### Purpose

### Design

### Participants

### Methods

*E*) and out-of-plane shear modulus (

*G*) in the cornea from experimental data was performed using a nearly incompressible transversely isotropic (NITI) medium material model assuming spatial isotropy of corneal tensile properties. Corneal samples were excised and parallel plate rheometry was performed to measure shear modulus,

*G*. Corneal samples were then subjected to strip extensometry to measure the Young’s modulus,

*E*.

### Main Outcome Measures

*E*) and shear (

*G*) moduli differing by more than an order of magnitude. These results show that AμT-OCE can quantify both moduli simultaneously with a noncontact, noninvasive, clinically translatable technique.

### Results

*E*= 12 ± 5 MPa and

*G*= 31 ± 11 kPa at 5 mmHg and

*E*= 20 ± 9 MPa and

*G*= 61 ± 29 kPa at 20 mmHg. Tensile testing yielded a mean Young’s modulus of 1 MPa – 20 MPa over a strain range of 1% to 7%. Shear storage and loss modulus (G′/G′′) measured with rheometry was approximately 82/13 ± 12/4 kPa at 0.2 Hz and 133/29 ± 16/3 kPa at 16 Hz (0.1% strain).

### Conclusions

## Keywords

#### Abbreviations and Acronyms:

AμT (acoustic microtapping), BSS (balanced saline solution), IOP (intraocular pressure), NITI (nearly incompressible transversely isotropic), OCE (OCT elastography)^{1}

^{,}If corneal shape is not optimal, then images formed on the retina are aberrated. Many methods assess corneal shape, but no clinical tools predict shape changes from interventions such as LASIK and collagen cross-linking therapies. Despite the overall success of these interventions over the last decades, outcomes remain unpredictable for an individual patient, and many procedures produce unexpected changes in visual acuity and can have additional side effects. To optimize outcomes, a personalized corneal biomechanical model based on quantitative maps of mechanical moduli and intraocular pressure (IOP)-induced changes in mechanical moduli is needed to predict final corneal shape.

^{3}

^{,}However, it cannot determine fundamental material parameters required for robust biomechanical models of corneal deformation. In particular, tonometry metrics depend on experimental conditions and often characterize deformation in response to a dynamic mechanical stimulus over a large region of the cornea and sclera. In addition, tonometry does not consider the highly nonlinear stress–strain relationship between corneal tissue and preload IOP, nor does it account for material anisotropy or variations in corneal thickness. Thus to date, no noninvasive tools can map corneal elasticity (accounting for its strong anisotropy) to provide the information needed for a personalized biomechanical model suitable for screening, surgical planning, and treatment monitoring.

^{5}

*E*(mainly determining its deformation and shape) and the out-of-plane shear modulus

*G*(mainly responsible for out-of-plane shearing). However, direct comparison between invasive mechanical tests and noncontact AμT-OCE, a necessary step in validating the method and helping to translate it into a clinical tool, was not performed.

## Methods

### Porcine Cornea Samples

### Acoustic Microtapping OCT Elastography

*E = 3*μ under the assumption of corneal tensile isotropy) and out-of-plane (

*G*) shear moduli (Fig 1C). The corneal thickness (

*h*) was determined using automated segmentation of the OCT image and served as a constraint on solutions to the dispersion relation.

### Parallel Plate Rheometry

*G*, assuming a homogenous cylindrical material (Fig 1E, F, corresponding equations in “Results” and additional detail provided in Supplemental Methods).

### Stress–Strain Extensometry

## Results

### Determination of Corneal Elastic Moduli with Acoustic Microtapping OCT Elastography

^{6}

*E = 3*μ (under the assumption of corneal tensile isotropy) and transverse shear modulus

*G*(detailed in Supplemental Methods). The stress response to an induced strain for such a material is described using Hooke’s law:

_{ij}denotes engineering stress, ε

_{ij}denotes engineering strain, τ

_{ij}denotes shear stresses, γ

_{ij}= 2ε

_{ij}denotes shear strains, and the subscripts

*x*,

*y*, and

*z*refer to standard Cartesian axes (a detailed description of elastic waves in NITI materials can be found in Pitre et al

^{6}

*G*. Thus, functional biomechanical models do not require estimating λ.

