# An original approach to generate periodic Representative Volume Elements with anisotropic heterogeneous microstructure: application to skeletal muscle

Aude Loumeaud, Philippe Pouletaut, Sabine Bensamoun, Daniel George, Simon Chatelin

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Aude Loumeaud, Philippe Pouletaut, Sabine Bensamoun, Daniel George, Simon Chatelin. An original approach to generate periodic Representative Volume Elements with anisotropic heterogeneous microstructure: application to skeletal muscle. Journal of Biomechanics, 2025, 191, pp.112897. 10.1016/j.jbiomech.2025.112897. hal-05212599

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# An original approach to generate periodic Representative Volume Elements with anisotropic heterogeneous microstructure: application to skeletal muscle

Aude Loumeaud<sup>a ID</sup>, Philippe Pouletaut<sup>b</sup>, Sabine F. Bensamoun<sup>b</sup>, Daniel George<sup>a</sup>, Simon Chatelin<sup>a,* ID</sup>

## ARTICLE INFO

### Keywords:

- Multiscale modeling

- Mouse skeletal muscle

- Extracellular matrix

- Fiber reinforced composites

- Representative Volume Element

## ABSTRACT

Skeletal muscle is an organ whose hierarchical, multiscale structure greatly influences the overall mechanical response. Complementary to mechanical experiments, finite element modeling is increasingly used to study the influence of its constituents across different scales. To develop such a multiscale model, particular attention must be paid not only to the scale transition, but also to the definition of the structure and its mechanical behavior at different scales (macroscopic, microscopic, submicron). One of the most effective approaches is to define a Representative Volume Element (RVE) including smaller scale components and their respective mechanical behavior laws, likely to be altered through pathologies.

In this study, an original approach for periodic RVE generation dedicated to multiscale modeling of the skeletal muscle is proposed. From optical microscopy cross-section images of mouse skeletal muscle and single fiber experiments, the RVE integrates parameters related to fiber type distribution, geometric and mechanical characteristics. The key features of this geometry are spatial periodicity, rounded edges and inclusion of experimentally measured probabilistic distributions of the extracellular matrix (ECM), slow and fast muscle fibers. Smooth variation of the mechanical properties between the muscle fibers and the ECM are implemented to avoid unrealistic and purely numerical stress accumulation at these interfaces through the definition of transition layers between the different microcomponents. By the inclusion of custom geometrical and material features, this original model allows the multiscale and multicomponent analysis of different muscle phenotypes and can also be used for other heterogeneous anisotropic materials such as fiber reinforced composites.

## 1. Introduction

Skeletal muscle is a hierarchical structure exhibiting specific constituents and mechanical properties at each scale: (1) the macroscopic scale, constituted of the skeletal muscle surrounded by epimysium; (2) the mesoscopic scale containing fascicles each wrapped into perimysium; (3) the microscopic scale including different types of muscle fibers each wrapped by endomysium; (4) the submicron scale, with myofibrils embedded in muscle fibers and collagen fibers embedded in epimysium, whereas costameres link the fibers and the endomysium (Mukund and Subramaniam, 2020). Each of these constituents influence the overall organ function and health and in particular its mechanical response to stress across these different scales (Böl et al., 2019; Kuravi et al., 2021b;

Pouletaut et al., 2023; Tatarenko et al., 2024). Finite element (FE) analysis is increasingly used to analyze the influence of muscle constituents at each scale (such as muscular fibers, fascicles, myofibrils, collagen fibers and others components of the extracellular matrix – ECM) on the overall mechanical response of the organ (Virgilio et al., 2015; Konno et al., 2022; Loumeaud et al., 2024; Sahani et al., 2024), allowing for the investigation of pathologies such as neuro-muscular diseases (Konno et al., 2021) or dystrophies (Sahani et al., 2024).

