# Myocardial Stiffness Quantification using Ultrasound Shear Wave Elastography and Reduced Modeling for Subject-specific Simulations

Camilla Ferrario, Jairo Rodríguez-Padilla, Maelys Venet, Olivier Villemain, Maxime Sermesant

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Camilla Ferrario, Jairo Rodríguez-Padilla, Maelys Venet, Olivier Villemain, Maxime Sermesant. Myocardial Stiffness Quantification using Ultrasound Shear Wave Elastography and Reduced Modeling for Subject-specific Simulations. FIMH 2025 - 13th Functional Imaging and Modeling of the Heart, Jun 2025, Dallas (TX), United States. pp.41-54, 10.1007/978-3-031-94562-5_5. hal-05123170

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Distributed under a Creative Commons Attribution 4.0 International License
Myocardial Stiffness Quantification using
Ultrasound Shear Wave Elastography and
Reduced Modeling for Subject-specific
Simulations

Camilla Ferrario<sup>1,3</sup>[0009-0007-5396-6342]*, Jairo Rodríguez Padilla<sup>1,3</sup>, Maelys Venet<sup>2,3</sup>, Olivier Villemain<sup>2,3</sup>, and Maxime Sermesant<sup>1,3</sup>

¹ Inria – Université Côte d'Azur, Epione Team, Sophia Antipolis, France

² Bordeaux University Hospital (CHU) – Department of Pediatric and Adult Congenital Cardiology, Pessac, France

³ Electrophysiology and Heart Modeling Institute, Institut Hospital-Universitaire (IHU)-Liryc - Fondation Bordeaux Université, Bordeaux, France

**Abstract.** Cardiac electromechanical models can significantly advance diagnosis, prognosis, and therapy planning. Despite their importance in evaluating myocardial function, key mechanical parameters such as myocardial stiffness (MS) are still missing in clinical practice. While ultrafast ultrasound imaging (UUI) provides reliable and reproducible data, its use in parameter estimation is still restricted, limiting the potential for comprehensive modeling. Integrating UUI data with reduced-order 0D models addresses these limitations by enabling more robust parameter estimation and improved data interpretation. The approach also significantly reduces computational costs compared to 3D simulations. This work presents a closed-loop 0D model calibrated using UUI-based shear wave elastography (SWE), a noninvasive technique providing detailed MS profiles across the cardiac cycle. By integrating MS, the model accounts for both active and passive contributions to myocardial function, with a primary focus on the active component. Modeled using the Bestel-Clement-Sorine (BCS) framework, it has been optimized with the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm and calibrated to healthy subject data. The aim is to develop a subject-specific tool for clinical studies supporting parameter calibration and integration into 3D simulations, while extending its application to pathological cases.

**Keywords:** Cardiac mechanics · reduced-order models · ultrafast ultrasound imaging · myocardial stiffness · patient-specific analyses

# 1 Introduction

Cardiovascular diseases are the leading cause of morbidity and mortality worldwide, making it essential to explore cardiac mechanics for accurate diagnosis and

<footnote>* Corresponding author: camilla.ferrario@inria.fr</footnote>
effective treatment. While 3D simulations offer detailed insights into the heart's multi-physics behavior, their high computational demands and need for extensive subjects data limit their use in time-sensitive clinical settings. Additionally, the parameter optimization required for these models is both computationally expensive and often ill-posed, highlighting the need for simplified yet physiologically relevant alternatives. Reduced-order models offer an efficient alternative by representing the heart and circulatory system as interconnected compartments governed by ordinary differential equations (ODEs). These models enable rapid simulations and sensitivity analyses, proving highly useful for subject-specific studies. However, the personalization process remains difficult due to the limited parameters observability in clinical data. Among these parameters, myocardial stiffness (MS) is central to cardiac function, evolving dynamically throughout the cardiac cycle and reflecting both passive tissue mechanics and active contraction. Despite its potential as a biomarker for cardiac function, MS remains challenging to quantify *in vivo* due to the lack of direct measurement techniques and the myocardium's complex mechanical properties—thin-walled, anisotropic, and actively contracting [3]. Emerging imaging modalities, such as ultrafast ultrasound imaging (UUI)-based shear wave elastography (SWE), offer a promising non-invasive approach for real-time MS tracking throughout the cardiac cycle. This study presents a closed-loop 0D cardiovascular model that integrates SWE-derived MS, potentially enhancing the diagnostic and predictive capabilities of reduced-order models. However, the extent to which MS directly reflects myocardial contractility remains debated [2, 7, 9, 11, 12], and its clinical translation is still limited [13]. The model's active stiffness component is calibrated using data from 20 healthy volunteers (HVs), employing a BCS-inspired framework to model ATP-driven sarcomere dynamics. Parameter estimation was performed using the CMA-ES optimization algorithm, ensuring an automated, reproducible fit to clinically measured MS profiles.

