# Finite Element Modeling of Compression Therapy in Lower Limb Lymphedema

Mateus Américo de Almeida, Aratz Garcia Llona, Maha Reda, Stéphane Avril

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Mateus Américo de Almeida, Aratz Garcia Llona, Maha Reda, Stéphane Avril. Finite Element Modeling of Compression Therapy in Lower Limb Lymphedema. International Journal of Numerical Methods in Biomedical Engineering, In press. hal-05381249

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# Finite Element Modeling of Compression Therapy in Lower Limb Lymphedema

Mateus Américo de Almeidaª, Aratz García-Llonaª, Maha Redaª, Stéphane Avrilª,\*

ªMines Saint-Etienne, INSERM, U 1059 Sainbiose, F - 42023, Saint-Etienne, France.

## Abstract

Patients with lower limb lymphedema experience lymphatic fluid accumulation and swelling, which can progress to fibrosis and fat deposition in the soft tissues, impacting patients physically, socially, and psychologically. Compression therapy is one of the main treatments for lymphedema, but its effects on lymphedematous soft tissues are not yet fully understood. In this study, we developed a finite element model of a lymphedematous leg including subcutaneous and muscle tissues, as well as skin and fascia cruris, to investigate the hydrostatic pressure distribution resulting from the interface pressure applied by a compression stocking. The results highlight the significant influence of the leg's external geometry on the interface pressure, and demonstrate the importance of modeling skin to accurately predict hydrostatic pressure distribution in the subcutaneous tissue, with a 3.5% reduction in leg volume observed after compression. The outcomes improve the understanding of the effects of compression therapy on lower limbs affected by lymphedema and support the development of adapted treatment strategies for patients.

**Keywords:** Lymphedema, lower limb, hyperelasticity, compression stocking, hydrostatic pressure, finite element.

## 1. Introduction

The lymphatic system plays a key role in maintaining interstitial fluid homeostasis, balancing fluid levels, and filtering lymph to detect and fight infections [1]. When there is impaired lymphatic drainage (either due to congenital underdevelopment of the lymphatic system, as seen in primary lymphedema, or due to damage or obstruction of the drainage pathways in secondary lymphedema [2-4]) fluid accumulates, leading to swelling and fibrosis [5]. In developed countries, most cases of lymphedema are secondary, particularly in women undergoing cancer treatments [6, 7]. Although the majority of cases occur in the upper limb [8], it is estimated that between 36% and 50% of women treated for gynecological cancer develop lower limb lymphedema [9, 10]. Additionally, venous hypertension has also been reported as a cause of the disease [11].

Lymphedema can entail excessive tissue swelling, infection, and mobility loss, requiring treatment tailored to the patient's condition and the physician's recommendations. To prevent the disease from worsening, two main treatments are available: surgical and non-surgical [12, 13]. While surgical treatments are typically invasive, non-surgical treatments effectively control fluid accumulation and prevent swelling, particularly those included in the Complex Decongestive Therapy [14]. The Complex Decongestive Therapy consists of two phases: the first phase focuses on reducing swelling to achieve maximum volume reduction of the lymphedematous limb, while the second phase aims to maintain this reduction over the long term [15]. Among the techniques of the second phase, the compression stockings combined with physical exercises constitute one of the most common and effective approaches [16], serving as a barrier against further limb volume increase and promoting lymphatic fluid drainage towards the upper body.

Even though compression stockings have proven effective in the treatment of lymphedema, their effects on soft tissues and lymphatic drainage are still not fully understood. Additionally, the morphological variation among lymphedematous patients makes analyzing compression therapy particularly complex. Among the solutions used to understand the behavior of biological tissues, finite element analysis is a valuable tool that has already been used to study the behavior of different parts of the human body [17, 18]. Regarding the human lower limb, finite element analysis has been used to identify the mechanical properties of lower limb soft tissues and the internal pressure distribution generated by compression stockings in cases of venous disease treatments [19-21].

<footnote>\*Corresponding author: Stéphane Avril (avril@emse.fr).</footnote>
Finite element analysis has also been used to study the effects of compression therapy on the upper limb lymphedema. In these studies, meshes of lymphedematous forearms were generated from limb surface data, and skin and subcutaneous tissues were modeled based on patients with mild lymphedema. This approach allowed the analysis of upper limb lymphedematous shape and the estimation of stress and strain fields resulting from compression therapy [22, 23]. However, there is a notable lack of studies applying finite element modeling to understand the effects of compression on the lymphedematous lower limb.

Among the studies that used finite element analyses to investigate compression therapy for lower limb venous diseases, the hydrostatic pressure has been employed as a parameter to analyze the effects of compression on the soft tissues in the region surrounding the veins [19–21], also called perivenous tissues. These studies considered only subcutaneous and muscle tissues in the leg and assumed that the perivenous hydrostatic pressure represents the pressure applied to the vessel walls, which is influenced by the interface pressure, the pressure that compression stockings apply to the surface of the lower limb.

Although the skin and fascia cruris are commonly not included in finite element models of the lower limb, their mechanical behaviors play important roles in its function. The skin tissue is essential for the body: it protects internal organs from the environment, regulates temperature, and is connected to the nervous system [24]. In the lower limb, volume changes also depend on the skin's capacity to stretch [25]. Furthermore, the fascia cruris surrounds the muscle tissue, influencing contraction and load transmission in the region [26].

In this study, we investigated the influence of the interface pressure on the pressure distribution within the soft tissues of a lymphedematous leg, particularly in the subcutaneous tissue where fluid accumulation is the most significant [27]. Additionally, we examined the inclusion of the skin and fascia cruris to understand their roles in pressure transmission within the leg. The study is organized as follows: the methodology used to create the finite element model of the lymphedematous leg is explained in Section 2; the results showing the behavior of the lymphedematous leg's soft tissues under compression are presented in Section 3; a discussion of these results is provided in Section 4; and the conclusions are given in Section 5.

