Cornea
The Location- and Depth-Dependent Mechanical Response of the Human Cornea Under Shear Loading Stephen R. Sloan Jr,1 Yousuf M. Khalifa,2 and Mark R. Buckley1 1 2
Department of Biomedical Engineering, University of Rochester, Rochester, New York, United States School of Medicine, Emory University, Atlanta, Georgia, United States
Correspondence: Stephen R. Sloan Jr, University of Rochester, Goergen Hall Room 321, Intercampus Drive, Rochester, NY 14627, USA; [email protected]. Submitted: June 10, 2014 Accepted: October 13, 2014 Citation: Sloan SR Jr, Khalifa YM, Buckley MR. The location- and depthdependent mechanical response of the human cornea under shear loading. Invest Ophthalmol Vis Sci. 2014;55:7919–7924. DOI:10.1167/ iovs.14-14997
PURPOSE. To characterize the depth-dependent shear modulus of the central and peripheral human cornea along the superior-inferior and nasal-temporal directions with a high spatial resolution. METHODS. Cylindrical explants from the central and peripheral corneas of 10 human donors were subjected to a 5% shear strain along the superior-inferior and nasal-temporal directions using a microscope-mounted mechanical testing device. Depth-dependent shear strain and shear modulus were computed through force measurements and displacement tracking. RESULTS. The shear modulus G of the human cornea varied continuously with depth, with a maximum occurring roughly 25% of the way from the anterior surface to the posterior surface. G also varied with direction in the superior region and (at some depths) was significantly higher for superior-inferior shear loading. In the anterior half of the cornea, the shear modulus along the nasal-temporal direction (GNT) did not vary with location; however, the superior region had significantly higher GNT in posterior cornea. In contrast, the shear modulus along the superior-inferior direction (GSI) was independent of location at all depths. CONCLUSIONS. This study demonstrates that the peak shear modulus of the human cornea occurs at a substantial distance within the corneal stroma. Depth-dependent differences between central and peripheral cornea possibly reflect the location-dependent mechanical environment of the cornea. Moreover, the cornea is not a transverse isotropic material, and must be characterized by more than a single shear modulus due to its dependence on loading direction. The material properties measured in this study are critical for developing accurate mechanical models to predict the vision-threatening morphological changes that can occur in the cornea. Keywords: cornea, biomechanics, shear, keratoconus
n corneal ectasia the cornea thins and steepens irregularly, and vision can be significantly impaired. Ectasia encompasses keratoconus, a degenerative disease of the cornea, and post laser-assisted in situ keratomileusis (LASIK) ectasia.1 Due to the significant clinical challenge posed by corneal ectatic disorders, much recent effort has gone into devising metrics to predict which patients are at highest risk for development and progression of keratoconus and post-LASIK ectasia.2 One possible approach to understanding and predicting keratoconus progression and post-LASIK corneal ectasia development is through analytical or computational mechanical models of the cornea3–21 that can account for its altered mechanical response to physiological stresses (i.e., from intraocular pressure) in the diseased state. In LASIK, the thickness of the corneal flap (i.e., the depth at which ablation occurs) generally varies between 48 and 187 lm.22 However, predicting how different choices of flap thickness affect corneal mechanics requires knowledge of corneal material properties with a high spatial resolution. For example, to distinguish between a 100-lm flap and a 150-lm flap, it is necessary to characterize the material properties of the cornea with a 50-lm resolution. Unfortunately, depth-dependent corneal mechanical properties have not yet been measured on this scale.
Another important consideration for corneal models is heterogeneity along the radial direction. Although corneal reshaping is performed in the central cornea, the cornea is an interconnected structure, and how LASIK affects corneal deformation will depend on corneal material properties in the periphery. Furthermore, mechanical models should consider the anisotropic (i.e., direction-dependent) structure and composition of the cornea that suggest an anisotropic mechanical response. That is, proper modeling of the cornea and post-LASIK alterations may require several material properties including multiple elastic moduli, Poisson’s ratios, and shear moduli. Thus, a key step in modeling the effects of LASIK, keratoconus or injury on corneal morphology under physiological stress is to characterize the location-, depth-, and direction-dependent material properties of the cornea, including the shear modulus G. In a recent study, torsional rheometry was used to measure the shear modulus of the human cornea as a function of depth (G(d/T)) in three partial thickness sections, yielding a spatial resolution of roughly 150 lm.23 However, it is still not known how the depth-dependent shear modulus profile of the cornea varies with direction or radial location. Moreover, G(d/T) may exhibit small-scale variations that are impossible to measure using partial thickness sectioning.
Copyright 2014 The Association for Research in Vision and Ophthalmology, Inc. www.iovs.org j ISSN: 1552-5783
7919
I
Depth-Dependent Mechanics of the Human Cornea Under Shear
IOVS j December 2014 j Vol. 55 j No. 12 j 7920
FIGURE 1. Image of a cornea after sections of the central and superior region were excised via 3-mm biopsy punches.
The objective of this study is to characterize the depthdependent shear modulus of the human cornea with a high spatial resolution in the central and peripheral regions and along the superior-inferior (SI) and nasal-temporal (NT) directions. We hypothesize that (1) the shear modulus of the human cornea varies continuously as a function of depth; (2) the shear modulus of the cornea depends on distance from the center of the cornea; and (3) the shear modulus of the human cornea depends on the direction of shear.
METHODS Study Design A total of 14 human corneas obtained from the Rochester Eye & Tissue Bank from deceased registered organ donors were used in this study. Donors ranged in age from 54 to 74 years old. Corneas were harvested within 16 hours of donor death with roughly 4 mm of scleral rim attached. Corneas were stored in corneal storage medium (Optisol, Chiron Ophthalmics, Irvine, CA, USA) at 378F until mechanical testing. All cornea testing was performed within 14 days of harvesting to ensure the corneal biomechanics remained similar to in vivo properties. Tenets of the Declaration of Helsinki were adhered throughout all experiments.
Preliminary Study of the Peripheral Cornea Four corneas were used for a preliminary study comparing the depth-dependent shear modulus in different peripheral locations (superior, inferior, nasal, and temporal) for shear in the superior-inferior direction.
Specimen Preparations and Cornea Mechanical Testing With Integrated Reflective Light and GreenFluorescence Imaging At the time of testing, corneas were removed from refrigeration and handled at room temperature. Anatomical orientation was determined by an experienced corneal surgeon, and noted on the surrounding sclera with a tissue-marking pen. Cylindrical sections were removed from the central and superior cornea regions with a 3-mm biopsy punch (Fig. 1) and marked with a tissue-marking pen to note axial orientation. The epithelium was not removed and remained attached throughout each
FIGURE 2. Schematic depicting the shear loading of a 3-mm cylindrical cornea section in the TDIS microscope-mounted mechanical testing device.
