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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.

1) 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.

2) 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.

3) 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.

References
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.

Vicky,

Thanks for taking the lead on the discussion of viscoelastic behaviors of soft tissues. I agree with you that soft tissues generally demonstrate very complicated behaviors. We have investigated the mechanical behaviors of pulmonary arterial (PA) tissues from calves for a few years. For equilibrium behaviors, arterial tissues are anisotropic and typically present J-shape stress-strain curves with initial low modulus followed by stiffening at the intermediate stretch level. This J-shape curve is believed due to collagen engagement. At the low stretch ratio, only elastin network carries load and collagen fibers do not, yielding a small modulus. At intermediate stretch ratio, collagen fibers engage, yielding a stiffening effect.

As in other soft tissues, arterial tissues also demonstrate the viscoelastic effect. We recently start to look at the viscoelastic behaviors of PA tissues. Our motivation is that we want to know how important the viscoelastic effect is in the physiological function of PA tissues. Although people are aware of viscoelastic effect in soft tissues, hyperelastic models are typically used in the investigation of arterial behaviors. One reason is that hyperelastic models require fewer material parameters, which is in general preferred. However, it is unclear if models with such a simplification can accurately produce the pressure-diameter (P-D) curves under the physiological conditions, as the ultimate goal of many constitutive studies of arterial tissues is to provide diagnostic capability when combined with in vivo P-D measurements. P-D behaviors in artery are a very complicated process that couples dynamic pressure variation and tissue finite deformation with the fluid inside. In PA tissues, they also couple with pressure change due to breathing. These strong coupling effects require a systematic approach. We have obtained some preliminary results on the viscoelastic properties of PA tissues and will report our results soon.

Jerry    

 

venapa's picture

Dear Vicky,

thank you for this discussion. Non linear viscoelasticity of tissues is a very interesting topic. I'd like to provide my small contribution to this field with a model applied to non linear viscoelasticity of ligaments and, in a second paper, to cartilage. The models is based on the main idea that the constituents of a composite material may exhibit different time-dependent properties.

The interested readers can see the details in the following two papers:

1) A Constituent-Based Model for the Nonlinear Viscoelastic Behavior of Ligaments

 

2) A Nonlinear Constituent Based Viscoelastic Model for Articular
Cartilage and Analysis of Tissue Remodeling Due to Altered
Glycosaminoglycan-Collagen Interactions

 

 

We are currently exploring the dynamic nanoindentation technique (sweeping a wide frequency range) as a possible experimental/theoretical tool to quantify the viscoelastic properties of the tissue at small length scales.

Best regards

Pasquale Vena

 

I have read the first paper.  I believe you used an Eyring type of flow rule with a stress activation term to model the nonlinearity. This was our approach also.  I am intrigued by the second paper you listed and will definitely take a look.  What frequency range are you planning to test. Please keep us updated.

  

Thao (Vicky) Nguyen
Assistant Professor
Mechanical Engineering Department
Johns Hopkins University
http://me.jhu.edu/~tnguy108

venapa's picture

Dear Vicky,

thank you very much for your reply.

We are currently running a project on indentation of cartilage at multiple length scales (from AFM-based indentation to microindentation up to classical macroscopic mechanical tests).

The AFM-based experiments will be run with harmonic loading, we are now reasoning on the optimum frequency span. We believe that the low frequency range will be explored first (with the purpose to determine visco-elastic properties of the tissues), and after that, frequency as high as hundreds of Hz will be swept. We also believe that high frequency tests will be largely affected by parameters like sample size (and possible interaction with eigenfrequency of the sample). All tests will be run in liquid environment.

Any suggestions are wellcome. 

Pasquale Vena

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