purpose. To determine the theoretical volumes of ablation for the laser treatment of spherical refractive errors in myopia and hyperopia.

methods. The cornea was modeled as a spherical shell. The ablation profiles for myopia and hyperopia were based on an established paraxial formula. The theoretical volumes of the ablated corneal lenticules for the correction of myopia and hyperopia were calculated by two methods: (1) mathematical approximation based on a simplified geometric model and (2) finite integration. These results were then compared for optical zone diameters of 0.5 to 11.00 mm and for initial radii of curvature of 7.5, 7.8, and 8.1 mm.

results. Referring to a simplified geometrical model, the volume of ablated corneal tissue was estimated to be proportional to the magnitude of treatment (*D*) and to the fourth power of the treatment diameter (*S* ^{4}). For refractive correction of myopia and hyperopia, volume estimations using our formula, *V* ≅ *D* · (*S*/9)^{4}, were similar to those obtained by finite integration for optical zone diameters of 0.5 to 8.5 mm and for corneal radii of curvature within the clinical range (7.5, 7.8, and 8.1 mm).

conclusions. The theoretical volume of corneal tissue ablated within the optical zone for spherical corrections can be accurately approximated by this simplified formula. This may be helpful in evaluating factors that contribute to corneal ectasia after LASIK for myopia and hyperopia. Treatment diameter (*S*) is the most important determinant of the volume of tissue ablation during excimer laser surgery.

^{ 1 }

^{ 2 }Although the depth of corneal ablation and the thickness of the residual stromal bed have been implicated as determinants of corneal stability, further studies are necessary to evaluate their exact significance. New models estimating the volume of the corneal tissue ablated by a laser refractive procedure may also be necessary to determine the influence of ablated volumes on corneal stability and the procedure’s outcome.

^{ 3 }in which the corneal surface is assumed to be spherical, and the optical power of the excised lenticule (

*D*) corresponds to the intended change in refraction. Calculation of the ablation profile for the correction of spherical myopia (M) can be performed according to the following general formula (Fig. 1)

*R*and

*R*

_{M}are the initial and final anterior radii of curvature, respectively (

*R*>

*R*

_{M}), and

*n*is the refractive index of the cornea.

*t*

_{M}(

*y*) expresses the depth of tissue removal in treating myopia as a function of the distance

*y*from the center of the treatment with an optical zone diameter of

*S*(in millimeters).

*t*

_{H}(

*y*) expresses the depth of tissue removal in treatment of hyperopia as a function of the distance

*y*from the center of the treatment zone (of diameter

*S*), and

*R*and

*R*

_{H}are the initial and final corneal anterior radii of curvature, respectively.

*D*diopters over an optical surface of

*S*millimeters is assimilated as a spherical cap of height

*t*

_{o}. The result of a spherical ablation for myopia on a theoretical flat corneal surface would thus be the sculpting of a crater with a shape and volume equal to those of a spherical cap.

*S*), and its height corresponds to the maximal depth of ablation

*t*

_{o}calculated by equation 6 . Thus, the volume of the spherical cap can be derived at by the following formula:

*t*

_{o}

^{2}is very much lower than

*S*

^{2},

*t*

_{o}

^{2}can be neglected, and the volume of the cap can be approximated by

*t*

_{o}by its expression as a function of

*D*and

*S*defined in equation 6 yields

*D*diopters, the following is used

*V*

_{M}is the approximate volume of tissue ablated for spherical correction in myopia (in cubic millimeters),

*D*is the intended change in myopia (in diopters), and

*S*is the diameter of the treatment zone (in millimeters).

*t*

_{o}is determined by equation 6 , and the volume of this dome can be calculated by using the formula for the volume of a spherical cap. The volume of ablated tissue

*V*

_{H}is obtained by subtracting the volume of the dome (which is equal to the volume of a spherical cap of height

*t*

_{o}and diameter

*S*) from the volume of the corresponding cylindrical segment

*V*

_{C}.

*x*-axis, the lenticule of corneal tissue ablated can be calculated using the general formula for the volume of solids of rotation and, by using mathematical integration, the volume

*V*

_{M}of the myopic ablated lenticule can be computed

*R*is the radius of the initial corneal surface,

*R*

_{M}is the radius of the final corneal surface, and

*S*is the optical zone diameter.

*M*

_{1}corresponds to the maximal depth of tissue ablation, and

*M*

_{2}is the length of the line segment from the apex of the postoperative anterior surface to the chord formed at the treatment zone diameter (Fig. 1) .

