# 3D-Shell Electromechanical Modeling of the Left Atrium

Oscar Ruz, Marina Vidrascu, Dominique Chapelle, Miguel Angel Fernández

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Oscar Ruz, Marina Vidrascu, Dominique Chapelle, Miguel Angel Fernández. 3D-Shell Electromechanical Modeling of the Left Atrium. Functional Imaging and Modeling of the Heart, Jun 2025, Dallas (TX), United States. Lecture Notes in Computer Science, 15672, Springer, 2025, 10.1007/978-3-031-94559-5_5. hal-05044627

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<figure><img src="image_1.png" /></figure>
# 3D-Shell Electromechanical Modeling of the Left Atrium

Oscar Ruz¹, Marina Vidrascu¹, Dominique Chapelle²[0000-0003-0794-412X], and
Miguel A. Fernández¹[0000-0003-4694-9118]

¹ Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, Inria, Paris, France

² Inria & LMS, École Polytechnique, CNRS, Institut Polytechnique de Paris, France
{oscar.ruz, marina.vidrascu, dominique.chapelle, miguel.fernandez}@inria.fr

**Abstract.** The thin-walled nature of the atrial wall can lead to numerical locking issues when using 3D models discretized with standard finite elements. In order to circumvent these issues, we introduce a comprehensive electromechanical model of the left atrium based on a 3D-shell formulation. The model integrates both the passive and active components of the atrial tissue, while blood flow and the mitral valve dynamics are described in a lumped parameter fashion. The resulting model is discretized with a finite element approach specifically designed to mitigate numerical locking. The effectiveness of the proposed approach is evaluated by comparing the numerical results with biomarkers reported in the literature, in the case of both healthy and pathological conditions.

**Keywords:** Cardiac mechanics · Left atrium · 3D-shell · Finite element method.

## 1 Introduction

The electromechanical simulation of the heart function has gained considerable attention over the last two decades, as it offers valuable insights into its biomechanical behavior, notably by providing access to mechanical quantities that cannot be measured through non-invasive methods. These simulations can also be used to investigate the response of the cardiac functions to different clinical options and can hence play a key role in the development of new therapies and medical devices (see [29, 33]).

Given the intricate contraction mechanism and the inherent geometrical complexity of the atria—stemming from their thin-walled structure and highly heterogeneous fiber architecture, both on the surface and through their thickness (see, e.g., [25, 36])—only a limited number of studies have explored their electromechanics. To the best of our knowledge, the only studies that comprehensively investigate the full electromechanics of the atrium, including both passive and contractile responses, are [26, 35, 2, 34, 18, 15]. Several of these studies have shown good agreement with key physiological indicators, such as the characteristic figure-eight shape of the pressure-volume loop. However, despite the valuable
insights they provide, none directly addresses the numerical difficulties associated with the thin-walled nature of the atrial wall.

Indeed, standard finite element approximations of 3D displacement-based solid mechanics models are known to suffer from severe robustness issues (artificial stiffening) whenever the thickness is small with respect to the other dimensions, the so-called numerical locking phenomenon. A widespread approach in solid mechanics to circumvent these issues, is to consider a reduced modeling approach based on shell theory in combination with an appropriate treatment of numerical locking (see, e.g., [5, 7, 9] and the reference therein). Particulary, 3D-shell elements with quadratic shape functions across the thickness are very appealing in practice, as they provide a good representation of the transverse strains, enable the use of general constitutive laws and facilitate the coupling with the surrounding media.

In this paper, we introduce an electromechanical model of the left atrium based on a 3D-shell formulation. In short, the proposed modeling approach builds on the 3D-shell model proposed in [7], and is suitable for the case of the left atrium, which it is characterized by its thin-walled structure. Additionally, we incorporate a Mixed Interpolation of Tensorial Components (MITC) reinterpolation technique, which aims to mitigate potential locking, a common issue in thin-walled structures like the atrium (see. e.g., [3]). We explore the capabilities of the proposed approach in a series of numerical examples under both physiological and pathological conditions.

The paper is organized as follows. In Section 2 we introduce the geometrical setting and the complete electromechanical model of the left atrium. Section 3 is devoted to the description and discussion of the numerical simulations under physiological and pathological conditions. Finally, a summary of the main conclusions is given in Section 5.