# 2 Materials and Methods

## 2.1 Shear Wave Elastography and Myocardial Stiffness Acquisition

SWE is a key application of UUI that tracks real-time shear wave (SW) propagation to capture short-lived cardiac events, such as rapid tissue dynamics. With frame rates exceeding 1,000 fps, SWE enables the visualization of myocardial motion with a temporal resolution of up to 0.1 ms, significantly outperforming conventional ultrasound systems [15]. This noninvasive technique quantifies MS (e.g. the resistance of cardiac tissue to deformation) by relating SW velocity to tissue elasticity [13]. Unlike indirect methods, UUI directly estimates stiffness, providing a more accurate representation of intrinsic myocardial properties [13]. However, applying SWE to the heart remains challenging due to the myocardium's anisotropic structure, viscoelasticity, and dynamic contraction [3]. These factors introduce inherent limitations, which are further discussed in Section 4. For details on the methodology, see [14, 15].
<figure><img src="image_2.png" /><figcaption>Fig. 1: (1a) Shear waves (SWs) are initiated by an external stimulus using an ultrasonic probe to induce localized displacement. (1b) Propagation is tracked during B-mode acquisition in short-axis (SA) and long-axis (LA) views, enabling the calculation of SW speed to assess myocardial stiffness (image adapted from [14]).</figcaption></figure>

The clinical protocol for SWE image acquisition remains unstandardized as the technique is still emerging. Our dataset includes shear wave measurements obtained using an acoustic radiation force (ARF)-based approach (Figure 1). An ultrasonic probe artificially generates a localized force in the antero-basal septum of 20 healthy volunteers at specific ECG phases, inducing SW propagation. The same probe then detects wave movement, enabling precise tracking of tissue dynamics. High-frame-rate Doppler echography further improves tissue characterization and diagnostic accuracy [14, 15]. Measurements were collected over 50–80 cardiac cycles per subject within the same session, spanning from early systole to late diastole.

The full dataset consists of waveforms from 20 healthy subjects (mean age = 9.78±5.27 years; sex ratio (M/F) = 1.22). The clinical curves exhibit variability (Figure 2), with some peaking sharply and others rising more gradually. This diversity highlights the multifactorial nature of MS, influenced by differences in sarcomere function, extracellular matrix composition, and overall myocardial structure [3].
<figure><img src="image_3.png" /><figcaption>Fig. 2: Myocardial stiffness (MS) clinical data for the cohort of 20 healthy volunteers (HVs).</figcaption></figure>

## 2.2 Reduced Model Description

To study MS efficiently while ensuring computational feasibility, we use a reduced-order modeling approach that simplifies 3D cardiac simulations while preserving key physiological dynamics. By leveraging kinematical and constitutive assumptions, this 0D framework retains essential cardiac function features while significantly reducing computational costs [4]. The cardiovascular model integrates two core components: (1) a lumped parameter model, based on the electro-hydraulic analogy, to simulate vascular system hemodynamics, and (2) an electromechanical model of the heart, implemented using the BCS approach [1] (Figure 3a). In the lumped model, resistances (R) represent viscous losses and compliances (C) capture the elastic storage capacity of the vessels. By solving a system of ODEs, the model generates flow-pressure dynamic curves, effectively capturing the cardiac cycle over seven beats to achieve a steady-state solution.

