# Unravelling anisotropic nonlinear shear elasticity in muscles: Towards a non-invasive assessment of stress in living organisms

Ha Hien Phuong Ngo, Ricardo Andrade, Juliette Lancelot, Aude Loumeaud, Corentin Cornu, Antoine Nordez, Simon Chatelin, Jean-Luc Gennisson

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## To cite this version:

Ha Hien Phuong Ngo, Ricardo Andrade, Juliette Lancelot, Aude Loumeaud, Corentin Cornu, et al. Unravelling anisotropic nonlinear shear elasticity in muscles: Towards a non-invasive assessment of stress in living organisms. Journal of the mechanical behavior of biomedical materials, 2023, 150, pp.106325. 10.1016/j.jmbbm.2023.106325. hal-04388240

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# Unravelling anisotropic nonlinear shear elasticity in muscles: Towards a non-invasive assessment of stress in living organisms

Ha Hien Phuong Ngo<sup>a</sup>, Ricardo J. Andrade<sup>b</sup>, Juliette Lancelot<sup>b</sup>, Aude Loumeaud<sup>c</sup>, Corentin Cornu<sup>a</sup>, Antoine Nordez<sup>b,d</sup>, Simon Chatelin<sup>c</sup>, Jean-Luc Gennisson<sup>a,*</sup>

## ARTICLE INFO

**JEL classification:**

Biological Sciences

Medical Sciences and Physical Sciences

Biophysics and Computational Biology

**Keywords:**

Shear wave elastography

Shear nonlinearity

Muscle force

Stress

Acoustoelasticity

## ABSTRACT

Acoustoelasticity theory describes propagation of shear waves in uniaxially stressed medium and allows the retrieval of nonlinear elastic coefficients of tissues. In transverse isotropic medium such as muscles the theory leads to 9 different configurations of propagating shear waves (stress axis vs. fibers axis vs. shear wave polarization axis vs. shear wave propagation axis). In this work we propose to use 4 configurations to quantify these nonlinear parameters *ex vivo* and *in vivo*. *Ex vivo* experiments combining ultrasound shear wave elastography and mechanical testing were conducted on *iliopsoas* pig muscles to quantify three third-order nonlinear coefficients A, H and K that are possibly linked to the architectural structure of muscles. *In vivo* experiments were performed with human volunteers on *biceps brachii* during a stretching exercise on an ergometer. A combination of the third order nonlinear elastic parameters was assessed. The knowledge of this nonlinear elastic parameters paves the way to quantify *in vivo* the local forces produced by muscle during exercise, contraction or movements.

## Significance statement

The present work presents for the first time the assessment of the third order nonlinear elastic coefficient of muscles *in vitro* and *in vivo* through an acoustoelasticity theory implementation in ultrasound elastography. This groundbreaking methodology holds the potential to revolutionize diagnostic and monitoring applications not only in pathological or post-traumatic skeletal muscles but also in other biological soft organs characterized by fiber reinforcement along a clearly identified main direction. This work paves the way to quantify *in vivo* individual muscles forces during contractions, movement or exercise.

## 1. Introduction

Tissue mechanical properties play a key role in governing cellular processes associated with morphogenesis, tissue homeostasis, and disease progression (Martinez-Vidal et al., 2021; Lopez et al., 2008; Guimaraes et al., 2020). The ability to measure the localized tissue mechanical properties using noninvasive imaging techniques within

living tissues is very important for gaining comprehensive insights into tissue growth, maturation, pathological processes, and identifying reliable biophysical biomarkers of disease. However, the noninvasive quantification of mechanical forces or stress within individual living tissues remains one of the biggest challenges in biomechanics, while the accurate quantification of forces would significantly enhance our understanding of *in vivo* mechanobiology (Zuela-Sopilniak and Lammerding, 2022; Lacourpaille et al., 2015; Szigyarto and Spitali, 2018).

Among the non-invasive medical imaging techniques ultrasound elastography has shown great promise in quantifying *in vivo* tissue stiffness including skeletal muscle (Bilston et al., 2019; Hug et al., 2015). Ultrasound elastography emerged in the 1990's and has been extensively used to assess soft tissue stiffness, which is significantly altered during several disease processes including cancer and fibrosis (Tapper and Parikh, 2023; Mann et al., 2020). Ophir et al. introduced the concept of mapping tissue strain by applying a stress using an ultrasound transducer array on the body's surface (Ophir et al., 1991). This technique, also known as static elastography, was primarily developed for breast cancer diagnosis (Garra et al., 1997). According to the Hooke's

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

<footnote><sup>b</sup> Mouvement Interactions Performance (MIP), University of Nantes, UR 4334, F-44000, Nantes, France</footnote>

<footnote><sup>c</sup> Engineering Science, Computer Science and Imaging Laboratory (ICube), University of Strasbourg, CNRS UMR 7357, Strasbourg, F-67000, France</footnote>

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

<footnote>* Corresponding author.
E-mail address: jean-luc.gennisson@universite-paris-saclay.fr (J.-L. Gennisson).</footnote>
law, strain (ε) is related to the Young's modulus (E), the most relevant stiffness measurement of a given material. However, because the stress (σ) applied to the medium cannot be quantified because it is operator-dependency, objectively quantifying tissue stiffness (E) through static elastography is challenging.

In 1995, Sarvazyan et al. (Sarvazyan et al. Jones, 1995) proposed to use shear wave propagation to quantify the Young's modulus (Sarvazyan et al. Jones, 1995). Specifically, in soft tissues modeled as linear, purely elastic, homogeneous, quasi-incompressible, the shear wave speed is directly linked to the Young's modulus through the equation E = 3μ = 3ρV₃², where ρ is the density of tissues (~1000 kg/m³) and μ the shear modulus. Over the years, numerous techniques have been developed to capture shear waves propagation (Nightingale et al., 2002; Gennisson et al., 2003, 2013; Sandrin et al., 2003; Palmeri et al., 2008; Urban et al., 2011; McAleavey et al., 2009).