# 2 Electromechanical model of the atria

## 2.1 Geometrical setting and curvilinear coordinates

Let $\Omega \subset \mathbb{R}^3$ be the reference configuration of the left atrial wall (see Figure 1(left)). Its boundary $\partial\Omega$ is assumed to be partitioned as $\partial\Omega = \Gamma_{pv} \cup \Gamma_{mv} \cup \Gamma_{endo} \cup \Gamma_{epi}$, where $\Gamma_{pv}$ corresponds to the pulmonary veins, $\Gamma_{mv}$ to the mitral valve and $\Gamma_{endo}$, $\Gamma_{epi}$ respectively denote the endocardium and epicardium of the left atrium. The mid-surface $S$ of $\Omega$ is assumed to be parametrized by a smooth enough mapping $\phi: \omega \subset \mathbb{R}^2 \to \mathbb{R}^3$, as $S = \phi(\omega)$, where $\omega$ denotes a reference domain in $\mathbb{R}^2$. We can hence define a couple of vectors

$$ \mathbf{a}_m(\xi^1, \xi^2) \stackrel{\text{def}}{=} \frac{\partial \phi}{\partial \xi^m}(\xi^1, \xi^2), \quad m = 1, 2, \quad \forall (\xi^1, \xi^2) \in \omega, $$

tangent to the coordinate curves. The vectors $\mathbf{a}_1$ and $\mathbf{a}_2$ are linearly independent at all points, forming a basis of the local plane tangent to $S$. The unit vector
normal to the tangential plane at any point in the mid-sruface S is given by

$$ \mathbf{a}_3 \stackrel{\text{def}}{=} \frac{\mathbf{a}_1 \times \mathbf{a}_2}{\|\mathbf{a}_1 \times \mathbf{a}_2\|} $$

The reference configuration of the atrial solid domain $\Omega$ is assumed to be given as the image of the mapping

$$ \mathbf{\Phi}: \omega \times (-1, 1) \to \mathbb{R}^3, \quad \Omega = \mathbf{\Phi}(\omega \times (-1, 1)), $$

defined as

$$ \mathbf{\Phi}(\xi^1, \xi^2, \xi^3) \stackrel{\text{def}}{=} \phi(\xi^1, \xi^2) + \xi^3 \frac{\epsilon(\xi^1, \xi^2)}{2} \mathbf{a}_3(\xi^1, \xi^2). \quad (1) $$

Here, the symbol $\xi^3$ corresponds to the curvilinear coordinate perpendicular to the mid-surface, and the function $\epsilon: \omega \to \mathbb{R}^+$ represents the local thickness of the atrial wall.

## 2.2 Electromechanical model

The complete mechanical model of the left atrium is derived from [8], from which now we summarize the main details. The passive behavior of the left atrium is represented by a combination of hyperelastic and viscous contributions. The cardiac tissue of the left atrium is treated as nearly incompressible and transversely isotropic, with a preferred direction along the fibers. The hyperelastic potential is given by the Demiray-Gasser energy density function (see [14, 16]):

$$ W_e \stackrel{\text{def}}{=} \frac{a_p}{2b_p} (e^{b_p(J_1-3)} - 1) + \frac{a_f}{2b_f} (e^{b_f(J_4-1)^2} - 1) + \kappa \log^2 J, $$

where $a_p, b_p, a_f, b_f$ and $\kappa$ are constitutive parameters that need to be calibrated to retrieve realistic physiological landmarks. Additionally, $J_1 \stackrel{\text{def}}{=} \text{tr}\mathbf{C}J^{-\frac{2}{3}}, J_4 \stackrel{\text{def}}{=} \mathbf{a}^T(\mathbf{C}J^{-\frac{2}{3}})\mathbf{a}$, are the reduced invariants of the right Cauchy-Green strain tensor $\mathbf{C} \stackrel{\text{def}}{=} \mathbf{F}^T\mathbf{F}$, while $J \stackrel{\text{def}}{=} \det\mathbf{F}$, with $\mathbf{F}$ denoting the deformation gradient tensor. The viscous effects are modeled using the following dissipation pseudo-potential,

$$ W_v = \frac{\eta}{2} \text{tr}(\dot{\mathbf{E}})^2, \quad \dot{\mathbf{E}} \stackrel{\text{def}}{=} \partial_t \mathbf{E}, $$

