Refractive Change Induced by the LASIK Flap in a Biomechanical Finite Element Model

B I O M E C H A N I C S S E C T I O N

Chaitanya Deenadayalu; Barzin Mobasher, PhD; Subby D. Rajan, PhD; Gary W. Hall, MD

ver recent years LASIK has become a popular surgi- ABSTRACT cal alternative for the correction of refractive errors such as myopia, hyperopia, and astigmatism. In this PURPOSE: To study the effect of varying four param- O eters on the refractive change induced by the LASIK technique, a hinged flap is created and folded back, flap. and the exposed stroma is photoablated using an excimer laser so that the curvature of the central cornea is reshaped, to METHODS: Using a variety of patient-specific data such improve uncorrected vision. as topography, pachymetry, and axial length, a finite Due to the inherent asymmetry of the corneal geometry, element model is built. The model is used in a non- a quantitative study of the topography is complex. Analyti- linear finite element analysis to determine the response and change in optical power of the cornea as cal studies have been carried out using simulated picosecond a function of a material property of the cornea (corneal laser keratomileusis involving two mathematical models. elasticity), flap diameter and thickness, and intraocular One was with a shape-subtraction paradigm, based on the pressure. assumption that the biomechanical response of the cornea is negligible. Another was a finite element method that took RESULTS: The central flattening or hyperopic shift oc- the cornea’s biomechanical response into account.1 Both of curred atop the flap in all four of the simulated eyes tested with the creation of the LASIK flap. Of the four these models overpredicted the curvature change, but the av- parameters tested, modulus of elasticity (Young’s mod- erage overprediction for the finite element model was by a ulus) had the most profound effect on the results of much smaller margin. The models used by Bryant et al1 look hyperopic shift, varying from 0.5 diopters (D) in at very high myopic corrections produced by the removal of the least elastic (stiffest) cornea to 2.0 D of an intrastromal lenticule of tissue. The comparative results hyperopic shift in the most elastic cornea. The depth of the lenticular cut was the second-most significant support the view that final corneal refractive changes can- parameter tested varying from 0.24 D at 100 microns not be determined by the shape of the ablated tissue alone. to 1.25 D at 275 microns of depth. Varying In another study, Djotyan et al2 constructed an analytically intraocular pressure demonstrated less difference, and solvable elastic shell model for biomechanical response of varying corneal flap diameter demonstrated the least the cornea to refractive surgery. The cornea was represented difference in induced refractive change on the model. The hyperopic shift was noted to be greater in as an axisymmetric shell with position-dependent thickness hyperopic than in myopic eyes (simulated) tested. and an estimated modulus of elasticity. First, the shape-sub-

CONCLUSIONS: Three-dimensional finite element From the Department of Civil Engineering, College of Engineering & Applied analysis modeling of actual patient data could lead to a Sciences, Arizona State University, Tempe, Ariz (Dennadayalu, Mobasher, better understanding of the biomechanical response of Rajan); and Gary Hall Laser Center, Phoenix, Ariz (Hall). corneal tissue to the lenticular flap creation and poten- This study received financial support from the Department of Civil Engineering, tially for ablation patterns produced by the excimer la- Arizona State University, Tempe, Ariz; and Alcon Laboratories Inc, Ft Worth, ser. Understanding these biomechanical responses may Tex. lead to greater predictability and improvement of visual outcomes. [J Refract Surg. 2006;22:xxx-xxx.] The authors are the owners of a patent on biomechanical corneal modeling. The authors thank Alcon Laboratories Inc and the State of Arizona (301 K Funds) for sponsoring the research work. Correspondence: Gary W. Hall, MD, Gary Hall Laser Center, 2501 N 32 St, Phoenix, AZ 85008. Tel: 602.957.6799; Fax: 602.957.0172; E-mail: [email protected] Received: May 11, 2004 Accepted: August 8, 2005