These advancements enabled ultrasound shear wave elastography technique to become a gold standard for various pathological processes associated with breast cancer and liver tissues (Berg et al., 2012; Deffieux et al., 2015). However, the addition of stiffness quantification to classical ultrasound imaging is insufficient to comprehensively characterize tissue mechanics in living tissues. For instance, breast cancers, such as carcinomas, can exhibit comparable stiffness to benign lesions like fibroadenoma (Chamming's et al., 2021). Furthermore, there is a critical need to quantify novel and non-invasive mechanical parameters that can aid in establishing an accurate and reliable clinical diagnosis of anisotropic soft tissues and to better understand mechanobiology processes in vivo. During the same time, magnetic resonance elastography (MRE) approaches have also made it possible in recent years to propose methods for characterizing the anisotropic properties of tissues (Van Houten et al., 1999; Qin et al., 2013; Babaei et al., 2021; Miller et al., 2018; Chatelin et al., 2016; Tweten et al., 2015), with applications to breast (Sinkus et al., 2005), muscle (Green et al., 2013) or brain tissue (Romano et al., 2012; Smith et al., 2020; McGarry et al., 2022).

Interestingly in 2007, Gennisson et al. developed the acousto-elasticity theory to quantify nonlinear shear elasticity in linear, homogeneous, quasi-incompressible and isotropic soft tissues (Gennisson et al., 2007). The experimental application of this theory combined static elastography (Ophir et al., 1991) and dynamic elastography (Bercoff et al., 2004) to quantify strain and the Young's modulus (Sujeptic shear imaging technique), respectively. The validation of the technique to estimate nonlinear shear modulus was demonstrated in various phantoms (Latorre-Ossa et al., 2012; Pagé et al., 2023; Urban et al., 2015) and finally applied in vivo in breast cancer assessment (Bernal et al., 2016). However, the acoustoelasticity theory, originally developed in vivo for breast tissue (Bernal et al., 2016), and all elastography techniques rely on a theoretical assumption that human tissues are fully isotropic. While this assumption holds true for liver and breast tissues, it cannot be the case for skeletal muscles due to their anisotropic nature.

Skeletal muscle has a hierarchical multiscale structure with a large number of muscle fibers, resulting in a global mechanical behavior that can be modeled as transverse isotropy, along a main fiber orientation direction. This consideration entails the existence of a symmetry axis along the fibers and this type of symmetry corresponds to a hexagonal

system also named transverse isotropic (TI) medium (Zimmer and Cost, 1970). The acoustoelasticity theory has recently been revisited in TI media and refinements were proposed to include the specificities of the TI symmetry that implies 9 different configurations where the principal direction of the TI medium, stress, polarization u→ and propagation k→ directions of the shear wave are either parallel (μ//) or perpendicular (μ⊥) to one another (Bied and Gennisson, 2021; Remeniéras et al., 2021). The resulting configurations are dependent on 3 elastic parameters of the second order (μ//, μ⊥, and the Young's modulus E// parallel to the main axis, respectively) and on 3 third order nonlinear elastic parameters (A, H, K) that are possibly linked to the architectural structure of muscles.

In this study we aimed to use the acoustoelasticity theory developed by Bied and Gennisson (2021) to quantify the third order elastic parameters that are linked to local stress to better characterize skeletal muscle mechanics in vivo. We initially demonstrated the feasibility of quantifying these parameters in vitro using a unique approach that combines specific ultrasound sequences to measure shear wave speed as a function of mechanical stress, assessed via a traditional mechanical testing method. This step provided an estimation of the magnitude of the third order nonlinear parameters expressed in units of kPa. In vitro only 3 configurations (out of nine) were required to calculate the nonlinear parameters. In vivo only two configurations were used to assess the nonlinear parameters. An experimental protocol was then developed with a specific ultrasound sequence to quantify, for the first time in humans, the nonlinear parameters coupled with an ergometer in a fusiform muscle (i.e., biceps brachii).

## 2. Acoustoelasticity theory

The acoustoelasticity theory is developed up to the 3rd order from the expression of the strain energy e of a compressible TI medium in terms of the Green-Lagrange strain tensor ε with a principal direction along $\vec{x}_p$ (Johnson, 1982; Spencer, 1984). In the case of quasi-incompressible TI medium which a principal direction along $\vec{x}_3$, this theory allows to show not only three linear elastic coefficients (E//, μ// and μ⊥), but also five third order elastic parameters of a TI medium (A, G, H, K) as follow (Bied and Gennisson, 2021):

$$e = \mu_{\perp} I_2 + \frac{E_{\parallel} - 3\mu_{\perp}}{2} \varepsilon_{33}^2 + 2(\mu_{\parallel} - \mu_{\perp})(\varepsilon_{13}^2 + \varepsilon_{23}^2) + \frac{A}{3} I_3 + (G + H + K) \varepsilon_{33}^3 \\ + (H + 2K) \varepsilon_{33}(\varepsilon_{13}^2 + \varepsilon_{23}^2) + K \varepsilon_{33}(\varepsilon_{11}^2 + \varepsilon_{22}^2 + 2\varepsilon_{12}^2) \quad (1)$$

where $\varepsilon_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial a_j} + \frac{\partial u_j}{\partial a_i})$. $u_i$ is the displacement of particles in the i direction, $I_2 = \text{Tr}(\varepsilon^2)$ and $I_3 = \text{Tr}(\varepsilon^3)$ are the second and third invariants of ε (Landau et al., 1986), E// is the Young's modulus along the principal direction. The whole theory leading to equation (1) is fully described in (Bied and Gennisson, 2021).

Then, by assuming elastic waves that propagates in a stressed medium and linearly polarized plane shear waves, acoustic wave equations are derived from the equation of motion of the medium's particles:

$$\rho_0 \frac{\partial^2 u_i^D}{\partial t^2} = \mu_{\perp} \frac{\partial^2 u_i^D}{\partial a_p^2} + 2 \left( \mu_{\perp} + \frac{A}{4} \right) \left( \frac{\partial u_i^S}{\partial a_p^2} \frac{\partial^2 u_i^D}{\partial a_p^2} + \frac{\partial u_p^S}{\partial a_p^2} \frac{\partial^2 u_i^D}{\partial a_p^2} \right) + (E_{\parallel} - 3\mu_{\perp}) \frac{\partial u_3^S}{\partial a_3} \frac{\partial^2 u_i^D}{\partial a_3^2} + (\mu_{\parallel} - \mu_{\perp}) \left\{ \delta_{i,3} \left[ \frac{\partial^2 u_3^D}{\partial a_1^2} + \frac{\partial^2 u_3^D}{\partial a_2^2} \right] + 2 \frac{\partial u_3^S}{\partial a_3} \left( \frac{\partial^2 u_3^D}{\partial a_1^2} + \frac{\partial^2 u_3^D}{\partial a_2^2} \right) \right\} \\ + \delta_{i,1} \left( \frac{\partial^2 u_i^D}{\partial a_3^2} + 2 \frac{\partial u_1^S}{\partial a_1} \frac{\partial^2 u_i^D}{\partial a_3^2} \right) + \delta_{i,2} \left( \frac{\partial^2 u_i^D}{\partial a_3^2} + 2 \frac{\partial u_2^S}{\partial a_2} \frac{\partial^2 u_i^D}{\partial a_3^2} \right) \right\} + \frac{H + 2K}{2} \left[ \delta_{i,3} \frac{\partial u_3^S}{\partial a_3} \left( \frac{\partial^2 u_3^D}{\partial a_1^2} + \frac{\partial^2 u_3^D}{\partial a_2^2} \right) + \delta_{i,1} \frac{\partial u_3^S}{\partial a_3} \frac{\partial^2 u_1^D}{\partial a_3^2} + \delta_{i,2} \frac{\partial u_3^S}{\partial a_3} \frac{\partial^2 u_2^D}{\partial a_3^2} \right] + K \left( \delta_{i,1} \frac{\partial u_3^S}{\partial a_3} \frac{\partial^2 u_1^D}{\partial a_2^2} + \delta_{i,2} \frac{\partial u_3^S}{\partial a_3} \frac{\partial^2 u_1^D}{\partial a_1^2} \right) \quad (2)$$
<caption>**Table 1**</caption>

Nonlinear elastodynamic equations for 4 of the 9 acoustoelastic configurations of a TI quasi-incompressible medium used in this present work. Refer to Fig. 1 and its legend for the definition of the configurations and variables $\vec{u}$ (polarization axis), $\vec{k}$ (propagation axis) and $\sigma$ (stress axis) (Bied and Gennisson, 2021).

<table><thead><tr><th>Shear wave</th><th>Stress axis</th><th>Config. #</th><th>Nonlinear elastodynamic equations</th><th>Eq.</th></tr></thead><tbody><tr><td>$\vec{u} \parallel \vec{x}_2 \vec{k} \parallel \vec{x}_3$</td><td>$\sigma_{33}$</td><td>3</td><td>$\rho_0 v_s^2 = \mu_{\parallel} - \frac{\sigma}{E_{\parallel}} \left(E_{\parallel} - \mu_{\perp} + \mu_{\parallel} + \frac{A}{4} + \frac{H+2K}{2}\right)$</td><td>(3)</td></tr><tr><td>$\vec{u} \parallel \vec{x}_3 \vec{k} \parallel \vec{x}_2$</td><td>$\sigma_{22}$</td><td>5</td><td>$\rho_0 v_s^2 = \mu_{\parallel} + \frac{\sigma}{2E_{\parallel}} \left[-E_{\parallel} - \mu_{\perp} + \mu_{\parallel} - \frac{\mu_{\parallel}E_{\parallel}}{\mu_{\perp}} + \frac{A}{4}\left(1 - \frac{E_{\parallel}}{\mu_{\perp}}\right) + \frac{H+2K}{2}\right]$</td><td>(4)</td></tr><tr><td>$\vec{u} \parallel \vec{x}_2 \vec{k} \parallel \vec{x}_1$</td><td>$\sigma_{11}$</td><td>7</td><td>$\rho_0 v_s^2 = \mu_{\perp} - \frac{\sigma}{2E_{\parallel}} \left(E_{\parallel} + 3\mu_{\perp} + \frac{A}{2} - K\right)$</td><td>(5)</td></tr><tr><td>$\vec{u} \parallel \vec{x}_2 \vec{k} \parallel \vec{x}_1$</td><td>$\sigma_{33}$</td><td>9</td><td>$\rho_0 v_s^2 = \mu_{\perp} + \frac{\sigma}{E_{\parallel}} \left(3\mu_{\perp} + \frac{A}{2} - K\right)$</td><td>(6)</td></tr></tbody></table>

with $\vec{u} = \vec{u}^S + \vec{u}^D$, the displacement of any particle in the medium with 2 components: a static term $\vec{u}^S$ due to the uniaxial stress and a dynamic term $\vec{u}^D$ induced by the propagation of the elastic wave, and $\delta_{m,n}$ the Kronecker delta. At last it results in 9 elastodynamic configurations (Config) and associated equations (Bied and Gennisson, 2021) of which the four presented in equations 3, 4, 5 and 6 are used in our present study to identify the nonlinear parameters (Table 1).

For the ex vivo part, from measurements and equations (3, 4 and 5), each equation can be rewritten in the form of a linear relationship expressing the shear wave speed as a function of the uniaxial stress:

$$\rho_0 v_s^2 = \mu_{(\perp, \parallel)} \pm k_{(3,5,7)} \sigma \quad (7)$$

Then the 3 nonlinear parameters (A, K and H) can be deduced from each configuration # (3, 5 and 7). By fitting the experimental results of the shear wave speed as a function of the stress, the slopes k (Guimarães et al., 2020; Lacourpaille et al., 2015; Bilston et al., 2019) can be measured and the nonlinear parameters founded from the following equations:

$$A = -4(\mu_{\perp}(2+k_3+2k_5)+\mu_{\parallel}), \quad (8)$$

$$K = - \left( E_{\parallel} + 3\mu_{\perp} + \frac{A}{2} + 2E_{\parallel}k_7 \right), \quad (9)$$

$$H = 2 \left( -k_3 E_{\parallel} - E_{\parallel} + \mu_{\perp} - \mu_{\parallel} + \frac{A}{4} - K \right), \quad (10)$$

For the in vivo part, a combination of the nonlinear parameters was deduced by using equations 3 and 6 from config #3 and config #9 respectively, and the parallel Young's modulus was deduced from the expression of the vertical shear wave mode in a quasi-incompressible transverse anisotropic medium at rest ($\sigma = 0$). In config #3 the propagation of shear wave as a function of the angle of the polarization axis with can be expressed as (Royer and Dieulesaint, 1999; Rouze et al., 2020; Ngo et al., 2023):

$$\rho_0 v_s^2(\theta) = \mu_{\perp} + 4 \left( \frac{E_{\parallel}}{E_{\perp}} - \mu_{\perp} - \mu_{\parallel} \right) \sin^2 \theta \cos^2 \theta \quad (11)$$

where $\theta$ is the angle between the shear wave polarization axis (on Fig. 1A, the $\vec{u}$ axis) and the vertical axis $x_1$, in the plane $x_1x_3$, and $E_{\perp}$ the Young' modulus perpendicular to the main axis ($x_3$ fiber axis). By assuming the medium as quasi-incompressible, equation (10) can be approximated to (Royer et al., 2011):

$$\rho_0 v_s^2(\theta) = \mu_{\perp} + \left( E_{\parallel} + \mu_{\perp} - 4\mu_{\parallel} \right) \sin^2 \theta \cos^2 \theta \quad (12)$$

