# Non-destructive assessment of multi-material micro-tissue mechanics reveals the critical role of rigidity gradients in tumour growth and pressure

Angélique Ala, Camille Douillet, Audrey Ferrand, Vincent Velay, Stéphane Segonds, Gaëlle Recher, Florian Bugarin

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Angélique Ala, Camille Douillet, Audrey Ferrand, Vincent Velay, Stéphane Segonds, et al. Non-destructive assessment of multi-material micro-tissue mechanics reveals the critical role of rigidity gradients in tumour growth and pressure. Acta Biomaterialia, 2026, 212, pp.383-398. 10.1016/j.actbio.2025.12.013. hal-05440875

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Full length article

# Non-destructive assessment of multi-material micro-tissue mechanics
reveals the critical role of rigidity gradients in tumour growth and pressure

Angélique Ala<sup>a, b</sup>, Camille Douillet<sup>b</sup>, Audrey Ferrand<sup>c, d</sup>, Vincent Lelay<sup>d</sup>, Stéphane Segonds<sup>a, d</sup>, Gaëlle Recher<sup>b, 1, *, e</sup>, Florian Bugarin<sup>a, 1, *, e</sup>

## ARTICLE INFO

### Keywords:

FEM

Identification parameters

Hydrogels mechanics

Modelisation

Tissue engineering

## ABSTRACT

Probing stiffness anisotropies in three-dimensional materials non-destructively is a major challenge in disciplines as diverse as aeronautics and medicine. While the former typically relies on various mechanical tests—such as tensile, compression, bending, and shear—performed on sample parts, the latter often employs acoustic techniques or wave propagation through matter. The choice of techniques depends on the size of the sample of interest and the desired resolution. In our case, to probe the mechanical properties of sub-millimetre micro-tissues, it is necessary to use methods with high resolution and as furtive as possible. We present a method based, 1/ on imaging the displacement of microbeads within a hydrogel resulting from the growth of a three-dimensional micro-tissue and, 2/ on finite element modelling of the deformations underlying bead displacements. This approach allows us to determine the elastic properties of the hydrogel and, in particular, to show that beyond a certain thickness, incomplete cross-linking of the hydrogel results in a stiffness gradient. We show that when the micro-tissue contacts with an immediately rigid alginate wall, the pressure exerted over time increases very rapidly, whereas when the micro-tissue encounters a substrate with a stiffness gradient, the pressure increase is more gradual. Uncovering this could provide a better understanding of the role of tumour microenvironment stiffness in metastatic escape processes.

**Statement of significance:** Understanding factors modulating tumours growth is crucial for developing better cancer treatments. This study introduces a non-destructive method to assess the stiffness of subcomponents of a tissue avatar, a question unwieldy to tackle. The authors show that small changes in stiffness of the tumour-mimic surrounding tissue can strongly affects how the tumour cell aggregate grows. This is an outreach in cancer biology because it connects the mechanical environment of tissues to cancer behaviour in a original way. It provides a powerful tool for studying 3D biological systems and could help design more suited materials for biomedical research and therapy. This work is relevant for the fields of cancer biology, biomaterials, and tissue engineering.

## 1. Introduction

Whether natural (embryos, organs, tumours) or bioengineered (spheroids, organoids, bioprinted tissues), three-dimensional multicellular assemblies develop within an enveloping environment. This environment can consist of neighbouring cells, surrounding tissues of varying stiffness, or hydrogels when cultured in vitro (e.g., Matrigel, alginate, GelMA, collagen [1,2]). Mechanical interactions within a tissue

influence collective cell behaviour (such as the “go-or-grow” decision [2, 3]) and modulate intercellular signalling pathways [4]. In addition, tissues exert stresses on their environment, which in turn exerts reciprocal stresses back onto them. According to the principle of action-reaction, both compartments experience the same stress, and cells must continuously adapt their behaviour to changes in their microenvironment [4–7]. Quantifying these reciprocal mechanical interactions is therefore essential to decipher how tissues respond under physiological

<footnote><sup>a</sup> ICA, Institut Clément Ader, Université de Toulouse; CNRS, UPS, INSA, ISAE-SUPAERO, IMT Mines Albi, 3 rue Caroline Aigle, Toulouse 31400, France</footnote>

<footnote><sup>b</sup> LP2N, Laboratoire Photonique Numérique et Nanosciences, Université de Bordeaux, Institut d'Optique Graduate School & CNRS UMR, Talence 5298, France</footnote>

<footnote><sup>c</sup> IRSD, Institut de Recherche en Santé Digestive, Université de Toulouse, INSERM, INRAE, ENVT, UPS, Toulouse, France</footnote>

<footnote><sup>d</sup> ICA, Institut Clément Ader, Université de Toulouse; CNRS, UPS, INSA, ISAE-SUPAERO, IMT Mines Albi, Campus Jarlard, Albi 81013, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail address: florian.bugarin@utoulouse.fr (F. Bugarin).</footnote>

<footnote n="1">¹ These authors contributed equally to this work.</footnote>
and pathological conditions, and to assess the consequences of altered mechanical reciprocity [8,9].

Several methods have been developed to map forces and stresses between tissues and their environment. For two-dimensional systems, these include traction force microscopy (TFM) [10–12], micropillar arrays [13,14], and monolayer stress microscopy [15,16]. In three dimensions, approaches such as 3D TFM [17–19], microbulge tests [20], or the insertion of deformable but incompressible oil droplets into tissues [21] allow stress measurements. However, these methods are limited when the extracellular matrix (ECM) is not purely elastic, making it difficult to recover forces directly from ECM deformations.

To overcome these limitations, a system that is tuneable, reproducible, and compatible with high-throughput all-optical measurements is needed. To address this challenge, we employ the Cellular Capsule Technology (CCT), which consists of encapsulating a cell suspension in hollow alginate spherical capsules [22–24] or tubes [25,26]. When fluorescent microbeads are incorporated into the alginate before gelation, they become embedded within the hydrogel mesh. As cells proliferate and form aggregates, the resulting growth deforms the alginate shell, displacing the beads according to the gel’s strain field. Using 3D time-lapse confocal microscopy, these displacements can be tracked.

To extract quantitative information from these measurements, assumptions must be made regarding the mechanical properties of the capsule (elastic, viscoelastic, or plastic behaviour) as well as its geometry. Alginate has been extensively characterized using rheology [27–30], elongation [31], and osmotic swelling assays [22], and is known to display nonlinear responses depending on the deformation regime [22,27]. Finite element modelling (FEM) [32] is particularly suited to account for such material properties, as it enables the simulation of complex 3D structures and provides both global and local stress and strain distributions in cases where analytical solutions are not available [33–36].

In this work, we combine CCT with live 3D time-lapse imaging to generate large datasets of encapsulated, fluorescently labelled cellular aggregates and their associated capsule deformations. These data are then integrated into a custom FEM framework designed to reconstruct strain and pressure fields exerted by the growing microtissue on its capsule. By accounting for capsule geometry, we not only predict pressure generation but also test new hypotheses regarding alginate’s behaviour under large deformations. Moreover, our methodology does not rely on regularization procedures, which are often required in existing numerical approaches. Our results reveal that pressure sensed and exerted by microtissues is strongly dependent on capsule stiffness: stiffer capsules produce steeper pressure increases (slope = 3.108 kPa), whereas capsules with a thickness gradient and softer inner layers yield lower pressure buildup (slope = 0.867 kPa). Altogether, we present an integrated optical and computational pipeline to quantify the mechanical properties of soft biomaterials and the resulting mechanical interactions between microtissues and their immediate environment.

## 2. Materials and methods

### 2.1. Caco-2 cells culture and alginate encapsulation

Caco-2 (DSMZ, acc169, gift from Audrey Ferrand) cells were cultured at 4000 to 8000 cells/cm2, passaged every 3 to 4 days, in DMEM (High Glucose, with L-Glutamine, with Sodium pyruvate, L0104–500, Dominique Dutscher) with 10 % FBS (FBS-16A, Capricorn Scientific), 1 % Penicillin-Streptomycin (P06–07,100, Pan Biotech), and 1 % MEM non-essential amino acids (11,140–050, Gibco), at 37 °C and 5 % CO2.

As previously described [22,23], we used the Cellular Capsule Technology (CCT) to incorporate Caco-2 cells in a basement membrane extract (BME) matrix (Cultrex BME001, R&D Systems) surrounded by a 3D spherical shell of alginate (AGI, I3G80). Unless stated otherwise, we use a cell formulation of 100 000 Caco-2 cells in 200 µL of DMEM with

50 % [v/v] BME maintained at 4 °C during encapsulation. Briefly, a microfluidic chip coextrudes the cell formulation inside an outer jet of alginate 2 % [w/v] supplemented with 1 % red-fluorescent microbeads (Sicastar®-redF plain 1 µm), both separated by solution of sorbitol (300 mM, 85,529, Sigma). The stream breaks into droplets that are collected in a CaCl₂ bath (100 mM, 7774–34–7, Themo scientific) gelling the alginate into a wrapping capsule. The cell-containing capsules are then cultured in suspension in cell culture medium at 37 °C and 5 % CO2.