where $\eta$ is the viscosity coefficient and $\mathbf{E} \stackrel{\text{def}}{=} \frac{1}{2}(\mathbf{C} - \mathbf{I})$ denotes the Green-Lagrange strain tensor. The contractile behavior is described by a chemically-regulated constitutive law of the cardiac myofiber mechanics, as proposed by [8, 23]. In this law, the fiber extension, $e_c$, is related to the active tension, $\tau_c$, and active stiffness, $k_c$, through the following set of equations:

$$ \begin{cases} \dot{k}_c = -(|u_e| + \alpha_d |\dot{e}_c|) k_c + n_0 k_0 |u_e|_+, \\ \dot{\tau}_c = -(|u_e| + \alpha_d |\dot{e}_c|) \tau_c + \dot{e}_c k_c + n_0 \sigma_0 |u_e|_+, \end{cases} $$
where $k_0$ is the maximum stiffness, while $\sigma_0$ accounts for the solid contractibility. The term $\alpha_d |\dot{e}_c|$ accounts for the bridges destruction upon rapid length changes. The parameter $0 \le n_0 \le 1$ is a reduction factor that accounts for the Frank-Starling effect, which describes the relation of the maximum active tension with the current strain of the sarcomeres. Finally, $|x|_+ \stackrel{\text{def}}{=} \max\{x, 0\}$ and $u_e$ denotes the chemical activation (or electrical input), which is here assumed to be given phenomenologically as an affine function of the transmembrane pontetial $V_m$, namely, $u_e = \alpha v_m + \beta$ in $\Omega$, where $\alpha$ and $\beta$ are given constants (see, e.g., [10]).

The passive and active behaviors are coupled through a nonlinear extension of the Hill-Maxwell rheological framework proposed in [8, 37], so that the second Piola-Kirchhoff stress tensor of the cardiac tissue $\Sigma$ is given by the constitutive relation

$$ \Sigma \stackrel{\text{def}}{=} \frac{\partial W_e}{\partial \boldsymbol{E}} + \frac{\partial W_v}{\partial \dot{\boldsymbol{E}}} + \frac{E_s(a^\mathrm{T}\boldsymbol{E}a - e_c)}{(1 + 2e_c)^2} \boldsymbol{a} \otimes \boldsymbol{a} \quad \text{in } \Omega, $$

where $\boldsymbol{a}$ is an unit vector parallel to the local fibre direction, and $E_s$ represents the Young modulus of the elastic component of the sarcomeres.

Standard finite element approximations of full 3D models of the left atrium face robustness issues due to the thin-walled nature of $\Omega$. To address this, we propose a reduced modeling approach for the atrial electromechanics based on a 3D shell formulation, which assumes a quadratic displacement in the transverse coordinate (see [7]). Therefore, for $t > 0$, the complete reduced atrial electromechanical model reads as follows: find the displacement $\boldsymbol{y}(t) \in \boldsymbol{V}^{\text{sh}}$, the velocity $\boldsymbol{u}(t) \in \boldsymbol{V}^{\text{sh}}$ with $\boldsymbol{u} = \partial_t \boldsymbol{y}$, the extension along the fibers $e_c : \Omega \times \mathbb{R}^+ \to \mathbb{R}$, the active stiffness $k_c : \Omega \times \mathbb{R}^+ \to \mathbb{R}^+$, the active tension $\tau_c : \Omega \times \mathbb{R}^+ \to \mathbb{R}$ and the internal pressure $P_{\text{at}} : \mathbb{R}^+ \to \mathbb{R}$, such that