Then by measuring the shear wave speed as a function of angle $\theta$ it is then possible to recover the Young's modulus parallel to fibers (Ngo et al., 2023). Regarding the nonlinear parameters, as in vivo only two configurations are accessible (configs #3 and #9), two parameters X and Y which are a combination of A, H and K, were defined from equations 3 and 6, respectively, to describe the nonlinear behavior in vivo:

$$X = \frac{A}{4} + \frac{H+2K}{2} \quad (13)$$

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** The 4 elastodynamic configurations used for ex vivo and in vivo measurements. The principal axis of the TI medium is chosen to be along ($x_3$). For each configuration, the directions $\vec{u}$ (polarization axis), $\vec{k}$ (propagation axis) of the shear wave are specified in red, as well as the orientation of the stress $\sigma$ in blue.</figcaption></figure>
$$Y = \frac{A}{2} - K \qquad (14)$$

# 3. Materials and methods

## 3.1. Ultrasound acquisitions

Ultrasound and elastography acquisitions were performed for both ex vivo and in vivo experiments, with a linear transducer array (SL18-5, 7.5 MHz central frequency, 256 elements, Hologic-Supersonic Imagine, Aix en Provence, France) driven by an ultrafast ultrasound device (Mach30, Hologic-Supersonic Imagine, Aix en Provence, France). Two distinct ultrasound sequences were used to get the shear modulus parallel ($\mu_{\parallel}$) or perpendicular ($\mu_{\perp}$) to the fiber muscle main axis, and the Young's modulus parallel ($E_{\parallel}$) to the fiber muscle main axis.

For the shear moduli, the commercial sequence included in the device was used to get the shear wave speed ($v_s$). It briefly consists in two steps: 1/generation of a radiation force by a focused ultrasonic beam that creates shear waves propagating perpendicularly to the ultrasound axis with a polarization parallel to the ultrasound axis, 2/detection of the propagation of the shear wave with an ultrafast imaging sequence (3500 frames.s⁻¹) (Bercoff et al., 2004). Then, by using a time-of-flight algorithm (Loupas et al., 1995; Deffieux, 2008), the shear velocity is retrieved for each pixel of the image, allowing the computation of a complete two-dimensional (2D) shear modulus map ($\mu$) by using the relationship:

$$\mu = \rho_0 v_s^2 \qquad (15)$$

Regarding the position of the transducer as a function of the main fiber muscle axis the shear modulus parallel (config. #3) and perpendicular (config. #9) was quantified.

For the Young's modulus a dedicated ultrasound sequence proposed and described by Ngo et al. (2023). This sequence was only used for the in vivo part in config #3. As for the commercial mode, it consists also on two steps: I) generation of the shear wave with a steered acoustic radiation force beam with an angle $\theta$ (from -20° to 20° by 5° step) regarding the vertical axis $x_1$. II) an ultrafast ultrasound acquisition steered with the same angle to detect the propagation of shear waves. In this particular sequence shear horizontal waves and shear vertical waves are generated, but in the imaging plane $-x_1x_3$ - plane) only shear vertical wave are measured. At rest, by knowing $\mu_{\parallel}$ and $\mu_{\perp}$ from the previous acquisition with the commercial ultrasound sequence, the Young's modulus parallel $E_{\parallel}$ to fibers can be retrieved from the change in the shear wave velocity as a function of $\theta$.

To quantify the strain $\epsilon$ during stretching an ultrasound image was

acquired at each step of the muscle lengthening. Then the displacement of the muscle in the axis of the fibers was calculated for each step by using a cross correlation algorithm (Latorre-Ossa et al., 2012). The original reference frame was partitioned into several sub-elements, each sized at 30 × 30 pixels (equivalent to 6 × 3.6 mm). For every sub-element in the reference frame, the algorithm performed a cross-correlation with an enlarged window (90 × 40 pixels, approximately 8 × 10.8 mm) from frames captured at various elbow extensions, centered at the same location. This iterative process was applied to each sub-element, generating a displacement matrix.

## 3.2. Ex vivo experiments

Six iliopsoas porcine muscles were excised from three healthy domestic pigs (Sus scrofa) with an age of 3-4 months and a weight of 23-42 kg. In accordance with the national and European laws on animal protection, no approval for the study was needed, since the muscles were harvested after sacrifice in a pig involved for scientific purposes in another protocol approved by the local Ethical Committee and authorized by French authorities (APAFIS #22589-20190726115576) at the IHU Strasbourg. Each iliopsoas muscle from the same pig was cut into different pieces: two longitudinal (parallel) and two to four perpendicular (transverse) to the fibers. Each sample has a similar rectangular shape (5 × 1 cm²) (Fig. 2). Ex vivo muscles were fixed to the stretching machine device (Instron 5944, Norwood, MA, USA) on the two ends by the system of jaws. The tests were carried out at room temperature, measured between 22 °C and 25 °C throughout, with regular hydration of samples on the tensile setup with physiological solution using a pipette. Three acoustoelastic configurations were set up to perform experiments. Fig. 2 shows a schematic illustration of sample extraction in the 3 configurations studied here, as well as the positioning of the samples on the traction machine.

For the config #3, the longitudinal samples were used. In that case the muscle fibers were aligned with the main axis of the traction device. The parallel Young's modulus ($E_{\parallel}$) was calculated by using the Hooke's law from the slope of the experimental stress-strain curve using a traction device. Stepwise stretches from 0 to 12% nominal strain were induced then by 1% steps. At each 1% strain step, the tensile test was carried out. For configs #5 and #7, the transverse samples were used. The stretching stress applied, and the shear modulus measured are thus perpendicular to the fibers and the US measurements parallel or perpendicular to fibers (depending on the configuration) were performed.

From the given stiffness maps in the linear elastic regime at each measurement, an averaged stiffness of a defined ROI was calculated. The

<figure><img src="image_4.png" /><figcaption>Fig. 2. (A) Example of porcine iliopsoas muscle with schematic diagram of extracted samples depending on the orientation of the muscular fibers. (B) Illustrative diagram of the three configurations tested, according to sample fiber orientation, on the tensile machine coupled to the ultrasound transducer array.</figcaption></figure>
shear wave speed curve (given by the mechanical device) was then plotted as a function of the nominal stress $\sigma$ (kPa) (stretching force F measured but the force load of the tensile machine divided by the initial cross section of the sample) and the slope of this relationship was calculated.

