# Objective assessment of cardiac function using patient-specific biophysical modeling based on cardiovascular MRI combined with catheterization

Maria Gusseva, Daniel Alexander Castellanos, Surendranath Veeram Reddy, Tarique Hussain, Dominique Chapelle, Radomír Chabiniok

<figure><img src="image_1.png" /></figure>

## To cite this version:

Maria Gusseva, Daniel Alexander Castellanos, Surendranath Veeram Reddy, Tarique Hussain, Dominique Chapelle, et al.. Objective assessment of cardiac function using patient-specific biophysical modeling based on cardiovascular MRI combined with catheterization. AJP - Heart and Circulatory Physiology, 2025, 329 (5), pp.H118-H1191. 10.1152/ajpheart.00232.2025. hal-05303143v2

HAL Id: hal-05303143

https://hal.science/hal-05303143v2

Submitted on 8 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Integrative Cardiovascular Physiology and Pathophysiology

# Objective assessment of cardiac function using patient-specific biophysical modeling based on cardiovascular MRI combined with catheterization

© Maria Gusseva,¹,²,⁴ Daniel A. Castellanos,³ Surendranath Veeram Reddy,⁴ Tarique Hussain,⁴ Dominique Chapelle,¹,² and ® Radomír Chabiniok⁴

¹Inria, Palaiseau, France; ²LMS, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau, France; ³Department of Cardiology, Boston Children's Hospital, Boston, Massachusetts, United States; and ⁴Division of Pediatric Cardiology, Department of Pediatrics, UT Southwestern Medical Center, Dallas, Texas, United States

**Abstract**

Synthesizing multimodality data, such as cardiovascular magnetic resonance imaging (MRI) combined with catheterization, into a single framework is challenging. Different acquisition systems are subjected to different measurement errors. Coupling clinical data with biomechanical models can assist in clinical data processing (e.g., model-based filtering of measurement noise) and quantify myocardial mechanics via metrics not readily available in the data, such as myocardial contractility. In this work, we use a biomechanical modeling with the aim 1) to quantitatively compare model- and data-derived signals, and 2) to explore the potential of model-derived myocardial contractility and distal resistance of the circulation ($R_d$) to robustly quantify cardiovascular physiology. We used 51 ventricular catheterization pressure and cine MRI volume data sets from patients with single-ventricle physiology and left and right ventricles of patients with repaired tetralogy of Fallot. Ventricular time-varying elastance (TVE) metrics and linear regression were used to quantify the relationship between the maximum value of TVE ($E_{max}$) and maximum time derivative of ventricular pressure [$max(dP/dt)$] in data- and model-derived pressure and volume signals at $P < 0.05$. Pearson's correlations were used to compare model-derived contractility and data-derived $E_{max}$ and $max(dP/dt)$, and model-derived $R_d$ and data-derived vascular resistance. All data and model-derived linear regressions were significant ($P < 0.05$). Model-derived $max(dP/dt)$ versus data-derived $E_{max}$ produced a higher coefficient of determination of linear regression models ($R^2$) than data-derived $max(dP/dt)$ versus data-derived $E_{max}$. Correlations demonstrated significant relationships between most data- and model-derived metrics. This work revealed the clinical value of biomechanical modeling to assist in clinical data processing by providing high-quality pressure and volume signals and to quantify cardiovascular pathophysiology.

**NEW & NOTEWORTHY** Combining cardiovascular MRI and catheterization data into a single framework provides important quantities of cardiovascular physiology. However, multimodality data are often subjected to measurement errors (e.g., time-dys synchrony between volume and pressure signals) limiting their clinical interpretation. This study demonstrates that coupling clini cal data with biomechanical modeling is an efficient way to filter measurement noise (providing, e.g., high-quality pressure- volume loops) and derive meaningful quantities of cardiovascular physiology.

biomechanical modeling; cardiovascular physiology; clinical data processing; myocardial contractility; time-varying elastance

INTRODUCTION

Cardiovascular magnetic resonance imaging (MRI) provides important insight into ventricular functionality, such as ejection fraction, stroke volume, or cardiac output. Cardiac catheterization provides complementary information about the ventricular loading conditions (e.g., end-diastolic and end-systolic pressure or time derivative of ventricular pressure). Combining cardiovascular MRI with catheterization data allows for the construction of pressure-volume loops (PV loops) and to obtain a more detailed description of ventricular

energetics in terms of stroke work (i.e., area encompassed within the PV loop) or ventricular and arterial elastance. These quantities can be used to evaluate cardiac functionality in various types of cardiovascular disease (1).

The PV loop is associated with the fundamental physiological relationship between the ventricular pressure $P(t)$ and volume $V(t)$ known as the ventricular time-varying elastance (TVE) and defined by $E(t) = P(t)/(V(t) - V_0)$—the concept introduced by Suga et al. (2). The hypothetical volume $V_0$ volume-axis intercept of the linearized end-systolic pressure-volume relationship is given by the volume-axis intercept of the

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /></figure>

linearized end-systolic pressure-volume relationship (ESPVR). ESPVR arises from the equilibrium between ventricular volume and internal myocardial stresses produced during ventricular systole (3). The maximum ventricular TVE, taking place at the end-systole (more precisely, at the outflow valve closure), is then given by $E_{max} = \text{ESP}/(\text{ESV} - V_0)$, where ESP and ESV are end-systolic pressure and volume, respectively, and is equivalent to the slope of linearized ESPVR. Physiological experiments demonstrated that this slope (and hence also the value of $E_{max}$) is sensitive to the changes in the cardiac inotropic state (4). Therefore, $E_{max}$ has been adopted as a surrogate measure of contractility (5).

Our previous clinical studies demonstrated that combining image and catheterization data with patient-specific biomechanical modeling can provide high-quality PV loops—additional markers of cardiovascular function supporting optimal clinical management (6, 7). In clinical settings, integrating multimodality data (e.g., MRI and catheterization) into a single framework is challenging due to *I*) difference in the heartbeat duration between MRI and catheterization, and 2) temporal delays inherent to each acquisition method. For example, in patients with prolonged QRS duration (e.g., right bundle branch block), MRI acquisition can be affected by temporal offsets in R-peak detection, with the S or R' wave sometimes identified instead. In addition, fluid-filled catheters can record pressure changes with a variable delay that depends on catheter size (8). Biomechanical modeling can mitigate these issues by reconstructing patient-specific PV loops from patients volumetric and pressure measurements, even when clinical signals are temporally misaligned or incomplete. The biomechanical model is formulated on physiological and biophysical principles of myocardial function. The parameters of the model are then calibrated using patient-specific measurements (such as arterial forward flow, ventricular mass and volumes, and ventricular pressure), providing a patient-specific simulation of a cardiac cycle. This approach not only generates high-quality PV loops but also provides additional functional biomarkers, such as myocardial contractility and distal vascular resistance, that are not readily available in clinical data (6, 7, 9).

Reproducible PV loops are particularly important for clinical management of patients with complex congenital heart disease. For example, our recent study demonstrated that in patients with congenitally corrected transposition of great arteries (ccTGA), biomechanical model-derived PV loops can provide robust biventricular functional assessment and yield additional insights into challenges posed by this complex pathophysiology, including left ventricular preparedness for a complex and risky cardiac surgery, named, double switch operation (10).

