Figure 1.
Scheme of the multizone corneal model.
Figure 1.
Scheme of the multizone corneal model.
Figure 2.
Detail of the corneal segmentation process. (a) octant definition; (b) circumferential definition; (c) detail of diameter definition.
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 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 4.
Scheme of the strain–stress field recovery in the physiological state with the displacements method.
Figure 5.
Flow chart of the iterative displacements method process.
Figure 5.
Flow chart of the iterative displacements method process.
Figure 6.
Scheme of the strain–stress field recovery in the physiological state with the prestress method.
Figure 6.
Scheme of the strain–stress field recovery in the physiological state with the prestress method.
Figure 7.
Flow chart of the iterative prestress method process.
Figure 7.
Flow chart of the iterative prestress method process.
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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).
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.
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 |
Table 2.
Anisotropic–hyperelastic material parameter definitions.
Table 2.
Anisotropic–hyperelastic material parameter definitions.
Material Constants | a1 (Pa) | a2 (Pa) | k1 (Pa) | k2 (−) |
---|
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.
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.
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.
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.
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 |