Shear properties of the cornea are important for understanding and modeling the mechanics of the tissue. The present study reports the transverse shear modulus of the human corneal stroma and characterizes the profile of the shear modulus through the depth. After controlling for axial strain, the anterior layer samples were significantly stiffer than the central and posterior layer samples, as hypothesized. Close examination of results for each donor in
Figure 7 reveals that, for a given compressive axial strain, the anterior layer modulus is two to five times greater than the central layer modulus, whereas the central and posterior layers are comparable. This trend matches the degree of lamellar interweaving seen in each third thickness of
Figure 1. The anterior third thickness shows many lamellae that run transversely to the tangent plane. The collagen fibrils that make up each lamella have high tensile stiffness and interwoven lamellae would be stretched in transverse shear deformation thereby engaging the fibrils in shear resistance. In the posterior third thickness, lamellae remain mostly parallel to the tangent plane and therefore would not be stretched during shearing deformation. More interweaving, with corresponding fibril engagement resisting the deformation, will imply a higher shear modulus. This could explain the larger moduli in the anterior layer samples. Average shear modulus values for each layer at the estimated physiological thickness provide an immediate starting point for choosing shear properties of a computational model where shear stiffness can vary with depth in the stroma (
Table 2).
Equilibrium axial compressive stress showed a power law relationship with axial strain, consistent with the findings of Olsen and Sperling.
19 Measured swelling pressure values for full thickness samples were generally lower than the mean relationship reported by Olsen and Sperling (
Fig. 9a). This is expected due to the different bath used during the tests. Optisol (Bausch & Lomb) contains dextran and chondroitin sulfate to help prevent swelling during storage which would lower swelling pressure.
20 Despite the lower swelling pressure values, the modulus values calculated at 0% axial strain for full thickness samples match well with the value calculated from Olsen and Sperling's data.
19 Swelling pressures for isolated layers of the cornea have not been reported before. Although isolated layer samples also exhibited a power law relationship, the equilibrium compressive axial stress and moduli are lower than expect based on the full thickness samples. Interactions between layers present in the intact cornea could have been lost when the layers were cut. Hence, the intact cornea would be stiffer than the effective modulus based on the isolated layers. Damage from the laser cuts, errors in axial strain calculation, and loss of macromolecules during layer isolation are also possible explanations for the low experimental values, which remain an open question.
As with all in vitro experiments on the cornea, swelling of the specimens is an important concern. Average CCT for human corneas is approximately 535 μm
21 so some swelling likely occurred before sample processing. Because the normal physiological CCT of the samples was not known, the axial strains calculated from pachymeter readings are an approximation with possible error due to swelling. The range of axial strain at which the tests were performed was designed to include a gap distance close to the physiological thickness of the sample. Swelling before the laser cuts is assumed to be uniform through the depth because all samples experienced only slight to moderate swelling (<20% CCT).
22 Therefore, errors in axial strain calculation due to swelling for thirds from the same cornea are the same. In an effort to minimize swelling, the cornea samples were stored and refrigerated in corneal storage medium (Optisol; Bausch & Lomb) except during the laser cutting procedure and preparation immediately before testing. Additionally, the epithelium and endothelium were left intact for all samples until final preparation because removal has been shown to cause swelling of the cornea in vivo.
16 Although corneal storage medium (Optisol; Bausch & Lomb) was used to prevent swelling and degradation of the corneal tissue, shear properties could change postmortem.
Figure 7 shows that the anterior isolated layer samples of Donors 3 and 4 (2 weeks postmortem) are stiffer than the anterior layer samples of Donors 1 and 2 (1 week postmortem). This could be explained by changes that occurred postmortem or varying degrees of anterior lamellar interweaving between individuals.
Estimated epithelium and endothelium thicknesses were used during the calculation of laser cut depths, introducing additional sources of error in axial strain calculation. A thick epithelium would cause a thin anterior third sample while a thin or partially removed epithelium would lead to a thicker anterior third sample than desired. By a similar argument applied to the endothelium, the posterior third could be thicker or thinner than expected. Because the central third was laser cut on both sides, the only error in axial strain calculation is from swelling. This could explain why the central thirds qualitatively have less variation than the other third types (
Fig. 6). Caution must be exercised when comparing measured values from different corneas at a given axial strain because samples can have different errors associated with the calculated axial strain from swelling as well as cut depths. Comparing the isolated layer samples to the full sample from Donor 2 shows that the full sample is stiffer than all the thirds for a given axial strain (
Fig. 8a). However, plotting the shear moduli for the isolated layer samples as a function of axial stress and including full sample values that fall in the same axial stress range indicates that full modulus values fall in between anterior and central values for a given axial stress (
Fig. 8b). In vivo measurements of certain biomechanical properties have shown little variation between properties of an individual's left and right eye.
23 Therefore, it is expected that full moduli values would fall in the range of isolated sample modulus values from the same donor. For all four donor pairs, plotting shear modulus values against axial stress satisfied this condition, while comparing with axial strain was inconsistent. Full thickness samples were tested over different calculated axial strain ranges and would be difficult to compare by axial strain. However,
Figure 5 exhibits the low variability of shear modulus for a given measured axial stress.
