# Evaluation of corneal elastic modulus based on Corneal Visualization Scheimpflug Technology

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## Abstract

### Background

Corneal biomechanical properties are important for the diagnosis of corneal diseases, individualized design and prognosis of corneal surgery. Clinical available devices such as Ocular Response Analyzer (ORA) and Corneal Visualization Scheimpflug Technology (Corvis ST) can provide corneal biomechanics related parameters, while corneal elastic modulus cannot be extracted directly from them at present. The aim of this study is to suggest a method to determine corneal elastic modulus based on the results of Corvis ST test according to Reissner’s theory on the relation between stress and small displacement in shallow spherical shell.

### Results

Five rabbits (10 eyes) and 10 healthy humans (20 eyes) were measured with Corvis ST to obtain the normal range of corneal elastic modulus. Results showed Corneal elastic modulus of rabbit was 0.16 MPa to 0.35 MPa, human corneal elastic modulus was 0.16–0.30 MPa. Rabbit corneas were also measured at different intraocular pressures (IOP), and results showed corneal elastic modulus, first applanation time (A1T) and stiffness parameter (SP-A1) were positively correlated with IOP. Deformation amplitude (DA), the second applanations time (A2T), and peak distance (PD) were negatively correlated with IOP. Finite element method was used to simulate the Corvis measurements according to the calculated elastic modulus and the simulated corneal apical displacements were agreement with experimental results in general.

### Conclusions

The method to determine corneal elastic modulus based on Corvis test according to the relationship between force and displacements of shallow spherical shell is convenient and effective.

## Keywords

Corneal Visualization Scheimpflug Technology (Corvis ST) Reisner’s theory Cornea Elastic modulus Intraocular pressure (IOP)## Background

Cornea is the soft tissue located in the outer layer of the eyeball. The transparent cornea provides 70% ocular refractive power [1]. Corneal biomechanical properties play an important role in maintaining the normal shape and function of cornea. Besides, the occurrence and development of ocular diseases such as keratoconus, myopia and pseudoexfoliation syndrome, the design and prognosis of corneal surgeries such as corneal transplantations, corneal refractive surgery and corneal collagen crosslinking surgery are also correlated to corneal biomechanics [1, 2, 3, 4, 5, 6, 7]. Therefore, more and more ophthalmologists and researchers expect to obtain human corneal biomechanics though simple clinical measurements.

Similar to most biological tissues, corneal biomechanics include its anisotropic, nonlinear elastic properties and viscoelastic properties. Corneal nonlinear elastic properties often be described as corneal tangent modulus under certain stress. While studies have found that cornea can be regarded as linear elastic material under physiological state [7], so the mechanical properties of cornea under physiological state may be described by elastic modulus, or Young’s modulus, as most of researchers concerned.

Ocular Response Analyzer (ORA) and Corneal Visualization Scheimpflug Technology (Corvis ST) are two of the most widely used devices for measurements of corneal biomechanical properties in clinic. In fact, the parameters provided by these devices such as corneal hysteresis (CH) and corneal resistance factor (CRF) by ORA measurements and corneal stiffness parameter (SP-A1), corneal deformation amplitude (DA) and first applanation time (A1T), etc. by Corvis measurements are descriptions of mechanical process of cornea under air-puff. These parameters are related to corneal biomechanics, intraocular pressure (IOP) and corneal geometrical parameters [8, 9, 10], so we call them corneal biomechanics related parameters. Although many researchers attempted to get the relationship between these parameters and mechanical properties such as Young’s modulus, there is still lack of acknowledged results at present [11, 12].

