# Study of the Influence of Boundary Conditions on Corneal Deformation Based on the Finite Element Method of a Corneal Biomechanics Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Patients

#### 2.2. Material Definition

**m**and

**n**are the fiber orientation vectors in the deformed state and

**1**is the identity tensor.

_{1}and a

_{2}, which bear in mind the extracellular matrix contribution, were considered to be equal in all the zones, whereas the fiber concentrations were implemented when considering the variation in parameter ${k}_{1}$, as per previous research works [27,28]. In this research work, the variation in collagen fiber concentrations in terms of the thickness was not considered. Parameter k

_{2}was constant according to the hypothesis that the behavior of collagen fibers in large strains is the same in all the corneal zones.

#### 2.3. Finite Element Model Definition

#### 2.4. Calculation of the Stress and Strain Field in the Measurement Phase

#### 2.4.1. Displacements Method

^{−9}m was considered.

#### 2.4.2. Prestress Method

^{−9}m) when considering the flow chart shown in Figure 7.

#### 2.5. Calculation of Displacements and the Stress Field in the Measurement Phase

## 3. Results and Discussion

#### 3.1. Stress-Free Geometry (SFG)

#### 3.2. Estimated Physiological Geometry (EPG)

#### 3.3. Stress Fields in the Estimated Physiological Geometry

#### 3.4. Strain Fields in the Physiological State

#### 3.5. Computational Time Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## List of Acronyms

EGP | Estimated physiological geometry |

SFG | Stress-free geometry |

IOP | Intraocular pressure |

${\mathrm{a}}_{\mathrm{i}}(i=\mathrm{1,2})$ | Material constant of the Mooney–Rivlin model |

k_{0} | Bulk factor |

k_{1} | Hyperelastic material parameter to consider the influence of collagen fibril stiffness. |

k_{2} | Hyperelastic material parameter to consider the influence of collagen fibrils in large extensions. |

$J$ | Jacobian of the deformation gradient tensor |

n_{max} | Maximum number of iterations |

$\overline{\mathbf{C}}$ | Isochoric Cauchy–Green deformation tensor |

F | Deformation gradient tensor |

F_{nm} | Deformation gradient tensor from step n to step m |

${\mathbf{m}}_{\mathbf{0}}$ | Initial preferential orientation of collagen fibril family 1. |

$\mathbf{m}$ | Deformed preferential orientation of collagen fibril family 1. |

${\mathbf{n}}_{\mathbf{0}}$ | Initial preferential orientation of collagen fibril family 2. |

$\mathbf{n}$ | Deformed preferential orientation of collagen fibril family 2. |

x | Deformed vector |

x_{n} | Deformed vector in the iteration n |

X | Reference vector |

X_{n} | Reference vector in the iteration n |

${\mathbf{u}}_{\mathbf{n}}$ | Displacement vector in the iteration n |

$\psi $ | Strain energy density |

${\psi}_{vol}$ | Volumetric strain energy density |

${\psi}_{iso}$ | Isotropic isochoric strain energy density |

${\psi}_{aniso}$ | Anisotropic isochoric strain energy density |

$\overline{{I}_{i}}$ (i = 1,9) | Invariants of the modified right Cauchy stress tensor |

ε | Tolerance |

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**Figure 2.**Detail of the corneal segmentation process. (

**a**) octant definition; (

**b**) circumferential definition; (

**c**) detail of diameter definition.

**Figure 3.**Multizone cornea definition: left side (

**a**,

**b**): zone definition; right side: detail of the global zone definition (

**c**) and detail of the regular mesh pattern (

**d**).

**Figure 4.**Scheme of the strain–stress field recovery in the physiological state with the displacements method.

**Figure 6.**Scheme of the strain–stress field recovery in the physiological state with the prestress method.

**Figure 8.**Estimated physiological geometry displacements (m) obtained for a grade III keratoconus cornea under fixed (

**top**) and pivoting (

**bottom**) boundary conditions. Detail of the boundary condition at the limbus (

**top fixed**,

**bottom pivoting**). IOP = 14 mmHg applied to the stress-free geometry (solid black line) obtained by the displacements method.

**Figure 9.**Estimated physiological geometry stresses (Pa) obtained for a grade III keratoconus cornea under fixed (

**top**) and pivoting (

**bottom**) boundary conditions. Detail of the boundary condition at the limbus (

**top fixed**,

**bottom pivoting**). IOP = 14 mmHg applied to the stress-free geometry (solid black line) obtained by the displacements method.