One of the critical points in the use of multiscale numerical models lies in modeling the transition from one anatomical scale to another. Mostly, multiscale FE analysis methods for skeletal muscles modeling are linking two different scales through the definition of a Representative Volume Element (RVE), which is defined at the lower scale

<footnote><sup>a</sup> University of Strasbourg, CNRS, Inserm, ICube, UMR 7357, Strasbourg, France</footnote>

<footnote><sup>b</sup> Université de technologie de Compiègne, CNRS, Biomechanics and Bioengineering, UMR 7338, Compiègne, France</footnote>

<footnote>* Corresponding author at: University of Strasbourg, CNRS, Inserm, ICube UMR 7357, IHU, 1 Place de l'hôpital, 67000 Strasbourg, France.
E-mail address: schatelin@unistra.fr (S. Chatelin).</footnote>
(Bargmann et al., 2018; Lamsfuss and Bargmann, 2022; Loumeaud et al., 2024). One of the most efficient way to define RVE consists in taking into account probabilistic distributions of the components at the lowest scale and to model muscle cross-sections as Voronoi diagrams (Honda, 1983, 1978). Such descriptions have been used in previous studies to generate numerical models of skeletal muscle at the microscopic scale (Loumeaud et al., 2024). However, few of them integrate parameters related to fiber types and geometric characteristics, which can be altered through a pathological process (Kammoun et al., 2017; Sahani et al., 2022).

In this study, we propose an original approach for periodic RVE generation dedicated to multiscale modeling of the skeletal muscle. Two successive processes are successively implemented: (1) the definition of a 2D geometry at the microscopic scale from the manual segmentation of a cross-sectional microscopy image of mouse skeletal muscle including statistical distribution of the anatomic components; (2) the generation of the 3D geometry, material and mesh controls in a specific FE code. Using Voronoi tessellations, we represent muscle fibers and ECM separated by an original transition layer representing costameres and transmembrane proteins (Peter et al., 2011). The diagram presents rounded-up edges, which purpose is twofold: (1) accurately represent muscle fibers and (2) decrease numerical errors due to the presence of sharp edges (Schmidt et al., 2009). In addition, we propose computing a gradient of mechanical properties during the FE simulation in the transition layer. By applying our pipeline to model murine muscle in uniaxial tensile tests simulation, we show that this approach is meant to (1) reproduce the macroscopic behavior of skeletal muscle and (2) integrate the costameres linking the components at the microscale, potentially involved in pathologies such as Duchenne Muscular Dystrophy (Virgilio et al., 2015).

## 2. Description of the RVE

The geometry is created in the Finite Element (FE) software ABAQUS® SIMULIA (Dassault Systèmes SE, Vélizy-Villacoublay, France). Two successive algorithms are implemented (Fig. 1). First, a 2D geometry is

obtained from the manual segmentation of microscopy image using Matlab software (Mathworks, Nattick, MA, USA). Then, the 3D geometry, material and mesh controls are generated using python script (all codes available upon request).

### 2.1. Generation of the 2D geometry

Our approach consists in considering first a simplified 2D muscle cross-section with the hypothesis that the muscle fibers are locally straight and parallel to each other within the material. The key features of this geometry are (1) its spatial periodicity, (2) its rounded aspect, and (3) its components distributions based on experimentally measured probabilistic distributions of the ECM width and fiber types. Lastly, we consider an additional component named “transition layer”, which separates the fibers and the ECM. Consequently, we assimilate the muscle cross section to a Voronoi tessellation.

Let us consider the metric space $\mathbb{R}^2$ with the Euclidian distance function $d$, and $(e_1, e_2)$ an orthonormal base of $\mathbb{R}^2$. Let $P_{i,l} \in [1, n]$ be a non-empty set of $n$ distinct points in the space $\mathbb{R}^2$, called Voronoi seeds. The Voronoi cell or polygon $F_i$ associated with the seed $P_i$ is the set of all points $X = (x_1, x_2)$ in $\mathbb{R}^2$ whose distance to $P_i$ is smaller than their distance to all seeds $P_j$ different from $P_i$. The resulting polygons $F_i$ are simple convex. The Voronoi diagram represents the partition of $\mathbb{R}^2$ using polygons $F_i$ (Eq. (1)).

$$F_i = \{(x_1, x_2) \in \mathbb{R}^2 | d((x_1, x_2), P_i) \le d((x_1, x_2), P_j) \forall j \ne i\} \quad (1)$$

Let us restrict ourselves to a nonempty, convex and continuous part of $\mathbb{R}^2$. We consider the square $[0; 1] \times [0; 1]$ where a set of seeds $S_i$ is defined. These are referred to as initial seeds. Considering the domain $[-1; 2] \times [-1; 2]$, the initial seeds are duplicated along $e_1$ and $e_2$. By computing a Voronoi diagram from the duplicated seeds $P_i$, a periodic arrangement of the geometry with respect to $e_1$ and $e_2$ can be obtained (Spyrou et al., 2019) (Fig. 1 Step 2). In the remainder of the article, we will refer to the polygons as fibers. In practice, this process for the

