# Investigation of the relationship between tensile viscoelasticity and unloaded ultrasound shear wave measurements in ex vivo tendon

Tobias Götschi, Yannick Schärer, Jean-Luc Gennisson, Jess G Snedeker

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Tobias Götschi, Yannick Schärer, Jean-Luc Gennisson, Jess G Snedeker. Investigation of the relationship between tensile viscoelasticity and unloaded ultrasound shear wave measurements in ex vivo tendon. Journal of Biomechanics, 2023, 146, pp.111411. 10.1016/j.jbiomech.2022.111411. hal-05144300

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<figure><img src="image_1.png" /></figure>
# Investigation of the relationship between tensile viscoelasticity and unloaded ultrasound shear wave measurements in ex vivo tendon

## Journal Article

### Author(s):

Götschi, Tobias; Schärer, Yannick; Gennisson, Jean-Luc; Snedeker, Jess Gerrit

<figure><img src="image_2.png" /></figure>

### Publication date:

2023-01

### Permanent link:

https://doi.org/10.3929/ethz-b-000588278

### Rights / license:

Creative Commons Attribution 4.0 International

### Originally published in:

Journal of Biomechanics 146, https://doi.org/10.1016/j.jbiomech.2022.111411
<figure><img src="image_3.png" /></figure>

# Investigation of the relationship between tensile viscoelasticity and unloaded ultrasound shear wave measurements in ex vivo tendon

Tobias Götschi<sup>a, b, *</sup>, Yannick Schärer<sup>b</sup>, Jean-Luc Gennisson<sup>c</sup>, Jess G. Snedeker<sup>a, b</sup>

## ARTICLE INFO

### Keywords:

Shear wave elastography

Tendon

Mechanical properties

Mechanical testing

## ABSTRACT

Mechanical properties of biological tissues are of key importance for proper function and *in situ* methods for mechanical characterization are sought after in the context of both medical diagnosis as well as understanding of pathophysiological processes. Shear wave elastography (SWE) and accompanying physical modelling methods provide valid estimates of stiffness in quasi-linear viscoelastic, isotropic tissue but suffer from limitations in assessing non-linear viscoelastic or anisotropic material, such as tendon. Indeed, mathematical modelling predicts the longitudinal shear wave velocity to be unaffected by the tensile but rather the shear viscoelasticity. Here, we employ a heuristic experimental testing approach to the problem to assess the most important potential confounders, namely tendon mass density and diameter, and to investigate associations between tendon tensile viscoelasticity with shear wave descriptors. Small oscillatory testing of animal flexor tendons at two baseline stress levels over a large frequency range comprehensively characterized tensile viscoelastic behavior. A broad set of shear wave descriptors was retrieved on the unloaded tendon based on high frame-rate plane wave ultrasound after applying an acoustic deformation impulse. Tensile modulus and strain energy dissipation increased logarithmically and linearly, respectively, with the frequency of the applied strain. Shear wave descriptors were mostly unaffected by tendon diameter but were highly sensitive to tendon mass density. Shear wave group and phase velocity showed no association with tensile elasticity or strain rate-stiffening but did show an association with tensile strain energy dissipation.

The longitudinal shear wave velocity may not characterize tensile elasticity but rather tensile viscous properties of transversely isotropic collagenous tissues.

## 1. Introduction

Mechanical properties of biological tissues are integral aspects to proper function and their alterations are associated with numerous pathologies including degenerative and cancerous processes (Fung and Skalak, 1982). Applying force to the tissue and observing the displacement response allows for a quantification of mechanical properties *in situ* (Parker et al., 2011). Of such nature, ultrasound (US) shear wave elastography (SWE) applies a single focused acoustic force to induce shear motion which then propagates radially as an impulsive wave. High frame-rate ultrasound enables visualization of this shear wave, who's descriptors are related to the tissue's mechanical properties

(Nightingale, 2012). In a linear elastic homogenous isotropic medium of infinite size, the shear modulus (G) is simply equated by: $G = c^2\rho$, where c is the shear wave velocity (SWV) and $\rho$ is the mass density of the medium. With the additional assumption of incompressibility, the Young's modulus is given by $E = 3G$ (Li and Cao, 2017). In the general case of linear viscoelasticity, shear wave speed and shear wave attenuation collectively – both retrieved as a function of wave temporal frequency – provide the complex-valued strain-rate dependent shear modulus (Nenadic et al., 2017). In structurally organized materials such as most biological tissues, anisotropy and non-trivial spatial wave diffraction additionally affect attenuation, rendering estimation of viscoelasticity-induced dissipation elusive (Bernard et al., 2017). In such

<footnote><sup>a</sup> Department of Orthopaedics, Balgrist University Hospital, University of Zurich, Switzerland</footnote>

