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Volume Estimation of Excimer Laser Tissue Ablation for Correction of Spherical Myopia and Hyperopia

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Volume Estimation of Excimer Laser Tissue Ablation for Correction of Spherical Myopia and Hyperopia Volume Estimation of Excimer Laser Tissue Ablation for Correction of Spherical Myopia and Hyperopia Damien Gatinel,1 Thanh Hoang-Xuan,1 and Dimitri T. Azar1,2 PURPOSE. To determine the theoretical volumes of ablation for stromal bed have been implicated as determinants of corneal the laser treatment of spherical refractive errors in myopia and stability, further studies are necessary to evaluate their exact hyperopia. significance. New models estimating the volume of the corneal METHODS. The cornea was modeled as a spherical shell. The tissue ablated by a laser refractive procedure may also be ablation profiles for myopia and hyperopia were based on an necessary to determine the influence of ablated volumes on established paraxial formula. The theoretical volumes of the corneal stability and the procedure’s outcome. ablated corneal lenticules for the correction of myopia and The primary focus of this work is to provide a formula to hyperopia were calculated by two methods: (1) mathematical approximate the volume of tissue ablation for spherical cor- approximation based on a simplified geometric model and (2) rection in myopia and hyperopia. In this study, we approached finite integration. These results were then compared for opti- the volume calculation by two approaches: finite integration cal zone diameters of 0.5 to 11.00 mm and for initial radii of and mathematical approximation. We compared the theoreti- curvature of 7.5, 7.8, and 8.1 mm. cal values of volumes of tissue removal in the treatment of spherical refractive errors predicted by our geometric approx- RESULTS. Referring to a simplified geometrical model, the vol- imation with the values given by finite integration. ume of ablated corneal tissue was estimated to be proportional This work is potentially useful to compare the amount of to the magnitude of treatment (D) and to the fourth power of tissue ablation after primary LASIK and retreatment proce- the treatment diameter (S4). For refractive correction of myo- Х dures. It may also contribute to our understanding of the pia and hyperopia, volume estimations using our formula, V factors that lead to keratectasia after LASIK surgery for myopia ⅐ 4 D (S/9) , were similar to those obtained by finite integration as well as factors that limit surgical correction of high hyper- for optical zone diameters of 0.5 to 8.5 mm and for corneal opia. radii of curvature within the clinical range (7.5, 7.8, and 8.1 mm). CONCLUSIONS. The theoretical volume of corneal tissue ablated MATERIALS AND METHODS within the optical zone for spherical corrections can be accu- rately approximated by this simplified formula. This may be Calculation of Ablation Profile for Spherical helpful in evaluating factors that contribute to corneal ectasia Correction in Myopia and Hyperopia after LASIK for myopia and hyperopia. Treatment diameter (S) is the most important determinant of the volume of tissue In both photorefractive keratectomy (PRK) and LASIK for spherical ablation during excimer laser surgery. (Invest Ophthalmol Vis refractive errors, flattening or steepening of the central corneal curva- Sci. 2002;43:1445–1449) ture due to tissue photoablation results in decreased or increased refractive power, respectively. Although the ablation profiles of avail- able excimer lasers are proprietary and may vary from one device to ecent concerns regarding the depth of tissue ablation with another, conventional ablation profiles for the correction of spherical Rthe excimer laser during laser in situ keratomileusis refractive errors rely on the theoretical pioneering work of Munnerlyn (LASIK) raise the general issue of understanding the biome- et al.3 in which the corneal surface is assumed to be spherical, and the chanical response of the cornea to keratorefractive surgery optical power of the excised lenticule (D) corresponds to the intended procedures. Models of the mechanical response of the cornea change in refraction. Calculation of the ablation profile for the correc- to the trauma inflicted by both the laser ablation and the flap tion of spherical myopia (M) can be performed according to the cut have recently been proposed to explain some of the dis- following general formula (Fig. 1) crepancies observed between the achieved and intended cor- 1,2 rections after refractive surgical procedures. Although the S 2 S 2 depth of corneal ablation and the thickness of the residual t ͑y͒ ϭ ͱR2Ϫͩ ͪ Ϫ ͱR 2 Ϫ ͩ ͪ ϩ ͱR 2 Ϫ y2 Ϫ ͱR2 Ϫ y2 (1) M 2 M 2 M with From the 1Rothschild Foundation, Paris, France; and the 2Massa- chusetts Eye and Ear Infirmary and Schepens Eye Research Institute, Harvard Medical School, Boston, Massachusetts. 