<figure><img src="image_4.png" /><figcaption>Fig. 1. FE model generation: workflow from a microscopic cross-section to a 3D RVE.</figcaption></figure>
generation of the 2D geometry is implemented based on several successive steps (Fig. 1):

### 2.1.1. Segmentation

The first step corresponds to manual segmentation of Voronoi seeds corresponding to the fibers on a cross-section image of mouse skeletal muscle from an experimental histological process (Fig. 1 Step 1). The images used in this study were acquired at magnification ×20 on a Leica microscope from 10 µm thin cross-sectional slices of mouse muscles (Kammoun et al., 2017). A fraction of this image is manually segmented by the user (Fig. 1. Step 1), corresponding to the domain [0; 1] × [0; 1]. The obtained seeds $S_i$ are duplicated in the domain [-1; 2] × [-1; 2].

### 2.1.2. Generation of the Voronoi diagram and subset selection

A Voronoi diagram is generated (Fig. 1 Step 2) from the duplicated seeds $P_i$ and modified to include ECM: for this purpose, ECM widths are measured on the muscle image using a custom code in Matlab software (Mathworks, Nattick, MA, USA). Then a kernel density estimation is applied to these measurements (Kotz and Balakrishnan, 1993) and used as probability distribution to draw values of ECM width for each edge in a specific subset of polygons $F_i$ of the diagram (Fig. 1 Step 3). The selected subset corresponds to the seeds $S_i$ and step 3 guarantees the diagram's periodicity.

### 2.1.3. Layers generation and fiber typing

The obtained polygons are shrunk by a constant value to obtain the transition layer. The polygons are now referred to as fibers and a type is also attributed to those fibers based on their area (Fig. 1 Step 4), corresponding to slow or fast fibers. The fiber type is drawn from a probability distribution of the fiber type knowing the area, based on the data published in (Kammoun et al., 2016).

### 2.1.4. Bounding box intersection

At each vertex, the geometry is sharp. Acute angles may affect the convergence and the results of the FE simulation (Schmidt et al., 2009). Thus, the geometry is rounded up at each vertex by removing a fixed area (parameter “A” in Fig. 1 Step 4) from the fiber and replacing the initial vertex by new vertices. The obtained geometry is intersected with a bounding box to generate the final geometry (Fig. 1 Step 5). To be able to apply periodic boundary conditions (Michel et al., 1999) to the 3D model in Abaqus (Omairey et al., 2019), edges on the opposite side of the RVE must display the same number of nodes at identical position on the edges. Thus, mesh seed numbers are assigned to opposite edges of the RVE boundary (Fig. 1 Step 6).

## 2.2. Definition of the 3D RVE

The 3D geometry is generated and dimensioned (1 unit = 1 µm). Sections are created and materials are assigned. Using the mesh seed assignments from the 2D algorithm (Fig. 1 Step 6), sets of edges and periodic mesh seed controls are generated using a specific script (python script implemented for ABAQUS, available upon request).

# 3. Application to murine soleus muscle

In this section, we provide an application of the pipeline to model the passive behavior of murine soleus skeletal muscle under uniaxial tension. For this purpose, we use histological and mechanical experimental data previously acquired within our group to create a multiscale model able to reproduce passive muscle behavior and predict ECM mechanical properties.

## 3.1. Geometry

### 3.1.1. Microscale geometry

To study murine soleus muscle, we segmented two subsets of the

same histological cross-section as shown in Fig. 2, with dimensions 93*93 µm each. One geometry was used to predict ECM mechanical properties, and the second geometry was used to evaluate reproducibility. Fiber type distributions were taken from (Kammoun et al., 2017) and ECM width distribution was obtained as described previously from the same image. Prior to this study, a parametric study was conducted to assess the influence of different algorithm parameters and provide references for our final choice (Table 1).

### 3.1.2. Macroscale geometry

The chosen soleus macroscale geometry is an idealized fusiform muscle geometry as shown in Fig. 3 C. Simplification consists in assuming 3 central axes of symmetry to reduce the model's size. The model is meshed with solid linear brick elements using reduced integration (C3D8R) elements.