The systemic and pulmonary circulations are represented by coupling a three-element *Windkessel (RCR)* model for the arterial circulation with a two-element (RC) model for the venous return. The right (RV) and left ventricles (LV) are modeled as spheres with radius $R_0$ and thickness $d_0$, assuming spherical symmetry for both geometry and constitutive properties [4]. This simplification preserves essential boundary conditions and incompressibility, ensuring physiological consistency in the representation of cardiac mechanics. While it involves a trade-off in accuracy, the 0D approach enables efficient parameter calibration, making it suitable for trend analysis and behavioral characterization. For further methodological details, we refer readers to [4, 5].
<figure><img src="image_4.png" /><figcaption>Fig. 3: Reduced-order cardiac and rheological models.</figcaption></figure>

## 2.3 Myocardial Stiffness Model

In this study, we extend the theoretical framework of Chapelle et al. [5] to model myocardial stiffness in a reduced-order setting. Their energy-preserving model connects sarcomere dynamics to myocardial mechanics using moment scaling techniques, effectively capturing the multiscale nature of cardiac biomechanics. In the context of continuum mechanics, constitutive equations for the myocardial tissue are represented as a Hill—Maxwell material [5].

Active force generation is driven by cross-bridge (XB) dynamics arising from acto-myosin (AM) interactions. The function f governs XB attachment when strain s ∈ [0, 1] and calcium levels exceed a threshold ([Ca²⁺] > C), while g describes XB detachment due to rapid length changes (α|dot_c|):

$$ 
\begin{align} 
f(t, s) &= k_{\text{ATP}} \mathbb{1}_{\{s \in [0,1]\}} \mathbb{1}_{\{[Ca^{2+}] > C\}}, \nonumber \\ 
g(t, s) &= \alpha |\dot{e}_c| + k_{\text{ATP}} \mathbb{1}_{\{s \notin [0,1]\}} \mathbb{1}_{\{[Ca^{2+}] > C\}} + k_{\text{SR}} \mathbb{1}_{\{[Ca^{2+}] < C\}} \tag{1}
\end{align} 
$$

here, $\mathbb{1}_{\{[Ca^{2+}] > C\}}$ is the indicator function, equal to 1 when $[Ca^{2+}] > C$, and 0 otherwise. The ATP-dependent cross-bridge formation rate $k_{\text{ATP}}$ reflects the energy-driven sarcomere activation and contraction, while the sarcoplasmic retic-
ulum (SR) calcium reuptake rate $k_{SR}$ controls calcium clearance during relaxation. These parameters regulate cross-bridge cycling in response to calcium transients, critically determining myocardial contractility and relaxation [5]. In the 0D framework, the collective behavior of XBs is characterized by the active stiffness $k_c = k_0 \int_{\mathbb{R}} n(s,t) ds$ and the active stress $\tau_c = k_0 \int_{\mathbb{R}} (s + s_0) n(s,t) ds$. Here, $\tau_c$ represents the force per unit area generated by the ensemble of XBs, with $k_0$ denoting the stiffness of an individual bridge, $n(s,t)$ the density of attached heads, and $s_0$ the pre-strain, accounting for the system's deformed state. Their dynamics are described by the following constitutive relations, formulated as a system of ODEs:

$$ 
\left\{
\begin{aligned}
\dot{k}_c &= -\left(k_{\text{ATP}} \mathbb{1}_{\{[Ca^{2+}] > C\}} + w k_{\text{SR}} \mathbb{1}_{\{[Ca^{2+}] < C\}} + \alpha |\dot{e}_c|\right) k_c \\
&\quad + n_0 k_0 k_{\text{ATP}} \mathbb{1}_{\{[Ca^{2+}] > C\}} \\
\dot{\tau}_c &= -\left(k_{\text{ATP}} \mathbb{1}_{\{[Ca^{2+}] > C\}} + w k_{\text{SR}} \mathbb{1}_{\{[Ca^{2+}] < C\}} + \alpha |\dot{e}_c|\right) \tau_c \\
&\quad + n_0 \sigma_0 \left(k_{\text{ATP}} \mathbb{1}_{\{[Ca^{2+}] > C\}} + k_c \dot{e}_c\right)
\end{aligned}
\right.
\quad (2) $$

where $n_0$ denotes the fraction of recruitable AM sites, $w$ modulates calcium reuptake, while $\sigma_0 = k_0(s_0 + \frac{1}{2})$ denotes the maximal active stress. Note that $\sigma_0$ represents the system's intrinsic capacity under full activation, while $\tau_c$ reflects the dynamically evolving stress during contraction. Both $\dot{k}_c$ and $\dot{\tau}_c$ respond to variations in $k_{\text{ATP}}$ and $k_{\text{SR}}$, reflecting myocardial behavior by accounting for $[Ca^{2+}]$ levels and relaxation dynamics in the tissue.