experiment. Superior specimens were taken from just inside (but as close as possible to) the visible limbus. The resulting radial distance from the center of the cornea to the center of the superior specimens was roughly 4.5 mm. To make the corneal keratocyte nuclei visible under fluorescence microscopy and enable strain analysis, buttons were stained for 30 seconds in a mixture containing 100 lL Optisol and 5 lL of 30 mM Acridine Orange (Sigma Aldrich, Inc., Allentown, PA, USA). Specimens were then shear tested according to the following protocol in an Optisol bath. Cylindrical sections were loaded into a tissue deformation imaging stage (TDIS; Harrick Scientific Products, Inc., Pleasantville, NY, USA) on custom fixture grips comprised of cantilevered plates. This device has been used extensively to characterize the location-dependent mechanical properties of several other collagenous connective tissues including articular cartilage and intervertebral disk.24–30 The orientation of the cylindrical sections in the grips was critical to determining direction-dependent shear properties, so careful attention was paid to ensure shear would occur in either the NT or SI axial direction. Two cylindrical sections (one central and one superior) from six donor corneas (total of 12 cylindrical sections) were shear tested in the NT direction, while two cylindrical sections (one central and one superior) from four donor corneas (total of eight cylindrical sections) were tested in the SI direction. The TDIS allowed for reflected light and fluorescence imaging with an inverted microscope (Olympus, Center Valley, PA, USA) simultaneous to mechanical testing (Fig. 2), which was necessary for tracking the depth-dependent corneal shear displacements. Prior to shear loading, corneas were compressed in the TDIS to a stress of 300 mm Hg (40 kPa) in order to avoid slipping between the grips and the specimens. Compressive stress was standardized instead of compressive strain due to varying corneal thickness from donor to donor. Compressed central and superior corneal specimens had thicknesses of 500 6 79 lm and 538 6 65 lm, respectively. For comparison, in the general population, the central cornea region has a mean thickness of 523 6 23 lm, while the superior region has a mean thickness of 597 6 34
IOVS j December 2014 j Vol. 55 j No. 12 j 7921
Depth-Dependent Mechanics of the Human Cornea Under Shear
FIGURE 3. (A) Representative green fluorescent microscopy image of a biopsy-punched cornea stained with Acridine Orange compressed to 40 kPa prior to shear loading. (B) Representative reflective light microscopy image of the TDIS fixture plates compressing a cornea section. Images were taken at 6.43 magnification.
lm under intraocular pressure.31 Before shear was applied, the specimen thickness was adjusted over a period of 10 minutes to maintain 300 mm Hg while adjusting for short-term stress relaxation. Although 10 minutes was insufficient to reach the equilibrium specimen thickness at 300 mm Hg (~375 lm), this protocol allowed for sufficiently rapid specimen testing to ensure that all buttons were tested within 14 days of harvesting. After the compression period, corneas were subjected to shear strain between the fixture grips in increments of 5 lm at a frequency of 0.2 Hz. Images were taken with both greenfluorescence (Fig. 3A) and reflective microscopy (Fig. 3B) after each shear increment, where fluorescence images were used to measure depth-dependent shear displacements of the cornea and reflective images were used to measure grip displacements. Shear strain was applied until a total displacement of 25 lm (approximately 5% shear) was reached.
Microscopy Image Analysis Green-fluorescence image sequences were used to track the displacements of fluorescently stained corneal nuclei and calculate depth-dependent GNT and GSI profiles (Fig. 4). Specifically, two-dimensional digital image correlation (2D DIC) software (Ncorr; www.ncorr.com) was used to calculate the location-dependent Lagrangian shear strain Exd(x,d) (Figs. 3, 4) for loading in the respective NT (GNT) or SI (GSI) direction. A subset size of 48 pixels (or 27 lm) and a subset spacing of 4 pixels were chosen for all image sequences. Since no out-of-plane motion of cell nuclei was observed, 2D DIC was deemed appropriate for image analysis. However, it is possible that for larger strains than those applied in this study, out-of-plane tissue motion may occur and 3D DIC may be required.14 In each specimen, Exd(x,d) was approximately uniform in x (Fig. 3a) and was therefore averaged across x to determine Exd(d). The displacements of the stationary grip were converted to an applied force F through calibration of the custom cantilever grips. Since the cross-sectional area A of the corneal buttons (~7 mm2) was found to be constant in depth before and during testing, it was assumed that the shear stress was also depth-independent. Therefore, G(d) was calculated according to G(d) ¼ (F/A)/Exd. Depth dependent mechanical properties are reported as a function of d/T, the ratio of depth to full corneal thickness, to allow for direct comparisons between the superior and central corneal regions. For statistical analysis, a two-way analysis of variance (ANOVA) with a Bonferroni posthoc test was performed across radial location and direction at each depth. A Bonferroni correction factor of 4 was used to establish significant differences at P 0.05/4. In addition, a one-way ANOVA with
Bonferroni posthoc analysis was performed across depth for each direction and radial location using a Bonferroni correction factor of 10 in order to establish significance at P 0.05/10.
RESULTS Since the shear modulus profile was not significantly different between peripheral locations (Supplementary Fig. S1), data from the superior location were chosen as representative of the periphery and compared with the central cornea. In general, characterization of corneal mechanical properties demonstrated that the shear strain and shear modulus G of the human cornea vary continuously with depth from the corneal surface to the endothelium. Shear strain recorded from NT direction shear loading in central cornea buttons (Fig. 4A) increased with depth in the deep stroma; however the minimum shear strain occurred at roughly 25% depth from the anterior surface. In superior cornea under NT shear (Fig. 4B), the depth dependent shear strain profile varied significantly from the central profile. Superior corneas exhibited maximum strain at approximately 70% depth toward the posterior, and the stroma most proximal to the posterior experienced strain similar to the anterior cornea. Strain profiles resulting from SI direction shear for both central and peripheral cornea were similar to that of the central region sheared in the NT direction, with strain increasing with depth beyond the minimum strain around 25% depth. Significant differences in GNT and GSI with depth were observed in both locations (Fig. 5). In fact, across all locations and shear directions, the maximum of G occurred at approximately 25% depth. Central cornea G was isotropic at all corneal depths (Fig. 6A), whereas superior cornea exhibited depth-dependent differences between GNT and GSI (Fig. 6B). GNT of the superior cornea in posterior regions near 80% depth
FIGURE 4. NT direction shear strain profiles of cornea section regions of interest (ROIs) plotted against d/T, where the top of the ROI is the corneal surface and the bottom is the endothelium. (A) Representative strain profile of a central specimen showing a roughly linear increase in strain from the surface to endothelium. (B) Representative strain profile of a superior specimen exhibiting a decrease then successive increase from the surface to endothelium.
Depth-Dependent Mechanics of the Human Cornea Under Shear
FIGURE 5. Shear modulus in the NT direction of central cornea exhibiting depth dependent mechanics. Data are mean 6 standard error of the mean, and ‡ denotes a significant difference in GNT between the noted data point and the posterior surface (P 0.05/10). G was also significantly depth-dependent in the SI direction and in the superior cornea (data not shown).
increased significantly over GSI. No other depths exhibited significant differences between shear directions in the superior cornea. GSI of the central and superior cornea was found to be independent of location (Fig. 6C), while the GNT between the two locations differed significantly at depths close to the endothelium (Fig. 6D). GNT of the superior cornea was significantly different than the central cornea at depths d/T > 0.8 and beyond. GNT at anterior depths was not significantly different between the two locations.