Due to the dynamic nature of each test, every shear modulus reported is the magnitude of the complex shear modulus recorded by the rheometer. The complex shear modulus is a dynamic, viscoelastic property that includes an elastic part and a dissipative part calculated by the rheometer from the amplitudes and phase lag between the measured stress and strain curves that are sinusoidal in time. The magnitude of the complex modulus will be higher than the static shear modulus, but the complex modulus is more straightforward to measure in the laboratory. It should also be noted that the tissue at the center of the sample undergoes a smaller shear strain than the outer edge. Shear modulus calculations by the rheometer use the assumption that the modulus does not depend on strain magnitude. Preliminary studies showed this is not the case, as with most biological tissues, so a small strain amplitude was used to minimize this effect.
When calculating the shear modulus from the torsional rheometry data it is assumed that the sample is a cylinder (uniform thickness and cross-section). The corneas cut with the femtosecond laser produced three layers of uniform thickness. An uncut cornea is thinnest at its center and thickens toward the limbus. Therefore, the full thickness cornea samples were not uniform thickness and the axial strain at the center of the sample was smaller than the axial strain at the perimeter of the 6 mm button during the test. This causes additional underestimation of the axial strain calculation superimposed on the error from swelling. Future studies could use the femtosecond laser to cut full thickness samples at a depth equal to the CCT to produce a uniform thickness sample that includes the entire stroma at its center. Laser cuts did not change any sample's curvature. Thus all ex vivo samples had a curvature that matched in vivo corneas. Consequently, all samples had to be flattened by the rheometer, introducing additional prestress in the tissue from bending that was neglected in the calculation of the shear properties. The in vivo cornea has a tensile prestress from intraocular pressure. Presence of an in-plane tensile prestress may produce a higher in vivo shear modulus when interwoven lamellae are present due to the nonlinear tensile behavior of collagen fibrils stretched in transverse shear deformation.
Mechanical properties of the stroma are of primary interest because the other layers are presumed to have little effect on the cornea's overall biomechanical properties. The tensile stiffness of the epithelium has been shown to be considerably lower than that of the stroma
24 and the thinner endothelial layer is expected to have similar properties. The epithelium and endothelium were removed from all samples to promote more rigid gripping of the sample in the rheometer. Bowman's layer and Descemet's membrane were not removed so their mechanical contribution needs to be considered. Bowman's layer, which is approximately 10 μm thick,
13 was present in the anterior and full samples. However, it is assumed to provide a negligible mechanical contribution because uniaxial tests of corneal strips have shown that the presence of Bowman's layer did not have significant effect.
25 Descemet's membrane, also approximately 10 μm thick,
13 was present in the full samples, but may or may not have been present in the posterior samples depending on the deepest cut with the laser. The presence of Descemet's membrane has been shown to have little mechanical effect in low pressure inflation tests so its mechanical contribution to the samples was also neglected.
26,27 Note that the mechanical contribution of these layers is ignored based on tensile data because shear property data has never been measured.
While numerous steps were taken to minimize damaging the samples during preparation, damage to lamellae and collagen fibrils is unavoidable. The raster pattern of the femtosecond laser leaves some residual tissue along the cut surface. Just before testing the layers were slowly pulled and wedged apart with forceps and a spatula, carefully dissecting the residual tissue. The tearing of tissue caused by separating the layers probably extends the damaged area of the tissue beyond the area of the laser cut. The anterior and posterior layers only had one surface that underwent the blunt dissection in contrast to both surfaces of the central third. In the posterior third, cuts followed the anterior curvature but lamellae follow the steeper posterior curvature due to the cornea becoming thicker toward the limbus; this resulted in undesired transversely cut lamellae that were not interwoven. The damage to lamellae and fibril integrity could lead to an underestimation of the tissue's mechanical shear properties due to fewer fibrils engaged during shear deformation. This could explain why shear modulus values of uncut full samples were comparable to the anterior layer rather than falling between anterior and posterior layer as expected.
The transverse shear stiffness of an individual lamella at low shear strain is conjectured to result from the gel properties of the interfibrillar electrolyte fluid interacting with the GAG fixed charges (Pinsky PM, et al., manuscript submitted, 2012).
28 The transverse shear modulus resulting from this molecular mechanism is expected to be small compared with the tensile modulus resulting from the direct engagement of the collagen fibrils in tension tests.
1 –3 Indeed, the magnitudes of the shear moduli are two to three orders lower than measured tensile moduli of the cornea.
1 –3 This interpretation also provides an explanation for the significant dependence of shear moduli on axial compression. As the tissue is compressed, the negative charges on the GAGs between and surrounding collagen fibrils move closer together, increasing the fixed charge density which has been shown in recent work to lead to an increased shear modulus.
28 Electrolyte properties of the matrix could be examined by performing shear tests while varying the ionic concentration of the bath.
Supported by National Institutes of Health Grant R01AR052861 (MEL), Stanford University Bio-X Interdisciplinary Initiatives Program (PMP), and Stanford Graduate Fellowship (SJP).
Disclosure:
S.J. Petsche, None;
D. Chernyak, Abbott Medical Optics, Inc. (E);
J. Martiz, International Refractive Consultants, LLC (E);
M.E. Levenston, None;
P.M. Pinsky, None
Presented in part at the annual meeting of the Association for Research in Vision and Ophthalmology, Fort Lauderdale, Florida, May 2011, and the American Society of Mechanical Engineers 2010 Summer Bioengineering Conference, Naples, Florida, June 2010.
The authors thank Abbott Medical Optics, Inc., for help in acquisition of the human corneas, James F. Nishimuta of the Soft Tissue Biomechanics Laboratory at Stanford University, for his help with the testing protocols, and Moritz Winkler and James V. Jester's group at the University of California, Irvine, for providing the second harmonic generated image (
Fig. 1).