The passive contribution of the myocardial connective tissue is modeled using a hyperelastic potential ($W_e$) and a viscous pseudo-potential ($W_v$) which describe the tissue's elasticity and resistance to deformation, respectively. We adopt an isotropic constitutive law based on the Mooney-Rivlin formulation, with $W_e$ defined using the right Cauchy-Green tensor $C = F^T F$. The deformation gradient $F$ describes local kinematics, and its determinant $J = \det(F)$ characterizes volumetric changes. Soft biological tissues, including the myocardium, are nearly incompressible (i.e., Poisson's ratio = 0.5), which imposes the incompressibility condition $J = 1$, implying $\det(C) = J_3 = 1$. As a result, the volumetric component of deformation can be neglected, making the strain energy depend solely on the deviatoric part of $C$, characterized by its first and second ($J_1$ and $J_2$) reduced invariants. Consequently, the hyperelastic potential can be expressed as:

$$ W_e(J_1, J_2) = C_1(J_1 - 3) + C_2(J_2 - 3), \quad (3) $$

where $C_1$ and $C_2$ are material parameters.

Despite employing an isotropic passive component, the overall behavior is non-isotropic—specifically transversely isotropic—due to the incorporation of the passive branch into the comprehensive rheological model, which accounts for additional active contributions along the muscle fiber direction [5]. Note that,
in the 0D spherical framework material properties are applied homogeneously across the chamber inherently averaging out spatial variations in anisotropy. This simplification is a necessary trade-off for computational efficiency, ensuring a physiologically relevant fiber-directed response while maintaining model tractability.

## 2.4 Translating Theoretical Stiffness Models into Clinical Parameter Estimation

Understanding MS requires bridging theoretical formulations with experimentally derived mechanical properties. Our approach integrates a rheological model designed to incorporate both the active and passive contributions to myocardial stiffness while preserving their distinct roles. The model consists of (1) an active BCS branch for sarcomere-driven force generation and (2) a passive branch representing extracellular matrix mechanics (Figure 3b). MS is expressed as an equivalent stiffness $k_{eq}$, combining these two components while allowing their effects to be evaluated independently:

$$k_{eq} = k_{active} + k_{passive} \quad (4)$$

This formulation provides a simplified yet physiologically meaningful representation of MS, enabling direct comparisons between simulated and clinical data.

*Active Myocardial Stiffness Hypothesis.* While both $k_{active}$ and $k_{passive}$ contribute throughout the cardiac cycle, their relative influence varies. In a healthy heart, the passive contribution remains minimal during systole to reduce energy dissipation [13], making the active component the primary determinant of myocardial mechanics. Given this, we assume $k_{active} \gg k_{passive}$, leading to $k_{eq} \approx k_{active}$ during systole. However, we acknowledge that neglecting the passive component across the full cycle is a limitation that may affect parameter identifiability. Future work will focus on explicitly incorporating $k_{passive}$ to better account for pathological states, where its contribution is more substantial.

*Parameter Influence on Simulated MS.* We define the active stiffness $k_c = k_0 \int_R n(s, t) ds$ as a proxy for the active branch stiffness $k_{active}$ of our rheological model (Eq. 4). In our setting, myocardial contraction and relaxation dynamics are governed by two key parameters: $k_{ATP}$ and $k_{SR}$, which modulate the evolution of $k_c$. By adjusting these parameters, we shape $k_c$ to reproduce the key stiffness dynamics observed in clinical data. Since our data consist of full temporal curves rather than isolated scalar values, our approach focuses on capturing the system's dynamics, addressing a observability problem where parameter inference must ensure the reconstructed trajectories align with physiological behavior.

It's worth noting that, while the constitutive equations (Eq. 2) define myocardial stiffness in terms of $k_0$ (individual bridge stiffness) and $\sigma_0$ (maximal active stress), these parameters act as scaling factors in the 0D model. They modulate the amplitude of $k_c$ without altering its dynamic evolution, as no changes in
curve shape or timing are observed. For this reason, their values—100 kPa and 64 kPa, respectively—were taken from literature rather than explicitly calibrated, ensuring a physiologically consistent yet computationally efficient representation of myocardial mechanics.

