# Simultaneous measurement of tensile and shear elasticity and anisotropy in human skeletal muscle tissue using steered ultrasound shear waves

Ricardo Andrade, Apolline Racapé, Mar Hernández-Secorún, Ha-Hien-Phuong Ngo, Alice Lemoine, Nicolas Etaix, Thomas Frappart, Christophe Fraschini, Jean-Luc Gennisson, Antoine Nordez

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Ricardo Andrade, Apolline Racapé, Mar Hernández-Secorún, Ha-Hien-Phuong Ngo, Alice Lemoine, et al.. Simultaneous measurement of tensile and shear elasticity and anisotropy in human skeletal muscle tissue using steered ultrasound shear waves. Acta Biomaterialia, 2025, 10.1016/j.actbio.2025.05.010. hal-05056264

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# Simultaneous measurement of tensile and shear elasticity and anisotropy in human skeletal muscle tissue using steered ultrasound shear waves

Ricardoj. Andrade<sup>a, *, †</sup>, Apolline Racapé<sup>a †</sup>, Mar Hernández-Secorún<sup>a †</sup>, Ha-Hien-Phuong Ngo<sup>b</sup>, Alice Lemoine<sup>c</sup>, Nicolas Etaix<sup>c †</sup>, Thomas Frappart<sup>c</sup>, Christophe Fraschini<sup>c</sup>, Jean-Luc Gennisson<sup>b, 1 †</sup>, Antoine Nordez<sup>a, d, 1, *</sup>

ARTICLE INFO

Keywords:

Elastography

Tissue elasticity imaging

Medical imaging

Ultrasonography

Muscle

Biomechanics

ABSTRACT

Load-bearing skeletal muscle tissues are reinforced by intricate networks of protein fibers aligned in preferential orientations, imparting direction-dependent mechanical properties (anisotropy). Characterizing this anisotropy *in vivo* is essential for understanding both normal and pathological muscle function, as well as structural integrity. However, current noninvasive techniques are limited in their ability to accurately measure the mechanical properties of anisotropic tissues such as skeletal muscle. Here, we used an innovative angle-resolved ultrasound elastography method, recently developed by our team, to simultaneously quantify tensile and shear elasticity and anisotropy, enabling comprehensive assessment of muscle biomechanics. We fully characterized the mechanical properties of the *biceps brachii* in fourteen healthy young adults under passive and active axial loadings, revealing distinct shear and tensile mechanical behaviors both along and across muscle fibers. Notably, tensile and shear moduli along the main fiber orientation were found to be uncoupled, while cross-muscle fiber measurements exhibited a consistent modulus ratio of 3.4 ± 0.2, regardless of axial loading conditions or intensities. These findings highlight the anisotropic nature of skeletal muscle and provide valuable insights into its *in vivo* mechanical behavior. Both shear and tensile anisotropy increased with muscle axial physiological loading, indicating that our method is sensitive to changes in anisotropic tissue mechanics. Lastly, we demonstrated that angle-resolved ultrasound shear wave elastography provides direct estimates of shear and tensile properties, offering significant promise for clinical applications, including neuromuscular disease diagnostics and monitoring, biomechanical modeling for predicting tissue responses to loading and therapies, and tissue engineering.

*Statement of significance:* : Conventional ultrasound shear wave elastography techniques overlook the anisotropy of skeletal muscles, leading to incomplete tissue mechanical characterization. In this study, we leveraged an innovative angle-resolved elastography method to assess tensile and shear elasticity, along with their anisotropic factors, of human muscle *in vivo*. For the first time, we reveal the intricate relationships between tensile and shear elasticities during active and passive physiological loading. This technique enhances our understanding of muscle mechanics and has promising clinical implications for muscle health and neuromuscular disease management, where tissue structural and mechanical properties are often altered. Additionally, it could help in developing constitutive models for muscle tissue and contribute to the design of tissue-engineered materials.

<footnote><sup>a</sup> Nantes Université, Movement - Interactions - Performance, MIP, UR 4334, F-44000 Nantes, France</footnote>

<footnote><sup>b</sup> Laboratoire d'imagerie biomédicale multimodale (BioMaps), University Paris-Saclay, CEA, CNRS UMR 9011, Inserm UMR 1281, Orsay, F-91401, France</footnote>

<footnote><sup>c</sup> Supersonic Imagine, Aix-en-Provence, France</footnote>

<footnote><sup>d</sup> Institut Universitaire de France (IUF), Paris, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: ricardo.andrade@univ-nantes.fr (Ricardoj. Andrade), antoine.nordez@univ-nantes.fr (A. Nordez).</footnote>

<footnote n="1">¹ Contributed equally.</footnote>
# 1. Introduction

Skeletal muscle is one of the major components of the human body, accounting for about 40 % of its mass and playing a key role in movement and posture, respiration, and basal metabolism [1]. This soft tissue is designed to contract, generating the forces required for voluntary movement, and to withstand high dynamic mechanical loads [2]. The mechanical strength and structural arrangement of the muscle tissue arise from intricate networks of cross-linked proteins, including collagen and elastin fibers, embedded in a hydrated matrix [3]. This extracellular matrix surrounds each myofiber, providing mechanical support [4,5] and regulating biochemical signaling [6,7]. From early development stages, physical constraints at the boundaries of the muscle tissue guide stress and strain along preferential directions, organizing muscle architecture across scales [8,9]. The resulting preferential fiber alignment leads to direction-dependent – i.e., anisotropic or transversely isotropic – mechanical properties that are fine-tuned to meet the functional demands of the soft and elastic muscle tissue [10]. Despite being a fundamental property of muscle, anisotropy remains largely understudied, primarily due to the lack of measurement techniques, both in vivo and ex vivo.

Muscle's elasticity is a key biomechanical property closely interwined with muscle electromechanics and structure, varying dynamically as the muscle contracts and relaxes during healthy human movement. Muscle activation, driven by electrochemical energy, increases muscle elastic modulus due to an increase in the number of force-generating cross-bridges [11,12]. Additionally, passive muscle lengthening induces an increase in muscle elastic modulus as a result of the nonlinear stress-strain behavior of passive components of the extracellular matrix and the myocytes [13], which, together, also define tissue elasticity in a resting muscle state. Capturing these properties noninvasively at the tissue level is essential for advancing our understanding of muscle function, as well as adaptation to use, disuse, injury, and neuromuscular diseases [14]. Furthermore, accurately characterizing dynamic mechanical properties in vivo may contribute to advance tissue engineering, where design must be guided by the intrinsic mechanical properties of the native tissue, often obtained from passive material tensile testing of ex vivo tissue samples [15-19].