DISCUSSION In this study, image-guided mechanical testing was used to characterize the shear modulus profile G(d/T) of the human cornea with a high spatial resolution. As hypothesized, G(d/T)
IOVS j December 2014 j Vol. 55 j No. 12 j 7922 varied continuously with depth. However, surprisingly, the global maximum of G(d/T) occurred not at the corneal surface, but at a depth of 25% of the corneal thickness. In addition, to our knowledge, this study describes the first measurements of the direction- and radial location-dependent shear modulus of the human cornea. Consistent with our hypotheses, the depthdependent shear modulus varied significantly between the central and peripheral cornea and between the SI and NT directions. Specifically, closer to the corneal surface, GNT was not dependent on radial location. But deeper in the cornea, GNT was significantly higher in the superior region. On the other hand, the depth-dependent shear modulus along the SI direction did not depend on location. We speculate that the increased shear modulus of the superior cornea (1) along the SI direction and (2) near the posterior cornea are adaptations to increased shear stress during blinking. Like bone according to Wolff’s law,32 collagenous soft tissues are known to stiffen in regions of increased loading. Blinking induces a shear stress on the cornea in the SI direction that lasts the longest in the superior cornea (i.e., closest to the eyelid). This altered loading environment may be responsible for the observed directionand radial-location dependent mechanical response of the cornea. Interestingly, tensile tests of corneal strips have also generally demonstrated an increased stiffness along the SI direction in the human and bovine eye and a similar mechanism was proposed.33–35 However, it is important to note that the corneal shear modulus measured in this study is a distinct material parameter from the tensile modulus and is not necessarily expected to exhibit the same direction-dependent behavior. On the other hand, the shear modulus is expected to be related to the interlamellar cohesive strength of the cornea, which, consistent with the findings of this study, is higher in the superior cornea than in the central cornea.36 The general decrease in the shear modulus of the human cornea with depth from the corneal surface observed in this study is consistent with previous work.23,37 However, the depth dependent shear moduli measured in this study are higher than those reported in a previous study,23 particularly
FIGURE 6. G versus fractional cornea depth. (A) GSI across all depths is not significantly different between the central and superior corneal locations. (B) GNT is significantly different between the central and superior regions at roughly 80% depth to the endothelium. (C) Central G is independent of shear direction, (D) while the superior G differs around 15% to 25% depth. Data are mean 6 standard error of the mean, * denotes significant differences between GNT and GSI (P 0.05/4), and † denotes significant differences between central and superior regions (P 0.05/4).
IOVS j December 2014 j Vol. 55 j No. 12 j 7923
Depth-Dependent Mechanics of the Human Cornea Under Shear close to the corneal surface. This is likely due to the fact that the cornea is a nonlinear, strain-stiffening tissue and our measurements were conducted at a higher total strain (5% vs. 1%). The primary limitation of this study is that the shear modulus of the cornea in different locations and directions was measured only at a single rate of deformation. Since the cornea is a viscoelastic tissue, its shear modulus is rate-, time-, and frequency-dependent. Therefore, a complete characterization of the shear mechanical response of the cornea requires knowledge of viscoelastic (dynamic) material properties that will be the subject of a future investigation. In addition, it is important to note that the powerful testing methodology used in this study requires a simple specimen geometry that enables a constant shear force to be transmitted through all depths of the tested explants. Therefore, tested corneal explants were cut into cylinders that did not retain the complete integrity of the intact tissue, and this reduced interconnectivity may have affected the measured material properties. However, extracting location-dependent material properties from an intact cornea using an inflation test38,39 or other technique is a more difficult process that requires sophisticated computational tools (e.g., inverse finite element modeling) and rigorous parametric studies to avoid erroneous, nonunique solutions. Another potential limitation of this study is the assumption that shear stress does not vary as a function of depth. Stress concentrations that develop near the shearing plates could lead to an overestimation of the shear modulus on the anterior and posterior surfaces of the cornea. However, according to Saint-Venant’s principle, this effect should be negligible away from the shearing plates. Stress concentrations may also occur on the interior or near the edges of the specimens, leading to differences in stress at different corneal depths. However, as described previously,25 similar tests on homogeneous phantoms yielded uniform strain profiles, suggesting that negligible boundary-associated stress concentrations are induced within the resolution of our measurements. A final limitation is that the compressive stress in the performed experiments exceeded physiological conditions. Because specimens were only allowed to relax for 10 minutes, equilibrium thickness and compressive stress were not reached. Under the experimental conditions used to test all specimens, ~ 90 minutes of relaxation is required to equilibrate the compressive stress (data not shown). Thus, although the thicknesses of the specimens tested in this study were comparable to physiological values, these specimens were tested under super-physiological swelling pressures (300 mm Hg vs. 55 mm Hg40) because mechanical equilibrium had not yet been reached. In particular, fluid had not yet fully exuded from the cornea—a poroelastic material—and was providing additional load support. However, an additional specimen tested at equilibrated physiologically conditions (a swelling pressure of 46 mm Hg and a thickness of 450 lm) exhibited a shear modulus profile with similar values and an identical shape to the profile obtained in the manuscript at 300 mm Hg (Supplementary Fig. S2; compare to Fig. 6C in the manuscript). This finding, in combination with the standardization of testing conditions for all specimens, suggests that comparisons across depth and location—the focus of the manuscript—would not be altered under a lower compressive stress. The location- and direction-dependent material properties measured in this study will be useful for analytical and computational (e.g., finite element) models designed to predict deformation and altered function of the cornea from refractive surgery, injury, or disease, each of which typically affect the cornea in a highly localized manner. For example, this study demonstrates that modeling the cornea as a transverse
isotropic material with a single shear modulus in all radial directions cannot fully capture its mechanical response. Instead, the cornea appears to be a heterogeneous, orthotropic material that requires at least nine separate material properties at each depth and location. Current investigations in our laboratory are aimed at measuring these additional parameters. Recent models that account for cornea’s heterogeneity in the radial direction20 and along its depth21 show good agreement with in vitro inflation experiments and may further benefit from the high spatial resolution data presented here. The general methodology used in these experiments could serve as a powerful testing platform for assessing the efficacy of corneal surgical interventions. For example, corneal collagen cross-linking (CXL), a novel therapy for corneal ectasia introduced in 2003, involves cross-linking collagen fibrils in the corneal stroma using riboflavin and ultraviolet light.41,42 Since keratoconus will severely alter G(d/T) in the cornea, the baseline findings of this study may be used as a standard to guide CXL. That is, post-CXL corneas may be assessed using image-guided mechanical testing and UV exposure times and riboflavin concentrations can be tuned to yield location-dependent material properties that match those described in this study as closely as possible.
Acknowledgments The authors thank Ryan Trombetta for technical assistance and useful discussions. Disclosure: S.R. Sloan Jr, None; Y.M. Khalifa, None; M.R. Buckley, None
References 1. Binder PS, Lindstrom RL, Stulting RD, et al. Keratoconus and corneal ectasia after LASIK. J Cataract Refract Surg. 2005;31: 2035–2038. 2. Randleman JB, Russell B, Ward MA, Thompson KP, Stulting RD. Risk factors and prognosis for corneal ectasia after LASIK. Ophthalmology. 2003;110:267–275. 3. Anderson K, El-Sheikh A, Newson T. Application of structural analysis to the mechanical behaviour of the cornea. J R Soc Interface. 2004;1:3–15. 4. Pandolfi A, Holzapfel GA. Three-dimensional modeling and computational analysis of the human cornea considering distributed collagen fibril orientations. J Biomech Eng. 2008; 130:061006. 5. Cabrera Fernandez D, Niazy AM, Kurtz RM, Djotyan GP, Juhasz T. Finite element analysis applied to cornea reshaping. J Biomed Opt. 2005;10:064018. 6. Alastrue V, Calvo B, Pena E, Doblare M. Biomechanical modeling of refractive corneal surgery. J Biomech Eng. 2006; 128:150–160. 7. Pandolfi A, Fotia G, Manganiello F. Finite element simulations of laser refractive corneal surgery. Eng Comput-Germany. 2009;25:15–24. 8. Carvalho LA, Prado M, Cunha RH, et al. Keratoconus prediction using a finite element model of the cornea with local biomechanical properties. Arq Bras Oftalmol. 2009;72: 139–145. 9. Elsheikh A. Finite element modeling of corneal biomechanical behavior. J Refract Surg. 2010;26:289–300. 10. Elsheikh A, Ross S, Alhasso D, Rama P. Numerical study of the effect of corneal layered structure on ocular biomechanics. Curr Eye Res. 2009;34:26–35. 11. Elsheikh A, Wang D. Numerical modelling of corneal biomechanical behaviour. Comput Methods Biomech Biomed Engin. 2007;10:85–95.