*Sensitivity Analyses.* Our experiments showed that increasing $k_{ATP}$ elevated peak $k_c$, enhancing contractility (inotropy) and reinforcing the link between calcium dynamics and myocardial stiffness [9, 10, 12]. Conversely, increasing $k_{SR}$ accelerated $k_c$ decay, improving relaxation (lusitropy) and reducing diastolic stiffness. Notably, neither parameter influenced the timing of peak $k_c$, indicating that their primary effect was on amplitude rather than temporal evolution (Figure 4). This analysis informed the selection of physiologically relevant parameter ranges for $k_{ATP}$ and $k_{SR}$, which were used to initialize the CMA-ES optimization. Given the stronger influence of $k_{ATP}$ on peak amplitude, its search range was more constrained, whereas $k_{SR}$ was allowed greater variability to accommodate different relaxation dynamics.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Sensitivity analysis of active stiffness $k_c$, where $k_{SR}$ and $k_{ATP}$ are alternately fixed at high or low values while varying the other.</figcaption></figure>
## 2.5 Computational Framework

**0D Model Implementation.** The code implements a 0D circulatory model solving a system of ODEs for closed-loop circulation, simulating cardiac function via pressure and volume curves over seven cardiac cycles to steady state. It includes 82 parameters (34 for the LV, 34 for the RV, and 6 for pulmonary and systemic circulation), managed using a Python class for logging and initialization. A Cython wrapper links Python to optimized C functions for efficient computations.

**CMA-ES Optimization for Parameter Fitting.** The CMA-ES algorithm was configured with a population size of 6 and a maximum of 50 iterations, optimizing two parameters ($k_{ATP}$, $k_{SR}$) within initial ranges of $k_{SR} \in [1, 30]$ and $k_{ATP} \in [0.5, 10]$. These ranges were refined based on our sensitivity analysis, which revealed that $k_{ATP}$ has a stronger influence on amplitude variations, necessitating a more restricted range. A (3-w,6)-aCMA-ES variant was used, employing weighted recombination to adapt the covariance matrix. Each iteration sampled candidates from a multivariate normal distribution, retaining the best-performing ones to refine the search.

The objective was to minimize discrepancies between simulated and clinical MS curves by optimizing a weighted cost function (Eq. 5) incorporating the mean absolute error (MAE), cross-correlation coefficient ($\rho$) and peak amplitude difference ($\Delta_{peak}$):

$$J(k_{ATP}, k_{SR}) = w_{error} \cdot \text{MAE} + w_{corr} \cdot (1 - \rho) + w_{peak} \cdot \Delta_{peak}, \quad (5)$$

where MAE = $\frac{1}{N} \sum_{i=1}^{N} |k_{c}^{sim}(t_{i}) - k_{c}^{clin}(t_{i})|$ quantifies the discrepancy between simulated and clinical myocardial stiffness values over $N$ time points, $\rho$ is the maximum cross-correlation coefficient between the simulated and clinical signals, and $\Delta_{peak} = |k_{c}^{sim,max} - k_{c}^{clin,max}|$ quantifies the difference between the peak stiffness values of the two signals. The weights $w_{error}$, $w_{corr}$, and $w_{peak}$ are user-defined and normalized to sum to 1. In cases with complex dynamics, where the point-by-point error remains high despite strong correlation, $w_{error}$ was assigned a lower weight to prevent it from disproportionately influencing the optimization. A stagnation detection mechanism is triggered up to three restarts with reinitialized parameter distributions when improvements plateaued over 50 iterations, preserving learned information to enhance convergence. The algorithm iteratively adjusts $k_{ATP}$ and $k_{SR}$ to minimize $J(k_{ATP}, k_{SR})$, with the final parameters yielding the lowest cost for an optimal fit to clinical MS dynamics.

Parallel computation across multiple CPU cores accelerated the process, which typically completed in 70.38 ± 1.54 seconds with 900 function evaluations.
## 3 Results

A quantitative evaluation of the CMA-ES optimization was performed to assess the accuracy and consistency of the MS model fitting across subjects. The optimized parameters, along with key error metrics, are summarized in Table 1. The results show that CMA-ES produced stable parameter estimates across all subjects, with an average $k_{SR} = 5.16 \pm 1.58$ kPa and $k_{ATP} = 3.98 \pm 1.76$ kPa.

