The role of 3-D collagen organization in stromal elasticity: a model based on X-ray diffraction data and second harmonic-generated images

Abstract

Examining the cross-section of the human cornea with second harmonic-generated (SHG) imaging shows that many lamellae do not lie parallel to the cornea's anterior surface but have inclined trajectories that take them through the corneal thickness with a depth-dependent distribution. A continuum mechanics-based model of stromal elasticity is developed based on orientation information extracted and synthesized from both X-ray scattering studies and SHG imaging. The model describes the effects of inclined lamella orientation by introducing a probability function that varies with depth through the stroma, which characterizes the range and distribution of lamellae at inclined angles. When combined with the preferred lamellar orientations found from X-ray scattering experiments, a fully 3-D representation of lamella orientation is achieved. Stromal elasticity is calculated by a weighted average of individual lamella properties based on the spatially varying 3-D orientation distribution. The model is calibrated with in vitro torsional shear experiments and in vivo indentation data and then validated with an in vitro inflation study. A quantitative explanation of the experimentally measured depth dependence of mechanical properties emerges from the model. The significance of the 3-D lamella orientation in the mechanics of the human cornea is demonstrated by investigating and contrasting the effects of previous modeling assumptions made on lamella orientation.

Collagen Structure and Mechanical Properties of the Human Sclera: Analysis for the Effects of Age

“…In these respects, WAXS presents a highly quantitative yet simplified view of the full tissue architecture. Recently, Petsche and Pinsky [62] modeled the 3D organization of the comeal stroma fiber structure by introducing a 3D distribution function. The spatial distribution of fibers was calibrated to out-of-plane shear measurements that are not available for scleral tissue.…”

A structural model for thein vivohuman cornea including collagen-swelling interaction

“…The form of W fibre ðC 0 Þ is based on the use of angular integration [3,59] where v is the unit sphere, r(X, A) is the distribution of fibre directions, n ¼ Ð v rðX, AÞdv is the normalization and the fibre stretching energy W stretch ðl 0 A Þ is described below. The function r(X, A) may be based on fibre orientation information obtained from X-ray diffraction and SHG imaging as described in [6]. A fibre direction A can be expressed in spherical coordinates with azimuthal angle u [ ½0, 2p and inclination angle f [ ½0, p: As the anterior and posterior surfaces have different curvatures, a local coordinate system is introduced that varies pointwise through the corneal thickness and matches both the anterior and posterior curvatures at those limits; details can be found in [6].…”

“…Following [6], we have eliminated the need to introduce analytical distribution functions and directly employ the raw X-ray data. These data provide scattering intensities versus orientation on a discrete grid of points over the cornea's anterior surface.…”

“…However, no fully three-dimensional and comprehensive model has yet been presented. It is noteworthy that current finite-element-based models for structural analysis of the cornea treat the interfibrillar fluid as an incompressible or nearly incompressible elastic solid [3][4][5][6]. While this approach is convenient, it cannot describe swelling behaviour or (steady-state) bulk compressibility.…”

On three- and two-dimensional fiber distributed models of biological tissues

“…In several tissues geometrically organized as structures, e.g., skin, corneas, irides, artery walls, and other shell-like or membrane like organs, the distribution of the collagen fibers assumes a prevailing two-dimensional configuration [2,20]. The attractive possibility of using membrane and shell theories to reduce the computational effort in view of evaluating the mechanical behavior of organs calls for the development of constitutive models characterized by planarity of the fiber micro-structural organization, see, e.g., [21,24].…”