# Corroboration in vivo mechanical strain distribution in soft tissues for pressure ulcer prevention: A comparative analysis between a simplified finite element analysis and experimental strain fields

Alexandre Segain, Helene Pillet, Stefano Zappalá, Giuseppe Sciumè,
Pierre-Yves Rohan

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Alexandre Segain, Helene Pillet, Stefano Zappalá, Giuseppe Sciumè, Pierre-Yves Rohan. Corroboration in vivo mechanical strain distribution in soft tissues for pressure ulcer prevention: A comparative analysis between a simplified finite element analysis and experimental strain fields. Journal of the mechanical behavior of biomedical materials, 2025, 168, pp.107017. 10.1016/j.jmbbm.2025.107017. hal-05070892

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# Corroboration *in vivo* mechanical strain distribution in soft tissues for pressure ulcer prevention: A comparative analysis between a simplified finite element analysis and experimental strain fields

Alexandre Segain<sup>a,∗</sup> <sup>†</sup>, Helene Pillet<sup>a</sup> <sup>†</sup>, Stefano Zappalá<sup>b,c</sup> <sup>†</sup>, Giuseppe Sciume<sup>d</sup>, Pierre-Yves Rohan<sup>a</sup> <sup>†</sup>

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

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## ARTICLE INFO

### Keywords:

Pressure ulcer

Constitutive modelling

Clinical relevance

Mechanical indices

Model corroboration

Digital volume correlation

## ABSTRACT

Basic research into the aetiology of pressure ulcers suggests that the concentration of mechanical strain in biological soft tissues is critical to their development. Direct measurement of *in vivo* strain is not compatible with clinical routine. To overcome this problem, several finite element models (FEM) have been proposed by the biomechanical community to estimate strain from imaging data. However, no direct experimental validation of the underlying relationships between mechanical loading and soft tissue strain distribution predicted by the model has been performed, and such validation evidence must be obtained prior to any clinical evaluation. Building on the experimental results obtained in N = 10 healthy volunteers (Zappalá et al., 2024), the relevance of the modelling hypotheses of the finite element model proposed in this study, which is based on a simplified geometric representation of the ischial region (Macron et al., 2020), was investigated. A methodology was proposed to estimate the different parameters needed to construct the model from the available MRI masks. The FEM was then used to estimate *in vivo* compressive and shear strains. The resulting strains were then compared with experimental data. The results show that the model assumptions lead to an overall overestimation of the compressive and shear strains in the muscle tissue, especially directly under the ischial tuberosity. Similarly, the model underestimates the strain in adipose tissue (mean error in shear strain of -0.2). This study highlights the fact that the assumptions usually made for the geometric modelling of muscle tissue (homogeneous soft tissue layer) lead to an incorrect estimation of peak strain localisation. Further work should be done to improve the representation of muscle tissue.

## 1. Introduction

Pressure ulcers (PUs) form as a result of prolonged exposure to mechanical loads that exceed the tolerance of the skin and underlying soft tissues. These loads may include prolonged periods of lying or sitting on support surfaces or the application of medical devices such as respiratory masks or prosthetic sockets. Although they represent a minority of PUs, Deep Tissue Injuries (DTI) are a particular threat because of the difficulty to detect their presence in the early stages. It is now widely accepted that there are at least two damage mechanisms responsible for the onset of DTIs: (i) ischemia/reperfusion damage initiated by

sustained moderate tissue deformations and (ii) direct cell damage initiated by excessive direct (local shear) deformation. One difficulty in preventing PUs has been quantifying the individual load tolerance of skin and underlying soft tissues.

Over the past two decades, research has sought to elucidate the risk factors associated with soft tissue injury in terms of the local mechanical environment within soft tissues. Of particular interest are the series of experiments conducted at the Eindhoven University of Technology, which involved the indentation of the tibialis anterior muscle of Brown-Norway rats. These experiments led to the identification of a damage threshold for healthy murine skeletal muscle tissue (Ceelen et al., 2008).

<footnote><sup>a</sup>Arts et Métiers Institute of Technology, Université Sorbonne Paris Nord, IBHGC - Institut de Biomécanique Humaine Georges Charpak, HESAM Université, F-75013, Paris, France</footnote>

<footnote><sup>b</sup>School of Computer Science and Informatics, Cardiff University, Cardiff, UK</footnote>

<footnote><sup>c</sup>Cardiff University Brain Research Imaging Centre (CUBRIC), School of Psychology, Cardiff University, Cardiff, UK</footnote>

<footnote><sup>d</sup>Arts et Metiers Institute of Technology, Univ. of Bordeaux, CNRS, Bordeaux INP, INRAE, I2M Bordeaux, Av D'Aquitaine, 33607, Pessac, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: alexandre.segain@ec-lyon.fr (A. Segain).</footnote>
Using dedicated organ-scale Finite Element models, it was shown that direct deformation damage was apparent only when a given maximum shear strain threshold was exceeded and that the amount of cell damage was correlated to the magnitude of the strain energy applied. It is likely that similar mechanisms play a role in humans and in other soft tissues, but with different strain thresholds. In that perspective, computational models of load-bearing soft tissue in humans have shown that bony prominences induce substantial strain concentrations (Linder-Ganz et al., 2007; Oomens et al., 2013; Luboz et al., 2014; Macron et al., 2018), which explains why these areas are vulnerable to ulceration.

However, contradictory results have been reported regarding the simulated response of the tissue under loading. Some authors have reported localisation of the peak of compressive and shear strain and stress in muscle tissue (Linder-Ganz et al., 2007; Then et al., 2009; Sopher et al., 2010; Macron et al., 2018) while other studies observed that the peak strain was located in the adipose tissue layer (Luboz et al., 2014). These differences lead to a high variability in the predicted outcome for pressure ulcer development and can be explained by the different modelling choices including the modelling of the interaction properties at the fat/muscle interface, assumption of incompressibility or quasi-incompressibility of the tissues, the anisotropy, pre-stress, and so forth.

