Biomechanics

Biomechanics
Rod Lakes, University of Wisconsin

Biomechanics Research: Bone, Ligament, Wood

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Bone
Bone rheology and adaptation mechanisms
The right image shows structure of a cross section of bovine plexiform bone; the left image, of a cross section of human Haversian bone under polarized light. bovine bone

The small dark circles in the human bone are cross sections of Haversian canals. The large circles are cross sections of osteons, which are large fibers about 200 microns across. Bovine plexiform has a laminated structure. Bone has a hierarchical structure in which each structural element itself contains structure.
Bone adaptation is a phenomenon in which human or animal bones slowly add or lose mass and alter their form in response to modifications from normal mechanical stimulus conditions. When subjected to prolonged exposure to sustained and semi-intensive cyclic loading in vivo , bone systems adapt by adding mass resulting in combined densification and thickening. Conversely, when living bone systems are continuously underloaded over extended periods (as in space travel; fixation; prolonged bed rest; or stress shielding from surgical implants) a fraction of the bone mass is resorbed resulting in diminished capacity from reduced bone densities and thinning. This suggests that the skeletal system senses changes in sustained mechanical load patterns and adapts itself to carry the predominant loads most efficiently using minimal bone mass.
While bone adaptation is well recognized, the specific mechanical stimuli that trigger and sustain it are not well characterized. Furthermore, the underlying biological mechanisms used by skeletal systems both to sense mechanical stimulus and to biologically adapt remain poorly understood. This poor understanding has forestalled progress on a number of positive clinical developments such as: treating bone and skeletal deformities; preventing and treating osteopenias; accelerating fracture healing; and optimizing implant designs. This new research program is thus directed toward obtaining a better fundamental understanding of bone adaptation.
Deformation driven fluid flow in bone is an increasingly hypothesized, but as yet untested, mechanism that appears quite capable of explaining the predominant characteristics of bone adaptation. Since compact bone has a hierarchical microstructure, such fluid flow occurs in bone on a spectrum of length scales during mechanical excitation. The specific challenge addressed in the research program is the development of computational models to accurately predict such flow, both how it is activated by mechanical stimulus, and how it might in turn activate the biological processes of bone adaptation. These models are developed and verified with the aid of well-conceived experiments in tissue viscoelasticity. From a scientific perspective, achievement of this objective led to an enhanced fundamental understanding of bone adaptation and new rheology experiments to verify this understanding.

Garner, E., Lakes, R. S., Lee, T., Swan, C. and Brand, R., "Viscoelastic dissipation in compact bone: implications for stress-induced fluid flow in bone", J. Biomech. Engineering, 122, 166-172 (2000). (Get pdf)
Buechner, P. M., Lakes, R. S., Swan, C., Brand, R. A., "A Broadband Viscoelastic Spectroscopic Study of Bovine Bone: Implications for Fluid Flow", Annals of Biomedical Engineering, 29, 719-728, August (2001). (Get pdf)
Lee, T., Lakes, R. S., and Lal, A., "Investigation of bovine bone by resonant ultrasound spectroscopy and transmission ultrasound", Biomechanics and Modeling in Mechanobiology, 1, 165-175, October (2002). (Get pdf)
Stewart, K. Brand, R., Swan, C. C., and Lakes, R. S., "Micromechanically based poroelastic modeling of fluid flow in Haversian bone" Journal of Biomechanical Engineering 125, 25-37, Feb. (2003). (Get pdf)
Buechner, P. M., and Lakes, R. S., "Size effects in the elasticity and viscoelasticity of bone", Biomechanics and Modeling in Mechanobiology, 1 (4), 295-301 (2003). (Get pdf)
Gururaja, S., Kim, H. J., Swan, C. C., Brand, R. A., and Lakes, R. S. "Modeling deformation-induced fluid flow in cortical bone's canalicular-lacunar system", Annals of Biomedical Engineering, 33, 7-25, Jan. (2005). (Get pdf)

Ligament: ligament viscoelasticity
Interrelation of creep and relaxation: a modeling approach for ligaments, R. Lakes and R. Vanderby, J. Biomechanical Engineering, 121, 612-615, Dec. (1999). (Get pdf)
Experimental tissue rheology data (Thornton, et al., 1997) show that relaxation proceeds more rapidly (a greater slope on a log log scale) than creep in ligament, a fact not explained by linear viscoelasticity. ligament fiber