Many studies [2, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] have put great concerns on how to extract corneal elastic modulus directly from Corvis and/or ORA parameters. Glass [13] and Han [14] regarded cornea as a simplified spring and dashpot system to simulate corneal response to air-puff. These studies can help researchers to understand corneal biomechanical related parameters preliminary, but cannot determine corneal biomechanical properties quantitatively due to too simple models. Besides, researches have simulated corneal response under air-puff pressure by finite element method [16, 17, 18, 19, 20, 21, 22] in order to evaluate corneal biomechanical properties based on different corneal constitutive model such as Fung’s model [18], Ogden model [19], fiber-dependent model [20] or other models [21, 22]. These studies provided approaches to obtain corneal traditional biomechanical properties based on Corvis results while these methods have a higher requirement for corneal geometrical models and are time consuming, which make it difficult to be used in clinic directly. Furthermore, Wang et al. [2] detected air puff-time curve and corneal apical displacement–time curve, from which two parameters: corneal tangent stiffness coefficient (*S*_{TSC}) and corneal energy absorbed area (*A*_{absorbed}) were determined. However, both of *S*_{TSC} and *A*_{absorbed} are not corneal elastic modulus or viscoelastic parameter. In the latest version of software of Corvis, Stiffness Parameter **(**SP-A1) is provided which is expressed by the ratio of the force and displacement at the first applanation state [23]. From the expression of SP-A1 we can find the SP-A1 was affected by IOP. These two parameters are significantly correlated with corneal geometrical parameters which lead to the difficulty to evaluate corneal elastic property accurately. In addition, Shih et al. [24] estimated corneal Young’s modulus in vivo by proposing modified Taber’s model to describe the relationship between the force applied on cornea and corneal apical displacement based on Taber’s results [25]. While in their paper cornea was regarded as hemisphere and the air-puff pressure was assumed as a point load at the apical point of cornea, which have discrepancies with corneal normal geometrical shape. Therefore, corneal biomechanical properties such as its elastic and viscoelastic properties have not been directly determined from Corvis parameters yet.

Considering relatively small deformation of the cornea and small range of the IOP, the cornea can be regarded as linear elastic material during Corvis measurements [7]. Motivated by the reported works [26, 27] on ultrasound indentation techniques, we applied Reissner’s theory [28] which described the stresses and small displacements in shallow spherical shells to determine corneal elastic modulus. The present study applied Reissner’s theory [28] to calculate corneal elastic modulus from Corvis measurements on human and rabbits in vivo. Besides, rabbit eye balls were also measured in vitro under different IOPs to evaluate the IOP’s influence on the calculated corneal elastic modulus. To validate the accuracy of the calculated elastic modulus, Corvis test was simulated using explicit finite element method based on the calculated corneal elastic modulus and corneal apical displacements were extracted to be compared with the experimental data. The significance of this work is to provide an effective method to determine corneal elastic modulus from the output data of Corvis test in clinic directly and may give possible guidance for the diagnosis of corneal disease and design of corneal surgery.

## Materials and methods

### Method to evaluate corneal modulus

*r*

_{p}. As

*r*

_{p}is small enough compared to the whole cornea, we considered there was no edge restraint at limbus. We used the reported air-puff force (Eq. 1) [2] instead of air-puff force at the gas nozzle as the energy loss from Corvis gas nozzle to corneal surface;

*r*

_{p}was replaced by the radius of first applanation area;

*R*is corneal radius which was extracted by detecting corneal anterior surface edge with threshold segmentation based on Canny operator and circular fitting of the edge. Besides, corneal apical displacements were also detected based on the results of corneal anterior edge detection.

*f*is as follows [28]:

*t*is the central corneal thickness; \(E\) is corneal elastic modulus and \(\mu\) is defined as Eq. 3:

*c*

_{1}and

*c*

_{5}were determined by the following equations from Eq. 4a to Eq. 4e:

*c*

_{5}= − 1. And then we can calculate corneal elastic modulus by the Eq. 5:

*r*

_{p}) between the air-puff and cornea is relative constant.

### Subjects and measurements

### Finite element method to validate corneal elastic modulus

### Statistical method

Kolmogorov–Smirnov (K–S) normality test was applied to test the normality of distribution of the rabbits’ and human Corvis parameters in vivo. One-way ANOVA analysis and Spearman correlation test was used to analyze the correlation between Corvis parameters, corneal elastic modulus and IOP. Regression analysis was carried out to establish the relationship between corneal elastic modulus and IOP. Bland–Altman diagram was used to compare the simulated and experimental corneal displacements. All of the statistical analysis was performed using SPSS 21.0 (International Business Machines Corporation, New York, United States of America) and MedCalc 13.0 (Ostend, Belgium), *p *< 0.05 was considered to be statistically significant.

