^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: SK SM. Performed the experiments: SK NB DP. Analyzed the data: SK NB CD SM. Contributed reagents/materials/analysis tools: SM. Contributed to the writing of the manuscript: SK NB SM. Designed and performed the simulations: SK NB. Designed and performed the experimental air-puff characterisation: CD DP.

Biomechanical properties are an excellent health marker of biological tissues, however they are challenging to be measured in-vivo. Non-invasive approaches to assess tissue biomechanics have been suggested, but there is a clear need for more accurate techniques for diagnosis, surgical guidance and treatment evaluation. Recently air-puff systems have been developed to study the dynamic tissue response, nevertheless the experimental geometrical observations lack from an analysis that addresses specifically the inherent dynamic properties. In this study a viscoelastic finite element model was built that predicts the experimental corneal deformation response to an air-puff for different conditions. A sensitivity analysis reveals significant contributions to corneal deformation of intraocular pressure and corneal thickness, besides corneal biomechanical properties. The results show the capability of dynamic imaging to reveal inherent biomechanical properties in vivo. Estimates of corneal biomechanical parameters will contribute to the basic understanding of corneal structure, shape and integrity and increase the predictability of corneal surgery.

The demand for measuring biomechanical properties of biological tissue in-vivo and non-invasively is high, because abnormal tissue biomechanics play a key role in a wide range of diseases. The stress distribution

In ophthalmology, ocular biomechanics are essential for basic research, clinical evaluation, prognosis and treatment. Pathological weakening of the cornea appears to be responsible for the corneal bulging, and dramatic visual degradation in keratoconus. Corneal collagen cross-linking is an emerging treatment to increase corneal stiffness in this disease

Today, most information regarding available corneal biomechanical properties was assessed ex vivo

In vivo approaches to measure corneal biomechanical properties include stepwise indentation with a cantilever

Studying the dynamic deformation following an air-puff has recently been proposed in different biomedical areas (skin

Air puff applanation of the cornea is a frequent technique in ophthalmology to measure intraocular pressure, yet requiring correction formulae to account for corneal thickness and stiffness. Recently, high speed Optical Coherence Tomography

Some studies have used forward biomechanical modeling of the corneal tissue to predict the corneal response to incisional surgery

In this study we present a finite element model that predicts the corneal deformation pattern upon air-puff ejection and which has allowed, for the first time to our knowledge, to retrieve both, the elastic and viscoelastic properties of the cornea. The model was validated with experimental data of porcine and human eyes and the sensitivity of corneal deformation to geometrical and biomechanical parameters was studied. Being able to retrieve dynamic material properties will open a new way for tissue characterization in vivo.

Corneal deformation was studied using a finite element model with a two-dimensional axis-symmetric geometry. Initial corneal curvature, thickness dimensions and the dynamic deformation response were available from previous experimental Scheimpflug cross-sectional images of the anterior eye segment (see

(a) Corneal geometrical data from Scheimpflug images are used to define the model geometry. Inverse modeling is performed to account for the effect of applying the IOP. (b) The temporal pressure profile measured experimentally and the spatial pressure profile obtained from CFD (Computation Fluid Dynamics) simulation are applied to the cornea as a function of time, location and current deformed shape. (c) The finite element model is solved for the current parameter set and simulation results are compared to the experimentally measured deformation. A step-wise optimization approach is used to find the parameter set that leads to the most similar deformation.

The Corvis system (Oculus, Wetzlar, Germany) uses high-speed Scheimpflug imaging to capture the spatio-temporal corneal deformation following an air-puff. Typically 140 images are taken during the ∼30 ms deformation event (i.e. at a speed of about 4330 images/sec) with a resolution of 640×480 pixels. The spatial and temporal deformation profiles are highly reproducible (RMS difference 0.015 mm and 0.012 mm, respectively).