$$ \left\{
\begin{aligned}
& \rho \int_{\Omega} \partial_t \boldsymbol{u} \cdot \boldsymbol{v} + \int_{\Omega} \boldsymbol{\Sigma} : D_{\boldsymbol{y}} \boldsymbol{E} \cdot \boldsymbol{v} + \int_{\Gamma_{\text{endo}}} J P_{\text{at}} \boldsymbol{F}^{-\mathrm{T}} \boldsymbol{n} \cdot \boldsymbol{v} = 0, \\
& \mu_c \dot{e}_c + \tau_c = E_s \frac{(\boldsymbol{a}^\mathrm{T} \boldsymbol{E} \boldsymbol{a} - e_c)(1 + 2\boldsymbol{a}^\mathrm{T} \boldsymbol{E} \boldsymbol{a})}{(1 + 2e_c)^3} \quad \text{in } \Omega \\
& \dot{k}_c = -(|u_e| + \alpha_d |\dot{e}_c|) k_c + n_0 k_0 |u_e|_+ \quad \text{in } \Omega, \\
& \dot{\tau}_c = -(|u_e| + \alpha_d |\dot{e}_c|) \tau_c + \dot{e}_c k_c + n_0 \sigma_0 |u_e|_+ \quad \text{in } \Omega, \\
& \int_{\Gamma_{\text{endo}}} J \boldsymbol{u} \cdot \boldsymbol{F}^{-\mathrm{T}} \boldsymbol{n} = Q_{\text{la}}(P_{\text{vs}}, P_{\text{l}}, P_{\text{at}}),
\end{aligned}
\right.
\quad (2) $$

for all $\boldsymbol{v} \in \boldsymbol{V}^{\text{sh}}$, with the 3D-shell reduced space given by

$$ \boldsymbol{V}^{\text{sh}} \stackrel{\text{def}}{=} \{ \boldsymbol{v} \in \boldsymbol{V} : \boldsymbol{v} = \boldsymbol{v}_0 + \boldsymbol{v}_1 \xi^3 + \boldsymbol{v}_2 (\xi^3)^2, \quad \boldsymbol{v}_i \in [H^1(\boldsymbol{S})]^3, \quad i = 0, 1, 2 \}, $$

where $\boldsymbol{V} \stackrel{\text{def}}{=} [H_{\Gamma_{\text{mv}}}^1(\Omega)]^3$ stands for the full space of admissible displacements and $\xi^3$ corresponds to the transverse coordinate defined in (1) through the geometrical mapping $\Phi$. The vector fields $\boldsymbol{v}_i$ for $i = 0, 1, 2$, are defined only in the mid-surface $\boldsymbol{S}$ and, hence, they are only functions of the tangential coordinates $(\xi^1, \xi^2)$. The internal pressure in the left atrium, denoted as $P_{\text{at}}$, is described by
a 0D lumped parameter model. In (2), the net blood flow in the left atrium $Q_{la}$ is given by the following relation

$$Q_{la}(P_{vs}, P_l, P_{at}) \stackrel{\text{def}}{=} \begin{cases} K_{vs}(P_{vs} - P_{at}) + K_{mv}(P_{at} - P_l) & \text{if } P_{at} \ge P_v, \\ K_{vs}(P_{vs} - P_{at}) & \text{otherwise,} \end{cases}$$

where $K_{vs}$ and $K_{mv}$ are the conductances for the flux entering from the veins and exiting to the left ventricle, respectively, while $P_{vs}$ and $P_l$, represent given pressures in the pulmonary veins and left ventricle, respectively.

## 2.3 Numerical approximation

The fully discrete method proposed for the approximation of (2) combines the energy preserving time-marching of [8] with a finite element spatial approximation based on prismatic 3D-shell elements and MITC re-interpolation (see [7, 27]). More precisely, the electromechanical problem (2) is discretized in time using a modified mid-point scheme. In this discretization, the momentum equation along with the 0D valve model are discretized using a mid-point scheme, while the active variables are discretized in time using a first-order backward Euler scheme, in order to preserve the energy of the system (see [8]). The reference domain of the left atrium $\Omega$ is approximated by a mesh $\mathcal{T}_h$ made of triangular prismatic elements, generated from the following data: (i) An approximation of the mid-surface $\mathcal{S}$ given in terms of a triangulation $\mathcal{T}_h^S$ made of linear triangles; (ii) An approximation of the unit normal vectors to $\mathcal{S}$ at each node of the triangulation $\mathcal{T}_h^S$; (iii) An approximation of the thickness of $\Omega$ associated to each node of the triangulation $\mathcal{T}_h^S$.