<footnote><sup>b</sup> Institute for Biomechanics, ETH Zurich, Switzerland</footnote>

<footnote><sup>c</sup> Université Paris-Saclay, Inserm, CNRS, CEA, Laboratoire d'Imagerie Biomédicale Multimodale (BioMaps), Service Hospitalier Frédéric Joliot, 4 place du général Leclerc, 91401 ORSAY, France</footnote>

<footnote>* Corresponding author at: Department of Orthopaedics, University of Zurich, Institute for Biomechanics, ETH Zurich, Balgrist Campus, Lengghalde 5, CH - 8008 Zurich, Switzerland.</footnote>

<footnote>E-mail address: tobias.goetschi@hest.ethz.ch (T. Götschi).</footnote>
an instance, the shear wave phase velocity alone can be fitted with an
appropriate material model (such as the Voigt model) to estimate
viscoelasticity parameters, where such model exists (Gennisson et al.,
2010; Parker et al., 2019). In structures with a diameter comparable to
the wavelength of the applied impulse (millimeters to centimeters in soft
tissue (Brum et al., 2014)) however, wave guidance and mode conver-
sion distort the measured phase velocities (Nenadic et al., 2011) which
prompted the development of specialized models specifically taking
these mechanisms into account (Brum et al., 2014; Maksuti et al., 2016;
Nenadic et al., 2014).

Among other connective tissues, tendon has been subject to US SWE
examinations in various scientific endeavors. Tendon degenerative
conditions are believed to manifest in reduction in tensile stiffness as
indicated by dynamometer-US measurements (Arya and Kulig, 2010;
Helland et al., 2013; Wiesinger et al., 2020) and in vitro experiments
(Sano et al., 1997). Accordingly, numerous (Dirrichs et al., 2016, 2018;
Götschi et al., 2022a; Ooi et al., 2016) but not all (Coombes et al., 2018;
Zhang and Fu, 2013) SWE investigations report a reduction in shear
wave velocity in tendinopathic tendons. Other investigations report ef-
fects of age (Hsiao et al., 2015), gender (Taş et al., 2017) and the type
(Gatz et al., 2020; Götschi et al., 2022b) and volume (Siu et al., 2016) of
physical activity on SWV. Many of these reports equate differences in
SWV with analogous differences in tensile stiffness, a presumption
which - for the reasons outlined herein - is far from obvious. The mea-
surement of tensile properties of tendons in situ using straightforward
measurement modalities thus remains an unmet need, and is an active
area of research across the fields of basic science, sports physiology and
musculoskeletal modelling.

It has been shown that in a linear transverse anisotropic medium
there is no analytical means to derive the tensile modulus along the fi-
ers in terms of SWV parallel thereto (Royer et al., 2011).

It is nevertheless conceivable that factors that govern tendon tensile viscoelasticity also affect its shear behavior and the current study was undertaken on that account. We employed a heuristic approach to the problem and performed a set of experiments to (1) assess the effects of the main confounders biasing shear wave descriptors and discuss appropriate adjustment strategies, and to (2) investigate potential associations between tendon tensile properties in the viscoelastic domain with shear wave descriptors.

## 2. Materials and methods

### 2.1. Tendon sample processing

In this investigation 19 digitorum flexor tendons from ~30 week old
bovines and 13 digitorum flexor tendons from adult horses, euthanized
for unrelated reasons, were used. Tendons from two species were used in
order to increase the natural variability of the retrieved parameters. The
tendons were wrapped in saline-soaked gauze and frozen within twelve
hours after euthanasia until the day of testing. At the day of testing, the
tendons were submerged in water at room-temperature for a minimum
of two hours to ensure complete hydration. For SWE measurements the
 unloaded tendon was submerged in US gel and the ultrasound trans-
ducer was positioned with a mounting arm at a distance of five milli-
meters from the tendon, parallel to its long axis. Adequate alignment
was ensured by observing and maximizing the fibrillar appearance of the
tendon on US B-mode (Götschi et al., 2021). Three SWE measurements
were then acquired, slightly repositioning the transducer before each
measurement to achieve independent image fields.

Mechanical testing was then performed on two subsections of each
tendon. For these tendon samples to yield representative data, the
sample must be uniform in cross-sectional area (CSA) along its length
and care must be taken not to damage any fibers during extraction. We
accomplished this by first identifying an appropriate set of fiber bundles
in the tendon cross section, applying a small cut accordingly at the
tendon end and then manually extracting the fiber bundle from the

tendon. We then cut this tissue section to a length of 40 mm for a clamp-
to-clamp sample length of ~ 30 mm. The sections were around 2 mm in
diameter. The sample cross-sectional area (CSA) was determined using a
custom-built laser caliper with a measurement error of less than 100 µm
by averaging three measurements at different longitudinal locations
(Fessel et al., 2014). Tissue mass density was determined by measuring
its weight in air and ethanol using buoyancy principle (Graf, 2004). Two
tendon sections were processed for each tendon and all parameters
pertaining to each subsection were averaged to yield tendon-specific
properties.

