**Purpose**:
To simulate deformation amplitude after laser-assisted in situ keratomileusis (LASIK), small incision lenticule extraction (SMILE), and photorefractive keratectomy (PRK) with finite element models.

**Methods**:
Finite element simulations of air-puff applanation on LASIK, SMILE, and PRK models were performed on a cohort of normal eyes, which had undergone refractive treatments. Short- and long-term wound healing responses were considered for SMILE and LASIK models based on evidence of microdistortions in Bowman's layer and crimping of collagen fibers. First, inverse simulations were performed to derive the preoperative properties of the cornea. Using these properties and planned refractive treatment, postoperative air-puff deformation amplitude was predicted and compared with the in vivo measurements.

**Results**:
The predicted postoperative corneal stiffness parameters agreed very well with in vivo values of SMILE, LASIK, and PRK eyes. Intraclass correlations (ICC) were greatest in PRK eyes (ICC > 0.95). This agreement was lower for peak deformation amplitude and peak deflection amplitude in SMILE and LASIK eyes (ICC < 0.9). In PRK eyes, peak deformation and deflection amplitude predictions were the best relative to in vivo magnitudes. Also, linear correlation (*r*) between in vivo measurement and predicted biomechanical parameters indicated strong agreement between them (SMILE: *r* ≥ 0.89, LASIK: *r* ≥ 0.83, PRK: *r* ≥ 0.87).

**Conclusions**:
The is the first study to present predictive simulations of corneal deformation changes after different procedures. Patient-specific preoperative corneal biomechanical properties and finite element models were a significant determinant of accurate postoperative deformation amplitude prediction.

^{1}This interweaving progressively becomes less through the depth of the stroma.

^{1}These structural features could be responsible for the greater tensile strength of the anterior stroma relative to the posterior stroma.

^{2}There are procedures that significantly alter the anterior stroma, such as laser-assisted in situ keratomileusis (LASIK) and photorefractive keratectomy (PRK). In PRK, a portion of the anterior stroma is ablated in comparison to LASIK. Small incision lenticule extraction (SMILE) is another refractive procedure, which leaves much of the anterior stroma intact compared to LASIK and PRK. Using theoretical models, it was concluded that corneas undergoing SMILE could be biomechanically stronger compared to LASIK and PRK postoperatively.

^{3,4}

^{5–7}Using dynamic air-puff applanation, two studies indicated a better biomechanical result after PRK than after LASIK.

^{5,6}Another study reported similar outcomes between PRK and LASIK.

^{3}However, a theoretical model predicted better biomechanical outcomes after SMILE and LASIK than after PRK.

^{3}Recent clinical data demonstrated equivalence between SMILE and LASIK with respect to biomechanical changes in the cornea after surgery.

^{8–10}Further, Corvis-ST (OCULUS Optikgerate Gmbh, Wetzlar, Germany) was an improved device over the Ocular Response Analyzer (Reichert, Inc., Depew, NY, USA) since it had a highly repeatable pressure profile and quantified the mechanical deformation of the cornea. Theoretical models showed that LASIK caused a greater increase in mechanical stress in the residual stromal bed than SMILE.

^{4}Therefore, this study investigated simulated air-puff applanation on LASIK, PRK, and SMILE finite element models, using the pressure profile generated by Corvis-ST.

^{11}In the finite element models, the transient air puff was exported from Corvis-ST dynamic Scheimpflug analyzer and spatially distributed on the anterior surface using fluid dynamics analysis.

^{11}

^{11}Three-dimensional (3-D) geometry of a patient cornea was created from Pentacam (OCULUS Optikgerate Gmbh). The Pentacam provided Cartesian coordinates of the anterior and posterior corneal surface, which were used for creating a 3-D volume. Epithelium thickness of the cornea was measured with RTVue (Optovue, Inc., Fremont, CA, USA). Finite element mesh was created with 8-noded linear hexahedral elements (TrueGrid; XYZ Scientific Applications, Inc., Livermore, CA, USA). A total of 3312 hexahedral elements were used to represent the corneal volume. Figures 1A and 1B show a cross section of the preoperative and postoperative mesh (with flap/cap) of the central cornea, respectively. PRK finite element mesh did not have any flap/cap zone. An anisotropic, hyperelastic, fiber-dependent material model with material incompressibility was chosen.

^{1}The material model accounted for the orthogonal arrangement of fibers in the central cornea, depth dependency of angular direction of the interweaving fibers, and reorientation of the in-plane peripheral collagen fibers to circumferential direction.