These finite element models are rarely verified (Sopher et al., 2010; Levy et al., 2013) or are verified indirectly using a comparison to experimental variables that do not represent the spatial distribution of the specific quantities of interest, such as the interface pressure instead of peak of local tissue shear strain (Linder-Ganz et al., 2008; Macron et al., 2018, 2020) or using physical phantoms. Yet such validation evidence must be obtained prior to any clinical evaluation. Moreover, it has been demonstrated in the literature that no particular correlation seems to exist between internal strains and external pressure distributions ((Candadai and Reddy, 1992)(in vitro) (Le et al., 1984)(in vivo) (Oomens et al., 2003; Linder-Ganz et al., 2007; Portnoy et al., 2011; Luboz et al., 2014; Macron et al., 2020) (in silico)). Therefore, the absence of experimental data, makes it challenging to assess the clinical relevance and robustness of the estimated quantities of interest, especially regarding the peak strains in the subdermal tissues.

Rare are the attempts in the literature to propose an experimental estimation of soft tissues' *in vivo* strains. MR Tagging methodologies were initially proposed for this purpose, but results were only reported on rats (Stekelenburg et al., 2007; Ceelen et al., 2008) and pigs (Solis et al., 2012). However, no such results were reported for human soft tissues, where measurement of strains often rely on a simple measure of thickness variation around anatomical landmarks (Linder-Ganz et al., 2007; Shabshin et al., 2010; Makhsous et al., 2011; Al-Dirini et al., 2017). However, this estimation appears unsatisfactory, particularly considering the additional component of rigid motion of the *gluteus maximus* muscle lodge occurring during the loading highlighted in various research papers (Sonenblum et al., 2015; Al-Dirini et al., 2016; Call et al., 2017; Zappalá et al., 2024). The observation of this rigid body motion suggests that the variation of thickness is not solely due to the compression of the different soft tissue layers and thus, that evaluating strains by simply measuring thickness variation is not sufficient. Attempts to estimate the 2D strain field from short ultrasound clips have been proposed in the literature (Doridam et al., 2018), however in this rare example the poor image quality in the ultrasound measurement create high dispersion in the estimation of the transverse displacement.

In a recently published study (Zappalá et al., 2024), the use of Digital Volume Correlation (DVC) was investigated for the first time on Magnetic Resonance Imaging (MRI) images of the buttock tissue. This allowed the estimation of the 3D compressive and shear strain fields during the loading of the buttock tissue in a semi-recumbent position (Al-Dirini et al., 2017) associated with Digital Volume Correlation. These novel data offer the community reference values for the comparison of predicted strain fields with experimental estimations. Building upon the aforementioned dataset, the objective of this contribution

is to estimate the discrepancy between the strain field estimated using a finite element model with a simplified methodology simplified as presented in (Macron et al., 2020) and an experimental estimation of the strain field given in (Zappalá et al., 2024).

## 2. Material and methods

### 2.1. Reference experimental results from the literature and simplified representation of the area of interest

#### 2.1.1. Experimental strain

The experimental displacement and strain fields from N = 10 healthy volunteers (Zappalá et al., 2024) were used to corroborate the relevance of the modelling hypotheses of the finite element model proposed in this study, which relies on a simplified geometrical representation of the ischial region (Macron et al., 2020).

Previously collected data from (Al-Dirini et al., 2017) was used in (Zappalá et al., 2024) to estimate the complex 3D deformation of the soft tissue under the ischial tuberosity of 10 healthy volunteers between different loading states using Digital Volume Correlation. The subjects were placed in a semi-recumbent position to mimic the sitting position in an MRI. Wooden plates placed under the buttock of the subjects were then removed progressively allowing to reduce the load transmitted to the left Ischial Tuberosity (IT) through the underlying soft tissues. 3 MRI acquisitions were then performed. The first one with the 3 planks placed under the ischial tuberosity (full weight bearing configuration), the second one after the removal of a 10 mm plate (partial weight bearing configuration) and a final one after the removal of all the plates (no weight bearing configuration). The post processing methodology proposed in (Zappalá et al., 2024) is composed of 3 main steps. First, a rigid transformation is estimated and applied to the images using the software Elastix (Klein et al., 2010). This is made to set the initial conditions of deformation. And thus, allows to compensate the changes in pelvis orientation between the different loading states. Subsequently, Elastix is employed to compute the elastic deformation of the soft tissues under the ischial tuberosity resulting in 3D displacement fields. Strain fields are then extracted by derivation of the displacement fields. Finally, the Gluteus Maximus muscle and adipose tissue under the Ischial Tuberosity are manually segmented, and an operator annotated the ischial tuberosity on the undeformed configuration. Further details on the methodology can be found in (Zappalá et al., 2024). The accuracy of the estimated displacement field was estimated using two simulated fields. An error of 0.99 mm was observed on the deformable registration on the 7.3 ± 3.15 mm simulated field and an error of 1.78 mm was observed on the deformable registration on the 19.37 ± 9.58 mm simulated field.

#### 2.1.2. Geometrical parametrization of the area of interest

As previously proposed in (Macron et al., 2020), only a local representation of the ischial tuberosity is considered instead of modelling the overall geometry of the pelvic region. This allows to reduce the overall scale of the finite element model and to focus on a fine representation of the areas where strain localizes.

Briefly, the Ischial Tuberosity (IT) is considered as a torus described by two radii of curvature. Soft tissues are represented as two parallelepiped whose height correspond to the patient specific thicknesses of the tissue layers. This parametrized geometry is then used in a Finite Element Model to estimate the distribution of the compressive and shear strains. The geometry is meshed using the algorithms available in the Finite Element Software Abaqus, using linear tetrahedral elements with a hybrid formulation (C3D4H). Boundary conditions are applied to the IT to allow only a vertical translation. A subject specific load is applied on the pelvis. Tie constraints are applied on the interfaces between the different regions (bone/muscle and muscle/fat interfaces). Finally, the contact between the skin and the support is modelled using a hard contact with a friction coefficient set to 0.4.