A TVE model predicts the conceptual link between the ventricular isovolumic phase (associated with maximum time derivative of ventricular pressure, $\max(dP/dt)$), and end of ejection (given by $E_{max}$) (11). Previous physiological works demonstrated the potential of $E_{max}$ and $\max(dP/dt)$ as clinical surrogate measures of myocardial contractility (12, 13). Biomechanical models, when coupled with patients' clinical data, provide patient-specific pressure and volume waveforms along with quantitative metrics of cardiovascular performance. In this work, we aim to further explore the clinical

value of model-derived pressure-volume signals and related physiological indices.

Specifically, we will *I*) quantitatively compare data- and model-derived PV loops using the metrics based on TVE to assess whether biomechanical model-derived PV loops can replace noisy clinical data waveforms, thereby improving objective assessment of cardiac function. Our secondary aims are to explore the relationships between: 2) model-derived contractility, data-derived $E_{max}$, and $\max(dP/dt)$; and 3) between model-derived distal resistance of the circulation and data-derived vascular resistance. This work aims to assess the validity of computational model-derived metrics as clinically relevant measures of cardiovascular physiology.

# MATERIALS AND METHODS

## Data

The study included 16 MRI and pressure datasets of patients with single-ventricle physiology referred to interventional cardiac MRI for pre-Fontan assessment (14), as well as 16 left and 19 right ventricular pressure-volume datasets of patients with repaired tetralogy of Fallot (rTOF) undergoing percutaneous pulmonary valve replacement (PVR). The data collections were performed under the ethical approvals of the Institutional Review Boards (IRB) of the UT Southwestern Medical Center, Dallas (STU-2020-0023) and the University of Texas, Austin (IRB 2020-06-0128). The IRBs waived the need for consent to use the anonymized retrospective data. In the single-ventricle cohort, pressure by catheterization and cine images by MRI (providing volumes of ventricles and wall thickness) were acquired simultaneously. A highly accelerated prospectively ECG-triggered cine balanced steady-state free precession (bSSFP) sequence was used (kT-SENSE acceleration factor 6; partial Fourier 0.625; slice thickness 10 mm; in-plane spatial resolution 2.4 mm). The data were acquired at baseline in 10 patients and immediately after administration of fluid boluses (either a single 5 mL/kg bolus or two consecutive 2.5 mL/kg boluses) in five patients. Out of these five patients, one patient received a second fluid bolus, and the data were collected again immediately afterward. In total, the single-ventricle cohort included 16 datasets. In patients with rTOF, the pressure and MRI data were acquired sequentially via right- and left-heart catheterization and cine MRI. Cine MRI was retrospectively ECG-gated with bSSFP sequence (SENSE = 2, spatial resolution $2 \times 2 \times 10$ mm). The target temporal resolution was ~30 ms in all acquisitions. The motion tracking algorithm (15) was used to extract time versus volume plots from cine MRI. For rTOF-left ventricular (LV) and -right ventricular (RV) pressure data, a semimanual digitization was performed using WebPlotDigitizer<fnref n="1" /> software and MATLAB to extract time versus pressure waveform. Single-ventricle pressure data were already available in digitized format.

## Biophysical Model of the Heart

Geometry and kinematics of the ventricular cavity are approximated by a sphere (3) with varying thickness *d* and inner radius *R* (Fig. 1). The biophysical function of the

<footnote n="1">¹https://automeris.io/docs/.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 1. Biomechanical model of a single ventricular cavity with wall thickness d and sphere inner radius R. The ventricular cavity is coupled to the atrio-ventricular (AV) valve and the ventricular outflow tract (VOT) via a system of diodes with forward and backward resistances ($R_{for}^{VOT}$ and $R_{back}^{VOT}$, respectively). The circulation system is given by a two-stage Windkessel. $P_{at}$, $P_{v}$, $P_{ar}$, $P_{d}$, and $P_{vs}$ are pressures in the atria, ventricle, artery, distal circulation, and venous system, respectively. $R_{for}^{VOT}$, forward resistance of ventricular outflow tract; $R_{back}^{VOT}$, backward resistance of ventricular outflow tract.</figcaption></figure>

myocardium, i.e., contraction and relaxation, is modeled by active and passive constitutive mechanical laws, as in full three-dimensional (3-D) heart model (16). The active mechanical law follows the biophysical principles of actin-myosin cross-bridge kinetics—originally proposed by Huxley’s sliding filament theory (17). Namely, the cross bridges formed by myosin heads are represented as assemblies of springs with active stress ($\tau_c$) and active stiffness ($k_c$) governing active contraction of the myocardium (18):

$$ \begin{cases} \dot{k}_c = -(|u| + \alpha|\dot{e}_{fib}|)k_c + n_0(e_{fib})k_0|u|_+ \\ \dot{\tau}_c = -(|u| + \alpha|\dot{e}_{fib}|)\tau_c + \dot{e}_{fib}k_c + n_0(e_{fib})\sigma_0|u|_+ \end{cases}, \quad (1) $$

where parameter $\sigma_0$ is an asymptotic active stress produced by the sarcomere (functional unit of muscle fiber) at a given inotropic level and under an optimal extension during ventricular systole. The parameter $\sigma_0$ is further referred to as myocardial contractility. The function $u(t)$ represents a summarized electrical activation based on the calcium kinetics that controls the activation ($u > 0$) and repolarization ($u \le 0$) of the myocardium. The parameter $\alpha$ is a myosin cross-bridge destruction rate upon a rapid sarcomeres’ length change (16). Function $n_0(e_{fib})$ (with values between 0 and 1) represents the Frank–Starling law (i.e., length-dependent property of the sarcomere contraction) that controls the fraction of recruited myosin heads at a given sarcomere extension ($e_{fib}$) (16), with $n_0(e_{fib}) = 1$ for the optimal sarcomere extension.

The tissue-level function is given by a nonlinear rheological law of Hill–Maxwell type (19). It contains a contractile component incorporating the aforementioned active behavior of the myocardium and producing stress only in the muscle fiber direction, and a transversely isotropic viscoelastic component, i.e., axisymmetric around the fiber direction, to model the properties of collagen-rich connective tissue in the myocardium. The viscoelastic properties of the myocardium are given by a passive constitutive law—with viscous pseudopotential and hyperelastic potential inspired from previous modeling works (20, 21).

Finally, to model an organ-level performance of the heart (i.e., cardiac beat), the principle of virtual work is used. The internal stresses within the myocardium, active (provided by a contraction of the sarcomeres) and passive (stiffness of viscoelastic component), are in mechanical equilibrium with the external pressure exerted on the endocardial surface and inertia forces.

The mechanics of the inlet and outlet valves of the ventricle, i.e., atrioventricular (AV) and aortic/pulmonary valve (inside ventricular outflow tract, VOT), respectively, are governed by a system of diodes with forward and backward resistance [forward resistance of atrioventricular valve ($R_{for}^{AV}$) and forward resistance of ventricular outflow tract ($R_{for}^{VOT}$), backward resistance of atrioventricular valve ($R_{back}^{AV}$), and backward resistance of ventricular outflow tract ($R_{back}^{VOT}$), respectively] (7, 22). The inward/outward ventricular flow is coupled with the pressures in the atrium, ventricle, and outflow artery. By modifying the values of forward/backward resistances of diodes, we can incorporate the required level of valvular stenosis/regurgitation in the system.