### 3.3. *In vivo* experiments

Three healthy participants (1 woman, 2 men; age $26 \pm 7$ years; height $174 \pm 17$ cm; weight $63 \pm 17$ kg), with no self-reported history of upper limb neuromuscular and musculoskeletal disorders, participated in this study. Prior to testing, written informed consent was obtained from all participants. The study was approved by the local Institutional Ethics Review Board (CERN, University of Nantes, France IRB #IORG0011023).

A single experimental session was performed. Ultrasound scans of the mid-portion of the biceps brachii (i.e., long head) were obtained from the right upper limb of each participant under passive conditions in several elbow postures. Participants were seated in a Con-Trex MJ isokinetic dynamometer (CMV AG, Dubendorf, Switzerland) chair, secured at the trunk with the shoulder approximately in the horizontal plane (i.e., abducted at $90^{\circ}$). The forearm and wrist were positioned in protonation and neutral flexion-extension positions respectively. The medial epicondyle of the humerus was aligned with the axis of rotation of the dynamometer (Fig. 3A). The dynamometer was used to perform passive elbow rotations and to measure the elbow angle and joint passive

torque. Surface electromyography (sEMG) electrodes (Zerowire, Cometa, Italy) were placed on the muscle belly of the biceps brachii and triceps brachii muscles. EMG was recorded to ensure that muscles remained passive throughout experimental session (specifically, ultrasound scans). sEMG and mechanical data were sampled at 5 kHz using a Vicon Nexus software (version 1.8.5, Oxford Metrics Ltd., UK). EMG signals were amplified and digitized (6–400 Hz bandwidth).

Prior to the ultrasound assessments, five passive elbow rotations were performed at $5^{\circ}\cdot s^{-1}$ to pre-condition the neuromusculoskeletal structures acting on the elbow joint (Nordez et al., 2008). Then, the testing session was divided in two steps.

First, we experimentally measured the slack angle of the biceps brachii as a surrogate measure of its slack length (Lacourpaille et al., 2013). The slack angle was defined as the joint angle under zero load (Lieber and Binder-Markey, 2021) which corresponds to the onset of muscle passive tension during a passive stretching. The biceps brachii passive tension was estimated by ultrasound shear wave elastography during a passive elbow extension performed at $2^{\circ}\cdot s^{-1}$, from $90^{\circ}$ flexion to maximal extension ($\sim 180^{\circ}$). Elbow extension axially loads (i.e., stretches) the biceps brachii muscle. The ultrasound transducer was carefully aligned parallel to the muscle fibres with minimal tissue compression using a custom-made probe holder. The muscle shear modulus was recorded at 1 Hz using an Aixplorer ultrasound scanner (version 6.1, Hologic-Supersonic Imagine, Aix-en-Provence, France) coupled with a linear transducer (SL15-4, 7.5 MHz central frequency, 256 elements, Hologic-Supersonic Imagine, Aix en Provence, France).

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** A) In vivo experimental setup for the measurement in human biceps brachii muscle. The yellow part represents the stretching direction from slack angle to $180^{\circ}$ extension. B) B-mode image of biceps brachii in which the ultrasound transducer is aligned parallel to the direction of the muscle fibers enabling the shear modulus measurements for the config #3. C) B-mode image of biceps brachii in which the ultrasound transducer is aligned perpendicular to the direction of the muscle fibers enabling the shear modulus measurements for the config #9. D) Apparent shear modulus measured using ultrasound shear wave elastography, as a function of stress for volunteer #3. Outcomes are depicted the tested configurations *in vivo*. The stress was estimated from the product between the Young's modulus parallel to the muscle fibers (measured using the steered pushing beams method) and the strain (quantified from cross-correlation algorithm performed in B-mode frames captured at various elbow extensions). The slopes were used to calculate *in vivo* X and Y parameters.</figcaption></figure>
Following this muscle loading cycle, the recorded shear wave elastography clip was exported in a DICOM file format. Using custom-made MATLAB scripts, a region of interest (ROI) was first defined on the first map as the largest muscle area. Then, this ROI was manually tracked frame-by-frame. Each frame was inspected for artefacts (i.e., saturation) and signal void areas, and the ROI was manually adjusted to be sure that artefacts were not included in the averaged shear modulus. The shear modulus data extracted from the ROI were then averaged, and this value was used for analysis. An analogic trigger signal originating from the Aixplorer ultrasound scanner at each elastography measurement allowed to synchronize the biceps brachii shear modulus data with elbow angle data. Then, the slack angle was visually determined from the raw shear modulus-elbow angle relationship (Le Sant et al., 2017) and this value was used for the following step.

Second, SWE data were acquired using a Mach30 ultrasound scanner (Hologic-Supersonic Imagine, Aix-en-Provence, France) and a linear transducer (SL18-5, 7.5 MHz central frequency, 256 elements, Hologic-Supersonic Imagine, Aix en Provence, France) driven by a custom sequences of steered pushing beams (Ngo et al., 2023). Both ultrasound scanners, Aixplorer and Mach30, are commercialized by the same manufacturer, utilize the same elastography technology and operational sequences. However, the Aixplorer scanner possesses the essential elastography trigger-out capability required for estimating the slack angle. The transducer was placed in the same location as in the previous step, using a custom-made holder to ensure a proper alignment with the main axis muscle fibres throughout all scans. Then, ultrasound data were acquired at various elbow angles. Assessments were conducted every 5° increments, starting at the slack angle (previously established in step 1), and continuing until 45° of elbow extension. After each ultrasound acquisition, the elbow angle was passively moved to the following increment elbow angle. Ultrasound transducer was then rotated 90° in respect to the muscle fibres and ultrasound acquisitions were repeated for each elbow angle (Gennisson et al., 2010). Real-time transducer rotation angle was measured by using an optical measurement system (NDI Polaris Vicra, Northern Digital Inc., Waterloo, ON, Canada) and retroreflective markers attached to the transducer.

Finally, two maximal voluntary isometric elbow flexion and extension were performed with the elbow in a neutral position (90°) to normalize the EMG signals. There was a 1-min rest interval between each maximal effort trial. For both steps 1 and 2, trials in which any increase in EMG activity was visually observed were immediately

discarded and repeated to ensure passive conditions were maintained throughout data collection. In addition, trials that exhibited EMG activity greater than 1% of maximal voluntary contraction during post-processing were discarded due to the potential effects of active muscle stiffness on measurements.