The mechanical behaviour of the subdermal soft tissues (adipose
tissue and muscle) is modelled using a hyperelastic first-order quasi-
incompressible Ogden law. The exponential coefficient $\alpha$ of the muscle
and adipose tissue is fixed to 5 and the shear modulus $\mu$ of both tissues is
calibrated so that the variation of thickness of the two tissues between
the loaded and unloaded states corresponds to the experimental
observation.

To sum up, 7 parameters are used to personalize the model: 4 geo-
metric parameters corresponding to the IT (R1, R2), and the soft tissue
thickness (eM, eF); 2 material parameters (μM, μF) and the force (F).

## 2.2. Construction of the simplified FE model from the MRI experimental data

As the experimental acquisitions from (Al-Dirini et al., 2017) differs
from the procedure used in (Macron et al., 2020), the parameters
describing the finite element model cannot be obtained as in the original
publication. A novel methodology used to estimate the 7 parameters (μF,
μM, R1, R2, eF, eM, F) is described below. In subsection 1.a, the
methodology used to construct a simplified geometric model from the
MRI masks provided in (Zappalá et al., 2024) is described. Then as no
force measurement is provided, subsection 1.b focuses on the estimation
of the load applying to the pelvis. Finally, subsection 1.c details the
methodology used to calibrate the material properties using the exper-
imental measurement of the thickness change of the different layers
(Al-Dirini et al., 2017).

a. Geometry

The measure of the different soft tissue layers' thicknesses in the
different configurations are provided in (Al-Dirini et al., 2017). The
tissue thicknesses in the Non-Weight Bearing configuration are used as
the initial tissue thickness of the model.

The segmentation masks of the pelvis in the different slices were
employed to estimate the two ischial radii of curvature. On each MRI
slice, a circle is fitted on the points located in a cubic bounding box of 40
pixels centred on the IT. The radius of curvature R1 is taken as the radius
of the circle fitted in the MRI slice containing the annotated coordinate
of the IT (Fig. 1a). The least square plane passing through the centres of
the obtained circles and an additional manually selected point on the
ischium is defined (green plane in Fig. 1b). This additional point is
necessary to ensure that the second radius of curvature is oriented in the
direction of the ischium. A least square circle passing through the pro-
jection of the centres of the circles on the plane is estimated. R2 is taken
as the radius of this circle (Fig. 1c).

In order to assess the quality of the estimated radii of curvature and
to compare it to the one obtained from the original method, a sensitivity

analysis of the estimated value of the radii of curvature to the orienta-
tion of the pelvis and to the position of MRI cutting planes was per-
formed. Simulated data were generated from a generic 3D model of the
pelvis. First, the model was oriented to mimic the semi-recumbent po-
sition. In the original paper (Al-Dirini et al., 2017) reported values of
sagittal angle (defined as the angle between the line passing through the
femoral head and the pubic symphysis and the horizontal line projected
in the sagittal plane) that ranges from 52.8° to 88°. Here 6 pelvis ori-
entations were considered each spaced from 5° ranging from 50° to 75°.

The 3D model of the pelvis was intersected by parallel cutting planes
spaced 10 mm apart to reproduce the MRI slices. One of the cutting
planes contained the ischial tuberosity which was annotated in accor-
dance with the experimental data set. The sensitivity of the estimated
radii of curvatures to the position of the cutting plane was investigated
by applying an offset in the x direction to the cutting planes. The pa-
rameters used for the sensitivity analysis are illustrated on Fig. 2. The
offset was varied from -5 mm to 5 mm in 0.5 mm steps.

Moreover, for each value of the offset and each orientation of the
pelvis, the methodology originally proposed in (Macron et al., 2020) was
also employed to estimate the difference between both methodologies.
Briefly the initial methodology estimates the first ischial radius of

<figure><img src="image_3.png" /><figcaption>Fig. 2. Definition of the different parameters used in the sensitivity analysis: pelvis orientations and cutting plane offset.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 1. Procedure to determine the radii of curvature. a) The least square circle passing through the lowest points of the mask of the pelvis is computed. Its radius is extracted and is R1. b) The same procedure is repeated on the 9 slices closest to the annotated IT. c) The least square plane passing through the extracted centres of circle and an additional point manually selected on the ischium is computed. In this plane, the least square circle passing through the centres of the circles is computed. Its radius is extracted and is R2.</figcaption></figure>
curvature by taking the smallest radius of least square circles obtained when cutting the 3D model of the pelvis by 18 vertical planes oriented with 10° between each. The second radius of curvature is the radius of the least square circle in the plane that is orthogonal to the first one.

Soft tissue geometry is meshed using default algorithms available in Abaqus, using linear tetrahedral elements with a hybrid formulation (C3D4H). The global mesh size was selected as 5 mm. A convergence study was performed and showed that doubling the number of elements resulted in a change in the maximal Green-Lagrange compressive strain that is in the range [0.05 %, 2.28 %] for all the subjects, excepting subject 5 where the observed variation was 4.37 %.