The arterial circulation is modeled by a two-stage Windkessel (23) with proximal and distal resistances and capacitances [proximal resistance of the circulation system ($R_p$), distal resistance of the circulation system ($R_d$), proximal capacitance of the circulation system ($C_p$) and distal capacitance of the circulation system ($C_d$), respectively] to represent mechanical properties of proximal and distal circulatory tree, respectively. Model-derived $R_d$ is assumed to be related to the periphery vascular resistance derived from clinical data, defined as the ratio of mean arterial pressure over cardiac output.

## Biomechanical Model Personalization

Patient-specific models for 51 datasets based on pressure and MRI data were built. Model parameters were calibrated manually, using a sequential calibration procedure as in our previous clinical works (6, 7). MATLAB implementation of the model required less than 30 s computation for a single
<figure><img src="image_7.png" /></figure>

cardiac beat. The overview of calibrated parameters is presented in Table 1. First, the measured arterial flow (proximal aorta/pulmonary artery) was imposed to the Windkessel system, and $R_p$, $R_d$, and $C_d$ were adjusted such that measured arterial peak systolic pressure (PSP) and end-diastolic pressure (EDP) were reached in the simulation. Since the model is not a closed-loop system, the venous pressure ($P_{vs}$) was prescribed from measured data. Then, the circulation system was coupled to the heart model. In the spherical model of the ventricle, sphere thickness and radius in reference configuration (i.e., theoretical volume at 0 mmHg ventricular pressure) were derived from MRI end-diastolic volume (EDV) and myocardial mass. Note that trabeculations were not included in the mass calculation. Reference volume of the ventricle was derived following Klotz's formula (24), $V_{ref} = EDV \times (0.6-0.006 \times EDP)$, where the pressure EDP is given in mmHg.

The electrical activation of the myocardium was prescribed as a piecewise linear continuous function $u$ (in Eq. 1), with the timings of QRS and ST intervals obtained from patient-specific electrocardiogram measurements (9). The onset of the ventricular electrical activation was given by the Q-wave of ECG, and the myocardium is assumed to be fully activated at the end of QRS complex. The ventricle is completely repolarized at the end of the T-wave (for details, we refer to Refs. 7, 9). The ventricle loading at the onset of activation is given by the pressure measured in the ventricle (end-diastolic pressure, EDP). This pressure was imposed in the ventricular cavity, and the myocardial stiffness was calibrated to reproduce the measured EDV in the model. Note that the model does not contain an active contraction model

for the atrial chamber; therefore, the atrial pressure ($P_{at}$) during passive ventricular filling and at end-diastole was directly associated with and prescribed in the ventricle (minimum ventricular pressure and EDP).

In patients with valvular abnormalities, the AV valve and VOT mechanics were calibrated by adjusting $R_{for}^{VOT}$, $R_{back}^{VOT}$, and $R_{back}^{AV}$ to achieve the level of measured regurgitation and/or stenosis on the corresponding valves.

Finally, myocardial contractility was calibrated to match measured ventricular PSP and end-systolic volume. The model calibration provided patient-specific PV loops and mechanical quantities of the heart and circulation, such as the myocardial contractility and stiffness, distal resistance of the circulation.

## Quantitative Assessment of Pressure-Volume Data

The original pressure and volume (P-V) data were synchronized in time by using patient-specific models as a template to identify temporal offset between the pressure and volume traces, as previously described (25). Subsequently, the time-synchronized clinical data were used.

The TVE (2) function was computed from data- and model-derived P-V waveform:

$$E(t) = (P(t))/((V(t) - V_0)) \quad (2)$$

where $V_0$ was assumed to be 0, similar to previous studies (4, 26).

The normalization of time-varying elastance curve [$E(t)$] by its characteristic parameters of peak amplitude ($E_{max}$) and the time to peak amplitude [time of the occurrence of $E_{max}$ ($t_{max}$)] has been shown to yield a curve with a unique

<table><caption>**Table 1.** Overview of biomechanical model parameters calibrated using clinical data of patients</caption><thead><tr><th></th><th>Model Segment</th><th>Calibrated Model Parameters</th><th>Patient Data Used for Calibration</th><th>Details of Calibration Procedure</th></tr></thead><tbody><tr><td>Step 1</td><td>Circulation model</td><td>Proximal resistance ($R_p$), distal resistance and capacitance ($R_d$ and $C_d$)</td><td>Forward arterial flow<br>Arterial pressure</td><td>MRI phase contrast flow is imposed, and $R_p$, $R_d$, $C_d$, are adjusted such that simulated arterial pressure corresponded to measured arterial ESP and EDP.</td></tr><tr><td>Step 2</td><td>Spherical heart model: geometry</td><td>None</td><td>Ventricular mass and EDV</td><td>Sphere thickness is derived from MRI-measured ventricular mass and prescribed to the spherical model.<br>Sphere radius at reference volume is derived from ventricular EDV and ventricular EDP via Klotz formula (24).</td></tr><tr><td>Step 3</td><td>Spherical heart model: passive law</td><td>Relative myocardial stiffness factor</td><td>EDV<br>Ventricular EDP</td><td>Ventricular EDP is imposed on cavity model, stiffness factor is adjusted such that simulated ventricular EDV corresponded to the measured ventricular EDV.</td></tr><tr><td>Step 4</td><td>Valve model</td><td>Forward ($R_{for}^{VOT}$) and backward ($R_{back}^{VOT}$) resistance of ventricular outflow tract<br>Backward resistance of AV valve ($R_{back}^{AV}$)</td><td>Ventricular-arterial pressure gradient<br>Arterial/AV valve regurgitation fractions</td><td>$R_{for}^{VOT}$, $R_{back}^{VOT}$, $R_{back}^{AV}$ are adjusted such that simulated ventricular-arterial pressure gradient or arterial regurgitation corresponded to the measured data.</td></tr><tr><td>Step 5</td><td>Spherical heart model: active law</td><td>Myocardial contractility</td><td>QRS, ESV<br>Ventricular pressure</td><td>Measured QRS and ST durations are prescribed to electrical activation function in the model. Myocardial contractility is adjusted such that simulated ventricular ESV and PSP corresponded to the measured data.</td></tr></tbody></table>

Model calibration is a sequential procedure (steps 1–5), with calibrated parameters from each step held constant in subsequent steps. In patients with no or negligible outflow tract stenosis or valvular regurgitation, step 4 is not required. AV, atrioventricular valve; EDP, end-diastolic pressure; EDV, end-diastolic volume; ESV, end-systolic volume; MRI, magnetic resonance imaging; PSP, peak systolic pressure; $R_{back}^{AV}$, backward resistance of atrioventricular valve; $R_{for}^{VOT}$, forward resistance of ventricular outflow tract; $R_{back}^{VOT}$, backward resistance of ventricular outflow tract. VOT, ventricular outflow tract.
shape independent of loading conditions and inotropic state in canine (2) and human (27) ventricles:

$$E^N(\bar{t}) = E(\bar{t} \cdot t_{\max})/E_{\max} \quad (3)$$

and

$$E(t) = E_{\max} \cdot E^N(\bar{t}) \quad (4)$$

where $\bar{t} = t/t_{\max}$ and $E^N(\bar{t})$ is the normalized TVE curve. According to the theory of TVE, the isovolumetric systole leads to a relationship between the maximum time derivative of ventricular pressure, $\max(dP/dt)$, and maximum time derivative of $E(t)$, $\max(dE/dt)$, assuming that the end-diastolic volume (EDV) is constant (11):

$$\max(dP/dt) = \max(dE/dt) \cdot \text{EDV}. \quad (5)$$

Differentiation of Eq. 4 yields the proportionality between $\max(dE/dt)$ and $E_{\max}$ (11):

$$\max(dE/dt) = (E_{\max}/t_{\max}) \cdot \max(dE^N(\bar{t})/d\bar{t}). \quad (6)$$