IOVS j December 2014 j Vol. 55 j No. 12 j 7924
Depth-Dependent Mechanics of the Human Cornea Under Shear 12. Gefen A, Shalom R, Elad D, Mandel Y. Biomechanical analysis of the keratoconic cornea. J Mech Behav Biomed Mater. 2009; 2:224–236. 13. Han Z, Sui X, Zhou D, Zhou C, Ren Q. Biomechanical and refractive behaviors of keratoconic cornea based on threedimensional anisotropic hyperelastic models. J Refract Surg. 2013;29:282–290. 14. Nguyen TD, Boyce BL. An inverse finite element method for determining the anisotropic properties of the cornea. Biomech Model Mechanobiol. 2011;10:323–337. 15. Pandolfi A, Manganiello F. A model for the human cornea: constitutive formulation and numerical analysis. Biomech Model Mechanobiol. 2006;5:237–246. 16. Roy AS, Dupps WJ. Effects of altered corneal stiffness on native and postoperative LASIK corneal biomechanical behavior: a whole-eye finite element analysis. J Refract Surg. 2009;25: 875–887. 17. Roy AS, Dupps WJ. Patient-specific computational modeling of keratoconus progression and differential responses to collagen cross-linking. Invest Ophth Vis Sci. 2011;52:9174–9187. 18. Roy AS, Dupps WJ. Patient-specific modeling of corneal refractive surgery outcomes and inverse estimation of elastic property changes. J Biomech Eng-T Asme. 2011;133. 19. Sanchez P, Moutsouris K, Pandolfi A. Biomechanical and optical behavior of human corneas before and after photorefractive keratectomy. J Cataract Refract Surg. 2014;40:905– 917. 20. Studer H, Larrea X, Riedwyl H, Buchler P. Biomechanical model of human cornea based on stromal microstructure. J Biomech. 2010;43:836–842. 21. Petsche SJ, Pinsky PM. The role of 3-D collagen organization in stromal elasticity: a model based on X-ray diffraction data and second harmonic-generated images. Biomech Model Mechan. 2013;12:1101–1113. 22. Durairaj VD, Balentine J, Kouyoumdjian G, et al. The predictability of corneal flap thickness and tissue laser ablation in laser in situ keratomileusis. Ophthalmology. 2000;107:2140–2143. 23. Petsche SJ, Chernyak D, Martiz J, Levenston ME, Pinsky PM. Depth-dependent transverse shear properties of the human corneal stroma. Invest Ophthalmol Vis Sci. 2012;53:873–880. 24. Buckley MR, Bergou AJ, Fouchard J, Bonassar LJ, Cohen I. High-resolution spatial mapping of shear properties in cartilage. J Biomech. 2010;43:796–800. 25. Buckley MR, Bonassar L, Cohen I. Localization of viscous behavior and shear energy dissipation in articular cartilage under dynamic shear loading. J Biomech Eng. 2013;135. 26. Buckley MR, Gleghorn JP, Bonassar LJ, Cohen I. Mapping the depth dependence of shear properties in articular cartilage. J Biomech. 2008;41:2430–2437. 27. Michalek AJ, Buckley MR, Bonassar LJ, Cohen I, Iatridis JC. Measurement of local strains in intervertebral disc annulus fibrosus tissue under dynamic shear: contributions of matrix
28.
29.
30.
31.
32. 33.
34.
35.
36.
37.
38.
39.
40. 41.
42.
fiber orientation and elastin content. J Biomech. 2009;42: 2279–2285. Michalek AJ, Buckley MR, Bonassar LJ, Cohen I, Iatridis JC. The effects of needle puncture injury on microscale shear strain in the intervertebral disc annulus fibrosus. Spine J. 2010;10: 1098–1105. Sevenler D, Buckley MR, Kim G, van der Meulen MC, Cohen I, Bonassar LJ. Spatial periodicity in growth plate shear mechanical properties is disrupted by vitamin D deficiency. J Biomech. 2013;46:1597–1603. Silverberg JL, Dillavou S, Bonassar L, Cohen I. Anatomic variation of depth-dependent mechanical properties in neonatal bovine articular cartilage. J Orthop Res. 2013;31:686– 691. Prospero Ponce CM, Rocha KM, Smith SD, Krueger RR. Central and peripheral corneal thickness measured with optical coherence tomography, Scheimpflug imaging, and ultrasound pachymetry in normal, keratoconus-suspect, and post-laser in situ keratomileusis eyes. J Cataract Refract Surg. 2009;35: 1055–1062. Wolf JH. [Julis Wolff and his ‘‘law of bone remodeling’’]. Orthopade. 1995;24:378–386. Boschetti F, Triacca V, Spinelli L, Pandolfi A. Mechanical characterization of porcine corneas. J Biomech Eng-T Asme. 2012;134:031003. Elsheikh A, Brown M, Alhasso D, Rama P, Campanelli M, Garway-Heath D. Experimental assessment of corneal Anisotropy. J Refract Surg. 2008;24:178–187. Jayasuriya AC, Scheinbeim JI, Lubkin V, Bennett G, Kramer P. Piezoelectric and mechanical properties in bovine cornea. J Biomed Mater Res A. 2003;66A:260–265. Smolek MK. Interlamellar cohesive strength in the vertical meridian of human eye bank corneas. Invest Ophthalmol Vis Sci. 1993;34:2962–2969. Randleman JB, Dawson DG, Grossniklaus HE, McCarey BE, Edelhauser HF. Depth-dependent cohesive tensile strength in human donor corneas: implications for refractive surgery. J Refract Surg. 2008;24:S85–S89. Elsheikh A, Anderson K. Comparative study of corneal strip extensometry and inflation tests. J R Soc Interface. 2005;2: 177–185. Elsheikh A, Geraghty B, Rama P, Campanelli M, Meek KM. Characterization of age-related variation in corneal biomechanical properties. J R Soc Interface. 2010;7:1475–1485. Dohlman CH, Hedbys BO, Mishima S. The swelling pressure of the corneal stroma. Invest Ophthalmol. 1962;1:158–162. Wernli J, Schumacher S, Spoerl E, Mrochen M. The efficacy of corneal cross-linking shows a sudden decrease with very high intensity UV light and short treatment time. Invest Ophthalmol Vis Sci. 2013;54:1176–1180. Wollensak G, Spoerl E, Seiler T. Riboflavin/ultraviolet-ainduced collagen crosslinking for the treatment of keratoconus. Am J Ophthalmol. 2003;135:620–627.