# 2. Materials and Methods

## 2.1. Mesh Generation

Magnetic Resonance Imaging (MRI) scans were acquired on a MAGNETOM SOLA 1.5T scanner (Siemens Health-ineers) using an axial T1-weighted protocol in the supine position. The study was conducted with informed consent from a 53-year-old female patient diagnosed with stage II primary lymphedema in the right lower limb. The patient was treated and monitored at the University Hospital Center of Nice, France. As shown in Figure 1, the MRI scans enabled the identification of subcutaneous, muscle, and bone tissues in the patient's legs.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Image from an MRI scan of a patient with stage II primary lymphedema in the right lower limb (used to build the finite element model) and a healthy left lower limb. The locations of the different tissues are indicated in both legs, and the subcutaneous tissue in the affected (right) leg is visibly thicker than in the healthy (left) one. Lymphatic fluid has accumulated in regions such as the one indicated by the yellow arrow.</figcaption></figure>

Subcutaneous tissue, muscles, and bones were segmented using the 3D Slicer software [28] that creates a surface for each tissue. Aiming to remove surface irregularities resulting from segmentation, VMTK [29] was used to treat the surface meshes with surface remeshing and surface smoothing. Then a volumetric mesh was constructed for
subcutaneous tissue and muscles in GMSH [30] subtracting tibia and fibula. Since it was not possible to segment the skin and fascia cruris, these two tissues were modeled using two-dimensional elements generated from the external surfaces of the subcutaneous and muscle tissues, respectively. Figure 2 shows the process of mesh generation using MRI scans.

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Process of mesh generation: (a) MRI scans of the right lymphedematous leg were obtained; (b) these scans were segmented; (c) the resulting surface meshes from the segmentation were remeshed and smoothed; subsequently, (d) the different tissues were assembled; (e) two- and three-dimensional meshes were created; and (f) the position of each tissue within the mesh was indicated in a cross-section of the leg.</figcaption></figure>

Finally, the volumetric mesh for the subcutaneous tissue and muscles was defined in Abaqus, the software used for the finite element analyses, using 4-node linear tetrahedral hybrid elements with constant pressure (C3D4H) [31]. And, the surface mesh for the skin and fascia cruris was defined using 3-node triangular membrane elements (M3D3) [31–33]. The mesh consists of 34,020 nodes, 171,538 C3D4H elements, and 28,539 M3D3 elements. Neighboring tissue element sets shared nodes, eliminating the need for contact laws to model their interactions. In addition, three additional meshes were created, either including or excluding the skin and fascia cruris. Altogether, four different models were analyzed, and their results are presented in Subsection 3.3.

## 2.2. Boundary Conditions

The bones are significantly stiffer than the surrounding soft tissues and the compression applied by the stockings does not affect them, which made it convenient to consider bone tissue as rigid media. Only their surfaces were considered, and the nodes on these surfaces were clamped to prevent displacement [19, 34].

The lower limb lymphedema finite element model neglected the thigh, considering only the region between the ankle and knee. A symmetry boundary condition constraining the longitudinal displacement ($u_z$) was applied to the top and bottom surfaces of the model to represent the continuation of the leg.

The interface pressure IP, which is the pressure applied by the compression stocking, was calculated using Laplace's law [35]:

$$IP(\theta, z) = \frac{T(z)}{R_c(\theta, z)} \qquad (1)$$

where $T$ is a distributed tension and $R_c$ is the curvature radius. Only the horizontal curvature was considered in this study.

Data from a tensile test [36] performed on a female medium-size semi-transparent stocking class 2 was provided by the manufacturer Sigvaris France, allowing for the calculation of the distributed tension $T$. In the test, the stocking was attached such that, in its initial state, its length was half of its perimeter. The stocking was then gradually stretched, and the resulting tension was recorded as the perimeter increased. The distributed tension
was calculated by dividing the measured tension values by the length of the pin (30 mm) that held the stocking during the test. The graduated compression stocking used has different compression levels for the ankle and calf regions, with higher compression applied at the ankle than at the calf.

One hundred sections along the length of the lymphedematous leg were defined, and their perimeters were computed. These perimeters were then compared with those obtained from the tensile test, and the corresponding distributed tension was assigned to each section height. The centroids of the skin elements were also computed, and their Z coordinates were used to determine the perimeter and the corresponding distributed tension at each height, allowing the distributed tension to be associated with each skin element. Additionally, a transition zone exists between the highest part of the ankle and the lowest part of the calf, where the distributed tension was calculated by linear interpolation from the ankle to the calf. Figures 3a-d illustrate the distributed tension for different heights of the leg.

Moreover, Figures 3e-h illustrate the curvature C of the elements on the surface of the patient's leg, which was initially calculated at the nodes as follows:

$$C(\theta, z) = \frac{1}{R_c(\theta, z)} \qquad (2)$$

The radius of curvature $R_c$ of the nodes on the surface of the leg is defined as:

$$R_c(\theta, z) = \frac{(r^2(\theta, z) + r'^2(\theta, z))^3}{r^2(\theta, z) + 2r'^2(\theta, z) - r(\theta, z)r''(\theta, z)} \qquad (3)$$

where r is an analytical expression for the contour of the leg at a given height, obtained from the Fourier series:

$$r(\theta, z) = a(z) + \sum_{j=1}^{n} [b_j(z) \cos(j\theta) + c_j(z) \sin(j\theta)] \qquad (4)$$

In these equations, $r, \theta, z$ represent the cylindrical coordinates of the nodes of the leg's surface; $a, b$, and $c$ are the Fourier coefficients; and $n$ is the degree of the Fourier series, estimated according to [37]. Additionally, $r'$ and $r''$ denote the first and second derivatives of r with respect to $\theta$.

Using Laplace's law from Equation 1, the interface pressure was calculated, and its distribution is displayed in Figures 3i-l. The elements with $R_c$ equal to or less than zero were assumed to have zero curvature, and the interface pressure for those elements was set to zero.

Figure 4 illustrates all boundary conditions applied to the model. The domain $\Omega$ represents the lymphedematous leg, and $\partial\Omega$ denotes its boundary.

