Journal Club March 2010: Viscoelasticity of Soft Tissues
It is widely accepted that collagenous soft tissues exhibit viscoelastic behavior, which includes time-dependent creep and stress relaxation, rate-dependence, and hysteresis in a loading cycle. The hysteresis is less sensitive than the stiffness to the loading rate, and this phenomenon is generally found in soft tissues and elastomers (Fung 1993). The experiments of Boyce et al. (2007 2008) to characterize the viscoelastic response of the cornea and sclera spanned three orders of loading rates. They observed at higher rates little rate-dependence in the stress-strain curve during loading but significant hysteresis during unloading.
Viscoelastic behavior is integral to the structural and protective function of many soft tissues, and changes in the viscoelastic properties may reflect the effects of disease. Recent uniaxial strip experiments have shown that the scleras of chick eyes with induced myopia (near-sightedness) creep at a higher rate and for a longer time than those of the controls Phillips et al. (2000). Similar experiments on the scleras of monkey eyes suffering from induced glaucoma also show an increased equilibrium (long-time) compliance and a longer stress relaxation time Downs et al. (2005). In the case of myopia, changes in the viscoelastic behavior are accompanied by macrostructural changes, including an enlarged scleral length and a thinner sclera, and microstructural changes, including smaller collagen fibril diameter, a narrower distribution of fibril diameters, and a decrease in the glycosaminoglycan content.
Given the importance of viscoelasticity to the physiological and pathophysiological function of soft tissues, it is important to develop methods to characterize the phenomena and to model its underlying mechanisms, both of which pose many challenges. The viscoelastic behavior of soft tissues, like those of engineering polymers, occurs over a broad spectrum of characteristic times that can span more than five orders of magnitude, which reflects the work of multiple time-dependent mechanisms of microstructural features spanning multiple length scales. However, their susceptibility to hydration, temperature, and other environmental effects prevent the use of standard time-temperature superposition techniques of dynamic mechanical analysis to span the time scales. Soft tissues can exhibit large characteristic times, which makes the equilibrium behavior difficult to characterize, and nonlinear time-dependent behavior, where the creep rate depends on the applied stress and the stress relaxation rate depends on the applied strain. Current strip methods require extensive preconditioning, where the tissue is cyclically loaded until a repeatable reference state is obtained. Preconditioning is necessary to characterize viscoelastic properties, particularly hysteresis, but it alters the material behavior from the in-vivo state.
To address these challenges for the cornea and sclera, Boyce et al. (2008) developed an inflation test method, which rigidly clamped the excised boundaries of the tissues, to prevent significant reorientation of the collagen fibrils during loading, and which subjected the tissues to physiological pressure loading. In addition, they designed a loading protocol which permits long periods of recovery between each loading cycle. These features eliminated the needed for preconditioning, in that the displacement response measured for identical successive pressure load-unload cycles and creep cycles were nearly identical through the day-long experiments. Methods to characterize the equilibrium behavior, include developing incremental pressure load-unload test, which have been applied successfully for polymers. However, very small pressure increments are needed, particularly at low pressures where the material is most compliant, along with long hold times to obtain full recovery after each load increment, all of which significantly increases the testing time.
The main challenge to modeling the viscoelastic behavior of soft tissues, is that the deformation mechanisms are poorly understood. Soft tissues are biphasic materials, with water comprising the majority of the weight. Thus, poroelasticity is an important mechanism at longer time scales for many tissues. The multiphasic/poroelastic approach has been applied successfully to model the time-dependent response of articular cartilage and other connective tissues in compression. However, it has been less successful in modeling the time-dependent behavior of soft fibrous tissues under uniaxial tension as demonstrated by Huang et al. (2001) for cartilage. Other mechanisms include the intrinsic viscoelastic behavior of the matrix, which arises from the chain dynamics of the long proteoglycan molecules (Bergstrom and Boyce 2001), and of the collagen fibers. The viscoelasticity of collagen fibrils is debatable based on experimental results on mitral valves anterior leaflets (Liao et al. 2007). The collagens are crosslinked to the proteoglycan matrix, and proteoglycan mediated fibrillar interactions such as fibril-fibril glide may be a significant viscoelastic mechanism (Liao and Vesely 2004).