### 2.2. Experimental determination of alginate young’s modulus in capsules

To experimentally determine the Young’s modulus of capsules in the case of a thin thickness, we conduct an osmotic swelling study. For this purpose, we consider an alginate capsule in which a Dextran solution is encapsulated, immersed in a Dextran bath. At the beginning of the experiment, $T_0$, the internal and external Dextran concentrations of the capsule are identical ($c_0$) and the average radius of the capsule is denoted as $R_0$. The external concentration is decreased by dilution, leading to a concentration decrease at $T_1$ (decrease from $c_0$ to $c$). Since the capsule is permeable to water but not to Dextran, water diffuses into the capsule, causing it to swell until the osmotic pressure $\Pi_0$ is balanced, which increases the initial radius by $\Delta R = R - R_0$ and decreases the initial thickness $h_0$. It is then possible to calculate the Young’s modulus using the following relation [22]:

$$ \frac{\Delta R}{R_0} = \frac{\Pi_0}{\Pi_c + 4E\left(\frac{h_0}{R_0}\right)} \quad (1) $$

With $\Pi_c = \Pi_0 + 3\beta c_0^2 + 6\gamma c_0^3$ ($\beta = 87$ and $\gamma = 5$). We perform this protocol on 8 different capsules and obtain an average Young’s modulus of $E = 94 \text{ kPa} \pm 28 \text{ kPa}$. This value will be used as the constant Young’s modulus for thin capsules and as the external Young’s modulus for thick capsules

### 2.3. Virtual model geometry

External and internal curves of the alginate capsule are based on the 2D Booth’s curve Lemniscate’s [37]. These lemniscates can be described using a rational parametrization defined by two parameters, a and b. The parameter a corresponds to the half-width at the widest point of the lemniscate, while b represents its half-height. The parametric equation is restricted to the interval $[0,\pi]$ (instead of $[0,2\pi]$) in order to obtain a quarter of the lemniscate. A revolution operation is applied along the y axis to obtain the capsule. Tip of internal lemniscate is smoothed to avoid geometrical irregularities. The dimensions of the capsule (internal height, internal half width, external height and external half width) are obtained from the experimental point cloud. External dimensions are directly taken from the dots positions. Internal dimensions are obtained by selecting the dots belonging to the internal surface, using an alpha-shape algorithm [38]. The thickness at the capsule bottom is also obtained directly on the capsule point cloud. The description of the dimensional measurements, as well as their values, are presented in the Supplementary Materials: Alginate capsules dimensions.

### 2.4. Finite elements choice

The capsule finite elements model is composed of 20-node hexahedrons (C3D20H) and 15-node prisms (C3D15H – wedge elements). Hexahedron elements are the best-performing volumetric elements. Prism are used to make connections and facilitate meshing where hexahedron cannot be used (top and bottom of the capsule). Quadratic elements are used because this gives a better representation of the capsule curved shape. Moreover, this enables a better analysis of the stresses distribution. A hybrid formulation is chosen because alginate is considered as an incompressible material (the volume cannot change
with loading). This avoids potential convergence problems during the simulation. We conducted a mesh convergence study to determine the mesh offering the best compromise between results accuracy and simulation cost. This analysis is based on various criteria: Von mises maximal Stress $\sigma_{11}^{\max}$s, $\sigma_{22}^{\max}$ and $\sigma_{33}^{\max}$. The complete analysis and results of this study are provided in the Supplementary Materials: Sensitivity analysis of the capsule mesh.

## 2.5. Choice of the material law

Alginate is a biocompatible, incompressible and elastic material. To study its material law, we use the following data [22]: Poisson's ratio is equal to 0.5 and elasticity depends on the deformation rate: for deformation below 10 %, alginate has purely linear elastic behaviour and above 80 % deformation, plasticity appears. For deformations between these two thresholds, alginate presents a non-linear stress-strain response and thus a non-linear elastic behaviour (eq. (7)).

We use a hyperelastic law to represent numerically the material behaviour. To use the software Simulia Abaqus, some information is needed to extend the unidirectional law (eq. (7)) in 3D We use the Marlow hyperelastic law [39], which considers that strain energy potential is independent of the second deviatoric invariant. The deviatoric behaviour is defined by $W_{dev}$, which is characterized by test data like uniaxial traction test [40]. The volumetric behaviour is defined by $W_{vol}$ if the material studied is compressible. The strain energy potential [41] is thus defined by:

$$W = W_{dev}(\bar{I}_1) + W_{vol}(J_{el}) \quad (2)$$

Where $\bar{I}_1$ is the first deviatoric invariant and $J_{el}$ is the elastic volume ratio:

$$J_{el} = \frac{J}{(1 + \varepsilon^{th})^{3}} \quad (3)$$

Where $\varepsilon^{th}$ is the linear thermal expansion strain and $J$ is the total volume ratio. In our case, alginate is an incompressible material with a Poisson's ratio equal to 0.5 so there is no volume variation during analysis. The strain energy potential is therefore defined solely by $W_{dev}$:

$$W_{dev} = \int_{0}^{\lambda_T^{-1}} T(\varepsilon) d\varepsilon \quad (4)$$

Where $T(\varepsilon)$ is the nominal uniaxial stress obtained by single uniaxial tension test and $\lambda_T$ the uniaxial stretch. The data table is constructed from Eq. (6). The minimum value of deformation rate is 0.1, because when deformation rate is lower, alginate behaves like a purely elastic material. The maximum value of $\varepsilon$ is 0.8, above plasticity and fracture appear. The value taken for the Young's modulus, $E$, is 94 kPa $\pm$ 28 kPa, which is determined with the osmotic pressure measurement with Dextran on experiments.

## 2.6. Boundary conditions and loads

Numerical boundary conditions are not the same than experimental ones. Experimental capsules rest on the bottom of a calcium bath with flat support linkage. However, this flat support is achieved randomly, with the capsule tip pointing upwards or downwards. At post-confluence times, when capsule sweeps due to pressure exerted by encapsulated cells within it, boundary conditions may change slightly. Swelling can cause a rotation of the capsule and changing the initial flat support into a different flat support. However, these rigid body movements are corrected during the experimental dots tracking with Fiji - ImageJ software. Thus, this has no impact on the final displacements. Numerical boundary conditions need to be set up to take possible experimental configurations in account while maintaining the same boundary conditions over

time to respect the modelling assumptions. To achieve this, the capsule is fully modelled, and the applied boundary conditions allow isostatic embedding. For this, the X-axis translation X axis is blocked, as is Z-axis. The Y-axis translation is blocked on the capsule tip. With these boundary conditions, the capsule can be freely deformed by the internal pressure, without being constrained by these conditions. However, they are very different than those implemented during experiments. To overcome this problem, numerical data are registered using our dedicated python algorithm. Rigid transformations are applied. This has no impact on deformation or on dots positions, since they are only translated or oriented without changing scale. About experimental load, there is no local dots displacement until cells have filled the capsule internal cavity. So, pressure can be modelled by a uniform internal pressure applied at confluence time.

## 2.7. Mass diffusion analysis

To represent the calcium ion diffusion through alginate capsule, a Mass diffusion analysis, on Abaqus, was set up (see section 6.9.1 of [42]). This analysis models the transient diffusion of one material through another one. To use it, it is necessary to use mass diffusion finite elements, so C3D15H become DC3D15 and C3D20H become DC3D20. The number of elements and nodes remains the same as for mechanical analysis. As alginate is 99 % water, characteristics used to simulate calcium ion diffusion through alginate are those of water (Diffusivity = $2.895 \times 10^{-9} m^2 s^{-1}$ at 35 °C) [43]. Calcium concentration in alginate at initial step is equal to zero, except on the external surface of the capsule, where the concentration is equal to one because capsule is immersed in calcium bath. A surface concentration flux is applied on the outer surface, inwards capsule. Mass diffusion analysis gives as result the evolution of calcium concentration in the alginate capsule. This concentration is between 0 (minimal concentration) and 1 (maximal concentration). To obtain a physical concentration profile, it is necessary to choose an increment to be retained for future analysis. The chosen increment should have a profile where the calcium concentration is highest on the outer surface of the capsule over a thickness of 5 to 10 µm. This ensures consistency with previous osmotic pressure tests. The chosen increment is the same for all thick capsules because time spend in the calcium bath is the same for all capsules. Calcium diffusion time is the same for all, but each have a different concentration profile, due to thickness, height and width differences between each capsule. It is then possible to consider this calcium concentration profile as a gradient of mechanical property. Where concentration is highest, the mechanical property is highest and vice versa. As the calcium ions diffusion makes alginate capsules more rigid, it is decided to treat this concentration gradient as a young's modulus gradient.

