The ability to surgically extract a refractive lenticule of stromal tissue using the small incision lenticule extraction (SMILE) technique has opened up further possible applications. It has been demonstrated that refractive lenticules can be cryopreserved successfully for 1 month in rabbits1,2 and as long as 5 to 6 months in humans.3 It has been suggested that these lenticules could be re-implanted as a method for restoring tissue in ectatic corneas or provide an opportunity for reversing the myopic correction in a patient progressing to presbyopia.1,2 Successful re-implantation was first demonstrated in rabbits.2
Alternatively, there is also the potential for implanting an allogenic lenticule obtained from a donor patient with myopia into a patient with hyperopia to correct the hyperopia, as originally proposed by Barraquer in 1980.4 The first case of this endokeratophakia procedure was performed in 2012 by Pradhan et al.5 This demonstrated that endokeratophakia was a viable procedure because the cornea remained clear during a 6-month follow-up period and there has been no change in corneal clarity at the recent 2-year follow-up (Kishore R. Pradhan, MD, unpublished data). However, only 50% of the intended +10.50 diopters (D) refractive correction was achieved because the implanted lenticule caused the cornea to bulge both anteriorly and posteriorly, whereas the intention was only to induce a curvature change on the anterior surface. The significant change to the posterior surface curvature meant that a large amount of the effect was lost given the similar refractive index between the stroma and aqueous humor in the anterior chamber. In another study, Sun et al.6 performed a procedure for patients with one myopic eye and one hyperopic eye in which a SMILE lenticule was extracted from the myopic eye and implanted under a LASIK flap in the hyperopic eye after performing an excimer laser ablation to account for the refractive difference. Following this case, Ganesh et al.3 reported the results of a series of 9 eyes in which cryopreserved lenticules from patients with myopia were re-implanted into patients with hyperopia, a procedure referred to as femtosecond laser-assisted intrastromal lenticule implantation. Although there was an undercorrection of the hyperopia in all eyes, the mean undercorrection of 21%3 was less than the 50% found in the first case report.5 This was likely due to differences between the amount of hyperopia treated, and factors such as the optical zone diameter.
Finite element modeling offers the potential for gaining an understanding of how different surgical interventions will affect the cornea before performing the procedure in vivo. This method has been applied to a variety of ophthalmic surgical procedures in recent years, including photorefractive keratectomy,7,8 LASIK,8,9 SMILE,9 arcuate keratotomy incisions,10 limbal relaxing incisions,7 corneal cross-linking,11 and intrastromal ring segment implantation.12
The aim of the current study was to use a finite element model13–15 of the eye to simulate the same endokeratophakia case reported by Pradhan et al.5 We were interested to see whether the observed changes to the posterior surface elevation would be reproduced by the finite element model simulation, and investigate the potential of such simulations for the prediction of surgical outcome.
Materials and Methods
The details of the endokeratophakia case have been described previously.5 Briefly, a −10.50 D myopic lenticule was extracted from a donor patient during a routine SMILE procedure using the VisuMax femtosecond laser (Carl Zeiss Meditec, Jena, Germany), and set aside in McCarey–Kaufman medium storage. The orientation of the lenticule was maintained throughout, with the refractive cut anterior and the planar cut posterior. Given the spherical rotationally symmetric geometry of the lenticule, rotation was not monitored. The endokeratophakia procedure was then performed for the patient with hyperopia. First, the femtosecond laser interface creation part of a SMILE procedure was performed by programming a −2.00-D correction. The parameters used were a cap thickness of 180 µm, an optical zone diameter of 6.25 mm, a side cut of 90°, and the two small incisions located at 150° (supertemporal) and 330° (inferonasal). The upper interface was separated in the normal fashion, but the lower interface of the lenticule was left unseparated to produce a lamellar pocket at a 180-µm depth with a 6.25-mm diameter. The donor lenticule was inserted into the space provided by the upper interface through the small incision using Kelman forceps, holding the donor lenticule lengthwise along a diameter. The donor lenticule was distended until it was flat and centered on the corneal vertex coincident with the axis of fixation.