The MAE of the optimized model was $3.57 \pm 1.22$, indicating a reasonable fit to the clinical data. Additionally, CMA-ES significantly reduced discrepancies in peak amplitude, achieving an average $\Delta_{peak} = 0.42 \pm 0.13$ kPa, ensuring a close match between the predicted and reference curves. The cross-correlation coefficient ($\rho = 0.27 \pm 0.02$) highlights the overall waveform alignment, while the total objective function cost was minimized to $J = 1.18 \pm 0.26$, reflecting the effectiveness of the optimization strategy.

Figure 5 presents representative examples of the reproduced MS dynamics for selected subjects. The optimized model captures the characteristic shape of the curves, maintaining physiological plausibility while adapting to individual subject variations. The comparison between the clinical data (dashed line) and the model predictions (solid blue line) highlights the accuracy of the fitting, particularly for simpler dynamics. However, more complex waveforms exhibit greater discrepancies, suggesting that additional refinement of the optimization framework or a more flexible model structure may be needed to better capture rapid transients and nonlinearities. Despite these challenges, the smoothness of the estimated stiffness evolution indicates that CMA-ES prevents overfitting while achieving a physiologically meaningful representation of the dynamics. These results confirm that CMA-ES provides a robust and reproducible parameter estimation framework, mitigating the variability inherent in manual tuning and ensuring consistent identification of model parameters across subjects.

Concluding with the scatter plot (Figure 6), we report the age and sex distribution of the analyzed cohort, which consists of healthy subjects ranging from 0 to 16 years old. At this stage, no clear correlation between MS and either age or sex could be identified. The lack of observable trends can be attributed to several factors: (i) the natural inter-subject variability in myocardial development during growth, (ii) the absence of extreme age groups such as neonates and elderly individuals, which limits broader age-dependent analyses, and (iii) the relatively small cohort size, which may not be sufficient to detect subtle trends. Given the ongoing developmental changes in myocardial structure and function, these results emphasize the need for further investigation into the biomechanical evolution of the growing heart.

## 4 Conclusion and Perspectives

This study presents a computationally efficient framework for integrating SWE-derived myocardial stiffness into a 0D circulatory model, providing a novel approach for assessing cardiac mechanical properties in healthy individuals. While
<table><caption>Table 1: Summary statistics (mean ± standard deviation) for parameters and error metrics across all subjects using the CMA-ES algorithm. $k_{SR}$ represents the calcium reuptake rate by the sarcoplasmic reticulum, while $k_{ATP}$ denotes the ATP-driven cross-bridge detachment rate. The reported error metrics include the mean absolute error (MAE), the difference in peak activation time ($\Delta_{peak}$), the correlation coefficient ($\rho$), and the cost function value ($J$).</caption><thead><tr><th>Parameters</th><th colspan="2">Error Metrics</th><th colspan="2">Other Metrics</th></tr></thead><tbody><tr><td>$k_{SR}$ [kPa]</td><td>$k_{ATP}$ [kPa]</td><td>MAE</td><td>$\Delta_{peak}$ [kPa]</td><td>$\rho$</td></tr><tr><td>5.16 ± 1.58</td><td>3.98 ± 1.76</td><td>3.57 ± 1.22</td><td>0.42 ± 0.13</td><td>0.27 ± 0.02</td></tr><tr><td></td><td></td><td></td><td></td><td>1.18 ± 0.26</td></tr></tbody></table>

the proposed methodology enables the inference of key cardiovascular indices that cannot be directly measured in vivo (such as cardiac contractility), certain limitations must be acknowledged.

On the modeling side, several simplifications were introduced to balance physiological relevance with computational feasibility. Our mechanical framework includes nonlinear viscoelasticity and anisotropy, yet assumes homogeneous material properties due to the intrinsic nature of 0D modeling. While the transversely isotropic model improves physiological relevance, it does not capture regional variations in anisotropy or time-dependent behaviors like hysteresis or frequency-dependent stiffness. Viscoelastic effects are approximated through damping without explicit modeling of their temporal complexity.

On the experimental side, shear wave elastography via UUI faces several technical challenges in the cardiac context. Myocardial structural features—such as wall thinness, fiber anisotropy, active contraction, and dynamic loading—complicate the interpretation of shear wave propagation and thus the stiffness estimates [3, 15]. Moreover, the high-frequency properties captured by SWE may not fully reflect the physiological responses relevant at cardiac timescales. Because imaging was performed segmentally, spatially localized stiffness values were obtained, limiting global myocardial assessment. Additionally, we lacked invasive pressure measurements to normalize cardiac loading conditions. Although limiting the study to healthy volunteers mitigates variability, this remains a constraint [2, 7, 11, 13].