### 2.2. Shear wave elastography acquisition

An ultrafast ultrasound scanner equipped with a linear 256 element,
5 – 18 MHz bandwidth ultrasound array transducer (SL 18–5) was used
(Aixplorer, Supersonic Imagine, Aix en Provence, France). The acoustic
impulse was aimed at the center of the tendon cross-section and con-
sisted of two simultaneous pushes spaced 5 mm apart axially which
results in a quasi-cylindrical wave front (Rouze et al., 2015) with a
bandwidth of 300 – 800 Hz (Gennisson et al., 2010). Immediately after
the acoustic impulse, the system was switched into echo mode and the
tissue’s strain response was imaged with 40 demodulated in-phase
quadrature (I/Q) frames at a pulse repetition frequency of 15 kHz.
Wave propagation was then obtained by calculating the phase angle
between consecutive I/Q signals. On the corresponding ultrasound
brightness-mode (B-mode) image the tendon was segmented manually
to mask all background signal. The axial displacement data was median-
aggregated over the axial dimension which yields a two-dimensional
time-lateral position representation of wave propagation. Five wave
propagations were acquired in rapid succession and again median-
aggregated to increase the signal to noise ratio. For lack of a suitable
model, no diffraction-induced amplitude correction was employed. The
two-dimensional discrete Fourier transform then provided the temporal-
spatial frequency spectrum of the impulse. Fig. 1A – 1C provides a visual
depiction of the applied procedure.

### 2.3. Shear wave elastography descriptors

Owing to the explorative nature of this study, we employed a broad
set of shear wave propagation descriptors. Propagation of the impulsive
wave envelope was characterized in terms of its velocity (Group veloc-
ity, $v_g$) and its amplitude decay (Impulse attenuation, $\alpha_g$). Fig. 1D rep-
resents the displacements of the shear wave laterally as a function of
time. The former was estimated by finding the velocity – the angle in the
time-lateral position representation – which maximizes the Radon
transform. Shear wave attenuation was parameterized by fitting an
exponential decay to the amplitude of the shear wave displacements
over the lateral position (Fig. 1G). All other shear wave descriptors were
obtained from the frequency-wavenumber spectrum of the shear wave
displacement (Fig. 1E). The wavenumber with maximum power per
temporal frequency band yields the phase velocity of each harmonic
component of the impulsive wave which displays a monotonic increase
with increasing frequency in tendon (Brum et al., 2014). This dispersive
behavior was parameterized by fitting the Voigt function predicting
viscoelastic shear wave phase velocity in an infinite isotropic medium to
the observed phase velocity-frequency relationship which yields
apparent shear elasticity ($\mu_c$) and viscosity ($\eta_c$), respectively (Gennisson
et al., 2010). Of different nature, we also fitted the phase velocity
dispersion model introduced by Brum et al. (Brum et al., 2014). This
model was developed specifically to be applied in tendon SWE mea-
surements and it is based on the assumption of transversely isotropic
linear elasticity of the tissue. Moreover, the model is aimed to take into
account the limited axial spatial extent of the structure and the resulting
guided wave propagation. As envisioned by the model’s devisers, a least
mean square optimization of the predicted and the observed phase ve-
locity yielded estimated elasticity tensor coefficients $c_{13}$ and $c_{55}$
<figure><img src="image_4.png" /><figcaption>Fig. 1. Visualization of the procedure for the shear wave elastography measurement and the extraction of the shear wave propagation descriptors. A: The ultrasound transducer was positioned parallel to the tendon long axis which was submerged in ultrasound coupling gel. B: The acoustic impulse was focused (grey ellipse) at the center of the tendon and the region of interest (ROI) was placed 3 mm to the right. C: The phase angle of consecutive I/Q ultrasound frames yields time-resolved axial displacement maps. D: Axial particle velocity in the time-lateral position representation yields group velocity. E: Two-dimensional frequency spectrum of the impulse propagation. F: Observed phase velocity as a function of temporal frequency and fitted phase velocity by Voigt function. G: Attenuation envelope decay of the propagating impulse and fitted exponential function. H: Temporal frequency spectrum of the propagating impulse. I: Spatial frequency spectrum of the impulse at maximum power frequency yields corresponding harmonic component attenuation. AU: Arbitrary units.</figcaption></figure>

(Fig. 1F). Constants and optimization constraints were set identically as in the paper of reference. Tendon mass density was assumed to be 1100 kg/m³ in both models.