An interrelation between creep and relaxation is therefore developed for ligament viscoelasticity based on a single-integral nonlinear superposition model. This interrelation differs from the convolution relation obtained by Laplace transforms for linear materials. We demonstrate via continuum concepts of nonlinear viscoelasticity that such a difference in rate between creep and relaxation phenomenologically occurs when the nonlinearity is of a strain-stiffening type, i.e. the stress-strain curve is concave up as observed in ligament. We also show that it is inconsistent to assume a Fung-type constitutive law (Fung, 1972) for both creep and relaxation in ligament viscoelasticity. Using published data of Thornton, et al., the nonlinear interrelation developed herein predicts creep behavior from relaxation data well (R greater than 0.998). Although data are limited and the causal mechanisms associated with viscoelastic tissue behavior are complex, continuum concepts demonstrated here appear capable of interrelating creep and relaxation with fidelity. The image shows ligament structure, after Vanderby.
The new research is primarily directed toward study of nonlinearly viscoelastic behavior of soft connective tissues, particularly the constitutive behavior representing ligament viscoelasticity. Because of their simplicity, ligaments are chosen as an experimental model. The research is organized as follows. (i) An experimental study is in progress, that defines viscoelastic behavior throughout the range of reversible ligament deformations. This study uses creep and relaxation plus recovery protocols to allow discrimination among various nonlinearities. (ii) A robust viscoelastic constitutive model is developed from experimental data. This model appropriately accounts for nonlinearities and interrelates creep and relaxation. (iii) The reversible deformation limits for levels of applied stress and strain (as a function of load-time) is experimentally identified from the data in (i), thus defining the onset of sub-failure tissue damage for a single overload. A sub-failure criterion for the onset of damage is then formulated. (iv) Creep and relaxation testing are performed on additional specimens to quantify the compromise in mechanical behavior after higher loadings and deformations produce greater levels of tissue damage (from a single overload). (v) The microtrauma and damage associated with irreversible deformations is morphologically characterized using scanning electron microscopy.
Nonlinear viscoelasticity is phenomenologically observed in all soft connective tissues. It is axiomatic then that any biomechanical study of these tissues must explicitly or implicitly take this behavior into account. Optimal soft tissue repair or replacement must viscoelastically mimic original tissue. Accurate and robust biomechanical models must reproduce this characteristic behavior. It is therefore sine qua non that nonlinear viscoelasticity be well defined in soft connective tissues. Commonly used viscoelastic descriptions have come into question. They appear inadequate to describe the different rates observed in creep and relaxation of ligaments. Since other soft connective tissues share compositional similarities (collagen fiber reinforced materials with relatively high levels of hydration produced by proteoglycans), it is likely that other soft connective tissues have similar constitutive issues that must be addressed in their viscoelastic descriptions. In addition, current descriptions do not allow comparison of studies performed under different experimental modalities and thereby render many previous studies quantitatively inaccessible. This biomechanical deficiency cannot be rectified with existing data, since with most tissues there never has been a study sufficiently complete to allow discrimination among nonlinearities. An experimental program is being conducted (with a ligament model) involving creep and relaxation experiments at different load and strain levels, with the goal of discriminating among nonlinear representations of ligament viscoelasticity and establishing a robust model.
A particular bonus in performing the above systematic study of nonlinear viscoelasticity throughout the range of reversible deformations is that it also identifies the boundaries of that range, i.e. the onset of tissue damage. This is another fundamental aspect of soft tissue biomechanics that has never been adequately defined. Despite a high clinical incidence, a biomechanical criterion for subfailure damage is largely unknown in soft connective tissues, and sub-failure damage is rarely considered as part of biomechanical studies.
Ligament viscoelasticity experiment
Provenzano, P., Lakes, R. S., Keenan, T, Vanderby, R. Jr., "Non-linear ligament viscoelasticity", Annals of Biomedical Engineering, 29, 908-914, Nov. (2001)
Abstract
Ligaments display time dependent behavior, characteristic of a viscoelastic solid, and are non-linear in their stress-strain response. Recent experiments (Thornton et al., 1997) reveal that stress relaxation proceeds faster than more rapidly than creep in medial collateral ligaments, a fact not explained by linear viscoelastic theory but consistent with non-linear theory by Lakes and Vanderby (1999). This study tests the following hypothesis. Non-linear viscoelasticity of ligament requires a description more general than the separable quasi-linear viscoelasticity (QLV) formulation commonly used. The experimental test for this hypothesis involves performing both creep and relaxation studies at various loading levels below the damage threshold. Freshly harvested, rat medial collateral ligaments were used as a model. Results consistently show a non-linear behavior in which the rate of creep is dependent upon stress level and the rate of relaxation is dependent upon strain level. Furthermore, relaxation proceeds faster than creep, consistent with the experimental observations of Thornton et al., (1997). The above results are not consistent with a separable QLV theory. Inclusion of these nonlinearities would require a more general formulation. lig plot