## b. Boundary Conditions (Loading)

No experimental measure of the load applied to the subject's buttock was available. In order to determine the load applied for each subject, it was necessary to consider the cushion placed under the knees and the upper trunk. This helps to maintain the semi-recumbent position. The load was estimated considering only the weight of the middle and lower trunk and the thigh of the subject. The load was then estimated using anthropometric parameters (de Leva, 1996) with the coefficient of mass of the mid trunk ($c_m^{mid trunk}$), lower trunk ($c_m^{lower trunk}$) and each thigh ($c_m^{thigh}$) are estimated to 16.33 %, 11.17 % and 14.16 %, respectively. It was assumed that the load was applied evenly on each IT and thus, the load applied by one IT on the soft tissue was considered as one half of the total load. The pertinence of this estimation is further discussed in the discussion.

$$F_{IT} = \frac{1}{2}(c_m^{mid trunk} + c_m^{lower trunk} + 2*c_m^{thigh}) * weight = \frac{1}{2} * 0.5582 * weight$$

## c. Material Properties

The mechanical behaviour of the soft tissues was modelled using a hyperelastic first order quasi-incompressible Ogden law. As in the original article, the exponential parameters $\alpha$ for the muscle and adipose tissue layers were fixed to 5. The shear modulus parameters $\mu$ for the muscle and adipose tissue layers was calibrated using an inverse method. Shear moduli were iteratively modified until the variation of thickness of the muscle and adipose tissue correspond to the values observed experimentally between the full weight bearing and the non-weight bearing situations and reported in (Al-Dirini et al., 2017). Optimisation was performed using the built-in Matlab fmincon function with an initial guess set to $\mu_{muscle} = 21.5 kPa$ and $\mu_{fat} = 31 kPa$. The cost function used for the optimisation (J) is the absolute error on the variation of thickness ($\Delta t$) between the unloaded and fully loaded position on each tissue computed as below.

$$J = abs(\Delta t_{muscle}^{experimental} - \Delta t_{muscle}^{numeric}) + abs(\Delta t_{fat}^{experimental} - \Delta t_{fat}^{numeric})$$

## 2.3. Comparison of the strain fields

In order to make the comparison between the compressive and shear strain fields easier, both numerical and experimental data (strain fields from (Zappalá et al., 2024)) were interpreted in the same reference frame, centred on the IT. Experimental data was linearly interpolated on the coordinates of the integration points of the numerical model. The distance between two consecutive nodes was $5.5 \pm 1.1$ mm in the adipose tissue and $5.7 \pm 1.2$ mm in the muscle tissue. The ROI for the comparison was selected as the intersection of the ROI of the finite element model – see (Macron et al., 2020) for further details – and the MRI masks. From this ROI, only the points located in a slice of 10 mm centred on the IT were selected.

The principal components of the logarithmic strain were extracted for each integration point of the model. Then the compressive and shear

strains were extracted using the following relationships

$$\begin{cases} \epsilon_{comp} = |\epsilon_{III}| \\ \epsilon_{shear} = |\epsilon_I - \epsilon_{III}| \end{cases}$$

Where $\epsilon_i$ denotes the ith principal Green-Lagrange strain calculated as $\epsilon_i = \frac{1}{2}(e^{LE_i} - 1)$ with $LE_i$ the ith principal component of the logarithmic strain tensor.

The error is considered as the difference between experimental and numerical strain field and was computed for the compressive and shear strain.

$$\begin{cases} E_{comp} = \epsilon_{comp}^{numeric} - \epsilon_{comp}^{experimental} \\ E_{shear} = \epsilon_{shear}^{numeric} - \epsilon_{shear}^{experimental} \end{cases}$$

Distributions of error on the shear and compressive strains are then considered.

In (Ceelen et al., 2008) experimental thresholds for the tissue damage have been identified as 0.75 for the Green-Lagrange shear strain and 0.45 for the Green-Lagrange compressive strain. The capability of the finite element model to predict pressure ulcer risk is also estimated by comparing the number of points that are above the identified thresholds for the compressive and shear strains between the numerical and the experimental strain fields.

# 3. Results

## 3.1. Construction of the simplified FE model from the MRI experimental data

### a. Geometrical modelling and sensitivity to pelvis orientations and cutting plane offset

The estimated values of the radii of curvature are presented in Table 1.

Subjects 5, 6, 7, 9 exhibited abnormally high values for the second radius of curvature. A visual examination of the obtained torus showed that the estimated coordinates of the second circle were outside the ischial branches for subjects 6 and 9, meaning that the estimated curvature of ischium is concave instead of convex (see Fig. 3). These subjects were thus excluded from further analysis.

The results of the sensitivity analysis are presented in Fig. 4. It can be observed that the estimated values of the radius of curvature R1, as determined by the methodology presented in this contribution are consistently higher than those estimated using the original methodology. Furthermore, the methodology presented in the original article from (Macron et al., 2020) displays a high sensitivity to the offset applied to the cutting planes.

For an offset comprised between -4mm and -1mm, the second radius of curvature obtained using the original methodology displays a low sensitivity to the pelvic tilt and the offset. In this range, the values of R2 estimated using the new methodology are 2 times higher.

### b. Constitutive parameters

The identified material properties for the adipose tissue and muscle layers estimated for the subjects in the dataset from (Zappalá et al., 2024) are presented in Table 2.

Except for subject 1, the shear modulus of the muscle tissue is higher than the shear modulus of the adipose tissue. Except subject 1, the median value of muscle shear modulus is 39.11 kPa and the one for the adipose tissue is 21.40 kPa. Boxplot on Fig. 5 shows the comparison of the muscle and adipose tissue.