Journal of Refractive Surgery Volume 22 February 2006 1 Refractive Change Induced by the L ASIK Flap/Deenadayalu et al

traction paradigm provided the postoperative geomet- The principal aim of any refractive treatment re- ric parameters of the cornea. The postoperative values gardless of the surgical technique is to attain a level of of the corneal modulus of elasticity, its range of varia- reliability and predictability. One of the more promis- tion, and its dependence on the geometric parameters ing approaches is to combine the current technology of of the removed corneal tissue were determined by best wavefront analysis with a biomechanical model. The fitting (the fitting parameter is the modulus of elasticity latter provides a methodology that can be used to pre- of the cornea) of predictions of the elastic shell mod- dict exactly how the components of the eye, especially el with LASIK nomograms from commercially avail- the cornea, react to the surgical plan on a patient-spe- able Bausch & Lomb (Rochester, NY) and VISX (Santa cific basis. One of the cornerstones of this biomechani- Clara, Calif) software for two different values of the cal approach to improve visual outcome is the finite ablation zone diameter. The elastic shell model then element method. considered the action of intraocular pressure (IOP) on The primary objective of our study was to scruti- the deformation of the anterior corneal surface and cal- nize and identify the relative importance that differ- culated the resulting principal curvatures of the cornea ent parameters might have on the refractive change and the correction of vision. induced by a LASIK flap. First understanding these Dupps and Roberts3 proposed a biomechanical mod- potential changes could be vital to predict a final vi- el of the intraoperative phototherapeutic keratectomy- sual outcome after the stromal ablation has occurred. induced central flattening. The peripheral stromal The specific objectives of this study were: 1) develop thickness was considered a feature of the biomechani- a methodology for creating a patient-specific finite ele- cal model and the acute postoperative flattening of the ment model of the eye using topography data, pachym- central cornea, ie, the hyperopic shift induced by the etry, axial length, and corneal dimensions; 2) develop procedure, was shown to rely on changes in peripheral a methodology for modeling the LASIK flap-creation stromal thickness rather than on the ablation pattern. surgery using the information from (1) in the form of a Roberts4 also presented a conceptual model that took finite element model with elements, nodes, boundary into account the corneal biomechanical response to ab- conditions, loads, and material behavior all defined; 3) lative surgery. The proposed model suggested that the develop a methodology for computing the power (or corneal shape change induced by the surgery is con- curvature) at various points on the anterior surface of tingent on structural modification along with ablation the cornea for both the pre- and postoperative eye; 4) profile and that it cannot be predicted with wavefront implement the developed methodologies in the form analysis alone. In LASIK (flap creation) without tissue of computer programs; and 5) study the sensitivity of removal, central flattening and peripheral steepening refractive changes induced by the LASIK flap by vary- were shown to occur. ing the different parameters used in the finite element Traditionally, the surgical outcome has been ana- model while using actual patient data. lyzed based on the solid material assumption. Katsube et al5 proposed a model that treated the cornea as a PATIENTS AND METHODS fluid-filled porous material. The solid and fluid con- The three-dimensional corneal topography data are stituents of the cornea were modeled separately and the used to construct the geometric model of a patient’s internal subatmospheric fluid pressure was deemed an eye. The geometric model established a reference con- important component of the mechanical loading, in ad- figuration of the cornea against which the response of dition to IOP. This model demonstrated higher amounts the structural change from the LASIK flap can be juxta- of unintended flattening (hyperopic shift) compared to posed to evaluate the resultant refractive change. Con- models based on solid material assumption. struction of the geometric model is accomplished by a Most biomechanical models of the cornea have been combination of curve and surface fitting procedures. simplified geometric models of sphere or an ellipsoid In this study, a commercially available corneal to- or an axisymmetric model. They have not been based pographer (Humphrey Atlas Eclipse Corneal Topo- on actual patient data. Overall, the material properties graphy System Model 995) is used in the form of have been assumed to be isotropic. The cornea has nu- 25 rings, each divided into 180 evenly spaced data merous layers, each layer with different geometrical points. The data are transformed into spatial carte- thickness parameters, exhibiting different material be- sian (X, Y, Z) coordinates. A typical topography map havior, all adding up to a complex material. Hence, a is shown in Figure 1. There is some amount of miss- biomechanically plausible computer model of the cor- ing data usually in the superior area of the cornea due nea will likely require an actual determination of each to the shadow of the upper lid and lashes. cornea’s material properties. Along each ray, a cubic polynomial is represented as