The value of $\max(dE^N(\bar{t})/d\bar{t})$ has been shown to be constant across species (11, 27). Therefore, an expression between ventricular isovolumetric and end of ejection phases can be rewritten as:

$$\max(dP/dt)/\text{EDV} = E_{\max}/t_{\max} \cdot k \quad (7)$$

where k is a constant. The reference value of k = 2.25 was assumed based on a previously published value of $\max(dE^N(\bar{t})/d\bar{t})$ for human left ventricles (27).

## Statistical Analysis

The quantitative value of the relationship between $\max(dP/dt)$ and $E_{\max}/t_{\max}$ in Eq. 7 was evaluated by fitting linear regression models through the origin at $P < 0.05$ to data- and model-derived P-V signals. Bland-Altman plots and Wilcoxon signed-rank tests at $P < 0.05$ were used to evaluate the difference between data- and model-derived $E_{\max}/t_{\max}$ and $\max(dP/dt)$.

Pearson's correlations were used to quantify the relationships between 1) model-derived contractility and data-derived $E_{\max}$, 2) model-derived contractility and data-derived $\max(dP/dt)$, 3) model-derived distal resistance of the circulation ($R_d$) and data-derived clinical vascular resistance, and 4) data-derived vascular resistance and model-derived ventricular $\max(dP/dt)$ divided by EDV at $P < 0.05$. Finally, the correlations between model-derived contractility, model-derived $E_{\max}$, and model-derived $\max(dP/dt)$ were studied.

# RESULTS

## Agreement between Data- versus Model-Derived PV Loops

Table 2 shows the indicators of cardiovascular function derived directly from clinical data (cardiac MRI, pressure catheter), and when the biomechanical model was used. Data- and model-derived ejection fraction, cardiac output, and stroke volume were not statistically different with $P > 0.05$ in all patient groups.

Figure 2 demonstrates an example of data- versus model-derived PV loops, and biophysical metrics of contractility—$E(t)$ and $dP/dt$. The PV loop-derived indicators demonstrated

that data- and model-derived stroke work was significantly different with $P < 0.05$ in rTOF-LV and rTOF-RV groups. Bland-Altman plot statistics demonstrated that mean bias ± standard deviation (SD) for data- versus model-derived percentage difference in $\max(dP/dt)$ were -28.0 ± 20.6%, -4.1 ± 19.0%, and 5.4 ± 12.8% for single-ventricle, rTOF-LV, and rTOF-RV patients, respectively (Fig. 3 and Supplemental Table S2). In the single-ventricle and rTOF-LV groups, data- versus model-derived $\max(dP/dt)$ were significantly different with $P < 0.05$. Data- versus model-derived $E_{\max}/t_{\max}$ was not significantly different in all patient groups. $E_{\max}$ and $t_{\max}$ in data- and model-derived PV loops were not statistically different with $P > 0.05$ in all patient groups. Supplemental Fig. S1 and Supplemental Table S1 show Bland-Altman statistics for the percentage difference between data- and model-derived $E_{\max}$ and $t_{\max}$.

## Relationship between $E_{\max}$ and $\max(dP/dt)$ in Data- versus Model-Derived PV Loops

Fitted linear regression models through the origin are shown in Fig. 4 and Supplemental Table S1. All data and model-derived linear regressions were significant with $P < 0.05$.

Data-derived $\max(dP/dt)$ versus data-derived $E_{\max}/t_{\max}$ linear regressions revealed $R^2 = 0.90$, $R^2 = 0.47$, and $R^2 = 0.52$ in single-ventricle, rTOF-LV, and rTOF-RV groups, respectively. Model-derived $\max(dP/dt)$ versus data-derived $E_{\max}/t_{\max}$ linear regressions showed stronger relationships than data-derived $\max(dP/dt)$ versus data-derived $E_{\max}/t_{\max}$ with $R^2 = 0.97$, $R^2 = 0.78$, and $R^2 = 0.86$ in single-ventricle, rTOF-LV, and rTOF-RV groups, respectively. In single-ventricle and rTOF-LV groups, coefficients of regression models ($k_{\text{fit}}$) of model-derived $\max(dP/dt)$ versus data-derived $E_{\max}/t_{\max}$ linear regressions was $k_{\text{fit}} = 2.12$ and $k_{\text{fit}} = 2.24$, respectively, approaching the reference $k = 2.25$ for human left ventricles (27).

Model-derived $\max(dP/dt)$ versus model-derived $E_{\max}/t_{\max}$ linear regressions showed the highest $R^2$ values in all patient groups (Fig. 4).

## Correlation between Data- and Model-Derived Indicators of Cardiovascular Function

Pearson's correlations demonstrated significant relationships between most data- and model-derived quantitative metrics of cardiovascular physiology in all patient groups (Fig. 5). The only nonsignificant correlation ($P > 0.05$) was in single-ventricle group between data-derived systemic vascular resistance and model-derived $\max(dP/dt)/\text{EDV}$ (Fig. 5).

Strong correlations were found between data-derived $E_{\max} \times \text{EDV}$ versus model-derived contractility rescaled by the ratio of sphere thickness/radius with $R > 0.81$ in all patient groups. Strong correlation between data-derived $\max(dP/dt)$ versus model-derived contractility rescaled by the ratio of sphere thickness/radius was revealed in single-ventricle and rTOF-RV groups with $R > 0.77$, moderate correlation in rTOF-LV group with $R = 0.52$. Strong correlations were shown between data-derived vascular resistance and model-derived distal resistance $R_d$ with $R > 0.88$ in all patient groups.
<figure><img src="image_8.png" /><figcaption>**Table 2.** Cardiac magnetic resonance imaging (MRI) derived indicators, pressure-volume-derived indicators, and biophysical model-derived indicators of cardiovascular function</figcaption></figure>