The estimated mechanical properties of 10th subject are the same for the muscle and adipose tissue layers.
<table><caption>**Table 1**

Ischial Radii of curvature obtained for the different subjects.</caption><thead><tr><th>Subject ID</th><th>1</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th></tr></thead><tbody><tr><td>R1 (mm)</td><td>9.71</td><td>5.02</td><td>10.97</td><td>10.67</td><td>11.25</td><td>9.18</td><td>8.75</td><td>9.38</td><td>8.93</td></tr><tr><td>R2 (mm)</td><td>67.11</td><td>51.81</td><td>57.53</td><td>200.49</td><td>567.15</td><td>250.21</td><td>63.11</td><td>165.98</td><td>52.91</td></tr></tbody></table>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** a) Estimated torus from subject 1. The torus has the expected shape. b) Estimate torus for subject 9. The torus appears with a negative curvature.</figcaption></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Sensitivity of the two ischial radii of curvature to the pelvic orientation and the position of the cutting planes. Plain lines correspond to the estimated radius of curvature using the current methodology and dashed lines correspond to the radius of curvature estimated using the methodology described in (Macron et al., 2020).</figcaption></figure>

<table><caption>**Table 2**

Optimal values of the muscle and adipose tissues shear moduli based on the measure of the compression.</caption><thead><tr><th>Subject ID</th><th>1</th><th>3</th><th>4</th><th>5</th><th>7</th><th>8</th><th>10</th></tr></thead><tbody><tr><td>μ<sub>muscle</sub> (kPa)</td><td>10.73</td><td>42.56</td><td>35.65</td><td>57.70</td><td>23.72</td><td>51.95</td><td>31.45</td></tr><tr><td>μ<sub>fat</sub> (kPa)</td><td>89.27</td><td>16.37</td><td>15.41</td><td>26.42</td><td>16.32</td><td>33.16</td><td>31.44</td></tr></tbody></table>

## 3.2. Comparison of the strain fields

Fig. 6 presents an illustrative example of the point-by-point comparison of the compressive and shear strain in the soft tissues below the

IT. The results obtained for all the subject is provided in Fig. 8. Both figures show that the finite element model tends to overestimate the compressive strain and shear strains in the muscle tissue directly below the ischial tuberosity and underestimates the strain in the adipose tissue.

Fig. 7 shows an example of the error distribution for the compressive and shear strain. This figure tends to confirm the observations made previously. Clearly showing that the shear stress is globally underestimated in the adipose tissue, when it is globally well estimated for the muscle tissue. In a similar approach, the compressive strain is globally overestimated in the muscle tissue, when it is globally underestimated in the adipose tissue.

The median and interquartile range of the error distribution are
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Comparison of muscle and adipose tissue shear moduli. The left box corresponds to the values of the shear modulus of muscle tissue and the right box corresponds to the values of the shear modulus of the adipose tissue.</figcaption></figure>

shown in Table 3. The numerical model underestimates the compressive and shear strains for all subjects. On average, the strain in adipose tissue is underestimated by 0.19 and the shear strain in adipose tissue is underestimated by 0.25.

The compressive and shear strains in the muscle tissue of subjects 3, 8 and 10 are in fairly good agreement. On the contrary, the compression and shear strains of subjects 4, 5 and 7 are underestimated by 0.21 and 0.14 on average.

The number of points above the risk thresholds for the strains are displayed in Table 4. Only subject 10 displays no regions above the thresholds identified in (Ceelen et al., 2008).

For all the subjects, the compressive strains in the muscle and adipose tissue estimated by the finite element model are always below the risk thresholds. Regarding the shear strain in the muscle tissue, the number of points where the strain estimated by the numerical model exceed the threshold is at least twice the number of points that exceed the threshold experimentally. On the contrary, the shear strains estimated in the adipose tissue by the numerical model are most of the time lower than the experimental, even when more than a thousand points are exceeding the threshold on the experimental measurements.

## 4. Discussion

The objective of this study was to compare the strain fields estimated by a simplified finite element model to an experimental reference. As far as the authors are aware of, this is the first time such an approach has been implemented for the corroboration of the predicted buttock tissue's deformation during sitting. The majority of the finite element models designed to estimate strains *in vivo* employ an indirect validation procedure, based on variables accessible from *in vivo* measurement that do not represent the whole spatial distribution of the deformation. The experimental data presented in (Zappalá et al., 2024) offers the first opportunity to validate directly the strain fields' distribution estimated from FE models and to question the clinical relevance of the modelling assumptions.

The seven parameters (the two radii of the IT, thicknesses and shear moduli of the muscle and adipose tissue, and the sitting load) required for the construction of the simplified FE model have been estimated from the experimental data from (Al-Dirini et al., 2017) using a new post processing procedure.

### 4.1. Estimation of patient specific morphology

A new methodology was proposed to estimate the radii of curvature of the IT from the segmented MR slices. The median value of the radius of curvature R1 obtained in this contribution was 9.38 mm, which is greater than the median identified as 7 mm in (Macron et al., 2020). As showed in Fig. 4, this difference is consistent with the one obtained when computing R1 from the two methodologies on the same experimental data. In contrast to the original contribution, the choice of the cutting plane is here imposed by the MRI slices. The initial methodology relies indeed on the intersection of the pelvis with vertical planes that are separated by a 10° angle. Thus, it allows to ensure that the estimated value of R1 is effectively the smallest radius of curvature. The methodology proposed in this contribution can only consider the MRI slices. In the general case, the plane of the MRI slices does not correspond to the plane with the smallest radius of curvature. Furthermore, differences in the radii of curvature can be explained by the different anatomical locations that are used to compute this radius of curvature. In (Macron et al., 2020), the radius of curvature was computed on the lowest point of the 3D model in the actual patient specific orientation while sitting. In this case, the patients were positioned in a semi-recumbent position. Thus, the pelvic tilt is higher, and the patient is lying on a posterior part of the ischium (see Fig. 2). It is anticipated that the differing anatomical locations will result in a discrepancy in the estimated radius of curvature, which may account for the observed disparity.

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Point by point comparison of the compressive strain. Left Mapping of the error on the compressive strain (No units). Right Mapping of the error on the shear strain (No units).</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Histogram of the error on compressive and shear strain field. Left error in the whole tissue; Right Error in each tissue layer: above adipose tissue and below muscle tissue.</figcaption></figure>

A sensitivity analysis was conducted, and the results are displayed on Fig. 4. The results showed that the current methodology for the estimation of the radii of curvatures is very sensitive to the offset between the closest MRI slice and the location of the IT. In addition, our method is very sensitive to the pelvic tilt when the offset is superior to -2mm for R1 and for all the offset values for R2. Thus, values estimated for the radii of curvature have to be interpreted with caution. In particular, it was demonstrated that the value of the radii of curvature have a high impact on the predicted value of the strain (Macron et al., 2020). A more robust procedure should be defined to estimate the second radius of curvature. However, as only the masks resulting from the segmentation of the MRI images are available, it was difficult to have a more reliable estimation of the radii of curvature.