* $\Gamma_{tibia}$ and $\Gamma_{fibula}$ correspond to the bone surfaces, where all displacement components are fixed:

$$\mathbf{u} = \mathbf{0} \qquad \text{on } \Gamma_{\text{tibia}}, \qquad \Gamma_{\text{tibia}} \subset \partial\Omega, \qquad (5)$$

$$\mathbf{u} = \mathbf{0} \qquad \text{on } \Gamma_{\text{fibula}}, \qquad \Gamma_{\text{fibula}} \subset \partial\Omega. \qquad (6)$$

* $\Gamma_{knee}$ and $\Gamma_{ankle}$ correspond to the top and bottom surfaces of the leg, near the knee and ankle, respectively, where symmetry conditions are imposed by constraining the longitudinal displacement:

$$u_z = 0 \qquad \text{on } \Gamma_{\text{knee}}, \qquad \Gamma_{\text{knee}} \subset \partial\Omega, \qquad (7)$$

$$u_z = 0 \qquad \text{on } \Gamma_{\text{ankle}}, \qquad \Gamma_{\text{ankle}} \subset \partial\Omega. \qquad (8)$$

* $\Gamma_{skin}$ corresponds to the skin surface, where the interface pressure (IP) is applied:

$$\sigma \cdot \mathbf{n} = -\text{IP} \cdot \mathbf{n} \qquad \text{on } \Gamma_{\text{skin}}, \qquad \Gamma_{\text{skin}} \subset \partial\Omega. \qquad (9)$$

## 2.3. Material Properties

The material properties of the lower limb soft tissues can take a wide range of values derived from different studies [19, 20, 38–48]. In our study, the soft tissues were considered homogeneous, isotropic and hyperelastic. Their mechanical behavior was modeled using the neo-Hookean strain energy density $U$ defined as follows:
<figure><img src="image_4.png" /><figcaption>**Figure 3:** Distributed tension in the lymphedematous leg from the (a) anterior, (b) lateral, (c) posterior, and (d) medial views; curvature distribution from the (e) anterior, (f) lateral, (g) posterior, and (h) medial views; and interface pressure distribution from the (i) anterior, (j) lateral, (k) posterior, and (l) medial views.</figcaption></figure>

$$U = C_{10}(\bar{I}_1 - 3) + \frac{1}{D_1}(J^{el} - 1)^2 \quad (10)$$

where $\bar{I}_1$ is the first isochoric strain invariant, $J^{el}$ is the elastic volume ratio, and $C_{10}$ and $D_1$ are stiffness parameters related to the shear and bulk moduli, respectively [31]. The parameter $C_{10}$ is also known as the elasticity coefficient, while $D_1$ is referred to as the compressibility coefficient.

The stiffness parameters used in the finite element model of the lymphedematous leg are presented in Table 1 and were used in previous studies conducted in our laboratory [45, 49]. The skin thickness was set to 1.6 mm, and the fascia cruris thickness was 0.924 mm [50, 51].

<table><caption>**Table 1:** Elasticity and compressibility coefficients for the soft tissues of the leg</caption><thead><tr><th>Tissue</th><th>C<sub>10</sub> (kPa)</th><th>D<sub>1</sub> (MPa<sup>-1</sup>)</th><th>References</th></tr></thead><tbody><tr><td>Skin</td><td>100</td><td>0.14</td><td>[40, 52]</td></tr><tr><td>Subcutaneous tissue</td><td>5</td><td>22.5</td><td>[37, 53]</td></tr><tr><td>Fascia cruris</td><td>100</td><td>22.5</td><td>[53, 54]</td></tr><tr><td>Muscle</td><td>3</td><td>28</td><td>[37, 53]</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Figure 4: Boundary conditions applied to the leg affected by lymphedema.</figcaption></figure>

## 2.4. Hydrostatic Pressure Calculation

The effect of compression exerted by the stocking on the soft tissues of the lower limb is analyzed based on the hydrostatic pressure distribution within them. The hydrostatic pressure was calculated using the equivalent pressure stress $p$ expression [31]:

$$p = -\frac{1}{3}\text{trace}(\boldsymbol{\sigma}) \qquad (11)$$

where $\boldsymbol{\sigma}$ is the Cauchy stress tensor. In this context, positive values indicate compression, while negative values indicate tension in the soft tissues of the leg.

## 2.5. Mesh and Material Parameters Sensitivity Analyses

### 2.5.1. Mesh Sensitivity

A mesh sensitivity analysis (Table 2) for the element size of the subcutaneous tissue was performed using the parameters from Table 1, the boundary conditions applied to the bones, top and bottom surfaces of the leg from Subsection 2.2, and a uniform pressure of 4 kPa applied as the interface pressure, which is within the range of pressures applied by the stocking.

<table><caption>Table 2: Mesh sensitivity analysis: comparison of total strain energy for different mesh sizes.</caption><thead><tr><td>Mesh Size (mm)</td><td>Number of Elements</td><td>Total Strain Energy (N·mm)</td></tr></thead><tbody><tr><td>2.0</td><td>420,455</td><td>164.78</td></tr><tr><td>3.0</td><td>200,077</td><td>164.31</td></tr><tr><td>4.0</td><td>126,608</td><td>164.14</td></tr></tbody></table>

The element size sensitivity analysis revealed that increasing the mesh density did not significantly affect the estimated total strain energy in the model. Therefore, an element size of 3.0 mm was chosen to capture a more detailed evolution of the hydrostatic pressure distribution in the subcutaneous tissue. The element size for the muscles, tibia, and fibula were set to 5 mm, which was sufficient to maintain a good element quality and avoid an overly distorted mesh, especially between the bones. Additionally, the minimum and maximum element sizes allowed during mesh generation were 2 mm and 5 mm, respectively. Furthermore, linear and quadratic elements
were compared for the subcutaneous and muscle tissues. Table 3 presents a mesh sensitivity analysis based on element order.