### 3.1.3. Scale transition and loads

Both scales were implemented in a FE simulation in ABAQUS® using the Direct Finite Element Square (DFE²) method (Tan et al., 2020). As shown in Fig. 3, uniaxial tension was applied along the muscle's longitudinal axis by constraining a small volume in displacement along Z, for a total displacement equal to 20 % of the total muscle length. Symmetry boundary conditions were imposed on the three surfaces aligned with the axes of symmetry.

## 3.2. Constitutive behavior

### 3.2.1. Constitutive modelling

Transversely isotropic hyperelastic materials are defined for the different microscopic components of the skeletal muscle. Let us define $F$, $C = F^T F$ as the deformation gradient, and the right Cauchy-Green deformation tensor, respectively. Let us define the 1st, 2nd and 4th invariants of $C$ as:

$$ \bar{I}_1 = \det(\bar{C}) \quad \text{where} \quad \bar{C} = J^{-\frac{2}{3}} \bar{C} \qquad (2) $$

$$ I_3 = J^2 = \det(C) \quad \text{with} \quad J = \det(F) \qquad (3) $$

$$ \bar{I}_4 = \mathbf{M}_i \cdot \bar{C} \mathbf{M}_i \qquad (4) $$

$\mathbf{M}_i$ defines the anisotropy directions in the material, computed in the plane X1-X3 with an angle $\pm \theta$ with respect to X3 (Fig. 4). We use the Holzapfel-Gasser-Ogden (HGO) model developed for fiber reinforced composites (Holzapfel et al., 2000), where the strain energy density function is written as:

$$ \Psi = C_{10}(\bar{I}_1 - 3) + \frac{K}{2}(J - 1)^2 + \frac{k_1}{2k_2} \sum_{i=1,2} \left( \exp\left[k_2(\bar{I}_4 - 1)^2\right] - 1 \right) \qquad (5) $$

With $k_1$, $k_2$ material parameters related to the collagen fiber reinforcement of the ECM.

### 3.2.2. Muscle fibers constitutive behaviour

We use experimental data acquired on ex vivo mechanical passive uniaxial tensile tests on isolated soleus muscle fibers (microscale) and on whole soleus muscles ( macroscale) (Tatarenko et al., 2024) of wildtype C57BL/6 mice. The mechanical properties of single muscle fibers were identified on the experimental data using using Nelder-Mead Simplex method (Lagarias et al., 1998) (fminsearch function, Matlab, Mathworks, Natick, MA, USA) and are listed in Table 2. A relationship between parameters C10 and K was established, corresponding to an equivalent Poisson's ratio of 0.49 to provide a tradeoff between quasi-incompressibility and numerical stability. The remaining mechanical parameters are linked to the ECM, which is scarcely studied in the literature. Moreover, the existing literature about ECM mechanical properties is very heterogeneous. Therefore, we use the macroscale experimental data to identify ECM behavior.
<figure><img src="image_5.png" /><figcaption>Fig. 2. Histological cross-section segmentation and corresponding geometries 1 and 2.</figcaption></figure>

<table><caption>**Table 1**

Characteristics and parameters for the generation of RVE based on segmentation 1 and 2.</caption><thead><tr><th>Criteria</th><th>RVE geometry 1</th><th>RVE geometry 2</th></tr></thead><tbody><tr><td><strong>Characteristics</strong></td><td></td><td></td></tr><tr><td>Number of fast fibers [-]</td><td>8</td><td>2</td></tr><tr><td>Number of slow fibers [-]</td><td>3</td><td>5</td></tr><tr><td>Number of mesh elements [-]</td><td>19,790</td><td>23,755</td></tr><tr><td>Fiber Volume Fraction (FVF) [-]</td><td>0.90</td><td>0.91</td></tr><tr><td><strong>Parameters for 2D geometry generation</strong></td><td></td><td></td></tr><tr><td>Area parameter [µm²]</td><td>0.052</td><td>0.052</td></tr><tr><td>Number of additional vertices to form a round</td><td>2</td><td>2</td></tr><tr><td><strong>Mesh parameters</strong></td><td></td><td></td></tr><tr><td>Mesh density for ECM and transition layer</td><td>30</td><td>30</td></tr><tr><td>Mesh density for muscle fibers</td><td>18</td><td>18</td></tr></tbody></table>