We further aimed to describe the frequency span with considerable excitation energy transfer by obtaining the full width at half maximum of the maximum power over temporal frequency (FWHM<sub>ω</sub>, Fig. 1F). Finally, the temporal frequency at maximum power and the corresponding phase velocity (c<sub>max</sub> p) and attenuation of the harmonic component (α<sub>max</sub> p) was assessed. The latter can be estimated by measuring FWHM in the wavenumber spectrum using the relationship (Nenadic et al., 2017) (Fig. 1I): α(ω) = FWHM(ω)×π/√3. The finite nature of the spatial and temporal observation, which manifests in a convolution of the true signal with a boxcar of the size of the observation window (Rouze et al., 2017), was not accounted for. Table 1 provides a summary of the shear wave descriptors analyzed in the study.

## 2.4. Oscillatory tensile testing

Comprehensive characterization of the dynamic mechanical response of each tendon sample was obtained using small amplitude oscillatory tensile testing (ElectroForce 3200, Bose Corporation, Minnesota, USA)). The tendon ends were wrapped with a glue-soaked gauze and fixated to the testing machine with clamps equipped with sandpaper. The sample was preconditioned by applying a linear stress ramp from 0 to 10 MPa at 0.01 N/s. The relatively high stress limit enabled assurance of adequate clamping fixation following further testing.

Sample stress was then reduced to 7.5 MPa and sinusoidal strain was applied with an amplitude of ± 0.05 mm at frequencies from 0.05 to 50 Hz and a sampling frequency of 10 kHz. This oscillatory test was repeated at 2.5 MPa. The baseline stress levels chosen account for stress-strain behavior in the linear and the low-stress non-linear (i.e. toe or heel) region, respectively. Sample zero length was defined as the clamp-to-clamp distance at 0.1 MPa. To account for the acceleration forces necessary to oscillate the system, its force response when the specimen is not loaded was determined first and then used to correct the acquired signal during testing. We computed the tensile modulus (The ratio between stress and strain amplitude, E) and the phase angle (θ) between input strain and the associated stress response at each applied frequency for both stress levels (Fig. 2). Tendon exhibits monotonic strain-rate stiffening at the tested frequencies and the tensile modulus as a function of oscillation frequency (ω) were parameterized using the function:

$E(ω) = E_1 + E_ω \log_{10} ω$ Where $E_1$ and $E_ω$ hence represent the elastic modulus intercept and strain-rate stiffening, respectively. Viscous energy loss during loading which manifests in force hysteresis can be adequately described by the strain-stress tangent phase angle. We found consistent monotonical increase in the tangent phase angle with increasing oscillation frequency and applied a linear fit where the intercept (tanθ<sub>0</sub>) quantifies strain-rate independent and the slope (tanθ<sub>ω</sub>) expresses strain-rate dependent energy dissipation.

$$tanθ(ω) = tanθ_0 + tanθ_ωω$$
<table><caption>**Table 1**
Shear wave descriptors used in the current study.</caption><thead><tr><th>Descriptor</th><th>Symbol</th><th>Units</th><th>Description</th></tr></thead><tbody><tr><td>Group velocity</td><td>v<sub>g</sub></td><td>m/s</td><td>Velocity of the envelope of the impulsive wave</td></tr><tr><td>Shear wave attenuation</td><td>α<sub>g</sub></td><td>m<sup>-1</sup></td><td>Attenuation of the impulsive wave group</td></tr><tr><td>Frequency at maximum power</td><td>ω<sub>max p</sub></td><td>Hz</td><td>Frequency with maximum power in the wavenumber-frequency representation</td></tr><tr><td>Phase velocity at maximum power</td><td>c<sub>max p</sub></td><td>m/s</td><td>Phase velocity at the frequency with maximum power in the wavenumber-frequency representation</td></tr><tr><td>Attenuation at maximum power</td><td>α<sub>max p</sub></td><td>m<sup>-1</sup></td><td>Dissipative energy loss of the frequency component with maximum power in the wavenumber-frequency representation</td></tr><tr><td>Frequency full width at half maximum</td><td>FWHM<sub>to</sub></td><td>Hz</td><td>Frequency span of the propagating wave</td></tr><tr><td>Voigt fit</td><td></td><td></td><td></td></tr><tr><td>Shear elasticity</td><td>μ<sub>c</sub></td><td>Pa</td><td>Shear elastic component of the Voigt function fitted on the phase velocity-frequency behavior of the impulsive wave</td></tr><tr><td>Shear viscosity</td><td>η<sub>c</sub></td><td>Pa*s</td><td>Dissipative component of the Voigt function</td></tr><tr><td>Guided wave propagation model (Brum et al., 2014)</td><td></td><td></td><td></td></tr><tr><td>Compression modulus</td><td>c<sub>13</sub></td><td>Pa</td><td>Compression modulus as estimated by fitting a wave propagation model accounting for the limited spatial extent of the sample</td></tr><tr><td>Shear modulus</td><td>c<sub>55</sub></td><td>Pa</td><td>Shear modulus as estimated by fitting a wave propagation model accounting for the limited spatial extent of the sample</td></tr></tbody></table>

