Biomechanical properties of the rat sclera obtained with inverse finite element modeling

Abstract

It is widely accepted that biomechanics plays an important role in glaucoma pathophysiology, but the mechanisms involved are largely unknown. Rats are a common animal model of glaucoma, and finite element models are being developed to provide much-needed insight into the biomechanical environment of the posterior rat eye. However, material properties of rat ocular tissues, including the sclera, are currently unknown. Since the sclera plays a major role in posterior ocular biomechanics, our goal was to use inverse finite element modeling to extract rat scleral material properties. We first used digital image correlation to measure scleral surface displacement during whole-globe inflation testing. We modeled the sclera as a nonlinear material with embedded collagen fibers and then fit modeled displacements to experimental data using a differential evolution algorithm. Subject-specific models were constructed in which 3 parameters described the stiffness of the ground substance and collagen fibers in the posterior eye, and 16 parameters defined the primary orientation and alignment of fibers within eight scleral sub-regions. We successfully extracted scleral material properties for eight rat eyes. Model displacements recreated general patterns of the experimental displacements but did not always match local patterns. The fiber directions and fiber concentration parameters were highly variable, but on average, fibers were aligned circumferentially and were more aligned in the peripapillary sclera than in the peripheral sclera. The material properties determined here will be used to inform future finite element models of the rat posterior eye with the goal of elucidating the role of biomechanics in glaucoma pathophysiology.

Appendix: Relationship between fiber concentration factor (\(k_{\text{f}}\)) and degree of alignment

We wished to compare our extracted $k_{\text{f}}$ values with the experimental data on degree of fiber alignment (DA) reported by Girard et al. (2011a) in the rat sclera, for which it was necessary to relate $k_{\text{f}}$ and DA. The DA parameter reported by Girard et al. (2011a) is defined by (note typographic error in that publication)

$${\text{DA}} = 1 - \frac{\text{OI}}{{{\raise0.7ex\hbox{$\pi $} \!\mathord{\left/ {\vphantom {\pi 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}}}$$

(10)

where the orientation index (Sacks et al. 1997), OI, is defined as

$$\int_{{\theta_{\text{p}} - {\text{OI}}}}^{{\theta_{\text{p}} + {\text{OI}}}} P\left( \theta \right){\text{d}}\theta = \frac{1}{2}$$

(11)

where $P\left( \theta \right)$ is the probability density function of fiber directions and satisfies

$$\int_{ - \pi /2}^{\pi /2} P\left( \theta \right){\text{d}}\theta = 1.$$

(12)

In other words, half of all fiber directions lie within − OI to + OI about a preferred direction, $\theta_{\text{p}}$. For DA, a value of 0 means that fiber direction is isotropic within a plane, and a value of 1 means that all fibers have an orientation of $\theta_{\text{p}}$. The von Mises distribution is defined as

$$P\left( \theta \right) = \frac{1}{{\pi I_{0} \left( {k_{\text{f}} } \right)}}\exp \left[ {k_{\text{f}} \cos 2\left( {\theta - \theta_{\text{p}} } \right)} \right]$$

(13)

where $I_{0}$ is the modified Bessel function of the first kind of order zero and $k_{\text{f}}$ is the fiber concentration factor. By substituting Eq. 13 into Eq. 11, and substituting $\varphi = \theta_{\text{p}} - \theta$, we obtain

$$\frac{ - 1}{{\pi I_{0} \left( {k_{\text{f}} } \right)}} = \int_{\text{OI}}^{{ - {\text{OI}}}} \exp \left( {k_{\text{f}} \cos 2\varphi } \right){\text{d}}\varphi = \frac{1}{2}$$

(14)

which can be rearranged to obtain an implicit relationship between $k_{\text{f}}$ and OI

$$\int_{0}^{\text{OI}} { \exp }\left( {k_{\text{f}} \cos 2\varphi } \right){\text{d}}\varphi = \frac{\pi }{4}I_{0} \left( {k_{\text{f}} } \right).$$

(15)

Although obtaining an analytical solution to Eq. 15 is difficult, one can iteratively find the value of OI that satisfies Eq. 15 for a given $k_{\text{f}}$ by plugging a value for $k_{\text{f}}$ into the term on the right and numerically evaluating the integral on the left with different values for OI until the equation is satisfied.