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Figure 1. Raw corneal topography data. z = g(r) with r as the independent variable. The cubic spline interpolation was used to estimate the values of z for the missing data points. In addition, the points de- scribing the sclera are mapped by extrapolating beyond the last corneal point on each ray. The transition from the cornea to the sclera has been modeled biomechanically with a change in elasticity at the transition, but it also involves a change in curvature at the limbus, which is unaccounted for in the current model. Thus the extrapo- lation of the scleral geometry points might potentially in- troduce an error. The scleral model was constructed for only half of the entire axial length of the eye, as the posterior half has little or no influence on the corneal model. Examples of completed points used in constructing the finite element model are shown in Figure 2. The finite element nodes are generated to be coinci- dent with the topographic data points. The structure is modeled with two layers of elements across the thickness. The nodes for the second layer of elements are B generated by projecting the nodes on the anterior sur- Figure 2. A) Cornea and sclera mapped with cubic spline interpolation face. The thickness of the cornea is obtained clinically and data points shown in Figure 1 and B) the coordinate system used in from ultrasonic pachymetry data at 33 points on the the mapping process. cornea, with the thickness at the remaining points estimated by linear interpolation. The cornea has a sub- elements are used to mesh the cornea, and linear hexa- stantially higher grid density than the sclera (Fig 3). hedral elements are used to mesh the sclera. The higher mesh density used in the cornea is required The hinged flap is modeled as a crack on the anterior to reach a good quantitative match to the experimental surface of the cornea (Fig 4). The flap starts from a speci- data. The finite element mesh (see Fig 3) of the mod- fied angle, extending counterclockwise to the specified el has 26,402 nodes (79,206 degrees-of-freedom) and angle at a specified radius along the surface (see Fig 2 16,200 elements, of which 9000 are used to model the for coordinate system). The arc length of the hinge used cornea. All of the degrees-of-freedom are assumed to is 4 mm in each eye. Note that the strain lessens within be fixed for the nodes at the periphery of the finite ele- the lenticule of the lamellar flap, which has been cut ment model. Linear triangular prism and hexahedral (see Fig 4). Note that Figure 4 shows strain in units of Journal of Refractive Surgery Volume 22 February 2006 3 Refractive Change Induced by the L ASIK Flap/Deenadayalu et al

Figure 3. Finite element mesh of the patient eye model. m/m. Duplicate nodes are introduced for all nodes present along the surface crack and beneath the flap. The interaction between the two layers, resulting from the flap, is simulated using spring elements that span the two nodes and have the same location in space before the incisions are made. The stiffness of each spring is computed using the following equation: E A 11 k = n (1) where E11 is the elastic modulus of the cornea in the direction perpendicular to the corneal plane, A is the area of the posterior surface of the hinged flap, and n is the number of spring elements that are in parallel. The area B A of the posterior surface of the flap is computed as Figure 4. Strain distribution with flap.

A = 2 r2 (2) the finite element analyses. The sclera is assumed to be linear isotropic and the ratio of the modulus of elasticity where r is the radius of the cornea and is the angle of the sclera to the in-plane modulus of elasticity of the described between the center of the cornea and the last cornea is assumed to be 2.5 throughout the study. corneal point on a ray. The purpose of these springs is The model is subjected to a nonlinear static struc- to keep the two surfaces from physically penetrating tural analysis using the commercial finite element each other. program ABAQUS (ABAQUS Inc, Providence, RI). The cornea is assumed to be orthotropic.6,7 The modu- The response of the structure at each node is used to lus of elasticity in the direction perpendicular to the cor- update the coordinates of the reference configuration neal plane, E , is assumed to be 0.125% of the in-plane and thereby map the postoperative topography. The 11 elastic modulus—E or E . The in-plane moduli values postoperative (X, Y, Z) locations of the nodes on the 22 33 are assumed to be equal, E = E . The corneal stiffness anterior surface of the cornea are computed as the sum 22 33 is primarily in-planar and the stiffness in the direction of the reference configuration and the nodal displace- perpendicular to the corneal plane is negligible. The in- ments ( X, Y, Z). The computation of curvature of plane Poisson’s ratio, , is assumed to be 0.49 (nearly the corneal first surface at finite element nodes is 23 ac- incompressible). The other two Poisson’s ratios, and complished by a method based on the least squares 12 ap- , are each assigned a value of 0.01. The shear moduli proach. In this method, the curvature is obtained by fit- 13 are assumed to be small relative to the in-plane modulus ting a sphere through a set of representative data points of elasticity. The (low) values of the material constants lying around the point in question. Because only the 4 journalofrefractivesurgery.com Refractive Change Induced by the L ASIK Flap/Deenadayalu et al