Let $\hat{K} \stackrel{\text{def}}{=} \hat{T} \times (-1, 1)$ be the reference unit prism, with $\hat{T}$ denoting the reference unit triangle in $\mathbb{R}^2$. Each element $K$ of the computational mesh $\mathcal{T}_h$ is defined as the image $K = \hat{\Phi}_{h,K}(\hat{K})$ of the local mapping $\hat{\Phi}_{h,K}: \hat{K} \to K \subset \mathbb{R}^3$ given by the relation

$$\hat{\Phi}_{h,K}(r, s, z) \stackrel{\text{def}}{=} \sum_{i=1}^{3} \lambda_i(r, s) \left( \mathbf{x}_T^{(i)} + z \frac{\epsilon_K^{(i)}}{2} \mathbf{a}_{3,T}^{(i)} \right) \quad \forall (r, s, z) \in \hat{K}, \quad (3)$$

where $\{\lambda_i\}_{i=1}^3$ stand for the 2D affine shape functions of the reference element $\hat{T}$, $\{\mathbf{x}_T^{(i)}\}_{i=1}^3$ denote the positions of the nodes of the mid-surface triangle $T \in \mathcal{T}_h^S$ associated to the prismatic element $K \in \mathcal{T}_h$ and $\{\mathbf{a}_{3,T}^{(i)}\}_{i=1}^3$, $\{\epsilon_T^{(i)}\}_{i=1}^3$ respectively represent the normal vectors to the mid-surface and the thickness associated to each node of the triangle $T$. For the spatial approximation of (2) we consider the following 3D-shell finite element approximation of the space $\mathbf{V}^{\text{sh}}$ (see [7]):

$$\mathbf{V}_h^{\text{sh}} \stackrel{\text{def}}{=} \left\{ \mathbf{v}_h \in \mathbf{V} : \mathbf{v}_h \circ \hat{\Phi}_{h,K} = \sum_{i=1}^{3} \lambda_i(r, s) \left( \mathbf{v}_0^{(i)} + \mathbf{v}_1^{(i)} z + \mathbf{v}_2^{(i)} z^2 \right) \right. \\ \left. \forall K \in \mathcal{T}_h, \mathbf{v}_j^{(i)} \in \mathbb{R}^3, \quad i = 1, 2, 3, \quad j = 0, 1, 2 \right\},$$
which corresponds to the tensorization of a $\mathbb{P}_1$ approximation in the tangent coordinates $(r, s)$ and a $\mathbb{P}_2$ approximation along the transverse coordinate $z$. This allows to build a 3D mesh based on a triangular surface mesh that captures the anatomical details. Furthermore, the tangential $\mathbb{P}_1$ approximation also facilitates the mechanical coupling with low-order approximations of blood flow in the left atrial cavity. Additionally, the MITC technique is considered to mitigate potential locking issues. Specifically, we consider the MITC3 approach [27] (see also [21, 22]) within each triangular layer of the prismatic element, where the in-layer strains are evaluated at the quadrature points using an interpolation of the values at the so-called tying points, while leaving the in-plane strains unmodified. Finally, the dynamics of the internal variables $(e_c, k_c, \tau_c)$ are spatially discretized using a collocation scheme at each quadrature point. These variables can be eliminated at the element level through a Schur complement procedure (see [37]).

# 3 Numerical experiments

In this section, we present the numerical results of the electromechanical simulation of the left atrium under physiological and pathological conditions.

## 3.1 Anatomical model, computational mesh and problem setting

The geometry of the left atrium is derived from the Zygote<fnref n="3" /> heart model, from which we extracted and refined the left atrium. Specifically, we smoothed the left appendage of the endocardial and epicardial surfaces in regions with sharp geometric variations, as well as the fold between the left appendage and the pulmonary vein, to prevent the generation of sharp elements. These geometric treatments were performed using the 3-Matic<fnref n="4" /> software. The computational meshes are made of one-layer prismatic elements, which were generated by extruding the surface of the epicardium in its normal direction toward the interior. The variation of the thickness over the atrium is considered by calculating the distance between the initial epicardium and endocardium surfaces, computed using the closest point projection method [4]. Moreover, the left atrium is assumed to be initially preloaded with a pressure of $P_{at} = 550$ Pa, corresponding to the beginning of its active contraction phase. To retrieve the stress-free configuration, we solve an inverse problem based on the static version of (2) using a simple fixed-point algorithm [17, 30]. The resulting mesh, which comprises 66,786 vertices and 43,957 prisms, is shown in Figure 1(middle).

