Finite element simulations of laser refractive corneal surgery

Abstract

We setup a mechanically based finite element model to evaluate the change in the shape of the human cornea induced by ablation of stromal tissue. By considering the deformability of the cornea, the model computes the change of the dioptric power resulting from ablative laser surgery. We use a previously developed 3-D finite element model of the human cornea (Pandolfi and Manganiello in Biomech Model Mechanobiol 5:237–246, 2006). The solid geometry is discretized into finite elements by an automatic procedure which recovers the unloaded configuration. The geometry is defined in parametric form and can be characterized by individual geometrical data when available. A two-fiber reinforced hyperelastic material model, which accounts for the organization of the anisotropic collagen structure, is adopted to describe the stromal tissue. For the simulation of laser refractive surgery of myopic and astigmatic eyes, a geometrical correction of the corneal profile is included into the code. We show two examples of application of the model to the reshaping of a myopic and an astigmatic eye. Numerical results provide the postoperative shape of the cornea, the corrected refractive power, and the distribution of the stress throughout the stromal tissue.

Appendix

The second Piola-Kirchhoff stress tensor S and the Cauchy stress tensor $\varvec{\sigma}$ are given by

$$ \user2{S} = 2 {\frac{\partial \Uppsi}{\partial \user2{C}}}, \qquad \varvec{\sigma} = J^{-1} \user2{F} \user2{S} \user2{F}^T. $$

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The stress tensor S decouples in the form

$$ \user2{S} = \user2{S}_{\rm vol} + \user2{S}_{\rm dev} = J p \user2{C}^{-1} + J^{-2/3} {\mathbb P} : \overline{\user2{S}}, $$

where

$$ p = {\frac{d \Uppsi_{\rm vol}}{d J}}, \qquad \overline{\user2{S}} = 2 \left[ {\frac{\partial \Uppsi_{\rm iso}}{\partial \overline{\user2{C}} }} + {\frac{\partial \Uppsi_{\rm aniso}}{\partial \overline{\user2{C}} }} \right], \qquad {\mathbb P} = {{\mathbb{I}}} - {\frac{1}{3}} \user2{C}^{-1} \otimes \user2{C}. $$

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