Over the past decade, ultrasound shear wave elastography (SWE) has become a leading noninvasive technique for biomechanical characterization of soft tissues in vivo. SWE measures the velocity of shear waves generated within a tissue by a focused ultrasound push beam. These waves propagate perpendicularly to the ultrasound focused beam axis and are captured by ultrafast ultrasound imaging (>1000 Hz) [20-22]. Based on linear, incompressible, homogeneous, purely elastic and isotropic material assumptions [22,23], the shear (μ) and Young's (E) moduli of some soft tissues, such as the breast and liver, can be directly estimated from the tissue density and shear wave velocity (E ≈ 3μ = ρV²) [24,25]. However, in transversely isotropic tissues such as skeletal muscle, theoretical models predict that shear elasticity and tensile elasticity, described by Young's modulus, are uncoupled [26-28]. Therefore, to fully characterize muscle tissue as an elastic, incompressible, transversely isotropic material, it is required to assess both the shear elasticity – comprising the along-muscle fiber shear modulus (μ∥) and the cross-muscle fiber shear modulus (μ⊥) – and the tensile anisotropy (XE), corresponding to the ratio of the Young's moduli along-muscle (E∥) and cross-muscle (E⊥) fibers [26-28,29]. However, conventional SWE methods are unable to provide direct measurements of tensile elasticity (Young's modulus) for transversely isotropic soft tissues [27]. In addition, it remains experimentally unknown whether muscle shear and tensile anisotropy are independent mechanical properties, as predicted by theoretical models governing shear wave propagation in fiber-reinforced transversely isotropic soft tissues [29,30].

Recently, our group validated the feasibility of exciting and visualizing tilted shear wave propagation using a steered ultrasound push

beams method (Fig. 1) [31]. This innovative angle-resolved technique enables the reconstruction of the shear vertical (SV) propagation mode using phase velocity analysis [27,28,32] and, thus, the direct quantification of tensile anisotropy ($X_E = \frac{E_\parallel}{E_\perp}$) of the targeted muscle in addition to the shear modulus along-muscle fibers ($\mu_\parallel$) [33]. Here, we comprehensively characterized, for the first time, local shear and tensile elasticity and anisotropy in an intact human muscle (biceps brachii) during physiological loading conditions, including both passive lengthening and isometric contractions. In line with the theory governing shear wave propagation in transversely isotropic soft tissues, we hypothesized that shear and tensile elasticity, as well as anisotropy factors, are uncoupled.

# 2. Methods

## 2.1. Participants

Fourteen healthy participants (5 women; age: 22.2 ± 3.5 years; body mass 65.4 ± 9.7 kg; height: 173.0 ± 8.6 cm) with no known history of neuromuscular diseases or injuries were enrolled in this study. All procedures were approved and overseen by the Institutional Review Boards of Nantes Université (CERN n°IRB: IORG0011023), and written consent was obtained from all participants prior to testing. All experiments were performed in accordance with the principles of the Declaration of Helsinki.

## 2.2. Ultrasound shear wave elastography with steered push beams: acquisition sequence and analysis

The ultrasound transducer was positioned either parallel (along-muscle) or perpendicular (cross-muscle) to the muscle fibers of the biceps brachii (long head). This muscle has a fusiform geometry, with fibers running parallel both to the muscle shortening direction and axis of symmetry. All measurements were performed at the mid-distance of the total muscle length previously identified by B-mode imaging. The ultrasound transducer was mounted in a dynamic probe fixation system (USONO, Eindhoven, Netherlands) to allow 90° rotation about the transducer's vertical axis (Fig. 2c–d).

The ultrasound steered push beam method, including the derivation of the elasticity tensor and shear wave propagation in a linear, incompressible ($\nu = 0.5$), transversely isotropic soft material has been previously described [31]. All shear wave elastography data were acquired with an ultrasound ultrafast system (Mach30, SuperSonic™ Imagine, Aix-en-Provence, France) and a linear ultrasound transducer (SL18–5, center frequency 7.5 MHz, 256 transducer elements, pitch 0.2 mm) controlled by a custom MATLAB (2022a, Mathworks, Natick, MA, USA) acquisition script [31]. The acquisition sequence framework is illustrated in Fig. 1. In the excitation stage (Fig. 1a), the steered ultrasound push beams were oriented with an angle $\varphi$ ranging from –18° to 18° with a gradual 3° increment (i.e., 13 $\varphi$ angles). The steered push beams were electronically created by small apertures with an oriented delay law of an angle $\varphi$. For each of the 13 $\varphi$ angles, four push lines, spaced 5 mm apart, were successively generated. In the imaging stage (Fig. 1b), the tracking ultrafast ultrasound sequence consisted of five tilted plane waves, oriented with the angle $\varphi$ of each push beam, ranging from $\varphi - 4°$ to $\varphi + 4°$, by 2° increments [34]. The tilted plane waves were used to improve the quality of the data [35]. These five plane waves were repeated for each push line. The ultrasound in-phase and quadrature signals (IQ) were obtained at a frame rate of 5 kHz for each push beam angle $\varphi$. These signals were then used to obtain the displacement field induced by shear wave propagation for each push beam $\varphi$. The total acquisition duration for the 13 pushing angles $\varphi$ was < 1 s.

All ultrasound data were analyzed in MATLAB using custom scripts. The fiber-to-transducer face angle $\alpha$ was initially measured on a B-mode image for each measurement. The plane wave imaging with delay and
<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic of steered ultrasound push beam sequence to generate and track the shear vertical (SV) propagation mode.</figcaption></figure>

(a) Thirteen steered push beams are electronically programmed with an angle $\varphi$ (ranging from -18° to 18°, with a 3° increment) to generate different focused acoustic radiation forces that create shear waves with a given propagation vector. (b) Ultrafast ultrasound plane waves enable the tracking of the propagating shear wave fronts. For each push beam $\varphi$, five steered plane waves (ranging from -4° to 4°, with a 2° increment) are used to track the respective propagating shear wave fronts. (c) The shear wave velocity is detected by applying a time-of-flight algorithm to the recorded shear wave propagation films. Typical examples of the propagation films (frames at 0.75 ms, 1.55 ms and 2.35 ms) for push beam angles $\varphi=-15°$, $\varphi=0°$, and $\varphi=15°$ are illustrated. (d) Shear wave velocity maps are reconstructed for each push beam. An ellipsoidal ROI (depicted in black and four points) is drawn across the different reconstructed shear wave velocity maps in the same muscle region across all the push beams $\varphi$. The shear wave velocity within each ROI/push beam angle is averaged and the elastic parameter $X_{E,\mu\perp}$ and the along-muscle fiber shear modulus $\mu_{\parallel}$ are calculated using the Eq. (1) for the SV propagation mode [$\rho V_{SV}^2 = \mu_{\parallel} + (X_{E,\mu\perp} - \mu_{\parallel})\sin^2(2\theta)$], as shown in the graphic. The angle $\theta$ corresponds to the difference between the push beam angle $\varphi$ and the fiber-to-transducer angle $\alpha$ previously measured on a B-mode figure.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Representative photographs of a participant during data collection and ultrasound transducer positioning to acquire elastography data along-muscle and cross-muscle fibers.</figcaption></figure>

