Finite element modeling of the cornea
View Patent ↗A system and method for simulating a corneal reconfiguration in response to LIOB uses a computer-programmed, finite element model. The model has a plurality of elements; with each element pre-programmed with coefficients based on diagnostic corneal data. Collectively the coefficients replicate biomechanical properties of the cornea. In use, designated biomechanical characteristics on a plurality of selected elements (i.e. selected coefficients) are minimized to simulate LIOB in an actual cornea. A computer then measures the resultant reconfiguration of the cornea model to assess an actual cornea's response to LIOB.
1. A system for simulating a reconfiguration of a model cornea which comprises:
a computer programmed with a finite element model, wherein the finite element model defines an anterior surface and a posterior surface for the cornea, with an axis perpendicular to the surfaces and passing through respective apexes of the surfaces, and further wherein the curvatures of the anterior and posterior surfaces are approximated by a respective conic section, and the finite element model includes a plurality of elements, wherein a first group of elements are pre-programmed with coefficients to simulate biomechanical characteristics for Bowman's capsule of the cornea and a second group of elements are pre-programmed with coefficients to simulate biomechanical characteristics for a stroma of the cornea, and further wherein the pre-programmed elements of the first and second groups are established for the finite element model according to diagnostic corneal data;
a computer means for reducing coefficients of the finite element model into a range approximately eighty to ninety-five percent of the preprogrammed value to minimize designated biomechanical characteristics on at least one selected element to create a reconfigured model cornea; and
a computer means for measuring the reconfigured model cornea.
2. A system as recited in claim 1 wherein the conic section for each surface is expressed as:
z
(
x
)
=
1
ⅇ
2
-
1
[
R
2
+
x
2
(
ⅇ
2
-
1
)
-
R
]
3. A system as recited in claim 2 where:
R for the anterior surface is approximately 7.86 mm;
R for the posterior surface is approximately 6.76 mm; and
e for the eccentricity of the cornea is 0.32.
4. A system as recited in claim 1 wherein each element is three dimensional and includes tensors having stress-strain coefficients.
5. A system as recited in claim 1 wherein the finite element model defines nine layers of elements wherein three of the layers include the first group of elements, and six of the layers include the second group of elements.
6. A system as recited in claim 1 wherein the first computer means simulates a cut inside the stroma, substantially parallel to the axis.
7. A system as recited in claim 1 wherein the first computer means simulates a cut inside the stroma, substantially perpendicular to the axis.
8. A system as recited in claim 1 wherein the finite element model is axisymmetric and is based on a nonlinearly elastic, slightly compressible, transversely isotropic formulation with an isotropic exponential Lagrangian strain-energy function based on:
W =½ C ( e Q −1)+ C compr ( I 3 InI 3 −I 3 +1)
and
Q=b ff E 2 ff +b xx ( E 2 cc +E 2 ss +E 2 cs +E 2 sc )+ b fx ( E 2 fc +E 2 cf +E 2 fs +E 2 sf )
Where:
I are invariants,
W is the strain potential (strain-energy function),
C is stress-scaling coefficient,
C compr is bulk modulus (kPa),
E is strain,
b ff is fiber strain exponent,
b xx is transverse strain component, and
b fx is fiber-transverse shear exponent.
9. A system as recited in claim 8 wherein the stress-scaling coefficient for Bowman's capsule (C Bowman ) is approximately five times greater than the stress-scaling coefficient for the stroma (C stroma ).
10. A system for simulating a reconfiguration of a model cornea which comprises:
a finite element model having a plurality of individual elements, wherein each element is pre-programmed with coefficients for biomechanical characteristics based on diagnostic corneal data to collectively replicate biomechanical properties of the cornea, and to represent the cornea in a first configuration wherein a first group of elements are pre-programmed elements to simulate biomechanical characteristics for Bowman's capsule of the cornea and a second group of elements are pre-programmed elements to simulate biomechanical characteristics for a stroma of the cornea, and further wherein the finite element model defines an anterior surface and a posterior surface for the cornea, with an axis perpendicular to the surfaces and passing through respective apexes of the surfaces, and further wherein the curvatures of the anterior and posterior surfaces are approximated by a respective conic section expressed as:
z
(
x
)
=
1
ⅇ
2
-
1
[
R
2
+
x
2
(
ⅇ
2
-
1
)
-
R
]
.
Where:
R for the anterior surface is approximately 7.86mm;
R for the posterior surface is approximately 6.76mm;
e for the eccentricity of the cornea is 0.32;
a computer means for reducing the coefficients of the finite element model into a range approximately eighty to ninety-five percent of the preprogrammed value to minimize designated biomechanical characteristics on a plurality of selected elements to create a reconfigured model cornea; and
a computer means for measuring the reconfigured model cornea.
11. A system as recited in claim 10 wherein the finite element model is axisymmetric and is based on a nonlinearly elastic, slightly compressible, transversely isotropic formulation with an isotropic exponential Lagrangian strain-energy function based on:
W =½ C ( e Q −1)+ C compr ( I 3 InI 3 −I 3 =1)
and
Q=b ff E 2 ff +b xx ( E 2 cc +E 2 ss +E 2 cs +E 2 sc )+ b fx ( E 2 fc +E 2 cf +E 2 fs +E 2 sf )
Where:
I are invariants,
W is the strain potential (strain-energy function),
C is stress-scaling coefficient,
C compr is bulk modulus (kPa),
E is strain,
b ff is fiber strain exponent,
b xx is transverse strain component,
b fx is fiber-transverse shear exponent, and
wherein the stress-scaling coefficient for Bowman's capsule (C Bowman ) is approximately five time greater than the stress-scaling coefficient for the stroma (C stroma ).
12. A system as recited in claim 10 wherein the finite element model defines nine layers of elements wherein three of the layers include the first group of elements and six layers include the second group of elements.
13. A system as recited in claim 10 wherein elements to be minimized are selected from the second group.
14. A method for using a computer to simulate a reconfiguration of a cornea which comprises the steps of:
creating a finite element model comprising a plurality of elements;
pre-programming the computer with a first plurality of the elements with coefficients to simulate biomechanical characteristics for Bowman's capsule of the cornea and a second plurality of the elements with coefficients to simulate biomechanical characteristics for a stroma of the cornea, wherein the pre-programmed elements are established for the finite element model according to diagnostic corneal data;
minimizing biomechanical characteristics on selected elements by reducing coefficients into a range approximately eighty to ninety-five percent of the preprogrammed value to create a reconfigured cornea;
measuring the reconfigured cornea in response to the minimizing step; and
repeating the minimizing and measuring steps, as required.
15. A method as recited in claim 14 wherein the finite element model defines an anterior surface and a posterior surface for the cornea, with an axis perpendicular to the surfaces and passing through respective apexes of the surfaces, and further wherein the curvatures of the anterior and posterior surfaces are approximated by a respective conic section expressed as:
z
(
x
)
=
1
e
2
-
1
[
R
2
+
x
2
(
e
2
-
1
)
-
R
]
.