## Results

### Results of rabbit corneal Corvis tests in vivo and in vitro

*p*> 0.2). In Table 1, SP-A1 is the corneal Stiffness Parameter, DA is the deformation amplitude, A1T and A2T are the time of the first and second applanation respectively, HCT is the highest indentation time, PD and HR is the peak distance and corneal inverse radius when the corneas arrive at corneal highest indentation.

Result of rabbits’ corneal Corvis test in vivo

CCT/μm | IOP/mmHg | SP-A1/mN mm | DA/mm | A1T/ms | A2T/ms | HCT/ms | PD/ms | HR/mm | |
---|---|---|---|---|---|---|---|---|---|

Mean | 384 | 7.5 | 16.916 | 1.17 | 6.35 | 21.93 | 17.48 | 4.65 | 5.16 |

Sd | 32 | 2.1 | 8.377 | 0.11 | 0.14 | 0.52 | 0.61 | 0.19 | 0.50 |

*r*= − 0.497, − 0.443, − 0.475;

*p*< 0.05) and A1T, SP-A1 were positively correlated with IOP (

*r*= 0.472, 0.444;

*p*< 0.05). While the correlation between HR and IOP was not significant (

*r*= − 0.108;

*p*= 0.375). In addition, we noticed that cornea has obvious vibration under IOP of 35 mmHg and the Corvis parameters showed poorer repeatability and higher standard deviation. Figure 7 shows that Corvis parameters and IOP (7–28 mmHg) were significantly correlated. So corneal elastic moduli at IOP of 7–28 mmHg were calculated in this study.

### Results of rabbit corneal elastic modulus evaluation

*r*= 0.417;

*p*= 0.001). The regression equation between corneal elastic modulus and IOP was:

From Fig. 8 we can see that although corneal elastic modulus was positively correlated with IOP overall, the elastic modulus was stable relatively at IOP of 14–28 mmHg, which may show the calculated corneal elastic modulus was less affected by IOP compared to SP-A1 (Fig. 7a).

### Results of the validation of the method by finite element analysis

### Results of human corneal Corvis tests and elastic modulus evaluation

Result of human corneal Corvis parameters and elastic modulus

CCT/μm | IOP/mmHg | SP-A1/mN mm | DA/mm | A1T/ms | A2T/ms | HCT/ms | PD/ms | HR/mm | E/MPa | |
---|---|---|---|---|---|---|---|---|---|---|

Mean | 522 | 14.6 | 122.528 | 1.02 | 7.18 | 21.05 | 16.86 | 4.82 | 6.96 | 0.22 |

Sd | 45 | 2.3 | 51.277 | 0.10 | 0.34 | 1.40 | 0.52 | 0.24 | 0.96 | 0.05 |

Correlation between corneal Corvis parameters and elastic modulus

SP-A1 | IOP | CCT | DA | A1T | HCT | PD | HR | |
---|---|---|---|---|---|---|---|---|

| 0.061 | 0.139 | − 0.838 | − 0.742 | 0.458 | − 0.387 | − 0.482 | − 0.402 |

| 0.822 | 0.526 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

## Discussion

In this study, we provide a method to determine corneal elastic modulus in vivo from the output data of Corvis measurements according to Reissner’s theory [28] which described the stresses and small displacements in shallow spherical shells. The rabbit and human corneas in vivo tests, as well as rabbit eyeballs in vitro tests by Corvis under different IOPs were carried out. The results showed that Corvis parameters such as corneal stiffness parameter, deformation amplitude, applanation time, and peak distance etc. were correlated with IOP. The corneal elastic modulus increased slightly with IOP under physiological IOP in vitro and the correlation between corneal elastic modulus and IOP in vivo was not significant. So, the most important innovation of this study is that a convenient and effective method to determine corneal elastic modulus in vivo based on Corvis test results was established. The method will provide important information for the diagnosis of some ocular diseases and the design of corneal surgery.

