Application of a Probabilistic Microstructural Model to Determine Reference Length and Toe-to-Linear Region Transition in Fibrous Connective Tissue

Abstract

This study shows how a probabilistic microstructural model for fibrous connective tissue behavior can be used to objectively describe soft tissue low-load behavior. More specifically, methods to determine tissue reference length and the transition from the strain-stiffening “toe-region” to the more linear region of the stress-strain curve of fibrous connective tissues are presented. According to a microstructural model for uniaxially loaded collagenous tissues, increasingly more fibers are recruited and bear load with increased tissue elongation. Fiber recruitment is represented statistically according to a Weibull probability density function (PDF). The Weibull PDF location parameter in this formulation corresponds to the stretch at which the first fibers begin to bear load and provides a convenient method of determining reference length. The toe-to-linear region transition is defined by utilizing the Weibull cumulative distribution function (CDF) which relates the fraction of loaded fibers to the tissue elongation. These techniques are illustrated using representative tendon and ligament data from the literature, and are shown to be applicable retrospectively to data from specimens that are not heavily preloaded. The reference length resulting from this technique provides an objective datum from which to calculate stretch, strain, and tangent modulus, while the Weibull CDF provides an objective parameter with which to characterize the limits of low-load behavior.