Constitutive Material Model
Biomechanically, corneal tissue is known for being nearly incompressible, having non-linear elastic characteristics, being highly inhomogeneous in-plane and over its thickness, and revealing a high degree of anisotropy. In the current study, we used a published biomechanical model,13–15 which used additive terms in a strain energy function to describe the tissue characteristics:
is the penalty function modelling incompressibility, Ψ̅m
is a non-linear adaptation of Hooke’s law representing the tissue matrix, and Ψ̅f1
are polynomial material functions16
modeling the main collagen fibers and cross-links, respectively. The probability distribution function Φ defines a realistic fiber distribution,17–20
by assigning weights to each fiber direction as a function of corneal depth. Material constants (C10
= 0.06 MPa, Υm
= 0.13, µm
= 24 MPa, Υk
= 0.08, µk
= 95 MPa) were determined using three sets of experimental data: one from button inflation21
and two (one superior-inferior strip, one superonasal-inferotemporal strip) from strip extensiometry.22
Patient-Specific Endokeratophakia Surgery Simulation
A patient-specific finite element model for the patient in the study was obtained with a four-step approach: (1) spatial elevation data of the front and back surface of the patient’s cornea was acquired with a Scheimpflug tomography system (Pentacam HR, Oculus, Wetzlar, Germany); (2) the geometrical information thus obtained was used to warp a spherical template cornea model to create a patient-specific finite element mesh, containing 35,000 elements and more than 44,000 nodes; (3) the initial stress distribution in the model was then computed with an iterative approach; and (4) the endokeratophakia surgery was simulated. The anterior and posterior surfaces, computed by the finite element model, were then compared to the corresponding postoperative surfaces to assess accuracy and reliability of finite element modeling. The details of steps 2 to 4 are described below.
Mesh Warping. In our earlier work, we showed that a model with patient-specific geometry of the human cornea can be obtained by warping a spherical finite element mesh such that its anterior and posterior surfaces match the respective surfaces of the tomography measurements.15 Thereby, the tomography surfaces are expressed as the coefficients obtained from Zernike expansion (up to the sixth order), and the inside mesh nodes proportionally follow the deformation of the respective surface nodes. This way, the template mesh was warped to match the patient’s cornea, without producing distorted elements (which is crucial for finite element analysis).
Calculation of Initial Stress Distribution. Because the Pentacam Scheimpflug camera measures corneal geometry in vivo, whereby the corneal tissue is under mechanical stress, the shape in the absence of acting forces is a priori not known. An iterative approach to calculate the initial stress distribution in the model, previously published,15 was employed in this study.
Surgery Simulation. The SMILE lamellar cut and the keyhole incision cuts were modeled precisely following the data by Pradhan et al.5 The two keyhole incisions were both 2.0 mm in length oriented at the 150° (superotemporal) and 330° (inferonasal) meridian. The SMILE lamellar cap cut was modeled with a diameter of 6.25 mm and a thickness of 180 µm (Figure 1).
(A) Optical coherence tomography (RTVue OCT; Optovue Inc., Fremont, CA) measurement 6 months after endokeratophakia procedure. (B) Section of the biomechanical model. The lenticule implant is shown in red. The lower figures are magnifications of the upper figures (original magnification ×3.5).
In a third simulation step, a model of the lenticular allograft was implanted into the prepared pocket, precisely reproducing the surgical procedure in silico. The lenticule had a diameter of 5.75 mm, a central thickness of 127 µm, a spherical power of −10.50 D, and an edge thickness of 3 µm (Figure 1). The allograft was simulated with the same stiffness as the cornea.
The results obtained from the finite element simulations were automatically imported and then analyzed in the user interface of the Optimeyes software (Integrated Scientific Services, Biel, Switzerland). The software uses an 8.0-mm region of interest for Zernike decomposition of the anterior and posterior corneal surface in the model, and the keratometric index n = 1.3375 for curvature calculation. Thereby, axial curvature was calculated as CS = (n-1)/R, where R is the normal distance between a surface point and the central axis of the cornea. From axial curvatures, corneal astigmatism was calculated as the difference between the steep and flat simulated keratometry values over a central annulus of 0.5 to 2.0 mm radius. Elevation data were calculated as the normal distance between the cornea and a reference surface, a best-fit sphere fitted over a central 8.0-mm diameter zone. Wavefront aberration data were obtained from optical path difference calculations with a 6.0-mm diameter analysis zone.