The fibers in the left atrium (see Figure 1(right)) are synthetically generated using an adaptation of the Laplace-Dirichlet Rule-Based method proposed in [36]. This approach is applied separately to the endocardial and epicardial surfaces, while the fiber orientations across the atrial wall thickness are computed using the expression proposed in [6].

<footnote n="3">³ https://go.zygote.com</footnote>

<footnote n="4">⁴ https://www materialise.com</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1: Left atrial domain (left), computational mesh (middle) and fiber directions (right).</figcaption></figure>

In this work, an electrophysiology simulation of the left atrium is used as input to the mechanical model through the electrical input $u_e$, without incorporating mechano-electrical feedback. Specifically, the electrical input is calculated in the reference configuration using the bidomain model [32] and the Courtemanche ionic model [13], which is well-suited for describing the electrical activity of the atrium. The electrical propagation is initiated near the Bachmann bundle. The boundary conditions are defined by fixing the system at the annuli of the pulmonary veins and the mitral valve. The external pressures $P_{vs}$ and $P_1$ are inferred from the numerical solution of a bi-ventricular electromechanical model. Both the physiological and pathological cases share the same physical properties, as detailed in Section A of the appendix. Each simulation comprises two complete cardiac cycles, each lasting $t = 0.85$ s, using a fixed time step of $\tau = 10^{-3}$ s. All the numerical computations have been performed with the FELiScE<fnref n="5" /> finite element library.

## 3.2 Physiological case

For illustration purposes, Figure 2 presents some snapshots of the simulated electrical activation and mechanical deformed configurations of the left atrium during the cardiac cycle.

Figure 3 presents (in solid lines) the time histories of both the atrial volume and pressure. The obtained pressure evolution remains within physiological ranges, reaching a minimum pressure of around $P_{at}^{min} = 1$mmHg at the end of the relaxation phase and a maximum pressure of $P_{at}^{max} = 10$mmHg at the end of the filling phase in the left atrium. The evolution of the volume, on the other hand, replicates the expected behavior under physiological conditions, showing a sustained decrease during contraction, followed by an increase during the filling phase, and a subsequent decrease after the mitral valve opens [24, 19]. Additionally, the pressure-volume loop (also presented in Figure 3) exhibits the characteristic eight-shape typically observed in the left atrium, with a larger

<footnote n="5">5 https://gitlab.inria.fr/felisce/felisce</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 2: Snapshots of the transmembrane potential (in color), driving the electrical activation, in the deformed configuration of the left atrium. Passive ejection (left), active contraction (middle), left atrium filling (right).</figcaption></figure>

stroke volume during the A-loop (atrial contraction phase) compared to the V-loop (ventricular filling phase). The obtained ejection fraction of the left atrium (LAEF) is 35%, to be compared with a commonly reported range of [45%-60%] (see, e.g., [38, 20, 28]).

<figure><img src="image_4.png" /><figcaption>Fig. 3: Biophysical landmarks left atrium. Pressures (left), volumes (middle), pressure-volume loops (right).</figcaption></figure>

## 3.3 Pathological case

The pathological condition is induced by applying a second electrical stimulus near the right pulmonary veins at 205 ms, during a phase when the atrium is not fully depolarized. This triggers a reentrant wave, while all other aspects of the problem remain unchanged from the physiological case.

Figure 3 shows (in dashed lines) the mechanical indicators of the left atrium under the pathological condition of atrial fibrillation. As expected, the atrium
loses its ability to contract effectively, leading to the absence of the atrial kick and
a subsequent shift in the onset of filling in the left atrium. Due to the irregular
electrical activity, the volume does not exhibit significant changes during the
hypothetical atrial kick phase, unlike in the physiological case where the volume
changes considerably. These changes in pressure and volume are reflected in the
pressure-volume loop, where the absence of the A-loop—indicative of a proper
atrial contraction—is clearly noticeable, which is consistent with the specialized
literature (see, e.g., [12, 31, 1, 28]).

## 4 Limitations

Among the limitations of the present work, it is worth noting that volumetric locking (resulting from material incompressibility) is not considered in this paper (atrial tissue is treated as a quasi-incompressible). Furthermore, the incorporation of incompressibility in shells models is known to be a delicate problem (see, e.g., [11]). Additionally, though numerous studies have demonstrated the advantages of shell elements over full 3D approximations for the simulation of thin-walled structures (see, e.g., [5, 3, 27]), the comprehensive numerical comparison between the proposed approach with standard 3D approaches will be the topic of future work.

