From Cole Eye Institute (Roy, Dupps), and the Department of Biomedical Engineering, Lerner Research Institute, Cleveland Clinic (Dupps), Cleveland, Ohio.
Supported in part by grant 1KL2RR024990 from the National Center for Research Resources (NCRR), a component of the National Institutes of Health (NIH) and NIH Roadmap for Medical Research, a Cleveland Clinic Foundation Innovations Grant, and a Research to Prevent Blindness Challenge Grant to the Department of Ophthalmology, Cleveland Clinic Lerner College of Medicine and Case Western Reserve University. Dr Dupps is a recipient of a Research to Prevent Blindness Career Development Award.
The authors have no financial interest in the materials presented herein.
Study concept and design (A.S.R., W.J.D.); data collection (A.S.R., W.J.D.); interpretation and analysis of data (A.S.R., W.J.D.); drafting of the manuscript (A.S.R., W.J.D.); critical revision of the manuscript (A.S.R., W.J.D.); statistical expertise (A.S.R., W.J.D.); obtained funding (A.S.R., W.J.D.); administrative, technical, or material support (A.S.R., W.J.D.)
Correspondence: William J. Dupps, Jr, MD, PhD, Cole Eye Institute, Cleveland Clinic, 9500 Euclid Ave, i-32, Cleveland, OH 44195. Tel: 219.444.8396; E-mail: email@example.com
One of the major challenges in refractive surgery is the accurate prediction of postoperative refractive error. Munnerlyn et al1 provided an essential framework for calculating the necessary pattern of tissue removal for photorefractive keratectomy (PRK) and analogous procedures such as LASIK. Empirical adjustments to this surgical algorithm have been necessary to compensate for error arising from the implicit assumption that the cornea is biomechanically and biologically inert. An increasing awareness of the limitations of a simple shape-subtraction model has sparked interest in better characterization of the biomechanical and reparative responses of the cornea responsible for phenomena such as refractive over- and undercorrection, keratectasia, corneal haze, and intraocular pressure (IOP) measurement artifacts.2–7
The finite element method (FEM) is a computational tool that can be used to represent the geometric, biomechanical, and biological characteristics of a structure. The cornea has been modeled primarily as a linear or non-linear hyperelastic and homogenous material with near incompressibility.8–12 Recent studies have considered the spatial variation of fibril orientation in biomechanical models to better account for directional differences in corneal elastic properties.13–15 These models have used simplified corneal geometries (spherical or ellipsoidal) constrained at the limbus to calculate the change in corneal curvature in eyes that have undergone LASIK. Finite element method simulations depend heavily on the material properties chosen for the cornea from a limited number of ex vivo measurements obtained under wide-ranging experimental conditions. Accordingly, a major limitation in modeling corneal behavior is the considerable variability in reported corneal elastic properties and uncertainty about the in vivo material properties of the eye under investigation.
In addition to uncertainty regarding the accuracy of material property assumptions in FEM modeling, questions about the impact and validity of simplifying assumptions about the boundary conditions of the cornea remain. Most computational models of the cornea treat the limbus as an infinitely stiff boundary incapable of motion. Some have argued that the limbus be understood as a flexible reinforcing structure that maintains corneal curvature in response to IOP changes under physiological conditions.16–18 A previous FEM analysis of postoperative LASIK shape included the cornea and only a partial segment of the sclera to model limbus displacements.10
The purpose of this study was to model the changes in corneal power in a representative computational model of the whole human eye. Specifically, the study focuses on 1) a sensitivity analysis of variable corneal hyperelastic properties on corneal stresses, displacements, and refractive power both before and after LASIK for myopia, 2) the effect of IOP changes on corneal curvature and power, and 3) the influence of an unconstrained limbus on these behaviors. The current model is unique in that it derives its geometry from magnetic resonance images (MRI) of a human eye19,20 and accounts for the biomechanical effects of the whole eye on corneal shape without a priori constraints on behavior at the corneoscleral junction. Although the model emphasizes the acute biomechanical effects of flap creation and stromal photoablation, the cohesive tensile strength of human LASIK wounds21 was incorporated to model LASIK after nominal wound healing has occurred.