<table><caption>**Table 3:** Mesh sensitivity analysis: comparison of total strain energy for different element orders.</caption><thead><tr><td>Element Order</td><td>Total Strain Energy (N·mm)</td></tr></thead><tbody><tr><td>Linear</td><td>189.24</td></tr><tr><td>Quadratic</td><td>189.74</td></tr></tbody></table>

The sensitivity analysis indicates that the total strain energy in a model using 4-node linear tetrahedral hybrid elements with constant pressure (C3D4H) and a model using 10-node quadratic tetrahedral hybrid elements with constant pressure (C3D10H) show no significant difference. Additionally, the global difference in hydrostatic pressure distribution across the entire leg between the two models is 3.07%. Therefore, to reduce computational cost, C3D4H elements were selected over C3D10H elements.

### 2.5.2. Material Properties Sensitivity

After choosing the size and order of the elements, a sensitivity study was conducted by varying the elasticity coefficients $C_{10}$ of the subcutaneous and skin tissues, as well as the compressibility coefficient $D_1$ of the subcutaneous tissue, while applying a uniform interface pressure of 4 kPa to investigate their influence on the computation of hydrostatic pressure in the subcutaneous tissue. All other parameters were taken from Table 1, and the boundary conditions applied to the bones, top and bottom surfaces of the leg from Subsection 2.2. The results of this sensitivity analysis are presented in Subsection 3.2.

## 3. Results

### 3.1. Hydrostatic Pressure Distribution in the Lymphedematous Leg

Figure 5 shows the distribution of hydrostatic pressure in the subcutaneous tissue and muscles across sections in different regions. In the region with the largest perimeter in the calf in Figure 5a, the curvature radius of the elements on the skin generated lower interface pressure compared to others with a smaller curvature radius, such as the ankle. Consequently, the hydrostatic pressure distribution is moderate compared to other regions. In Figure 5b, taken from the middle of the transition region between the ankle and the calf, the posterior surface of the leg is flat, and the interface pressure applied there is either small or zero. This resulted in low hydrostatic pressure values, implied low compression and tension (negative hydrostatic pressure) in the soft tissue. The ankle region in Figure 5c was affected by a higher interface pressure due to the small curvature radius and presented a higher hydrostatic pressure distribution in the subcutaneous and muscle tissues. However, the tibia and fibula acted as a barrier to pressure propagation in the muscle tissue between them. The high hydrostatic pressure in the surrounding areas pushed the tissue between the bones, resulting in negative hydrostatic pressure values in this region, indicating that the tissue was under tension.

The low interface pressure applied to the posterior surface of the leg (Figure 3k) resulted in minimal compression or, in some regions, tension development within the inner soft tissue, as indicated by the blue area under the calf in Figure 5d. Conversely, the anterior view of the ankle experienced higher interface pressure (Figure 3i), leading to an increased hydrostatic pressure distribution in this region. This is evident in Figure 5d, at the bottom of the leg, in front of the empty area representing the tibia.

In summary, the inner soft tissues deformed and generated a hydrostatic pressure distribution within the leg. Regions with smaller curvature radii on the surface exhibit higher hydrostatic pressure distributions inside, which is consistent with Laplace's law. On the other hand, areas with larger curvature radii, flat, and concave surfaces exhibit low, negligible, or even negative hydrostatic pressure distribution in the inner soft tissues.

Figure 6 illustrates the cross-sectional area reduction caused by the graduated compression stocking in the deformed leg compared to its initial state, showing the greatest reduction in the ankle, followed by the calf region, and the smallest reduction in the transition area between the ankle and the calf. The total volume reduction of the leg, considering only the subcutaneous and muscle tissues, excluding the bones, after compression was 3.5%.
<figure><img src="image_6.png" /><figcaption>**Figure 5:** Hydrostatic pressure distribution in the subcutaneous and muscle tissues: (a) in the calf, (b) in the middle of the transition region between the ankle and calf, (c) in the ankle, and (d) along the lymphedematous leg.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 6:** Cross-sectional area reduction along the leg height in the deformed leg compared to its initial state. The normalized height is zero at the level of the ankle and one at the level of the knee.</figcaption></figure>

## 3.2. Sensitivity Analysis of Skin and Subcutaneous Tissue Elasticity Coefficients

Figure 7 the results from the sensitivity analysis explained in Subsection 2.5.2.

The hydrostatic pressure within the subcutaneous tissue is inversely proportional to its elasticity coefficient, resulting in lower hydrostatic pressure as the coefficient increases. In Figure 7a, the hydrostatic pressure is lower when $C_{10} = 20$ kPa compared to $C_{10} = 4$ kPa. This trend is also observed in the median hydrostatic pressure, which decreases as the elasticity coefficient of the subcutaneous tissue increases. Additionally, the cross-sectional area reduction along the leg is greater for the lowest value of subcutaneous tissue $C_{10}$.

The elasticity coefficient of the skin similarly affects the hydrostatic pressure in the subcutaneous tissue: higher coefficients correspond to lower hydrostatic pressure values. Figure 7b shows that increasing the skin's $C_{10}$ from 1 kPa to 100 kPa reduces both the hydrostatic pressure in the subcutaneous tissue and the cross-sectional area reduction.