## 5 Conclusions

We have introduced a comprehensive electromechanical model for the left atrium incorporating a 3D-shell formulation with an MITC treatment of numerical locking phenomena. The numerical simulations showed that the proposed electromechanical model can reasonably reproduce key physiological biomarkers, including the characteristic eight-shape of the pressure-volume loop and a mechanical response consistent with the behavior of the left atrium as reported in the literature. The simulations of the left atrium under pathological conditions revealed a loss of its contractile capacity, which is consistent with existing literature and further highlights the robustness of the proposed electromechanical model. A natural extension of this work is the incorporation of a fluid-structure interaction model of the left atrium, which may provide a more realistic representation of its mechanical behavior, and allow for a detailed analysis of the hemodynamics within the atrial cavity.

**Acknowledgments.** The authors wish to thank Philippe Moireau (École Polytechnique) for the valuable discussions on cardiac modeling. OR was financially supported by the Vicerrectoria de Investigación, Desarrollo e Innovación (VRIDEI) of the Universidad de Santiago de Chile, and the National Agency for Research and Development (ANID) Doctorado Becas Chile/2020 - 72210101 of the Government of Chile. MF and MV were partially supported by the French National Research Agency (ANR) through the SIMR project (ANR-19-CE45-0020).
**Disclosure of Interests.** The authors have no competing interests to declare that are relevant to the content of this article.

A Bio-physical parameters of the simulations reported in Section 3

Tables 1 provide the values of the mechanical parameters while Table 2 reports the values for the electrophysiology parameters (physiological and pathological cases).

<table><caption>Table 1: Parameters of the passive behavior, active behavior and reduced valve model (units in m³ Pa⁻¹ s⁻¹).</caption><tr><td>a<sub>p</sub> (Pa)</td><td>6 · 10<sup>3</sup></td><td>α<sub>d</sub></td><td>3.0</td><td></td></tr><tr><td>b<sub>p</sub> (Pa)</td><td>5.5</td><td>σ<sub>0</sub> (Pa)</td><td>8.5 · 10<sup>5</sup></td><td>K<sub>mv</sub> | 9 · 10<sup>-7</sup></td></tr><tr><td>a<sub>f</sub> (Pa)</td><td>12 · 10<sup>3</sup></td><td>E<sub>s</sub>(Pa)</td><td>3 · 10<sup>5</sup></td><td>K<sub>vs</sub> | 1 · 10<sup>-7</sup></td></tr><tr><td>b<sub>f</sub></td><td>11.75</td><td>k<sub>0</sub> (Pa)</td><td>1 · 10<sup>5</sup></td><td></td></tr><tr><td>κ (Pa)</td><td>4 · 10<sup>5</sup></td><td>μ<sub>c</sub> (N s m<sup>-1</sup>)</td><td>50.</td><td></td></tr><tr><td>η (N s m<sup>-1</sup>)</td><td>300</td><td></td><td></td><td></td></tr></table>

<table><caption>Table 2: Parameters of the electrophysiological simulations (units in S cm⁻¹).</caption><thead><tr><th></th><th>Tissue</th><th>Banchmann bundle</th><th>Fossa ovalis</th></tr></thead><tbody><tr><td>σ<sub>i</sub><sup>t</sup></td><td>2.2 · 10<sup>-4</sup></td><td>1.045 · 10<sup>-3</sup></td><td>2.2 · 10<sup>-4</sup></td></tr><tr><td>σ<sub>i</sub><sup>l</sup></td><td>2.2 · 10<sup>-3</sup></td><td>1.639 · 10<sup>-3</sup></td><td>2.2 · 10<sup>-3</sup></td></tr><tr><td>σ<sub>e</sub><sup>t</sup></td><td>9.0 · 10<sup>-4</sup></td><td>4.275 · 10<sup>-3</sup></td><td>9.0 · 10<sup>-4</sup></td></tr><tr><td>σ<sub>e</sub><sup>l</sup></td><td>2.2 · 10<sup>-3</sup></td><td>1.435 · 10<sup>-3</sup></td><td>2.07 · 10<sup>-4</sup></td></tr></tbody></table>

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