(a) The participant is seated in an isokinetic dynamometer chair with the trunk secured and the shoulder abducted at 90°. The elbow angle was passively rotated from 70° to 180° in 5° increments for the axial lengthening condition. For the isometric contractions, the participant's elbow was held at a 90° flexion, while they maintained targeted myoelectric activity via visual feedback (from 0 % to 20 % of maximal voluntary isometric contraction) for ultrasound shear wave elastography acquisitions. (b) Placement of the EMG electrodes and visual feedback provided during data collection for the isometric contractions. (c) Ultrasound transducer positioning for ultrasound-based measurements along-muscle fibers. The lower panel shows a typical B-mode image of the *biceps brachii* (outlined in yellow) in a longitudinal view, with muscle fibers aligned parallel to the ultrasound transducer. (d) Ultrasound transducer positioning for ultrasound-based measurements cross-muscle fibers. The lower panel shows a typical transverse B-mode image in this cross-sectional view of the *biceps brachii* long head muscle outlined in yellow.

sum beamforming with specular reconstruction [36], which accounts for the fiber-to-transducer angle, was adopted to reconstruct each frame. The particle velocity field was then calculated offline for each push beam angle and push line using a time-of-flight algorithm [37,38] to the recorded shear wave propagation films [20] (Fig. 1c). For each push beam angle, we then performed a concatenation across the four push lines [37] to create a representative shear wave velocity map of the medium (e.g., elastography maps illustrated in Fig. 1d).

The along-muscle fiber shear modulus ($\mu_{\parallel}$) and the elastic parameter $X_{E,\mu\perp}$, defined as the product of tensile anisotropy ($X_E$) and the shear modulus perpendicular to the muscle fibers ($\mu_{\perp}$), were first quantified using the data acquired with the probe aligned along the fiber's main shortening direction. This was achieved fitting the shear wave velocity data (i.e., one value per push beam angle $\varphi$) with the Eq. (1), governing the SV propagation mode:
$$ \rho V_{SV}^2 = \mu_{\parallel} + (X_{E} \cdot \mu_{\perp} - \mu_{\parallel}) \sin^2(2\theta) \quad (1) $$

where $\rho$ is the density assumed to be constant (1100 kg.m$^{-3}$); $\mu_{\parallel}$ and $\mu_{\perp}$ are the shear modulus along-muscle and cross-muscle fibers respectively; and $X_E$ is the tensile anisotropy corresponding to the ratio of the Young's moduli along-muscle ($E_{\parallel}$) and cross-muscle ($E_{\perp}$) fibers, and $\theta$ is the angle formed by the polarization vector (or push beam angle $\varphi$) and the transducer-to-fiber orientation ($\alpha$). The theoretical framework relating these elastic constants to shear wave elastography measurements has been extensively described in previous studies [35,39,40]. The same region of interest (ROI), an ellipse with a radius of ~6 mm, was used for each steered push beam (Fig. 1d and Fig. 4c). A coordinate system ensured consistent measurements within the exact muscle region across all push beam angles $\varphi$.

Cross-muscle fibers shear modulus ($\mu_{\perp}$) was obtained from measurements performed with the ultrasound transducer oriented perpendicular to the fusiform muscle fibers, and directly estimated using the Eq. (2) for the shear horizontal (SH) propagation mode:

$$ \rho V_{SH}^2 = \mu_{\parallel} \cos^2 \beta + \mu_{\perp} \sin^2 \beta \quad (2) $$

where $\beta$ represents the orientation of the ultrasound transducer relative to the muscle fibers. For this configuration, we applied the same image processing pipeline; however, only the shear wave velocities from the push beam angle perpendicular to skin ($\varphi = 0^{\circ}$) were used to calculate the shear modulus. In this transducer-to-fiber's configuration, $\beta = 90^{\circ}$, and Eq. (2) simplifies to:

$$ \rho V_{SH}^2 = \mu_{\perp} \quad (3) $$

Therefore, no fitting procedure was required, and $\mu_{\perp}$ was obtained directly. Fig. 3 illustrates the fusiform structure of the fiber-reinforced human biceps brachii muscle (Fig. 3a) and depicts its mechanical anisotropy. This is shown by the distinct particle velocities along and across the muscle fibers (Fig. 3b), which result in different shear wave velocities, as exemplified by the measurements taken at a push beam angle of $\varphi = 0^{\circ}$ (Fig. 3c).

Based on these measurements, the shear anisotropy was directly calculated as the ratio of the shear moduli along-muscle fiber ($\mu_{\parallel}$) and cross-muscle fiber ($\mu_{\perp}$), as follows [41]:

$$ X_{\mu} = \frac{\mu_{\parallel}}{\mu_{\perp}} \quad (4) $$

where $\mu_{\parallel}$ is derived from Eq. (1) and $\mu_{\perp}$ from Eq. (3).

The use of the steered push beams method within a rotational transducer approach (Fig. 2c-d) applied to the fusiform biceps brachii muscle further enabled the direct quantification of the tensile anisotropy, given by:

$$ X_E = \frac{X_E \cdot \mu_{\perp}}{\mu_{\perp}} \quad (5) $$

where the elastic parameter $X_E \cdot \mu_{\perp}$ was calculated from Eq. (1), while the cross-muscle fibers shear modulus ($\mu_{\perp}$) was obtained from Eq. (3). This approach has been previously validated against material tensile testing in ex vivo porcine muscle [31].

Finally, using these parameters, the Young's modulus along-muscle ($E_{\parallel} = 4 \times X_E \cdot \mu_{\perp} - \mu_{\perp}$) [42,43] and cross-muscle ($E_{\perp} = \frac{E_{\parallel}}{X_E}$) fibers were then computed to provide a detailed and individual characterization of the directional tensile moduli. These steps are illustrated in Fig. 4.