A major limitation of the procedure is that it led to the exclusion of two subjects (20 % of the data). As illustrated in Fig. 3, the radii of curvature for these subjects are in the wrong direction. Multiple reasons can explain this limitation. First, the segmented masks that were provided displayed an almost flat geometry near the IT. In addition, the distance between two consecutive MR slices is large. Thus, imprecisions on the estimation of the centre of the fitted circles may have a consequent impact on the estimated curvature of the ischium. It is expected that improvement of the pelvis segmentation, closer slices and a better strategy for the estimation of the radii of curvature could help to improve the model reliability. However, only the segmented masks were shared due to ethical issues.

## 4.2. Load estimation

Measure of loads during MRI acquisitions is a major challenge for the biomechanical community and MR compatible load measurement

devices are not frequent in the literature (Stekelenburg et al., 2006; Petre et al., 2008; Then et al., 2009; Moerman et al., 2011; Tan et al., 2011; Nelissen et al., 2017; Trebbi et al., 2021). Absence of such a measure is a major limitation of the present study, because the estimated load is used both in the inverse procedure and in the estimation of the strain fields. The methodology used to estimate the loading relies on strong assumptions. It assumes, first that the loading is perfectly vertical, and that no shear forces are applied to the skin. However, when looking at the experimental setup proposed in (Al-Dirini et al., 2017), it can be expected that frictional forces are exerting at the level of the pelvis due to the semi recumbent position. Moreover, the fact that, the whole weight of thighs, upper and mid trunks affected the deformation of the buttock without considering its partial redistribution over the different supports is also a strong assumption. A part of these forces should be distributed on the other supports under the knees and the back. In addition, even if the subject is lying on two wooden planks that are at the same level, it is unlikely that the load was distributed evenly between both ischial tuberosity due to small pelvic rotation. However, estimation of this imbalance between the load repartition is very difficult without any correct measurement of the load.

## 4.3. Estimation of the constitutive parameters

As often in the literature (Then et al., 2009; Al-Dirini et al., 2016; Macron et al., 2018; Rayward et al., 2023), an inverse FE procedure was proposed to estimate soft tissues' constitutive parameters.

Muscle tissue shear modulus was found higher than the one from adipose tissue for most of the subject, which is consistent with observations done elsewhere in the literature (Avril et al., 2010; Silber and Then, 2013; Affagard et al., 2015; Al-Dirini et al., 2016; Macron et al.,
<figure><img src="image_12.png" /><figcaption>Fig. 8. Response map and histogram for all the subjects.</figcaption></figure>

<table><caption>**Table 3**

Median and Interquartile range for the different error distribution.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="2">Subject 1</th><th colspan="2">Subject 3</th><th colspan="2">Subject 4</th><th colspan="2">Subject 5</th><th colspan="2">Subject 7</th><th colspan="2">Subject 8</th><th colspan="2">Subject 10</th></tr><tr><th>Med</th><th>IQR</th><th>Med</th><th>IQR</th><th>Med</th><th>IQR</th><th>Med</th><th>IQR</th><th>Med</th><th>IQR</th><th>Med</th><th>IQR</th><th>Med</th><th>IQR</th></tr></thead><tbody><tr><td rowspan="2">Muscle</td><td>Compressive</td><td>0,10</td><td>0,14</td><td>-0,02</td><td>0,16</td><td>-0,17</td><td>0,22</td><td>-0,17</td><td>0,19</td><td>-0,28</td><td>0,13</td><td>0,02</td><td>0,12</td><td>0,05</td><td>0,08</td></tr><tr><td>Shear</td><td>0,03</td><td>0,38</td><td>-0,05</td><td>0,23</td><td>-0,12</td><td>0,30</td><td>-0,13</td><td>0,25</td><td>-0,17</td><td>0,23</td><td>-0,04</td><td>0,20</td><td>0,00</td><td>0,17</td></tr><tr><td rowspan="2">Adipose</td><td>Compressive</td><td>-0,11</td><td>0,05</td><td>-0,12</td><td>0,05</td><td>-0,12</td><td>0,16</td><td>-0,29</td><td>0,11</td><td>-0,26</td><td>0,19</td><td>-0,25</td><td>0,08</td><td>-0,16</td><td>0,16</td></tr><tr><td>Shear</td><td>-0,41</td><td>0,17</td><td>-0,13</td><td>0,27</td><td>-0,20</td><td>0,21</td><td>-0,30</td><td>0,16</td><td>-0,31</td><td>0,27</td><td>-0,21</td><td>0,14</td><td>-0,18</td><td>0,15</td></tr></tbody></table>
<table><caption>**Table 4**
Number of points above the risk thresholds from (Ceelen 2008).</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="2">Subject 1</th><th colspan="2">Subject 3</th><th colspan="2">Subject 4</th><th colspan="2">Subject 5</th><th colspan="2">Subject 7</th><th colspan="2">Subject 8</th><th colspan="2">Subject 10</th></tr><tr><th>Exp</th><th>Num</th><th>Exp</th><th>Num</th><th>Exp</th><th>Num</th><th>Exp</th><th>Num</th><th>Exp</th><th>Num</th><th>Exp</th><th>Num</th><th>Exp</th><th>Num</th></tr></thead><tbody><tr><td rowspan="2">Muscle</td><td>Compressive</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>127</td><td>0</td><td>1850</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Shear</td><td>221</td><td>2188</td><td>0</td><td>744</td><td>0</td><td>0</td><td>0</td><td>0</td><td>59</td><td>480</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td rowspan="2">Adipose</td><td>Compressive</td><td>0</td><td>0</td><td>1063</td><td>0</td><td>293</td><td>0</td><td>612</td><td>0</td><td>1235</td><td>0</td><td>229</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Shear</td><td>2</td><td>0</td><td>106</td><td>48</td><td>0</td><td>0</td><td>81</td><td>0</td><td>666</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

2020; Hatt et al., 2023).