^{6}

*G*(Fig 1G).

*E*) was 12 ± 5 MPa at 5 mmHg and 20 ± 9 MPa at 20 mmHg. The mean transverse shear modulus (

*G*) was 31 ± 11 kPa at 5 mmHg and 61 ± 29 kPa at 20 mmHg. The inflation pressure placed the cornea in a state of pre-stress, which varied in magnitude according to the IOP. This result demonstrated that both in-plane tensile and out-of-plane shear moduli increase with increasing IOP, consistent with the nonlinear material properties of the cornea. Note that both Young’s and shear moduli changed by 98 ± 29% and 67 ± 15%, respectively, relative to the value measured at 5 mmHg as the pressure increased to 20 mmHg (see Supplemental Methods). The mean ± standard deviation of corneal thickness (assuming refractive index n = 1.389) was 0.76 ± 0.10 mm at 5 mmHg and thinned on average by 50 μm as the pressure was increased. In all samples, the endothelium appeared intact.

### Determination of the Out-of-Plane Shear Modulus with Parallel Plate Rheometry

*t*is time, and ${\gamma}_{\theta {z}_{0}}$ is the peak shear strain amplitude. Collagen-rich tissue generally is linearly elastic at less than 1%; thus, 0.1% peak shear strain was applied during the frequency sweep. The shear stress $(\tau {\theta}_{z})$ is described by:

*G*′)) and out-of-phase (loss (

*G*′′)) stress-strain relationships. Frequency-dependent storage and loss moduli of the complex

*G*value in the NITI model (detailed in Supplemental Methods) were determined over a range of 0.16 to 16 Hz. Sample thickness was recorded based on the parallel plate gap distance.

### Determination of Corneal Young’s Modulus with Stress–Strain Extensometry

*xx*) direction and the corresponding stress–strain relationship described by:

*E*measured in tensile testing over a range of engineering strain from 1% to 10% was 1.4 to 40 MPa (Supplemental Methods). The orientation of the strip lay in the xy-plane and the direction of the loading stress was uniaxial along the nasotemporal axis. The thickness of each sample was measured before extension testing in each sample. Corneal strips on average were 0.77 ± 0.09 mm thick and 5.9 ± 0.4 mm wide, resulting in a cross-sectional area of 4.6 mm

^{2}.

*xx*-loading direction in an NITI material,

*E*is sensitive only to the in-plane elastic modulus (detailed in Supplemental Methods). Based on this analysis, AμT-OCE provides a functionally equivalent measure of the elastic moduli probed in both shear rheometry (

*G*) and tensile extension (

*E*), but in a single, noncontact acquisition that can be performed noninvasively using intact whole-globe eyes.

### Comparison between Methods

*E*and shear modulus

*G*differ by approximately an order of magnitude, consistent with previous results.

*G*value from OCE with the storage modulus (

*G*′) in rheometry.

## Discussion

*E*and

*G*, simultaneously under physiologic loading conditions.

^{6}

*E*, was in good correspondence with literature results obtained by destructive ex vivo inflation and tensile tests, whereas the out-of-plane shear modulus

*G*, being a few orders of magnitude smaller than

*E*, reasonably matched literature data on the shear modulus obtained by rheometry.

^{11}

^{12}

^{, }

^{13}

^{, }

^{14}

^{, }

^{15}

^{, }

^{16}

^{, }

^{17}

^{, }).

*G*(0.3–9 kPa

^{19}

^{, }

^{20}

^{, }

^{21}

^{24}

^{, }

^{25}

^{, }

^{26}

*μ*and

*G*, which are in good agreement with moduli obtained with mechanical tests. However, as noted in the Supplemental Methods, the NITI model is simplified by assuming corneal tensile isotropy, that is, isotropic Young’s modulus (

*E*), although tensile and shear deformations are driven by different moduli.

