Calibration of the shear wave speed-stress relationship in ex vivo tendons
Introduction
Knowledge of tissue loading is inherently important to the study of musculoskeletal disorders. This is particularly true for tendon, which transmits muscle forces to the skeleton, and thereby facilitates human movement. Unfortunately, it remains challenging to assess in vivo tendon loading during movement. Modeling approaches can provide estimates of tendon loading based on measured kinematics and external forces, but rely on many assumptions regarding muscle coordination, musculoskeletal geometry, and tissue morphology (Buchanan et al., 2004, Crowninshield and Brand, 1981, Delp et al., 2007, Erdemir et al., 2007, Seireg and Arvikar, 1975). Direct measurement is more favorable, but conventional approaches (e.g., buckle transducers (Fleming and Beynnon, 2004, Komi et al., 1987), optic fiber technique (Finni et al., 1998, Komi et al., 1996)) are highly invasive, and thus have limited applicability for human use.
We recently introduced a technique for non-invasive measurement of tendon loading based on shear wave propagation speed (Martin et al., 2018). Our technique is related to ultrasound shear wave elastography (SWE), which uses ultra-high framerate imaging to track tissue shear wave speed, and subsequently computes an estimate of tissue elasticity (Bercoff et al., 2004). Prior studies have shown that tendon shear wave speed increases markedly when the tendon is stretched (Aubry et al., 2013, DeWall et al., 2014, Hug et al., 2013, Slane et al., 2017). This phenomenon has most often been attributed to strain-stiffening behavior, where wave speed increases are caused by a load-dependent increase in tendon shear elastic modulus. However, a tensioned beam model reveals that bulk shear wave speed in tendon exhibits a direct dependence on tensile stress that is independent of nonlinearity in the shear modulus, and that this effect may be dominant as loading is increased (Martin et al., 2018). Thus, shear wave speed could potentially serve as a non-invasive indicator of tendon loading.
Certain aspects of the tendon wave speed-stress relationship require further study to confirm that wave speed can be a dependable indicator of tendon loading. Tendons are known to be viscoelastic, and so it is unclear whether tendon stresses estimated from wave speed will be sensitive to loading rate. This is important because functional activities generally impose a range of loading rates on in vivo tendons. Additionally, it must be determined whether tendon-specific calibration is preferable to a generic relationship for estimating absolute tendon stresses from wave speed. It is theorized that differences between in vivo calibrations may arise due to anatomical differences in the adjacent medium, including tissue and fluid, which can add to the effective mass of the tendon and affect wave propagation speed (Martin et al., 2018).
Here we addressed these issues in the simplified case of ex vivo porcine digital flexor tendons. The purpose of this work was threefold. First, we evaluated whether the wave speed-stress relationship was independent of loading rate, such that a sensor based on this phenomenon would be applicable for use during movements at varying rates. Next, we examined the potential for predicting tendon stress based on calibrations determined from tendon-specific and group-compiled data. Finally, we assessed how the adjacent medium affected the tendon wave speed-stress relationship in a simple ex vivo case as a first step toward understanding the in vivo case.
Section snippets
Tensioned beam model
Tendon shear wave propagation was modeled using a tensioned Timoshenko beam (Ginsberg, 2001) assuming locally linear elastic material properties. At higher vibration frequencies, shear motion becomes dominant, and shear wave speed (c) is dependent on a shear correction factor (k′) based on the finite shape of the cross-section, the tangential shear modulus (μ), the axial stress acting on the tendon cross-section (σ), and the effective density of the tendon (ρeff) (Martin et al., 2018):
Sensitivity of wave speed to loading rate
Shear wave speed was not significantly dependent on loading rate (p = 0.565, Fig. 3). Wave speed was significantly dependent on tendon stress (p < 0.001), and no rate-by-stress interaction effects were observed (p = 0.998).
Linearity of the squared wave speed-stress relationship
The relationship between squared wave speed and stress was highly linear for all individual trials in isolation (r2 = 0.98 ± 0.03). Tendon-specific data showed similar correlation (r2 = 0.97 ± 0.02). When data from all trials on all tendons were compiled and considered
Discussion
The objective of this study was to examine the suitability of a tensioned beam model to describe the relationship between axial stress and shear wave speed in ex vivo tendons. The high observed correlation between stress and wave speed squared is consistent with model predictions. The constant of proportionality is the tendon effective density, which was shown to be relatively constant across specimens and across a range of loading rates. These data demonstrate that tendon shear wave speed is
Acknowledgements
The authors would like to thank Ryan DeWall and Laura Slane for technical assistance and Dan Segalman for discussions of modeling and theory. The work presented herein was supported by the National Science Foundation (United States) GRFP fellow (DGE-1256259) fellowship (J.M.), National Institutes of Health (United States) grants HD092697 and EB024957, and the University of Wisconsin Hilldale Undergraduate/Faculty Research Fellowship. The study sponsors had no direct involvement in the work
Conflict of interest statement
Three of the authors (J.A.M., D.G.T., M.S.A.) are co-inventors on a pending patent application for technology relating to the methods described herein. The other authors declare no conflict of interest.
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Sensitivity of the shear wave speed-stress relationship to soft tissue material properties and fiber alignment
2022, Journal of the Mechanical Behavior of Biomedical MaterialsCitation Excerpt :Regarding the second key finding, the simulated relationship between shear wave speed and axial stress was generally consistent with a simple tensioned beam model (Martinet al., 2018) for tissues with highly aligned fibers. Further, shear wave speeds measured using the finite element model were in range with those from previous tensiometry studies, where shear wave speeds ranged from 20 to 90 m/s in tendons (Martin et al., 2019) and from 20 to 160 m/s in ligaments (Blank et al., 2020a) during tensile loading with axial stresses ranging from 0 to 10 MPa. The c2-σ relationship was highly linear (R2 = 0.99), with the slope of this relationship well represented by the density of the model (Fig. 7i).
In vivo assessment of material properties of muscles and connective tissues around the knee joint based on shear wave elastography
2020, Journal of the Mechanical Behavior of Biomedical MaterialsCitation Excerpt :However, there were some limitations. Firstly, because tendons and ligaments are layered tissues, the speed at which the shear waves propagate through these tissues is affected by the thickness of the tissue itself, and the properties of surrounding tissues (Martin et al., 2018, 2019; Sadeghi et al., 2020; Helfenstein-Didier et al., 2016). In the future, precise methods will be applied to obtain more accurate information on the mechanical properties of layered tissues.