Materials and Methods
Preoperative LASIK Geometry
An axi-symmetric model of the whole eye was created. The geometric profile of the cornea, sclera, iris, and lens were obtained from MRI of the whole eye.19,20 The images were digitized and processed in a computer-aided design package to obtain the coordinates of the profiles approximated as smoothed splines (ADINA v8.4; ADINA R&D Inc, Watertown, Mass). Sigal et al22,23 provided a comprehensive summary of Young’s modulus, Poisson’s ratio, and thicknesses of various tissue components of the human eye, including the sclera, choroid, retina, iris, and ciliary body. The lens and zonules were also modeled, as they may impose elastic restraints on limbus displacement that impact corneal curvature. Using the thickness values reported by Sigal et al,22,23 the geometries of individual ocular tissue components were defined by superimposing these thicknesses on the geometric profiles of the cornea and sclera. For simplification, it was assumed that the sclera, choroid, and retina have a uniform thickness.24 Previously reported geometric dimensions of the optic nerve head region22,23 were used to construct the geometries of lamina cribrosa, postoperative laminar tissue, cup depth, and pia mater. The thickness of the iris was obtained from Huang et al.25 It was also assumed that the ciliary body had the same thickness as the iris.24 A summary of the geometric thicknesses of the ocular structures is provided in Table 1. Figure 1A shows the geometry of the whole eye used for the analyses.
Table 1: Thicknesses of Tissue Layers
Figure 1. A) Whole-Eye Model Acquired from in Vivo Magnetic Resonance Imaging Showing the Various Tissue Components and the Cornea with LASIK Flap Detail (inset). B) Finite Element Model Mesh in the Central and Paracentral Zone of the Cornea. the LASIK Flap Can Be Seen as a Single Layer.
Postoperative LASIK Geometry
The LASIK flap was modeled as a hingeless, 8-mm diameter flap of uniform thickness (125 μm) in each model. Photoablation was simulated by reducing the corneal thickness profile within the central 6-mm diameter optical zone to obtain a 2.00-, 4.00-, and 6.00-D refractive change according to Munnerlyn’s equations.1 The maximum depth of tissue removed was 25.3, 50.4, and 75.4 μm for a refractive change of 2.00, 4.00, and 6.00 D, respectively. The lamellar wound was modeled as a thin layer, 10 μm in thickness and 8 mm in diameter (inset of Fig 1A).
Material Properties of Tissues
Because a 2.00-D model was used, the cornea was modeled as an isotropic, hyperelastic, and incompressible material using the Mooney-Rivlin stress versus stretch ratio equation.9 Uniaxial stress-stretch ratio data derived from human donor corneas were used.26,27 Two different material stress-strain data sets26,27 were used to compare corneal behavior over a range of tissue properties that might be encountered in corneas representative of the sample populations from which these data were derived (ie, donors without frank corneal disease). Stress-strain data can be obtained from strip and inflation testing, the latter providing a more isotropic response than the former.28 As one of the goals of this study was to analyze the effect of varying corneal elasticity on corneal biomechanics, two independent sets of stress-strain data were used to represent patient-to-patient variation in elasticity without any bias to the testing methods.
The stress-stretch ratio data from Wollensak et al26 (hereafter designated “W”) represents lower corneal stiffness than that measured by Hoeltzel et al27 (designated “H”) using similar methods. The LASIK wound was also modeled as a hyperelastic and incompressible material. The cohesive tensile strength of the flap wound has been reported to vary from 2.4% to 30% of the corresponding value for the native corneal tissue based on ex vivo measurements by Schmack et al.21 In our models, the wound elastic material constants were fixed at 2.4% of native tissue material constants for W and H to simulate the weaker wound conditions that predominate across the flap interface. The material properties of the other tissues used in this study are summarized in Table 2. Thus, only the corneal and flap wound elastic properties were varied among the models whereas the properties of the remaining tissues were held constant. A summary of the analyses and associated nomenclature is provided in Table 3. For mnemonic purposes, H properties can be thought of as representing “high” stiffness whereas W properties can be remembered as relatively “weak.”