*E =*3μ. The comparison between tensile and AμT-OCE measurements for

*E*suggests that its OCE-based value may be overestimated. If the relationship

*E =*2μ (corresponding to δ = –2μ) had been used, both methods would have demonstrated very close agreement. Thus, the in-plane Young’s modulus of cornea is likely closer to its lower limit, rather than to its highest possible value, as presented here. However, additional studies should be performed to confirm this observation. Nevertheless, we emphasize that using a simplified NITI material model rather than a simple isotropic model is an important step in quantifying anisotropic properties in the cornea because it separates the effects of μ from

*G*, which are much greater than any possible variations of

*E*introduced by δ ≠ 0.

^{16}

^{,}

^{28}

^{, }

^{29}

^{, }

^{30}

^{, }This suggests that for normal IOPs (less than 25 mmHg in porcine cornea), corneal microstructure can be approximated with the NITI model (i.e., symmetric for any direction in the xy-plane).

^{32}

^{33}

^{,}

## Acknowledgments

## Supplementary Data

- Supplemental Methods

## References

- The anatomy and physiology of cornea.
*Keratoprostheses Artif Corneas Fundam Surg Appl.*2015; 37: 19-25 - Corneal structure and transparency.
*Prog Retin Eye Res.*2015; 49: 1-16 - Risk factors for visual field progression in treated glaucoma.
*Arch Ophthalmol.*2011; 129: 562-568 - Concepts and misconceptions in corneal biomechanics.
*J Cataract Refract Surg.*2014; 40: 862-869 - Optical coherence elastography in ophthalmology.
*J Biomed Opt.*2017; 22: 1-28 - Nearly-incompressible transverse isotropy (NITI) of cornea elasticity: model and experiments with acoustic micro-tapping OCE.
*Sci Rep.*2020; 10: 12983 - Corneal thickness in pigs measured by ultrasound pachymetry in vivo.
*Scand J Lab Anim Sci.*2008; 35: 39-43 - Group index of the human cornea at 13-μm wavelength obtained in vitro by optical coherence domain reflectometry.
*Opt Lett.*2004; 29: 83 - Collagen: structure and mechanics, an introduction.Springer, Boston, MA2008: 1-13
- Translating ocular biomechanics into clinical practice: current state and future prospects.
*Curr Eye Res.*2015; 40: 1-18 - Influence of analytical methods versus clamping procedure on biomechanical response of cornea through experimental strip tests.
*Mater Today Proc.*2021; 44: 4375-4380 - Material properties from air puff corneal deformation by numerical simulations on model corneas.
*PLoS One.*2016; 11e0165669 - Mechanical characterization of porcine corneas.
*J Biomech Eng.*2012; 134031003 - Stress-strain measurements of human and porcine corneas after riboflavin-ultraviolet-A-induced cross-linking.
*J Cataract Refract Surg.*2003; 29: 1780-1785 - A comparison of biomechanical properties between human and porcine cornea.
*J Biomech.*2001; 34: 533-537 - Mechanical anisotropy of porcine cornea and correlation with stromal microstructure.
*Exp Eye Res.*2009; 88: 1084-1091 - Effect of UVA/riboflavin collagen crosslinking on biomechanics of artificially swollen corneas.
*Invest Ophthalmol Vis Sci.*2017; 59: 764-770 - Biomechanical property analysis after corneal collagen cross-linking in relation to ultraviolet A irradiation time.
*Graefes Arch Clin Exp Ophthalmol.*2011; 249: 1223-1227 - Viscoelastic shear properties of the corneal stroma.
*J Biomech.*2014; 47: 723-728 - Assessment of UVA-riboflavin corneal cross-linking using small amplitude oscillatory shear measurements.
*Invest Ophthalmol Vis Sci.*2016; 57: 2240-2245 - Corneal resistance to shear force after UVA-riboflavin cross-linking.
*Invest Ophthalmol Vis Sci.*2013; 54: 5059-5069 - Mechanisms of residual stress in soft tissues.
*J Biomech Eng.*2009; 131: 1-5 - Automatized patient-specific methodology for numerical determination of biomechanical corneal response.
*.*2016; 44: 1753-1772 - Lamellar orientation in human cornea in relation to mechanical properties.
*J Struct Biol.*2005; 149: 1-6 - Application of structural analysis to the mechanical behaviour of the cornea.
*J R Soc Interface.*2004; 1: 3-15 - Elastic modulus and collagen organization of the rabbit cornea: epithelium to endothelium.
*Acta Biomater.*2014; 10: 785-791 - Measuring mechanical wave speed, dispersion, and viscoelastic modulus of the cornea using optical coherence elastography.
*Opt Express.*2019; 27: 16635 - Acoustic micro-tapping for non-contact 4D imaging of tissue elasticity.
*Sci Rep.*2016; 6: 38967 - An inverse finite element method for determining the anisotropic properties of the cornea.
*Biomech Model Mechanobiol.*2011; 10: 323-337 - Revealing anisotropic properties of cornea at different intraocular pressures using optical coherence elastography.
*Opt Elastography Tissue Biomech III.*2016; 9710: 97100T - Investigating elastic anisotropy of the porcine cornea as a function of intraocular pressure with optical coherence elastography.
*J Refract Surg.*2016; 32: 562-567 - Proper material properties are required for the finite element method.
*Arch Ophthalmol.*2006; 124: 1064-1065 - Constitutive modelling of cornea tissue; influence of three-dimensional collagen fiber microstructure.
*J Biomech Eng.*2021; 143031002 - Cornea modelling.
*Eye Vis.*2020; 7: 1-15