# 3. Results

Statistical analyses are detailed in the Statistical Methods subsection.

## 3.1. Experimental based finite element model and implementation of an inverse method to monitor volumetric deformation and compute tissue exerted pressure

The CCT [22,23] is a microfluidic technique developed to encapsulate cells and ECM inside a hollow alginate shell. The device consists of a 3D-printed microfluidic chip from which solutions are extruded concentrically [23] (Fig. 1A). The outermost solution is a mix of 2 % alginate with dispersed 1 µm diameter fluorescent microbeads (1 %, corresponding to 4.8 $\times$ 10$^5$ beads/mL, Sicastar, RedF, see Materials & Methods section). The intermediate sorbitol solution (200 mM) prevents alginate reticulation inside the chip. The innermost solution is the cell suspension, composed of fluorescently labelled (LifeAct::eGFP) Caco2 cells and ECM (100,000 cells in 200 µL DMEM and 200 µL CultrexBME,
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Caco2 filled alginate capsules 3D+time imaging and diagram of the inverse method used to compute internal pressure. (A) Schematic of the CCT. View of the microfluidic chip with the three solutions: alginate (AL) and fluorescent beads; intermediate sorbitol solution (IS) and the cell suspension (CS). After extrusion, capsules gel in the calcium bath. (B) Bright Field snapshots of encapsulated Caco2 cells over 12 days. (C) Corresponding fluorescent confocal maximum intensity projections of 3D z-stacks, total projection thickness along the z-axis: 237.6 µm (bead in magenta and tissue labelled with LifeAct::eGFP actin in cyan). (D) Corresponding fluorescent confocal maximum intensity projection of the microbeads at the equatorial plane of encapsulated cells. Total projection thickness along the z-axis: 6.6 µm. (E) Schematic of the inverse method used. The FE model is built by taking dimensions on the experimental point cloud (dots position) and by using the Booth lemniscate's equations. Experimental data are registered on the mesh with an ICP registration. The aim of the inverse method is to find θ by minimizing ε with Nelder Mead algorithm [44].</figcaption></figure>
see Materials & Methods section). The jet breaks in the air and the capsules land in a calcium bath (100 mM), where gelling occurs almost instantaneously.

Alginate is a hydrogel that is porous to nutrients and optically transparent, making the capsules compatible with light microscopy. Deformations induced by cells in the alginate shell can therefore be monitored and quantified, providing insights into the pressure exerted by the encapsulated tissue. Bright-field time-lapse microscopy (Fig. 1B) allows imaging and measuring shell deformation in the focal plane using detection algorithms [22]. However, these analyses are limited to the equatorial plane and only provide a rough description of average deformation, thus missing potential spatial anisotropy.

To overcome this limitation, fluorescent beads were added to the alginate and combined with 3D time-lapse microscopy. This provides 4D datasets of both cell aggregate growth and the resulting deformation of the alginate shell (Fig. 1C).

The aim of this section is to describe our approach to calibrate the parameters of a Finite Element (FE) model based on the behaviour of the capsule subjected to cellular aggregate growth. This involves three key steps: (i) acquiring experimental data, (ii) creating a geometric model to simulate the observed system, and (iii) updating the model parameters through optimisation by comparing experimental and numerical data.

The first step (i) involves tracking the displacements of fluorescent beads embedded in the alginate capsule using a 3D microscopy system and ImageJ software (Fijiyama [45] and Trackmate [46] plugins). This allows for the mapping of the bead displacement field resulting from the pressure exerted by the growing cellular aggregates on the capsule.

To achieve step (ii), we developed an FE model representing the capsule geometry with an outer radius ranging from 100 µm to 300 µm [24]. The initial dimensions (outer/inner heights, outer/inner widths, and capsule bottom thickness) of this model are measured from the initial positions (i.e., before deformation) of the fluorescent beads (Fig. 1D, part I – step 1). These measurements then determine the mathematical parameters of a Booth lemniscate (see definition in Materials & Methods section: Virtual Model Geometry) representing the 2D profile of the capsule (Fig. 1D, part I – step 2). The final FE model is obtained by revolving this 2D profile around the Y-axis and creating the finite elements using a Python algorithm (Fig. 1D, part I – step 3). By imposing a displacement on the nodes of the FE mesh located on the inner wall of the capsule, we can simulate the pressure exerted by the cellular aggregate inside the capsule over time.

The third step (iii) (Fig. 1D part III) can be divided into two phases: writing an error function, representing the discrepancy between the numerical model and the observations, and the process of minimizing this error by updating the model parameters.

The writing of the error function, denoted as $\epsilon$, is based on the normalized comparison of the n positions of the n microbeads. Let $U_j^{exp}$ be the experimental position of bead j and $U_j^{exp} = (U_j^{exp})_{j=1...n}$ be the set of experimental positions of all n beads. Similarly, let $U_j^{num}$ be the numerically simulated position of bead j and $U_j^{num} = (U_j^{num})_{j=1...n}$ be the set of numerically simulated positions of all n beads.

The first phase consists of aligning the observations and the FE model of the capsule in the same 3D reference frame using an ICP algorithm [47] between the mesh nodes and the 3D positions of the beads (Fig. 1D part II). By associating each bead with the centroid of a finite element, we obtain a set of experimental $U_j^{exp}$ and simulated $U_j^{num}$ 3D points in the same 3D reference frame.

The second phase involves introducing a dependency parameter into each cloud $U_j^{num}$ to update the numerical model. As a first approach, we consider the triplet of parameters E (Young's modulus), $\nu$ (Poisson's ratio), and P (internal pressure) as the optimization variable $\theta$. Indeed, numerically, the simulated positions of the microbeads $U_j^{num}$ directly depend on the pressure P applied to the internal wall by the confluent tissue, as well as the Poisson's ratio $\nu$ and the Young's modulus of the

alginate E.

However, optimizing with the parameter $\theta$ containing three quantities (E, $\nu$, and P) is very challenging due to the complexity of the models used. We therefore assume an incompressible material. Next, we then conducted an experimental plan to evaluate the influence of E and P on the variability of the maximum Von Mises stress [48] (quantifying the intensity of stresses experienced by the capsule) on $U_j^{num}$ to verify if it is possible to further reduce the size of the optimization parameter $\theta$.

Simulations were performed with values varying by $\pm 30\%$ around $E = 94\,\text{kPa}$ and $P = 4\,\text{kPa}$, which are the extreme values presented in Table 2. We then carried out four simulations to calculate the numerical positions of the beads $U_j^{num}$, combining the minimum and maximum values of E and P, and then calculated $\sigma_{VM}^{moy}$ the average (in kPa) of the maximum Von Mises stress.

The results (cf. Fig. 2C) indicate that an increase in Young's modulus from 66 kPa to 122 kPa leads to a 13% decrease in $\sigma_{VM}^{moy}$, while a variation in pressure from 2.81 kPa to 5.19 kPa results in a 185% increase. The impact of P on the variability of $U_j^{num}$ is thus approximately 12 times greater than that of E. It therefore appears that it is possible to set the Young's modulus E to a given estimated value without introducing bias into the simulation results, as its influence is negligible compared to that of the pressure.

Through osmotic pressure swelling measurements (see Material & Methods section), the Young's modulus is set to $E = 94\,\text{kPa}$ in this model. The optimization parameter $\theta$ is thus reduced to the pressure P, which constitutes our sole decision variable, as its variation leads to different deformed meshes and, consequently, different point clouds $U_j^{num}(P)$.