Experimental corneal deformation data following an air-puff were taken from a previous study in porcine eyes ex vivo and human eyes in vivo and ex vivo

A pressure sensor (MPX2301DT1, Freescale Seminconductor Inc, Tempe, AZ, USA) was used to measure the central temporal pressure distribution of the air-puff at 11-mm distance (typical position of the cornea) from the air-tube. The temporal pressure profile was fit by a linear function for pressure increase (slope = 5.38 mmHg/ms; r = 0.9883; p<0.001) as well as for pressure decrease (slope = −8.38 mmHg/ms; r = 0.9897; p = 0.03). Although the overall air-puff duration was 27.50 ms, only 20.63 ms (i.e. 97% of the air-pressure) contributed effectively to the corneal deformation.

In order to determine the geometry-dependent spatial shape of the air-puff at different stages of the corneal deformation, the maximum air speed (115 m/s) was estimated by Bernoulli’s equation at the stagnation point of the airflow:_{stagnation} is the measured absolute stagnation pressure of the air puff, and p_{atmosphere} is the static atmospheric pressure. This value is comparable with the air speed measured previously

To retrieve the spatial air pressure distribution, an axis-symmetric computational fluid dynamics (CFD) simulation was performed using the finite volume method (FVM) implemented in the Fluent module of the ANSYS Release 14.0 software package.

Prior experimental data show a correlation between the maximum deformation amplitude of the cornea and the peak-to-peak distance.

Mesh of cells of the modeled air volume (left) and streamlines (right) colored by the flow velocity distribution.

Case n° | Apex indentation[mm] | Peak distance[mm] | Number of cells in mesh | Average cell size[mm^{2}] |

0 | 0 | 0 | 5600 | 0.0464 |

1 | 0.5 | 3 | 5754 | 0.0452 |

2 | 0.6 | 4 | 5703 | 0.0457 |

3 | 0.8 | 5 | 5858 | 0.0446 |

4 | 1 | 5.3 | 5858 | 0.0446 |

5 | 1.5 | 5.5 | 6925 | 0.0378 |

The air-puff emitting tube and the cornea were considered as wall type boundary conditions (i.e. the cornea was simulated as a rigid body not considering fluid-structure interaction). A velocity inlet was modeled at the end of the air tube and a pressure outlet around the cornea. The initial flow velocity of the air-puff (115 m/s) was calculated from the stagnation pressure, i.e. the peak central pressure measured in the air-puff characterization. As inertial effects are expected to dominate over viscous effects, turbulent flow was assumed and described using the Reynolds stress model. The Reynolds averaged momentum equations for the mean velocity are:

In the differential stress model,

The pressure distribution along the corneal surface was obtained for each of the six deformed corneal geometries.

(a) Experimentally measured temporal air-puff profile; (b) Results from CFD simulation showing the air-puff as a function of apex indentation and location along the cornea (horizontal distance from the apex).

In order to account for measurement inaccuracies of the maximal pressure and hence the estimation of the air speed the spatial pressure function was normalized and a scaling factor P_{max} introduced. This was necessary as the sensor, due to the geometry of its housing, tends to overestimate the pressure in high-speed dynamic measurements. In the structural finite element simulations, the applied surface pressure was a function of apex indentation, distance from the apex and time.

A 2D axis-symmetric eye model was defined corresponding to the different corneal conditions and boundaries (see

Human (virgin) | Porcine (cross-linked) | |

Corneal thickness (µm) | 558 | 211 |

Anterior curvature (mm) | 8.03 | 8.06 |

Posterior curvature (mm) | 6.86 | 7.54 |

Corneal diameter (mm) | 10 | 12 |

Scleral diameter (mm) | 19.5 | 23.5 |

The corneal tissue was modeled by a linear viscoelastic material. Thereby only the shear response was considered, as it is typically dominant over the volumetric response.

Then the fluid pressure load vector

Biomechanical parameters of the sclera and limbus (see ^{3} for a 700 µm corneal thickness and scaled according to the thickness values in the experiments. A material damping of 10 µs was used.

For the simulations of the ex vivo whole globe the sclera was fixed mimicking the fixation of the eye in the eye-holder in the experiments. For the in vivo condition the eye globe was damped along on the vertical symmetry axis representing the ocular muscles and other surrounding fatty tissue. It was assumed that those external damping factors can be summarized in a single vertical damping element, while horizontal damping effects were neglected. The vertical damping was implemented by a mass-less longitudinal spring-damper, which was modeled by a uniaxial tension-compression element, defined by a spring constant (5·10^{6} N/m) and a damping coefficient (1.0). The intraocular pressure was applied in the model to the interior surfaces of cornea, limbus and sclera according to the experimental data (15, 20, 25 and 35 mmHg).