## 2.5. Statistical analysis

Goodness-of-fit of the parameterization of the tensile mechanical behavior was quantified by computing the coefficient of determination (R²) between observed and fitted data. Reliability in estimating tendon tensile viscoelasticity and mass density based on two tendon subsections was quantified by the intra-class correlation coefficient (ICC(2,1)) based on a two-way random effects model assessing the absolute agreement of an average-measure approach (Shrout and Fleiss, 1979). ICC values were classified as poor (≤0.2), fair (0.21–0.4), moderate (0.41–0.6), good (0.61–0.8), and very good (>0.8) (Altman, 1991). SWE measurement repetitions after repositioning of the ultrasound probe were analyzed with regards to reliability analogously. Intra- and inter-domain associations were quantified using bivariate spearman rank correlation test. Statistical significance was set at α = 0.05. The analysis was conducted in MATLAB (The Math Works, Inc., 2020a).

# 3. Results

## 3.1. Tendon tensile viscoelasticity characterization

Log-linear parameterization of tendon elasticity in response to oscillation frequency yielded a mean ± SD coefficient of determination of R² = 0.95 ± 0.09. Stress–strain tangent phase angle was fitted with a coefficient of determination of R² = 0.71 ± 0.16. Fig. 3 provides an example of the performed fits.

Reliability in estimating global tendon tensile viscoelasticity based on two sections ranged from moderate to very good (Table 2).

We found strong correlations among the various measures of tensile viscoelasticity (Fig. 4).

## 3.2. Mass density and tendon diameter

Tendon mass density ranged from 1060 to 1250 kg/m³ (Mean ± SD: 1110 ± 350 kg/m³) with measurements of tendon sample pairs from the same explant yielding an ICC of 0.700 (95 %CI: 0.288; 0.875). Tendon diameter ranged from 3.5 to 11.6 mm (Mean ± SD: 7.1 ± 2.2).

## 3.3. Shear wave elastography descriptors

Shear wave elastography yielded very good reliability in all assessed descriptors (Table 3).

Shear wave elastography descriptors showed strong intra-domain associations (Fig. 5).

## 3.4. Inter-domain associations

We found no association between shear wave group velocity and tendon tensile stiffness or strain rate-stiffening, at either applied tissue baseline stress level. Group velocity was however associated with tendon tensile hysteresis. These associations were even stronger for the phase velocity at maximum power (c<sub>max p</sub>). On the other hand, tendon stiffness as well as strain rate-stiffening was associated with the frequency span of substantial acoustic excitation (FWHM<sub>to</sub>). Except for the Voigt-estimated shear elasticity (μ<sub>c</sub>), tendon diameter had no significant effect on shear wave descriptors. Almost all shear wave descriptors showed a significant association with tendon mass density (Fig. 6).

Fig. 7 visualizes the nature of the association for a selection of the assessed parameters. Scatterplots of all assessed associations are provided in the supplementary material (Figure S1).

# 4. Discussion

When subjected to low strain-rate loading, tendon and ligament compliance enables and accommodates a wide range of joint movements (Biewener, 2008). High strain-rate loading both requires and results in stiffening of these passive structures thereby protecting the joint from overextension and minimizing the required muscular contraction range (Biewener, 2008). Tendon elastic behavior limits dissipation of strain energy which prevents excessive heat accumulation (Lichtwark and

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Schematic representation of the oscillatory tensile testing. A: Sinusoidal elongation with an amplitude of 0.05 mm was applied at two stress levels with frequencies from 0.05 to 50 Hz. B: The phase angle between input strain and associated stress response quantifies dissipative energy loss during deformation.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Exemplary visualization of the parameterization of the tensile viscoelastic behavior at 7.5 MPa of tensile stress. A: Tendon tensile elasticity was parameterized with a log-linear fit as a function of oscillation frequency (R² of the displayed example: 0.882). B: Tensile hysteresis as a function of oscillation frequency was parameterized by fitting a linear function to the observed tangent phase difference angle (R² of the displayed example: 0.997).</figcaption></figure>

<caption>**Table 2**</caption>

Descriptive statistics and associated reliability in estimating tendon tensile viscoelasticity properties based on two samples of the same explant. ICC: Intra-class correlation coefficient. CI: Confidence interval.