The relationship between stress and displacement of shallow spherical shell corneal model has been used in evaluation corneal elastic modulus based on corneal indentation experiments with 3–5 mm diameter cylindrical indenter in recent years [26, 27], and the results showed the possibility of this method to obtain corneal elastic modulus. We noticed that the estimation of tangent modulus in the literatures mentioned above [26, 27] was carried out according to Roark’s formula [29]. In this formula elastic modulus was calculated by interpolation method and the error cannot be ignored when *μ* is not small enough. If we regard Corvis tests as an indentation experiment for the cornea, a method to determine corneal elastic modulus in vivo can be established since corneal deformation and air-puff force can be obtained directly in Corvis tests. As *r*_{p} is not small enough, we calculated *μ* and solved *c*_{1} for every test according to Eq. 4 [28] instead of interpolation method. Compared to corneal indentation tests reported by Wang et al. [26], our method based on noncontact Corvis tests which are safer. Compared to the inverse finite element method, this method is more convenient and feasible as we do not need to establish complex corneal geometrical model and conduct abundant calculation. Besides, the simulated displacements using inverse finite element method showed a well coincidence with the experimental displacements, which remind us that the calculated corneal elastic moduli have same effectivity with inverse finite element when taking the error of corneal apical displacements between simulated results and experimental results as objective function. As air-puff force acts on an area around corneal apex, our method is closer to real experiments than Taber’s model in which Corvis test was regarded as a concentrated force acted on corneal apex.

The correlation between corneal Corvis parameters and IOP showed that DA, A2T, PD were negatively correlated with IOP; A1T and SP-A1 were positively correlated with IOP. Similar research was made by Metzler [30] and Bao et al. [31]. The range of Corvis parameters of rabbit cornea are consistent on the whole and the correlation between Corvis parameters and IOP in our results is coincident with their results at IOP of 7–28 mmHg. While the trend of variation is opposite at IOP of 28 mmHg to 35 mmHg. The possible reason maybe that the IOP was monitored at vitreous body in their research while at anterior chamber in our study, and the pressure difference caused the different Corvis parameters values at the same IOP. And the results showed that Corvis parameters present certain instability at anterior chamber pressure of 28 mmHg to 35 mmHg. It reminds us to be attention when evaluate corneal biomechanics of patients with high IOP by Corvis. This hint from Corvis measurements of eyeballs should be further verified from clinical data.

To the authors’ knowledge, there was few studies have reported Corvis parameters of rabbit cornea in vivo. The coefficient of variation (sd/mean) of Corvis parameters showed a good consistency with the results reported by researches about the Corvis parameters of normal human cornea, which indicated that the Corvis parameters of rabbit cornea has a similar repeatability with human Corvis parameters. From Eq. 6 and Fig. 8, rabbit corneal elastic modulus ranged from 0.16 to 0.60 MPa when IOP increased from 7 to 35 mmHg. Corneal elastic moduli were usually obtained from corneal uniaxial tensile test or corneal inflation tests, and the range of normal rabbit corneal elastic modulus ranged from 0.1 to 0.36 MPa in different studies [7, 24, 32, 33, 34, 35]. Corneal elastic modulus extracted from this study has the same magnitude with the results reported by the literature. Besides, corneal elastic modulus was found positively correlated with IOP, and corneal inflation tests of human cornea showed a similar positive correlation between corneal elastic modulus and IOP [36]. These results remind us cornea present nonlinear elastic properties under IOP of 7–35 mmHg. It has been known that Except IOP, corneal thickness and corneal curvature radius maybe also the main factors when determine corneal elastic modulus. From Eq. (3) and Eq. (5), we have considered the influence of corneal thickness and curvature radius in the computation of the elastic modulus. To analyze whether our calculated elastic modulus has screen out the influence of corneal thickness and curvature radius, we calculated the correlation between elastic modulus and thickness, curvature radius. Results showed the correlation between corneal elastic modulus and corneal thickness (*r *= 0.041, *p *= 0.823), corneal curvature radius (*r *= 0.115, *p *= 0.531) in vivo were not significant, in vitro results also showed our calculated elastic modulus was less affected by corneal thickness (*r *= 0.476, *p *= 0.086) and curvature radius (*r *= 0.253, *p *= 0.384). So, we considered that the method proposed in this study to extract corneal elastic modulus based on results of Corvis tests is effective for rabbit cornea.