1 1 D ϭ ͑n Ϫ 1͒ ⅐ ͩ Ϫ ͪ (2) Supported by the New England Corneal Transplant Research R RM Fund, a Research to Prevent Blindness Lew R.Wasserman Merit Award, and the Massachusetts Lions Eye Research Award, Northborough, MA (DTA). where R and RM are the initial and final anterior radii of curvature, Ͼ Submitted for publication August 29, 2001; revised December 7, respectively (R RM), and n is the refractive index of the cornea. tM(y) 2001; accepted December 7, 2001. expresses the depth of tissue removal in treating myopia as a function Commercial relationships policy: N. of the distance y from the center of the treatment with an optical zone The publication costs of this article were defrayed in part by page diameter of S (in millimeters). charge payment. This article must therefore be marked “advertise- The maximal depth of ablation (t ) occurs at the center of the ment” in accordance with 18 U.S.C. §1734 solely to indicate this fact. o treatment zone (y ϭ 0) and is calculated by Corresponding author: Dimitri T. Azar, Director, Corneal, External Disease, and Refractive Surgery Service, Massachusetts Eye and Ear Infirmary, 243 Charles Street, Boston, MA 02114; S 2 S 2 [email protected]. t ϭ ͱR2 Ϫ ͩ ͪ Ϫ ͱR 2 Ϫ ͩ ͪ (3) o 2 M 2 Investigative Ophthalmology & Visual Science, May 2002, Vol. 43, No. 5 Copyright © Association for Research in Vision and Ophthalmology 1445 Downloaded from iovs.arvojournals.org on 09/24/2021 1446 Gatinel et al. IOVS, May 2002, Vol. 43, No. 5 FIGURE 1. Diagram of laser ablation, for myopia, of a spherical surface Ͼ with initial radius of curvature R and final radius of curvature RM (RM R). The depth of ablation tM(y) is the distance between the initial and ϭ final surfaces (equation 1). It varies between a maximum to at y 0 ϭ and zero at y S/2. Along the x-axis, M1 corresponds to the maximal depth of tissue ablation and M2 corresponds to interception of the chord joining the edges of the treatment zone. To correct hyperopia (H), the surface has to be steepened over the treatment zone (Fig. 2). The ablation profile for the correction of spherical hyperopia is ͑ ͒ ϭ ͱ 2 Ϫ 2 Ϫ ͱ 2 Ϫ 2 ϩ Ϫ IGURE tH y R y RH y RH R (4) F 3. Schematic diagram of laser ablation of depth to and diam- eter S, corresponding to a myopia correction of D (diopters), is applied to a flat surface (R ϭϱ). The volume of the cap of ablation is estimated where t (y) expresses the depth of tissue removal in treatment of H from equation 7 to be D(S/9)4, as in equation 11. hyperopia as a function of the distance y from the center of the treatment zone (of diameter S), and R and RH are the initial and final corneal anterior radii of curvature, respectively. 1 The maximal depth of ablation occurs at the external limit of the t Ϸ S2D .(6) o 8͑n Ϫ 1͒ optical zone and is determined by An initially flat surface has a radius of curvature infinitely high and S 2 S 2 ϭ ͱ 2 Ϫ ͩ ͪ Ϫ ͱ 2 Ϫ ͩ ͪ ϩ Ϫ no optical power. The ablated volume needed to induce a power of D to R RH RH R:(5) 2 2 diopters over an optical surface of S millimeters is assimilated as a spherical cap of height to. The result of a spherical ablation for myopia Mathematical Approximation of the Volume of on a theoretical flat corneal surface would thus be the sculpting of a Tissue Ablated for Spherical Myopic Corrections crater with a shape and volume equal to those of a spherical cap. The simplified equation used to estimate the maximal depth of the Figure 3 represents the cap in cross section. The diameter of the ablation is3 cap is equal to the diameter of the treatment zone (S), and its height corresponds to the maximal depth of ablation to calculated by equation 6. Thus, the volume of the spherical cap can be derived at by the following formula: 1 S 2 V ϭ ␲t ͫ3ͩ ͪ ϩ t 2ͬ (7) cap 6 o 2 o 2 2 2 Because to is very much lower than S , to can be neglected, and the volume of the cap can be approximated by 1 S 2 1 V Ϸ ␲t 3ͩ ͪ Ϸ ␲t S2 .(8) cap 6 o 2 8 o Replacing to by its expression as a function of D and S defined in equation 6 yields 1 V Ϸ ␲DS4 (9) cap 64,000͑n Ϫ 1͒ FIGURE 2. Diagram of laser ablation for hyperopia. The initial radius of curvature R is greater than the final radius of curvature RH. The depth ϭ ϭ of ablation increases from zero at y 0toto at y S/2. Along the (in cubic millimeters). Since x-axis, H and H are the intercepts of the chords formed at the 1 2 ␲ 1 4 treatment edges of the untreated and treated corneal surfaces, respec- Х ͩ ͪ (10) tively.
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