<table><thead><tr><th>Parameter</th><th>Symbol used</th><th>Stress level [MPa]</th><th>Mean ± SD</th><th>Units</th><th>ICC</th><th>ICC 95 % CI</th></tr></thead><tbody><tr><td rowspan="2">Elastic modulus intercept</td><td rowspan="2">E<sub>1</sub></td><td>2.5</td><td>379 ± 67.1</td><td>MPa</td><td>0.567</td><td>-0.028; 0.819</td></tr><tr><td>7.5</td><td>676 ± 107 MPa</td><td>MPa</td><td>0.624</td><td>0.107; 0.843</td></tr><tr><td rowspan="2">Strain rate- stiffening</td><td rowspan="2">E<sub>o</sub></td><td>2.5</td><td>20.4 ± 5.61</td><td>MPa/log<sub>10</sub>Hz</td><td>0.689</td><td>0.263; 0.870</td></tr><tr><td>7.5</td><td>26.4 ± 6.64</td><td>MPa/log<sub>10</sub>Hz</td><td>0.755</td><td>0.419; 0.898</td></tr><tr><td rowspan="2">Hysteresis intercept</td><td rowspan="2">tanθ<sub>0</sub></td><td>2.5</td><td>59.1e-3 ± 9.38e-3</td><td rowspan="2"></td><td>0.911</td><td>0.788; 0.963</td></tr><tr><td>7.5</td><td>52.3e-3 ± 7.97e-3</td><td>0.719</td><td>0.334; 0.883</td></tr><tr><td rowspan="2">Strain rate- dependent hysteresis increase</td><td rowspan="2">tanθ<sub>o</sub></td><td>2.5</td><td>1.45e-3 ± 0.198e-3</td><td>Hz<sup>-1</sup></td><td>0.742</td><td>0.387; 0.892</td></tr><tr><td>7.5</td><td>1.25e-3 ± 0.229e-3</td><td>Hz<sup>-1</sup></td><td>0.697</td><td>0.281; 0.873</td></tr></tbody></table>

Cresswell, 2013; Wilson and Goodship, 1994) and increases movement efficiency (Biewener, 2008; Lichtwark and Barclay, 2010). Small amplitude oscillatory testing and subsequent parameterization allows a comprehensive characterization of the viscoelastic properties of tendon that enable these functions. For each input sinusoidal strain signal, amplitude and phase difference of the corresponding stress response is dependent on the frequency-specific stiffness and hysteresis, respectively. Tendon stiffness responded in a linear fashion to the logarithm of the applied frequency within the range of applied frequencies. Dissipation of strain energy or hysteresis – by definition, the ratio between storage and loss modulus (tanθ) – increased linearly albeit with a small slope with increasing oscillation frequency. Strain energy dissipation was found to be lower at the higher of the two assessed baseline stress levels.

The accuracy in experimentally determining tendon tensile viscoelasticity is affected by various confounding factors. For instance, anatomical heterogeneity including non-uniform cross-sectional area as well as deformations and slippage at the clamping site can falsify measurements (Firminger and Edwards, 2021). Our approach of extracting and testing discrete fiber bundles circumvented some of these potential confounders. Testing two fiber bundles per tendon allowed an estimation of the measurement validity by inspecting the correlation between sample pairs.

In situ characterization of tendon tensile viscoelasticity would be of value in the context of basic and sports science alike. For instance, degenerative tendon conditions have been linked with altered tensile behavior (Wang et al., 2012; Wiesinger et al., 2020) but the study of

these pathomechanisms is impeded by the limited capability of available in situ measurement methods. Measuring voluntary joint torque and assessing the corresponding change in tendon length theoretically enables estimation of tendon tensile viscoelasticity. Numerous factors however, including limitations in human neuromuscular control, considerable noise in tendon length measurements and limited temporal resolution renders the technique to be applicable in the quasi-static context only, ignoring strain-rate dependent effects (Lichtwark and Cresswell, 2013; Peltonen et al., 2013; Seynnes et al., 2015). Furthermore, these measurements are cumbersome which limits their applicability for large scale testing.

By avoiding some of these limitations, US SWE may offer a practical approach to gain information on tendon mechanical characteristics in the viscoelastic domain. In isotropic media, viscoelastic properties can readily be derived by inspecting frequency-dependent phase velocity and amplitude attenuation (Kazemirad et al., 2016; W. Urban et al., 2012). Such models however, disregard mode conversion and waveguide effects potentially distorting the observed behavior. By modelling the interaction of the impulsive wave with the medium’s borders and the surrounding material, the effect of limited axial diameters can additionally be accounted for (Lu et al., 2021; Maksuti et al., 2016; Nenadic et al., 2011; Tanter et al., 2009). However, our data indicate that shear wave descriptors of tendon as a transverse isotropic structure were barely affected by even the highly variable sample diameters across our sample set (range: [3.5, 11.6 mm], SD: 2.2 mm). On the contrary, fitting the observed phase velocity with an analytical model that was specifically derived to correct for diameter-induced phase velocity dispersion
Intra-domain associations