<table><thead><tr><th></th><th>Single-Ventricle<br>(n = 16)</th><th>rTOF-LV<br>(n = 16)</th><th>rTOF-RV<br>(n = 19)</th></tr></thead><tbody><tr><td colspan="4"><strong>Cardiac MRI indicators</strong></td></tr><tr><td>EF, %</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>42 ± 10</td><td>59 ± 6</td><td>52 ± 10</td></tr><tr><td>Model</td><td>42 ± 10</td><td>59 ± 6</td><td>52 ± 9</td></tr><tr><td>CO (L/min/m²)</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>3.76 ± 1.12</td><td>2.95 ± 0.72</td><td>4.22 ± 1.05</td></tr><tr><td>Model</td><td>3.77 ± 1.13</td><td>2.91 ± 0.70</td><td>4.19 ± 1.04</td></tr><tr><td>Stroke volume, mL/m²</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>44 ± 13</td><td>40 ± 9</td><td>57 ± 12</td></tr><tr><td>Model</td><td>44 ± 13</td><td>39 ± 9</td><td>56 ± 12</td></tr><tr><td colspan="4"><strong>Pressure-volume indicators</strong></td></tr><tr><td>Stroke work, J</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>0.305 ± 0.107</td><td>0.513 ± 0.246</td><td>0.359 ± 0.194</td></tr><tr><td>Model</td><td>0.305 ± 0.100</td><td>0.539 ± 0.248*</td><td>0.411 ± 0.233*</td></tr><tr><td>E<sub>max</sub>, mmHg/mL</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>1.91 ± 1.13</td><td>2.17 ± 0.76</td><td>0.62 ± 0.25</td></tr><tr><td>Model</td><td>1.95 ± 1.13</td><td>2.08 ± 0.73</td><td>0.59 ± 0.24</td></tr><tr><td>t<sub>max</sub>, s</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>0.26 ± 0.05</td><td>0.31 ± 0.03</td><td>0.27 ± 0.05</td></tr><tr><td>Model</td><td>0.28 ± 0.03</td><td>0.32 ± 0.04</td><td>0.27 ± 0.03</td></tr><tr><td>E<sub>g</sub>, mmHg/mL</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>2.99 ± 0.95</td><td>1.87 ± 0.67</td><td>0.83 ± 0.45</td></tr><tr><td>Model</td><td>2.89 ± 0.91*</td><td>1.91 ± 0.72</td><td>0.82 ± 0.44</td></tr><tr><td>E<sub>g</sub>/E<sub>max</sub></td><td></td><td></td><td></td></tr><tr><td>Data</td><td>1.81 ± 0.74</td><td>0.88 ± 0.27</td><td>1.34 ± 0.43</td></tr><tr><td>Model</td><td>1.71 ± 0.68*</td><td>0.93 ± 0.23</td><td>1.37 ± 0.32</td></tr><tr><td>max(dP/dt), mmHg/s</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>849 ± 260</td><td>1,208 ± 299</td><td>559 ± 189</td></tr><tr><td>Model</td><td>1,094 ± 196*</td><td>1,324 ± 220</td><td>538 ± 202</td></tr><tr><td>Vascular resistance, (*1e8 Pa·s/m³)</td><td></td><td></td><td></td></tr><tr><td>Data</td><td>1.94 ± 0.61</td><td>1.47 ± 0.47</td><td>0.29 ± 0.14</td></tr><tr><td>Biophysical model-derived indicators</td><td></td><td></td><td></td></tr><tr><td>Contractility, kPa</td><td></td><td></td><td></td></tr><tr><td>Model</td><td>97 ± 34</td><td>80 ± 17</td><td>65 ± 16</td></tr><tr><td>Stiffness (relative to the reference data) (6)</td><td></td><td></td><td></td></tr><tr><td>Model</td><td>1.61 ± 0.27</td><td>1.53 ± 0.13</td><td>1.75 ± 0.30</td></tr><tr><td>Distal resistance of circulation, (*1e8 Pa·s/m³)</td><td></td><td></td><td></td></tr><tr><td>Model</td><td>1.64 ± 0.54</td><td>1.13 ± 0.37</td><td>0.19 ± 0.20</td></tr></tbody></table>

Values are means ± SD. CO, cardiac output; E<sub>a</sub>, arterial elastance; EF, ejection fraction; E<sub>max</sub>, maximum value of time-varying elastance; max(dP/dt), maximum time derivative of ventricular pressure; MRI, magnetic resonance imaging; t<sub>max</sub>, the time of the occurrence of E<sub>max</sub>. *Significant difference between corresponding data and model-derived indicators at *P* < 0.05.

In data-derived vascular resistance versus model-derived max(dP/dt)/EDV correlations, in rTOF-LV the correlation was strong with *R* = 0.82, in rTOF-RV the correlation was moderate with *R* = 0.69, and in single-ventricle group the correlation was weak and not significant with *R* = 0.44 (*P* > 0.05).

Correlations between the model-derived contractility and model-derived E<sub>max</sub> or max(dP/dt) were also significant and are shown in Supplemental Fig. S2.

## DISCUSSION

In this work, we assessed the value of utilizing a biophysical model of the heart to derive PV loops and physiological indicators for the clinical evaluation of patients' pathophysiology. We used metrics based on TVE to investigate the quantitative difference between data- and model-derived P-V traces in patients with various types of heart pathologies. Both data- and model-derived P-V traces produced statistically significant relationships between ventricular

isovolumetric [max(dP/dt)] and end-of-ejection phases (E<sub>max</sub>/t<sub>max</sub>)—in accordance with the prediction of the TVE theory (11, 27).

Our results, therefore, suggest that the level of ventricular elastance at the end of systole (i.e., given by E<sub>max</sub>) is proportional to the maximum rate of stress development during isovolumetric phase [i.e., given by max(dP/dt)], as is expected from theory (ESPVR and time-varying elastance). Previous physiological studies reported similar findings: E<sub>max</sub> was found to be the main determinant of ventricular max(dP/dt) under different loading conditions and inotropic states (28) and a strong correlation between E<sub>max</sub> and max(dP/dt) was discovered by Morimont et al. (29).

### Model Validation

The linear regression relationships were quantitatively stronger for model-derived P-V traces, suggesting the validity of the model-derived PV loops. Data- and model-derived E<sub>max</sub>/t<sub>max</sub> was not statistically different, indicating the ability of the model to accurately represent patient-specific
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 2.** Data- and model-derived pressure-volume loops, time-varying elastance, and the time derivative of ventricular pressure (dP/dt) for a single-ventricle patient (black lines), and left (red) and right (blue) ventricles of a patient with repaired tetralogy of Fallot (rTOF-LV and rTOF-RV, respectively).</figcaption></figure>

features present in the measured data. In addition, the model-derived max(dP/dt) was more accurately reflective of data-derived Emax/tmax than the data-derived max(dP/dt). This, perhaps surprising, result suggests that the model can provide a more reliable assessment of dP/dt than direct measurement, likely because the latter is limited by the fidelity of pressure measurement (8) and digitization technique.

Model calibration does not rely on complete time-varying pressure or volume waveforms, but on discrete patient-specific data points: ventricular EDV, EDV, EDP, and ESP; arterial EDP and ESP; and electrical activation input (QRS and ST durations). Although these input values constrain the model parameters, the resulting pressure and volume waveforms are generated independently of the time-varying profiles of data-derived waveforms. Thus, the output of the model is calculated at the temporal resolution of ≤0.001 s (as in the presented simulations), resulting in a high-quality continuous signal. This approach enables the model to act as a data-filtering tool by producing synthetic (modeled) pressure and volume waveforms that contain patient-specific features of a given subject. Such synthetic signals have the potential to replace noisy or missing clinical signal and support more robust functional evaluation of cardiovascular system.