^{11}The hyperelastic material model was represented by free energy density (

*ψ*):

*I*

_{1}= C : 1 and

*I*

_{3}= det[

*C*] were the 1st and 3rd invariants of the deformation tensor.

*C*was the right Cauchy-Green deformation tensor and det[

*C*] represented determinant of the tensor. The isotropic energy density of the matrix (

*ψ*) was described by:

_{m}*I*

_{1},

*I*

_{3}was the determinant of deformation gradient tensor, and

*C*

_{1}s were the material constants.

*D*

_{1}was the bulk modulus to enforce incompressibility and was assumed to be 10

^{−6}. The fiber energy density (

*ψ*) was described by:

_{f-plane}*ψ*represented the energy density of in-plane lamellar collagen fibers with stretch,

_{f-plane}*λ*equal to

_{f-plane}*A*= [cos

*θ*, sin

*θ*, 0]

^{T}was the local direction vector of the fibers.

*k*

_{1}and

*k*

_{2}were the material constants.

*D*(

_{plane}*θ*) represented a weighted average of the energy density of the fiber families at each integration point of the element. It also represented the change in the preferred direction of the fibers from orthogonal in the central cornea to circumferential near the limbus.

*ψ*represented the energy density of crosslink fibers between the lamellae with stretch,

_{f-cross}*λ*equal to

_{f-cross}*B*was the direction vector of the crosslink fibers.

*B*was determined by taking a cross-product of

*A*with the surface normal and then rotating it out-of-plane around

*A*by an angle

*ξ*. The angle

*ξ*was assumed to be a function of depth and was modeled as follows

^{11}:

*s*was the nondimensional local thickness. The angle

*ξ*was evaluated at each element centroid.

*D*(

_{plane}*θ*) was kept equal to

*D*(

_{cross}*θ*). From the above equations, the Cauchy stress was determined by:

*F*was the deformation gradient tensor. The epithelium was modeled as an isotropic, hyperelastic, incompressible material [

*c*

_{1}= 5 kPa (Pascal) and

*c*

_{2}= 0.0 kPa.

^{11}The inverse finite element model derived the in situ corneal (

*C*

_{1},

*C*

_{2},

*k*

_{1}, and

*k*

_{2}) and extracorneal [Kz (N/m), μ (Pa.sec), and m (gm)] material properties. The inverse model minimized the difference between the measured displacements of the anterior edge of the cornea and calculated displacements of the same edge from the finite element simulations. The inverse model was designed such that the corneal and extracorneal properties were governed by the corneal deformation (reported as deflection amplitude by Corvis-ST) and whole-eye movement, respectively.

^{11}Here, deformation amplitude was the arithmetic sum of corneal deformation and whole-eye movement.

^{11}An iterative method (Levenberg–Marquardt algorithm) for minimization was adopted. The finite element simulations were performed in ABAQUS (Dassault Systèmes Americas Corporation, Waltham, MA, USA). The material model was incorporated in the simulations using ABAQUS material subroutine (UMAT). The inverse calculations were performed using a custom script written in Python (v2.7.3). ABAQUS simulations and Python scripts were executed simultaneously in a multithreaded workstation.

^{12}Figure 1C shows a schematic of a cross section of the cornea (epithelium and stroma). The figure shows the epithelium, flap in LASIK (or cap in SMILE), peripheral cornea outside the flap/cap and the residual stromal bed. For simplicity, the stroma was further subdivided into three zones in the axial direction for better visualization of the angular orientation of the interweaving fibers. As shown in Figure 1C, the angular direction of the interweaving fibers decreased through the depth of the stroma; that is, zone 1 (anterior stroma) was the stiffest followed by zones 2 and 3.

^{1}Different surgical procedures were simulated as follows.

^{12}

^{13}This increase in microdistortions could be indicative of compression of the cap due to mechanical extraction of the lenticule and mismatch between the lenticular surfaces.

^{13}This directly implied that the collagen fibers in the cap were probably under reduced tensile stress and underwent some crimping

^{14}since the microdistortions were greater than in the preoperative state even 3 months after SMILE (Fig. 2). Therefore, the cap was assumed to have only isotropic material properties (

*C*

_{1}and

*C*

_{2}were the same as preoperative,

*k*

_{1}and

*k*

_{2}were set to zero) in the finite element model assuming short-term wound healing. In the long-term finite element model, the cap material properties were set the same as preoperative anisotropic, hyperelastic properties with the assumption that incision in the anterior stroma did not cause a significant decrease in the cap's strength. Further, the postoperative material properties and fiber distributions of the peripheral cornea outside the cap zone were assumed to be the same as preoperative state.