The error function $\epsilon(P)$ is then defined as the root mean square, over the n time steps j, of the normalized discrepancies between the experimental $U_j^{exp}$ and simulated $U_j^{num}(P)$ positions:

$$ \epsilon(P) = \sum_{j=0}^{n} \left( \frac{\| U_j^{exp} - U_j^{num}(P) \|}{\| U_j^{exp} \|} \right)^2 \quad (5) $$

Finally, a minimization algorithm applied to $\epsilon(P)$ finally allows obtaining the value of the pressure $P^* = \min_P \epsilon(P)$ for which the numerical model best simulates the actual displacements of the microbeads. The algorithm used is the Nelder-Mead [44] with $P_0 = 0$ as the initial guess

## 3.2. FEM capsule model validation with numerical analysis

To numerically evaluate the robustness of the inverse method, we conducted an analysis to estimate the sensitivity of our approach to measurement noise arising from the tracking of experimental microbeads. For this purpose, we used a reference pressure of $P^{ref} = 5\,\text{kPa}$ (with the Young's modulus still fixed at $E = 94\,\text{kPa}$) and generated, from the initial observed positions of the microbeads, a synthetic cloud of n positions $U_j^{ref} = (U_j^{num}(P^{ref}))_{j=1...n}$. Noise was then added to the coordinates of the cloud $U_j^{ref}$ according to a Gaussian distribution centered at 0 with different variance values $\sigma^2$ ([0.0, 0.5, 1.0, 1.5, 2.0] in $\mu\text{m}$). This resulted in a cloud $U_j^{ref}$ which we substituted for our experimental cloud $U_j^{exp}$ in (1) to obtain a discrepancy function $\epsilon_{\sigma}(P)$. Minimizing this function allowed us to obtain a pressure $P_{\sigma}^{ref}$ and the associated error $\Delta P = |P^{ref} - P_{\sigma}^{ref}|$, quantifying the influence of noise on our inverse method. For each variance value, we generated 100 simulations of the cloud $U_j^{ref}$, calculated the corresponding pressure values $P_{\sigma}^{ref}$, averaged them, and plotted them as a function of $\sigma^2$ in Fig. 2D. The results show that the slope of the trend line is less than 0.01, which is 0.2% of $P_{\sigma}^{ref}$, indicating very low sensitivity to noise. This demonstrates that the proposed inverse method is particularly numerically robust to measurement noise related to the extraction of the 3D positions of the
<figure><img src="image_4.png" /><figcaption>B</figcaption></figure>

<table><caption>C</caption><thead><tr><th>Simulation index</th><th>E [kPa]</th><th>P [kPa]</th></tr></thead><tbody><tr><td>1</td><td>122</td><td>5.19</td></tr><tr><td>2</td><td>122</td><td>2.81</td></tr><tr><td>3</td><td>66</td><td>5.19</td></tr><tr><td>4</td><td>66</td><td>2.81</td></tr></tbody></table>

<table><caption>D</caption><thead><tr><th></th><th>σ<sub>VM</sub> average [kPa]</th><th>Rate of change</th></tr></thead><tbody><tr><td>E 66 kPa</td><td>44.71</td><td rowspan="2">-13%</td></tr><tr><td>E 122 kPa</td><td>38.99</td></tr><tr><td>P 2.81 kPa</td><td>21.76</td><td rowspan="2">185%</td></tr><tr><td>P 5.19 kPa</td><td>61.94</td></tr></tbody></table>

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

**Fig. 2.** Computing numerical analysis reveals the importance of pressure over Young's modulus. (A) Schematic of the inverse method with the numerical analysis step. (B) Values of input parameters used for the design experiment. (C) Variation of Von Mises (VM) stress according to Young's modulus and pressure values for alginate capsule model. (D) Study of the noise sensitivity of the alginate capsule model when pressure varies. Left: violin plot of the distribution of results. On the right: averaged results and the slope of the linear regression, which characterize the cost function sensitivity to noise.

microbeads.

### 3.3. Pressure exerted on the inner wall of alginate shells by cell aggregates

With our model numerically validated, we conducted a study on six different capsules, identified as A to F (see Fig. 3). To standardize the processing scales, we define, for each alginate capsule, a reference time $T_0$ corresponding to the last acquisition moment before confluence, at which the internal pressure is assumed to be zero. The uniaxial material law for alginate, assumed to be isotropic and homogeneous, is as follows:

$$ \left\{ \begin{array}{l} \sigma(\varepsilon) = E\varepsilon + A\varepsilon^2 \\ E_{\text{eff}}(\varepsilon) = E \times (1 + a\varepsilon) \end{array} \right. \quad \text{avec} \quad \left\{ \begin{array}{l} E: \text{Young's modulus} \\ E_{\text{eff}}: \text{effective elastic modulus} \\ A \text{ and } a: \text{material parameters} \\ \varepsilon: \text{strain} \\ \sigma: \text{stress} \end{array} \right. \quad (6) $$

Here $E_{\text{eff}}$ is dependent on the strain $\varepsilon$ in the regime of large deformations. The non-linearity of stress with respect to strain indicates hyperelastic behaviour; accordingly, we model the material law of alginate using the Marlow law [39] (see Material and Methods section). This law is a tabulated law, calculated from pairs of values $(\varepsilon, (\varepsilon))$ obtained using the system of Eq. (6).

The inverse method derived from these assumptions allows us to calculate, in a common reference frame, the evolution of the pressure exerted by the cells on the shell of each of the six alginate capsules. Curve 1 in Fig. 3A shows this evolution for capsule A. The images above illustrate, on the left, the growth of the encapsulated tissue in confocal microscopy and, on the right, the distribution of von Mises stresses estimated by the FE model. This comparison highlights the similarity between the deformation of the FE model and that of the alginate capsule deformed by the cell aggregate. It also reveals that the stress is maximal at the bottom of the capsule, where the thickness is minimal, and zero inside the tip.

As illustrated in Fig. 3B, the FE model also allows for a detailed 3D
<figure><img src="image_6.png" /></figure>

C

<table><thead><tr><th></th><th></th><th>Capsule A</th><th>Capsule B</th><th>Capsule C</th><th>Capsule D</th><th>Capsule E</th><th>Capsule F</th></tr></thead><tbody><tr><th rowspan="3" scope="rowgroup">Small strain</th><th scope="row">Theoretical relation [kPa]</th><td>4.290</td><td>2.905</td><td>2.964</td><td>2.940</td><td>2.085</td><td>-</td></tr><tr><th scope="row">Finite Element model [kPa]</th><td>4.375</td><td>2.813</td><td>2.938</td><td>3</td><td>2.375</td><td>-</td></tr><tr><th scope="row">Rate of change</th><td>1.99%</td><td>3.18%</td><td>0.90%</td><td>2.06%</td><td>13.92%</td><td>-</td></tr></tbody></table>

D

<table><thead><tr><th></th><th></th><th>Capsule A</th><th>Capsule B</th><th>Capsule C</th><th>Capsule D</th><th>Capsule E</th><th>Capsule F</th></tr></thead><tbody><tr><th rowspan="3" scope="rowgroup">Large strain</th><th scope="row">Theoretical relation [kPa]</th><td>10.736</td><td>-</td><td>9.361</td><td>-</td><td>9.605</td><td>10.683</td></tr><tr><th scope="row">Finite Element model [kPa]</th><td>10.117</td><td>-</td><td>11.934</td><td>-</td><td>11.341</td><td>12.031</td></tr><tr><th scope="row">Rate of change</th><td>5.76%</td><td>-</td><td>27.48%</td><td>-</td><td>18.08%</td><td>12.62%</td></tr></tbody></table>

**Fig. 3.** Pressure exerted by cells aggregate on the inner wall of alginate shell. (A) Pressure applied by tissue on the internal surface of the thin capsule over confluence times. The pressure unit is kPa. Day 0 is the reference day of the study, the pre-confluence times. Above the curve, there are couples of images. On the left of each couple, this is the microscopic image of the capsule A with fluorescent dots (magenta) and cellular aggregate (cyan). On the right, this is the representation of the VM stress map in the capsule, over time. (B) Types of results available using FE model: radial displacement field (µm), radial strain map, Von Mises stress map (kPa) and radial stress map (kPa). (C) Comparison table for small strain (ε < 10%) between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different alginate capsules, named from A to F. Pressures are expressed in kPa. (D) Comparison table for large strain (10% < ε < 80%) between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different alginate capsules, named from A to F. Pressures are expressed in kPa.

analysis in the form of maps (radial displacements, radial deformation, VM stress, and radial stress) of the alginate capsule's response to stresses. Tables 3 and 3 compare these results to the theoretical relationship, established for capsules assumed to be spherical, given by:

$$P = \frac{2E}{1 - \nu} \times \frac{h}{R} \times \frac{U(R)}{R} \quad \text{avec} \quad \begin{cases} h: \text{wall thickness} \\ R: \text{average equatorial radius} \\ U(R) \text{ variation in radius} \end{cases} \quad (7)$$

These tables distinguish between small deformations (ε < 10%) in

Fig. 3C and large deformations (10% < ε < 80%) in Fig. 3D. To ensure the consistency of Eq. (7) with the values of ε in the domain of large deformations, the comparison with our model was performed by replacing the Young's modulus E = 94 kPa with the effective modulus E<sub>eff</sub>. In the domain of small deformations, the numerical simulation produces results close to the theoretical formula, confirming the consistency of our model. In the domain of large deformations, the discrepancies with the theoretical values are greater, which can be explained by different geometric assumptions. Indeed, our model computes non-spherical capsules with distinct internal and external tips,
leading to variability in the wall thickness. Given that our inverse method relies on 3D displacements within this wall (whose thickness is no longer constant), the obtained pressure values necessarily differ, explaining the observed discrepancies.

