Viscoelastic relaxation spectra are essential for predicting and interpreting the mechanical responses of materials and structures. For biological tissues, these spectra must usually be estimated from viscoelastic relaxation tests. Interpreting viscoelastic relaxation tests is challenging because the inverse problem is expensive computationally. We present here (1) an efficient algorithm and (2) a quasi-linear model that enable rapid identification of the viscoelastic relaxation spectra of both linear and nonlinear materials. We then apply these methods to develop fundamental insight into the mechanics of collagenous and fibrotic tissues.
The first algorithm, which we term the discrete spectral approach, is fast enough to yield a discrete spectrum of time constants that is sufficient to fit a measured relaxation spectrum with an accuracy insensitive to further refinement. The algorithm fits a discrete spectral generalized Maxwell (Maxwell-Wiechert) model, which is a linear viscoelastic model, to results from a stress-relaxation test. The discrete spectral approach was tested against trial data to characterize its robustness and identify its limitations and strengths. The algorithm was then applied to identify the viscoelastic response of reconstituted collagen and engineered fibrosis tissues, revealing that cells actively adapted the ECM, and that cells relax at multiple timescales, including one that is fast compared to those of the ECM.
The second algorithm, which we term the discrete quasi-linear viscoelastic (DQLV) approach, is a spectral extension of the Fung quasi-linear viscoelastic (QLV) model, a standard tool for characterizing biological materials. The Fung QLV model provides excellent fits to most stress-relaxation data by imposing a simple form upon a material\u27s temporal relaxation spectrum. However, model identification is challenging because the Fung QLV model\u27s “box” shaped relaxation spectrum, predominant in biomechanics applications, because it can provide an excellent fit even when it is not a reasonable representation of a material\u27s relaxation spectrum. The DQLV model is robust, simple, and unbiased. It is able to identify ranges of time constants over which the Fung QLV model\u27s typical box spectrum provides an accurate representation of a particular material\u27s temporal relaxation spectrum, and is effective at providing a fit to this model. The DQLV spectrum also reveals when other forms or discrete time constants are more suitable than a box spectrum. After validating the approach against idealized and noisy data, we applied the methods to analyze medial collateral ligament stress-relaxation and sinusoidal excitation data and identify the strengths and weaknesses of an optimal Fung QLV fit.
Taken together, the tools in this dissertation form a comprehensive approach to characterizing the mechanics of viscoelastic biological tissues, and to dissecting the micromechanical mechanisms that underlie a tissue\u27s viscoelastic responses