Figure 5. LASIK design parameters.

in this fashion, the model does not attempt to measure the effect of the flap on the peripheral cornea, either TABLE anatomically or refractively. Summary of Parameters The least squares method is used and the problem is Examined for Each Eye posed as follows: Corneal Flap Flap Intraocular Elasticity Thickness Diameter Pressure Find (R, a, b, c) Case E (MPa) (µm) (mm) (mmHg) n 1 2 sf 2 2 i 2 i 2 to minimize f = ∑ [R [(xi a) (y b) (z c) ]] (3) i=1 2 8 160 9 15 where R is the radius of the fitted sphere, a, b, and c are 3 10 the coordinates of the center of the fitted sphere, and n 4 12 is the number of representative data points. Once R is 5 160 obtained, the power, D, expressed in terms of diopters, 6 8 275 9 15 is evaluated as 7 440 D = 376/R (4) 8 8 9 8 160 9 15 where the constant, 376, is a factor based on the difference 10 10 between the refractive indices of the air and the cornea. 11 15 The parameters that are selected for the study are 12 8 160 9 18.33 as follows (Fig 5). Two clinical and one exaggerated thickness were selected for consideration. 13 21.67 a) The in-planar modulus of elasticity of the cornea— 14 25 varying from 2 MPa to 12 MPa.7 b) Diameter of the flap—varying from 8 to 10 mm. c) Flap thickness—varying from 100 to 275 µm (ie, 100 RESULTS µm, 160 µm, 275 µm). In all four modeled eyes, a hyperopic shift occurred d) Intraocular pressure—varying from 15 to 25 mmHg. with the creation of a LASIK flap. The amount of hy- The Table summarizes the parameters examined. peropic shift noted in each modeled eye increased in The refractive change was derived from the average relation to increased elasticity, flap thickness, and IOP. change of corneal power in the central 3-mm optical It did not change appreciably in relation to flap diam- zone. Two patients (four eyes) were selected for the eter. From this model, it is not possible to determine study with the following manifest spherical equivalent whether the hyperopic shifts occurred as a result of refractions: Patient 1: right eye 1.75 sphere, left eye relative thickening of the peripheral cornea, changes 1.75 sphere and Patient 2: right eye 5.25 sphere, in the stress/strain relationship of the corneal colla- left eye 5.00 0.25 170. gen fibrils, or some combination of both. However, the model does show the relative sensitivity of refractive

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Figure 6. Mean postoperative refractive change as a function of param- Figure 7. Surface elevation (in µm) maps before (above) and after eter. (below) cutting flap in a myopic eye for very elastic (2 MPa) and moderately elastic (8 MPa) cornea. changes to each of these four variables. The clinical correlation of these sensitivities needs to be validated. change in flap diameter appears to exert no influence on the postoperative refractive change (see Fig 6). EFFECT OF MODULUS OF EL ASTICITY A series of finite element simulations is carried EFFECT OF FLAP THICKNESS out with a variety of corneal elasticity from very high The central portion of the optical zone steepens as (E = E = 2 MPa), gradually changing to a relatively the thickness of the 9-mm diameter flap increases from 11 22 stiff cornea (low elasticity) (E = E = 12 MPa), hold- 100 to 275 µm. However, the peripheral cornea steep- 11 22 ing the other parameters to constant values as reported ens to an even larger extent with the deep lamellar in the Table. Peripheral steepening and central flatten- cut, creating a relative flattening of the central cornea ing are observed from the flap formation in each case (hyperopic shift). This hyperopic shift increases from (hyperopic shift). The amount of corneal deformation, 0.24 D at 100 µm depth to 1.25 D at 275 µm depth or hyperopic shift, is proportional to elasticity and (see Fig 6). Figure 9 shows the more marked elevation decreases as stiffness increases. The variation of the of the postoperative cornea in the 275 µm flap when postoperative change in curvature with the modulus compared to the 100 µm flap in the myopic eye. Com- of elasticity of the cornea is observed to be approxi- parison of the circumferential strain distribution on mately linear in each of the four simulated eyes that the posterior corneal surface on a myopic eye shows a were examined (Fig 6). The effect the corneal elasticity deeper lamellar cut, which also creates a higher strain has on the postoperative elevation maps is depicted in within the remaining stromal bed. Figures 7 and 8 for a myopic and hyperopic eye. The resultant refractive change varied from less than 0.5 D EFFECT OF INTRAOCUL AR PRESSURE in the stiffest cornea (E11= E22 = 12 MPa) to more than The magnitudes of the peripheral corneal steepening 2.25 D in the most elastic cornea (E11 = E22 = 2 MPa). increase as the pressures increase from 15 to 25 mmHg. However, for the same elasticity, more hyperopic shift The central hyperopic shift increases from 0.75 D at occurred on the hyperopic simulated eyes than the 15 mmHg to 1.25 D at 25 mmHg (see Fig 6). myopic simulated eyes. DISCUSSION EFFECT OF FLAP DIAMETER From our results of four simulated eyes studied The flap diameter varied from 8 to 10 mm. Using in the finite element analysis model presented, the a moderately elastic cornea (E = E = 8 MPa), the parameter that had the most effect on the refractive 11 22 6 journalofrefractivesurgery.com Refractive Change Induced by the L ASIK Flap/Deenadayalu et al