## 2.3. Experimental protocol

Ultrasound scans of the mid-portion of the long head of the biceps brachii were conducted on the right upper limb of each participant during a single experimental session of approximately three hours. The session was divided into two phases: initially, measurements were performed at different myoelectric levels of submaximal isometric muscle contractions, followed by scans conducted at various passive muscle lengths. For both phases, participants were seated in an isokinetic dynamometer chair (System 3; Biodex Medical Systems, NY, USA), secured at the trunk with the shoulder abducted at 90°. The forearm and wrist were positioned in pronation, and firmly strapped to avoid any displacements during the elbow rotations and muscle isometric contractions. The medial epicondyle of the humerus was aligned with the axis of rotation of the dynamometer and used to estimate the elbow angle (Fig. 2a). The dynamometer was used to passively move the forearm and set the targeted elbow angle for SWE measurements. Prior

<figure><img src="image_5.png" /><figcaption>Fig. 3. Skeletal muscle mechanical anisotropy.</figcaption></figure>

(a) B-mode ultrasound images of the biceps brachii muscle acquired along-muscle and cross-muscle fibers. (b) Propagation of shear wave fronts for each transducer-to-muscle configuration, demonstrating faster shear wave propagation along-muscle compared to cross-muscle fibers. (c) Typical shear wave velocity maps corresponding to the respective propagation films for both transducer-to-muscle configurations. The region of interest (ROI) used for shear wave velocity quantification is indicated by a dotted white ellipse within the maps.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Flowchart depicting the assessment of elasticity and anisotropy parameters.</figcaption></figure>

A schematic representation outlining the methodology used to quantify elasticity and anisotropy parameters, including key steps and processes involved in data acquisition and analysis.

each set of SWE measurements (i.e., muscle contraction or passive lengthening), five passive elbow rotations (5°/s) were performed between 70° and 180° as a pre-conditioning of the neuromusculoskeletal structures acting on the elbow joint [44].

**Muscle submaximal isometric contractions.** Following a standardized warm-up, participants were asked to perform two maximal voluntary isometric contractions (MVIC) in flexion with the elbow joint positioned in a neutral flexion-extension position (90°) for 3 s with 1 min rest interval in between. Elbow angle, torque and EMG-RMS data were simultaneously acquired at 1 kHz and real-time displayed using a Biopac system (MP35, Biopac Systems Inc., Goleta, CA, USA). A pair of surface electrodes (Kendall 100 Series Foam Electrodes, Covidien, Massachusetts, USA) was placed over the biceps brachii (long head) following the Surface EMG for Non-Invasive Assessment of Muscles guidelines [45]. The electrodes were placed immediately medially and aligned parallel to the ultrasound probe (Fig. 2b). The root mean square (RMS) of the EMG signals (RMS-EMG) was calculated in real time using a 300 ms moving average window. All EMG signals were amplified (gain = 1000) and band-pass filtered between 5 and 500 Hz, with a band-stop at 50 Hz. The maximum RMS-EMG across trials was considered as the best performance and then used to determine the submaximal contractions for the ultrasound scans. After 5 min of rest, visual EMG-RMS feedback was provided in real time to the participants. They were carefully instructed to reach and maintain a plateau (Fig. 2b) for approximately 10 s at 5%, 10%, 15% and 20% of maximal EMG-RMS registered during the MVIC with a 2 min rest interval in between. The order of the submaximal contractions was randomized for all participants. The elastography sequence was launched once a stable plateau was visually attained. An additional measurement was performed at a full resting state (0% of EMG-RMS). All measurements were conducted along and across the muscle fibers using the steered ultrasound push beam sequence, in a randomized order across participants. Exact myoelectric activity, expressed as a percentage of maximal EMG-RMS MVIC, at the time of elastography measurements was as follows: 0.4% ± 0.4%, 5.8% ± 1.2%, 11.0% ± 1.5%, 15.5% ± 2.4%, and 20.7% ± 2.6% for the

along-muscle fibers orientation; and 0.5% ± 0.5%, 6.3% ± 1.2%, 11.4% ± 1.8%, 15.9% ± 2.4%, and 21.5% ± 2.6% for cross-muscle fibers. No significant differences in myoelectric activity were observed between along-muscle and cross-muscle fiber acquisitions at equivalent targeted contraction intensities (two-tailed, paired t-tests: *p* > 0.05).

**Muscle passive axial lengthening.** Ultrasound data were acquired at various elbow angles, from 70° (initial position) to 180° (full extension) in 5° increments. At 70° the long head of the biceps brachii is in an axially unloaded state – i.e., under the muscle slack angle [46]. Elastography measurements were performed along-muscle and cross-muscle fibers in a randomized order at the same elbow angles. The isokinetic dynamometer was used to passively rotate the elbow angle at 2°/s (Fig. 2a). The elastography sequence was immediately launched once the elbow reached the targeted ankle angle. To mitigate the impact of viscoelastic stress relaxation on the SWE measurements [47], the elbow was promptly returned to the initial flexion angle (70°) after each elastography acquisition (~ 5–6 s). Then, the elbow was passively rotated into the next measurement angle for the subsequent elastography scan. This procedure was repeated until 180°, representing a total of 23 passive measurements. EMG-RMS was recorded as per the contraction phase to ensure that muscles were passive during elastography acquisitions and elbow rotation [48]. All muscles were passive throughout all elastography measurements. Specifically, RMS-EMG was inferior to 1.5% for all scans regardless the elbow angle and, thus, may not have played a role in the elastography estimates during passive axial lengthening [48].

## 2.4. Statistical analysis

Statistical analyses were performed using one-dimensional statistical parametric mapping (SPM) allowing the analysis of continuous data, especially the muscle elastic modulus as a function of elbow angle (axial passive lengthening) and isometric contraction intensity. A custom MATLAB script was used to conduct SPM analyses implementing functions from the open source spm1d library (www.spm1d.org, version
M.0.4.7), as described by Pataky [49]. Prior to any inferential procedure, data normality was confirmed with the built-in function 'spm1d.stats.normality.sw' for the Shapiro-Wilk test. Using SPM, a two-tailed, paired t-test was conducted to compare the tensile versus shear anisotropy factors across all elbow angles in the axial lengthening experimental experiment or myoelectric intensities for the isometric contraction experiment. A similar procedure was implemented to compare the Young's modulus versus shear modulus, both along-muscle and cross-muscle fibers. Associations between Young's and shear moduli were quantified using bivariate Pearson rank correlation test for both lengthening (70° to 180°, in 20° increments for ease of interpretation) and isometric contraction (all intensity levels, from 0% to 20% of MVIC) experiments. A similar analysis was performed to explore the associations between tensile and shear anisotropy factors. The magnitude of the difference between cross-muscle fiber shear modulus and Young's modulus was assessed by the slope of the linear regression model for both passive axial lengthening and isometric contraction conditions. To facilitate comparisons, the mean ± standard deviation (s. d.) increase in Young's and shear moduli was calculated (in kPa) between 90° and 180° for the passive lengthening condition, and between 0% (elbow flexed at 90°) and 20% MVIC for the isometric contraction experiment. The significance level for all statistical comparisons was set to p < 0.05. All graphics were performed in GraphPad Prism (version 9.0; GraphPad Software, Inc., CA, US). Descriptive data are presented as mean ± standard deviation.

