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Customized Finite Element Modelling of the Human Cornea.

Simonini I, Pandolfi A - PLoS ONE (2015)

Bottom Line: Corneal elevation maps of five human eyes were taken with a rotating Scheimpflug camera combined with a Placido disk before and after refractive surgery.Patient-specific solid models were created and discretized in finite elements to estimate the corneal strain and stress fields in preoperative and postoperative configurations and derive the refractive parameters of the cornea.Patient-specific models can be used as indicators of feasibility before performing the surgery.

View Article: PubMed Central - PubMed

Affiliation: Dipartimento di Matematica, Politecnico di Milano, Milano, Italy.

ABSTRACT

Aim: To construct patient-specific solid models of human cornea from ocular topographer data, to increase the accuracy of the biomechanical and optical estimate of the changes in refractive power and stress caused by photorefractive keratectomy (PRK).

Method: Corneal elevation maps of five human eyes were taken with a rotating Scheimpflug camera combined with a Placido disk before and after refractive surgery. Patient-specific solid models were created and discretized in finite elements to estimate the corneal strain and stress fields in preoperative and postoperative configurations and derive the refractive parameters of the cornea.

Results: Patient-specific geometrical models of the cornea allow for the creation of personalized refractive maps at different levels of IOP. Thinned postoperative corneas show a higher stress gradient across the thickness and higher sensitivity of all geometrical and refractive parameters to the fluctuation of the IOP.

Conclusion: Patient-specific numerical models of the cornea can provide accurate quantitative information on the refractive properties of the cornea under different levels of IOP and describe the change of the stress state of the cornea due to refractive surgery (PRK). Patient-specific models can be used as indicators of feasibility before performing the surgery.

No MeSH data available.


Related in: MedlinePlus

Fiber organization in the top layer of the finite element discretization of the cornea model.(a) Mean orientation of the two sets of fibers. (b) Map of the von Mises coefficient b for the statistical distribution of the orientation. High values of b define highly oriented set of fibers. Low values of b define nearly isotropic orientations.
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pone.0130426.g008: Fiber organization in the top layer of the finite element discretization of the cornea model.(a) Mean orientation of the two sets of fibers. (b) Map of the von Mises coefficient b for the statistical distribution of the orientation. High values of b define highly oriented set of fibers. Low values of b define nearly isotropic orientations.

Mentions: The first term, ψvol, accounts for volume changes, and it is assumed to be dependent on the determinant of the deformation gradient F, or jacobian J = det F. For incompressible materials, this term must be considered as a penalty contribution to contrast undesired changes of the volume. The expression of ψvol is:ψvol(J)=14K(J2−1−2logJ),(11)where K is the bulk modulus. The term ψiso describes the behaviour of the isotropic aspects of the material behaviour, including the underlying matrix and the portion of randomly distributed fibrous reinforcement. As usual, ψiso is assumed to be dependent on the first and second invariants of the modified Cauchy-Green deformation tensor where . We select an expression corresponding to the Mooney-Rivlin modelψiso(I¯1,I¯2)=12μ1(I¯1−3)+12μ2(I¯2−3),(12)with μ = μ1 + μ2 the shear modulus. The third term ψaniso describes the anisotropic behavior, including the effects of the microstructure of the fibrils. It usually depends on the modified tensor and on particular vectors or tensors describing the intrinsic structure of the material. In the present calculations we refer to the particular model described in [37], of the formψaniso(I¯4,1*,I¯4,2*)=∑F=1212k1Fk2Fexp[k2F(I¯4,F*−1)2](1+KF*σI4,F2),(13)where k1F are the stiffness parameters and k2F are dimensionless rigidity parameters. Overall, the material model for the cornea needs the assignment of seven material parameters (K,μ1,μ2,k11,k12,k21,k22). The material model requires also the definition of the spatial distribution of the main orientation aF of two sets of fibrils and of the concentration parameter bF(ρ,θ,z), that describes the spatial dispersion of the fibrils about the main orientation. It is now well known that the organization of the fibrils in the cornea follows a particular pattern with dominant orientation in the nasal-temporal (NT) and superior-inferior (SI) directions [40, 41]. The variability of the interlacing and of the dispersion of the fibrils orientation across the thickness has been recently elucidated [35, 36]. According to recent findings, the organization of the collagen fibrils in the deep cornea is better modeled by a planar distribution of the fibrils orientation than a fully tridimensional distribution. In the present calculations, though, we kept using the model already employed and validated in previous studies, since here we are interested in the geometrical aspects of the corneal modeling, while the investigation on the material model will be object of an on-going study. In the outermost layer of the cornea model, fibrils are organized according to the model described in [17, 34]. In the central part of the model, fibrils follow the NT and SI orientation, and progressively rotate the orientation while moving towards the limbus, where the main set of fibrils runs circumferentially. A secondary, more compliant, set of fibrils runs in the radial direction to guarantee the correct mechanical behavior of the shell, Fig 8A. Everywhere, the spatial orientation of collagen fibrils follows a transversely isotropic and π-periodic, normalized, von Mises distribution:ρ(Θ)=12πIexp(bcos2Θ),I=1π∫0πexp(bcos2Θ)dΘ,(14)with a concentration parameter b distributed according to Fig 8B.