Table 2: Material Properties of Ocular Tissues
Table 3: Summary of Finite Element Models
Boundary Conditions and Finite Element Method
The displacement boundary conditions are shown in Figure 1A. To maintain symmetry, the axis was restrained along the radial direction (r) while the posterior end of the optic nerve head region was restrained in the axial direction (z). To assess the importance of changes in corneal boundary conditions, two additional simulations were performed with only the cornea using the two different corneal elastic properties (W and H) in a fixed-limbus model. A uniform IOP (after preliminary simulations of intraocular flow revealed spatial IOP differentials between the anterior and posterior much less than 1 mmHg)24 ranging from 1 to 30 mmHg was applied on the surfaces bounding the anterior and posterior chamber of the eye (blue arrows in Fig 1). The model was meshed and solved in a commercial finite element package (ADINA v8.4). The 2.00-D, axi-symmetric, bilinear, quadrilateral element was used. The convergence criterion for residuals of the energy and displacement norm was set at 10−4. Mesh sensitivity analysis was performed and the total numbers of elements used in the final analyses were 15,815 and 17,108 in the pre- and postoperative LASIK models, respectively.
Apical and Tangential Power Calculations
The apical power is given by Fa = (nc–1)/Ra, where Ra and nc are the apical radius of curvature and refractive index of the cornea, respectively.33 The anterior surface was approximated by a conic equation about the z-axis (the elevation axis) and up to radius (r) of 1.5 mm: .32 Here, e is the eccentricity. The local radius of curvature and the tangential power is determined by: and F = (nc – 1)/R, respectively.33 These equations were chosen for analyzing the central 3-mm diameter zone of the cornea, as they give an accurate estimate of power in the central optical zone.33
The following results focus primarily on the effect of varying corneal stiffness and inclusion or exclusion of whole-eye properties on corneal displacements and associated changes in corneal power in myopic LASIK. The optical effects of isolated changes in IOP and corneal stiffness are also evaluated. The change in axial power at the corneal apex as a function of IOP before and after LASIK is summarized for all attempted corrections (2.00, 4.00, and 6.00 D), whereas corneal displacements, stresses, and tangential power changes are analyzed before and after LASIK with an attempted correction of 6.00 D.
Displacement of Anterior Corneal Surface
Figure 2 shows corneal displacement magnitudes sampled at the center (point 1), paracenter (point 2), and limbus (point 3) for preoperative LASIK (W) (Fig 2A) and postoperative LASIK (6D-W) (Fig 2B), respectively. Each plot also shows the circumferential stretch ratio analyzed at the same points. The circumferential stretch ratio is a measure of strain along the arcuate path of a meridional corneal cross-section approximated by the corneal curvature and corresponding to the predominant central stromal collagen orientation.5 The stretch ratio is used mathematically to relate stress to strain in soft tissues similar to the cornea. A high stretch ratio indicates a high strain and vice versa. In previous studies assuming a constrained limbus with no allowance for lateral or axial limbal displacement, the central cornea showed more displacement than paracentral cornea.8–15 A similar trend is observed in the current model when the limbus is constrained and only the cornea is modeled (preoperative LASIK [W] cornea-only model, represented by open data markers and dotted lines in Fig 2A). At an IOP of 15 mmHg and 30 mmHg for the preoperative LASIK (W) cornea-only model, displacements at the center are 18 and 40.7 μm greater than the paracenter, respectively. Thus, constraining the limbus and excluding extracorneal structures from the model has a significant impact on the spatial pattern, magnitudes of displacement, and power change in simulated LASIK.
Figure 2. Displacement and Stretch Ratio as a Function of Intraocular Pressure (IOP) at the Center, Paracenter, and Limbus in A) a Preoperative LASIK Low-Stiffness (W) Whole-Eye Model and Cornea-Only (fixed Limbus) Model, and B) a Postoperative LASIK 6D-W Whole-Eye Model. Increasing IOP Favors Paracentral Corneal Displacement in Preoperative Whole-Eye Models. Predominant Displacements Are in the Central Cornea in Unoperated Cornea only when the Cornea Is Modeled Without Consideration of Extracorneal Structures. In a Lower Stiffness Cornea, 6.00-D Myopic LASIK Causes a Shift in the Region of Predominant Displacement to the Corneal Center.
Figure 2 illustrates that corneal displacements increase with increasing IOP and paracentral displacements are greater than central and limbal displacements in the unoperated whole-eye model with lower corneal stiffness. For the preoperative LASIK (W) model, displacements at the paracenter are 2 and 6 μm greater than at the center at IOP of 15 and 30 mmHg, respectively. For the 6D-W model, however, central displacements predominate and are 16 and 22 μm greater than paracentral displacements at IOP of 15 and 30 mmHg, respectively. Increased displacements in the postoperative LASIK state are due to the structural loss of ablated tissue and presence of a flap interface wound that is considerably weaker than the native cornea. Anteriorly directed paracentral displacements are expected to favor corneal flattening whereas central displacements favor central steepening. In models with lower corneal stiffness, the circumferential strain ([Circumferential Stretch Ratio – 1] × 100) was no more than 3% in both the preoperative LASIK (W) and 6D-W states (see Fig 2).
