**Purpose.**:
To measure the transverse shear modulus of the human corneal stroma and its profile through the depth by mechanical testing, and to assess the validity of the hypothesis that the shear modulus will be greater in the anterior third due to increased interweaving of lamellae.

**Methods.**:
Torsional rheometry was used to measure the transverse shear properties of 6 mm diameter buttons of matched human cadaver cornea pairs. One cornea from each pair was cut into thirds through the thickness with a femtosecond laser and each stromal third was tested individually. The remaining intact corneas were tested to measure full stroma shear modulus. The shear modulus from a 1% shear strain oscillatory test was measured at various levels of axial compression for all samples.

**Results.**:
After controlling for axial compression, the transverse shear moduli of isolated anterior layers were significantly higher than central and posterior layers. Mean modulus values at 0% axial strain were 7.71 ± 6.34 kPa in the anterior, 1.99 ± 0.45 kPa in the center, 1.31 ± 1.01 kPa in the posterior, and 9.48 ± 2.92 kPa for full thickness samples. A mean equilibrium compressive modulus of 38.7 ± 8.6 kPa at 0% axial strain was calculated from axial compression measured during the shear tests.

**Conclusions.**:
Transverse shear moduli are two to three orders of magnitude lower than tensile moduli reported in the literature. The profile of shear moduli through the depth displayed a significant increase from posterior to anterior. This gradient supports the hypothesis and corresponds to the gradient of interwoven lamellae seen in imaging of stromal cross-sections.

^{ 1 –4 }However, even in the most simple of material models there are at a minimum two elastic constants that must be measured to characterize the three-dimensional elastic behavior of the material. This most simple case is called isotropic elasticity

^{ 5 }and occurs when the material properties under investigation exhibit no dependence on direction during testing. In this case, the material is fully characterized by the Young's modulus and the shear modulus. The shear modulus naturally measures the resistance of the tissue to shearing strains. In fact, the microstructure of the corneal stroma suggests that its elasticity cannot be isotropic. The parallel arrays of collagen fibrils within each lamella and the layering of lamellae one on another imply that the transverse (anterior-posterior) properties of the tissue will be different from the in-plane properties. To address this anisotropy and other considerations, increasingly complex elasticity models have been introduced with the goal of achieving greater fidelity to the full three-dimensional behavior of the tissue.

^{ 6 –8 }More complex models always have more than two intrinsic elastic constants that need to be measured. When such models are extended to nonlinear behavior, yet more constants must necessarily be introduced (Petsche SJ, et al., manuscript submitted, 2012).

^{ 6,8 }

^{ 5 }is the simplest possible model that can reasonably be applied to the corneal stroma. Materials exhibiting transverse isotropy have a single plane of material isotropy (the corneal tangent plane) and properties in this plane will be different from properties measured orthogonally (through the corneal thickness). In this case, characterization of the material elasticity requires the measurement of five independent elastic constants: the in-plane Young's modulus (related to the tensile modulus) and transverse Young's modulus, the in-plane and transverse Poisson's ratios, and the transverse shear modulus, denoted G.

^{ 5 }To appreciate the role of the transverse shear modulus, consider the anterior and posterior surfaces of a circular stromal button subjected to relative torsional twisting about an axis perpendicular to the surfaces (see Fig. 4). The tissue will become deformed in a state of pure transverse shear strain and the resistance of the tissue will be dependent only on the transverse shear modulus G. Experimental measurement of the shear properties of human corneas are missing from the extant literature. A thesis by Nickerson

^{ 9 }discusses the use of torsional rheometry to measure shear properties of porcine cornea. Standard inflation and strip testing

^{ 1 –4 }do not introduce shearing deformations and these tests therefore give no information about shear stiffness.

^{ 10 }The interweaving appears maximal at the anterior surface and significantly reduces toward the posterior. Recent images by Jester et al.

^{ 11,12 }using second harmonic generated imaging confirm this assessment. Figure 1 shows the central part of a full human cornea cross-section created from many second harmonic generated images and in which distinct interweaving in the anterior third may be discerned by the through-thickness trajectory of many of the lamellae. It is also noted that scanning electron microscopy and transmission electron microscopy images show that lamellae become wider and thicker toward the posterior of the stroma.

^{ 13 }X-ray scattering studies have demonstrated that the collagen associated with preferred directions measured in the limbal plane also varies with depth through the cornea.

^{ 14 }After using a femtosecond laser to cut the stroma into thirds through the thickness, Abahussin et al.

^{ 14 }showed that lamellae exhibit preferred angular distributions in the posterior third but transition toward a more uniform distribution in the anterior third.