However, the methodology used in this contribution resulted in high values for the shear modulus for the muscle and adipose tissues. Typical values found in the literature range indeed between 0.5 kPa and 6 kPa (Avril et al., 2010; Affagard et al., 2015; Al-Dirini et al., 2016; Moerman et al., 2017; Macron et al., 2020), when the median value of muscle shear modulus is 39.11 kPa and the one for the adipose tissue is 21.40 kPa in this contribution. Multiple factors can explain these high values. First, the calibration procedure proposed relies on a previous contribution (Macron et al., 2020) where the values of the exponential coefficients $\alpha$ are set to 5 for the muscle and adipose tissues. However, several papers reporting results using procedures with varying exponential coefficients identified rather different values for this parameter. In (Al-Dirini et al., 2016) the exponential coefficient of the muscle tissue was 4.6 and the one from the adipose tissue was 16.2. It is expected that those low values of the exponential coefficient force the muscle and adipose tissue's shear moduli to be higher than usual, to compensate the absence of stiffening. Secondly, the calibration procedure is based only on the 1D change in thickness of the soft tissues between two configurations. In the literature, the cost function for the optimisation is based on more complete data such as the displacement of the muscle and adipose tissue contours (Linder-Ganz et al., 2007; Oomens et al., 2013; Moerman et al., 2017; Zeevi et al., 2018), or include several loading states to construct the force/displacement curves (Then et al., 2009). The paucity of the data used for the calibration of the material parameters might lead to multiple sets of parameters with the same cost function value. (Fougeron et al., 2024) recently demonstrated that the force displacement curve alone is not sufficient to identify a single set of constitutive parameters when muscle and adipose tissue layers are modelled using an Ogden law. In this contribution, only two points of the curve were known, one in the non-weight bearing and one in the full weight bearing configurations. It is thus expected that the degraded information will lead to even more difficulties to identify a unique set of values. Also, the error made during the estimation of other parameters affecting the model such as the tissue thicknesses or the radii of curvature or the force used for the calibration have a strong impact on the identified material properties. As the different methodologies used for the estimation of these parameters all rely on strong assumptions, it is probable that error propagates, and the estimated constitutive parameters might not be very precise.

Material properties estimated for subject 1 differ significantly from the other subjects. These differences might be explained by the small variation of tissue thickness in the adipose tissue between the unloaded and loaded states compared to the thickness variation observed in the muscle tissue (2.4 mm in the adipose tissue and 22 mm in muscle tissue) (Al-Dirini et al., 2017). In addition, the measurement performed in this previous study is a 1D measurement at the level of the ischial tuberosity and do not consider other bony structures that might also distribute the load. Moreover, the measurement is taken as the vertical distance between the landmark and the skin. Slight motion and pelvic rotations might also have happened between two MRI acquisitions and can perturb the measurement of this displacement -see the negative displacement of the fat between the unloaded and loaded states in subject 9 (Al-Dirini et al., 2017).

In addition, several anatomical structures were not included in the modelling such as the skin or fascia. Other modelling assumptions such

as the value of the Poisson's ratio used to model the quasi-incompressibility or the modelling of the muscle tissue as a homogeneous layer of isotropic material might influence significantly the material properties used in this contribution and explain differences that are observed between values issued from *in vivo* measurements and *ex vivo* acquisitions.

### 4.4. Corroboration of the estimated strain field with an experimental reference

Mesh convergence study showed that the result is almost independent from the mesh for all the subjects excepted subject 5. For this subject, relatively high variations were observed on the value of the maximal Green-Lagrange compressive strain (4.37 %) when the number of elements was doubled. This variation was considered as acceptable for the present study. In particular, it was checked that this variation did not induced changes in the localisation of the maximum strain and that the strain is still overestimated in the region of the maximal load.

As it can be seen in the histograms (Fig. 7), the calibration of the model using only the compression values resulted globally in an overestimation of the compressive and shear strains in the muscle directly under the IT. This result is in accordance with the modelling choices done for the construction of the model. As in many other contributions (Linder-Ganz et al., 2007; Luboz et al., 2014; Moerman et al., 2017; Zeevi et al., 2018), the muscle is here modelled as a single homogeneous tissue layer and no relative motion is allowed between the muscle and adipose tissue layers. The variation of muscle thickness in the model is thus entirely explained by the compression of the muscle tissue, whereas experimental data suggests an additional component rigid body motion of the muscular lodges during the loading (Sonenblum et al., 2015; Al-Dirini et al., 2016; Call et al., 2017; Zappalá et al., 2024). This highlights the fact that such modelling of the geometry is not appropriate to capture the complexity of the deformation observed *in vivo* in a semi recumbent position. However, a finer modelling of the geometry and behaviour of the differentiated muscular lodges would require a deeper understanding of the *in vivo* mechanics of such deformation. This would further inform on the choices concerning the boundary conditions that would be applied to the model, such as the coordinate of the attachment points, the pre-tension applied on the muscle at the level of the tendons and the interaction that occurs at the interface between different muscular lodges and between the muscle and adipose tissue layers *in vivo*.