## 3. Results

### 3.1. Mechanical characterization of skeletal muscle during axial lengthening and isometric contraction

Averaged raw data measured with the steered ultrasound push beam method cross-muscle ($\mu_{\perp}$) and along-muscle ($X_E \cdot \mu_{\perp}$ and $\mu_{\parallel}$) fibers are illustrated in Fig. 5. Both shear modulus along-muscle fibers ($\mu_{\parallel}$) and the elastic parameter $X_E \cdot \mu_{\perp}$ increased non-linearly with the increasing of the stretching intensity (i.e., higher elbow angles, Fig. 5a), and with the increase in isometric contraction intensity (i.e., greater myoelectric activity, Fig. 5b). The shear modulus cross-muscle fibers ($\mu_{\perp}$) remained relatively unaltered in response to increased axial passive lengthening, and registered a minimal increase, when compared to shear modulus along-muscle fibers ($\mu_{\parallel}$), in response to the isometric contraction loading. All acquired shear wave elastography data were successfully analyzed for all participants.

### 3.2. Changes in muscle Young's and shear moduli with active and passive loading

From the angle-resolved steered push beams measurements of the human muscle, we directly estimated the Young's modulus along-muscle (Fig. 6a-b) and cross-muscle (Fig. 7a-b) fibers. The Young's modulus along-muscle fibers largely increased with the increase in axial loading for both lengthening (+24.3 ± 17.1 kPa) and isometric contraction (+213.0 ± 89.1 kPa) experiments; while the Young's modulus cross-muscle fibers only showed minimal increase compared to that obtained along muscle (+1.2 ± 2.1 kPa for axial lengthening; and +27.6 ± 16.3 kPa for isometric contraction). The shear modulus along-muscle fibers increased with the passive lengthening (+8.7 ± 4.0 kPa) and isometric contraction (+72.1 ± 15.8 kPa); while the shear modulus cross-muscle fibers did not change with stretching (0.1 ± 0.6 kPa) and increased minimally with the isometric contraction load (7.1 ± 5.0 kPa).

As expected, the Young's modulus along-muscle ($E_{\parallel}$) and cross-muscle ($E_{\perp}$) fibers differed significantly ($p < 0.001$) for both muscle conditions across all active or passive loading intensities (Supplementary Data File, Fig. 1a-b). Similar relationships were found for the relationship between the Young's and shear moduli along-muscle and cross-muscle fibers (Supplementary Data File, Fig. 1c-d).

### 3.3. Relationship between Young's modulus and shear modulus

We found a very strong correlation between cross-muscle fiber Young's and shear moduli, with Pearson correlation coefficients ranging from 0.95 to 0.99 in the axial passive lengthening (Fig. 7c) experiment and from 0.94 to 0.99 in the isometric contraction condition (Fig. 7d).

The ratio between the along-muscle fibers Young's and shear moduli was clearly dependent on the type and intensity of the axial loading (Fig. 8a-b), indicating that Young's modulus and shear modulus along-muscle fibers are independent parameters. In contrast, our findings demonstrate that the cross-muscle fibers Young's modulus registered an order of magnitude between 3 and 4 the shear modulus ($3\mu_{\perp} \le E_{\perp} \le 4\mu_{\perp}$) regardless the type and the intensity of the axial loading (Fig. 8c-d). Specifically, linear regression modelling indicated a magnitude difference of 3.2 (95 % Confidence Intervals: 3.2 to 3.3; $R^2$: 0.95) for the passive lengthening and 3.5 (95 % Confidence Intervals: 3.5 to 3.6; $R^2$: 0.99) for the isometric contraction experiment.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Data acquired using the steered push beam sequence along-muscle (elastic parameter $X_E \cdot \mu_{\perp}$ and shear modulus $\mu_{\parallel}$) and cross-muscle fibers (shear modulus $\mu_{\perp}$).</figcaption></figure>

(a) Averaged data during passive axial lengthening of the biceps brachii, plotted as a function of elbow angle. Higher angles correspond to greater elbow extension and, consequently, increased axial passive load. (b) Averaged data during active loading, expressed as a percentage of contraction intensity. Active loading is represented as a percentage of the maximal voluntary isometric contraction (MVIC). $\mu_{\perp}$, cross-muscle shear modulus; $X_E \cdot \mu_{\perp}$, elastic parameter combining tensile anisotropy and cross-muscle fiber; $\mu_{\parallel}$ along-muscle fiber shear modulus.
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 6. Relationship between Young's modulus and shear modulus along-muscle fibers of the *biceps brachii* during passive axial lengthening (a, c) and isometric contractions (b, d).</figcaption></figure>

(a) Mean differences between Young's modulus and shear modulus during passive lengthening, plotted as a function of elbow angle (from 70° flexion to 180° maximal extension). Higher angles correspond to greater axial passive loads or stretching. (b) Mean differences between Young's modulus and shear modulus for a range of isometric contraction intensities, expressed as a percentage of the myoelectric activity of the *biceps brachii* during maximal isometric voluntary contraction (MVIC). Data range from 0 % (resting muscle, elbow angle at 90°) to 20 %, in 5 % increments. (c) Correlations between Young's modulus and shear modulus obtained along-muscle fibers are shown for a range of elbow angles during the passive lengthening condition. (d) Correlations between Young's modulus and shear modulus along-muscle fibers are shown for all isometric contraction intensions. (a, b) Significant differences were observed between Young's modulus and shear modulus for all elbow angles and contraction intensities (*p* < 0.001; SPM with two-tailed paired t-test). Data are shown as mean ± s.d. (c, d) Correlations are given by r<sub>p</sub>, two-tailed test. Linear regression lines are shown for each angle and contraction intensity with corresponding color. *p* < 0.05 was considered statistically significant; ns*p* > 0.05; \*p* < 0.01; \*\*p* < 0.001.