### 3.2.3. ECM constitutive behaviour

Very few studies have quantified the mechanical properties of ECM, even more scarcely using the HGO model parameters (Gasser et al., 2006; Holzapfel et al., 2000; Kailash et al., 2024; Lanzl et al., 2024). Amongst those studies, even fewer consider murine ECM (Dupres et al., 2025; Lanzl et al., 2024). Since these large deformation models are so underdeveloped, we propose ranges of values of 1.1–200 kPa for parameter C10, 0.56–997 kPa for parameter k1 and 0.1–525 for

parameter k2. Parameter $\theta$ defining the orientation of the collagen fibers within the ECM in regard to the main muscle axis is admitted to be in range 55° to 59° in healthy muscle, confirming observations ranging from 34° to 55° in compression and 55° to 73° in tension in bovine skeletal muscles (Purslow and Trotter, 1994). Since we only consider uniaxial extension, we used a value of 55° for parameter $\theta$. Parameter space sampling was conducted using Latin HyperCube Sampling (Shields and Zhang, 2016; Stein, 1987) and criterion to stop the

<figure><img src="image_6.png" /><figcaption>Fig. 4. Definition of material orientation and applied loads.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. (A) Experiment conducted on isolated soleus muscle, (B) illustration of the boundary conditions and (C) representation of the macroscale geometry.</figcaption></figure>
<table><caption>**Table 2**
Material parameters identified for fast and slow fibers and implemented for multiscale model.</caption><thead><tr><th>Material</th><th>Parameters</th><th>Values</th></tr></thead><tbody><tr><td rowspan="5">Fast fiber</td><td>C<sup>sf</sup><sub>10</sub></td><td>37.6 kPa</td></tr><tr><td>K<sup>sf</sup></td><td>3740 kPa</td></tr><tr><td>k<sup>sf</sup><sub>1</sub></td><td>6.00 kPa</td></tr><tr><td>k<sup>sf</sup><sub>2</sub></td><td>1.17</td></tr><tr><td>θ<sup>sf</sup></td><td>0.0°</td></tr><tr><td rowspan="5">Slow fiber</td><td>C<sup>sf</sup><sub>10</sub></td><td>53.8 kPa</td></tr><tr><td>K<sup>sf</sup></td><td>5350 kPa</td></tr><tr><td>k<sup>sf</sup><sub>1</sub></td><td>8.58 kPa</td></tr><tr><td>k<sup>sf</sup><sub>2</sub></td><td>1.17</td></tr><tr><td>θ<sup>sf</sup></td><td>0.0°</td></tr></tbody></table>

identification algorithm was a threshold of 0.02 for normalized RMSE value between experimental and numerical values.

### 3.2.4. Transition layer constitutive behaviour

The transition layer component represents costameres. As it is a continuous linkage from muscle fibers to ECM, we propose to represent it as an isotropic Neo-Hookean material, with mechanical properties varying between bounds formed by the fibers and ECM materials following the function in Eq. (6). The function describes the evolution of a mechanical parameter P over a local direction orthogonal to the nearest muscle fiber boundary. P is computed as a function of distance x from the muscle fiber boundary. The mechanical parameter P varies between bounds P<sub>Fib</sub> and P<sub>ECM</sub>, respectively representing the muscle fiber (x = 0) and the ECM (x = 1) values for this parameter.

$$P(x) = P_{\text{Fib}} - (P_{\text{Fib}} - P_{\text{Mat}})^* \frac{1}{1 + e^{-5*x}} \quad (6)$$

The Abaqus subroutine is adapted from (Castillo-Méndez and Ortiz, 2022). For the transition layer component, the algorithm (1) localizes the points in the geometry, and (2) computes the distance of these points to the nearest fiber (Fig. 1). From this distance, a gradient in the mechanical properties is computed and applied to all mechanical parameters in the transition layer following Eq. (6).