As human cornea is too precious to get its biomechanical properties, human corneal elastic modulus was usually obtained from biomechanical tests in vitro of cadaver eyes or transplant cornea at present. Wollensak et al. [37] measured human corneal elastic modulus by corneal uniaxial tensile tests and got that the elastic modulus of human cornea was 0.8 MPa. Elsheikh et al. [36, 38] measured corneal biomechanical properties by corneal inflation experiments and got the normal range of human corneal elastic modulus was 0.16–0.8 MPa. Human corneal elastic modulus extracted from this study was 0.14–0.30 MPa, which was coincident with above experiments results at the magnitude. Besides, with the development of corneal biomechanical tests such as Corvis test and corneal indentation test, corneal biomechanical properties can be obtained by inverse finite element methods or other methods based on these results conveniently, and the evaluated corneal elastic modus ranged from 0.05 to 0.4 MPa [2, 24, 34, 39], which also showed a well coincidence with our results. Therefore, the method proposed in this study to evaluate corneal elastic modulus was feasible and effective.

In the latest version of Corvis software, corneal stiffness parameter (SP-A1) has been provided. Besides, Wang [2] has also extracted a parameter termed *S*_{TSC} to reflect corneal elastic properties. Both of the two parameters reflect corneal stiffness. From Fig. 7a, SP-A1 increased significant with IOP while corneal elastic modulus was stable relatively (Fig. 8) at IOP of 14–28 mmHg. From Table 3 we can see the correlation between SP-A1and corneal elastic modulus was not significant. These results indicated that SP-A1 was more influenced by IOP than corneal elastic modulus, and it is not enough to evaluate corneal elastic property according to SP-A1. In this study, *S*_{TSC} was also calculated and results showed *S*_{TSC} increased significantly with IOP. The calculated corneal elastic modulus from Corvis measurements in vivo also showed less significant correlation with IOP (*r *= 0.139, *p *= 0.526) than SP-A1 and *S*_{TSC} (*r *= 0.429, 0.437; *p *< *0.05*), which indicated that corneal elastic modulus calculated in this study may be less affected by IOP when the IOP is in physiological range than SP-A1 and *S*_{TSC}. Even the principle of the stronger correlation between S_{TSC}, SP-A1 and IOP, and the weaker correlation between corneal elastic modulus and IOP in our study need to undertake further analysis, our calculated results showed that our method is more reliable.

Corneal elastic modulus has been reported to be lower in keratoconus patients [2] and shown increased after corneal refractive surgery [7]. Corneal biomechanical properties also changed significantly after corneal transplantation [4, 5, 6]. One could calculate corneal elastic modulus, for example, based on our method from Corvis results thoroughly. Calculated corneal elastic modulus may provide important information for the diagnosis of keratoconus, the prevention of corneal ectasia after corneal refractive surgery. In addition, from Figs. 7 and 8, corneal elastic modulus showed similar trend with A1T and SP-A1, and opposite trend with DA, A2T, PD, when IOP increased from 7 to 35 mmHg. The significant correlation between Corvis parameters and corneal elastic modulus (Table 3) showed similar results for Corvis measurements in vivo. These indicated that one can estimate whether corneal elastic modulus is in normal range according to Corvis parameters roughly. This study makes it possible to diagnose abnormal corneas more timely and effectively based on the combination of corneal morphological examination results and corneal elastic modulus.

The limitation of this study is that cornea was regarded as a spherical shell and the ununiform corneal thickness and curvature were ignored which may have influence on the evaluation of corneal elastic modulus, while the simulated corneal apical displacements based on the calculated elastic modulus showed a well coincidence with the experimental results, which indicated that our simplification was acceptable; In addition to corneal elastic modulus, corneal viscoelastic and nonlinear elastic properties are also important aspects of corneal biomechanics, so study should be made on the method to determine corneal nonlinear elastic and viscoelastic properties based on Corvis tests in the future, in which the concrete results of finite element calculation, such as deformation amplitude, flattening time and peak distance will be used to match with the experimental data.