Soft tissues are also anisotropic materials, and there have been significant advances in modeling the fiber-derived anisotropic, large deformation, viscoelastic behavior of soft tissues. Earlier approaches, such as Holzapfel et. al (2000), Bischoff et al. (2004), and Merodio and Rajagopal (2007), specify anisotropic viscoelastic constitutive relations at the tissue level. An alternative approach attributes the viscoelastic behavior to the matrix and fiber constituents, then use homogenization techniques to determine the constitutive behavior of the tissue. Lanir (1983) developed a fully three-dimensional, anisotropic, quasilinear viscoelastic model for fibrous tissues that homogenized the quasilinear viscoelastic behavior of a wavy fiber over a distribution of fiber orientations (see also Bischoff (2006). Nguyen et al. (2007 2008) developed an anisotropic viscoelastic model for the cornea that assumed a nonlinear viscoelastic flow relation for the collagen lamellae. These were homogenized over a continuous distribution of the lamellar orientations to construct an anisotropic flow rule for the cornea. In the latter approach, the effects of anisotropy arise directly from the collagen fiber architecture, which can be determined independently from mechanical experiments by histologically.
I have presented a brief discussion of the experimental and modeling challenges and current approaches to addressing these challenges in studying the viscoelastic behavior of soft tissues. The discussion cites a few representative publications, but I would like to highlight the following three on viscoelastic deformation mechanisms of soft tissues. I welcome discussions on your experiences in this area and on your perceived future challenges to the study of soft tissues viscoelasticity.
Bergstrom, J., Boyce, M., 2001. Constitutive modeling of the time-dependent and cyclic loading of elastomers and application to soft biological tissues. Mechanics of Materials 3, 523–530.
Bischoff, J. E., 2006. Reduced parameter formulation for incorporating viscoelasticity into tissue level biomechanics. Annals of Biomedical Engineering 34, 1164–1172.
Bischoff, J. E., Arruda, E. M., Grosh, K., 2004. A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue. Biomech Model Mechanobiol 3, 56–65.
Boyce, B. L., Jones, R. E., Nguyen, T. D., Grazier, J. M., 2007. Stress-controlled viscoelastic tensile response of bovine cornea. J. Biomech. 40, 2367–2376.
Boyce, B. L., Jones, R. E., Nguyen, T. D., Grazier, J. M., 2008. Full-field deformation of bovine cornea under constrained inflation conditions. Biomaterials 28, 3896–3904.
Downs, J. C., Suh, J. K. F., Thomas, K. A., Belleza, A. J., Hart, R. T., Burgoyne, C. F., 2005. Viscoelastic material properties of the peripapillary sclera in normal and early-glaucoma monkey eyes. Invest. Ophthalmol. Vis. Sci. 46, 540–546.
Fung, Y. C., 1993. Biomechanics: mechanical properties of living tissues. Springer-Verlag, New York, NY.
Huang, C. Y., Mow, V. C., Ateshian, G. A., 2001. The role of flow-independent viscoelasticity in the biphasic tensile and compressive reponse of articular cartilage. J. Biomech. Eng. 123, 410–417.
Lanir, Y., 1983. Constitutive Equations for Fibrous Connective Tissues. J. Biomech. 16, 1–12.
Liao, J., Vesely, I., 2004. Relationship between collagen fibrils, glycosaminoglycans, and stress relaxation in mitral valve chordae tendineae. Journal of Biomedical Engineering 2004, 977–983.
Liao, J., Yang, L., Grashow, J., Sacks, M. S., 2007. The relation between collagen fibril kinematics and mechanical properties in the mitral valve anterior leaflet. J. Biomech. Eng. 129, 78–87.
Merodio, J., Rajagopal, K. R., 2007. On constitutive equations for anisotropic nonlinearly viscoelastic solids. Mathematics and Mechanics of Solids 12, 131–147.
Nguyen, T. D., Jones, R. E., Boyce, B. L., 2007. Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites. Int. J. Solids Struct. 44, 8366-8389.
Nguyen, T. D., Jones, R. E., Boyce, B. L., 2008. A nonlinear anisotropic viscoelastic model for the tensile behavior of the corneal stroma. J. Biomech. Eng. 130, 041020–1.
Phillips, J. R., Khalaj, M., McBrien, N. A., 2000. Induced myopia associated with increased scleral creep in chick and tree shrew eyes. Invest. Ophthalmol. Vis. Sci. 41, 2028–2043.