### 3.4. Implementation of the young's modulus variation with a thick alginate shell

Up to this point, our modelling was based on capsules with a thickness of <10 µm. Based on calcium diffusion kinetics reported in the literature [49–51], complete cross-linking is expected for a thickness at the capsule bottom of approximately 10–15 µm, under our encapsulation conditions. We therefore refer to capsules with a bottom-wall thickness below this value as thin (homogeneously cross-linked) and those above this value as thick. We hypothesize that for thicknesses greater than 15 µm, calcium ions diffusion leads to a weaker internal cross-linking, resulting in a gradient of Young's modulus from the outer to the inside. We therefore investigate whether integrating a variable Young's modulus, dependent on calcium ion concentration, into our FE model would better reproduce experimental results.

As a first step, we demonstrate that the assumption of a constant E in the EF model is not valid with respect to experimental data. For this purpose, we will use experimental data on the position of microbeads at the pre-confluence time $T_0$ and at the $n_T$ post-confluence times $T_i$. From the pre-confluence data, we select a portion of the capsule (red partition, Fig. 4A1), divided into $n_c$ of equal thickness and a similar number of points. The number of layers corresponds to the elements in the thickness of the FE model (e.g., 7 layers for 7 elements, Fig. 4A2).

This segmentation allows the separate analysis of the displacements of the different layers (internal and external) and thus provides insights into a possible variation in the Young's modulus. The points of the inner surface are then identified and interpolated to reconstruct this surface (Fig. 4A3). For each layer j, we calculate $d_{Ti,j}^{exp}$, the average distance of its points to this surface at the post-confluence time $T_i$ (Figs. 4A4-5-6).

Initially, the corresponding numerical distances, $d_{Ti,j}^{num}$, are simulated using our FE model, assuming a constant Young's modulus of 94 kPa and applying the average experimental displacements of the inner surface at each time $T_i$. By subtracting, for each layer j and time $T_i$, the experimental and numerical distances, we obtain $\Delta d_{Ti,j} = d_{Ti,j}^{exp} - d_{Ti,j}^{num}$. We assume that this difference is zero at the reference time $T_0$. The results are shown in Fig. 4B, where the objective is to achieve $\Delta d_{Ti,j} \approx 0$ for each post-confluence curve. However, at times $T_2$ and $T_3$, significant deviations of –1 µm to –4 µm are observed, indicating that the FE model does not accurately describe the material law of alginate for thick-walled capsules.

Since the assumption of a constant Young's modulus is no longer valid for capsules with thicknesses greater than 15 µm, we integrated a simulation of the calcium ion concentration gradient in the alginate via a mass diffusion analysis using ABAQUS (see Material and Methods). This simulation allows for a linear variation of the ionic concentration across the capsule thickness as a function of E. We denote this concentration ratio by $\alpha$, varying between 0 (low concentration) and 1 (maximum concentration). This relationship is described by the following equation:

$$E_{\text{thick capsule}} = E_{\text{min}} + (E_{\text{max}} - E_{\text{min}}) \times \alpha \quad (8)$$

Where $E_{min}$ depends on the studied capsule (values given in Table 4) and $E_{max}$ is set to 94 kPa (Material and Methods, 2.2. Experimental determination of alginate Young's modulus in capsule). This modelling approach allows us to directly incorporate the effect of the calcium ion diffusion profile into the estimation of stiffness variations across the capsule thickness.

To adapt the diffusion profile to the considered thicknesses (>15 µm) while ensuring consistency with the modelling of thinner capsules (<10 µm) where the Young's modulus was constant, we halt the numerical

diffusion when the maximum concentration thickness reaches 10 µm. Each capsule then exhibits a distinct concentration profile, dependent on its geometry, despite identical diffusion times. Fig. 4F illustrates these profiles, where the maximum concentration (in red) remains within the defined range. As we assume that $E_{ext}$, the Young's modulus corresponding to this concentration, is 94 kPa, the numerical distances $d_{Ti,j}^{num}$ and thus the deviations $\Delta d_{Ti,j}$, depend solely on $E_m$, the Young's modulus associated with the minimum concentration. To estimate this value for each capsule, we minimize the average deviation between the experimental and numerical distances, considering all $n_T$ post-confluence times and $n_c$ layers:

$$E_{\text{int}} = \min_{n_c} \frac{1}{n_c n_P} \sum_{i=1}^{n_T} \sum_{j=1}^{n_c} |\Delta d_{Ti,j}(E_m) - \Delta d_{T_0,j}(E_m)| \quad (9)$$

The obtained values of $E_{int}$ are presented in Table 4. Their heterogeneity can be explained by the differences in diffusion and geometry specific to each capsule. We then compared the FE model with a constant Young's modulus to the one incorporating its variations by analysing the deviations for all the studied capsules (Table 4). For the six capsules, the divergence is systematically lower with the variable modulus model, reducing the overall deviation by up to 70 % compared to the constant model. In the model of Fig. 4B for $\Delta d_{Ti,j}$, Fig. 4C plots $\Delta d_{Ti,j}(E_{int})$, where the goal is to achieve $\Delta d_{Ti,j}(E_{int}) \approx 0$ for each post-confluence curve. This graphical representation of the results shows a significantly reduced deviation between each post-confluence time and the reference time $T_0$. Unlike Fig. 4B, where the curves deviated from $T_0$, they cluster around this reference here, indicating better agreement with the experimental data. Graphs corresponding to the other five thick capsules (G-K) are presented in the Supplementary Materials: Study of Young's modulus variation across the thickness of thick capsules. For future studies on thick-walled capsules, we will therefore consider the Young's modulus to be directly related to the diffusion gradient.

### 3.5. Numerical analysis to validate the addition of young's modulus variation in the capsule model

With the addition of the Young's modulus variation locally modifying the material law of the alginate capsules, we replicated the analyses from Fig. 2 by applying them to the FE model incorporating this variation. Similar to the sensitivity analysis presented in Tables 2 and 2, simulations were conducted with values varying by ±30% around $E_{ext} = 94$ kPa, $E_{int} = 50$ kPa, and $P = 4$ kPa, corresponding to the extreme values in Table 5.

The results presented in Table 5 quantify the impact of each parameter on the stress. As before, the internal pressure significantly influences the stress more than the internal and external Young's moduli. We thus draw a similar conclusion: for the remainder of the study, we will fix $E_{ext}$ at 94 kPa and use the previously estimated values for $E_{int}$, as their influence is negligible compared to that of the internal pressure.

The reproduction of the robustness study (Fig. 2D) on the inverse method incorporating the Young's modulus variation with respect to errors in the extraction of microbead positions was conducted with a minimum internal Young's modulus set to $E_{int} = 12.188$ kPa (the smallest Young's modulus value obtained previously) and a pressure of $P^{ref} = 5$ kPa. The results, presented in Fig. 5C, are visualized using a violin plot illustrating the distribution of $\Delta P$ for each of the 100 trials performed per $\sigma^2$ value. The linear regression graph allows for direct evaluation of the sensitivity to the inserted noise on the microbead positions. The slope of this regression, equal to 1e-5, represents <1 % of $P^{ref}$, confirming that the algorithm remains insensitive to noise in this configuration.
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

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

<table><thead><tr><th></th><th>Thick<br>capsule G</th><th>Thick<br>capsule H</th><th>Thick<br>capsule I</th><th>Thick<br>capsule J</th><th>Thick<br>capsule K</th><th>Thick<br>capsule L</th></tr></thead><tbody><tr><th>E<sub>cst</sub> deviation [µm]</th><td>0.541</td><td>2.452</td><td>1.014</td><td>0.829</td><td>0.860</td><td>1.004</td></tr><tr><th>E<sub>grad</sub> deviation [µm]</th><td>0.417</td><td>0.895</td><td>0.523</td><td>0.707</td><td>0.279</td><td>0.276</td></tr><tr><th>Rate of change</th><td>-22.85%</td><td>-63.51%</td><td>-48.42%</td><td>-14.71%</td><td>-67.57%</td><td>-72.53%</td></tr></tbody></table>