For whole-eye preoperative LASIK (H) and 6D-H models, displacements are considerably smaller due to stiffer corneal properties. At an IOP of 15 mmHg, the displacement of the center in preoperative LASIK (H) (Fig 3A) is 25.6 μm less than corresponding central displacement in preoperative LASIK (W) (see Fig 2A). In contrast to the lower stiffness W model, the limbus (point 3) shows greater displacement than both the center and paracenter for all IOP above 15 mmHg in both preoperative LASIK (H) and 6D-H. Due to the greater resistance to corneal deformation imparted by a stiffer cornea, myopic LASIK does not lead to substantial increases in limbal displacement when comparing preoperative LASIK (H) (see Fig 3A) and 6D-H (Fig 3B). Similar to lower stiffness (W) pre- and postoperative LASIK models, paracentral cornea is displaced more than central cornea in both preoperative LASIK (H) and 6D-H models. For the preoperative LASIK (H) model, this displacement difference between the paracentral and central cornea is 30.4 μm at a pressure of 15 mmHg and 58.5 μm at 30 mmHg. For 6D-H, the corresponding differences are 23.6 μm and 53 μm. Circumferential strains are also considerably lower in the preoperative LASIK (H) and 6D-H models (maximum of 1%) than in the lower stiffness models and are therefore also sensitive to corneal material properties. The preoperative LASIK (H) yields strain values comparable to the 1% circumferential strains measured in whole donor globe inflational studies.17
Figure 3. Displacement and Stretch Ratio as a Function of Intraocular Pressure (IOP) at the Center, Paracenter, and Limbus. A) Preoperative LASIK (H) and Cornea-Only (fixed Limbus) and B) 6D-H Postoperative LASIK Models. The Stiffer Hyperelastic Behavior of the Cornea Favors Paracentral and Limbal Displacement over Central Corneal Displacement in Both the Unoperated and Postoperative LASIK States, but only when Extracorneal Structures Are Considered.
Change in Anterior Chamber Depth
Figure 4 illustrates the impact of corneal material properties on the axial anterior chamber depth, measured from the posterior cornea to the anterior lens surface. Figure 4A shows that in a lower stiffness corneal model, increasing IOP produces a deepening of the anterior chamber that is explained in part by forward displacement of the posterior corneal surface. This effect is exacerbated by a 6.00-D LASIK treatment. In Figure 4B, a contrasting trend is seen in the high corneal stiffness model in which the anterior chamber depth decreases with increasing IOP and with 6.00-D myopic LASIK, owing now to a posterior shift in the corneal endothelial surface. Due to increasing IOP, both the lens and the cornea move forward in all models. However, in the stiffer model, the forward lens displacement is larger than the corneal displacement and this leads to a decrease in anterior chamber depth. In the weaker model, the corneal forward displacement is larger than the lens displacement and this leads to an increase in anterior chamber depth. These trends follow from the behaviors noted above and are of relevance to the observation that excessive posterior corneal “vaulting” in LASIK may be a predictor of ectasia that is precipitated by an abnormally weak cornea.34 Thus, in this whole-eye FEM model, a cornea with lower stiffness shows more displacement in the paracentral cornea than the central and limbal zones and a cornea with high stiffness shows more displacement near the limbus.
Figure 4. Change in Anterior Chamber Depth (ACD). A) Weaker Cornea Before and After 6.00-D LASIK and B) Stiffer Cornea Before and After 6.00-D LASIK. IOP = Intraocular Pressure
Stress and Displacement Distribution in the Cornea
In both preoperative LASIK (W) (Fig 5A) and 6D-W (Fig 5C) models loaded to an IOP of 15 mmHg, displacements are greatest in the paracentral cornea as shown numerically in Figure 2A. Owing to differential central ablation, myopic LASIK results in a central shift in the location of peak displacement (shown by the color contour shift in the figure) towards the axis. Of specific interest is the effective stress distribution22,23 (von Mises stress) in preoperative LASIK (W) (Fig 5B) and 6D-W postoperative LASIK (Fig 5D) models near the anterior surface. A distinct stress discontinuity is seen in the 6D-W stress contour plot due to the presence of the interface. In both preoperative LASIK (H) (Fig 6A) and 6D-H (Fig 6D) models, displacement contour plots at an IOP of 15 mmHg show peak displacement near the limbus as the cornea is stiffer than the posterior segment of the globe in these models. Concentrated stresses develop in all models in the central posterior cornea after 6.00-D LASIK. The stress distributions in Figures 6B and 6D for the stiffer cornea (H) are similar to those in Figures 5B and 5D, although the magnitudes are higher.