Finally, the results of the sensitivity analysis on the compressibility coefficient $D_1$ of the subcutaneous tissue in
<figure><img src="image_8.png" /><figcaption>**Figure 7:** Sensitivity study of the hydrostatic pressure distribution in the subcutaneous tissue when (a) varying the elasticity coefficient of the subcutaneous tissue, (b) varying the elasticity coefficient of the skin, and (c) varying the compressibility coefficient of the subcutaneous tissue. Figures on the left show the hydrostatic pressure distribution, while the figures on the right show the corresponding cross-sectional area reduction compared to the initial state of the leg.</figcaption></figure>

Figure 7c showed that increasing $D_1$ decreases the hydrostatic pressure in the subcutaneous tissue, similar to the case of $C_{10}$. For $D_1 = 0 \text{ MPa}^{-1}$, the total volume reduction of the leg was 1.87% compared to the initial state. For $D_1 = 11.25 \text{ MPa}^{-1}$ and $D_1 = 22.5 \text{ MPa}^{-1}$, the reductions were 3.10% and 4.09%, respectively. However, unlike $C_{10}$, $D_1$ is proportional to the cross-sectional area reduction: the higher the $D_1$, the greater the reduction in cross-sectional area along the leg.
### 3.3. Hydrostatic Pressure Distribution in Different Models

Four different finite element models were compared to identify the influence of skin and fascia cruris tissues on the hydrostatic pressure generated in the subcutaneous tissue. Model I has subcutaneous and muscle tissues; model II has subcutaneous, fascia cruris and muscle tissues; model III has skin, subcutaneous and muscle tissues; and model IV has skin, subcutaneous, fascia cruris and muscle tissues. The mesh details are presented in Table 4, where model I was built exclusively with C3D4H elements, while models II, III and IV also included M3D3 elements. The material parameters were taken from Table 1 and the interface pressure from Figures 3i-l.

<table><caption>**Table 4:** Mesh details of model I (subcutaneous and muscle tissues); model II (subcutaneous, fascia cruris, and muscle tissues); model III (skin, subcutaneous, and muscle tissues); and model IV (skin, subcutaneous, fascia cruris, and muscle tissues).</caption><thead><tr><th></th><th>Model I</th><th>Model II</th><th>Model III</th><th>Model IV</th></tr></thead><tbody><tr><td>Number of nodes</td><td></td><td>34,020</td><td></td><td></td></tr><tr><td>Number of C3D4H elements</td><td></td><td>171,538</td><td></td><td></td></tr><tr><td>Number of M3D3 elements</td><td>—</td><td>7,676</td><td>20,863</td><td>28,539</td></tr></tbody></table>

The hydrostatic pressure distribution in the subcutaneous and muscle tissues of the models, with and without skin and fascia cruris, was computed and is shown in Figure 8. The hydrostatic pressure distribution in the skin and fascia cruris was not considered, even in the models that include these tissues.

Sections cutting through the region with the largest perimeter of the calf in the models are shown in Figures 8a-d. Model I that includes only the subcutaneous tissue and muscles (Figure 8a) exhibits the highest hydrostatic pressure distribution among all models. When the fascia cruris is added in model II (Figure 8b), there is a reduction in hydrostatic pressure, particularly in the muscle tissue. This reduction is most noticeable in the vertical region between the tibia and fibula. In the model III shown in Figure 8c, where only the skin tissue is added to the initial model, the hydrostatic pressure distribution in the medial region (lower left region) decreases, along with a further reduction in the vertical region between the bones. Finally, when both skin and fascia cruris are included simultaneously in model IV (Figure 8d), the reduction in hydrostatic pressure in the vertical region between the bones and in the medial region (lower left region) is more pronounced compared to the other models.

The sections of the transition region between the ankle and the calf in Figures 8e-h shows a similar behavior to that of the sections of the calf. In this sections, the medial and lateral regions (left and right regions) exhibit the greatest hydrostatic pressure distribution in the subcutaneous and muscle tissues for the models without skin (I and II) in Figures 8e-f. However, in the models with skin (III and IV) in Figures 8g-h, there was an important reduction in the hydrostatic pressure distribution in the medial and lateral regions.

Finally, the hydrostatic pressure distribution in the ankle is shown in Figures 8i-l for the subcutaneous and muscle tissues. As in other parts of the leg, the addition of skin and fascia cruris caused a reduction in the hydrostatic pressure distribution, with the model I containing subcutaneous and muscle tissues (Figure 8i) showing the highest values of pressure, and the model IV with skin, subcutaneous, fascia cruris, and muscle tissues (Figure 8l) showing the lowest hydrostatic pressure distribution.

The comparison in Figure 9a shows that in the models considering the skin, the spread and median of hydrostatic pressure in the subcutaneous tissue are smaller than those without it. On the other hand, the spread and median remain relatively similar between the two models with skin, as well as between the two models without skin. Figure 9b presents the cross-sectional area reduction in each of the four models compared to the initial state of the leg, with model I showing the highest reduction in the ankle and calf regions, and model IV the lowest reduction in these same regions.

Moreover, the addition of skin and fascia cruris influences the overall shape of the hydrostatic pressure distribution, with the skin having the greatest impact. This is highlighted in Table 5, where the global relative difference $D_{global}$ in hydrostatic pressure distribution in the subcutaneous tissue was calculated as follows:

$$D_{global} = \frac{\|\mathbf{d}_{ref} - \mathbf{d}_{mod}\|_{2}}{\|\mathbf{d}_{ref}\|_{2}} \qquad (12)$$

where $\mathbf{d}_{ref}$ is the original vector, and $\mathbf{d}_{mod}$ is the modified vector.

The difference in volume reduction was calculated for the four finite element models and is shown in Table 6, comparing them to the initial state of the leg. The models with skin exhibited a lower volume reduction than those without it.
<figure><img src="image_9.png" /><figcaption>**Figure 8:** Hydrostatic pressure distribution in the subcutaneous and muscle tissues of the lymphedematous leg: (a–d) in the calf, (e–h) in the middle of the transition region between the ankle and calf, and (i–l) in the ankle; for model I (a, e, i) with subcutaneous and muscle tissues; model II (b, f, j) with subcutaneous, fascia cruris, and muscle tissues; model III (c, g, k) with skin, subcutaneous, and muscle tissues; and model IV (d, h, l) with skin, subcutaneous, fascia cruris, and muscle tissues.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 9:** (a) Comparison of the hydrostatic pressure distribution in the subcutaneous tissue among different finite element models. (b) Cross-sectional area reduction along the leg length for the four models compared to the leg's initial state. The models comprise: (I) subcutaneous and muscle tissues; (II) subcutaneous, fascia cruris, and muscle tissues; (III) skin, subcutaneous, and muscle tissues; and (IV) skin, subcutaneous, fascia cruris, and muscle tissues.</figcaption></figure>

### 3.4. Adaptation of Interface Pressure to Geometrical Changes

In the compression therapy with stockings, the pressure applied to the surface of the leg changes as the soft tissues deform until equilibrium is reached. To replicate this behavior in the proposed finite element model, multiple static simulations were performed, updating the node coordinates and interface pressure according to the leg's deformation after each simulation. The criterion for stopping the updates was a global relative difference of 3% between the interface pressure distribution of the last two simulations, which was achieved after eighteen iterations. Figure 10 shows the spread of interface pressure between the first and last simulations, indicating that the median remained around 4 kPa in both cases. Additionally, the range of hydrostatic pressure obtained in the final simulation was smaller, showing reduced variation in the interface pressure values.