### 3.3. Results

Several sets of parameters allowed to reproduce the macroscopic behavior of the whole soleus muscle (Table 3). Both geometries display similar behavior and satisfying goodness of fit with the experimental data. Normalized RMSE errors are below 0.02 for geometry 1 and below 0.06 for geometry 2. Behavior simulated using geometry 2 is less rigid in all cases (Fig. 5), showcasing the influence of microscopic geometry

<table><caption>**Table 3**
Material parameters identified for ECM.</caption><thead><tr><th>Set</th><th>Parameters</th><th>Values</th></tr></thead><tbody><tr><td rowspan="5">ECM set 1</td><td>C<sup>m</sup><sub>10</sub></td><td>36.0 kPa</td></tr><tr><td>K<sup>m</sup></td><td>3760 kPa</td></tr><tr><td>k<sup>m</sup><sub>1</sub></td><td>15.1 kPa</td></tr><tr><td>k<sup>m</sup><sub>2</sub></td><td>80.0</td></tr><tr><td>θ<sup>m</sup></td><td>±55.0°</td></tr><tr><td rowspan="5">ECM set 2</td><td>C<sup>m</sup><sub>10</sub></td><td>36.0 kPa</td></tr><tr><td>K<sup>m</sup></td><td>3760 kPa</td></tr><tr><td>k<sup>m</sup><sub>1</sub></td><td>5.00 kPa</td></tr><tr><td>k<sup>m</sup><sub>2</sub></td><td>100</td></tr><tr><td>θ<sup>m</sup></td><td>±55.0°</td></tr></tbody></table>

variations.

## 4. Discussion

The developed algorithm proposes a pipeline for the integration of histological and mechanical experimental data to develop custom RVE of skeletal muscle. Particularly, different fiber types are integrated, allowing to model different muscle types. However, some limitations are introduced in the algorithm, such as the hypothesis of parallel fibers at the microscale, which does not allow to capture full anisotropy of the skeletal muscle (Kuravi et al., 2021). These fibers are divided into two groups, fast and slow, thus ignoring further subdivisions for this study. The algorithm allows to distinguish between fiber types I, IIA, IIB and IIX; however, we lack experimental data at the microscale to be able to distinguish the mechanical properties of each subgroup. Moreover, hyperparameters such as the geometrical smoothing parameter “A” and the gradient function present in the transition layer are not linked to measured biological phenomena. Therefore, further developments should include measurements of geometrical smoothing on histological cross-section images. Furthermore, the transition layer constitutive modeling can be changed to be more representative of costameres, as modeled in (Virgilio et al., 2015; Zhang and Gao, 2014).

The study presented here showcases a small number of geometries and mechanical solicitations. It would be interesting to study muscle behavior in different loading conditions, such as in-plane and out-of-plane shear loadings, which are naturally occurring in skeletal muscles and responsible for force transmission (Lieber and Binder-Markey, 2021), however we lack experimental data to evaluate our model. Similarly, although active and multiphysics skeletal muscle are present in the literature (Bleiler et al., 2019; Spyrou et al., 2019), this study focuses solely on passive hyperelastic mechanical behavior of the skeletal muscle, due to a lack of experimental data at both scales. Further experimental tests should focus on the viscous and active behavior at both scales.

Moreover, two different RVE geometries do not provide the same macroscopic behavior through the multiscale modeling approach. This effect can be due to the difference of Fiber Volume Fraction (FVF) between the two geometries or result from a more complex effect of microcomponent organization. Such effects of component spatial distributions can be investigated as a perspective of this study.

## 5. Conclusion

We propose a novel algorithm for 3D periodic RVE generation of skeletal muscle at the microscopic scale. This algorithm includes custom features that allow the representation of different muscle phenotypes, while providing features to address numerical issues. This algorithm will be applied to skeletal muscles both in health and in disease condition. Moreover, although this development is specific to skeletal muscle, it is compatible with other long fiber reinforced materials exhibiting a high fiber volume fraction (Omairey et al., 2019; Riano and Joliff, 2019).

### CRediT authorship contribution statement

**Aude Loumeaud:** Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Conceptualization.
**Philippe Pouletaut:** Writing – review & editing, Writing – original draft, Validation, Conceptualization.
**Sabine F. Ben-samoun:** Writing – review & editing, Writing – original draft, Supervision, Funding acquisition, Conceptualization.
**Daniel George:** Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Methodology, Conceptualization.
**Simon Chatelin:** Writing – review & editing, Writing – original draft, Validation, Supervision, Project administration, Methodology, Funding acquisition, Data curation, Conceptualization.
Comparison of simulated Soleus WT muscle traction using DFE² muscle model
and experimental results

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Engineering stress–strain curves of simulated muscle using geometries 1 (dashed line) and 2 (full line) and parameters sets 1 and 2, compared to the experimental uniaxial tension data from (Tatarenko et al., 2024).</figcaption></figure>

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.

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