## Conclusions

From this study we can conclude that the method we proposed to determine corneal elastic modulus based on Reisner’s theory is convenient and effective, and the calculated corneal elastic modulus was less influenced by IOP. The clinical significance in the prognosis of eye surgery or in patients with keratoconus remains to be proved by abundant clinical data. In the future, we may calculate corneal elastic modulus of normal and keratoconus patients according to this method to explore its value in diagnosis of early keratoconus.

## Notes

### Authors’ contributions

LL: Corresponding author; conception and design of the study; XQ: acquisition of the data, drafting the article; LT: analysis and interpretation of the data; HZ: analysis of the data, revise the article; XC: acquisition of the data. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

The datasets during and/or analyzed during the current study available from the corresponding author on reasonable request.

### Consent for publication

Not applicable.

### Declaration

This work was financially supported by the National Natural Science Foundation of China (No. 31370952, 31470914, 31600758); Beijing Natural Science Foundation (No. 7174287); Beijing Nova Program (No. Z181100006218099); Beijing Municipal Administration of Hospitals’ Youth Programme (No. QMS20170204).

National Natural Science Foundation of China No. 31370952) was used for purchase and breeding the rabbits; No. 31470914 was used for recruitment of subjects; No. 31600758 was used for Corvis measurements. Beijing Natural Science Foundation (No. 7174287), Beijing Nova Program (No. Z181100006218099), Beijing Municipal Administration of Hospitals’ Youth Programme (No. QMS20170204) were used for other Ophthalmic examination, finite element simulation and interpretation of the results.

### Ethics approval and consent to participate

Ethical approval was granted for this study by Capital Medical University, all participants were informed consent.

