Peripapillary and posterior scleral mechanics--part I: development of an anisotropic hyperelastic constitutive model

Abstract

The sclera is the white outer shell and principal load-bearing tissue of the eye as it sustains the intraocular pressure. We have hypothesized that the mechanical properties of the posterior sclera play a significant role in and are altered by the development of glaucoma-an ocular disease manifested by structural damage to the optic nerve head. An anisotropic hyperelastic constitutive model is presented to simulate the mechanical behavior of the posterior sclera under acute elevations of intraocular pressure. The constitutive model is derived from fiber-reinforced composite theory, and incorporates stretch-induced stiffening of the reinforcing collagen fibers. Collagen fiber alignment was assumed to be multidirectional at local material points, confined within the plane tangent to the scleral surface, and described by the semicircular von Mises distribution. The introduction of a model parameter, namely, the fiber concentration factor, was used to control collagen fiber alignment along a preferred fiber orientation. To investigate the effects of scleral collagen fiber alignment on the overall behaviors of the posterior sclera and optic nerve head, finite element simulations of an idealized eye were performed. The four output quantities analyzed were the scleral canal expansion, the scleral canal twist, the posterior scleral canal deformation, and the posterior laminar deformation. A circumferential fiber organization in the sclera restrained scleral canal expansion but created posterior laminar deformation, whereas the opposite was observed with a meridional fiber organization. Additionally, the fiber concentration factor acted as an amplifying parameter on the considered outputs. The present model simulation suggests that the posterior sclera has a large impact on the overall behavior of the optic nerve head. It is therefore primordial to provide accurate mechanical properties for this tissue. In a companion paper (Girard, Downs, Bottlang, Burgoyne, and Suh, 2009, "Peripapillary and Posterior Scleral Mechanics--Part II: Experimental and Inverse Finite Element Characterization," ASME J. Biomech. Eng., 131, p. 051012), we present a method to measure the 3D deformations of monkey posterior sclera and extract mechanical properties based on the proposed constitutive model with an inverse finite element method.

Figure 1

Polar representation of the semi-circular von-Mises distribution describing in-plane collagen fiber alignment. In this case, the preferred fiber orientation θ_p is equal to zero degrees. When the fiber concentration factor k is equal to zero, the collagen fibers have an isotropic distribution in a plane tangent to the scleral wall. As k increases, the collagen fibers align along the preferred fiber orientation θ_p. Note that the distributions were plotted on a circle of unit one to ease visualization.

Figure 2

An idealized FE model of the posterior hemisphere of an eye. The fiber orientation was defined such that θ_p = 0° represents the circumferential orientation (i.e. preferred fiber orientation tangent to the scleral canal boundary) and θ_p = 90° the meridional orientation (i.e. preferred fiber orientation perpendicular to the scleral canal boundary). The ONH is considered as the posterior hemisphere's pole, which includes the lamina cribrosa and retinal ganglion cell axons.

Figure 3

Schematics of the scleral canal and the ONH in the undeformed and deformed configurations to explain how the four output values were calculated. The four output values considered are scleral canal expansion, scleral canal twist, posterior laminar deformation and scleral canal z-displacement.

Figure 4

Model verification for a biaxial extension test on an 8-noded hexahedral element where symmetry conditions were applied. Here, the preferred fiber orientation is aligned along the y-axis and collagen fibers are confined within the xy-plane. When the fiber concentration factor k is equal to zero (isotropy in the xy-plane), Cauchy stresses in x- and y-directions are equal, for both the analytical and finite element solutions. When the fiber concentration factor is large (k ≫ 1), the sclera behave like a transversely isotropic material. Notice the good agreement between the analytical and finite element solutions.

Figure 5

Effects of the preferred fiber orientation θ_p and fiber concentration factor k on scleral canal expansion and posterior laminar deformation at 45 mm Hg. Each curve has a specific k value, where k = 0, 2, or 4 (dashed lines) and k = 1, 3, or 5 (solid lines).

Figure 6

Effects of the preferred fiber orientation θ_p and fiber concentration factor k on scleral canal z-displacement and scleral canal twist at 45 mm Hg. Each curve has a specific k value, where k = 0, 2, or 4 (dashed lines) and k = 1, 3, or 5 (solid lines).

Figure 7

Effects of the preferred fiber orientation θ_p under acute elevation of IOP. Scleral deformations were exaggerated and displayed for three idealized cases with θ_p = 0°, θ_p = 45° and θ_p = 90°, respectively. The fiber concentration factor k was equal to five for all three cases, and deformations were exaggerated ten times to emphasize the scleral deformation patterns observed.