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Visualization of all assessed tensile intra-domain associations from dynamic mechanical testing of tendon samples. Spearman rank correlations are indicated for significant (p < 0.05) relationships among tendon tensile properties. Two nodes are connected with red or blue lines indicating positive or negative correlations respectively, with the thickness of the line indicating the strength of the correlation. The superscript on the mechanical parameters refers to the applied tissue baseline stress level. Positive correlations exist between elastic tangent modulus (E₁) and strain rate stiffening modulus (E₀), while these elastic properties negatively correlate to the viscoelastic parameters of hysteresis (tanθ₀) and strain rate dependent hysteresis (tanθ₀). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 3**</caption>

Descriptive statistics and associated reliability of the shear wave elastography descriptors assessed in this study.

<table><thead><tr><th>Descriptor</th><th>Symbol used</th><th>Mean ± SD</th><th>Units</th><th>ICC</th><th>ICC 95 % CI</th></tr></thead><tbody><tr><td>Group velocity</td><td>v<sub>g</sub></td><td>10.60 ± 3.69</td><td>m/s</td><td>0.983</td><td>0.969; 0.991</td></tr><tr><td>Shear wave attenuation</td><td>α<sub>g</sub></td><td>89.0 ± 29.8</td><td>m<sup>-1</sup></td><td>0.933</td><td>0.881; 0.965</td></tr><tr><td>Frequency at maximum power</td><td>ω<sub>max p</sub></td><td>479 ± 66.6</td><td>Hz</td><td>0.926</td><td>0.867; 0.961</td></tr><tr><td>Phase velocity at maximum power</td><td>c<sub>max p</sub></td><td>7.72 ± 0.378</td><td>m/s</td><td>1.000</td><td>1.000; 1.000</td></tr><tr><td>Attenuation at maximum power</td><td>α<sub>max p</sub></td><td>125 ± 15.4</td><td>m<sup>-1</sup></td><td>0.974</td><td>0.953; 0.986</td></tr><tr><td>Frequency full width at half maximum</td><td>FWHM<sub>ω</sub></td><td>866 ± 263</td><td>Hz</td><td>0.972</td><td>0.949; 0.985</td></tr><tr><td>Voigt fit</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Shear elasticity</td><td>μ<sub>c</sub></td><td>3.95e3 ± 3.62e3</td><td>Pa</td><td>0.902</td><td>0.824; 0.949</td></tr><tr><td>Shear viscosity</td><td>η<sub>c</sub></td><td>34.8 ± 13</td><td>Pa*s</td><td>0.972</td><td>0.950; 0.986</td></tr><tr><td>Guided wave propagation model</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Compression modulus</td><td>c<sub>13</sub></td><td>3.36e9 ± 2.39e5</td><td>Pa</td><td>0.946</td><td>0.903; 0.972</td></tr><tr><td>Shear modulus</td><td>c<sub>55</sub></td><td>3.71e5 ± 2.78e5</td><td>Pa</td><td>0.938</td><td>0.889; 0.968</td></tr></tbody></table>

yielded SWE descriptors (c₁₃, c₅₅) that were most affected by the varying diameter. We speculate that the cause of this behavior is the hierarchical architecture and the resulting internal structural heterogeneity of the tendon. Given that the tendon cross-section is organized into discrete subunits of varying size, knowledge of the tendon diameter alone appears to be insufficient to account for the guided wave phenomena that take place. Nevertheless, we conclude that, given the heuristic approach of the current study and within the tendon diameter range studied herein, application of pulse propagation models with explicit spatial

dimensions (Helfenstein-Didier et al., 2016; Lu et al., 2021; Maksuti et al., 2016; Nenadic et al., 2011; Tanter et al., 2009) is neither required nor helpful in the current context.

Although shear wave speed is frequently used as a proxy for longitudinal stiffness of transversely isotropic structures such as the tendon (Karatekin et al., 2018; Zhang et al., 2016, 2014), theoretical modelling predicts no such relationship (Pelivanov et al., 2019; Royer et al., 2011). Our experimental data clearly support this postulate in that we did not find correlations between shear wave group / phase velocity and tendon stiffness at either stress level. This finding is corroborated by previous work which found no correlation between the tensile elastic modulus of healthy rabbit Achilles tendons and shear wave group velocity (Martin et al., 2015). Similarly, we did not find any relationship between shear wave attenuation and tensile dissipative behavior.