Figures 11a–d complete the information in Figure 10, showing that the interface pressure used in the final simulation is more uniform than in the first simulation (Figures 3i–l), especially in the posterior view of the leg, as seen in Figure 11c. The interface pressure calculated and used in the last simulation does not have zero values; consequently all elements of the skin experience compression. Moreover, the local absolute difference $\Delta_{abs,i}$ between the first and last simulations, shown in Figures 11e–h, was calculated for each element on the skin surface as follows:
<table><caption>**Table 5:** Comparison of the global relative difference of hydrostatic pressure in the subcutaneous tissue between the models: (I) subcutaneous and muscle tissues; (II) subcutaneous, fascia cruris, and muscle tissues; (III) skin, subcutaneous, and muscle tissues; and (IV) skin, subcutaneous, fascia cruris, and muscle tissues.</caption><thead><tr><th>Model</th><th>I</th><th>II</th><th>III</th><th>IV</th></tr></thead><tbody><tr><th>I</th><td>—</td><td>3.16%</td><td>16.74%</td><td>16.51%</td></tr><tr><th>II</th><td>—</td><td>—</td><td>16.58%</td><td>15.86%</td></tr><tr><th>III</th><td>—</td><td>—</td><td>—</td><td>3.91%</td></tr><tr><th>IV</th><td>—</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

<table><caption>**Table 6:** Difference in volume reduction between the lymphedematous leg before and after compression for the models: (I) subcutaneous and muscle tissues; (II) subcutaneous, fascia cruris, and muscle tissues; (III) skin, subcutaneous, and muscle tissues; and (IV) skin, subcutaneous, fascia cruris, and muscle tissues.</caption><thead><tr><th>Model</th><th>Volume Reduction</th></tr></thead><tbody><tr><td>I</td><td>4.20%</td></tr><tr><td>II</td><td>4.07%</td></tr><tr><td>III</td><td>3.60%</td></tr><tr><td>IV</td><td>3.50%</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Figure 10:** Comparison of the interface pressure median and spread of the first and the last simulations using the model IV containing skin and fascia cruris.</figcaption></figure>

$$ \Delta_{\text{abs},i} = |d_{\text{ref},i} - d_{\text{mod},i}| \qquad (13) $$

where $d_{\text{ref},i}$ is the original value for the i-th element, and $d_{\text{mod},i}$ is the modified value for the i-th element. The region with the largest changes in the applied interface pressure is the posterior region of the leg (Figure 11g), which was less compressed in the first simulation.

The local absolute differences helped to identify the regions with more changes in the interface pressure applied to the surface of the leg between the first and last simulations. To have a more general understanding of this difference, the global relative difference was estimated using Equation 12 and found to be 13.89%, indicating significant changes in the distributed pressure values.

The effects of updating the interface pressure on the shape of the lower limb and its hydrostatic pressure distribution are presented in Figure 12. Due to the update of node coordinates, which accounts for the deformation of the leg during each simulation, and the update of the interface pressure, after eighteen simulations, the geometry of the leg exhibited a wrinkled surface, as shown in Figure 12a. Comparing Figures 12b-e with Figures 5, it is evident that the inner soft tissues are more compressed after the updates, and the hydrostatic pressure distribution becomes more homogeneous. On the other hand, the superficial layer of the subcutaneous tissue shows a higher variation
<figure><img src="image_12.png" /><figcaption>**Figure 11:** Interface pressure distribution in the (a) anterior, (b) lateral, (c) posterior, and (d) medial views of the leg; and local absolute difference in interface pressure between the first and last simulations in the (e) anterior, (f) lateral, (g) posterior, and (h) medial views of the leg.</figcaption></figure>

in hydrostatic pressure, as can be seen in the sections of Figures 12b-d.

<figure><img src="image_13.png" /><figcaption>**Figure 12:** (a) Deformed geometry of the leg after the last simulation. Hydrostatic pressure distribution in the subcutaneous and muscle tissues from the last simulation: (b) in the calf, (c) in the middle of the transition region between the ankle and calf, (d) in the ankle, and (e) along the lymphedematous leg. Local absolute difference in hydrostatic pressure between the first and last simulations: (f) in the calf, (g) in the middle of the transition region between the ankle and calf, (h) in the ankle, and (i) along the lymphedematous leg.</figcaption></figure>

The local absolute difference of hydrostatic pressure between the first and last simulations was also calculated and is shown in Figure 12f-i. The region with the greatest difference between the results of the two simulations is the superficial layer of the subcutaneous tissue. The global relative difference is 15.77%.

Figure 13 presents the hydrostatic pressure and cross-sectional area reduction after updating the interface pressure. The spread of the hydrostatic pressure distribution in the last simulation (Figure 13a) appears smoother than in the first simulation, while the median remains around 3 kPa in both cases. The last simulation also shows a greater cross-sectional area reduction than the first simulation (Figure 13b). Additionally, the total volume reduction of
the leg in the last simulation is 3.91% greater than in the first simulation and 7.27% greater than in the initial state of the leg.

