**Purpose.**:
A novel nanoindentation technique was used to biomechanically characterize each of three main layers of the cornea by using Hertzian viscoelastic formulation of creep, the deformation resulting from sustained-force application.

**Methods.**:
The nanoindentation method known as mechanical interferometry imaging (MII) with <1-nm displacement precision was used to observe indentation of bovine corneal epithelium, endothelium, and stroma by a spherical ferrous probe in a calibrated magnetic field. For each specimen, creep testing was performed using two different forces for 200 seconds. Measurements for single force were used to build a quantitative Hertzian model that was then used to predict creep behavior for another imposed force.

**Results.**:
For all three layers, displacement measurements were highly repeatable and were well predicted by Hertzian models. Although short- and long-term stiffnesses of the endothelium were highest of the three layers at 339.2 and 20.2 kPa, respectively, both stromal stiffnesses were lowest at 100.4 and 3.6 kPa, respectively. Stiffnesses for the epithelium were intermediate at 264.6 and 12.2 kPa, respectively.

**Conclusions.**:
Precise, repeatable measurements of corneal creep behavior can be conveniently obtained using MII at mechanical scale as small as one cell thickness. When interpreted in analytical context of Hertzian viscoelasticity, MII technique proved to be a powerful tool for biomechanical characterization of time-dependent biomechanics of corneal regions.

^{ 1 –4 }has been recently popularized as a noninvasive, in vivo method, but decoupling of intraocular pressure from mechanical properties of the cornea remains a quandary. Laboratory methods include uniaxial tensile testing,

^{ 5 }ultrasound propagation,

^{ 6,7 }atomic force microscopy,

^{ 8 }and whole-globe pressurization

^{ 9 –14 }for determination of corneal elasticity

^{ 7,15 –17 }and hysteresis,

^{ 1,4,18 }both in vivo

^{ 19,20 }and in vitro.

^{ 5,16 }Some of these methods, however, produce data that are only relative or in an idiosyncratic framework difficult to integrate into modern mechanical engineering analyses. Attempts to interpret, in inappropriate context, data obtained by idiosyncratic methods can lead to severely flawed conclusions.

^{ 21 }

^{ 22 –25 }On any scale, indentation data can be interpreted in the Hertzian viscoelastic framework suitable for general mathematical characterization appropriate to approaches such as finite element analysis (FEA).

^{ 26 }These models, which have general validity in the mechanics of materials irrespective of microscopic and molecular composition, accurately captured the viscoelastic behavior exhibited by each corneal layer.

^{ 27 }The interferometer consists of an optical microscope with a Michelson interference objective that allows for the observation of not only lateral (

*x*and

*y*) specimen features with typical optical resolution (1.16 μm) but also height (

*z*) at resolution below 1 nm through tracking of optical interference fringes.

^{ 28 }The Michelson interferometer consists of a beam splitter and reference mirror. Specimens were placed in air at 100% relative humidity, in a chamber covered by a rubber membrane. The interferometer imager includes a 640 × 480-pixel charge coupled device detector array that when combined with a 20× objective produces a 315 × 240-μm field of view and 500-nm spatial sampling. Positions of reflective spherical indenters were measured with respect to the sample chamber bottom and were corrected for the effect of dispersion in liquid using a group velocity at 535-nm wavelength and

*Ng*= 1.33 band pass at 30 nm.

^{ 29 }

*h–t*) data were acquired in four to five different trials at each of two different forces per specimen. Displacement during the trials did not significantly change indenter force, because indentation displacement was negligible relative to overall distance from the indenters to the magnet.

*R*toward an asymptotic displacement over time. The present study employed the viscoelastic formulation of Oyen

^{ 26 }incorporating Hertzian contact mechanics as derived in the Appendix. Briefly, creep was described by a function consisting of two exponential terms fit to the data using the Levenberg–Marquardt nonlinear least-squares method.

^{ 24 }Similar to our previous microindentation study,

^{ 30 }material constants

*C*

_{i}were computed. Ideal creep testing would load the specimen instantaneously, which of course is experimentally impossible. Experimentally, loading was linearly increased as rapidly as possible as a ramp. The mathematical model incorporates a ramp correction factor (RCF) that accounts for finite loading time. Short-term (

*E*

_{0}) and long-term (

*E*

_{∞}) elastic stiffnesses were determined from the fitted viscoelastic model with computed material parameters.

Parameter | Epithelium | Endothelium | Stroma |
---|---|---|---|

C_{0}, m^{2}/N | 1.16E-04 | 7.11E-05 | 4.12E-04 |

C_{1}, m^{2}/N | 6.02E-05 | 2.14E-05 | 1.13E-04 |

τ_{1}, sec | 31.76 | 60.74 | 4.17 |

C_{2}, m^{2}/N | 4.98E-05 | 4.54E-05 | 2.84E-04 |

τ_{2}, sec | 61.76 | 95.75 | 91.32 |

E_{0}, kPa | 271.6 | 344.8 | 106.0 |

E_{∞}, kPa | 12.98 | 21.11 | 3.64 |

^{2}. Coefficients of determination for all four different trials exceeded 0.94. The RMSE between the four data sets for the 19.0-μN higher loading and model predictions based on fits from one experimental trial for the 5.5-μN loading was also small at 0.35, 0.32, 0.24, and 0.27 μm

^{2}, respectively. Coefficients of determination for all four trials exceeded 0.98. For endothelium, Figure 2 shows creep responses of nine different specimens indented with loadings of 632 (five specimens) and 1310 (four specimens) pN, compared with prediction from the model based on fits from one of the trials using the lower loading. For both loadings, the RMSE between data and model predictions for all trials was <1.8 × 10

^{−4}μm

^{2}. Coefficients of determination for fits to data exceeded 0.96 for both loadings.