Figure 5. Contour Plots of Displacement and Effective Stress at Intraocular Pressure (IOP) = 15 mmHG. A) Displacement of Preoperative LASIK (W), B) Displacement of 6D-W, C) Effective Stress of Preoperative LASIK (W), and D) Effective Stress of 6D-W. Central and Paracentral Displacements Are Higher than the Limbus. The Flap Results in Larger Displacements Centrally. Displacement Is in Millimeters, Stress Is N/m2.
Figure 6. Contour Plots of Displacement and Effective Stress at Intraocular Pressure (IOP) = 15 mmHg. A) Displacement of Preoperative LASIK (H), B) Displacement of 6D-H, C) Effective Stress of Preoperative LASIK (H), and D) Effective Stress of 6D-H. Due to the Stiffer Cornea, the Limbus Shows the Maximum Displacement. Displacement Is in Millimeters, Stress Is N/m2.
Change in Power (Apical and Tangential Power) as A Function of Corneal Stiffness
The optical and clinical relevance of changes in corneal geometry can be better illustrated by calculating and comparing corneal power in pre- and postoperative LASIK models. Figure 7A shows the apical corneal power for the preoperative LASIK (W) state and 2D-W, 4D-W, and 6D-W postoperative LASIK states. The power decreases with increasing IOP (see Fig 7A). After an attempted correction of 2.00 D, the power decreases from 46.60 D at an IOP of 10 mmHg to 44.40 D at 30 mmHg, a decrease of 2.20 D (see Fig 7A). After a 6.00-D correction, the power decreased from 43.40 D to 41.60 D, a decrease of 1.80 D. Similarly, the power decreases with increasing IOP in all stiffer cornea models (H), but at a rate that reflects the linear elastic material property assumptions of the sclera rather than the nonlinear properties of the cornea. For a 2.00-D and 6.00-D correction, the power decreases by 3.20 D and 3.70 D, respectively. Thus, the stiffer cornea imparts corneal behavior that is more influenced by displacements at the corneoscleral junction and increasingly representative of the properties of the surrounding sclera.
Figure 7. A) Apical Power of the Lower Stiffness Cornea (preoperative LASIK [W], 2D-W, 4D-W, 6D-W) and Stiffer Cornea (preoperative LASIK [H], 2D-H, 4D-H, 6D-H) as a Function of Intraocular Pressure (IOP). Solid Lines Represent W, Dotted Lines Represent H. B) Difference in Apical Power Between Stiff and Lower Stiffness Cornea, ie, H–W at Iop of 15 mmHg. As IOP Increases, the Weaker Cornea (W) Flattens More than the Stiffer Cornea (H). However, the Flap Causes a Reversal in Trend with the Weaker Cornea Flattening Less.
Inspection of the trends in Figure 7A along a single IOP value (eg, 15 mmHg) illustrates the impact of corneal material properties on apical corneal power with different magnitudes of attempted correction. The stiffer cornea is more resistant to central corneal biomechanical flattening than the lower stiffness cornea up to an attempted myopic correction of 6.00 D. Stated differently, given the same flap geometry, corneal thickness, and ablation pattern geometry, the cornea with lower stiffness undergoes greater central power reduction. In Figure 7B, the magnitude of the apical power difference at 15 mmHg IOP is plotted against attempted correction and is shown to decrease approximately linearly from 0.84 D prior to LASIK to 0.17 D after 6.00-D LASIK for moderate myopia. When the trend is extrapolated to a higher 8.00-D myopic correction with a thinner postoperative cornea, the difference will be −0.06 D at 15 mmHg (see Fig 7B). This implies that for higher myopic corrections, the stiffer cornea will be flatter than a lower stiffness cornea at physiologic IOP. This highlights the sensitivity of corneal power to tissue properties before and after LASIK.