A pressure load was applied on the element edges of the anterior corneal surface according to the spatial pressure distribution of the air-puff at the different indentation depths. This was necessary because the fluid dynamics characteristics change significantly as the cornea deforms. Furthermore the pressure variation with time (measured experimentally with the pressure sensor) was considered by multiplying the current pressure with the normalized temporal pressure profile.

In the first load step the IOP was applied to the model. Then, in the next step the load modeling the air-puff was applied. In order to achieve sufficient temporal resolution, this step was divided into 49 sub-steps, so the applied pressure was updated every 625 µs.

A multiple step optimization approach (schematic shown in

Pre-optimization step (0), where the maximal air-puff pressure is adjusted; Multi-step optimization, comprising adjustment of initial elastic modulus and corneal geometry (I); adjustment of viscoelastic parameters (II) and further refinement of the elastic modulus and geometry (I); adjustment of the anterior and posterior elastic moduli (III), followed by a further refinement of the elastic moduli and geometry (I). In human eyes and additional step was incorporated to account for damping by ocular muscles and external tissue (IV). In the illustration the size of the loops is positively correlated with the dominance of the parameter adjusted therein. The resulting parameter values, shown in blue, represent an example of the output parameters at each step for human eyes ex vivo and in vivo.

(a) Elastic properties dominate the maximal indentation depth. (b) Viscoelastic properties dominate the amount of hysteresis when the air-pressure has decreased to zero. (c) The ratio between anterior and posterior stiffness dominates the distance between corneal apex and bending points.

In the beginning, the elasticity modulus was the only variable considered to account for the deformation behavior. Then the biomechanical description of the material model was refined in consecutive steps, adding viscoelastic properties and a difference between anterior and posterior corneal stiffness. Thereby the fact that two deformation profiles (temporal, spatial) were available, and that the optimization variables had differential effects on each, reduced the amount of possible local minima. The multi-step procedure allowed us to obtain better comparable parameter sets across the different tested conditions.

In order to define the individual steps of the optimization procedure, we studied the pattern of influence for each optimization variable on the temporal and on the spatial corneal deformation profiles. Changing a given variable subsequently, while keeping the other parameters constant allowed then identifying systematic effects.

The different patterns were subsequently used to optimize the geometrical and biomechanical parameter set of the FE-model, as illustrated in _{max} was adjusted in the range from 65.8 mmHg (115 m/s) to 111 mmHg (156 m/s) using the deformation profiles of the cross-linked porcine cornea at 15, 20, 25 and 35 mmHg. This led to a value of P_{max} = 72.5 mmHg, which was then kept constant for all conditions (see “pre-optimization” in

In the first step of the optimization a very simplified material model - linear elasticity - was assumed. It included the adjustment of geometrical parameters and the elastic modulus: the initial anterior radius of curvature, central corneal thickness and elastic modulus were adjusted so that after applying the intraocular pressure, both the modeled and experimental corneal geometry within the 8-mm diameter central zone and the maximal indentation depth were identical.

In the second step the material model was refined adding viscoelastic properties expressed by a Prony and a relaxation time constants. This second step was iteratively coupled with the first step in order to correct for the viscoelastic component, the effects of which were initially accounted for by the previously adjusted elastic modulus (see central part in

The third step provides a further refinement the model accounting for physiological differences in the anterior and posterior collagen interweaving, which result in an in-depth variation of corneal stiffness. This difference in stiffness between the anterior and posterior corneal regions was introduced in order to adjust the spatial corneal deformation profile (see

For the in-vivo human cornea a fourth step was necessary, which included the adjustment of the two elements (elastic and viscous parameters) of the external damping element (see “extra step for in vivo eyes” in

After selecting the model parameter set that best represented the virgin human cornea in vivo condition, a sensitivity analysis was performed in order to determine the geometrical and mechanical parameters that dominate the shape and amount of corneal deformation following the air-puff. Seven parameters were selected (corneal thickness, stiffness, curvature, density, IOP and two viscoelastic constants) and changed consecutively within physiological or pathological ranges. For each parameter variation the effect on the overall predicted corneal deformation was determined by analyzing the coordinates of the deformed shape, including changes in the maximal corneal indentation and peak distance. Evaluation of the factors that dominate the corneal deformation is highly relevant in the clinical practice, where diagnostics is currently performed from geometrical deformation parameters.