## Physiological and Biomechanical Implications

We further investigated the validity of the model-derived indices to quantify cardiovascular physiology. Our results revealed strong correlations between all data- and model-

derived metrics. First, the model-derived contractility was well correlated with the data-derived Emax and max(dP/dt). Model-derived contractility is a measure of the level of stress developed by the sarcomeres during ventricular systole. Thanks to the multiscale formulation of the model, the sarcomere-level contraction is translated into the whole organ (i.e., pumping) performance of the heart. Therefore, the model-derived contractility can provide additional clinically meaningful insight into myocardial mechanics.

Model-derived distal resistance of the circulation was strongly correlated with data-derived vascular resistance. Interestingly, rTOF-LV and RV groups demonstrated a significant relationship between data-derived vascular resistance and ventricular max(dP/dt) suggesting a coupling between ventricular performance and vasculature. This could be a sign of myocardial resilience. Single-ventricle group did not show a significant relationship between distal vascular resistance and ventricular max(dP/dt). This may be due to a loss of ventricular vascular coupling as has been implicated in this group (30). One of the steps in translating novel model-derived indices into clinical practice is to validate them against existing biomarkers. Our results demonstrated that the model has the potential to quantify cardiovascular physiology in accordance with the physiological equivalents of associated biomarkers.

In the rTOF-RV group, linear regression coefficients ($k_{fit}$) for all data- and model-derived relationships were lower compared with that in single-ventricle and rTOF-LV patient
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 3.** Bland–Altman plots for the ratio of $E_{max}/t_{max}$ and maximum value of time derivative of ventricular pressure [$\max(dP/dt)$] for single-ventricle patients (black), and left (red) and right (blue) ventricles of patients with repaired tetralogy of Fallot (rTOF-LV and rTOF-RV, respectively). Solid horizontal lines are the mean of the percentage difference between data and the model, top and bottom dashed horizontal lines represent the limits of agreement at 95% confidence interval (±1.96 SD). $E_{max}$, maximum value of time-varying elastance [$E(t)$].</figcaption></figure>

groups. This suggests that for a given $E_{max}$ rTOF-RVs generate lower levels of $\max(dP/dt)$. This could be caused by a difference in geometry (crescent-shape RV) and possibly also a different level of afterload between the ventricles, and the fact that the $k_{ref}$ value was obtained in the experiments with human systemic left ventricles (as we pointed out in the MATERIALS AND METHODS) (27). These important aspects of the LV and RV differences will be studied in our future research.

## Clinical Implications

Biomechanical modeling offers a robust framework to combine multimodality data (e.g., when pressure measurements are combined with cine MRI, computed tomography, and echocardiography) into a single cohesive physiological assessment. This approach addresses a long-standing limitation in heart disease diagnostic evaluation, where catheter-derived pressures and MRI-derived volumes are often considered separately, by generating high-quality patient-specific PV loops. Importantly, model-derived PV loops provide clinical insights beyond that given by EDP or volumetric data alone. For example, model-derived myocardial contractility enables novel, clinically meaningful stratification of the effect of combined and independent pulmonary valve regurgitation and stenosis

on RV mechanics in patients with rTOF (7). In patients with ccTGA, model-derived contractility quantified biventricular mechanical response to an in silico double switch operation, offering a more mechanically grounded assessment of patients readiness for a complex double switch operation (10).

Patient-specific biomechanical modeling has the potential to advance noninvasive reconstruction of PV loops. Recent studies showed promising results in evaluating LV physiology from noninvasive PV loops reconstructed from cine MRI and brachial cuff pressure measurements using the TVE model (31–33). The method presented here can extend this capability by producing synthetic P-V traces for a wide range of pathologies (including, for instance, valvular disease, which would be a limitation of the TVE model) from noninvasive discrete measurements of end-systolic/-diastolic volumes, phase contrast flow through the artery, heart rate, and max/min pressures in the artery.

Overall, current modeling techniques can facilitate larger retrospective cohort studies using historical datasets, enabling new mechanistic insights across diverse pathologies and informing optimal clinical management strategies. Broader adoption of this framework to catheter and MRI protocols could allow other centers to combine invasive and imaging-based measurement into a unified
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 5.** Correlation between data- and model-derived indicators of cardiovascular function. First row: data-derived $E_{max}$ (mmHg/mL) and model-derived contractility (kPa) (indexed by ventricular radius and wall thickness); second row: data-derived max(dP/dt) (mmHg/s) and model-derived contractility (kPa); third row: data-derived vascular resistance (i.e., ratio of mean arterial pressure over cardiac output) and model-derived resistance of distal circulation $R_d$; fourth row: data-derived vascular resistance and model-derived ventricular max(dP/dt) (mmHg/s) divided by EDV (mL). EDV, end-diastolic volume; max(dP/dt), maximum time-derivative of ventricular pressure; R, Pearson correlation coefficient at *P* < 0.05. Correlation for single-ventricle systemic circulation data-derived vascular resistance vs. model-derived max(dP/dt)/EDV had *P* > 0.05, all other correlations were significant with *P* < 0.05. dP/dt, time-derivative of ventricular pressure; $E_{max}$, maximum value of time-varying elastance [$E(t)$]; max(dP/dt), maximum time-derivative of ventricular pressure.</figcaption></figure>

they are substantially more demanding computationally and technically (42).

Catheterization using a conductance catheter is a gold standard technique for the acquisition of PV loops. A future objective would be to perform a similar study with such a catheter to further validate model-derived P-V loops and established $E_{max}/t_{max}$ versus max(dP/dt) relationship.

## Conclusion

Physiological experiments provide a link between the ventricular elastance and the inotropic state of the ventricle (the level of active stress generated by the myocardium during ventricular contraction). The maximum ventricular elastance and max(dP/dt) represent surrogate measures of myocardial contractility: the former related to ESPVR and time-

varying elastance, the latter directly derived from ventricular pressure. This work validated an important property of the time-varying elastance by demonstrating a significant relationship between the maximum ventricular elastance and max(dP/dt), representing the ventricular end of ejection and isovolumetric phases, respectively. This was demonstrated on ventricular pressure and volume traces both obtained directly from clinical measurements, as well as those provided by the model coupled to the clinical data.

In comparison to the measured P-V waveforms, the model-derived P-V signals provided quantitatively stronger relationships between ventricular elastance and max(dP/dt). Therefore, the model can assist in clinical data processing by complementing noisy data traces and improving objective assessment of cardiac function. In our work, the model-derived dP/dt was superior as a
<figure><img src="image_15.png" /></figure>

physiological marker than the noisier value of dP/dt provided directly by the measurement system. This work demonstrates the validity of novel model-derived indices to quantify cardiovascular physiology in clinical settings. The proposed approach integrates catheter and MRI protocols into unified framework, enabling broader use of catheter and imaging data across the centers worldwide and enhancing diagnostic accuracy of a wider spectrum of heart disease. This study contributes to the increased understanding of multimodality cardiovascular exams (MRI and possibly also other cardiac imaging techniques) and their links to cardiac (patho)physiology.

## DATA AVAILABILITY

Clinical data that have been used are confidential.

## SUPPLEMENTAL MATERIAL

Supplemental Figs. S1 and S2 and Supplemental Tables S1 and S2: https://doi.org/10.6084/m9 figshare.29998306.v1.

## ACKNOWLEDGMENTS

The authors acknowledge Dr. Philippe Moireau, Ecole Polytechnique and Inria France, for the development of the cardiac simulation software CardiacLab used in this work.