A key area for future refinement is the current omission of a passive mechanical branch throughout the full cardiac cycle, which impacts parameter identifiability. Moving forward, we aim to incorporate an explicit representation of $k_{passive}$, thereby improving the model's ability to capture diastolic mechanics—a feature especially relevant when extending to pathological conditions [3, 13, 15].

Despite these limitations, this framework offers a rapid and informative calibration tool. Contraction and relaxation parameters ($k_{ATP}$ and $k_{SR}$) are fundamental across electromechanical models, and capturing their inter-subject variability supports personalization even in 3D cardiac simulations. By efficiently extracting subject-specific mechanical properties, the 0D model aids parameter tuning in high-fidelity frameworks.
<figure><img src="image_6.png" /><figcaption>Fig. 5: Best-fitting active stiffness ($k_c$) curves obtained after parameter optimization for 9 out of the 20 healthy volunteers in the dataset. Each curve represents the optimized model output using subject-specific values of the relaxation ($k_{SR}$) and contraction ($k_{ATP}$) rates, expressed in the format $HV_{ID}(k_{SR}, k_{ATP})$. The dashed grey line corresponds to the clinical data, while the blue line represents the time-aligned model prediction.</figcaption></figure>

Importantly, these preliminary results align with recent literature. While the precise relationship between intracellular calcium concentration, cross-bridge dynamics, and active force development remains under investigation, multiple experimental and computational studies have highlighted a strong correlation between them [6, 9, 10]. Variations in intracellular calcium concentration are closely associated with changes in the myocardium's ability to generate active force—ultimately influencing contractility [10]. In this context, the parameters $k_{ATP}$ and $k_{SR}$ regulate the kinetics of contraction and relaxation, respectively, by shaping how XBs are recruited and deactivated in response to the calcium signal. Although they do not directly influence intracellular calcium concentration, they govern the mechanical response of the tissue to it—modulating both the amplitude and timing of active force and active stiffness development.
<figure><img src="image_7.png" /><figcaption>Fig. 6: Scatter plot of subject-specific values for the relaxation rate $k_{SR}$ and the contraction rate $k_{ATP}$. Each point is labeled as $HV_{ID}(k_{SR}, k_{ATP})$, where triangles represent male subjects and circles represent female subjects. Dashed lines indicate the mean parameter values across the cohort: $k_{SR} = 5.16 \pm 1.58$ kPa and $k_{ATP} = 3.98 \pm 1.76$ kPa.</figcaption></figure>

**Acknowledgements** The authors are grateful to Nicolas Drettakis for his valuable help and insightful discussions throughout the development of this work.

This work has received funding from the French government through the National Research Agency (ANR) under several programs: *PEPR Digital Health ChroniCardio* (22-PESN-0015), *3IA Côte d'Azur* and *IA Cluster* (ANR-19-3IA-0002 and ANR-23-IACL-0001), and *IHU Liryc* (ANR-10-IAHU-04). Additional support was provided by France 2030 and the European Union (Next Generation EU) through the *MediTwin* project. The authors are also grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.

**Competing Interests** The authors have no competing interests to declare that are relevant to the content of this article.
References

1. Bestel, J., Clément, F., Sorine, M.: A biomechanical model of muscle contraction. In: Medical Image Computing and Computer-Assisted Intervention (MICCAI). pp. 1159–1161 (2001)

2. Caenen, A., Bézy, S., Pernot, M., Nightingale, K.R., Vos, H.J., Voigt, J.U., Segers, P., D'hooge, J.: Ultrasound shear wave elastography in cardiology. JACC: Cardiovascular Imaging **17**(3), 314–329 (2024).
https://doi.org/10.1016/j.jcmg.2023.12.007

3. Caenen, A., Pernot, M., Nightingale, K.R., Voigt, J.U., Vos, H.J., Segers, P., D'hooge, J.: Assessing cardiac stiffness using ultrasound shear wave elastography. Physics in Medicine & Biology **67**(2), 02TR01 (2022).
https://doi.org/10.1088/1361-6560/ac404d, https://doi.org/10.1088/1361-6560/ac404d