<figure><img src="image_14.png" /><figcaption>**Figure 13:** (a) Comparison of the hydrostatic pressure distribution in the subcutaneous tissue resulting from the first and last simulations of the updating process; (b) Cross-sectional area reduction along the leg length in the first and last simulations compared to the leg's initial state.</figcaption></figure>

# 4. Discussion

This study highlights the biomechanical effects of compression therapy on the lower limb, particularly in the subcutaneous tissue, where the undrained fluid mainly accumulates [27]. The proposed lower limb finite element model includes the effect of the skin and fascia crusis tissues on the hydrostatic pressure distribution, an aspect that has not been considered in previous studies that investigated the compression therapy in healthy legs for venous diseases applications [19–21]. Another important aspect taken into account is the volume of the leg affected by lymphedema. Previous models have used a healthy leg to obtain the geometry, whereas in our study, we considered the lymphedematous leg which had a significant larger volume than the healthy one (44.52%). This volume difference directly impacts the superficial shape of the leg and the thickness of the subcutaneous tissue.

The calculation of the interface pressure provided by the compression stocking is done through the use of Laplace's law, which depends on the curvature radius of the element on the surface of the skin. A compression in the range of 2.7 to 5.3 kPa has been shown to reduce chronic edema [55]. In our model, the interface pressure applied by the compression stocking reached values up to 6.8 kPa, with a median around 4 kPa, indicating that it is sufficient to reduce leg volume. Additionally, the swelling generated by the fluid accumulation in the lower limb of patients with lymphedema modifies the morphology and the external surface of the leg. Tissue swelling creates flat zones with small curvature and also zones with larger curvature, impacting in the interface pressure applied in those regions. This is why a lymphedematous leg can increase the complexity of lower limb finite element models. As an example, the flat regions such as the posterior of the leg received less pressure from the compression stocking than regions with larger curvatures such as the anterior of the ankle as shown in Figures 3k and 3i, respectively. Consequently, the compression caused by the interface pressure was transmitted to the soft tissues inside the leg, where regions with higher interface pressure on the surface presented higher hydrostatic pressure in the inner tissues under it such as in the anterior and medial (lower left zone) regions in Figure 5c, and regions with lower interface pressure on the surface presented lower hydrostatic pressure distribution in the inner tissues under it such as the posterior region in Figure 5b. These variation created positive and negative hydrostatic pressure in the soft tissues. As mentioned in Subsection 2.4, positives values of hydrostatic pressure indicates tissue under compression, while negative values indicates tissue under tension. When the soft tissues of the lymphedematous leg are under compression, they deform such that tissues from regions of higher compression moved toward adjacent regions of lower compression, pushing the less-compressed tissues outward and creating tension in certain areas of the leg, such as the region under the calf in Figure 5d.

The soft tissues are typically modeled as incompressible materials, with their bulk modulus much larger than their shear modulus [56–59]. Moreover, it is known that volume reduction in a leg under compression depends mostly on fluid transport [60–63], when lymphatic fluid is drained from the soft tissues. Nevertheless, our finite element model
of a lymphedematous leg adopted a simplified approach, considering only the solid mechanics part of the problem and treating the soft tissues of the lower limb, particularly the subcutaneous tissue and muscles, as compressible. Having this in mind, the volume reduction of the leg can be regulated by modifying the compressibility coefficient $D_1$. Using the $D_1$ values defined for the subcutaneous tissue and muscle in Table 1, the compression stocking generated a 3.5% volume reduction in the lymphedematous leg. The literature says that when the compression therapy is combined with physical exercise, such as walking, edema is more easily drained, reducing the amount of fluid accumulated in the leg. This phenomenon has been reported in a study where lymphedematous patients wore bandages, resulting in volume reduction ranging from 1.16% to 4.05% [64]. The reduction obtained in our model falls within the range observed in this clinical study. Volume reduction has also been investigated in healthy subjects wearing compression stockings, with reductions between 4.4% and 7.7%, depending on the compression level of the stockings [62, 65]. Since fluid accumulates primarily in the subcutaneous tissue, an inverse identification approach could be used to adjust the subcutaneous tissue's $D_1$ value to match volume reductions observed during compression therapy. However, there is currently a lack of experimental data in the literature specifically associating compression stockings with volume reduction in lymphedematous legs.

When lower limb lymphedema is not treated, the disease can progress, worsening the patient's condition due to the development of tissue fibrosis [66]. Fibrosis increases soft tissue stiffness, consequently, the skin and subcutaneous tissue become stiffer as the pathology evolves [67, 68]. This stiffness affects the effectiveness of compression therapy. According to the results presented in Figures 7a and 7b, increasing the stiffness of either the skin or the subcutaneous tissue, by increasing their elasticity coefficients $C_{10}$, reduces the transmission of pressure from the compression stocking to the subcutaneous tissues. This condition makes compression therapy less effective and may require the use of a higher compression level stocking, which can negatively impact the patient's comfort and well-being.

Skin tissue is an essential component of finite element models of lower limb lymphedema, as it is also subject to swelling and fibrosis and the fascia cruris plays a significant role in the lower limb by providing structural support to internal tissues. However, previous models of the lower limb often do not account for these two soft tissues. Since both were included in our model, it was important to assess the influence of each on the hydrostatic pressure distribution within the leg's soft tissues. The results from Subsection 3.3 show that the skin and fascia cruris form a barrier, reducing the transmission of compression to the subcutaneous tissue and muscles. The fascia cruris primarily protects the muscle tissue, while the skin affects subcutaneous and muscle tissues. As shown in Table 5, models including the skin (Models III and IV) have a greater influence on the hydrostatic pressure distribution in the subcutaneous tissue compared to those without it (Models I and II). Consequently, including skin tissue is crucial for understanding the effects of compression therapy. Models without it tend to overestimate hydrostatic pressure distributions, as can be seen by analyzing Figures 8 and 9a.