### 3.4. *In vivo* tensile and shear anisotropy estimation of human skeletal muscle

Our methodological approach enabled the simultaneous quantification of both tensile and shear anisotropy. Both anisotropy factors increased with the increase in the passive axial lengthening. In the absence of passive tension (i.e., at or below the *biceps brachii* slack length ~95° [46]), tensile anisotropy was consistently – albeit not significant – greater than the shear anisotropy. With the increasing of axial passive muscle lengthening, the shear anisotropy became significantly greater than the tensile anisotropy, especially from 170° to 180° of elbow extension (*p* < 0.001; Fig. 9a). For the isometric contraction experiment, the shear and tensile anisotropy factors differed significantly from 5 % to 20 % of the myoelectric activity expressed as percentage of maximal isometric voluntary contraction (*p* < 0.001; Fig. 9b). Specifically, shear anisotropy was greater than the tensile anisotropy with the increase of axial shortening loading. At rest (i.e., elbow angle positioned at 90°), average tensile anisotropy typically expressed higher values than the shear anisotropy, but the difference was not significant (*p* > 0.05). For both experimental conditions, shear and tensile anisotropy factors were moderately to strongly significantly correlated across the range of

passive (Fig. 9c; Pearson correlations ranged from 0.64 to 0.95) or active (Fig. 9d; Pearson correlations ranged from 0.71 to 0.91) loading intensities.

### 4. Discussion

Skeletal muscle elasticity and anisotropy are fundamental properties of its physiological performance and structural integrity [13,50]. In this study, we provide the first comprehensive *in vivo* characterization of human skeletal muscle mechanics using angle-resolved shear wave propagation. Our findings confirm that skeletal muscle is a highly mechanically anisotropic soft tissue and unravel, for the first time, the intricate interplay between shear and tensile elasticity under muscle physiological loading conditions. Specifically, we demonstrated that the relationship between tensile and shear elasticity along-muscle fibers follows a complex pattern and is clearly influenced by the magnitude of axial muscle load. This supports our hypothesis that the tensile elasticity (or Young's modulus) derived from the steered ultrasound pushed beam method [31] is independent of the shear modulus that could be measured using conventional SWE methods. Moreover, we showed that cross-muscle fiber tensile elasticity is directly proportional to shear
<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 7. Relationship between Young's modulus and shear modulus cross-muscle fibers of the biceps brachii during passive axial lengthening (a, c) and isometric contractions (b, d).</figcaption></figure>

(a) Mean differences between Young's modulus and shear modulus during passive lengthening, plotted as a function of elbow angle (from 70° flexion to 180° maximal extension). Higher angles correspond to greater axial passive loads or stretching. (b) Mean differences between Young's modulus and shear modulus for a range of isometric contraction intensities, expressed as a percentage of the myoelectric activity of the biceps brachii during maximal isometric voluntary contraction (MVIC). Data range from 0% (resting muscle, elbow angle at 90°) to 20%, in 5% increments. (c) Correlations between Young's modulus and shear modulus cross-muscle fibers are shown for a range of elbow angles during the passive lengthening condition. (d) Correlations between Young's modulus and shear modulus obtained cross-muscle fibers are shown for all isometric contraction intensions. (a, b) Significant differences were observed between Young's modulus and shear modulus for all elbow angles and contraction intensities (*p* < 0.001; SPM with two-tailed paired t-test). Data are shown as mean ± s.d. (c, d) Correlations are given by rp, two-tailed test. Linear regression lines are shown for each angle and contraction intensity with corresponding color. *p* < 0.05 was considered statistically significant; **p* < 0.001.

elasticity obtained in the same direction, with a consistent magnitude difference of 3.4 ± 0.2. This finding provides experimental evidence of a direct and proportional link between these properties in the anisotropic skeletal muscle tissue. Lastly, we observed that both shear and tensile anisotropy are sensitive to mechanical load, and differ significantly during passive and active physiological muscle loading.

In incompressible and isotropic soft tissues, a straightforward relationship exists between shear modulus (μ) and elastic modulus (E), enabling the estimation elastic modulus from shear wave speed (V) and tissue density (ρ): E ≈ 3μ = ρV². This simplistic approach, commonly implemented in conventional SWE scanners, has traditionally been used to infer the “apparent” tensile elasticity in transversely isotropic tissues such as skeletal muscles [51], tendons [52,53] and peripheral nerves [54]. However, theoretical models suggest that the relationship between the shear and elastic moduli in transversely isotropic tissues, such as skeletal muscles, is more complex [26,55]. Unlike conventional elastography methods, angle-resolved elastography enables the direct quantification of tensile elasticity (E∥, E⊥) and tensile anisotropy

(X_E = E_∥ / E_⊥) in transversely isotropic tissues [31]. Our experimental

results (Figs. 6 and 8a-b) strongly support the theoretical predictions, demonstrating that relationship between the Young's and shear moduli along-muscle fibers is intricate and depends on the tissue's tensile state. These findings challenge the interpretation of the "apparent" shear modulus reported in earlier studies, which may represent a less relevant mechanical property compared to Young's modulus. Additionally, as SWE primarily measures shear wave velocity, a conservative approach would be to report only shear wave velocity values. Our findings experimentally demonstrate that along-muscle fiber Young's modulus (E) cannot be directly estimated from shear modulus (μ) measurements, as it is done in some standard clinical scanners using the equation: E = 3μ. The anisotropic nature of skeletal muscle should be considered, and muscle-related studies should report shear modulus when the scan is performed along or perpendicular to the fibers. For pennate muscles, proper ultrasound transducer alignment with the fiber's direction is not possible, leading to the measurement of an "apparent" shear modulus [56]. Despite growing evidence for the independence between of Young's modulus and shear elasticities, the direct measurement of Young's modulus in human skeletal muscle has remained largely unexplored, primarily due to the absence of noninvasive methods for in vivo assessment [27].
<figure><img src="image_16.png" /><figcaption>b. Along-muscle: Isometric Contraction

Ratio $E_{\parallel}/\mu_{\parallel}$ (a.u.)

Myoelectric activity (% of MVIC)</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>c. Cross-muscle: Passive Lengthening</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>d. Cross-muscle: Isometric Contraction</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 8. Ratios between Young's modulus and shear modulus along-muscle and cross-muscle fibers.</figcaption></figure>

(a, b) Ratios between Young's modulus and shear modulus along-muscle fibers during (a) passive lengthening and (b) isometric contraction. (c, d) Ratios between Young's modulus and shear modulus cross-muscle fibers during passive lengthening (c) and contraction (d). The average data (black line), standard error of the mean (gray shaded area), and individual data points (colored outlined circles, with each color representing a specific participant) are shown.

A notable finding of this study was the strong correlation between the shear modulus measured cross-muscle fibers and the Young's modulus perpendicular to the muscle fibers (Fig. 7). Remarkably, this relationship was consistent across all participants and testing conditions (Fig. 8c-d). This provides the first experimental evidence that, in a transversely isotropic biological soft tissue such as muscle, the cross-muscle fibers shear modulus is directly related to the Young's modulus ($3\mu_{\perp} \le E_{\perp} \le 4\mu_{\perp}$), corroborating the theoretical biophysical models [26]. These results suggest that tensile elasticity perpendicular to muscle fibers can be reliably and direct estimated from shear wavefront velocity measurements propagating across the muscle's axis of symmetry, using conventional and clinically available SWE methods.

