Viscoelastic Models for Ligaments and Tendons

Collagenous tissues such as ligaments and tendons are viscoelastic materials. They exhibit a slow continuous increase in strain over time, or creep, when subjected to a constant stress and a slow continuous decrease in stress over time, or stress relaxation, when subjected to a constant strain. Moreover, the loading and unloading stress-strain curves are different when the tissues are subjected to cyclic loading, showing hysteresis and softening phenomena. The micro-structural origin of the viscoelasticity of these tissues is still unknown and the subject of debate among experts in biomechanics. Therefore, formulating viscoelastic models by accounting for the mechanical contributions of the structural components of these tissues can help in understanding the genesis of viscoelasticity.

A nonlinear viscoelastic modeling framework has been developed to describe the elastic and viscoelastic properties of ligaments and tendons by considering their main structural components, the collagen fibers and proteoglycan-rich matrix. The mathematical models derived within this framework can illustrate the tensile behavior, stress relaxation and creep by as suming that the collagen fibers are elastic and the surrounding proteoglycan-rich matrix is viscoelastic. The collagen fibers are represented by linear elastic springs that are engaged to support load at different values of the tissue's strain according to a Weibull distribution function. The mechanical contribution of the matrix is introduced via a Maxwell-type viscoelastic element arranged in parallel with the collagen fibers. According to the proposed mathematical framework, both the collagen fibers and the proteoglycan-rich matrix are responsible for resisting tensile loads. However, the collagen fibers play a significant role in creep while the proteoglycan-rich matrix has a dominant role in stress relaxation. The model parameters that define the stress relaxation and strain stiffening phenomena are estimated by using published experimental on rabbit medial collateral ligaments and are then used to predict creep.

The above modeling framework has been also extended to capture the in uence of preconditioning on the mechanical properties of ligaments and tendons. The stress softening and decrease in hysteresis that are observed during successive loading cycles in preconditioning are assumed to be determined by a decrease in the elastic properties of the collagen fibers and proteoglycan-rich matrix. Preliminary data collected on stress relaxation and preconditioning on rat medial collateral ligaments by collaborators are used to evaluate the model parameters and analyze its predictions.

The elastic and viscoelastic properties of single collagen fibers are studied by formulating a nonlinear viscoelastic framework by accounting for their main components: microfibrils, cross-links and proteoglycan-rich matrix. The model illustrates tensile behavior and stress relaxation of a single collagen fiber by assuming that the microfibrils and the cross-links are elastic and the surrounding proteoglycan-rich matrix is viscoelastic. The mechanical contribution of the microfibrils is included via a linear elastic spring while the cross-links are represented by linear elastic springs that progressively fail at different values of the tissue's strain according to an exponential distribution function. The matrix is defined by linear dashpots arranged in parallel with each single spring that represents an individual cross-link. The viscous properties of the matrix associated with the unbroken and broken cross-links are assumed to have different values. In the model formulation, the microfibrils and the cross-links are assumed to determine the elastic response of the fibers while the proteoglycan-rich matrix determines the stress relaxation. Microfibrils, cross-links and the proteoglycan-rich matrix are responsible for resisting the loading force during tensile behavior. Experimental data collected by performing incremental stress relaxation tests by other investigators on reconstituted rat tail tendons are used to estimate the parameters in the model and evaluate its performance.