^{2}, respectively, while all coefficients of determination for fits to data exceeded 0.98. For four trials at higher loading of 30 μN, RMSE was 0.92, 0.70, 1.25, and 1.35 μm

^{2}, respectively, while all coefficients of determination consistently exceeded 0.96. Short- and long-term stiffnesses computed from viscoelastic models based on data from one specimen were 271.7, and 13.0 kPa, respectively, for the epithelium; 344.8 and 21.1 kPa, respectively, for endothelium; and 106.0 and 3.6 kPa, respectively, for stroma. Although the model predicted results accurately over two different loading levels, it was more accurate at lower than higher loading (Fig. 2).

*P*≥ 0.9) and 20.2 ± 0.71 kPa (

*P*≥ 0.8), respectively, while stroma exhibited the lowest short- and long-term stiffnesses at 100.4 ± 6.04 kPa (

*P*≥ 0.9) and 3.6 ± 0.29 kPa (

*P*≥ 0.8). Corresponding stiffnesses for epithelium were 264.6 ± 12.6 kPa (

*P*≥ 0.9) and 12.6 ± 0.73 kPa (

*P*≥ 0.9), respectively.

^{ 30 }

^{ 9 }reported that the epithelium's contribution to overall corneal stiffness is negligible, our finding shows that the intrinsic stiffness of the relatively thin epithelial layer considerably exceeded that of the thicker stroma. The stroma contributes more to overall corneal mechanical behavior than does the epithelium, because the stroma is much thicker than the epithelium, not because the stroma is intrinsically stiffer. Compressive modulus and tensile modulus can also differ because the cornea is not an isotropic material; cornea may be orthotropic or anisotropic. Hence, the discrepancy between our findings and those of Elsheikh et al. could also be due to the difference in the nature of the measured modulus.

^{ 8 }reported the mean stiffness of Descemet's membrane to range from 20 to 80 kPa, averaging 50 ± 17.8 kPa, while that of the anterior basement membrane ranged from 2 to 15 kPa and averaged 7.5 ± 4.2 kPa. Since the endothelial layer as tested in the present study included Descemet's membrane, the present finding of long-term endothelial layer stiffness of 20.2 ± 0.71 kPa falls within the range reported by Last et al.

^{ 14 }estimated whole corneal elastic modulus to be 158 kPa, which is comparable to the short-term stiffness that we calculated for stroma at 100 kPa. Considering the stiffness of the relatively thin epithelium and endothelium at 265 and 339 kPa, respectively, the current stiffness measurements are comparable to those of Forster et al. The corneal stiffness of 24.5 ± 5.7 kPa reported by Sjøntoft and Edmund is in the same order of magnitude of the long-term stiffness reported in the present study.

^{ 17 }Since they calculated corneal stiffness using linear elastic theory, their reported value represents static Young's modulus, equivalent to the long-term stiffnesses reported in the current investigation. In addition, the corneal elastic coefficient reported by Edmund

^{ 16 }was 24.9 ± 5.5 kPa, which is in the same order of magnitude as the current determinations.

^{ 31 }have provided insights into mechanical behavior of incised cornea, only stromal properties were considered. The FEA investigation by Uchio et al.

^{ 32 }neglected viscous corneal properties, precluding evaluation of time-dependent behavior. By differentiating corneal layers and assigning corresponding time-dependent mechanical properties, realistic time-dependent behavior of the entire cornea can be captured in FEA.

^{30}Rearranging force

*P*, which is a function of Poisson ratio ν, probe sphere radius

*R*, and elastic modulus

*E*, the governing equation can be determined as a displacement

*h*. Equation 1 can be written as a function of shear modulus G with incompressible assumption that Poisson ratioν = 0.5. As suggested by Oyen,

^{26}this corresponds to a Boltzmann integral equation when (P/2G) is replaced by a viscoelastic integral operator for creep

^{33}as in equation 3. where

*J*(

*t*) is the material creep function and

*u*is a dummy variable of integration for time. Ramp loading from 0 to maximum load at linearly increasing (ramp) rate

*k*can be described by equations 4 and 5: where

*t*

_{R}is rise time to peak load.

*P*

_{max}=

*ktR*, the solution for creep due to spherical indentation is given by equation 7:

^{24,34}the generalized creep function (equation 7) can be related to the shear creep function, allowing computation of the instantaneous

*E*

_{0}and long-time

*E*

_{∞}stiffnesses from material creep constants (

*C*), as shown in equations 9 and 10.

_{i}