This study presents, to our knowledge, the first comprehensive data set on the local tensile and shear anisotropy of a skeletal muscle *in vivo* under passive and active mechanical loading conditions. While the tensile anisotropy of passive muscle tissue has been well-established in *ex vivo* studies through material tensile testing along and across the muscle fiber directions [10,57–60], such characterizations are limited to passive muscle properties of isolated samples in vitro. These (*ex vivo*) approaches often overlook the influence of neighboring muscles and connective tissues, which may contribute to physiological lateral force transmission during isometric contraction or passive lengthening [61]. Additionally, they typically assess the mechanical properties of the whole tissue sample rather than those of specific capturing localized regions. Only few studies have successfully measured shear anisotropy *in vivo* using conventional SWE [e.g., 45]. More recently, experimental SWE frameworks have attempted to characterize muscle as an elastic,

incompressible, transversely isotropic material to assess tensile anisotropy. However, these methods rely on complex transducer rotation protocols involving time-consuming computational simulations [62] or less accurate transducer tilting [63] to quantify the shear vertical propagation mode and derive tensile elasticity and anisotropy. For example, Knight et al. (2022) demonstrated that tensile anisotropy $X_E$ can be estimated through Green's function simulations from SWE data acquired *in vivo* muscle using a 360° rotational approach to capture both shear horizontal and shear vertical wavefronts. On a single participant and passive muscle state, they observed that $X_E$ ($4.7 \pm 1.4$) was greater than the shear anisotropy $X_{\mu}$ ($2.1 \pm 0.9$), leading to the conclusion that $X_E$ represents an additional independent parameter and a potential biomarker for muscle health characterization. Under axially unloaded conditions, such as at the biceps brachii slack angle (i.e., elbow joint positioned at ~95° [46]), our findings align with those of Knight et al. (2022), with $X_E$ being higher than $X_{\mu}$. However, our reported values ($X_E = 1.8 \pm 0.4$ and $X_{\mu} = 1.5 \pm 0.3$) were lower, potentially reflecting differences in muscle type (vastus lateralis versus biceps brachii) and muscle length or extent of tensile loading during SWE measurements, making direct comparisons challenging. It is also important to note that the definitions of tensile anisotropy $X_E$ and shear anisotropy $X_{\mu}$ used in this study aligns with some previous works [39] but slightly differ from some others [64–66]. In some studies [e.g., 65], anisotropy derived by ultrasound shear wave elastography was defined as $X_{\mu} = \frac{\mu_{\perp} - \mu_{\parallel}}{\mu_{\perp}}$ and $X_E = \frac{E_{\parallel} - E_{\perp}}{E_{\perp}}$. With these equations, a value of zero is obtained for an isotropic material. In contrast, herein we define these parameters as the ratio of
<figure><img src="image_20.png" /><figcaption>Fig. 9. Relationship between shear and tensile anisotropy factors of the human biceps brachii in vivo.</figcaption></figure>

Relationships are illustrated for both passive axial lengthening (a, c) and isometric contractions (b, d) conditions. (a) Mean differences between shear anisotropy and tensile anisotropy during passive lengthening, plotted as a function of elbow angle (from 70° flexion to 180° maximal extension). Higher angles correspond to greater axial passive loads or stretching. (b) Mean differences between shear anisotropy and tensile anisotropy across a range of isometric contraction intensities, expressed as a percentage of the myoelectric activity of the biceps brachii during maximal isometric voluntary contraction (MVIC). Data range from 0 % (resting muscle, elbow angle at 90°) to 20 %, in 5 % increments. (c) Correlations between shear anisotropy and tensile anisotropy for a range of elbow angles during the passive lengthening condition. (d) Correlations between shear anisotropy and tensile anisotropy obtained cross-muscle fibers are shown for all isometric contraction intensions. (a, c) Significant differences were observed in shear and tensile anisotropy between 170° and 180° during the passive lengthening condition, and between 5 % MVIC and 20 % MVIC during the contraction condition (*p* < 0.001; SPM with two-tailed paired *t*-test). Data are shown as mean ± s.d. (c, d) Correlations are given by *r*<sub>p</sub>, two-tailed test. Linear regression lines are shown for each angle and contraction intensity with corresponding color. *p* < 0.05 was considered statistically significant; *p* < 0.01; **p** < 0.001.

moduli (i.e., X<sub>μ</sub> = μ<sub>⊥</sub>/μ<sub>∥</sub>; X<sub>E</sub> = E<sub>E</sub>/E<sub>∥</sub>), which results in a value of one for isotropic materials. While the values are different, the interpretation of results would be identical with both ways.

Collectively, these results underscore the potential of our unique
mechanical characterization to provide valuable insights into *in vivo*
muscle mechanics and represent an important step towards clinical
translation of these ultrasound parameters. Moreover, as shear wave
propagation is inherently influenced by the tensile state of the muscle,
including passive axial lengthening or active shortening [51], it
remained unclear how tensile anisotropy X<sub>E</sub> and shear anisotropy X<sub>μ</sub>
respond to such axial stresses that characterize muscle function. In this
study, we demonstrated that both X<sub>E</sub> and X<sub>μ</sub> are dependent on the axial
mechanical loading, whether passive or active. This finding highlights
the critical importance of standardizing the mechanical loading condi-
tions in future research and clinical settings and suggests that manipu-
lating loading intensities could yield deeper insights into muscle
function and mechanics, including the tension-compression asymmetry
*in vivo* [67].

Another important finding is that, in contrast to the unloaded state,

the shear anisotropy was significantly greater than the tensile anisotropy when applying both passive and active axial loading (Fig. 9). Although the mechanisms underlying this complex behavior cannot be conclusively determined from this *in vivo* and noninvasive study, some plausible explanations can be speculated. In the unloaded or relaxed state, the extracellular matrix and connective tissue contribute minimally to the basal resistance because the fibers are not stretched or tensioned. However, as axial loading increases, either through isometric contraction or passive lengthening, the muscle fibers become more tensioned. This increases their own elastic modulus but also tightens the extracellular matrix and connective tissues around them, leading to an increase in the interactions between the fibers and the surrounding extracellular matrix. Any shear deformation could therefore require more stress to overcome these interactions and could lead to greater longitudinal shear modulus than tensile. Since transverse shear and Young's moduli remained relatively constant across loading conditions, it may explain the larger shear anisotropy. Future investigations are needed to better understand the interplay between extracellular matrix and fibers in the shear and tensile behaviors. As shearing is a common mode of muscle injury – causing rupture of the myofibers and their basal lamina and
mysial sheaths as well as the nearby capillaries [68] – shear anisotropy could represent an important biomechanical property for assessing force transmission [69] and predicting susceptibility to muscle injuries.