<table><thead><tr><th></th><th>Thick capsule G</th><th>Thick capsule H</th><th>Thick capsule I</th><th>Thick capsule J</th><th>Thick capsule K</th><th>Thick capsule L</th></tr></thead><tbody><tr><th>E<sub>min</sub> [kPa]</th><td>12.188</td><td>14</td><td>58.75</td><td>54.25</td><td>47.188</td><td>13.25</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Fig. 4.** Importance of implementing a Young's modulus variation when the alginate shell is thick. (A) Schematic of method used to obtain experimental data for the concentration gradient study. 1 – Partition selection on the experimental point cloud. 2- Layers creation on the dots partition. 3 – Inner surface interpolation. 4 – Measurement of the distance between the dots and the inner surface, layer by layer. 5 – Calculation of the average distance from each layer to the inner surface using the distances previously measured. 6 – Overview of average distances. (B) Graph showing the distance difference to the inner surface between Fe model and experimental data with constant Young's modulus for each layer at each time. (C) Graph showing the distance difference to the inner surface between Fe model and experimental data with Young's modulus gradient for each layer at each time for capsule L. (D) For each thick capsule studied, named from G to L, table of deviations between: constant Young's modulus model and experimental data (first line); Young's modulus gradient model and experimental data (second line). Third row shows the rate of decrease involved using a modulus gradient model. (E) Table of minimum internal Young's modulus values for each thick-walled capsule. (F) On the left: schematic of calcium ion diffusion through the alginate capsule. On the right of the legend: Numerical modelling of calcium concentration profiles in different thick-walled capsules. These profiles are linked to Young's modulus. The legend shows the ion concentration within the capsule, ranging from 0 (minimum internal modulus) to 1 (maximum external modulus).</figcaption></figure>
A

<table><thead><tr><th>Simulation index</th><th>E<sub>ext</sub> [kPa]</th><th>E<sub>int</sub> [kPa]</th><th>P [kPa]</th></tr></thead><tbody><tr><td>1</td><td>122</td><td>64.89</td><td>5.19</td></tr><tr><td>2</td><td>122</td><td>64.89</td><td>2.81</td></tr><tr><td>3</td><td>122</td><td>35.11</td><td>5.19</td></tr><tr><td>4</td><td>122</td><td>35.11</td><td>2.81</td></tr><tr><td>5</td><td>66</td><td>64.89</td><td>5.19</td></tr><tr><td>6</td><td>66</td><td>64.89</td><td>2.81</td></tr><tr><td>7</td><td>66</td><td>35.11</td><td>5.19</td></tr><tr><td>8</td><td>66</td><td>35.11</td><td>2.81</td></tr></tbody></table>

B

<table><thead><tr><th></th><th>σ<sub>VM</sub> average [kPa]</th><th>Rate of change</th></tr></thead><tbody><tr><td>E<sub>ext</sub> 66 kPa</td><td>20.05</td><td rowspan="2">-13%</td></tr><tr><td>E<sub>ext</sub> 122 kPa</td><td>16.80</td></tr><tr><td>E<sub>int</sub> 35.11 kPa</td><td>18.28</td><td rowspan="2">2%</td></tr><tr><td>E<sub>int</sub> 64.39 kPa</td><td>18.57</td></tr><tr><td>P<sub>2.81</sub> kPa</td><td>11.67</td><td rowspan="2">116%</td></tr><tr><td>P<sub>5.19</sub> kPa</td><td>25.18</td></tr></tbody></table>

C

<figure><img src="image_11.png" /><figcaption>**Fig. 5.** Numerical analysis on thick-walled capsule model. (A) Values of parameters used for the design experiment of mechanical parameter on results for thick-walled alginate capsule model. (B) Variation of VM stress according to Young's modulus and pressure values for thick-walled alginate capsule model. (C) Noise sensibility of the thick-walled alginate capsule model when pressure varies. Left: violin plot of the results distribution, right: averaged values and linear regression, corresponding to the cost function sensitivity to noise.</figcaption></figure>

### 3.6. Pressure exerted on the inner wall of thick alginate shells by cell aggregates

With our model incorporating the variation of the Young's modulus numerically validated, we extended the study conducted on thin capsules (Fig. 3) to six thick-walled capsules, labelled G to L, with internal moduli corresponding to the values in Table 4. Graph 6A shows the evolution of the pressure exerted by the cells on capsule H. As before, pairs of images positioned above the curve illustrate the growth of the encapsulated tissue at the same time points. To the right of each pair of images, the FE simulations display the Von Mises stresses induced by the cellular push. The results confirm the trends observed for the thin capsules: the stress is maximal at the bottom of the capsule and minimal at the tip. Tables 6 and 6 compare the pressure values between the FE model and the formula for thick-walled capsules [22]:

$$P = \frac{4E(R_{\text{out}}^3 - R_{\text{int}}^3)u(R_{\text{out}})}{3R_{\text{out}}^4} \text{avec} \begin{cases} R_{\text{int}}: \text{internal radius of the capsule} \\ R_{\text{out}}: \text{external radius of the capsule} \\ u(R_{\text{out}}): \text{variation of } R_{\text{out}} \text{ between } T_0 \text{ and } T_i, \end{cases} \quad (10)$$

This relationship has limitations for thick-walled capsules because the assumption of constant tangential stress across the entire thickness is no longer valid. Table 6 focuses on the regime of small deformations, while Table 6 addresses large deformations. Each capsule can transition between these two regimes depending on the post-confluence time. For example, capsule G immediately enters the regime of large deformations at the first post-confluence time. Similarly, capsule H initially

experiences small deformations on the first day after confluence before shifting to the regime of large deformations in the following days. In both regimes, the pressure estimated by the theoretical law differs significantly from that obtained by the FE model, with discrepancies reaching up to 90 %. Specifically, the theoretical law tends to overestimate the pressure compared to the FE model. This divergence is primarily explained by the incorporation of the Young's modulus variation in the numerical model, whereas the theoretical relation assumes a constant modulus.

Tables 1, 7, 8, 9 and 10.

### 3.7. Behavioural differences between thin and thick-walled capsules

We compare the impact of capsule thickness on the pressure exerted by the encapsulated tissue. Graph 7A shows the evolution of the average pressure for six samples of thin and thick capsules, along with their standard deviation (blue and pink areas, respectively). We calculated the average pressure values and standard deviation at each post-confluence time step. The blue area on the curve for thin capsules represents this standard deviation, while the pink area corresponds to the standard deviation for thick-walled capsules. It is immediately apparent that the tissue encapsulated in thin capsules exerts significantly higher pressure on the inner wall than the tissue encapsulated in thick-walled capsules. This provides a biological insight into these tissues, which react differently depending on the capsule thickness.

The analysis reveals a significantly higher pressure in the thin capsules. Fig. 7B illustrates these differences in cellular behaviour: the Von Mises stress is significantly higher in thin capsules than in thick capsules
<figure><img src="image_12.png" /><figcaption>B</figcaption></figure>

<table><caption>C</caption><thead><tr><th></th><th></th><th>Thick capsule G</th><th>Thick capsule H</th><th>Thick capsule I</th><th>Thick capsule J</th><th>Thick capsule K</th><th>Thick capsule L</th></tr></thead><tbody><tr><td rowspan="3">Small strain 1</td><td>Theoretical relation [kPa]</td><td>-</td><td>9.561</td><td>4.097</td><td>4.391</td><td>2.506</td><td>2.416</td></tr><tr><td>Finite Element model [kPa]</td><td>-</td><td>0.781</td><td>1.375</td><td>0.688</td><td>0.281</td><td>0.234</td></tr><tr><td>Rate of change</td><td>-</td><td>91.83%</td><td>66.44%</td><td>84.34%</td><td>88.78%</td><td>90.30%</td></tr><tr><td rowspan="3">Small strain 2</td><td>Theoretical relation [kPa]</td><td>-</td><td>-</td><td>7.238</td><td>9.626</td><td>10.243</td><td>5.146</td></tr><tr><td>Finite Element model [kPa]</td><td>-</td><td>-</td><td>5.019</td><td>2.784</td><td>3.952</td><td>0.809</td></tr><tr><td>Rate of change</td><td>-</td><td>-</td><td>30.66%</td><td>71.07%</td><td>61.42%</td><td>84.29%</td></tr></tbody></table>

<table><thead><tr><th></th><th></th><th>Thick capsule G</th><th>Thick capsule H</th><th>Thick capsule I</th><th>Thick capsule J</th><th>Thick capsule K</th><th>Thick capsule L</th></tr></thead><tbody><tr><td rowspan="3">Large strain 1</td><td>Theoretical relation [kPa]</td><td>17.769</td><td>16.933</td><td>-</td><td>23.669</td><td>-</td><td>15.289</td></tr><tr><td>Finite Element model [kPa]</td><td>2.344</td><td>2.207</td><td>-</td><td>4.629</td><td>-</td><td>3.073</td></tr><tr><td>Rate of change</td><td>86.81%</td><td>86.97%</td><td>-</td><td>80.44%</td><td>-</td><td>79.90%</td></tr><tr><td rowspan="3">Large strain 2</td><td>Theoretical relation [kPa]</td><td>20.073</td><td>21.065</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Finite Element model [kPa]</td><td>3.105</td><td>4.262</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Rate of change</td><td>84.53%</td><td>79.77%</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="3">Large strain 2</td><td>Theoretical relation [kPa]</td><td>25.828</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Finite Element model [kPa]</td><td>5.745</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Rate of change</td><td>77.76%</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