We did however find an association between unloaded SWV and tendon tensile dissipative behavior. Attenuation and SWV showed a negative and a positive association with the mass density of the tendon, respectively. Given that the main constituents of tendon are collagen (~15%) and water (~60%) with regards to mass fraction (McKee et al., 2019), tendon mass density is likely primarily modulated by its collagen content. Following this line of thought, high collagen content may yield high shear elasticity, thereby decreasing viscous energy dissipation as manifested in a lower attenuation coefficient. On the same account, high collagen content potentially results in an increase in shear stiffness leading to higher SWV. Although beyond the scope of this investigation, the observation that tendon mass density was not predictive for its tensile stiffness may indicate that tendon stiffness is primarily dependent on structural organization, relative abundance of the different collagen types and the amount of cross-linking (Zhang et al., 2021). On the same note, high tendon mass density – and proportionally lower water content – was associated with low shear wave attenuation. The observed associations of FWHM<sub>ω</sub> with both tendon stiffness and hysteresis warrants closer investigation. To the best of our knowledge this quantity has not yet been reported in a comparable context. We speculate FWHM<sub>ω</sub> to be affected by the internal tendon structure such as fibrillar diameter and -spacing. Future studies may further investigate
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Spearman rank correlations with $p < 0.05$ among the shear wave elastography descriptors. Strong intra-domain associations were present. For instance, high shear wave group velocity was associated with low shear wave attenuation of the frequency component with maximum power.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Spearman rank correlations with $p < 0.05$ among shear wave descriptors, assessed confounding variables and tensile viscoelastic properties at the two assessed stress levels of 2.5 and 7.5 MPa. Interestingly, shear wave group ($v_g$) or phase velocity ($c_{max p}$) showed no association with tendon stiffness at any stress level ($E_1$ and $E_\infty$) but significant associations were present with tensile hysteresis ($\tan\theta_0$ and $\tan\theta_\infty$). The spectral range of the propagating wave on the other hand was associated with both tendon stiffness and hysteresis.</figcaption></figure>

the effect of specific tendon structural properties, such as collagen cross linking, on the currently used – or similar – shear wave descriptors.

This study carries limitations worth considering. SWE was conducted on isolated tendons where the ultrasound transducer and the sample were separated by a small layer of ultrasound gel, whereas in vivo, tendons will be covered by different tissue layers. However, provided the shear stiffness of the neighboring tissue is low compared to the tendon's, this change in surrounding medium has a negligible effect on the acquired descriptors (Brum et al., 2014; Couade et al., 2010). Measuring tendon tensile tangent modulus showed lower repeatability

compared to the determination of the tendon's viscous behavior. The observation of an absence of association between tendon tensile stiffness and most SWE descriptors should therefore be interpreted with appropriate caution. Tendon tensile tangent modulus varies with the applied strain with low magnitude at small deformations, followed by a gradual increase reaching a region of invariance, ending in sudden failure (Fessel and Snedeker, 2009). Sampling the tangent modulus at two stress levels only therefore does not provide an exhaustive representation of the stress–strain behavior of the sample. Given the strong correlations between tensile parameters at the two stress levels however, one might
Inter-domain associations

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Scatterplots of selected inter-domain associations. We found no association between shear wave group velocity (top row) with tendon stiffness ($E_1$ and $E_\omega$) but with tendon strain dissipative behavior ($\tan\theta_0$ and $\tan\theta_\omega$). The full width at half maximum of the frequency power spectrum on the other hand, showed an association with both components of tendon mechanical properties. $\sigma$: Spearman rank correlation coefficient. p: P-value.</figcaption></figure>

assume that viscoelasticity properties are highly interdependent among the entire strain range. Whereas tensile testing discrete fiber bundles has operational advantages as outlined above potential between-bundle interactions in whole organ deformation will naturally evade this method.

In conclusion, the presented data provides insights in the functional and mechanistical implications of SWE measurements in transversely isotropic collagenous soft tissue. Our study highlights the limited utility of SWE for assessing the tensile elastic properties of tendon as non-isotropic medium, yet indicated that it may be useful for characterizing viscous behaviors. Moreover, our results warrant further investigations of selected SWE descriptors towards *in situ* assessment of the viscoelastic properties of these tissues, potentially in combination with other measurement modalities.

## Funding Information

The author(s) received no financial support for the research, authorship, and/or publication of this article.

## Conflict of interest statement

JLG is scientific consultant for the company: Supersonic Imagine.

## Conflict of Interest

The authors declare no conflict of interest. JLG is scientific consultant for the company: supersonic imagine.

## CRediT authorship contribution statement

**Tobias Götschi:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Yannick Schärer:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Data curation.
**Jean-Luc Gennisson:** Writing – review &

editing, Supervision, Methodology, Conceptualization.
**Jess G. Schärer:** Writing – review & editing, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

We gratefully acknowledge Marcelo Looser's contribution to this work. Imaging was performed with equipment maintained by the Swiss Center for Musculoskeletal Imaging, SCMI, Balgrist Campus AG, Zürich.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jbiomech.2022.111411.

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