## GRANTS

This work was supported by the Inria-UT Southwestern Associated Team Tetralogy of Fallot and Modeling of Diseases (TOFMOD). It was also funded in part from by the W.B. & Ellen Gordon Stuart Trust, The Communities Foundation of Texas, and a Pogue Family Distinguished Chair. Research reported in this publication was supported by Children's Health.

## DISCLAIMER

The content is solely the responsibility of the authors and does not necessarily represent the official views of Children's Health.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

M.G., D.A.C., T.H., D.C., and R.C. conceived and designed research; S.V.R. performed experiments; M.G. and D.A.C. analyzed data; M.G., D.A.C., S.V.R., T.H., D.C., and R.C. interpreted results of experiments; M.G. prepared figures; M.G. drafted manuscript; M.G., D.A.C., S.V.R., T.H., D.C., and R.C. edited and revised manuscript; M.G., D.A.C., S.V.R., T.H., D.C., and R.C. approved final version of manuscript.

## REFERENCES

1. Hiremath G, Batlivala S, Callahan R, Thatte N, Rockefeller T, Nawaytou H, Reddy SV, Hussain T, Chabiniok R, Butts R, Vettukattil J, Aregullin EO, Aldweib N, Burkhoff D, Brener MI. Clinical applications of pressure-volume assessment in congenital heart disease. *J Soc Cardiovasc Angiogr Interv* 2: 100599, 2023. doi:10.1016/j.jscai.2023.100599.

2. Suga H, Sagawa K, Shoukas AA. Load independence of instantaneous pressure-volume ratio of canine left ventricle and effects of epinephrine and heart-rate on ratio. *Circ Res* 32: 314–322, 1973. doi:10.1161/01.res.32.3.314.

3. Caruel M, Chabiniok R, Moireau P, Lecarpentier Y, Chapelle D. Dimensional reductions of a cardiac model for effective validation and calibration. *Biomech Model Mechanobiol* 13: 897–914, 2014. doi:10.1007/s10237-013-0544-6.

4. Sagawa K. The end-systolic pressure-volume relation of the ventricle: definition, modifications and clinical use. *Circulation* 63: 1223–1227, 1981. doi:10.1161/01.cir.63.6.1223.

5. Burkhoff D, Mirsky I, Suga H. Assessment of systolic and diastolic ventricular properties via pressure-volume analysis: a guide for clinical, translational, and basic researchers. *Am J Physiol Heart Circ Physiol* 289: H501–H512, 2005. doi:10.1152/ajpheart.00138.2005.

6. Le Gall A, Vallée F, Pushparajah K, Hussain T, Mebazaa A, Chapelle D, Gayat E, Chabiniok R. Monitoring of cardiovascular physiology augmented by a patient-specific biomechanical model during general anesthesia. A proof of concept study. *PLoS One* 15: e0232830, 2020. doi:10.1371/journal.pone.0232830.

7. Gusseva M, Hussain T, Friesen CH, Moireau P, Tandon A, Patte C, Genet M, Hasbani K, Greil G, Chapelle D, Chabiniok R. Biomechanical modeling to inform pulmonary valve replacement in tetralogy of fallot patients after complete repair. *Can J Cardiol* 37: 1798–1807, 2021. doi:10.1016/j.cjca.2021.06.018.

8. de Vecchi A, Clough RE, Gaddum NR, Rutten MC, Lamata P, Schaeffter T, Nordsletten DA, Smith NP. Catheter-induced errors in pressure measurements in vessels: an in-vitro and numerical study. *IEEE Trans Biomed Eng* 61: 1844–1850, 2014. doi:10.1109/TBME.2014.2308594.

9. Ruijsink B, Zugaj K, Wong J, Pushparajah K, Hussain T, Moireau P, Razavi R, Chapelle D, Chabiniok R. Dobutamine stress testing in patients with Fontan circulation augmented by biomechanical modeling. *PLoS One* 15: e0229015, 2020. doi:10.1371/journal.pone.0229015.

10. Gusseva M, Thatte N, Castellanos DA, Hammer PE, Ghelani SJ, Callahan R, Hussain T, Chabiniok R. Biomechanical modeling combined with pressure-volume loop analysis to aid surgical planning in patients with complex congenital heart disease. *Med Image Anal* 10: 103441, 2025. doi:10.1016/j.media.2024.103441.

11. Little WC. The Left-ventricular dp/dtmax-end-diastolic volume relation in closed-chest dogs. *Circ Res* 56: 808–815, 1985. doi:10.1161/01.res.56.6.808.

12. Starling MR, Kirsh MM, Montgomery DG, Gross MD. Mechanisms for left ventricular systolic dysfunction in aortic regurgitation: importance for predicting the functional response to aortic valve replacement. *J Am Coll Cardiol* 17: 887–897, 1991. doi:10.1016/0735-1097(91)90870-f.

13. Bogaard MD, Houthuizen P, Bracke FA, Doevendans PA, Prinzen FW, Meine M, van Gelder BM. Baseline left ventricular dP/dtmax rather than the acute improvement in dP/dtmax predicts clinical outcome in patients with cardiac resynchronization therapy. *Eur J Heart Fail* 13: 1126–1132, 2011. doi:10.1093/eurjhf/hfr094.

14. Veeram Reddy SR, Arar Y, Zahr RA, Gooty V, Hernandez J, Potersnak A, Douglas P, Blair Z, Greer JS, Roujol S, Forte MNV, Greil G, Nugent AW, Hussain T. Invasive cardiovascular magnetic resonance (ICMR) for diagnostic right and left heart catheterization using an MR-conditional guidewire and passive visualization in congenital heart disease. *J Cardiovasc Magn Reson* 22: 20–11, 2020. doi:10.1186/s12968-020-0605-9.

15. Genet M, Stoeck CT, Von Deuster C, Lee LC, Kozerke S. Equilibrated warping: Finite element image registration with finite strain equilibrium gap regularization. *Med Image Anal* 50: 1–22, 2018. doi:10.1016/j.media.2018.07.007.

16. Chapelle D, Le Tallec P, Moireau P, Sorine M. Energy-preserving muscle tissue model: formulation and compatible discretizations. *Int J Mult Comp Eng* 10: 189–211, 2012. doi:10.1615/IntJMultCompEng.2011002360.

17. Huxley AF. Muscle structure and theories of contraction. *Prog Biophys Biophys Chem* 7: 255–318, 1957. doi:10.1016/S0096-4174(18)30128-8.

18. Bestel J, Clément F, Sorine M. A biomechanical model of muscle contraction. *4th International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI 2001).* Utrecht, Netherlands, Oct 14–17, 2001, p. 1159–1161.
<figure><img src="image_16.png" /></figure>

19. Hill AV. The heat of shortening and the dynamic constants of muscle. *Proceedings of the Royal Society of London Series B-Biological Sciences*. The Royal Society, 1938, vol. 126, p.136-195.

20. Holzapfel GA, Ogden RW. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. *Philos Trans A Math Phys Eng Sci* 367: 3445-3475, 2009. doi:10.1098/rsta.2009.0091.

21. Hadjicharalambous M, Chabiniok R, Asner L, Sammut E, Wong J, Carr-White G, Lee J, Razavi R, Smith N, Nordsletten D. Analysis of passive cardiac constitutive laws for parameter estimation using 3D tagged MRI. *Biomech Model Mechanobiol* 14: 807-828, 2015. doi:10.1007/s10237-014-0638-9.