*R*is the radius of the initial corneal surface,

*R*

_{H}is the radius of the final corneal surface,

*S*is the optical zone diameter, and

*H*

_{1}and

*H*

_{2}are the lengths of the line segment from the anterior surface apex to the chord formed at the treatment zone diameter of the initial and final corneal surfaces, respectively (Fig. 2) .

*R*of 7.5, 7.8, and 8.1 mm). The refractive index of the stroma (

*n*) was set at 1.377.

*R*) of 7.8 mm and optical zone (

*S*) of 6.0 mm. For similar magnitude of treatments, the ablation volume necessary for spherical correct in myopia is slightly greater than that for spherical correction in hyperopia.

*R*) of 7.8 mm and optical zone diameter (

*S*) of 4 to 8 mm are shown in Table 1 . Volume approximations were within ±2.0% in all calculations for hyperopia and myopia with

*S*less than 7 mm. Similarly, Figure 6 illustrates the accuracy of our formulas for volume approximation for myopia and hyperopia given by equation 11 for

*R*of 7.5, 7.8, and 8.1 mm. Comparison of the ablated corneal volumes, as determined by the finite integration method, with the volume approximations of equation 11 , shows that our formula provides an acceptable estimate of the ablated volume for spherical myopia and hyperopia for optical zone diameters up to 9 mm. Beyond 9 mm (

*S*≥ 9.5 mm), our formula tends to underestimate the ablated volume compared with the finite integration calculations, especially for treatment of myopia. As expected, the higher the initial radius of curvature (i.e., the flatter the initial corneal surface), the better is the approximation provided by our formula.

^{2}in normal and keratoconus corneas, suggesting that keratoconus is not true ectasia, in which the total surface area increases, but rather is a specialized type of warpage.

^{ 4 }No similar investigation has been undertaken of keratectasia after refractive lamellar surgery.

^{ 5 }

^{ 6 }Geggel and Talley

^{ 7 }have reported a case of keratectasia that occurred after 6.6 D of myopia correction with an estimated posterior corneal thickness of 160 μm. No evidence of forme fruste keratoconus or unusually thin cornea was noted to explain the occurrence of this complication. Based on retrospective analysis of similar cases of corneal ectasia after high magnitudes of myopia treatment, several safety guidelines have been proposed, including leaving a minimal residual stromal bed thickness of 250 μm and at least 50% of the initial corneal thickness.

^{ 6 }

^{ 8 }and PRK,

^{ 9 }

^{ 10 }

^{ 11 }in which the 250-μm rule was not violated, suggesting that factors other than residual bed thickness could play a causative role. Argento et al.

^{ 10 }and Pallikaris et al.

^{ 11 }have postulated that the amount of tissue removed may be another factor influencing the development of ectasia. The use of equations 11 13 14 and 17 in similar studies will be valuable in determining the specific situations in which the volume of ablated tissue is a major contributing factor to keratectasia after LASIK surgery.

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**

**Figure 5.**

**Figure 5.**

Optical Zone Diameter (mm) | Finite Integration (Hyperopia) | Volume Approximation | Finite Integration (Myopia) |
---|---|---|---|

4 | 0.0345 | 0.039 | 0.0356 |

4.5 | 0.0558 | 0.0625 | 0.0581 |

5 | 0.0859 | 0.0953 | 0.0904 |

5.5 | 0.1274 | 0.1395 | 0.1355 |

6 | 0.1828 | 0.1975 | 0.197 |

6.5 | 0.2556 | 0.2721 | 0.2794 |

7 | 0.3496 | 0.366 | 0.3883 |

7.5 | 0.4692 | 0.4823 | 0.5304 |

8 | 0.6198 | 0.6243 | 0.7144 |

**Figure 6.**

**Figure 6.**

*Invest Ophthalmol Vis Sci*

*.*2000;41(4)S679.Abstract nr 3614

*J Refract Surg*

*.*2000;16:407–413. [PubMed]

*J Cataract Refract Surg*

*.*1988;14:46–52. [CrossRef] [PubMed]

*Arch Ophthalmol*

*.*2000;118:1179–1186. [CrossRef] [PubMed]

*J Cataract Refract Surg*

*.*1998;24:190–195. [CrossRef] [PubMed]

*Ophthalmology*

*.*1999;106:406–410. [CrossRef] [PubMed]

*J Cataract Refract Surg*

*.*1999;25:582–586. [CrossRef] [PubMed]

*Ophthalmology*

*.*2001;108:1415–1422. [CrossRef] [PubMed]

*J Cataract Refract Surg*

*.*2000;26:872–878. [CrossRef] [PubMed]

*J Cataract Refract Surg*

*.*2001;27:1440–1448. [CrossRef] [PubMed]

*J Cataract Refract Surg*

*.*2001;27:1796–1802. [CrossRef] [PubMed]