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Provenzano, P., Heisey, D., Hayashi, K., Lakes, R. S., and Vanderby, R. Jr., "Subfailure damage in ligament: a structural and cellular evaluation", J. Applied Physiology, 92: 362-371, (2002).
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Provenzano, P., Lakes, R. S., Corr, D. T., and Vanderby, R. Jr, "Application of nonlinear viscoelastic models to describe ligament behavior", Biomechanics and Modeling in Mechanobiology,, 1: 45-57, (2002).
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Oza, A., Lakes, R. S. and Vanderby, R., "Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal" Rheologica Acta, 42, 557-568 (2003).
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Manley E. Jr, Provenzano, P. P., Heisey D., Lakes R., Vanderby R. Jr., "Required test duration for group comparisons in ligament viscoelasticity: a statistical approach", Biorheology, 40(4): 441-50 Apr. (2003). pdf

Hingorani, R., Provenzano, P., Lakes, R. S., Escarcega, A., and Vanderby, R., Jr. "Experimental evaluation of nonlinear viscoelastic behavior in rabbit ligament", 40th annual meeting, Society of Engineering Science, Ann Arbor, MI, 15 Oct. (2003). Get pdf or image.

Hingorani, R., Provenzano, P., P., Lakes, R. S., Escarcega, A., Vanderby, R. Jr., "Nonlinear viscoelasticity in rabbit medial collateral ligament", Annals of Biomedical Engineering, 32, 306-312, Feb. (2004). Get pdf.
Results show that within the physiologically relevant region of ligament behavior, the rate of relaxation is strain dependent in the rabbit MCL, with the rate of relaxation decreasing with increasing tissue strain. The rate of creep is stress dependent in the rabbit MCL, with the rate of creep decreasing with increasing stress. Nonlinearities in rates of creep and relaxation cannot be robustly modeled by the QLV formulation (quasi-linear viscoelasticity).

Jensen, K. T. Dwyer, K. W., Lakes, R. S., and Vanderby, R., Jr., "The rate of viscoelastic recovery is faster than the rate of creep", 50th ORS, paper 0046, March (2004). Get pdf. This phenomenon cannot be described by either linear viscoelasticity or quasilinear viscoelasticity (QLV).

Oza, A., Vanderby, R. and Lakes, R. S., "Generalized solution for predicting relaxation from creep: application to ligament" International Journal of Mechanical Sciences, 48, (6) 662-673 June (2006)
Creep and relaxation are two viscoelastic phenomena that are easily interrelated for a linearly viscoelastic material, but interrelationships are complex for nonlinearly viscoelastic materials. We use a single-integral nonlinear superposition principle to relate creep and relaxation, where the kernel is assumed to be a nonseparable product of strain and time. Herein, we develop time dependence as general power laws with up to four terms for creep compliance and relaxation modulus. Higher-order formulations give better results for ligament in terms of curve fitting and prediction of relaxation from creep. This is illustrated by a comparison between a two- and a three-term formulation on the experimental data of rabbit medial collateral ligaments. Also, an interrelation between several aspects of creep and relaxation is presented for arbitrarily high order, and the nature of high-order interrelation is discussed. The generality of the method makes it suitable to phenomenologically model many complex materials, to predict complex behaviors and to therefore reduce the amount of testing for robust material characterization.

We thank the National Science Foundation for support.

Wood: wood mechanics
Wood, with S. Cramer, D. Kretschmann

The overall hypotheses of this research are that: The mechanical properties of earlywood compared to latewood are dramatically different in loblolly pine plantation wood.
The mechanical properties and their variations of earlywood and latewood can be quantified and characterized.
These mechanical properties of earlywood and latewood vary with location within a tree and conditions of growth.
Ultimately, the properties of earlywood and latewood can be used to explain previously unexplained variation in wood product performance.
Among other things, we are conducting research into the mechanical properties of slices of individual growth rings, using our micromechanics apparatus for broadband viscoelastic spectroscopy.

Kretschmann, D. E., Schmidt, T. W., Lakes, R. S., and Cramer, S. M., "Micromechanical measurement of wood substructure properties", Society of Experimental Mechanics Annual Conference, Milwaukee, WI, June 10-11 (2002). Get pdf.
Cramer, S., Kretschmann, D., Lakes, R. And Schmidt, T. "Mesostructure Elastic Properties in Loblolly Pine," Proceedings of the 4th Plant Biomechanics Conference, Michigan State University, East Lansing, MI, 21-25 July (2003). Get pdf
Kretschmann, D. E., Cramer, S. M., Lakes, R. S., and Schmidt, T. W. "Selected mesostructure propertes in loblolly pine from Arkansas plantations", Chapter 12, p. 147-170, in Characterization of the Cellulosic Cell Wall, ed..D. D. Stokke, L. H. Groom, Blackwell, Oxford, UK, (2006).
Cramer, S. M.,, Kretschmann, D. E., Lakes, R. S., and Schmidt, T. W. "Earlywood and latewood elastic properties in loblolly pine", Holzforschung, 59, 531-538 (2005).

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