22. Sainte-Marie J, Chapelle D, Cimrman R, Sorine M. Modeling and estimation of the cardiac electromechanical activity. *Computers & structures* 84: 1743-1759, 2006. doi:10.1016/j.compstruc.2006.05.003.

23. Stergiopulos N, Westerhof BE, Westerhof N. Total arterial inertance as the fourth element of the windkessel model. *Am J Physiol Heart Circ Physiol* 276: H81-H88, 1999. doi:10.1152/ajpheart.1999.276.1.H81.

24. Klotz S, Hay I, Dickstein ML, Yi GH, Wang J, Maurer MS, Kass DA, Burkhoff D. Single-beat estimation of end-diastolic pressure-volume relationship: a novel method with potential for noninvasive application. *Am J Physiol Heart Circ Physiol* 291: H403-H412, 2006. doi:10.1152/ajpheart.01240.2005.

25. Gusseva M, Castellanos DA, Greer JS, Hussein MA, Hasbani K, Greil G, Veeram Reddy SR, Hussain T, Chapelle D, Chabiniok R. Time-synchronization of interventional cardiovascular magnetic resonance data using a biomechanical model for pressure-volume loop analysis. *J Magn Reson Imaging* 57: 320-323, 2023. doi:10.1002/jmri.28216.

26. Stevenson D, Revie J, Chase JG, Hann CE, Shaw GM, Lambermont B, Ghuysen A, Kolh P, Desaive T. Beat-to-beat estimation of the continuous left and right cardiac elastance from metrics commonly available in clinical settings. *Biomed Eng Online* 11: 73-14, 2012. doi:10.1186/1475-925X-11-73.

27. Senzaki H, Chen CH, Kass DA. Single-beat estimation of end-systolic pressure-volume relation in humans. A new method with the potential for noninvasive application. *Circulation* 94: 2497-2506, 1996. doi:10.1161/01.cir.94.10.2497.

28. Monge Garcia MI, Jian Z, Settels JJ, Hunley C, Cecconi M, Hatib F, Pinsky MR. Performance comparison of ventricular and arterial dP/dtmax for assessing left ventricular systolic function during different experimental loading and contractile conditions. *Crit Care* 22: 1-12, 2018. doi:10.1186/s13054-018-2260-1.

29. Morimont P, Lambermont B, Desaive T, Janssen N, Chase G, D'Orio V. Arterial dP/dt max accurately reflects left ventricular contractility during shock when adequate vascular filling is achieved. *BMC Cardiovasc Disord* 12: 13-16, 2012. doi:10.1186/1471-2261-12-13.

30. Godfrey ME, Rathod RH, Keenan E, Gauvreau K, Powell AJ, Geva T, Prakash A. Inefficient ventriculoarterial coupling in Fontan patients: a cardiac magnetic resonance study. *Pediatr Cardiol* 39: 763-773, 2018. doi:10.1007/s00246-018-1819-6.

31. Sjöberg P, Seemann F, Arheden H, Heiberg E. Non-invasive quantification of pressure-volume loops from cardiovascular magnetic

resonance at rest and during dobutamine stress. *Clin Physiol Funct Imaging* 41: 467-470, 2021. doi:10.1111/cpf.12718.

32. Seemann F, Arvidsson P, Nordlund D, Kopic S, Carlsson M, Arheden H, Heiberg E. Noninvasive quantification of pressure-volume loops from brachial pressure and cardiovascular magnetic resonance. *Circ Cardiovasc Imaging* 12: e008493, 2019 [Erratum in *Circ Cardiovasc Imaging* 18: e000087, 2025]. doi:10.1161/CIRCIMAGING.118.008493.

33. Edlund J, Arvidsson PM, Nelsson A, Smith JG, Magnusson M, Heiberg E, Steding-Ehrenborg K, Arheden H. Noninvasive assessment of left ventricular pressure-volume relations: inter-and intraobserver variability and assessment across heart failure subtypes. *Am J Cardiol* 184: 48-55, 2022 [Erratum in *Am J Cardiol* 236: 98, 2025]. doi:10.1016/j.amjcard.2022.09.001.

34. Mehmel H, Stockins B, Ruffmann K, Von Olshausen K, Schuler G, Kübler W. The linearity of the end-systolic pressure-volume relationship in man and its sensitivity for assessment of left ventricular function. *Circulation* 63: 1216-1222, 1981. doi:10.1161/01.cir.63.6.1216.

35. Brown KA, Ditchey RV. Human right ventricular end-systolic pressure-volume relation defined by maximal elastance. *Circulation* 78: 81-91, 1988. doi:10.1161/01.cir.78.1.81.

36. Gusseva M, Hussain T, Hancock Friesen C, Greil G, Chapelle D, Chabiniok R. Prediction of ventricular mechanics after pulmonary valve replacement in tetralogy of Fallot by biomechanical modeling: a step towards precision healthcare. *Ann Biomed Eng* 49: 3339-3348, 2021. doi:10.1007/s10439-021-02895-9.

37. Ruijsink B, Zugaj K, Pushparajah K, Chabiniok R. Model-based indices of early-stage cardiovascular failure and its therapeutic management in Fontan patients. In: *Functional Imaging and Modeling of the Heart: 10th International Conference, FIMH 2019*. Springer, 2019, vol. 10, p. 379-387.

38. Lumens J, Delhaas T, Kirn B, Arts T. Three-wall segment (TriSeg) model describing mechanics and hemodynamics of ventricular interaction. *Ann Biomed Eng* 37: 2234-2255, 2009. doi:10.1007/s10439-009-9774-2.

39. Sermesant M, Chabiniok R, Chinchapatnam P, Mansi T, Billet F, Moireau P, Peyrat JM, Wong K, Relan J, Rhode K, Ginks M, Lambiase P, Delingette H, Sorine M, Rinaldi CA, Chapelle D, Razavi R, Ayache N. Patient-specific electromechanical models of the heart for the prediction of pacing acute effects in CRT: a preliminary clinical validation. *Med Image Anal* 16: 201-215, 2012. doi:10.1016/j.media.2011.07.003.

40. Niederer SA, Plank G, Chinchapatnam P, Ginks M, Lamata P, Rhode KS, Rinaldi CA, Razavi R, Smith NP. Length-dependent tension in the failing heart and the efficacy of cardiac resynchronization therapy. *Cardiovasc Res* 89: 336-343, 2011. doi:10.1093/cvr/cvq318.

41. Gurev V, Lee T, Constantino J, Arevalo H, Trayanova NA. Models of cardiac electromechanics based on individual hearts imaging data: image-based electromechanical models of the heart. *Biomech Model Mechanobiol* 10: 295-306, 2011 [Erratum in *Biomech Model Mechanobiol* 10: 307, 2011]. doi:10.1007/s10237-010-0235-5.

42. Chabiniok R, Moireau P, Lesault PF, Rahmouni A, Deux JF, Chapelle D. Estimation of tissue contractility from cardiac cine-MRI using a biomechanical heart model. *Biomech Model Mechanobiol* 11: 609-630, 2012. doi:10.1007/s10237-011-0337-8.