Customized Finite Element Modelling of the Human Cornea.

Simonini I, Pandolfi A - PLoS ONE (2015)

Fiber organization in the top layer of the finite element discretization of the cornea model.(a) Mean orientation of the two sets of fibers. (b) Map of the von Mises coefficient b for the statistical distribution of the orientation. High values of b define highly oriented set of fibers. Low values of b define nearly isotropic orientations.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4476710&req=5

pone.0130426.g008: Fiber organization in the top layer of the finite element discretization of the cornea model.(a) Mean orientation of the two sets of fibers. (b) Map of the von Mises coefficient b for the statistical distribution of the orientation. High values of b define highly oriented set of fibers. Low values of b define nearly isotropic orientations.
Mentions: The first term, ψvol, accounts for volume changes, and it is assumed to be dependent on the determinant of the deformation gradient F, or jacobian J = det F. For incompressible materials, this term must be considered as a penalty contribution to contrast undesired changes of the volume. The expression of ψvol is:ψvol(J)=14K(J2−1−2logJ),(11)where K is the bulk modulus. The term ψiso describes the behaviour of the isotropic aspects of the material behaviour, including the underlying matrix and the portion of randomly distributed fibrous reinforcement. As usual, ψiso is assumed to be dependent on the first and second invariants of the modified Cauchy-Green deformation tensor where . We select an expression corresponding to the Mooney-Rivlin modelψiso(I¯1,I¯2)=12μ1(I¯1−3)+12μ2(I¯2−3),(12)with μ = μ1 + μ2 the shear modulus. The third term ψaniso describes the anisotropic behavior, including the effects of the microstructure of the fibrils. It usually depends on the modified tensor and on particular vectors or tensors describing the intrinsic structure of the material. In the present calculations we refer to the particular model described in [37], of the formψaniso(I¯4,1*,I¯4,2*)=∑F=1212k1Fk2Fexp[k2F(I¯4,F*−1)2](1+KF*σI4,F2),(13)where k1F are the stiffness parameters and k2F are dimensionless rigidity parameters. Overall, the material model for the cornea needs the assignment of seven material parameters (K,μ1,μ2,k11,k12,k21,k22). The material model requires also the definition of the spatial distribution of the main orientation aF of two sets of fibrils and of the concentration parameter bF(ρ,θ,z), that describes the spatial dispersion of the fibrils about the main orientation. It is now well known that the organization of the fibrils in the cornea follows a particular pattern with dominant orientation in the nasal-temporal (NT) and superior-inferior (SI) directions [40, 41]. The variability of the interlacing and of the dispersion of the fibrils orientation across the thickness has been recently elucidated [35, 36]. According to recent findings, the organization of the collagen fibrils in the deep cornea is better modeled by a planar distribution of the fibrils orientation than a fully tridimensional distribution. In the present calculations, though, we kept using the model already employed and validated in previous studies, since here we are interested in the geometrical aspects of the corneal modeling, while the investigation on the material model will be object of an on-going study. In the outermost layer of the cornea model, fibrils are organized according to the model described in [17, 34]. In the central part of the model, fibrils follow the NT and SI orientation, and progressively rotate the orientation while moving towards the limbus, where the main set of fibrils runs circumferentially. A secondary, more compliant, set of fibrils runs in the radial direction to guarantee the correct mechanical behavior of the shell, Fig 8A. Everywhere, the spatial orientation of collagen fibrils follows a transversely isotropic and π-periodic, normalized, von Mises distribution:ρ(Θ)=12πIexp(bcos2Θ),I=1π∫0πexp(bcos2Θ)dΘ,(14)with a concentration parameter b distributed according to Fig 8B.

Bottom Line: Corneal elevation maps of five human eyes were taken with a rotating Scheimpflug camera combined with a Placido disk before and after refractive surgery.Patient-specific solid models were created and discretized in finite elements to estimate the corneal strain and stress fields in preoperative and postoperative configurations and derive the refractive parameters of the cornea.Patient-specific models can be used as indicators of feasibility before performing the surgery.

View Article: PubMed Central - PubMed

Affiliation: Dipartimento di Matematica, Politecnico di Milano, Milano, Italy.

ABSTRACT

Aim: To construct patient-specific solid models of human cornea from ocular topographer data, to increase the accuracy of the biomechanical and optical estimate of the changes in refractive power and stress caused by photorefractive keratectomy (PRK).

Method: Corneal elevation maps of five human eyes were taken with a rotating Scheimpflug camera combined with a Placido disk before and after refractive surgery. Patient-specific solid models were created and discretized in finite elements to estimate the corneal strain and stress fields in preoperative and postoperative configurations and derive the refractive parameters of the cornea.

Results: Patient-specific geometrical models of the cornea allow for the creation of personalized refractive maps at different levels of IOP. Thinned postoperative corneas show a higher stress gradient across the thickness and higher sensitivity of all geometrical and refractive parameters to the fluctuation of the IOP.

Conclusion: Patient-specific numerical models of the cornea can provide accurate quantitative information on the refractive properties of the cornea under different levels of IOP and describe the change of the stress state of the cornea due to refractive surgery (PRK). Patient-specific models can be used as indicators of feasibility before performing the surgery.

No MeSH data available.


Related in: MedlinePlus