**Figure 10.**Total distance (m) from the measured physiological geometry to obtain the stress-free geometry. Corneal G1 model (IOP = 15 mmHg). Embedded boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 11.**Total distance (m) from the measured physiological geometry to obtain the stress-free geometry. Corneal G1 model (IOP = 15 mmHg). Pivoting boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 12.**Total distance (m) from the measured physiological geometry to obtain the stress-free geometry. Corneal G3 model (IOP = 14 mmHg). Embedded boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 13.**Total distance (m) from the measured physiological geometry to obtain the stress-free geometry. Corneal G3 model (IOP = 14 mmHg). Pivoting boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 14.**Total distance (m) from the stress-free geometry to obtain the estimated physiological geometry. Corneal G1 model (IOP = 15 mmHg). Embedded boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 15.**Total distance (m) from the stress-free geometry to obtain the estimated physiological geometry. Corneal G1 model (IOP = 15 mmHg). Pivoting boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 16.**Total distance (m) from the stress-free geometry to obtain the estimated physiological geometry. Corneal G3 model (IOP = 14 mmHg). Embedded boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 17.**Total distance (m) from the stress-free geometry to obtain the estimated physiological geometry. Corneal G3 model (IOP = 14 mmHg). Pivoting boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 18.**The Von Mises stress field (Pa) for the G1 corneal model (IOP = 15 mmHg) stress-free geometry. Embedded boundary conditions (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 19.**The Von Mises stress field (Pa) for the G1 corneal model (IOP = 15 mmHg) stress-free geometry. Pivoting boundary conditions (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 20.**The Von Mises stress field (Pa) for the G3 corneal model (IOP = 14 mmHg) stress-free geometry. Embedded boundary conditions (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 21.**The Von Mises stress field (Pa) for the G3 corneal model (IOP = 14 mmHg) stress-free geometry. Pivoting boundary conditions (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 22.**The Von Mises stress field (Pa) for the G1 corneal model (IOP = 15 mmHg) stress-free geometry (

**Upper**: embedded boundary condition;

**lower**: pivoting boundary condition).

**Figure 23.**The Von Mises stress field (Pa) for the G3 corneal model (IOP = 14 mmHg) stress-free geometry (

**Upper**: embedded boundary condition;

**lower**: pivoting boundary condition).

**Figure 24.**The Von Mises strain field (m/m) for the G1 corneal model (IOP = 15 mmHg) and embedded boundary conditions (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 25.**The Von Mises strain field (m/m) for the G1 corneal model (IOP = 15 mmHg) and pivoting boundary conditions. (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 26.**The Von Mises strain field (m/m) for the G3 corneal model (IOP = 14 mmHg) and embedded boundary conditions (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 27.**The Von Mises strain field (m/m) for the G3 corneal model (IOP = 14 mmHg) and pivoting boundary conditions (

**Upper**: displacements method;

**lower**: prestress method).

**Figure 28.**Comparison of the computational time (s) of the iterative process for the embedded boundary condition. (A total of 11,640 SOLID 186 elements with the mixed u-P formulation and reduced integration with large displacements, 10-substep solution and two core processors).

**Figure 29.**Comparison of the computational time (s) of the iterative process for the pivoting boundary condition. (A total of 11,640 SOLID 186 elements with the mixed u-P formulation and reduced integration with large displacements, 10-substep solution and two core processors).

**Figure 30.**Comparison of computational time (s) depending on the boundary conditions for the displacements method. (A total of 11,640 SOLID 186 elements with the mixed u-P formulation and reduced integration with large displacements and 10-substep solution).

**Figure 31.**Influence of number of core processors on computational time (pivoting constraint) (11,640 SOLID 186 elements with the mixed u-P formulation and reduced integration with large displacements and 10-substep solution).

**Table 1.**Main clinical characteristics of the eyes used during simulations. A-K, Amsler–Krumeich keratoconus grade (G1: grade I; G2: grade II; G3: grade III; G4: grade IV); IOP, intraocular pressure; mean K, mean keratometry; AXL, axial length; CDVA, corrected-distance visual acuity.

A-K | IOP (mm) | Mean K (D) | Age (y) | Gender | Eye | AXL (mm) | Manifest Sphere (D) | Manifest Cylinder (D) | Cylinder Axis (°) | Spherical Equivalent (D) | Decimal CDVA |
---|---|---|---|---|---|---|---|---|---|---|---|

G1 | 15 | 47.29 | 49 | M | OS | 21.81 | 0.5 | −0.5 | 100 | 0.62 | 1.05 |

G2 | 13 | 51.59 | 55 | F | OD | 24.1 | 2 | −5.5 | 85 | −0.40 | 0.55 |

G3 | 14 | 53.77 | 26 | F | OS | 25.39 | −0.5 | −2.5 | 120 | −2 | 0.65 |

G4 | 17 | 69.1 | 33 | M | OD | 23.61 | 0 | −2.5 | 60 | −1 | 0.15 |

Material Constants | a_{1} (Pa) | a_{2} (Pa) | k_{1} (Pa) | k_{2} (−) |
---|---|---|---|---|