## Article info

### Publication history

### Footnotes

*Supplemental material available at* www.ophthalmologyscience.org.

Disclosure(s):

All authors have completed and submitted the ICMJE disclosures form.

The author(s) have made the following disclosure(s): R.W.: Consultant – Carl Zeiss Meditec, Inc.

Supported in part by the National Institutes of Health, Bethesda, Maryland (grant nos.: R01-EY026532, R01-EY024158, R01-EB016034, R01-CA170734, R01-AR077560, and R01-HL093140); Life Sciences Discovery Fund (no.: 3292512); the Coulter Translational Research Partnership Program; Research to Prevent Blindness, Inc., New York, New York (unrestricted grant); the Department of Bioengineering, University of Washington, Seattle, Washington; and the National Science Foundation (graduate fellowship no.: DGE-1256082 [M.A.K.]). This material was based on work supported by the National Science Foundation Graduate Research Fellowship Program (grant no.: DGE-1256082).

The authors declare that all data from this study are available within the article and its supplemental material. Raw data for the individual measurements are available on reasonable request.

HUMAN SUBJECTS: No human subjects were included in this study.

Nonhuman animals were used in this study. No patient-level consent or institutional review board approval were required. All research adhered to the tenets of the Declaration of Helsinki.

Author Contributions:

Conception and design: Kirby, Pelivanov, O’Donnell, Shen

Analysis and interpretation: Kirby, Liou, Pelivanov, O’Donnell, Shen

Data collection: Kirby, Pitre, Li, Shen

Obtained funding: Wang, O'Donnell, Shen

Overall responsibility: Kirby, Pitre, Liou, Li, Wang, Pelivanov, O’Donnell, Shen

### Identification

### Copyright

### User license

Creative Commons Attribution – NonCommercial – NoDerivs (CC BY-NC-ND 4.0) |## Permitted

### For non-commercial purposes:

- Read, print & download
- Redistribute or republish the final article
- Text & data mine
- Translate the article (private use only, not for distribution)
- Reuse portions or extracts from the article in other works

## Not Permitted

- Sell or re-use for commercial purposes
- Distribute translations or adaptations of the article

Elsevier's open access license policy