^{15,16}First, the preoperative material properties of each eye were determined with the inverse finite element method described earlier. The programmed refractive error (sphere and cylindrical error) was simulated in the finite element mesh with aspheric profile. Then, postoperative deformation amplitude of each eye was simulated with its respective surgical model, that is, LASIK, PRK, or SMILE. The peak deformation amplitude and Kc (constant) was determined from the computed postoperative deformation amplitude. The “short-term wound healing” assumption was applied to the postoperative finite element simulations. These were compared with the same derived from in vivo measurement of postoperative deformation amplitude. In the second phase, a case example of one eye from each group with similar preoperative IOP and central corneal thickness (CCT) was chosen. In other words, validation was conducted both for a cohort of eyes and a set of individual eyes. For the cohort of eyes, root mean square error (RMSE) was calculated for each parameter within a group, for example, RMSE of Kc (constant) for SMILE eyes was square root of sum of squares of difference between in vivo and predicted magnitude of Kc (constant) divided by total number of eyes that underwent SMILE.

*P*> 0.05). Table 2 shows the in vivo preoperative and postoperative Kc (mean), Kc (constant), peak deformation amplitude, and peak deflection amplitude. The predicted magnitudes of the above variables are also shown in Table 2. There was excellent agreement between measured and predicted magnitudes of Kc (mean) and Kc (constant) since intraclass correlation (ICC) was greater than 0.9 for all procedures. Overall, prediction of peak deformation amplitude and deflection amplitude was best in PRK eyes (ICC ∼ 0.9), which could be due to absence of flap or cap in PRK. However, the difference between in vivo preoperative and in vivo postoperative peak deformation and deflection amplitude was similar between all the eye groups (Table 2). RMSE of Kc (mean) and Kc (constant) were significantly greater than the difference between the mean values of in vivo and predicted postoperative magnitudes (Table 2). This explained the high ICC for Kc (mean) and Kc (constant) for SMILE, LASIK, and PRK eyes. Interestingly, RMSE of peak deformation and deflection amplitude were within repeatability of these parameters.

^{17,18}Further, PRK eyes had the least decrease in magnitude of Kc (mean) and Kc (constant) after surgery; for example, in vivo mean Kc (mean) and mean Kc (constant) decreased by 6.35 and 5.04 N/m only (Table 2). In contrast, LASIK and SMILE caused a greater decrease in stiffnesses (Table 2).

*r*) were 0.95, 0.94, 0.87, and 0.90 for Kc (mean), Kc (constant), peak deformation amplitude, and peak deflection amplitude, respectively, for all eyes (

*n*= 36). The slopes of the linear regressions were 0.93, 0.91, 0.81, and 0.70, respectively (

*P*< 0.001 for all).

- LASIK eye:
*C*_{1}= 80.13 kPa,*C*_{2}= 2.52 MPa,*k*_{1}= 25.1 kPa,*k*_{2}= 430.9, Kz = 1267 N/m, μ = 4.61 Pa.sec and m = 0.03 gm. - SMILE eye:
*C*_{1}= 82.75 kPa,*C*_{2}= 1.98 MPa,*k*_{1}= 55.3 kPa,*k*_{2}= 637.1, Kz = 1775 N/m, μ = 3.07 Pa.sec and m = 0.05 gm. - PRK eye:
*C*_{1}= 68.4 kPa,*C*_{2}= 2.62 MPa,*k*_{1}= 41.9 kPa,*k*_{2}= 706.4, Kz = 1379 N/m, μ = 3.72 Pa.sec and m = 0.04 gm.

^{2–4}SMILE leaves most of the anterior stroma intact, which incidentally is the stiffest region of the stroma. Thus, SMILE caused the least biomechanical changes in the cornea in theoretical models.