# 4. Results

## 4.1. Ex vivo experiments

The very first results lie in the ex vivo quantification of the non-linear coefficients, a necessary preliminary part the understanding of the in vivo results presented below. To do this, by an original coupling, the mechanical curves in traction are acquired simultaneously with the measurements by ultrasound elastography (Fig. 4A). In Fig. 4B is presented a typical stress-strain relationship for a porcine skeletal muscle (Iliopsoas) samples (N = 6) where the fibers were oriented in 3 different configurations: (Martinez-Vidal et al., 2021) config #3: fibers and ultrasound probe parallel to the main axis of the mechanical device (Fig. 1A); (Lopez et al., 2008) config #5: fibers perpendicular and ultrasound probe parallel to the main axis of the mechanical device (Fig. 1B); (Guimaraes et al., 2020) config #7: fibers and ultrasound probe perpendicular to the main axis of the mechanical device (Fig. 1C). The change of slope of the stress-strain relationship allowed to define the linear elastic regime and hyperelastic regime and was defined at 8% (Fig. 4B). For each step, the slope and the nominal stress values were collected as well as the shear wave speed. Shear wave speed was quantified by using an ultrasound elastography sequence (Bercoff et al., 2004). This ultrasound technique allows to investigate the change in the stiffness map as a function of the applied stress (Fig. 4D).

In Table 2 are presented results for six fresh ex vivo muscles extracted from three pigs. The slopes of the relationship of the apparent shear modulus (shear modulus at each stress step $\mu_{app}(\sigma) = \rho.V_s^2(\sigma)$) as a function of the stress evolved between (0.27 ± 0.06 and 3.71 ± 0.22) as well as the corresponding coefficient of determination $R^2$. The Young's moduli extracted from the mechanical testing $E_{//}$ and $E_{\perp}$ are given in kPa as well as the shear moduli $\mu_{//}$ and $\mu_{\perp}$ and quantified by shear wave elastography at 0 stress. By using the acoustoelasticity theory described above and equations (8)–(10) the nonlinear elastic moduli $A$, $H$, and $K$ are quantified in kPa.

<figure><img src="image_6.png" /><figcaption>Fig. 4. A) Experimental setup of the shear wave speed acquisition during mechanical testing. B) Typical stress-strain relationship on one muscle for the three investigated configurations with the acoustoelasticity theory to quantified elastic nonlinear parameters. C) Shear wave speed maps of the muscle at different strain step (from 0 to 8%). D) Apparent shear modulus as a function of the nominal stress (measured using the force load of the tensile machine) for the three configurations. Each quantified slope was used to recover the nonlinear elastic parameters.</figcaption></figure>
<caption>**Table 2**</caption>

Results obtained for 6 muscles extracted from 3 pigs. Slopes $k_{3,5,7}$ of the shear wave speed as a function of the stress. Young's moduli//and $\perp$ quantified by mechanical testing. Shear moduli//and $\perp$ quantified by shear wave elastography at 0 stress. Resulting nonlinear elastic parameters, A, H, K.

<table><caption>**Table 3**</caption><thead><tr><th></th><th>PIG #1</th><th>Right muscle</th><th>PIG #2</th><th>Right muscle</th><th>PIG #3</th><th>Right muscle</th></tr></thead><tbody><tr><td>k<sub>3</sub></td><td>0.27 ± 0.06</td><td>0.84 ± 0.03</td><td>0.78 ± 0.03</td><td>0.79 ± 0.11</td><td>1.55 ± 0.32</td><td>1.24 ± 0.14</td></tr><tr><td>k<sub>5</sub></td><td>1.98 ± 0.58</td><td>2.80 ± 0.44</td><td>1.13 ± 0.36</td><td>2.08 ± 0.24</td><td>0.95 ± 0.13</td><td>3.71 ± 0.22</td></tr><tr><td>k<sub>7</sub></td><td>2.42 ± 0.96</td><td>2.52 ± 0.55</td><td>0.93 ± 0.06</td><td>0.70 ± 0.03</td><td>1.51 ± 0.11</td><td>1.70 ± 0.07</td></tr><tr><td>R<sup>2</sup>(k<sub>3</sub>)</td><td>0.62</td><td>0.73</td><td>0.62</td><td>0.82</td><td>0.93</td><td>0.77</td></tr><tr><td>R<sup>2</sup>(k<sub>5</sub>)</td><td>0.82</td><td>0.70</td><td>0.77</td><td>0.84</td><td>0.76</td><td>0.92</td></tr><tr><td>R<sup>2</sup>(k<sub>7</sub>)</td><td>0.81</td><td>0.78</td><td>0.75</td><td>0.60</td><td>0.89</td><td>0.83</td></tr><tr><td>E<sub>||</sub> [kPa]</td><td>27.39 ± 0.61</td><td>28.97 ± 1.68</td><td>31.58 ± 1.92</td><td>32.42 ± 2.21</td><td>31.95 ± 5.91</td><td>27.62 ± 5.42</td></tr><tr><td>E<sub>⊥</sub> [kPa]</td><td>10.02 ± 1.24</td><td>9.82 ± 1.16</td><td>9.16 ± 1.23</td><td>9.15 ± 3.42</td><td>10.91 ± 2.13</td><td>13.02 ± 1.67</td></tr><tr><td>μ<sub>||</sub> [kPa]</td><td>19.90 ± 3.35</td><td>9.98 ± 1.07</td><td>7.35 ± 1.26</td><td>4.53 ± 0.46</td><td>8.16 ± 0.94</td><td>8.44 ± 0.76</td></tr><tr><td>μ<sub>⊥</sub> [kPa]</td><td>9.40 ± 1.60</td><td>7.37 ± 1.86</td><td>4.17 ± 0.90</td><td>3.40 ± 0.43</td><td>4.37 ± 0.48</td><td>3.87 ± 0.39</td></tr><tr><td>A [kPa]</td><td>-313.85 ± 99.14</td><td>-288.73 ± 93.9</td><td>-113.47 ± 35.69</td><td>-112.64 ± 21.82</td><td>-127.91 ± 24.36</td><td>-198.78 ± 28.65</td></tr><tr><td>K [kPa]</td><td>-31.23 ± 11.38</td><td>-52.72 ± 0.64</td><td>-46.1 ± 5.87</td><td>-31.69 ± 2.37</td><td>-77.6 ± 20.04</td><td>-33.75 ± 14.56</td></tr><tr><td>H [kPa]</td><td>128.82 ± 63.99</td><td>137.98 ± 41.9</td><td>30.13 ± 3.34</td><td>1.37 ± 8.9</td><td>48.62 ± 0.76</td><td>34.01 ± 10.69</td></tr></tbody></table>

Second order elastic parameters and nonlinear elastic parameters of the biceps brachii for 3 volunteers.