The Location- and Depth-Dependent Mechanical Response of the Human Cornea Under Shear Loading Stephen R. Sloan Jr,1 Yousuf M. Khalifa,2 and Mark R. Buckley1 1 2
Department of Biomedical Engineering, University of Rochester, Rochester, New York, United States School of Medicine, Emory University, Atlanta, Georgia, United States
Correspondence: Stephen R. Sloan Jr, University of Rochester, Goergen Hall Room 321, Intercampus Drive, Rochester, NY 14627, USA; [email protected]. Submitted: June 10, 2014 Accepted: October 13, 2014 Citation: Sloan SR Jr, Khalifa YM, Buckley MR. The location- and depthdependent mechanical response of the human cornea under shear loading. Invest Ophthalmol Vis Sci. 2014;55:7919–7924. DOI:10.1167/ iovs.14-14997
PURPOSE. To characterize the depth-dependent shear modulus of the central and peripheral human cornea along the superior-inferior and nasal-temporal directions with a high spatial resolution. METHODS. Cylindrical explants from the central and peripheral corneas of 10 human donors were subjected to a 5% shear strain along the superior-inferior and nasal-temporal directions using a microscope-mounted mechanical testing device. Depth-dependent shear strain and shear modulus were computed through force measurements and displacement tracking. RESULTS. The shear modulus G of the human cornea varied continuously with depth, with a maximum occurring roughly 25% of the way from the anterior surface to the posterior surface. G also varied with direction in the superior region and (at some depths) was significantly higher for superior-inferior shear loading. In the anterior half of the cornea, the shear modulus along the nasal-temporal direction (GNT) did not vary with location; however, the superior region had significantly higher GNT in posterior cornea. In contrast, the shear modulus along the superior-inferior direction (GSI) was independent of location at all depths. CONCLUSIONS. This study demonstrates that the peak shear modulus of the human cornea occurs at a substantial distance within the corneal stroma. Depth-dependent differences between central and peripheral cornea possibly reflect the location-dependent mechanical environment of the cornea. Moreover, the cornea is not a transverse isotropic material, and must be characterized by more than a single shear modulus due to its dependence on loading direction. The material properties measured in this study are critical for developing accurate mechanical models to predict the vision-threatening morphological changes that can occur in the cornea. Keywords: cornea, biomechanics, shear, keratoconus
n corneal ectasia the cornea thins and steepens irregularly, and vision can be significantly impaired. Ectasia encompasses keratoconus, a degenerative disease of the cornea, and post laser-assisted in situ keratomileusis (LASIK) ectasia.1 Due to the significant clinical challenge posed by corneal ectatic disorders, much recent effort has gone into devising metrics to predict which patients are at highest risk for development and progression of keratoconus and post-LASIK ectasia.2 One possible approach to understanding and predicting keratoconus progression and post-LASIK corneal ectasia development is through analytical or computational mechanical models of the cornea3–21 that can account for its altered mechanical response to physiological stresses (i.e., from intraocular pressure) in the diseased state. In LASIK, the thickness of the corneal flap (i.e., the depth at which ablation occurs) generally varies between 48 and 187 lm.22 However, predicting how different choices of flap thickness affect corneal mechanics requires knowledge of corneal material properties with a high spatial resolution. For example, to distinguish between a 100-lm flap and a 150-lm flap, it is necessary to characterize the material properties of the cornea with a 50-lm resolution. Unfortunately, depth-dependent corneal mechanical properties have not yet been measured on this scale.
Another important consideration for corneal models is heterogeneity along the radial direction. Although corneal reshaping is performed in the central cornea, the cornea is an interconnected structure, and how LASIK affects corneal deformation will depend on corneal material properties in the periphery. Furthermore, mechanical models should consider the anisotropic (i.e., direction-dependent) structure and composition of the cornea that suggest an anisotropic mechanical response. That is, proper modeling of the cornea and post-LASIK alterations may require several material properties including multiple elastic moduli, Poisson’s ratios, and shear moduli. Thus, a key step in modeling the effects of LASIK, keratoconus or injury on corneal morphology under physiological stress is to characterize the location-, depth-, and direction-dependent material properties of the cornea, including the shear modulus G. In a recent study, torsional rheometry was used to measure the shear modulus of the human cornea as a function of depth (G(d/T)) in three partial thickness sections, yielding a spatial resolution of roughly 150 lm.23 However, it is still not known how the depth-dependent shear modulus profile of the cornea varies with direction or radial location. Moreover, G(d/T) may exhibit small-scale variations that are impossible to measure using partial thickness sectioning.
Copyright 2014 The Association for Research in Vision and Ophthalmology, Inc. www.iovs.org j ISSN: 1552-5783
7919
I
Depth-Dependent Mechanics of the Human Cornea Under Shear
IOVS j December 2014 j Vol. 55 j No. 12 j 7920
FIGURE 1. Image of a cornea after sections of the central and superior region were excised via 3-mm biopsy punches.
The objective of this study is to characterize the depthdependent shear modulus of the human cornea with a high spatial resolution in the central and peripheral regions and along the superior-inferior (SI) and nasal-temporal (NT) directions. We hypothesize that (1) the shear modulus of the human cornea varies continuously as a function of depth; (2) the shear modulus of the cornea depends on distance from the center of the cornea; and (3) the shear modulus of the human cornea depends on the direction of shear.
METHODS Study Design A total of 14 human corneas obtained from the Rochester Eye & Tissue Bank from deceased registered organ donors were used in this study. Donors ranged in age from 54 to 74 years old. Corneas were harvested within 16 hours of donor death with roughly 4 mm of scleral rim attached. Corneas were stored in corneal storage medium (Optisol, Chiron Ophthalmics, Irvine, CA, USA) at 378F until mechanical testing. All cornea testing was performed within 14 days of harvesting to ensure the corneal biomechanics remained similar to in vivo properties. Tenets of the Declaration of Helsinki were adhered throughout all experiments.
Preliminary Study of the Peripheral Cornea Four corneas were used for a preliminary study comparing the depth-dependent shear modulus in different peripheral locations (superior, inferior, nasal, and temporal) for shear in the superior-inferior direction.
Specimen Preparations and Cornea Mechanical Testing With Integrated Reflective Light and GreenFluorescence Imaging At the time of testing, corneas were removed from refrigeration and handled at room temperature. Anatomical orientation was determined by an experienced corneal surgeon, and noted on the surrounding sclera with a tissue-marking pen. Cylindrical sections were removed from the central and superior cornea regions with a 3-mm biopsy punch (Fig. 1) and marked with a tissue-marking pen to note axial orientation. The epithelium was not removed and remained attached throughout each
FIGURE 2. Schematic depicting the shear loading of a 3-mm cylindrical cornea section in the TDIS microscope-mounted mechanical testing device.