Analysis of the number of points exceeding the risk threshold defined in the literature (Ceelen et al., 2008) available in Table 4 shows that the model fails to estimate at risk compressive in that are observed experimentally. This suggests that the numerical model fails to represent the patient specificity of the compressive strain distribution and thus fail to identify at risk situations. Comparison of the shear strain fields in the muscle tissue (Fig. 8) showed that the model generally overestimates the strains. This was reflected on the at-risk strains are predicted by the models. Models predicted indeed bigger risks zone for subjects 1 and 7, and predict at risk strains for subject 3, whereas the experimental data do not present at risk regions for this subject.

On the contrary, for the adipose tissue layer, experimental data displays at risk strains for almost all the subject for compressive and shear strain. The high experimental shear strains may be explained by
the semi recumbent position in the MRI device that is not properly represented in the numerical model as only a vertical displacement is applied to load the tissue. This leads to a global underestimation of the compressive and shear strain in the adipose tissue, which is then reflected on the prediction of at risk strains. The model is indeed not able to predict all the compressive strains that are above the threshold. Regarding the shear strain, the model is not able to reproduce at-risk strains observed experimentally for subjects 1,5 and 7 and underestimates the size of the at risk zone for subject 3.

This analysis questions the results regularly reported in the literature. As in this contribution, most of the studies report a maximum peak strain that is located in the muscle tissue directly under the ischial tuberosity (Linder-Ganz et al., 2007; Then et al., 2009; Sopher et al., 2010; Macron et al., 2018). However, data presented in this contribution highlight that current modelling assumptions lead to contradictory outcome predictions. Thus, FE models presented in the literature might in general overestimate the risk of onset of Deep Tissue Injuries (DTI). Moreover, a closer look to the error on the strain maps (Fig. 6) show that, in addition to the global overestimation in the muscle, peaks of error exist at the bony interface directly under the ischial tuberosity. Current theory about the understanding of the DTI formation relies partly on the results of these FE models. Thus, it is often assumed in the literature that DTI initiate deeply in the muscle tissue at the bony interface. However, results from this study showed that this is where the strains are overestimated. Animal experiment showed that most of the tissue damage was observed in the muscle tissue but not directly at the bony interface (Stekelenburg et al., 2006; Ceelen et al., 2008). Thus, results from this study are in accordance with experimental observations and questions the current understanding of the DTI formation. Moreover, several experimental studies imaging the buttock soft tissue under a sitting load (Sonenblum et al., 2015; Al-Dirini et al., 2016; Call et al., 2017) reported a sliding of the muscle tissue during the loading of the IT.

Mapping of the error on compressive strain in subject 7 (Fig. 8) displays an overall underestimation of this parameter in the muscle and adipose tissue. However, the numerical and experimental variation of the tissue thicknesses are consistent. Moreover, the experimental strain field displays a homogeneous value of the Cauchy compressive strain with values around -0.5, which would correspond to a total compression of the tissue. These values tend to show that the experimental strain fields obtained with digital volume correlation are also to be interpreted with caution. Some experimental and post processing uncertainties could indeed lead to uncertainties on the strain fields obtained experimentally, especially because of the large differences between the loading configurations. However, only few studies reported uncertainty estimation for the use of DVC to estimate *in vivo* deformation in soft tissue. In (Zappalá et al., 2024) the error on the displacement field is estimated on synthetic data to be 0.99 mm during the transition from partial to full WB and 1.78 mm during the transition from no weight bearing to partial weight bearing, close from the order of magnitude of the thickness changes measured in (Al-Dirini et al., 2017). This study highlights that the global lack of validation visible in the literature may introduce inaccuracies regarding the localisation of compressive and shear strain occurring in the tissue. The modelling assumptions that are done in (Macron et al., 2020) to simplify the geometry leads to a localisation of the maximal compressive and shear strains in the muscle tissue as it is predicted by most finite element models presented in the literature (Linder-Ganz et al., 2007; Then et al., 2009; Sopher et al., 2010; Levy et al., 2013; Macron et al., 2018). However, experimental data rather show that the localisation of maximal compressive and shear strain occurs in the adipose tissue. More effort should be put into verifying and validating finite element models of the pelvic region as often recommended (Anderson et al., 2007; Burkhart et al., 2013; Oefner et al., 2021).

### 4.5. Perspective work and recommendations

Even if the work done in this contribution aimed to the comparison of the predicted strain fields with an experimental reference, it is not a proper validation of the strain fields. Several strong assumptions were done both for the construction and the comparison of the predicted strain field to the experimental reference. In addition, the experimental reference used in this research relies also on a novel methodology. More effort is needed to provide clean experimental data that will allow both to confirm the displacement fields provided in (Zappalá et al., 2024) and to build a FE model with reliable data that would allow a more robust comparison of the strain fields.

This contribution is a first step toward the validation of finite element models used for the *in vivo* estimation of strain in muscular and adipose tissues. Corroboration of the strain field highlights that choices usually made for the modelling of the muscle tissue allow a good representation of the variation of the different layer's thicknesses under the ischial tuberosity. However, these modelling choices lead often to an overestimation of the compressive and shear strain in the muscle tissue region under the ischial tuberosity and a global underestimation of the compressive and shear strain in the adipose tissue region. Further work is thus required on the improvement of the modelling and in particular, on acquisition of data *in vivo* to guide the different modelling choices and propose reliable models for the investigation of pressure ulcer risk thresholds.

### CRediT authorship contribution statement

**Alexandre Segain:** Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis. **Helene Pillet:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition, Conceptualization. **Stefano Zappalá:** Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Investigation, Data curation. **Giuseppe Sciume:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology. **Pierre-Yves Rohan:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition, Conceptualization.

### Declaration of competing interest

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

### Acknowledgements

The authors of the manuscript would like to sincerely thank Rami Al-Dirini for his work on the original acquisition of the experimental data which made the work reported in this article feasible. Also, the authors wanted to thank the HPC CASSIOPEE from the Arts et Métiers Institute of Technology.

This research was founded by the Fondation de l'Avenir under the grant ID AP-RM-23-036.

### Data availability

Data will be made available on request.

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