pulse on the flat surface at height y. The ablation profile on the spherical surface is given by Ndp(θ,y) (θ indicates the incidence angle). Dividing the two profiles for the same y value, we would obtain the ablation efficiency factor, dp(θ,y)/ dp (0,y), denoted as dp(F(θ)). The geometry of spherical surfaces results in ablation efficiency changes from the centre to the periphery. The exposure on the sphere is given by : F = F 0cosθ(1-R), R being the reflectivity, with F0=120mJ/cm2 . The key for studying possible deviations of Lambert-Beer’s law is to analyse dp(F(θ)) as a function of ln(F/Fth) with Fth, being 80mJ/cm2 .
Figure 1 shows the experimental efficiency factor, dp(F(θ)), as a function of ln(F/Fth) and the best linear fit. The departure from the linear fit is greater at higher exposure levels. Differences between experimental data and best linear fit are significant for visual performance. We can compute the difference in refractive power between experimental and linear fit data assuming for computations on cornea (see procedure in Ref. ) the values from fittings here obtained for PMMA. In this case, the average difference of refractive power is 0.8D, reducing the effective visual acuity, contrast sensitivity and exceeding the quarter-aberration criterion . A degree of myopia lower than -12D would provide a lower refractive-power difference. One important point shown in this analysis is that the range of exposure variations represents a real situation during the ablation. Most analyses of Lambert-Beer’s law  have used a broad range of exposure variations that do not correspond to the real situation during surgery. In light of this analysis of the experimental data, we believe that new experiments and models on PMMA and corneal ablations are needed to explain shape differences. These experiments and models should not assume Lambert-Beer’s law, but should consider different exposure levels and include dynamic optical properties (for polymers and corneas) that occur during ablation [6,7] with the aim of minimizing failures of shape prediction that interfere with a total emmetropization of the eye.
Authors wish to thank C. Dorronsoro and other authors for providing the data on PMMA.
References and links
1. J. R. Jiménez, R. G. Anera, J. A. Díaz, and F. Pérez-Ocön, “Corneal asphericity after refractive surgery when the Munnerlyn formula is applied,” J. Opt. Soc. Am. A 21, 98–103 (2004). [CrossRef]
2. R. G. Anera, C. Villa, J. R. Jiménez, R. Gutiérrez, and L. Jiménez del Barco, “Differences between real and predicted corneal shapes after aspherical corneal ablation,” Appl. Opt. 44, 4528–4532 (2005). [CrossRef]
3. J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, and L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).
4. J. R. Jiménez, F. Rodríguez-Marín, R. G. Anera, and L. Jiménez del Barco, “Deviations of Lambert-Beer’s law affect corneal refractive parameters after refractive surgery,” Opt. Express 14, 5411–5417 (2006). [CrossRef]
5. 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]
7. B. T. Fisher and D. W. Hahn, “Development and numerical 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]