Besides postoperative geometrical shape, the deformed finite element model also provides full-field biomechanical stress information for every simulation step as part of the software package. The average stress (and standard deviation) was calculated from the simulation within the central 3.0-mm zone for the cap and the stromal bed below the cap, for the untreated cornea and after the implantation of the lenticule. The biomechanical change of the surgical treatment was calculated as the percentage change between preoperative and postoperative stress values.
Axial curvature maps and indices calculated on the surface of the preoperative finite element model were compared to preoperative clinical data. Steep and flat astigmatism axis curvature were reported as 44.50 and 41.80 D in the Pentacam software, whereas the same indices calculated on the patient-specific finite element model were 44.23 and 41.79 D. The Pentacam software indicated corneal astigmatism of 2.70 D @ 167.5°, which compared well with corneal astigmatism of 2.44 D @ 167° on the finite element model. Figure 2 shows the preoperative axial curvature map and the equivalent map calculated using the model.
Comparing preoperative (left) axial curvatures in the Pentacam software (Pentacam HR, Oculus, Wetzlar, Germany), and (right) axial curvatures calculated from the preoperative finite element model. The shaded peripheral area indicates extrapolated data in the right image, meaning that only the central 8.0 mm are reliable information. The color scale is the same for both images.
Figure 3 shows the postoperative axial curvature map and the equivalent map calculated using the model. Central average curvature (central 4.0-mm diameter zone) was 48.01 D clinically and 48.23 D in the simulation. Corneal astigmatism was 3.01 D clinically and 2.88 D in the simulation. Central corneal thickness was 675 µm clinically and 685 µm in the simulation. Figure 4 shows the postoperative posterior best-fit sphere elevation map and the equivalent map calculated using the model. As observed in the clinical case, the predicted posterior best-fit sphere elevation map can also be seen to bulge inward, in both cases by approximately 40 µm.
(Left) The postoperative endokeratophakia procedure anterior axial curvature map. (Right) The predicted anterior axial curvature map from finite element simulations, reproducing the surgical treatment in silico. The figure shows that the simulation closely reproduces postoperative clinical anterior corneal curvatures. The color scale ranges from 10.0 to 90.0 diopters.
(A) Postoperative endokeratophakia procedure posterior elevation map (best-fit sphere). (B) Predicted posterior elevation map from finite element simulations, reproducing the surgical treatment in silico. Interestingly, the simulation closely reproduces postoperative posterior corneal elevation. It almost mimics the clinically observed posterior bulging. The color-scale ranges from −150 to +150 µm.
Figure 5 shows the postoperative total corneal wavefront map using a 6.0-mm analysis zone, and the equivalent map calculated using the model. Postoperative spherical aberration was 0.332 µm and higher-order root mean square (third order or greater) was 0.110 µm. In comparison, the simulation predicted spherical aberration of 0.197 µm and higher-order root mean square at 0.083 µm.
(A) Postoperative endokeratophakia procedure total corneal wavefront map. (B) Predicted total corneal wavefront map from finite element simulations, reproducing the surgical treatment in silico. The figure shows that the simulation closely reproduces postoperative clinical wavefront aberration of total cornea. The color-scale ranges from −50 to +50 µm.
Surgically induced changes in corneal biomechanics can be best expressed as changes in stress distribution between the preoperative and postoperative simulations. Comparing the stress distribution before and after the simulated endokeratophakia procedure, the stress averaged within the central 3.0-mm diameter zone increased by 159.94% ± 73% in the cap, and decreased by 32.41% ± 21% in the stromal bed below the lenticule (Figure 6).
The finite element simulation model shows an increase (red) in stress in the cap above the implant of approximately 160%, and a decrease (blue) in the stromal bed below it of 32%. Equivalent stresses with a scale from 0.000 (blue) to 0.035 MPa (red). Interestingly, stresses were highest in the central anterior cornea, anteriorly of the thickest part of the implanted graft.
In the current study, the anterior and posterior corneal curvature changes observed after an endokeratophakia procedure were reproduced with close agreement using a finite element model of the cornea. This provides some validation for the use of this model in evaluating corneal refractive surgical procedures. Specifically, this model might therefore be used to obtain biomechanical simulations of endokeratophakia procedures with different parameters, including varying the implanted lenticule dimensions (refractive power, optical zone), implantation depth, and location of the small incisions, and will be the subject of further study.