### Funding

This work was financially supported by the National Natural Science Foundation of China (Nos. 31370952, 31470914, 31600758); Beijing Natural Science Foundation (No. 7174287); Beijing Nova Program (No. Z181100006218099); Beijing Municipal Administration of Hospitals’ Youth Programme (No. QMS20170204).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- 1.Viswanathan D, Kumar NL, Males JJ, Graham SL. Relationship of structural characteristics to biomechanical profile in normal, keratoconic, and crosslinked eyes. Cornea. 2015;34(7):791–6.CrossRefGoogle Scholar
- 2.Wang LK, Tian L, Zheng YP. Determining in vivo elasticity and viscosity with dynamic Scheimpflug imaging analysis in keratoconic and healthy eyes. J Biophotonics. 2016;9(5):454–63.CrossRefGoogle Scholar
- 3.Krysik K, Dobrowolski D, Polanowska K, Lyssek-Boron A, Wylegala EA. Measurements of corneal thickness in eyes with pseudoexfoliation syndrome: comparative study of different image processing protocols. J Healthc Eng. 2017;2017:4315238.CrossRefGoogle Scholar
- 4.Jankowska-Szmul J, Dobrowolski D, Krysik K, Kwas J, Nejman M, Wylegala E. Changes in technique and indications for keratoplasty in poland, 1989 to 2014: an analysis of corneal transplantations performed at saint barbara hospital, trauma center, sosnowiec, poland. Transplant proc. 2016;48(5):1818–23.CrossRefGoogle Scholar
- 5.Hugo J, Granget E. Ho Wang Yin G, Sampo M, Hoffart LJ, Intraocular pressure measurements and corneal biomechanical properties using a dynamic Scheimpflug analyzer, after several keratoplasty techniques, versus normal eyes. Fr Ophtalmol. 2018;41(1):30–8. https://doi.org/10.1016/j.jfo.2017.06.006
**(Epub 2017 Nov 28)**.CrossRefGoogle Scholar - 6.Maeda N, Ueki R, Fuchihata M, Fujimoto H, Koh S, Nishida K. Corneal biomechanical properties in 3 corneal transplantation techniques with a dynamic Scheimpflug analyzer. Jpn J Ophthalmol. 2014;58(6):483–9. https://doi.org/10.1007/s10384-014-0344-2
**(Epub 2014 Sep 5)**.CrossRefGoogle Scholar - 7.Zhang H, Khan MA, Zhang D, Qin X, Lin D, Li L. Corneal biomechanical properties after fs-lasik with residual bed thickness less than 50% of the original corneal thickness. J Ophthalmol. 2018;2018(5):1–10.Google Scholar
- 8.Wang W, He M, He H, Zhang C, Jin H, Zhong X. Corneal biomechanical metrics of healthy Chinese adults using corvis st. Cont Lens Anterior Eye. 2017;40(2):97–103.CrossRefGoogle Scholar
- 9.Nemeth G, Szalai E, Hassan Z, Lipecz A, Flasko Z, Modis L. Corneal biomechanical data and biometric parameters measured with Scheimpflug-based devices on normal corneas. Int J Ophthalmol. 2017;10(2):217–22.Google Scholar
- 10.Wu Y, Tian L, Huang YF. In vivo corneal biomechanical properties with corneal visualization Scheimpflug technology in Chinese population. Biomed Res Int. 2016;2016:7840284.Google Scholar
- 11.Vellara HR, Patel DV. Biomechanical properties of the keratoconic cornea: a review. Clin Exp Optom. 2015;98(1):31–8.CrossRefGoogle Scholar
- 12.Pinero DP, Alcon N. Corneal biomechanics: a review. Clin Exp Optom. 2015;98(2):107–16.CrossRefGoogle Scholar
- 13.Glass DH, Roberts CJ, Litsky AS, Weber PA. A viscoelastic biomechanical model of the cornea describing the effect of viscosity and elasticity on hysteresis. Invest Ophthalmol Vis Sci. 2008;49(9):3919–26.CrossRefGoogle Scholar
- 14.Han Z, Tao C, Zhou D, Sun Y, Zhou C, Ren Q, Roberts CJ. Air puff induced corneal vibrations: theoretical simulations and clinical observations. J Refract Surg. 2014;30(3):208–13.CrossRefGoogle Scholar
- 15.Zhang H, Qin X, Cao X, Zhang D, Li L. Age-related variations of rabbit corneal geometrical and clinical biomechanical parameters. Biomed Res Int. 2017;2017:3684971.Google Scholar
- 16.Elsheikh A. Finite element modeling of corneal biomechanical behavior. J Refract Surg. 2010;26(4):289–300.CrossRefGoogle Scholar
- 17.Elsheikh A, Joda A, Abass A, Garway-Heath D. Assessment of the Ocular Response Analyzer as an instrument for measurement of intraocular pressure and corneal biomechanics. Curr Eye Res. 2015;40(11):1111–9.CrossRefGoogle Scholar
- 18.Montanino A, Angelillo M, Pandolfi A. Modelling with a meshfree approach the cornea-aqueous humor interaction during the air puff test. J Mech Behav Biomed Mater. 2018;77:205–16.CrossRefGoogle Scholar
