An Improved Method to Analyze the Stress Relaxation of Ligaments Following a Finite Ramp Time Based on the Quasi-Linear Viscoelastic Theory

Abstract

The quasi-linear viscoelastic (QLV) theory proposed by Fung (1972) has been frequently used to model the nonlinear time- and history-dependent viscoelastic behavior of many soft tissues. It is common to use five constants to describe the instantaneous elastic response (constants A and B) and reduced relaxation function (constants C, τ1, and τ2) on experiments with finite ramp times followed by stress relaxation to equilibrium. However, a limitation is that the theory is based on a step change in strain which is not possible to perform experimentally. Accounting for this limitation may result in regression algorithms that converge poorly and yield nonunique solutions with highly variable constants, especially for long ramp times (Kwan et al. 1993). The goal of the present study was to introduce an improved approach to obtain the constants for QLV theory that converges to a unique solution with minimal variability. Six goat femur-medial collateral ligament-tibia complexes were subjected to a uniaxial tension test (ramp time of 18.4 s) followed by one hour of stress relaxation. The convoluted QLV constitutive equation was simultaneously curve-fit to the ramping and relaxation portions of the data r2>0.99. Confidence intervals of the constants were generated from a bootstrapping analysis and revealed that constants were distributed within 1% of their median values. For validation, the determined constants were used to predict peak stresses from a separate cyclic stress relaxation test with averaged errors across all specimens measuring less than 6.3±6.0% of the experimental values. For comparison, an analysis that assumed an instantaneous ramp time was also performed and the constants obtained for the two approaches were compared. Significant differences were observed for constants B, C, τ1, and τ2, with τ1 differing by an order of magnitude. By taking into account the ramping phase of the experiment, the approach allows for viscoelastic properties to be determined independent of the strain rate applied. Thus, the results obtained from different laboratories and from different tissues may be compared.