Figure 8. Surface elevation (in µm) maps before (above) and after Figure 9. Surface corneal elevation (in µm) maps before (above) and (below) cutting flap in a hyperopic eye for very elastic (2 MPa) and mod- after (below) cutting flap in a myopic eye. erately elastic (8 MPa) cornea. change in power distribution was the modulus of elas- corneal biomechanical response to LASIK may be es- ticity. Patient eyes (simulated) with a highly elastic sential to achieving optimal visual performance. cornea (modulus of elasticity = 2 MPa) could undergo as much as 2.0 D hyperopic shift with just the intro- REFERENCES duction of the flap. 1. Bryant MR, Marchi V, Juhasz T. Mathematical models of picosecond laser keratomileusis for high myopia. J Refract Surg. The second most sensitive parameter was the depth 2000;16:155-162. of the cut. However, within the clinical range of depth 2. Djotyan GP, Kurtz RM, Fernandez DC, Juhasz T. An analytically used for LASIK flaps, the hyperopic shift was 1.0 D. solvable model for biomechanical response of the cornea to re- Higher IOP induced a greater hyperopic shift than fractive surgery. J Biomech Eng. 2001;123:440-445. lower IOP. The diameter of the LASIK flap appeared 3. Dupps WJ, Roberts C. Effect of acute biomechanical changes to have little effect on the induced hyperopic shift. A on corneal curvature after photokeratectomy. J Refract Surg. 2001;17:658-669. hyperopic eye with the same corneal stiffness as the 4. Roberts C. Biomechanics of the cornea and wavefront-guided myopic eye experienced a greater hyperopic shift with laser refractive surgery. J Refract Surg. 2002;18:S589-S592. the introduction of the flap. 5. Katsube N, Wang R, Okuma E, Roberts C. Biomechanical re- 8 Pallikaris et al have shown that the flap cut chang- sponse of the cornea to phototherapeutic keratectomy when es the eye’s higher order aberrations and the present treated as a fluid-filled porous material. J Refract Surg. 2002;18: S593-S597. study may explain that observation. In addition, there is a growing body of evidence that the combination 6. Pinsky PM, Datye DV. A microstructurally-based finite element model of the incised human cornea. J Biomech. 1991;24:907-922. of wavefront technology and corneal topography pro- 7. Rajan SD, Mobasher B, Sun CH, Wooton JW. An integrated vides the surgeon with the optimal information on the system for design of keratorefractive surgeries. Presented at: vision characteristics of the eye.4 The effectiveness of ASCRS/ASOA Symposium on Cataract, IOL and Refractive this combination in predicting better visual outcomes Surgery; April 1995; San Diego, Calif. and reducing higher ocular aberration may be in- 8. Pallikaris IG, Kymionis GD, Panagopoulou SI, Siganos CS, The- odorakis MA, Pallikaris AI. Induced optical aberrations follow- creased by tying these to the finite element simulation ing formation of a laser in situ keratomileusis flap. J Cataract of the LASIK procedure. A better understanding of the Refract Surg. 2002;28:1737-1741.

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