- 19.Lago MA, Ruperez MJ, Martinez-Martinez F, Monserrat C, Larra E, Guell JL, Peris-Martinez C. A new methodology for the in vivo estimation of the elastic constants that characterize the patient-specific biomechanical behavior of the human cornea. J Biomech. 2015;48(1):38–43.CrossRefGoogle Scholar
- 20.Sinha Roy A, Dupps WJ Jr, Roberts CJ. Comparison of biomechanical effects of small-incision lenticule extraction and laser in situ keratomileusis: finite-element analysis. J Cataract Refract Surg. 2014;40(6):971–80.CrossRefGoogle Scholar
- 21.Shih PJ, Cao HJ, Huang CJ, Wang IJ, Shih WP, Yen JY. A corneal elastic dynamic model derived from Scheimpflug imaging technology. Ophthalmic Physiol Opt. 2015;35(6):663–72.CrossRefGoogle Scholar
- 22.Whitford C, Studer H, Boote C, Meek KM, Elsheikh A. Biomechanical model of the human cornea: considering shear stiffness and regional variation of collagen anisotropy and density. J Mech Behav Biomed Mater. 2015;42:76–87.CrossRefGoogle Scholar
- 23.Roberts CJ, Mahmoud AM, Bons JP, Hossain A, Elsheikh A, Vinciguerra R, Vinciguerra P, Ambrosio R Jr. Introduction of two novel stiffness parameters and interpretation of air puff-induced biomechanical deformation parameters with a dynamic Scheimpflug analyzer. J Refract Surg. 2017;33(4):266–73.CrossRefGoogle Scholar
- 24.Shih PJ, Huang CJ, Huang TH, Lin HC, Yen JY, Wang IJ, Cao HJ, Shih WP, Dai CA. Estimation of the corneal young’s modulus in vivo based on a fluid-filled spherical-shell model with Scheimpflug imaging. J Ophthalmol. 2017;2017:5410143.Google Scholar
- 25.Taber LA. Large deflection of a fluid-filled spherical shell under a point load. J Appl Mech. 1982;49(1):l2l–l28.CrossRefGoogle Scholar
- 26.Wang LK, Huang YP, Tian L, Kee CS, Zheng YP. Measurement of corneal tangent modulus using ultrasound indentation. Ultrasonics. 2016;71:20–8.CrossRefGoogle Scholar
- 27.Ko MW, Leung LK, Lam DC, Leung CK. Characterization of corneal tangent modulus in vivo. Acta Ophthalmol. 2013;91(4):e263–9.CrossRefGoogle Scholar
- 28.Reissner E. Stresses and small displacements of shallow spherical shells. J Math Phys. 1946;25(1–4):80–5.MathSciNetCrossRefGoogle Scholar
- 29.Nailen R. Roark’s formulas for stress and strain. New York: McGraw-Hill; 2003.Google Scholar
- 30.Metzler KM, Mahmoud AM, Liu J, Roberts CJ. Deformation response of paired donor corneas to an air puff: intact whole globe versus mounted corneoscleral rim. J Cataract Refract Surg. 2014;40(6):888–96.CrossRefGoogle Scholar
- 31.Bao F, Deng M, Wang Q, Huang J, Yang J, Whitford C, Geraghty B, Yu A, Elsheikh A. Evaluation of the relationship of corneal biomechanical metrics with physical intraocular pressure and central corneal thickness in ex vivo rabbit eye globes. Exp Eye Res. 2015;137:11–7.CrossRefGoogle Scholar
- 32.Bao F, Deng M, Zheng X, Li L, Zhao Y, Cao S, Yu A, Wang Q, Huang J, Elsheikh A. Effects of diabetes mellitus on biomechanical properties of the rabbit cornea. Exp Eye Res. 2017;161:82–8.CrossRefGoogle Scholar
- 33.Wollensak G, Iomdina E, Dittert DD, Herbst H. Wound healing in the rabbit cornea after corneal collagen cross-linking with riboflavin and UVA. Cornea. 2007;26(5):600–5.CrossRefGoogle Scholar
- 34.Zhang B, Gu J, Zhang X, Yang B, Wang Z, Zheng D. Biomechanical measurement of rabbit cornea by a modified scheimpflug device. J Ophthalmol. 2016;2016(4):1–6.Google Scholar
- 35.Zhang D, Sun TF, Zhang HX, Li L. The simulation study on the deformation of rabbit cornea after refractive surgery with different cutting thickness. J Mech Med Biol. 2017;17(8):1750118.CrossRefGoogle Scholar
- 36.Elsheikh A, Wang D, Brown M, Rama P, Campanelli M, Pye D. Assessment of corneal biomechanical properties and their variation with age. Curr Eye Res. 2007;32(1):11–9.CrossRefGoogle Scholar
- 37.Wollensak G, Spoerl E, Seiler T. Stress-strain measurements of human and porcine corneas after riboflavin-ultraviolet-A-induced cross-linking. J Cataract Refract Surg. 2003;29(9):1780–5.CrossRefGoogle Scholar
- 38.Elsheikh A, Wang D, Pye D. Determination of the modulus of elasticity of the human cornea. J Refract Surg. 2007;23(8):808–18.CrossRefGoogle Scholar
- 39.Kling S, Bekesi N, Dorronsoro C, Pascual D, Marcos S. Corneal viscoelastic properties from finite-element analysis of in vivo air-puff deformation. PLoS ONE. 2014;9(8):e104904.CrossRefGoogle Scholar

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