The most notable finding of the original case report5 was that only 50% of the refractive correction was achieved due to a large proportion of the curvature change occurring on the posterior corneal surface. This was closely matched by the finite element model simulation. The simulation also indicated that stresses became highest in the central cornea anterior to the implant and lowest in the posterior cornea, which follows from the stiffness gradient of the stromal tissue being greatest anteriorly and smallest posteriorly. This seems to be the most likely cause for the cornea to give way on the posterior side and produce the observed inward bulging of the posterior surface, as seen clinically and in the simulation. In the study by Ganesh et al.,3 there was an undercorrection of the hyperopia in all eyes; however, the mean undercorrection of 21%3 was less than the 50% reported in the first case report.5 Ganesh et al. suggested that this difference might be due to the lower amount of hyperopia treated (mean +5.39 D) and the use of a larger optical zone for the implanted lenticule (between 6.00 and 6.80 mm compared to 5.75 mm in the original case report). This is supported by the fact that the largest undercorrection of 31% occurred in the eye in which a +10.00 D lenticule was implanted—the next highest lenticule power was 6.80 D. We intend to test this hypothesis by using the model described in the current study to investigate the effect of changing the lenticule power, lenticule diameter, and implantation depth.
One weakness of the current model is that Bowman’s layer is not included as a distinct layer. Bowman’s layer was demonstrated to have different biomechanical properties to stromal tissue by Last et al.,23 who measured the Young’s modulus of Bowman’s layer (109.8 kPa) to be a factor of three greater than for the anterior stroma (33 kPa). However, inhomogeneity over the thickness profile of the cornea was included in the simulation, because anterior stromal tissue was modeled to be stiffer than posterior stromal tissue.19 In addition, previous studies have reported posterior corneal surface bulging after insertion of hydrogel implants.24,25 These studies are summarized by Arffa,26 who described that when inserting an artificial implant or tissue lens intrastromally, the posterior corneal surface will change shape rather than the anterior corneal surface unless Bowman’s layer is completely severed (Figure 7). Although Bowman’s layer may lessen change on the anterior surface, the endokeratophakia cases reported by Pradhan et al.5 and Ganesh et al.3 showed that the majority of the change does occur on the anterior surface. In further studies, we intend to incorporate Bowman’s layer into the finite element model as a distinct layer to investigate whether severing Bowman’s layer as described will indeed increase the effectiveness of a tissue lens implantation procedure.
(A) Arffa describes that an inserted implant would cause posterior bulging rather than anterior deformation. (B) Endokeratophakia cases reported in the literature show that there is indeed change on the posterior surface elevation. However, the majority of deformation still takes place on the anterior surface, suggesting that Bowman’s layer also deforms. (Reprinted with permission from Arffa RC. Optics of lamellar refractive keratoplasty. In: Tasman W, ed. Duane’s Clinical Ophthalmology Volume 1 [CD-ROM]. Hagerstown, MD: Lippincott, Williams & Wilkins; 2006; chap 64.)
Finite element modeling appears to offer a viable option for evaluating the potential impact of corneal refractive surgical procedures such as endokeratophakia. This has the advantage of testing any number of different surgical parameters to optimize the procedure before actually treating patients, and might even be used as a patient-specific planning aid in daily clinical practice in the future.
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- Angunawela RI, Riau AK, Chaurasia SS, Tan DT, Mehta JS. Refractive lenticule re-implantation after myopic ReLEx: a feasibility study of stromal restoration after refractive surgery in a rabbit model. Invest Ophthalmol Vis Sci. 2012;53:4975–4985. doi:10.1167/iovs.12-10170 [CrossRef]
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- Arffa RC. Optics of lamellar refractive keratoplasty. In: Tasman W, ed. Duane’s Clinical Ophthalmology Volume 1 [CD-ROM]. Hagerstown, MD: Lippincott, Williams & Wilkins; 2006; chap 64. Available at: http://www.eyecalcs.com/DWAN/pages/v1/v1c064.html. Accessed December 18, 2014.