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TECHNICAL PAPERS: Soft Tissue

# A Cellular Solid Model of the Lamina Cribrosa: Mechanical Dependence on Morphology

[+] Author and Article Information
E. A. Sander

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 and Department of Biomedical Engineering, Tulane University, New Orleans, LA 70118

J. C. Downs, C. F. Burgoyne

Devers Eye Institute, Portland, OR 97232

R. T. Hart

Department of Biomedical Engineering, Tulane University, New Orleans, LA 70118

E. A. Nauman1

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, Department of Biomedical Engineering, Tulane University, New Orleans, LA 70118, and Weldon School of Biomedical Engineering, Purdue University, West Lafayette, IN 47907enauman@purdue.edu

1

Author to whom all correspondence should be addressed, 585 Purdue Mall, West Lafayette, IN 47907-2088.

J Biomech Eng 128(6), 879-889 (Jun 16, 2006) (11 pages) doi:10.1115/1.2354199 History: Received January 04, 2005; Revised June 16, 2006

## Abstract

The biomechanics of the optic nerve head (ONH) may underlie many of the potential mechanisms that initiate the characteristic vision loss associated with primary open angle glaucoma. Therefore, it is important to characterize the physiological levels of stress and strain in the ONH and how they may change in relation to material properties, geometry, and microstructure of the tissue. An idealized, analytical microstructural model of the ONH load bearing tissues was developed based on an octagonal cellular solid that matched the porosity and pore area of morphological data from the lamina cribrosa (LC). A complex variable method for plane stress was applied to relate the geometrically dependent macroscale loads in the sclera to the microstructure of the LC, and the effect of different geometric parameters, including scleral canal eccentricity and laminar and scleral thickness, was examined. The transmission of macroscale load in the LC to the laminar microstructure resulted in stress amplifications between 2.8 and $24.5×IOP$. The most important determinants of the LC strain were those properties pertaining to the sclera and included Young’s modulus, thickness, and scleral canal eccentricity. Much larger strains were developed perpendicular to the major axis of an elliptical canal than in a circular canal. Average strain levels as high as 5% were obtained for an increase in IOP from $15to50mm$ Hg.

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## Figures

Figure 1

The conformal mapping of an ellipse in the z plane to the unit circle in the ζ plane. The region bounded by the ellipse represents the lamina cribrosa and is mapped onto the region λ1∕2<∣ζ∣<1 on the unit circle. The ellipse also contains two critical points: z=±2cλ1∕2 that form the inner annulus and are not part of the solution. The region outside the ellipse represents the sclera and maps to the outside of the unit circle. The outward normal, n̂, makes an angle η with the x axis and an angle θ with the real axis. Figure adapted from Hardiman (18).

Figure 2

The geometry of a regular octagonal unit cell. (a) The lamina cribrosa is idealized as a repeating array of unit cells. (b) The unit cell is composed of plates of thickness, t, depth, hLC, and either length, L, or, 1∕2L. The angle θ is 45°. Macroscale (M) stresses are applied to the faces of the unit cell. (c) Macroscale stresses are translated to microscale (m) stresses in the unit cell. (d) The unit cell symmetry allows a simple deconstruction and analysis of the microscale mechanics. Because the stresses in E1 and E2 are much greater than the IOP in E3, loading in the anterior-posterior direction is neglected. (a) The pictured LC has a vertical diameter of 1.6mm and the expanded area has a length of 243μm. Adapted from Quigley (27).

Figure 3

(a) The macroscale modulus,EM, was normalized by the unit cell microstructural modulus,Em, and decreases with increasing porosity. (b) The macroscale Poisson’s ratio, νM, increases with increasing porosity. The horizontal (c) and vertical (d) macroscale stress in the lamina cribrosa, normalized by IOP, versus unit cell porosity. The macroscale stresses decrease with increasing porosity because the macroscale modulus also decreases. Stress in the E1 direction, which is perpendicular to the major axis of scleral canal, increases with increasing eccentricity for a given porosity. Stress in the E2 direction at low- to mid-porosities decreased. When a∕b=1, the stress is equivalent in both directions. Stress also increases as the microstructural modulus of the unit cell increases and the lamina cribrosa becomes stiffer relative to the sclera. The value of Es was maintained at 4100kPa and Em altered accordingly.

Figure 4

Effect of α on the macroscale stress. For Es∕Em=1, changes in α, the ratio of the laminar thickness over the scleral thickness, shift the macroscale stress accordingly. (a) and (b) The scleral thickness remains constant, hs=0.95mm, while the laminar thickness is varied. σ1M increases the smaller this ratio becomes and with increasing eccentricity. σ2M also increases the smaller α becomes but is reduced with increasing eccentricity. (c) and (d) The scleral thickness is varied while the laminar thickness remains constant, hLC=0.3167mm. For this case, decreasing α affects the macroscale stresses differently; increasing the scleral thickness reduces the macroscale stress. This occurs because the scleral stress decreases with increasing scleral thickness (law of Laplace).

Figure 5

Microscale stress dependence on porosity, canal eccentricity, and relative stiffness. The microscale stress in both the horizontal (a) and vertical (b) plates is strongly affected by eccentricity and increases with an increasing microscale modulus. For the most part, stresses also increase with porosity, except for σ1m, which slightly decreases when the canal is eccentric and the scleral modulus is either equal to or less than the microstructural modulus. (c) The highest stress occurs in the oblique plates and is a combination of the axial stress and the bending stress at the plate surface. No bending occurs when a∕b=1 (circle). (d) Shear stresses develop when the scleral canal deviates from a circle. Solid lines −a∕b=2. Dotted line a∕b=1. Note the difference in scales on (c) and (d).

Figure 6

Macroscale strain developed from a change in IOP from 15to50mm Hg. Unit cell strain was dependent on porosity, canal eccentricity, and microscale modulus. Higher strains occur in the E1 direction when the scleral canal deviates from a circle.

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