<table><thead><tr><th>Volunteer</th><th>#1</th><th>#2</th><th>#3</th></tr></thead><tbody><tr><td>E<sub>||</sub> (kPa)</td><td>20.53 ± 1.27</td><td>23.20 ± 1.35</td><td>24.25 ± 3.03</td></tr><tr><td>μ<sub>||</sub> (kPa)</td><td>5.13 ± 1.21</td><td>5.77 ± 0.92</td><td>4.27 ± 0.45</td></tr><tr><td>μ<sub>⊥</sub> (kPa)</td><td>1.60 ± 0.27</td><td>2.12 ± 0.31</td><td>2.41 ± 0.87</td></tr><tr><td>k<sub>3</sub></td><td>0.05 ± 0.03</td><td>0.49 ± 0.11</td><td>0.17 ± 0.04</td></tr><tr><td>k<sub>9</sub></td><td>0.03 ± 0.01</td><td>0.03 ± 0.02</td><td>0.02 ± 0.01</td></tr><tr><td>R<sup>2</sup>(k<sub>3</sub>)</td><td>0.68</td><td>0.96</td><td>0.95</td></tr><tr><td>R<sup>2</sup>(k<sub>9</sub>)</td><td>0.77</td><td>0.57</td><td>0.88</td></tr><tr><td>X (kPa)</td><td>-25.01 ± 2.06</td><td>-38.19 ± 4.43</td><td>-30.23 ± 2.82</td></tr><tr><td>Y (kPa)</td><td>-4.22 ± 1.11</td><td>-5.57 ± 1.70</td><td>-6.75 ± 3.07</td></tr></tbody></table>

## 4.2. *In vivo* biceps brachii experiments

*In vivo* the easiest configurations to quantify the nonlinear parameters are config #3 (Fig. 1A) and #9 (Fig. 1D). The shear wave speed was quantified within 3 healthy volunteers on biceps brachii (Fig. 3A). To produce the stress of the arm of the volunteers was placed in an ergometer to control the angle of the elbow. The arm was extended from 90° to ~180° (from rest to full extension of the arm). At rest the Young's modulus parallel to fibers as well as both shear moduli were measured. For each step the shear wave speed was measured parallel and perpendicular to fibers as well as the strain induced in the muscle due to the extension of the arm (Fig. 3B). Then as already proposed in Latorre-Ossa et al., 2012), since the Young's modulus is estimated, the stress was retrieved for each step by applying the Hooke's law. Results are presented in Table 3, the slopes of the relationship of the apparent shear modulus as a function of the stress evolved between (0.02 ± 0.01 and 0.49 ± 0.11) as well as the corresponding coefficient of determination R². The Young's moduli E<sub>||</sub> in kPa extracted from the dedicated ultrasound sequence proposed by Ngo et al. (2023) as well as the shear moduli μ<sub>||</sub> and μ<sub>⊥</sub> and quantified by shear wave elastography at 0 stress. By using the acoustoelasticity theory described above and equations (13) and (14) a combination X and Y, of the nonlinear elastic moduli A, H, and K is quantified in kPa.

## 5. Discussion

In this study we demonstrated, for the first time, the feasibility of assessing the nonlinear shear elasticity of (anisotropic) skeletal muscles through a combination of mechanical testing and ultrasound shear wave elastography. Prior to the application of this completely non-invasive approach in humans (in vivo biceps brachii muscle), we characterized the third-order nonlinear elastic properties in vitro (iliopsoas muscle). In addition, this allowed us to validate the acoustoelasticity theoretical framework for skeletal muscles. The successful assessment of these

nonlinear properties represents a promising step towards the noninvasive quantification of mechanical forces or stress in living organisms. Furthermore, our comprehensive approach opens numerous potential applications in both research and clinical use, which can offer insights into better understanding and diagnosing muscle-related pathological processes and mechanobiology.

For all *in vitro* experiments and for all muscle configurations, we obtained the slope of the apparent shear modulus μ<sub>app</sub>(σ) = ρ<sub>0</sub>V<sub>s</sub><sup>2</sup>(σ) as a function of the wave propagation direction were obtained for each applied stress. As expected by theory, and while operating under small strain levels (linear elastic behavior as investigated by the usual elastography methods), the shear wave speed evolved linearly with stress. Subsequently, by combining different configurations (#3, #5, and #7, corresponding to equations 3, 4, and 5, respectively) with the E<sub>||</sub> measurement obtained through mechanical tests, we successfully quantified the nonlinear parameters *in vitro*, validating the theory and enabling the first-ever measurement of these three independent nonlinear elastic parameters: A, K, and H. The reliability of the approach is also demonstrated by the fact that two muscles (left and right) extracted from the same animal show very similar values. It should be noted that, even in the presence of different elastic parameters between the left and right muscles of the same animal, which are quite likely intra-individual differences, the nonlinear elasticity parameters remain quite similar. Moreover, one can notice that even each parameter varies greatly across muscles they keep the same sign (as an example H varies from 1.37 kPa to 137.98 kPa but is always positive). One can also notice that each parameter has the same order of magnitude across pigs, minus hundreds of kPa, minus tens of kPa and plus tens of kPa for A, K and H, respectively. While our *in vitro* setting provides valuable insights, it is essential to acknowledge some limitations such as the conservation method after sampling. To improve reproducibility and reduce the margins of error of the results, a thermostatic bath regulated at 37 °C could be used. This would not only have enabled the tests to be carried out at 37 °C (instead of room temperature between 22 °C and 25 °C), but also to maintain sample hydration. However, this option was not possible here due to the need to position the ultrasound probe close to the tested sample throughout the tensile test. Even if temperature has a significant effect on viscous properties in small deformations or hyperelastic properties in large deformations (as well as on the active properties of muscle tissue), we can reasonably estimate here that the influence of temperature on passive elastic properties in small deformations and on the results of this study remains limited.