^{3,4}However, clinical evaluation of biomechanical changes cannot be performed in terms of the mechanical stresses and displacements

^{19}as these parameters cannot be measured in patients yet. Air-puff applanation is the only available technique to clinically evaluate these procedures. Therefore, this study focused exclusively on expected deformation response of the cornea after simulated LASIK, PRK, and SMILE coupled with air-puff applanation. The following were the key outcomes from this study:

- When a cohort of eyes was measured with Corvis-ST before and after surgery, PRK eyes had the least decrease in stiffness parameters. Also, LASIK and SMILE caused a much greater decrease in stiffness parameters (Table 2). However, the change in in vivo peak deformation and deflection amplitude was similar between the cohorts (Table 2). This highlighted the need for patient-specific prediction of deformation amplitude using patient-specific finite element models since theoretical models, devoid of patient-specific material properties, predicted SMILE to cause the least change in corneal stiffness.
^{3,4} - PRK eyes had the best agreement between in vivo and predicted postoperative value of stiffness, peak deformation amplitude, and peak deflection amplitude. SMILE and LASIK eyes also had excellent agreement for stiffness parameters (Table 2). Flap or cap could have reduced the level of agreement between in vivo and predicted postoperative value of peak deformation amplitude and peak deflection amplitude in LASIK and SMILE eyes.
- Figures 4A and 4B show the accuracy of determination of deformation amplitude using the finite element simulations and assumptions of wound healing (crimping of collagen fibers) in one eye from each cohort (SMILE, LASIK, and PRK). Such modeling tools could be integrated with Corvis-ST in future versions of the device.

^{13}The results showed that air-puff deformation response of the cornea was highly dependent on the preoperative material properties of the cornea with the assumption of short-term wound healing response. Further, for the same treatment, a thicker flap or cap and theoretical models predicted slightly greater stiffness postoperatively.

^{3,4}Clinical results of comparative studies between LASIK and PRK with the Corvis-ST generally indicate a stiffer biomechanical response after PRK than predicted by the model.

^{5–7}This indicated the importance of greater degree of fibrotic scars and haze formation in PRK than in LASIK,

^{20,21}which could have resulted in some biomechanical compensation to removal of the stiffest region of the stroma. Clinical results of comparative studies between LASIK and SMILE using the Corvis-ST indicated similar biomechanical changes.

^{8–10}This study provided an explanation for these observations and demonstrated the limitation in using device deformation variables such as peak deformation amplitude to compare SMILE and LASIK. Our inverse simulation method of comparing postoperative outcomes may yield better segregation of biomechanical responses after SMILE and LASIK. Future inverse models could benefit with a continuum mechanics approach to simulate the biological stiffening effect after PRK, though this would be a challenging task.

^{22}However, the distortions were greater than preoperative magnitudes in the LASIK eyes after surgery.

^{22}This led us to conclude that crimping in the collagen may not have returned to preoperative levels in the LASIK eyes even after longer healing time. Further long-term follow-up would be required to confirm this trend, but 6-month results lend credence to our modeling assumptions. Another limitation of the study was that the effect of hydration on the elastic strength of the cornea was not evaluated. The deformation amplitude was a dynamic but fast measurement. The deformation amplitude is primarily determined by elastic properties of the cornea.

^{11,15,16,23}However, the net stress in the fiber would be a vector summation of the stress due to IOP and fluid pressure. If hydration was significantly altered after surgery, then postoperative deformation amplitude could differ significantly from the model predictions; that is, ICC could be lower than 0.9 for all variables.

^{11}Further, postoperative epithelium thickness is no longer as uniform as preoperative thickness, and this introduced an approximation to the true thickness of the postoperative stroma. Most current OCT devices limit epithelium thickness reports to the central 6-mm cornea only. Thus, the data were insufficient for inclusion in patient-specific simulations, where the corneal diameter was significantly greater. These limitations could also explain the difference between postoperative corneal stiffnesses derived from in vivo deformation amplitude and the same estimated from simulation results.

^{4}The same was not observed in the LASIK model.

^{4}Thus, SMILE left the residual cornea biomechanically stiffer than LASIK.

^{4}However, the relation between stress and peak deformation amplitude was not linear; for example, a 10% increase in the stress in the RSB cannot be considered as a 10% increase in peak deformation amplitude. The results from this study show that the alteration in stresses in the postoperative models resulted in minor changes in simulated deformation amplitude. This was in agreement with recent clinical studies comparing biomechanics of SMILE and LASIK with Corvis-ST.

^{8–10}To conclude, this is the first simulation study to show the predicted deformation amplitude after simulated LASIK, SMILE, and PRK, using novel structural perturbations that may be representative of the in vivo state of the cornea after surgery. Future studies need to investigate alternate analysis techniques or newer measurement techniques to quantify the viscous contribution to in vivo tissue stiffness.

**M. Francis**, None;

**P. Khamar**, None;

**R. Shetty**, None;

**K. Sainani**, None;

**R.M.M.A. Nuijts**, None;

**B. Haex**, None;

**A. Sinha Roy**, None

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