A Structurally Based Stress-Stretch Relationship for Tendon and Ligament

Abstract

We propose a mechanical model for tendon or ligament stress-stretch behavior that includes both microstructural and tissue level aspects of the structural hierarchy in its formulation. At the microstructural scale, a constitutive law for collagen fibers is derived based on a strain-energy formulation. The three-dimensional orientation and deformation of the collagen fibrils that aggregate to form fibers are taken into consideration. Fibril orientation is represented by a probability distribution function that is axisymmetric with respect to the fiber. Fiber deformation is assumed to be incompressible and axisymmetric. The matrix is assumed to contribute to stress only through a constant hydrostatic pressure term. At the tissue level, an average stress versus stretch relation is computed by assuming a statistical distribution for fiber straightening during tissue loading. Fiber straightening stretch is assumed to be distributed according to a Weibull probability distribution function. The resulting comprehensive stress-stretch law includes seven parameters, which represent structural and microstructural organization, fibril elasticity, as well as a failure criterion. The failure criterion is stretch based. It is applied at the fibril level for disorganized tissues but can be applied more simply at a fiber level for well-organized tissues with effectively parallel fibrils. The influence of these seven parameters on tissue stress-stretch response is discussed and a simplified form of the model is shown to characterize the nonlinear experimentally determined response of healing medial collateral ligaments. In addition, microstructural fibril organizational data (Frank et al., 1991, 1992) are used to demonstrate how fibril organization affects material stiffness according to the formulation. A simplified form, assuming a linearly elastic fiber stress versus stretch relationship, is shown to be useful for quantifying experimentally determined nonlinear toe-in and failure behavior of tendons and ligaments. We believe this ligament and tendon stress-stretch law can be useful in the elucidation of the complex relationships between collagen structure, fibril elasticity, and mechanical response.

Anisotropy in hypoelastic soft-tissue mechanics, I: Theory

“…The use of a distribution density function for the handling of fiber splay in connective tissues was introduced by Lanir [1983]. Since then single-family fiber splays have been adopted by Hurschler et al [1997] for tendon and ligament, Billiar and Sacks [2000] and Freed et al [2005] for heart valves, and Usyk et al [2000] for ventricular myofibers; double-family fiber splays have been adopted by Gasser et al [2006] for arteries; and triple-family fiber splays have been adopted by Nash and Hunter [2000] for myocardial laminae. In the papers by Freed et al [2005] and Gasser et al [2006], the distribution function describing anisotropy was extracted from the overall constitutive formulation yielding a material or structural tensor.…”

A constitutive model for fibrous tissues considering collagen fiber crimp

“…The basic approach proposed in this work is similar to that of others; see, e.g., [19,23,21,36]. However, the underlying tissue characterization is different since the statistical distribution can be measured [30].…”

“…Among these, a number of structural models have been proposed that characterize the fiber recruitment process by assuming that either the fiber length, [19,38,39,22,36], or the stretch at which the fibers engage [23,21] are statistically distributed. These models use unbounded distributions, which seem non-physiological for quantities such as the fiber length and the stretch.…”

“…Their wavy appearance in the reference configuration motivates the assumption that they are unable to sustain compressive loads. For the modeling of the tensile behavior, we follow the work of others and consider that a given fiber carries load only after unfolding, assuming that the force necessary to perform this 5 A c c e p t e d m a n u s c r i p t is negligible [22,23]. Once the fiber starts to bear load, it is assumed that it follows Hooke's law until rupture [31].…”

“…There is, thus, a distribution in either the stretch of the fibers or their lengths. This idea has also been adopted subsequently by means of constitutive models to describe the mechanical response of, e.g., arterial walls ( [21] with ideas from [22,14]) or tendons and ligaments [23], just to name a few. All these constitutive models, however, assume unbounded statistical distributions for the fiber length (or stretch), which is a bounded quantity.…”