**Fig. 6.** Pressure exerted by cells aggregate on the inner wall of thick-walled capsules. (A) Pressure applied by tissue on the alginate shell of the thick-walled capsule over confluence times. The pressure unit is kPa. Day 0 is the reference day of the study, the pre-confluence times. Above the curve, there are couples of images. On the left of each couple, this is the microscopic image of the capsule A with fluorescent dots (magenta) and cellular aggregate (blue). On the right, this is the representation of the VM stress map in the capsule, over time. (B) Comparison table for small strain between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different thick capsules, named from G to L. Pressures are expressed in kPa. (C) Comparison table for large strain between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different thick capsules, named from G to L. Pressures are expressed in kPa.

for the same post-confluence time. We previously showed that thick capsules exhibit a variation in Young's modulus across their thickness, whereas thin capsules can be assumed to have a constant modulus. We then wanted to compute the thin capsules by applying a similar gradient,

which logically would lead to an almost constant Young's Modulus value considering the very small distance for the calcium ions to diffuse through. Fig. 7C highlights these differences by illustrating the distribution of gradients for an identical diffusion time. The analysis reveals
<table><caption>**Table 1**
Values of input parameters used for the design experiment.</caption><thead><tr><th>Simulation index</th><th>E [kPa]</th><th>P [kPa]</th></tr></thead><tbody><tr><td>1</td><td>122</td><td>5.19</td></tr><tr><td>2</td><td>122</td><td>2.81</td></tr><tr><td>3</td><td>66</td><td>5.19</td></tr><tr><td>4</td><td>66</td><td>2.81</td></tr></tbody></table>

<table><caption>**Table 2**
Variation of Von Mises (VM) stress according to Young's modulus and pressure values for alginate capsule model.</caption><thead><tr><th></th><th>σ<sub>VM</sub> average [kPa]</th><th>Rate of change</th></tr></thead><tbody><tr><td>E 66 kPa</td><td>44.71</td><td>-13 %</td></tr><tr><td>E 122 kPa</td><td>38.99</td><td></td></tr><tr><td>P 2.81 kPa</td><td>21.76</td><td>185 %</td></tr><tr><td>P 5.19 kPa</td><td>61.94</td><td></td></tr></tbody></table>

that this gradient is much less pronounced in thin capsules, For thick capsules, this gradient shows that the maximum diffusion zone (in red) extends approximately 10 µm, comparable to the thickness of the bottom of thin capsules.

## 4. Statistical methods

Experimental data are reported as mean ± standard deviation (SD), with error bars in figures representing SD. For numerical simulations, parameter values (E = 94 kPa, P = 4 kPa) were varied by ±30 % to assess their influence on von Mises stress (Fig. 2C). Noise sensitivity was evaluated through repeated simulations and quantified by linear regression of the cost function slope (Fig. 2D).

Discrepancies between experimental and numerical bead positions were quantified by root mean square error (RMSE; Eq. (5)). Model optimisation was performed using the Nelder-Mead algorithm [44],

<table><caption>**Table 7**
Values of parameters used for the design experiment of mechanical parameter on results for thick-walled alginate capsule model.</caption><thead><tr><th>Simulation index</th><th>E<sub>ext</sub> [kPa]</th><th>E<sub>int</sub> [kPa]</th><th>P [kPa]</th></tr></thead><tbody><tr><td>1</td><td>122</td><td>64.89</td><td>5.19</td></tr><tr><td>2</td><td>122</td><td>64.89</td><td>2.81</td></tr><tr><td>3</td><td>122</td><td>35.11</td><td>5.19</td></tr><tr><td>4</td><td>122</td><td>35.11</td><td>2.81</td></tr><tr><td>5</td><td>66</td><td>64.89</td><td>5.19</td></tr><tr><td>6</td><td>66</td><td>64.89</td><td>2.81</td></tr><tr><td>7</td><td>66</td><td>35.11</td><td>5.19</td></tr><tr><td>8</td><td>66</td><td>35.11</td><td>2.81</td></tr></tbody></table>

<table><caption>**Table 8**
Variation of VM stress according to Young's modulus and pressure values for thick-walled alginate capsule model.</caption><thead><tr><th></th><th>σ<sub>VM</sub> average [kPa]</th><th>Rate of change</th></tr></thead><tbody><tr><td>E<sub>ext</sub> 66 kPa</td><td>20.05</td><td>-13 %</td></tr><tr><td>E<sub>ext</sub> 122 kPa</td><td>16.80</td><td></td></tr><tr><td>E<sub>int</sub> 35.11 kPa</td><td>18.28</td><td>2 %</td></tr><tr><td>E<sub>int</sub> 64.39 kPa</td><td>18.57</td><td></td></tr><tr><td>P 2.81 kPa</td><td>11.67</td><td>116 %</td></tr><tr><td>P 5.19 kPa</td><td>25.178</td><td></td></tr></tbody></table>

with $P_0 = 0$ as the initial condition. Reproducibility was assessed across six independent capsules, and variability between them was used to evaluate robustness of pressure estimates.

## 5. Discussion

The mechanical behaviour of encapsulated tissues is markedly influenced by capsule wall thickness. Our results show that tissues encapsulated in thin-walled capsules exert significantly higher pressure on the inner surface, compared to those in thick-walled capsules. This

<caption>**Table 3**</caption>

Comparison table for small strain ($\epsilon$ < 10%) between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different alginate capsules, named from A to F. Pressures are expressed in kPa.

<table><caption>**Table 4**</caption><thead><tr><th></th><th></th><th>Capsule A</th><th>Capsule B</th><th>Capsule C</th><th>Capsule D</th><th>Capsule E</th><th>Capsule F</th></tr></thead><tbody><tr><td rowspan="3">Small strain</td><td>Theoretical relation [kPa]</td><td>4.290</td><td>2.905</td><td>2.964</td><td>2.940</td><td>2.085</td><td>-</td></tr><tr><td>Finite Element model [kPa]</td><td>4.375</td><td>2.813</td><td>2.938</td><td>3</td><td>2.375</td><td>-</td></tr><tr><td>Rate of change</td><td>1.99 %</td><td>3.18 %</td><td>0.90 %</td><td>2.06 %</td><td>13.92 %</td><td>-</td></tr></tbody></table>

Comparison table for large strain (10% < $\epsilon$ < 80%) between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different alginate capsules, named from A to F. Pressures are expressed in kPa.

<table><caption>**Table 5**</caption><thead><tr><th></th><th></th><th>Capsule A</th><th>Capsule B</th><th>Capsule C</th><th>Capsule D</th><th>Capsule E</th><th>Capsule F</th></tr></thead><tbody><tr><td rowspan="3">Large strain</td><td>Theoretical relation [kPa]</td><td>10.736</td><td>-</td><td>9.361</td><td>-</td><td>9.605</td><td>10.683</td></tr><tr><td>Finite Element model [kPa]</td><td>10.117</td><td>-</td><td>11.934</td><td>-</td><td>11.341</td><td>12.031</td></tr><tr><td>Rate of change</td><td>5.76 %</td><td>-</td><td>27.48 %</td><td>-</td><td>18.08 %</td><td>12.62 %</td></tr></tbody></table>

For each thick capsule studied, named from G to L, table of deviations between: constant Young's modulus model and experimental data (first line); Young's modulus gradient model and experimental data (second line). Third row shows the rate of decrease involved using a modulus gradient model.

<table><caption>**Table 6**</caption><thead><tr><th></th><th>Thick capsule G</th><th>Thick capsule H</th><th>Thick capsule I</th><th>Thick capsule J</th><th>Thick capsule K</th><th>Thick capsule L</th></tr></thead><tbody><tr><td>E<sub>est</sub> deviation [µm]</td><td>0.541</td><td>2.452</td><td>1.014</td><td>0.829</td><td>0.860</td><td>1.004</td></tr><tr><td>E<sub>grad</sub> deviation [µm]</td><td>0.417</td><td>0.895</td><td>0.523</td><td>0.707</td><td>0.279</td><td>0.276</td></tr><tr><td>Rate of change</td><td>-22.85 %</td><td>-63.51 %</td><td>-48.42 %</td><td>-14.71 %</td><td>-67.57 %</td><td>-72.53 %</td></tr></tbody></table>

Table of minimum internal Young's modulus values for each thick-walled capsule.

<table><thead><tr><th></th><th>Thick capsule G</th><th>Thick capsule H</th><th>Thick capsule I</th><th>Thick capsule J</th><th>Thick capsule K</th><th>Thick capsule L</th></tr></thead><tbody><tr><th>E<sub>min</sub> [kPa]</th><td>12.188</td><td>14</td><td>58.75</td><td>54.25</td><td>47.188</td><td>13.25</td></tr></tbody></table>
<caption>**Table 9**</caption>

Comparison table for small strain between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different thick capsules, named from G to L. Pressures are expressed in kPa.