4. Caruel, M., Chabiniok, R., Moireau, P., Lecarpentier, Y., Chapelle, D.: Dimensional reductions of a cardiac model for effective validation and calibration. Biomechanics and Modeling in Mechanobiology **12**(4), 815–829 (2013).
https://doi.org/10.1007/s10237-012-0446-5

5. Chapelle, D., Le Tallec, P., Moireau, P., Sorine, M.: An energy-preserving muscle tissue model: Formulation and compatible discretizations. International Journal for Multiscale Computational Engineering **10**(2), 189–211 (2012).
https://doi.org/10.1615/IntJMultCompEng.2011002360

6. Kimmig, F., Caruel, M., Chapelle, D.: Varying thin filament activation in the framework of the huxley'57 model. International Journal for Numerical Methods in Biomedical Engineering **38**(10), e3636 (2022).
https://doi.org/10.1002/cnm.3636

7. Nejad, N.S., Klein, M.D., Mirsky, I., Lown, B.: Assessment of myocardial contractility from ventricular pressure recordings. Cardiovascular Research **5**(1), 15–23 (1971).
https://doi.org/10.1093/cvr/5.1.15, https://academic.oup.com/cardiovascres/article/5/1/15/340483

8. Piersanti, R., Regazzoni, F., Salvador, M., Corno, A.F., Dede', L., Vergara, C., Quarteroni, A.: 3d–0d closed-loop model for the simulation of cardiac biventricular electromechanics. Computer Methods in Applied Mechanics and Engineering **391**, 114607 (2022).
https://doi.org/10.1016/j.cma.2022.114607

9. Pinzauti, F., Pertici, I., Reconditi, M., Narayanan, T., Stienen, G.J.M., Piazzesi, G., Lombardi, V., Linari, M., Caremani, M.: The force and stiffness of myosin motors in the isometric twitch of a cardiac trabecula and the effect of the extracell- lular calcium concentration. The Journal of Physiology **596**(13), 2581–2596 (2018).
https://doi.org/10.1113/JP275579, https://doi.org/10.1113/JP275579, pMCID: PMC6023834

10. Regazzoni, F., Poggesi, C., Ferrantini, C.: Elucidating the cellular determinants of the end-systolic pressure-volume relationship of the heart via computational mod-elling. arXiv preprint arXiv:2411.06115 (2024), https://arxiv.org/abs/2411.06115

11. Sagawa, K., Suga, H., Shoukas, A.A., Bakalar, K.M.: End-systolic pressure/volume ratio: A new index of ventricular contractility. The American Journal of Cardiology **40**, 748–753 (1977).
https://doi.org/10.1016/0002-9149(77)90192-8, https://doi.org/10.1016/0002-9149(77)90192-8

12. Tanter, M., Pernot, M., Couade, M., Mateo, P., Crozatier, B., Fischmeister, R.: Real-time assessment of myocardial contractility us- ing shear wave imaging. Journal of the American College of Cardiology **58**(1), 65–72 (2011).
https://doi.org/10.1016/j.jacc.2011.02.042, https://doi.org/10.1016/j.jacc.2011.02.042
13. Villalobos Lizardi, J.C., Baranger, J., Nguyen, M.B., Asnacios, A., Malik, A., Lumens, J., Mertens, L., Friedberg, M.K., Simmons, C.A., Pernot, M., Villemain, O.: A guide for assessment of myocardial stiffness in health and disease. Nature Cardiovascular Research 1(1), 8–22 (2022). https://doi.org/10.1038/s44161-021-00007-3, https://doi.org/10.1038/s44161-021-00007-3

14. Villemain, O., Baranger, J., Friedberg, M.K., Papadacci, C., Dizeux, A., Messas, E., Tanter, M., Pernot, M., Mertens, L.: Ultrafast ultrasound imaging in pediatric and adult cardiology: Techniques, applications, and perspectives. JACC Cardiovasc Imaging 13(8), 1771–1791 (2020). https://doi.org/10.1016/j.jcmg.2019.09.019

15. Villemain, O., Correia, M., Mousseaux, E., Baranger, J., Zarka, S., Podetti, I., et al.: Myocardial stiffness evaluation using noninvasive shear wave imaging in healthy and hypertrophic cardiomyopathic adults. JACC Cardiovasc Imaging 12(7 Pt 1), 1135–1145 (2019). https://doi.org/10.1016/j.jcmg.2018.02.002