ANSYS APDL structural mechanics code (ANSYS, Inc., Canonsburg, PA) was used for the mechanical simulations and the FLUENT module for the CFD simulations. The analysis of the deformed corneal shape was performed in Matlab (The MathWorks, Natick, MA).

The finite element model could well reproduce average experimental data from a previous study

The corneal response upon air-puff ejection was simulated for different IOPs (ranging from 15 to 35 mmHg, i.e. covering the IOP range from physiological values to those found in severe glaucoma). We found that the maximum corneal apex indentation was 1.13 mm for the lowest IOP and 0.46 mm for the highest IOP. Air-puff maximum pressure and, to a lesser extent, differences in stiffness between the anterior and posterior cornea were found to play a major role in the predicted corneal deformation. ^{21} in cross-linked porcine corneas ex vivo for different IOPs. The reconstructed elasticity moduli from the model were: E_{anterior} = 25.5 MPa; E_{posterior} = 0.85 MPa (see _{fem}−0.676 mm versus Δ_{exp}−0.666 mm) with increased IOP from 15 to 35 mmHg and the decreased peak distance, i.e. horizontal distance between the corneal bending points at maximal deformation (Δ_{fem}−2.04 mm versus Δ_{exp}−2.34 mm) are well reproduced by the model.

Dotted lines represent experimental corneal deformations and continuous lines simulated corneal deformations Panels (a, b) show data for porcine corneas: simulated and experimental data at different IOPs. Panels (c, d) show data for human corneas: simulated response with and without ocular muscle damping, compared to in vivo experimental deformations measured in patients and ex vivo deformations measured in an enucleated whole globe.

Human | Porcine | |

Condition | virgin | cross-linked |

Elasticity modulus (Mpa)posterior cornea | 0.40 | 0.85 |

Difference between anterior and posterior | 1.12 | 30 |

Relaxation time (ms) | 10 | 1.5 |

Relative modulus | 0.3 | 0.7 |

Muscle spring constant (Nm) | 350 | - |

Muscle damping constant (kg/s) | 100 | - |

The relation between corneal indentation and eye globe compression (i.e. general deformation from a spherical to an elliptical eye shape resulting from scleral deformation) depended on the IOP, but also on the difference between the anterior and posterior corneal stiffness. At 15 mmHg, corneal deformation contributed 90.4% to the overall indentation, while at 35 mmHg only 44.9%. We also found that increasing the difference between anterior and posterior corneal stiffness produced larger corneal deformation (i.e. for E_{anterior}/E_{posterior} = 30: corneal deformation = 90.4%; for E_{anterior}/E_{posterior} = 1: corneal deformation = 83.7%).

The viscoelastic effect was evidenced by the remaining deformation (between 18.7 and 24.1% of the overall deformation at different IOPs), which gradually decreased with time after the air-puff event. The viscoelasticity of the cross-linked porcine corneas ex vivo was described by a relative modulus of 0.7 and a relaxation time constant of 1.5 ms, and contributed with 46% to the maximal apex indentation.

Defining a damping element for the fixation of the ocular globe allowed us to describe the effect of ocular muscles and fatty tissue. Corneal deformation in vivo (average across 9 eyes) was compared to simulations where damping of the entire ocular globe was considered, while corneal deformation ex vivo was compared to simulations without damping (see

We found that the retro-bulbar tissue contributed 0.112 mm to the apex displacement and induced a hysteresis (additional to the viscoelastic hysteresis of the cornea) in the temporal profile in the region of indirect air-puff response.

A sensitivity analysis was performed for the human eye in vivo (i.e. with damping) in order to evaluate which geometrical and biomechanical parameters determine the corneal deformation with an air-puff.