^{ 15 }On acquisition, the central corneal thickness (CCT) of each cornea was measured with a pachymeter (IOPac, Reichert Inc., Depew, NY). The right cornea from each donor pair was mounted in an artificial anterior chamber (Barron; Katena Products, Inc., Denville, NJ) and divided into three layers through the thickness (Fig. 2a) with lamellar cuts from a femtosecond laser (iFS; Abbott Medical Optics, Inc., Santa Ana, CA). The left donor corneas were left intact and maintained in corneal storage medium (Optisol; Bausch & Lomb). To minimize swelling, the epithelium and endothelium were not removed until immediately before mechanical testing.

^{ 16 }

^{ 17 }and 25 μm

^{ 18 }, respectively) from the CCT measurement and dividing by three. The thicknesses of Bowman's layer and Descemet's membrane were neglected in this calculation. Therefore, the first (posterior) laser cut was set at a depth equal to the stromal thickness plus the epithelium thickness beneath the anterior surface, while the second (central) and third (anterior) laser cuts were one and two stromal third thicknesses shallower, respectively. These three cuts were performed with an 8.5-mm diameter raster area, and a final 8 mm diameter, full thickness circular cut was made perpendicular to the other cuts. The raster pattern of the laser left some residual tissue bridging across the cutting plane which allowed each cornea to remain in one piece when placed back into corneal storage medium (Optisol; Bausch & Lomb) after the cuts.

^{ 19 }found a linear relationship between human corneal swelling pressure and sample thickness when plotted on a double logarithmic scale. Therefore, their data were fit to a power equation of the form

*SP*=

*aT*

^{ b }, where

*SP*is swelling pressure,

*T*is sample thickness, and

*a*and

*b*are constants. Sample thickness can be related to axial strain with the equation

*T*=

*T*

_{0}(1 + ε), where

*T*

_{0}is the reference thickness and ε is the axial strain. Combining these equations gives a relationship between swelling pressure and axial strain:

*SP*=

*a*[

*T*

_{0}(1 + ε)]

^{ b }. The derivative of equilibrium axial compressive stress (swelling pressure) with respect to strain gives the negative of the equilibrium compressive modulus as a function of strain. For each sample, equilibrium axial stress-strain data were fit to the above equation with the reference thickness taken as the same value used to calculate axial strain. The compressive modulus at 0% strain was then calculated from the derivative of the resulting fit equation.

*P*< 0.05. Initial CCTs of right and left corneas were compared using a paired

*t*-test.

^{ 16 }; Minitab Inc., State College, PA) was used to perform an analysis of covariance (ANCOVA) on the log transformed shear modulus data with layer (anterior, central, and posterior) as a fixed factor and donor as a random factor. The shear modulus data were log transformed to achieve a homogeneous variance. Bonferroni pairwise comparison tests were used to compare third types.

*P*= 0.17). The targeted stromal third thicknesses used to define the depths of the laser cuts thus ranged from 150 μm to 190 μm.

*P*< 0.001) and donor variability (

*P*= 0.033). Anterior samples were stiffer than central samples (

*P*< 0.0001) and central samples were stiffer than posterior samples (

*P*< 0.0001).

*R*

^{2}value of the power fit used to calculate each modulus are collected in Table 3. For 15 of the 16 samples, the power fit produced an

*R*

^{2}value greater than 0.94 with nine samples greater than 0.99. Compressive moduli spanned a wide range from 0.79 to 49 kPa. The average full thickness sample compressive modulus of 38.7 ± 8.6 kPa matches well with the compressive modulus of 38.2 kPa calculated at normal thickness from the average power fit reported by Olsen and Sperling

^{ 19 }based on 45 human corneas. Sample type had a significant effect on equilibrium compressive modulus (

*P*< 0.001) while donor did not show a significant effect (

*P*= 0.425). Full samples were stiffer than all isolated layers (

*P*< 0.001) but there were no significant differences among isolated layers (

*P*> 0.999).

Donor | Full | Anterior | Central | Posterior |
---|---|---|---|---|

1 | 49.0 (0.98) | 3.80 (0.98) | 7.65 (0.99) | 10.9 (0.99) |

2 | 42.5 (0.99) | 1.80 (0.99) | 3.26 (0.99) | 3.11 (0.99) |

3 | 31.7 (0.99) | 11.2 (0.99) | 3.82 (0.96) | 0.79 (0.78) |

4 | 31.4 (0.96) | 10.1 (0.99) | 6.26 (0.95) | 1.90 (0.94) |

Mean | 38.7 | 6.72 | 5.25 | 4.17 |

SD | 8.60 | 4.62 | 2.07 | 4.58 |

^{ 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.

^{ 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.

^{ 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.

^{ 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.

^{ 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.

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