Figure 8 shows the achieved myopic correction expressed as the apical power change from the preoperative LASIK condition to the postoperative LASIK state for 2.00-, 4.00- and 6.00-D corrections in stiff (H) and lower stiffness (W) corneas. Central flattening is seen in all models, but the lower stiffness cornea (solid line) demonstrates absolute undercorrection in every case as well as undercorrection relative to a stiffer cornea (dashed line). However, for 2.00-D and 4.00-D correction, the stiffer cornea shows slight overcorrection in terms of apical cornea power with an increasing tendency toward secondary hyperopia at higher IOP. The lower stiffness cornea (solid line) demonstrates a different trend, with greater tendency toward residual myopia both at higher levels of correction and at higher IOP. Thus, what is seen here is the effect of both changing flap geometry and corneal thinning on achieved corneal power after LASIK as compared to trends discussed in Figure 7B. These trends and their relevance to postoperative LASIK ectasia and topographic behavior after refractive surgery during IOP elevations are addressed further in the Discussion section.
Figure 8. Achieved Myopic Correction as a Function of Intraocular Pressure (IOP), Expressed as the Change in Apical Power from the Pre- to Postoperative LASIK Condition. Solid Lines Represent Change in the Lower Corneal Stiffness (W) Model. Dotted Lines Represent Change in the Stiffer Corneal (H) Model.
Figures 9 and 10 show the tangential (instantaneous) power before and after 6.00-D LASIK at 15 mmHg and after an IOP increase to 30 mmHg in the postoperative LASIK model for the lower and higher stiffness corneas, respectively. The power decreases indicate central flattening from preoperative LASIK (W) in Figure 9A to postoperative LASIK (6D-W) in Figure 9B at 15 mmHg. This is shown more clearly by way of the difference map in Figure 9D, which indicates a greater tangential power decrease in the center (5.20 D) than in the periphery (4.86 D) of the 3-mm diameter optical zone. With an increase in IOP from 15 to 30 mmHg in 6D-W, there is a further decrease in power, indicating enhanced flattening with IOP increase after LASIK. This is shown in the difference map in Figure 9E. For the stiffer cornea, the power decreases indicating flattening from preoperative LASIK (H) in Figure 10A to postoperative LASIK (6D-H) in Figure 10B at 15 mmHg. This is shown in the difference map (Fig 10D). This flattening effect seen in the stiffer cornea was similar to that in the weaker cornea from before to after LASIK. With an increase in IOP from 15 to 30 mmHg in 6D-H, there is a decrease in power, more in the center than in the periphery. This is shown in the difference map in Figure 10E. Further, with IOP increase from 15 to 30 mmHg, the magnitude of corneal flattening for 6.00-D myopic correction is more in the stiffer model than the weaker corneal model (see Figs 9E and 10E).
Figure 9. Tangential Power Maps for Preoperative LASIK (W), 6D-W at 15 and 30 mmHg, and Their Difference Map (postoperative–Preoperative LASIK).
Figure 10. Tangential Power Maps for Preoperative LASIK (H), 6D-H at 15 and 30 mmHg, and Their Difference Map (postoperative–Preoperative LASIK). Compared to Weaker Cornea and Postoperative LASIK, the Stiffer Cornea (H) Flattens Less than the Weaker Cornea as the Intraocular Pressure Increases from 15 to 30 mmHg.
In this study, we present a finite element model of the whole globe and investigate the sensitivity of LASIK-mediated corneal refractive power changes to corneal elasticity. The geometry and material properties of the entire globe were incorporated into the analysis to model limbus displacement without a priori constraints on limbus motion. The lower stiffness corneal models (W) showed more displacement in the paracentral and central zones whereas the stiffer corneal models (H) demonstrated more displacement near the limbus. Compared with a fixed-limbus model where only the cornea is considered, the whole-eye models showed significantly higher displacements near the paracentral zone. Some past studies have analyzed the effect of limbus motion without considering the effect of whole eye biomechanics on corneal curvature and found a difference of less than 1% between peak displacement at the apex.9,10,13,14 The presence of a LASIK flap interface with nominal healing and myopic photoablation patterns led to greater corneal displacements and curvature changes in the central and paracentral zone coupled with stress concentrations near the wound layer. These results show that variations in native corneal elasticity, localized reductions in corneal elastic strength associated with the LASIK flap, and limbus motion in the context of whole-eye biomechanics are important determinants of corneal power change and could represent clinical sources of outcome variability not accounted for in laser vision correction algorithms.
