Elsevier

Journal of Biomechanics

Volume 90, 11 June 2019, Pages 9-15
Journal of Biomechanics

Calibration of the shear wave speed-stress relationship in ex vivo tendons

https://doi.org/10.1016/j.jbiomech.2019.04.015Get rights and content

Abstract

It has recently been shown that shear wave speed in tendons is directly dependent on axial stress. Hence, wave speed could be used to infer tendon load provided that the wave speed-stress relationship can be calibrated and remains robust across loading conditions. The purpose of this study was to investigate the effects of loading rate and fluid immersion on the wave speed-stress relationship in ex vivo tendons, and to assess potential calibration techniques. Tendon wave speed and axial stress were measured in 20 porcine digital flexor tendons during cyclic (0.5, 1.0 and 2.0 Hz) or static axial loading. Squared wave speed was highly correlated to stress (r2avg = 0.98) and was insensitive to loading rate (p = 0.57). The constant of proportionality is the effective density, which reflects the density of the tendon tissue and additional effective mass added by the adjacent fluid. Effective densities of tendons vibrating in a saline bath averaged 1680 kg/m3 and added mass effects caused wave speeds to be 22% lower on average in a saline bath than in air. The root-mean-square error between predicted and measured stress was 0.67 MPa (6.7% of maximum stress) when using tendon-specific calibration parameters. These errors increased to 1.31 MPa (13.1% of maximum stress) when calibrating based on group-compiled data from ten tendons. These results support the feasibility of calculating absolute tendon stresses from wave speed squared based on linear calibration relationships.

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):c2=k'μ+σρ

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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