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Glaukom und das menschliche Auge als biomechanische Mehrskalenstruktur

R. Grytz und G. Meschke

Motivation and background

This research project is concerned with the ocular disease glaucoma, which is the second leading cause of blindness in the world. Glaucoma is characterized by a progressive, irreversible loss of optic nerve cells, which are responsible for carrying visual information from the retina to the brain. An elevated intraocular pressure level is the most relevant risk factor for the development and progression of glaucoma. The mechanism whereby nerve cells die is still unknown. However, it is known that cell death is initiated at the level of the lamina cribrosa, which is characterized by a complex collagen network with several openings through which axon bundles pass. A leading hypothesis relates mechanical induced micro-structural changes of the lamina cribrosa to interruptions of the axoplasmatic flow within the axons. The biomechanical environment of the micro-structure at the lamina cribrosa may play a significant role in retinal ganglion cell loss in glaucomatous optic neuropathy. A realistic biomechanical simulation of the human eye structure at multiple scales opens a promising perspective to predict a certain risk for the development of glaucoma.

Anatomy of the human eye (top); sieve-like collagen network of the lamina cribrosa (bottom).

Computerbasierte Homogenisierung.

Numerical multi-scale simulations of the human eye

The biological tissues existing in the human eye such as the sclera, the cornea and the lamina cribrosa are characterized by heterogeneities on one or another spatial scale and can undergo very large elastic strains. These tissues constitute shell-like structures at the macroscopic scale, where the physical material directions follow curvilinear paths. The use of computational homogenization schemes together with a formulation of the continua in curvilinear coordinates is a prerequisite for realistic biomechanical multi-scale simulations of shell-like soft tissues. The solution of this type of multi-scale problems within the framework of computational homogenization schemes based upon the same physical space at both scales would need different representative volume elements (RVEs) at each macroscopic point. In contrast, introducing different physical spaces at micro- and macro-scale the same initial RVE can be used for every macroscopic point. This strategy is employed in this project. In this case, however, the rotation between both spaces has to be considered when transferring tensor variables from one scale to another.

Constitutive framework for crimped collagen fibrils in human eye tissues

Organized collagen fibrils form fibrous networks that introduce strong anisotropic and highly nonlinear attributes into the constitutive response of eye tissues. In order to provide reliable biomechanical simulations of the human eye the goal is to incorporate micro-structural information into a hyperelastic constitutive formulation for crimped collagen fibrils. The model is based on observations that collagen fibrils embedded in a soft matrix crimp into a smooth 3D pattern when unloaded. The presented model is derived from the nonlinear axial force-stretch relationship of an extensible helical spring including the fully extension of the spring as a limit case. This helical spring model is introduced as a fiber-matrix constitutive formulation into an incompressible finite shell element considering statistical distributed collagen fibril orientations. The incompressibility constraint is enforced through elimination of displacement and strain variables. The ability of the present numerical model to reproduce the biomechanical response of individual human eye shells to different intraocular pressure levels opens a promising perspective to predict a certain risk for the development of glaucoma.

A fit of the helical spring model to data taken from strip extensiometry experiments on porcine corneas.

Deformations (5x exaggerated) of the human eye model under intraocular pressure (40 mmHG) with two- (left) and three-dimensional distributed collagen fibril orientations (right).

In-plane shear stresses of the lamina cribrosa for intraocular pressure (40 mmHG) with two- (left) and three-dimensional distributed collagen fibril orientations (right).