experiment. Superior specimens were taken from just inside (but as close as possible to) the visible limbus. The resulting radial distance from the center of the cornea to the center of the superior specimens was roughly 4.5 mm. To make the corneal keratocyte nuclei visible under fluorescence microscopy and enable strain analysis, buttons were stained for 30 seconds in a mixture containing 100 lL Optisol and 5 lL of 30 mM Acridine Orange (Sigma Aldrich, Inc., Allentown, PA, USA). Specimens were then shear tested according to the following protocol in an Optisol bath. Cylindrical sections were loaded into a tissue deformation imaging stage (TDIS; Harrick Scientific Products, Inc., Pleasantville, NY, USA) on custom fixture grips comprised of cantilevered plates. This device has been used extensively to characterize the location-dependent mechanical properties of several other collagenous connective tissues including articular cartilage and intervertebral disk.24–30 The orientation of the cylindrical sections in the grips was critical to determining direction-dependent shear properties, so careful attention was paid to ensure shear would occur in either the NT or SI axial direction. Two cylindrical sections (one central and one superior) from six donor corneas (total of 12 cylindrical sections) were shear tested in the NT direction, while two cylindrical sections (one central and one superior) from four donor corneas (total of eight cylindrical sections) were tested in the SI direction. The TDIS allowed for reflected light and fluorescence imaging with an inverted microscope (Olympus, Center Valley, PA, USA) simultaneous to mechanical testing (Fig. 2), which was necessary for tracking the depth-dependent corneal shear displacements. Prior to shear loading, corneas were compressed in the TDIS to a stress of 300 mm Hg (40 kPa) in order to avoid slipping between the grips and the specimens. Compressive stress was standardized instead of compressive strain due to varying corneal thickness from donor to donor. Compressed central and superior corneal specimens had thicknesses of 500 6 79 lm and 538 6 65 lm, respectively. For comparison, in the general population, the central cornea region has a mean thickness of 523 6 23 lm, while the superior region has a mean thickness of 597 6 34
IOVS j December 2014 j Vol. 55 j No. 12 j 7921
Depth-Dependent Mechanics of the Human Cornea Under Shear
FIGURE 3. (A) Representative green fluorescent microscopy image of a biopsy-punched cornea stained with Acridine Orange compressed to 40 kPa prior to shear loading. (B) Representative reflective light microscopy image of the TDIS fixture plates compressing a cornea section. Images were taken at 6.43 magnification.
lm under intraocular pressure.31 Before shear was applied, the specimen thickness was adjusted over a period of 10 minutes to maintain 300 mm Hg while adjusting for short-term stress relaxation. Although 10 minutes was insufficient to reach the equilibrium specimen thickness at 300 mm Hg (~375 lm), this protocol allowed for sufficiently rapid specimen testing to ensure that all buttons were tested within 14 days of harvesting. After the compression period, corneas were subjected to shear strain between the fixture grips in increments of 5 lm at a frequency of 0.2 Hz. Images were taken with both greenfluorescence (Fig. 3A) and reflective microscopy (Fig. 3B) after each shear increment, where fluorescence images were used to measure depth-dependent shear displacements of the cornea and reflective images were used to measure grip displacements. Shear strain was applied until a total displacement of 25 lm (approximately 5% shear) was reached.
Microscopy Image Analysis Green-fluorescence image sequences were used to track the displacements of fluorescently stained corneal nuclei and calculate depth-dependent GNT and GSI profiles (Fig. 4). Specifically, two-dimensional digital image correlation (2D DIC) software (Ncorr; www.ncorr.com) was used to calculate the location-dependent Lagrangian shear strain Exd(x,d) (Figs. 3, 4) for loading in the respective NT (GNT) or SI (GSI) direction. A subset size of 48 pixels (or 27 lm) and a subset spacing of 4 pixels were chosen for all image sequences. Since no out-of-plane motion of cell nuclei was observed, 2D DIC was deemed appropriate for image analysis. However, it is possible that for larger strains than those applied in this study, out-of-plane tissue motion may occur and 3D DIC may be required.14 In each specimen, Exd(x,d) was approximately uniform in x (Fig. 3a) and was therefore averaged across x to determine Exd(d). The displacements of the stationary grip were converted to an applied force F through calibration of the custom cantilever grips. Since the cross-sectional area A of the corneal buttons (~7 mm2) was found to be constant in depth before and during testing, it was assumed that the shear stress was also depth-independent. Therefore, G(d) was calculated according to G(d) ¼ (F/A)/Exd. Depth dependent mechanical properties are reported as a function of d/T, the ratio of depth to full corneal thickness, to allow for direct comparisons between the superior and central corneal regions. For statistical analysis, a two-way analysis of variance (ANOVA) with a Bonferroni posthoc test was performed across radial location and direction at each depth. A Bonferroni correction factor of 4 was used to establish significant differences at P 0.05/4. In addition, a one-way ANOVA with
Bonferroni posthoc analysis was performed across depth for each direction and radial location using a Bonferroni correction factor of 10 in order to establish significance at P 0.05/10.
RESULTS Since the shear modulus profile was not significantly different between peripheral locations (Supplementary Fig. S1), data from the superior location were chosen as representative of the periphery and compared with the central cornea. In general, characterization of corneal mechanical properties demonstrated that the shear strain and shear modulus G of the human cornea vary continuously with depth from the corneal surface to the endothelium. Shear strain recorded from NT direction shear loading in central cornea buttons (Fig. 4A) increased with depth in the deep stroma; however the minimum shear strain occurred at roughly 25% depth from the anterior surface. In superior cornea under NT shear (Fig. 4B), the depth dependent shear strain profile varied significantly from the central profile. Superior corneas exhibited maximum strain at approximately 70% depth toward the posterior, and the stroma most proximal to the posterior experienced strain similar to the anterior cornea. Strain profiles resulting from SI direction shear for both central and peripheral cornea were similar to that of the central region sheared in the NT direction, with strain increasing with depth beyond the minimum strain around 25% depth. Significant differences in GNT and GSI with depth were observed in both locations (Fig. 5). In fact, across all locations and shear directions, the maximum of G occurred at approximately 25% depth. Central cornea G was isotropic at all corneal depths (Fig. 6A), whereas superior cornea exhibited depth-dependent differences between GNT and GSI (Fig. 6B). GNT of the superior cornea in posterior regions near 80% depth
FIGURE 4. NT direction shear strain profiles of cornea section regions of interest (ROIs) plotted against d/T, where the top of the ROI is the corneal surface and the bottom is the endothelium. (A) Representative strain profile of a central specimen showing a roughly linear increase in strain from the surface to endothelium. (B) Representative strain profile of a superior specimen exhibiting a decrease then successive increase from the surface to endothelium.
Depth-Dependent Mechanics of the Human Cornea Under Shear
FIGURE 5. Shear modulus in the NT direction of central cornea exhibiting depth dependent mechanics. Data are mean 6 standard error of the mean, and ‡ denotes a significant difference in GNT between the noted data point and the posterior surface (P 0.05/10). G was also significantly depth-dependent in the SI direction and in the superior cornea (data not shown).
increased significantly over GSI. No other depths exhibited significant differences between shear directions in the superior cornea. GSI of the central and superior cornea was found to be independent of location (Fig. 6C), while the GNT between the two locations differed significantly at depths close to the endothelium (Fig. 6D). GNT of the superior cornea was significantly different than the central cornea at depths d/T > 0.8 and beyond. GNT at anterior depths was not significantly different between the two locations.