The soft tissues of the lower limb deform to adapt to the compression of the stocking until they reach a final deformed geometry. In our study, the proposed solution to replicate this behavior using static simulations required eighteen iterations to achieve the final geometry. The updates to the node coordinates and the interface pressure applied to the skin influenced the hydrostatic pressure distribution in the soft tissues, particularly in the superficial layer of the subcutaneous tissue, where the greatest difference was observed between the first and the final simulations. This happened because the updates amplified the wrinkling of the skin tissue, visible in Figure 12a, that is associated to the use of membrane elements. When a membrane structure is subjected to compression, it tends to buckle [31]. Previous studies have shown that skin buckling can result in the formation of wrinkles [69, 70], which also depends on the thickness of the tissue [71]. This behavior of the skin can be related to the stage of lymphedema and the presence of fibrosis. In our model, the neighboring tissues shared nodes, and no contact law was implemented to represent the interaction between the skin and the subcutaneous tissue. As a consequence, the skin wrinkling led to a non-uniform hydrostatic pressure distribution in the superficial layer of the subcutaneous tissue.

Regarding the assumptions made in the study, the soft tissues were modeled as homogeneous and isotropic to reduce computational cost, and their mechanical behavior was simulated using the neo-Hookean formulation [19, 45, 72]. However, it is known that these tissues are heterogeneous and anisotropic [73, 74]. This, combined with their non-linear behavior and the complexity of the mesh of the lymphedematous leg, could lead to unfeasible finite element analyses. In addition, the coefficients $C_{10}$ and $D_1$, representing elasticity and compressibility, respectively, in the neo-Hookean strain energy density were obtained from healthy subjects. In lymphedematous tissues such as skin and subcutaneous tissue, mechanical properties may differ due to fluid accumulation and fibrosis. Although there is a lack of data in the literature concerning the material parameters of lymphedematous soft tissues in the lower limb, this led to the assumption that their $C_{10}$ and $D_1$ values are the same as those of healthy tissues.

The boundary conditions applied in the model also depend on assumptions. Since the movement of the fibula relative to the tibia is very small [75, 76], the bones were considered fixed with respect to each other, and the displacements of the nodes on their surfaces were constrained in all directions. This constraint can influence the hydrostatic pressure distribution between the bones, particularly at the ankle level, where the distance between
the tibia and fibula is smaller. Furthermore, symmetry was imposed on the top and bottom surface of the leg, constraining the longitudinal displacement of their nodes. This condition has an effect on the hydrostatic pressure distribution in the elements on the vicinity of these nodes, because when the soft tissues are compressed they deform and also relocate in the longitudinal direction depending on the pressure applied in different regions of the leg. To avoid convergence problems, this symmetry boundary condition was necessary on those surfaces, allowing to model just the region of interest of the lower limb where the interface pressure is applied. Moreover, the interface pressure used to compress the leg, calculated with Laplace's law, considered only the compression along the circumference of the leg in the XY plane. This is because the compression stocking does not exert significant pressure along the vertical (Z-axis) direction of the leg.

The parameter used to analyze the effect of compression on the lymphedematous leg was the equivalent pressure stress, which represents the hydrostatic pressure in the soft tissues of the leg and has also been used in previous studies [19–21]. The objective of the finite element analysis performed and of future investigations is to demonstrate that the proposed solid finite element model can be used to predict the pressure transmitted to the lymphatic vessel walls as a function of the interface pressure applied by the compression stocking on the leg surface, by estimating the hydrostatic pressure. When pressure is applied to the surface of a limb, it can be transmitted to the walls of blood vessels either through solid tissue or interstitial fluid [77]. A similar mechanism is likely present in the case of lower limb compression therapy for lymphedema, where external pressure is transmitted to the walls of lymphatic vessels, aiding the drainage of lymphatic fluid from the interstitial spaces within the soft tissues. Therefore, knowing the hydrostatic pressure in the vicinity of the lymphatic vessels can help for predicting the pressure exerted on their walls.

The model we proposed can still be improved. One possible enhancement is the inclusion of muscular activity, as it is known that patients who remain standing and those who walk while wearing compression stockings exhibit different levels of leg volume reduction [62, 65]. This difference occurs because the calf muscles act as a pump, pushing fluid upward through the lower limb [78]. Additionally, incorporating a lymphatic vessel into the finite element model of the lymphedematous leg could provide a better understanding of how hydrostatic pressure transmitted through the soft tissues influences the lymphatic vessel function and fluid drainage. Another important aspect of lymphedema is the accumulation of lymphatic fluid in the skin tissue. Therefore, further investigations modeling the skin with three-dimensional elements could offer valuable insights into the finite element analysis. Another limitation of the study is that only the solid mechanics part of the problem was considered, when it is known that volume reduction is mainly caused by the fluid drainage from the soft tissues. A more realistic approach would be to model the soft tissues of the lower limb as poroelastic media [63] to take into account the fluid accumulated in the leg and its drainage towards the upper part of the lower limb. Additionally, the findings presented in this study will be compared to results from a poroelastic model of the leg, such as the one developed by our group [79], which analyzed compression therapy from a fluid mechanics perspective.

## 5. Conclusion

In this study, we proposed a finite element model to simulate the compression therapy with stocking in a stage II lower limb lymphedema. The model investigates the transmission of pressure from the compression stocking to the soft tissues of the leg. The results demonstrated the influence of the leg's external geometry on the pressure applied by the stocking, the importance of modeling skin tissue to accurately estimate pressure transmission to the subcutaneous tissue, and the need to update node coordinates and interface pressure to account for tissue deformation during compression. Since the same compression stocking may produce different effects in different individuals, future work will involve extending the methodology to other lymphedematous legs with varying geometries to better understand the role of subcutaneous tissue thickness. Additionally, a lymphatic vessel will be incorporated into the model, and the influence of muscle activity on the hydrostatic pressure distribution in the leg's soft tissues will be explored.

## Acknowledgments

The project was supported by the Symphonies Project funded by BPI France.
**Consent**

The MRI scans used in this study were obtained with the informed consent of the patient, provided by the University Hospital Center of Nice, France.

**Ethics Statement**

The authors have nothing to report.

**Conflicts of Interest**

The authors declare that there are no conflicts of interest related to this article.

**Data Availability Statement**

The data and codes that support the findings of this study are available on request from the corresponding author.

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