In this study, the fusiform architecture of the *biceps brachii* muscle allowed direct measurement of the cross-muscle fiber shear modulus ($\mu_{\perp}$), enabling the calculation of tensile anisotropy ($X_E$) from the elastic parameter $X_{E,\mu_{\perp}}$ assessed via the steered ultrasound push beam method. While this angle-resolved method enhances the evaluation of shear modulus in pennate muscle geometries [33], achieving precise perpendicular alignment between the transducer and pennate muscle fibers remains technically challenging with a 2D imaging approach. Consequently, the cross-muscle fiber shear modulus ($\mu_{\perp}$) cannot be directly measured in pennate geometries, limiting the application of this method for calculating tensile anisotropy or longitudinal Young's modulus. Interestingly, our results reveal a strong correlation between the cross-muscle fiber shear modulus ($\mu_{\perp}$) and the transverse Young's modulus ($E_{\perp}$), characterized by a direct proportionality factor of 3.4 ± 0.2. This relationship identifies the elastic parameter $X_{E,\mu_{\perp}}$ as a robust predictor of the Young's modulus along muscle fibers ($E_{\parallel}$). Although associated errors remain to be quantified, the steered ultrasound push beam method shows promising potential for estimating tensile elasticity in pennate musculature in addition to the shear elasticity corrected to the fiber's orientation [33].

There is considerable potential for ultrasound SWE in identifying and monitoring tissue pathologies, as evidenced in liver fibrosis and breast cancer [24,70]. The recent technological advances enabling the assessment of both local shear and tensile elasticity, as well as inherent anisotropy in skeletal muscle tissue, offer promising applications in evaluating muscle function and structural tissue changes. Our angle-resolved elastography approach detects changes in tensile elasticity and anisotropy in response to both passive and active physiological loading in healthy muscle (Figs. 6 and 9). While conventional SWE has already shown promise in measuring the muscle apparent shear modulus to estimate individual muscle forces [51] and detect changes in various conditions, including myopathies [71], dystrophies [72,73], and post-stroke spasticity [74], further characterization of muscle tissue as transversely isotropic material – including tensile elasticity and anisotropy – may improve its clinical applicability (e.g., in conditions that profoundly impact tissue mechanics and fibrillar organization, including sarcopenia or myopathies [75,76]). Additionally, the comprehensive characterization of muscle mechanics described in this study shows that the angle-resolved elastography method provides valuable *in vivo* insights into human muscle behavior under axial loading, which could support the development of more accurate numerical simulations of individual muscle function. These findings also hold significant implications for tissue engineering, where replicating the native elastic and anisotropic properties of muscle tissue has been a long-lasting challenge [15–17,77].

Our study has certain limitations. The methodology employed does not account for nonlinear elasticity or viscosity effects. In particular, viscosity can induce dispersion in shear wave elastography data, potentially biasing shear wave group velocity estimates depending on the rheological model applied [78,79]. Additionally, the steered ultrasound push beam method assume linear elasticity in transversely isotropic soft material, despite skeletal muscle's well-documented nonlinear mechanical behavior [42,80,81]. However, the low loading conditions in this study – limited by the anatomical constraints of biceps brachii stretching (e.g., elbow extension restricted by joint configuration) and the low isometric contraction intensities tested (up to 20 % MVIC) – likely minimize the nonlinear effects on our findings. Our team is actively working to incorporate these variables to better model muscle tissue as an incompressible, viscoelastic, nonlinear, transversely isotropic material [e.g., 38,68]. Furthermore, structures such as the humerus, blood vessels, adipose tissue, and intramuscular fascia within the imaging field may influence shear wave propagation. To mitigate

these effects, we carefully avoided some of these structures from the imaging plane during data acquisition and ROIs, during the post-processing, for accurate muscle shear wave velocity quantification. Lastly, the ability to generalize this tissue characterization to other muscles may be constrained by muscle size. When the muscle body thickness approaches the shear wavelength, wave-guiding behavior and boundary effects may influence the measured waves [82].

## 5. Conclusion

The present work provides, to our knowledge, the most comprehensive, non-destructive, mechanical characterization of human skeletal muscle *in vivo* during physiological loadings. Using angle-resolved ultrasound shear wave propagation, we captured both the tensile and shear anisotropy of skeletal muscle and confirmed that skeletal muscle tissue has a strongly direction-dependent mechanical behavior which depends significantly on the amount of tissue (active or passive) tensile load. In addition, we present, for the first time, individual quantification of the Young's moduli along-muscle and cross-muscle fibers in a skeletal muscle *in vivo*. This enabled us to comprehensively describe the relationships between shear and tensile elasticity across the muscle fiber's orientation. We found that the Young's modulus along-muscle fibers is an independent mechanical parameter, essential to fully and accurately characterize muscle mechanical properties. However, in a plane perpendicular to the muscle fibers, the Young's modulus is directly and proportionally linked to the shear modulus, leading to significantly different shear and tensile anisotropy. Taken together, these experiments reveal unique measurements of skeletal muscle tensile elasticity, including tensile anisotropy and the Young's modulus along-muscle fibers, that can directly be obtained by steered ultrasound push beams method. This angle-resolved method can be used to quantify local tissue mechanical properties of anisotropic soft tissues (e.g., muscles, peripheral nerves, tendons) in health and disease and has potential to contribute to advances in tissue engineering.

## Availability of data

Datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

## CRediT authorship contribution statement

Ricardoj. Andrade: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Apolline Racapé: Writing – review & editing, Investigation, Formal analysis. Mar Hernández-Secorún: Writing – review & editing, Investigation. Ha-Hien-Phuong Ngo: Writing – review & editing, Software, Methodology. Alice Lemoine: Writing – review & editing, Software. Nicolas Etaix: Writing – review & editing, Supervision, Software, Methodology. Thomas Frappart: Writing – review & editing, Supervision, Software, Methodology. Christophe Fra-schini: Writing – review & editing, Supervision, Software, Resources. Jean-Luc Gennisson: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Funding acquisition, Conceptu-alization. Antoine Nordez: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Resources, 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.
Acknowledgements

This was funded by ANR INNOVAN (ANR-19-CE19-0017). RJA was supported by the AFM-Téléthon (Grant ID#28643). We thank Maceo Renault for his help with data collection. We extend our gratitude to all participants who generously volunteered for the study.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.actbio.2025.05.010.

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