DISCUSSION In this study, image-guided mechanical testing was used to characterize the shear modulus profile G(d/T) of the human cornea with a high spatial resolution. As hypothesized, G(d/T)
IOVS j December 2014 j Vol. 55 j No. 12 j 7922 varied continuously with depth. However, surprisingly, the global maximum of G(d/T) occurred not at the corneal surface, but at a depth of 25% of the corneal thickness. In addition, to our knowledge, this study describes the first measurements of the direction- and radial location-dependent shear modulus of the human cornea. Consistent with our hypotheses, the depthdependent shear modulus varied significantly between the central and peripheral cornea and between the SI and NT directions. Specifically, closer to the corneal surface, GNT was not dependent on radial location. But deeper in the cornea, GNT was significantly higher in the superior region. On the other hand, the depth-dependent shear modulus along the SI direction did not depend on location. We speculate that the increased shear modulus of the superior cornea (1) along the SI direction and (2) near the posterior cornea are adaptations to increased shear stress during blinking. Like bone according to Wolff’s law,32 collagenous soft tissues are known to stiffen in regions of increased loading. Blinking induces a shear stress on the cornea in the SI direction that lasts the longest in the superior cornea (i.e., closest to the eyelid). This altered loading environment may be responsible for the observed directionand radial-location dependent mechanical response of the cornea. Interestingly, tensile tests of corneal strips have also generally demonstrated an increased stiffness along the SI direction in the human and bovine eye and a similar mechanism was proposed.33–35 However, it is important to note that the corneal shear modulus measured in this study is a distinct material parameter from the tensile modulus and is not necessarily expected to exhibit the same direction-dependent behavior. On the other hand, the shear modulus is expected to be related to the interlamellar cohesive strength of the cornea, which, consistent with the findings of this study, is higher in the superior cornea than in the central cornea.36 The general decrease in the shear modulus of the human cornea with depth from the corneal surface observed in this study is consistent with previous work.23,37 However, the depth dependent shear moduli measured in this study are higher than those reported in a previous study,23 particularly
FIGURE 6. G versus fractional cornea depth. (A) GSI across all depths is not significantly different between the central and superior corneal locations. (B) GNT is significantly different between the central and superior regions at roughly 80% depth to the endothelium. (C) Central G is independent of shear direction, (D) while the superior G differs around 15% to 25% depth. Data are mean 6 standard error of the mean, * denotes significant differences between GNT and GSI (P 0.05/4), and † denotes significant differences between central and superior regions (P 0.05/4).
IOVS j December 2014 j Vol. 55 j No. 12 j 7923
Depth-Dependent Mechanics of the Human Cornea Under Shear close to the corneal surface. This is likely due to the fact that the cornea is a nonlinear, strain-stiffening tissue and our measurements were conducted at a higher total strain (5% vs. 1%). The primary limitation of this study is that the shear modulus of the cornea in different locations and directions was measured only at a single rate of deformation. Since the cornea is a viscoelastic tissue, its shear modulus is rate-, time-, and frequency-dependent. Therefore, a complete characterization of the shear mechanical response of the cornea requires knowledge of viscoelastic (dynamic) material properties that will be the subject of a future investigation. In addition, it is important to note that the powerful testing methodology used in this study requires a simple specimen geometry that enables a constant shear force to be transmitted through all depths of the tested explants. Therefore, tested corneal explants were cut into cylinders that did not retain the complete integrity of the intact tissue, and this reduced interconnectivity may have affected the measured material properties. However, extracting location-dependent material properties from an intact cornea using an inflation test38,39 or other technique is a more difficult process that requires sophisticated computational tools (e.g., inverse finite element modeling) and rigorous parametric studies to avoid erroneous, nonunique solutions. Another potential limitation of this study is the assumption that shear stress does not vary as a function of depth. Stress concentrations that develop near the shearing plates could lead to an overestimation of the shear modulus on the anterior and posterior surfaces of the cornea. However, according to Saint-Venant’s principle, this effect should be negligible away from the shearing plates. Stress concentrations may also occur on the interior or near the edges of the specimens, leading to differences in stress at different corneal depths. However, as described previously,25 similar tests on homogeneous phantoms yielded uniform strain profiles, suggesting that negligible boundary-associated stress concentrations are induced within the resolution of our measurements. A final limitation is that the compressive stress in the performed experiments exceeded physiological conditions. Because specimens were only allowed to relax for 10 minutes, equilibrium thickness and compressive stress were not reached. Under the experimental conditions used to test all specimens, ~ 90 minutes of relaxation is required to equilibrate the compressive stress (data not shown). Thus, although the thicknesses of the specimens tested in this study were comparable to physiological values, these specimens were tested under super-physiological swelling pressures (300 mm Hg vs. 55 mm Hg40) because mechanical equilibrium had not yet been reached. In particular, fluid had not yet fully exuded from the cornea—a poroelastic material—and was providing additional load support. However, an additional specimen tested at equilibrated physiologically conditions (a swelling pressure of 46 mm Hg and a thickness of 450 lm) exhibited a shear modulus profile with similar values and an identical shape to the profile obtained in the manuscript at 300 mm Hg (Supplementary Fig. S2; compare to Fig. 6C in the manuscript). This finding, in combination with the standardization of testing conditions for all specimens, suggests that comparisons across depth and location—the focus of the manuscript—would not be altered under a lower compressive stress. The location- and direction-dependent material properties measured in this study will be useful for analytical and computational (e.g., finite element) models designed to predict deformation and altered function of the cornea from refractive surgery, injury, or disease, each of which typically affect the cornea in a highly localized manner. For example, this study demonstrates that modeling the cornea as a transverse
isotropic material with a single shear modulus in all radial directions cannot fully capture its mechanical response. Instead, the cornea appears to be a heterogeneous, orthotropic material that requires at least nine separate material properties at each depth and location. Current investigations in our laboratory are aimed at measuring these additional parameters. Recent models that account for cornea’s heterogeneity in the radial direction20 and along its depth21 show good agreement with in vitro inflation experiments and may further benefit from the high spatial resolution data presented here. The general methodology used in these experiments could serve as a powerful testing platform for assessing the efficacy of corneal surgical interventions. For example, corneal collagen cross-linking (CXL), a novel therapy for corneal ectasia introduced in 2003, involves cross-linking collagen fibrils in the corneal stroma using riboflavin and ultraviolet light.41,42 Since keratoconus will severely alter G(d/T) in the cornea, the baseline findings of this study may be used as a standard to guide CXL. That is, post-CXL corneas may be assessed using image-guided mechanical testing and UV exposure times and riboflavin concentrations can be tuned to yield location-dependent material properties that match those described in this study as closely as possible.
Acknowledgments The authors thank Ryan Trombetta for technical assistance and useful discussions. Disclosure: S.R. Sloan Jr, None; Y.M. Khalifa, None; M.R. Buckley, None
References 1. Binder PS, Lindstrom RL, Stulting RD, et al. Keratoconus and corneal ectasia after LASIK. J Cataract Refract Surg. 2005;31: 2035–2038. 2. Randleman JB, Russell B, Ward MA, Thompson KP, Stulting RD. Risk factors and prognosis for corneal ectasia after LASIK. Ophthalmology. 2003;110:267–275. 3. Anderson K, El-Sheikh A, Newson T. Application of structural analysis to the mechanical behaviour of the cornea. J R Soc Interface. 2004;1:3–15. 4. Pandolfi A, Holzapfel GA. Three-dimensional modeling and computational analysis of the human cornea considering distributed collagen fibril orientations. J Biomech Eng. 2008; 130:061006. 5. Cabrera Fernandez D, Niazy AM, Kurtz RM, Djotyan GP, Juhasz T. Finite element analysis applied to cornea reshaping. J Biomed Opt. 2005;10:064018. 6. Alastrue V, Calvo B, Pena E, Doblare M. Biomechanical modeling of refractive corneal surgery. J Biomech Eng. 2006; 128:150–160. 7. Pandolfi A, Fotia G, Manganiello F. Finite element simulations of laser refractive corneal surgery. Eng Comput-Germany. 2009;25:15–24. 8. Carvalho LA, Prado M, Cunha RH, et al. Keratoconus prediction using a finite element model of the cornea with local biomechanical properties. Arq Bras Oftalmol. 2009;72: 139–145. 9. Elsheikh A. Finite element modeling of corneal biomechanical behavior. J Refract Surg. 2010;26:289–300. 10. Elsheikh A, Ross S, Alhasso D, Rama P. Numerical study of the effect of corneal layered structure on ocular biomechanics. Curr Eye Res. 2009;34:26–35. 11. Elsheikh A, Wang D. Numerical modelling of corneal biomechanical behaviour. Comput Methods Biomech Biomed Engin. 2007;10:85–95.