To this point, the model predicts different degrees of over- or undercorrection after myopic LASIK depending on corneal hyperelastic properties and the magnitude of attempted correction. In low and moderate myopic corrections, the model predicts (see Fig 8) mild overcorrection (secondary hyperopia) in stiffer corneas. This model behavior is consistent with the clinical observation that surgeon offsets are often required to avoid overcorrection of myopia, particularly in older patients who may have stiffer corneas.4,35,36 This behavior is consistent with a biomechanical model of corneal curvature change proposed by Dupps and Roberts4,37 and expanded here. In this model, central corneal flattening has been proposed to occur due to relaxation of lamellar tension peripheral to the flap side-cut and ablation zone, stress-stiffening in the remaining stress-bearing layers of the residual stromal bed, and centripetal (laterally directed) strains. These changes collectively favor central flattening, which is observed in the current FEM analysis. The current FEM model invokes a solid mechanics approach and does not incorporate a mechanism for peripheral thickening of the cornea, which might favor further central flattening due to localized stromal imbibition of fluid.37
With higher attempted corrections and deeper photoablation, a tendency toward greater residual myopia in lower stiffness corneas (see Fig 8), or lower amounts of secondary hyperopia in stiffer corneas, is predicted by the model. In an eye bank tissue study of successively deeper phototherapeutic keratectomies, Litwin et al38 demonstrated that deeper ablations might lead to relative central weakening and a shift toward central steepening that partially offsets the flattening tendency. In living eyes, a tendency toward myopic regression on the order of 0.50 D after LASIK has been reported by several investigators, with regression averaging approximately 1.00 D in attempted myopic corrections greater than 10.00 D.39–42
Although epithelial and stromal remodeling are probably important in myopic regression, it is also possible that regression represents a protracted viscoelastic change in the geometry of the cornea due to focal weakening and stress concentration that is greater in higher myopic corrections. In our hyperelastic computational model with weaker corneal properties (see Fig 8), myopic undercorrections of 0.50 D and 1.00 D can be seen for attempted corrections of 4.00 D and 6.00 D, respectively, at physiological IOP. In the stiffer model, the tendency shifts away from myopic undercorrection and toward overcorrection due to greater resistance of the central cornea to deformation relative to the limbus. Weaker corneal properties favor trends toward increasing stresses in the posterior/central cornea (see Figs 5 and 6), forward corneal displacement, and relative central steepening with deeper myopic corrections that may ultimately contribute to postoperative LASIK instability or ectasia. The clinical observation that a history of previous LASIK enhancement is a risk factor for ectasia43 may stem not simply from the risk of additional tissue removal but from the tendency, demonstrated here, for weaker corneas to appear under-corrected after primary LASIK. The threshold beyond which progressive viscoelastic and plastic deformation occurs is a major function of corneal properties,34,37 and these results suggest the importance of patient-to-patient variability in corneal elastic properties as a driver of patient outcomes in refractive surgery.
The current study was not specifically designed to model the behavior of the cornea in the pathologically weak or artificially stiffened states. However, our sensitivity analysis provides initial insight into differences in corneal behavior that might be expected due solely to differences or changes in global corneal elastic properties. To the extent that corneal stiffness is lower than normal in keratoconus (as demonstrated by Andreassen et al44 in whole-button specimens), the sensitivity analysis suggests that maximum displacements will occur in the central and paracentral cornea, where topographic cones are clinically most prevalent. This behavior is predicted in the case of weaker corneas without corneal thinning as a prerequisite.