<table><caption>**Table 10**</caption><thead><tr><th></th><th></th><th>Thick capsule G</th><th>Thick capsule H</th><th>Thick capsule I</th><th>Thick capsule J</th><th>Thick capsule K</th><th>Thick capsule L</th></tr></thead><tbody><tr><td rowspan="3">Small strain 1</td><td>Theoretical relation [kPa]</td><td>-</td><td>9.561</td><td>4.097</td><td>4.391</td><td>2.506</td><td>2.416</td></tr><tr><td>Finite Element model [kPa]</td><td>-</td><td>0.781</td><td>1.375</td><td>0.688</td><td>0.281</td><td>0.234</td></tr><tr><td>Rate of change</td><td>-</td><td>91.83 %</td><td>66.44 %</td><td>84.34 %</td><td>88.78 %</td><td>90.30 %</td></tr><tr><td rowspan="3">Small strain 2</td><td>Theoretical relation [kPa]</td><td>-</td><td>-</td><td>7.238</td><td>9.626</td><td>10.243</td><td>5.146</td></tr><tr><td>Finite Element model [kPa]</td><td>-</td><td>-</td><td>5.019</td><td>2.784</td><td>3.952</td><td>0.809</td></tr><tr><td>Rate of change</td><td>-</td><td>-</td><td>30.66 %</td><td>71.07 %</td><td>61.42 %</td><td>84.29 %</td></tr></tbody></table>

Comparison table for large strain between pressure obtained with the theoretical relation and pressure obtained with the FE model. We work on 6 different thick capsules, named from G to L. Pressures are expressed in kPa.

<table><thead><tr><th></th><th></th><th>Thick capsule G</th><th>Thick capsule H</th><th>Thick capsule I</th><th>Thick capsule J</th><th>Thick capsule K</th><th>Thick capsule L</th></tr></thead><tbody><tr><td rowspan="3">Large strain 1</td><td>Theoretical relation [kPa]</td><td>17.769</td><td>16.933</td><td>-</td><td>23.669</td><td>-</td><td>15.289</td></tr><tr><td>Finite Element model [kPa]</td><td>2.344</td><td>2.207</td><td>-</td><td>4.629</td><td>-</td><td>3.073</td></tr><tr><td>Rate of change</td><td>86.81 %</td><td>86.97 %</td><td>-</td><td>80.44 %</td><td>-</td><td>79.90 %</td></tr><tr><td rowspan="3">Large strain 2</td><td>Theoretical relation [kPa]</td><td>20.073</td><td>21.065</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Finite Element model [kPa]</td><td>3.105</td><td>4.262</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Rate of change</td><td>84.53 %</td><td>79.77 %</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="3">Large strain 2</td><td>Theoretical relation [kPa]</td><td>25.828</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Finite Element model [kPa]</td><td>5.745</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Rate of change</td><td>77.76 %</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

difference arises from the distinct elastic properties of the capsules: thin capsules are homogeneously cross-linked and present uniformly high stiffness, whereas thick-walled capsules are exhibiting a gradient in stiffness, with lower rigidity near the capsule center, increasing outwardly.

The distinction between “thin” and “thick” capsules was not defined a priori but emerged from the behaviour of the material and the model. Calcium diffusion drives alginate gelation. Based on published gelation kinetics, a theoretical fully cross-linked thickness of approximately 10–15 µm is expected after 30 s in a 100 mM CaCl₂ bath, corresponding to our encapsulation protocol. Below this threshold, cross-linking is complete and the rigidity is homogeneous. Above this threshold, a diffusion gradient of cross-linking arises. This is consistent with early studies [50] reporting full gelation of 3.3 mm beads after 2 h in 50 mM CaCl₂ bath (approximatively 12.5 µm/min) and later work [51] showing a gelation of 50 µm/min at 500 mM CaCl₂. The numerical model also confirms this interpretation: introducing a rigidity gradient in capsules thinner than 10 µm does not change the predicted mechanical response, but capsule with a greater thickness display a Young’s modulus gradient that significantly changes pressure evolution.

The Von Mises stress distribution support this observation, indicating higher stress levels in thin capsules at equivalent post-confluence time points. This aligns with the assumption that thin capsules, due to their limited thickness, allow for more homogeneous calcium diffusion, resulting in more complete cross-linking and a nearly constant Young’s modulus. In contrast, thick capsules display a mechanical gradient, consistent with the diffusion-limited cross-linking described before. This is evidence in Fig. 7C where the calcium diffusion gradient is more pronounced in thick capsules, and the maximum cross-linked zone (~10 µm) corresponds closely to the total thickness of thin capsules.

The FE model, validated across these conditions, provides reliable estimations of pressure fields and stress distribution in 3D For thin capsules, it agrees well with theoretical values under small deformation, suggesting that the material can be reasonably modelled as isotropic and homogeneous. However, for thick capsules, discrepancies with theoretical models arise when assuming a constant Young’s modulus, which justifies the need to incorporate gradient-based mechanical properties.

Beyond the results presented here, our approach opens interesting perspectives regarding pathological behaviours associated with tumour compression. It has been shown that when spheroids grow up to being

compressed by the alginate capsule, they exhibit a specific feature of cancer early metastasis that is the translation of the “grow or go” principle. Specifically, after compression and further release of colon cancer spheroids from the alginate capsule, surface cells escape from the aggregate and spread on the substrate [22]. Similarly, another study demonstrated that compression triggers cancer-associated fibroblasts to spread and engulf the tumour spheroid [52]. These observations confirm that the CCT and the quantitative model we propose represents a relevant model for investigating early mechanisms of metastasis upon mechanical compression.

Finally these finding emphasize the critical role of the mechanical environment in modulating tissue behaviour, a concept of relevance not only for encapsulated tissue studies but also in broader biological systems. For example, in developmental biology, embryos in many species develop within protective shells like the zona pellucida (mammals), chorion (zebrafish), or vitelline membrane (Xenopus) [53–55]. Modeling such systems requires capturing the complex interaction between growing tissues and their mechanical confinement. While particle-based models [56,57] offer biologically relevant cell-scale behaviour, they remain computationally intensive and less adapted for simulating continuous materials like alginate. The finite element method (FEM), in contrast, offers a mechanically rigorous and computationally efficient framework suited for such materials.

## 6. Conclusion and outlook

In this work, we developed a finite element model of encapsulated tissue mechanics validated using experimental data and inverse method. The model accurately captures the interaction between cells and their surrounding capsule, accounting for pressure, stress, and material nonlinearities. For thick capsules, the inclusion of mechanical gradients was essential to reproduce observed behaviours, highlighting the limitations of classical homogeneous models such as those in [22]. Our results confirm that capsule thickness and rigidity critically influence tissue pressure and deformation. While our study focused and idealised symmetrical capsules, future work could incorporate geometrical irregularities such as hernias or non-uniform thickness. Finally, this model provides a basis for future integration with cellular numerical models [36] and could be extended to other biological systems requiring accurate simulation of mechanical confinement, such as embryonic
<figure><img src="image_13.png" /><figcaption>**Fig. 7.** Differences between thin capsules and thick capsules. (A) Averaged pressure evolution over post-confluence times for thin and thick-walled capsules. The blue surface represents the standard deviation of thin capsules (6 capsules) pressures. The rose surface represents the standard deviation of thick-walled capsules (6 capsules) pressures. (B) Differences in Von Mises stress between thin capsules (left) and thick-walled capsules (right). (C) Differences in calcium ion concentration gradient between thin capsules (left) and thick capsules (right).</figcaption></figure>

development [53–55], thereby contributing to the design of numerical twins in biology [56,57].

Funding

This collaborative work was funded by ANR Molière ANR-21-CE45-0028 (agreement grant number R21171BB).

CRediT authorship contribution statement

**Angélique Ala:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Camille Douillet:** Writing – review & editing, Writing – original draft,

Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Audrey Ferrand:** Writing – review & editing, Writing – original draft, Validation, Project administration, Investigation, Funding acquisition, Conceptualization. **Vincent Velay:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Stéphane Segonds:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Methodology, Investigation, Formal analysis, Conceptualization. **Gaëlle Recher:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Florian Bugarin:** Writing – review & editing, Writing – original draft, Visualization, Validation,
Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, 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

We wish to thank Julien Laussu, Deborah Michel, Léa Magne, Dimitri Hamel, Pierre Nassoy and Adeline Boyreau for critical discussions and suggestions on this work. The microscopy was done in the Imaging Center a service unit of the CNRS-INSERM and Bordeaux University, member of the national infrastructure France BioImaging supported by the French National Research Agency (ANR-10-INBS-04). We also thank Plateforme Voxel for technical assistance.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.actbio.2025.12.013.

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