IOVS j December 2014 j Vol. 55 j No. 12 j 7924
Depth-Dependent Mechanics of the Human Cornea Under Shear 12. Gefen A, Shalom R, Elad D, Mandel Y. Biomechanical analysis of the keratoconic cornea. J Mech Behav Biomed Mater. 2009; 2:224–236. 13. Han Z, Sui X, Zhou D, Zhou C, Ren Q. Biomechanical and refractive behaviors of keratoconic cornea based on threedimensional anisotropic hyperelastic models. J Refract Surg. 2013;29:282–290. 14. Nguyen TD, Boyce BL. An inverse finite element method for determining the anisotropic properties of the cornea. Biomech Model Mechanobiol. 2011;10:323–337. 15. Pandolfi A, Manganiello F. A model for the human cornea: constitutive formulation and numerical analysis. Biomech Model Mechanobiol. 2006;5:237–246. 16. Roy AS, Dupps WJ. Effects of altered corneal stiffness on native and postoperative LASIK corneal biomechanical behavior: a whole-eye finite element analysis. J Refract Surg. 2009;25: 875–887. 17. Roy AS, Dupps WJ. Patient-specific computational modeling of keratoconus progression and differential responses to collagen cross-linking. Invest Ophth Vis Sci. 2011;52:9174–9187. 18. Roy AS, Dupps WJ. Patient-specific modeling of corneal refractive surgery outcomes and inverse estimation of elastic property changes. J Biomech Eng-T Asme. 2011;133. 19. Sanchez P, Moutsouris K, Pandolfi A. Biomechanical and optical behavior of human corneas before and after photorefractive keratectomy. J Cataract Refract Surg. 2014;40:905– 917. 20. Studer H, Larrea X, Riedwyl H, Buchler P. Biomechanical model of human cornea based on stromal microstructure. J Biomech. 2010;43:836–842. 21. Petsche SJ, Pinsky PM. The role of 3-D collagen organization in stromal elasticity: a model based on X-ray diffraction data and second harmonic-generated images. Biomech Model Mechan. 2013;12:1101–1113. 22. Durairaj VD, Balentine J, Kouyoumdjian G, et al. The predictability of corneal flap thickness and tissue laser ablation in laser in situ keratomileusis. Ophthalmology. 2000;107:2140–2143. 23. Petsche SJ, Chernyak D, Martiz J, Levenston ME, Pinsky PM. Depth-dependent transverse shear properties of the human corneal stroma. Invest Ophthalmol Vis Sci. 2012;53:873–880. 24. Buckley MR, Bergou AJ, Fouchard J, Bonassar LJ, Cohen I. High-resolution spatial mapping of shear properties in cartilage. J Biomech. 2010;43:796–800. 25. Buckley MR, Bonassar L, Cohen I. Localization of viscous behavior and shear energy dissipation in articular cartilage under dynamic shear loading. J Biomech Eng. 2013;135. 26. Buckley MR, Gleghorn JP, Bonassar LJ, Cohen I. Mapping the depth dependence of shear properties in articular cartilage. J Biomech. 2008;41:2430–2437. 27. Michalek AJ, Buckley MR, Bonassar LJ, Cohen I, Iatridis JC. Measurement of local strains in intervertebral disc annulus fibrosus tissue under dynamic shear: contributions of matrix
28.
29.
30.
31.
32. 33.
34.
35.
36.
37.
38.
39.
40. 41.
42.
fiber orientation and elastin content. J Biomech. 2009;42: 2279–2285. Michalek AJ, Buckley MR, Bonassar LJ, Cohen I, Iatridis JC. The effects of needle puncture injury on microscale shear strain in the intervertebral disc annulus fibrosus. Spine J. 2010;10: 1098–1105. Sevenler D, Buckley MR, Kim G, van der Meulen MC, Cohen I, Bonassar LJ. Spatial periodicity in growth plate shear mechanical properties is disrupted by vitamin D deficiency. J Biomech. 2013;46:1597–1603. Silverberg JL, Dillavou S, Bonassar L, Cohen I. Anatomic variation of depth-dependent mechanical properties in neonatal bovine articular cartilage. J Orthop Res. 2013;31:686– 691. Prospero Ponce CM, Rocha KM, Smith SD, Krueger RR. Central and peripheral corneal thickness measured with optical coherence tomography, Scheimpflug imaging, and ultrasound pachymetry in normal, keratoconus-suspect, and post-laser in situ keratomileusis eyes. J Cataract Refract Surg. 2009;35: 1055–1062. Wolf JH. [Julis Wolff and his ‘‘law of bone remodeling’’]. Orthopade. 1995;24:378–386. Boschetti F, Triacca V, Spinelli L, Pandolfi A. Mechanical characterization of porcine corneas. J Biomech Eng-T Asme. 2012;134:031003. Elsheikh A, Brown M, Alhasso D, Rama P, Campanelli M, Garway-Heath D. Experimental assessment of corneal Anisotropy. J Refract Surg. 2008;24:178–187. Jayasuriya AC, Scheinbeim JI, Lubkin V, Bennett G, Kramer P. Piezoelectric and mechanical properties in bovine cornea. J Biomed Mater Res A. 2003;66A:260–265. Smolek MK. Interlamellar cohesive strength in the vertical meridian of human eye bank corneas. Invest Ophthalmol Vis Sci. 1993;34:2962–2969. Randleman JB, Dawson DG, Grossniklaus HE, McCarey BE, Edelhauser HF. Depth-dependent cohesive tensile strength in human donor corneas: implications for refractive surgery. J Refract Surg. 2008;24:S85–S89. Elsheikh A, Anderson K. Comparative study of corneal strip extensometry and inflation tests. J R Soc Interface. 2005;2: 177–185. Elsheikh A, Geraghty B, Rama P, Campanelli M, Meek KM. Characterization of age-related variation in corneal biomechanical properties. J R Soc Interface. 2010;7:1475–1485. Dohlman CH, Hedbys BO, Mishima S. The swelling pressure of the corneal stroma. Invest Ophthalmol. 1962;1:158–162. Wernli J, Schumacher S, Spoerl E, Mrochen M. The efficacy of corneal cross-linking shows a sudden decrease with very high intensity UV light and short treatment time. Invest Ophthalmol Vis Sci. 2013;54:1176–1180. Wollensak G, Spoerl E, Seiler T. Riboflavin/ultraviolet-ainduced collagen crosslinking for the treatment of keratoconus. Am J Ophthalmol. 2003;135:620–627.