In addition, the process of sampling the longitudinal and transverse specimens for the tensile testing machine was carried out manually. Enhancing the cutting technique to achieve a fully homogenous and regular shape could have further enhance the precision of the mechanical testing. Directly acquiring the forces at the traction machine could have further been useful. Although this data might have been redundant,
it would have added further validation to our results. Moreover, due to the specified minimum and maximum dimensions required for the tensile machine, any configurations other than the three described here were not considered for *ex vivo* testing. Testing the remaining five elastodynamic configurations (Bied and Gennisson, 2021), which were not explored in our work, would provide further confirmation of the results. However, the *in vitro* experiments were conducted on a limited number of specimen samples to minimize the number of sacrificed animals and comply with European ethical guidelines (the 3R rules: recycle, reuse, reduce). The validation of this approach could also be considered by other methods, such as numerical simulations or magnetic resonance elastography. Notably, these preliminary results demonstrate the possibility of quantifying nonlinear shear elasticity in anisotropic tissues such as skeletal muscle by coupling mechanical testing and shear wave elastography.

Subsequently, for the very first time, we successfully estimated a combination of the nonlinear elastic parameters *in vivo*, which paves the way for a wide range of biomechanical and clinical applications. Notably, our investigation focused on a fusiform muscle model, the *biceps brachii*, utilizing ultrasound shear wave elastography to assess the shear modulus both parallel and perpendicular to the main direction of the muscle fibers (Gennisson et al., 2010). In this specific setting, we were limited to only two elastodynamic configurations (#3 and #9, eq. 3 and 6), during which we evaluated shear modulus and stress during axial stretching. The applied stress on the muscle resulted from a passive extension of the elbow without significant myoelectric activity, which corresponds to the best approach to control the applied stress *in vivo*. The stress was calculated based on the local muscle strain applied during two elbow angles during the passive stretching, which, when combined with Young's modulus measurement, allowed us to retrieve the stress through the Hooke's law (Latorre-Ossa et al., 2012). This allowed to obtain an estimation of localized – in the plane of the shear modulus measurements – and individual mechanical stress of the targeted muscle, as opposed to the traditional joint torque value provided by dynamometers. Importantly, knowing the nonlinear properties of muscle allows to understand the inherent stress in each muscle independently.

Several limitations and potential are as for improvement warrant discussion, with one critical aspect being the need for estimating $E_{//}$. This value can be readily quantified through the previously described *in vitro* tests. One viable *in vivo* approach to quantify $E_{//}$ involves the utilization of a dedicated ultrasound sequence under conditions of zero stress (Ngo et al., 2023). This approach is useful since it allows also to investigates Young's modulus in pinnated muscle but remains complicated to interpret during contraction. Moreover, there are some limitations in the assumed characteristics of muscle tissue mechanics that have to be noted, with the assumption of uniaxial fiber alignment resulting in transverse isotropy, even though this is commonly assumed for this type of muscle, as well as tissue quasi-incompressibility. Added to this is the difficulty of accessing the stress exerted on muscle tissue, unlike *ex vivo* testing, and of overcoming this by reconstructing nonlinear properties from force measurements. The same shear wave elastography sequences were employed in both human and *ex vivo* tests, making it unlikely that they contributed significantly to the observed differences between the two types of tests. These disparities are more likely attributed to variations in the methods used to estimate the mechanical stress. In the *ex vivo* measurements, stress was directly measured from mechanical testing, while in the *in vivo* tests, it was indirectly estimated as the product of $E_{//}$ (Young's modulus, derived from the steered pushing beams method) and $\varepsilon_{//}$ (strain). In the latter case, strain was computed using a cross-correlation algorithm applied across multiple elbow angles B-mode images reflecting muscle stretching. These differences in stress estimation methods could potentially result in alterations to the slopes of the experimental stress-shear modulus curves, consequently affecting the calculated X and Y values. Moreover, viscoelastic stress relaxation could have affected the shear modulus used in the stress-shear modulus relationships, from where the slopes were quantified. However, all

ultrasound measurements were performed at small deformation levels which may have only a minimal effect on the shear modulus (Freitas et al., 2015). At last the large standard deviation of the nonlinear parameters A, H and K can come from the cumulative errors made on each parameter estimated (i.e. shear wave speed, Young's modulus estimated by ultrasound, stress and strain calculation). This would be improved in future development.

Despite these assumptions, the values and variations obtained *in vivo* are in good agreement with the behavior expected by our theory and already observed *ex vivo*, confirming the reliability of the values obtained despite the absence of other existing *in vivo* methods that could serve as a comparison or validation reference.

## 6. Conclusion

Taken together, our work provides a promising proof of concept for accurate and robust noninvasive nonlinear characterization of skeletal muscles *in vivo*, while introducing a novel innovative approach for quantifying localized mechanical forces in living tissues. This ground-breaking methodology holds the potential to revolutionize diagnostic and monitoring applications not only in pathological or post-traumatic skeletal muscles but also in other biological soft organs characterized by fiber reinforcement along a clearly identified main direction. By unlocking new avenues for noninvasive mechanical stress assessment, our research may pave the way for advancements in biomechanics and facilitate a deeper understanding of tissue mechanics and mechanobiology in living organisms.

### CRediT authorship contribution statement

**Ha Hien Phuong Ngo:** Writing – original draft, Software, Methodology, Investigation, Funding acquisition, Data curation. **Ricardo J. Andrade:** Writing – review & editing, Validation, Formal analysis, Data curation, Conceptualization. **Juliette Lancelot:** Software, Funding acquisition, Data curation. **Aude Loumeaud:** Funding acquisition, Data curation. **Corentin Cornu:** Validation, Software. **Antoine Nordez:** Writing – review & editing, Project administration, Methodology, Conceptualization. **Simon Chatelin:** Writing – review & editing, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Jean-Luc Gennisson:** Writing – review & editing, Writing – original draft, Validation, Supervision, Project administration, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: The authors declare no competing interest: Jean-Luc Gennisson reports a relationship with Supersonic Imagine that includes: consulting or advisory.

### Data availability

Data will be made available on request.

### Acknowledgments

This work was funded by the ANR INNOVAN (ANR-19-CE19-0017), by PEPS CNRS fundings (MUSCLOR 2022) and by the France Life Imaging (ANR-11-INBS-0006 grant from the French "Investissements d'Avenir" program). These developments were supported by IdEx Unistra (ANR-10-IDEX-0002) with the IRIS technology platform and by SFRI (STRAT'US project, ANR-20-SFRI-0012) under the framework of the French Investments for the Future Program. The authors also thank the team of the preclinical department of the IHU Strasbourg for their
logistic support.

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