Peripapillary and posterior scleral mechanics--part II: experimental and inverse finite element characterization

Abstract

The posterior sclera likely plays an important role in the development of glaucoma, and accurate characterization of its mechanical properties is needed to understand its impact on the more delicate optic nerve head--the primary site of damage in the disease. The posterior scleral shells from both eyes of one rhesus monkey were individually mounted on a custom-built pressurization apparatus. Intraocular pressure was incrementally increased from 5 mm Hg to 45 mm Hg, and the 3D displacements were measured using electronic speckle pattern interferometry. Finite element meshes of each posterior scleral shell were reconstructed from data generated by a 3D digitizer arm (shape) and a 20 MHz ultrasound transducer (thickness). An anisotropic hyperelastic constitutive model described in a companion paper (Girard, Downs, Burgoyne, and Suh, 2009, "Peripapillary and Posterior Scleral Mechanics--Part I: Development of an Anisotropic Hyperelastic Constitutive Model," ASME J. Biomech. Eng., 131, p. 051011), which includes stretch-induced stiffening and multidirectional alignment of the collagen fibers, was applied to each reconstructed mesh. Surface node displacements of each model were fitted to the experimental displacements using an inverse finite element method, which estimated a unique set of 13 model parameters. The predictions of the proposed constitutive model matched the 3D experimental displacements well. In both eyes, the tangent modulus increased dramatically with IOP, which indicates that the sclera is mechanically nonlinear. The sclera adjacent to the optic nerve head, known as the peripapillary sclera, was thickest and exhibited the lowest tangent modulus, which might have contributed to the uniform distribution of the structural stiffness for each entire scleral shell. Posterior scleral deformation following acute IOP elevations appears to be nonlinear and governed by the underlying scleral collagen microstructure as predicted by finite element modeling. The method is currently being used to characterize posterior scleral mechanics in normal (young and old), early, and moderately glaucomatous monkey eyes.

Figure 1

Schematic showing a cross-section of the custom-built pressurization apparatus. The posterior scleral shell was first mounted onto the plastic ring, and then clamped slightly above the equator by moving the vertical stage toward the clamping stage. PBS outflow was interrupted after PBS filled the posterior shell cavity and IOP reached 5 mm Hg. The scleral surface was imaged as IOP was increased from 5 to 45 mm Hg.

Figure 2

Example of raw speckle fringes (one of four illumination directions from the ESPI sensor) on a rubber balloon for an IOP increase from 10 to 10.2 mm Hg. The distance between the fringes is proportional to the 3D displacement of the balloon surface.

Figure 3

(A) Location of the twenty thickness measurement sites, shown as black dots on the outer surface of the posterior sclera. For each location, the 20 MHz ultrasound transducer was positioned perpendicularly to the scleral surface, and the corresponding voltage echo signal was recorded in order to extract scleral thickness. (B) An example of a voltage echo signal obtained from the oscilloscope for one measurement site. $\frac{1}{2}{t}_i$is the time taken by an acoustic wave to travel from the transducer/sclera interface to the sclera/air interface.

Figure 4

(A) FE model of the monkey posterior scleral shell of a left eye showing one regionalization pattern. Regions (1–4) are the peripheral sclera, regions (5–8) are the peripapillary sclera, and region (9) is the ONH. (B) Same FE mesh as (A), but with a different regionalization pattern. S, superior; N, nasal; I, inferior; T, temporal.

Figure 5

One sub-region of the peripapillary sclera of a left monkey eye, showing how collagen fiber alignment was defined. For each hexahedral element, the local unit vector i was constructed with element edge information. The local unit vector j was constructed as being perpendicular to the unit vector i and tangent to the scleral surface. Unit vectors i and j define the plane in which the collagen fibers lie. Although each element of this sub-region share the same preferred fiber orientation θ_p, the unit vector associated with θ_p will be different for each element in the global coordinate system. For example, if the preferred fiber orientation is equal to zero for this sub-region (θ_p = 0°), collagen fibers will be oriented along the unit vector i (for k ≠ 0), corresponding to a circumferential organization.

Figure 6

Comparison of experimentally-measured (exp) and model-predicted (mod) displacements (in µm) for both eyes for three IOP ranges (5 to 10, 10 to 30, and 30 to 45 mm Hg). Model displacements were simultaneously fitted to the experimental data obtained at 7, 10, 20, 30 and 45 mm Hg. Both experimental and model displacements are small for the 30–45 mm Hg IOP range, demonstrating a high degree of nonlinearity.

Figure 7

The main modeling results are shown as color maps for both eyes at an IOP of 30 mm Hg. Scleral thickness maps were derived from experimental ultrasound measurements at 5 mm Hg, and show that the peripapillary sclera is much thicker than the peripheral sclera. An inverse relationship between scleral thickness and scleral tangent modulus ${\mathrm{𝕔}}_{1111}^{\prime }$was observed, indicating that thinner sclera is likely to be associated with a higher tangent modulus. The structural stiffness along the preferred fiber orientation, S₁ , helps visualize this inverse relationship. Finally, maximum principal strain was concentrated around the scleral canal.

Figure 8

Tangent Modulus ${\mathrm{𝕔}}_{1111}^{\prime }$versus scleral thickness for both eyes at an IOP of 30 mm Hg, which illustrates the concept of inverse relationship between two quantities. A thin sclera (the peripheral sclera) has a tendency to be associated with larger${\mathrm{𝕔}}_{1111}^{\prime }$, and a thick sclera (the peripapillary sclera) with smaller${\mathrm{𝕔}}_{1111}^{\prime }$. Notice small R² values obtained from linear regression analyses, possibly due to the scatteredness of the data points.