Central, N-T and S-I zones | 40,000 | −10,000 | 50,000 | 200 |

Transition zones | 40,000 | −10,000 | 37,500 | 200 |

Central oblique zones | 40,000 | −10,000 | 25,000 | 200 |

Limbus | 40,000 | −10,000 | 50,000 | 200 |

**Table 3.**Total distance (m) from the measured physiological geometry to obtain the stress-free geometry.

Stress-Free Geometry (m) | Displacements Method | Prestress Method | |||
---|---|---|---|---|---|

Geometry | IOP (mmHg) | Embedded | Pivoting | Embedded | Pivoting |

G1 | 15 | 2.53·10^{−4} | 2.37·10^{−4} | 2.57·10^{−4} | 2.41·10^{−4} |

G2 | 13 | 2.87·10^{−4} | 4.23·10^{−4} | 2.96·10^{−4} | 4.35·10^{−4} |

G3 | 14 | 3.99·10^{−4} | 3.45·10^{−4} | 4.22·10^{−4} | 3.88·10^{−4} |

G4 | 17 | 4.78·10^{−4} | 6.71·10^{−4} | 4.98·10^{−4} | 7.42·10^{−4} |

**Table 4.**Statistical analysis of the difference in the total displacements between the iterative displacements method and the iterative prestress method for the calculated stress-free geometry.

G1 | Embedded | Pivoting |
---|---|---|

Maximum (μm) | 8.36 | 7.47 |

Mean (μm) | 2.92 | 3.07 |

Standard deviation (μm) | 1.83 | 1.41 |

G3 | Embedded | Pivoting |

Maximum (μm) | 43.54 | 43.54 |

Mean (μm) | 13.18 | 15.59 |

Standard deviation (μm) | 11.35 | 10.53 |

**Table 5.**Total distance (m) from the stress-free geometry to obtain the estimated physiological geometry.

Estimated Physiological Geometry (m) | Displacements Method | Prestress Method | |||
---|---|---|---|---|---|

Geometry | IOP (mmHg) | Embedded | Pivoting | Embedded | Pivoting |

G1 | 15 | 2.53·10^{−4} | 2.37·10^{−4} | 2.54·10^{−4} | 2.38·10^{−4} |

G2 | 13 | 2.87·10^{−4} | 4.23·10^{−4} | 2.90·10^{−4} | 4.28·10^{−4} |

G3 | 14 | 3.99·10^{−4} | 3.45·10^{−4} | 4.17·10^{−4} | 3.64·10^{−4} |

G4 | 17 | 4.78·10^{−4} | 6.71·10^{−4} | 4.89·10^{−4} | 7.08·10^{−4} |

**Table 6.**Statistical analysis of the difference in the total displacements between the iterative displacements method and the iterative prestress method for the calculated estimated physiological geometry.

G1 | Embedded | Pivoting |
---|---|---|

Maximum (μm) | 0.12 | 0.14 |

Mean (μm) | 0.04 | 0.05 |

Standard deviation (μm) | 0.03 | 0.03 |

G3 | Embedded | Pivoting |

Maximum (μm) | 0.11 | 0.11 |

Mean (μm) | 0.04 | 0.05 |

Standard deviation (μm) | 0.03 | 0.03 |

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**MDPI and ACS Style**

Gómez, C.; Piñero, D.P.; Paredes, M.; Alió, J.L.; Cavas, F.
Study of the Influence of Boundary Conditions on Corneal Deformation Based on the Finite Element Method of a Corneal Biomechanics Model. *Biomimetics* **2024**, *9*, 73.
https://doi.org/10.3390/biomimetics9020073

**AMA Style**

Gómez C, Piñero DP, Paredes M, Alió JL, Cavas F.
Study of the Influence of Boundary Conditions on Corneal Deformation Based on the Finite Element Method of a Corneal Biomechanics Model. *Biomimetics*. 2024; 9(2):73.
https://doi.org/10.3390/biomimetics9020073

**Chicago/Turabian Style**

Gómez, Carmelo, David P. Piñero, Manuel Paredes, Jorge L. Alió, and Francisco Cavas.
2024. "Study of the Influence of Boundary Conditions on Corneal Deformation Based on the Finite Element Method of a Corneal Biomechanics Model" *Biomimetics* 9, no. 2: 73.
https://doi.org/10.3390/biomimetics9020073