Conversely, corneal stiffening by collagen cross-linking techniques used for treatment of corneal ectasias45–48 can be modeled by selectively altering the corneal properties within the whole-eye model. In the thinnest postoperative LASIK corneas (≥6.00 D myopic corrections), stiffening reduced central corneal strain and shifted the focus of maximum strain toward the limbus, ultimately favoring central corneal flattening and a hyperopic optical shift. This aspect of the model’s behavior, which is reflected throughout the simulations and in the apical power trends in Figures 7A and 7B, provides an explanatory mechanism for the previously unexplained observation that central corneal power not only stabilizes but typically decreases significantly in thin, keratoconic eyes after riboflavin/ultraviolet A (UVA) collagen cross-linking.45–48 The observation that this flattening may continue for months after photochemical stiffening may be a result of gradual viscoelastic adjustments to the altered distribution of stresses imposed by selective stiffening of the cornea relative to the surrounding sclera. Further model adaptations to account for the localized nature of corneal property changes in keratoconus and collagen cross-linking are needed to address this issue more completely, but the magnitude of hyperopic shift predicted to occur with global corneal stiffening is similar to that reported after clinical UVA/riboflavin cross-linking.47
The current model is based on a two-dimensional, isotropic approach and does not account for three-dimensional astigmatic effects or meridional differences in geometry or material properties. Similar to previous studies,8–15 the model uses corneal elasticity data measured in grossly normal but excised donor corneas. The hyperelastic data used for this study were selected on the basis of their quality and the similarity of the conditions under which they were obtained, and they are thought to represent a range of values that might be encountered as a result of normal patient-to-patient variation. However, the average age of donor tissue is higher than the average age of refractive surgery patients, and the extent to which the stiffness of the tissue is representative of properties in normal living corneas is not known. Although this uncertainty could affect the accuracy of the model when behavior is compared in absolute terms to empirical data, several known clinical trends in corneal behavior have been reproduced by the model with reasonable fidelity in this report. The properties of individual corneal layers could also contribute significantly to the biomechanical response after LASIK and need to be evaluated in future studies. Our ongoing efforts include considering the three-dimensional pre- and postoperative biomechanical responses using clinical topographic measurements, incorporating fibril orientation into the hyperelastic model, and characterizing the optical consequences of biomechanical responses in patient-specific models.
We assume for the purposes of these simulations that the material properties of the cornea are not immediately altered by LASIK (with the exception of the flap interface) and do not account for possible alterations in hyperelastic constants as a result of LASIK. Our model does not capture the clinical phenomenon of increased risk of myopic overcorrection at higher attempted corrections, an effect that may depend more on stromal dehydration and increased ablative efficiency during protracted treatments than on corneal material properties.4 Also, the cornea’s hydrophilic matrix and porous microstructure confer viscoelastic properties that are not fully characterized by an elastic model4,7 and that may be important in the time-dependence of corneal responses. A recent study has shown that corneal biomechanics cannot be completely delineated from scleral deformation in an in vivo state.49 Our study strongly supports a role for the sclera as a factor in corneal refractive outcomes. Although we have modeled the sclera as a linearly elastic structure, future studies might benefit from parameterization of sclera elasticity in finite element models using hyperelastic properties. The question of whether non-linear sclera elasticity would produce non-linear displacement of the limbus and the related question of its importance in modeling remains unanswered; one ex vivo study has demonstrated a linear pressure-volume relation (or change in radius) in whole-eye globes, which suggests that the intact whole-eye behaves as a linear elastic material.50 Our initial simplifying assumptions have allowed straightforward sensitivity analyses of corneal hyperelastic material properties and limbal boundary conditions that have confirmed the importance of each. These are important steps toward extending the whole-eye model to the three-dimensional anisotropic and viscoelastic realms.
Finite element models have been proposed as tools for refractive surgery planning, but this and other work suggests a need for accurate in vivo characterization of corneal geometry and material properties. Such capabilities are important because of 1) the likelihood of inter-individual variability in corneal elastic properties and 2) the unsuitability of ex vivo elastic properties as a surrogate for individual in vivo properties.4,7 The current analysis also suggests that the behavior of the corneoscleral limbus could be important in predicting optically important differences in the corneal response to LASIK and to simple IOP alterations before or after LASIK. Several technologies for in vivo tissue elasticity characterization are under investigation and could play an important part in improving the safety and predictability of refractive surgery.
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Thicknesses of Tissue Layers
|Lamina cribrosa thickness at the axis||3023,24|
|Optic nerve head cup depth*||33022,23|
|Lens thickness at the center||4.44 mm19,20|
Material Properties of Ocular Tissues
|Tissue||Material Constants (MPa)|
| Wollensak et al26||C1=0.555, C2=−0.514,26,27|
| Hoeltzel et al27||C1=10.005, C2=−1014,26,27|
|Post-lamina neural tissue||0.0322,23|
Summary of Finite Element Models
|Before LASIK||Before – LASIK (W) & (H)|
| 2.00 D||2D-W, 2D-H|
| 4.00 D||4D-W, 4D-H|
| 6.00 D||6D-W, 6D-H|