Geometrical/biomechanical parameters | Change within the physiologic range | Δ peak distance (mm) | Δ apex indentation (mm) |

Thickness (mm) | Δ 100 µm | −0.7168 | 0.2011 |

Stiffness (MPa) | Δ 0.2 MPa | −0.7218 | 0.3764 |

Relative modulus (no unit) | Δ 1 | 0.8929 | 0.1000 |

Relaxation time (ms) | Δ 10 ms | 0.4300 | −0.2324 |

IOP (mmHg) | Δ 40 mmHg | −2.472 | 0.2320 |

Curvature (mm) | Δ 1 mm | 0.1875 | −0.0163 |

Density (10^{6}g/m3) |
Δ 500 kg/m3 | 0.0008 | 0.0000 |

New imaging acquisition techniques allow capturing the dynamic geometrical deformation response of the cornea following an air-puff. To our knowledge, we have presented for the first time numerical estimates of corneal elastic and viscoelastic parameters, based on Scheimpflug imaging of the spatio-temporal dynamic corneal deformation and sophisticated Finite Element Modeling. The simulation reproduces with high accuracy the corneal deformation patterns, while the estimated biomechanical parameters are independent of geometry. Overall, cross-linked porcine corneas were 9.38 times more rigid than virgin human corneas. The much higher stiffness of the anterior cornea than of the posterior in cross-linked corneas can be primarily attributed to the cross-linking treatment - literature reports an increase in corneal stiffness between 72% and 329% after CXL

In addition, by simulating different boundary conditions, the movement of the cornea in response to the air puff could be isolated from the additive effects of scleral deformation and movement of the whole eye observed in the Scheimpflug images.

The dynamic response

Time scales are not only relevant when comparing different measurement systems, but also in regard to the physiological interpretation. Elastic properties, i.e. static properties, are determined by the collagen matrix and depend on the microstructural arrangement as well as the number of cross-links. These material properties determine the long-term corneal resistance against the IOP. Viscoelastic properties, i.e. time-dependent properties, describe the rearrangement of the extracellular matrix (mainly due to diffusion of water) upon changes in stress. Generally the shorter the time scale where material properties are analyzed, the harder the material and the larger the viscoelastic contribution. Thereby the time scale of interest for corneal material properties depends on the time scale of the treatment or pathology: For refractive surgery short-term viscoelasticity is probably more important, while in keratoconus long-term viscoelasticity is of interest. Future air-puff systems may be provided with alternatives to allow a viscoelastic analysis at different time scales.

Although the sensitivity analysis showed a correlation between corneal rigidity and corneal deformation, geometrical parameters, such as the central corneal thickness, as well as IOP, also showed a large effect. This means that all factors need to be considered and entered as input parameters in the simulation, which increases the complexity of the finite element simulation. Comparison of the gradients obtained from sensitivity analysis with clinical data - based on thickness changes after LASIK surgery, and over a physiological IOP range (personal communication by Oculus) - show good agreement.

Although the multi-step optimization approach has allowed a systematic retrieval of the model parameters, a potential limitation of this method might be the identification of a local rather than a global minimum. Nevertheless, the characteristic pattern how individual parameters changed the cost-function as well as the fact that a single parameter set (CXL porcine corneas) allowed an accurate reproduction of the corneal response at different IOPs (15–35 mmHg), suggesting that the solution is robust and likely to be unique.

A potentially strong limitation of this study is that linear material properties were assumed. Experimental diagrams from uniaxial stress-strain testing suggest a linear stress-strain relation up to about 5–6% strain

We believe that finite element modeling is an extremely valuable tool for the analysis of in vivo dynamic corneal deformation. Inverse simulation together with state of the art imaging systems will allow the analysis of (in particular the viscoelastic) corneal biomechanical properties (viscoelastic properties in particular) in a clinical setting, facilitating diagnosis of corneal pathologies, patient screening for refractive surgery and evaluation of treatment efficacy such as after cross-linking.

We acknowledge Oculus for providing access to the Corvis ST system and we thank Nicolas Alejandre and Pablo Perez-Merino for their help in the acquisition of human and porcine eyes, respectively.

^{nd}ed), (2007) New York, NY, Springer.