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The end-systolic elastance (\(E_{\text {es}}\)) – the slope of the end-systolic pressure-volume relationship (ESPVR) at the end of ejection phase – has become a reliable indicator of myocardial functional state. The estimation of \(E_{\text {es}}\) by the original multiple-beat method is invasive, which limits its routine usage. By contrast, non-invasive single-beat estimation methods, based on the assumption of the linearity of ESPVR and the uniqueness of the normalised time-varying elastance curve \(E^N(t)\) across subjects and physiology states, have been applied in a number of clinical studies. It is however known that these two assumptions have a limited validity, as ESPVR can be approximated by a linear function only locally, and \(E^N(t)\) obtained from a multi-subject experiment includes a confidence interval around the mean function. Using datasets of 3 patients undergoing general anaesthesia (each containing aortic flow and pressure measurements at baseline and after introducing a vasopressor noradrenaline), we first study the sensitivity of two single-beat methods—by Sensaki et al. and by Chen et al.—to the uncertainty of \(E^N(t)\). Then, we propose a minimally-invasive method based on a patient-specific biophysical modelling to estimate the whole time-varying elastance curve \(E^{\text {model}}(t)\). We compare \(E^{\text {model}}_{\text {es}}\) with the two single-beat estimation methods, and the normalised varying elastance curve \(E^{N,\text {model}}(t)\) with \(E^{N}(t)\) from published physiological experiments.

International audience; We consider a previously proposed general nonlinear poromechanical formulation, and we derive a linearized version of this model. For this linearized model, we obtain an existence result and we propose a complete discretization strategy - in time and space - with a special concern for issues associated with incompressible or nearly-incompressible behavior. We provide a detailed mathematical analysis of this strategy, the main result being an error estimate uniform with respect to the compressibility parameter. We then illustrate our approach with detailed simulation results and we numerically investigate the importance of the assumptions made in the analysis, including the fulfillment of specific inf-sup conditions.

International audience; We follow a formal homogenization approach to investigate the effects of mechanical deformations in electrophysiology models relying on a bidomain description of ionic motion at the microscopic level. To that purpose, we extend these microscopic equations to take into account the mechanical deformations, and proceed by recasting the problem in the framework of classical two-scale homogenization in periodic media, and identifying the equations satisfied by the first coefficients in the formal expansions. The homogenized equations reveal some interesting effects related to the microstructure - and associated with a specific cell problem to be solved to obtain the macroscopic conductivity tensors - in which mechanical deformations play a non-trivial role, i.e. do not simply lead to a standard bidomain problem posed in the deformed configuration. We then present detailed numerical illustrations of the homogenized model with coupled cardiac electrical-mechanical simulations - all the way to ECG simulations - albeit without taking into account the abundantly-investigated effect of mechanical deformations in ionic models, in order to focus here on other effects. And in fact our numerical results indicate that these other effects are numerically of a comparable order, and therefore cannot be disregarded.

International audience; Considering a large class of muscle contraction models accounting for actin-myosin interaction, we present a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we propose a complete discretization strategy for which we are also able to obtain discrete versions of the thermodynamic balances and other properties. Our major objective is to show how the thermodynamics of such models can be tracked after discretization, including when they are coupled to a macroscopic muscle formulation in the realm of continuum mechanics. Our approach allows to carefully identify the sources of energy and entropy in the system, and to follow them up to the numerical applications.

International audience; The recent achievements in the accelerometer-based seismocardiography field indicate a strong potential for this technique to address wide variety of clinical needs. Recordings from different locations on the chest can give a more comprehensive observation and interpretation of wave propagation phenomena than a single-point recording, can validate existing modeling assumptions (such as the representation of the sternum as a single solid body), and provide better identifiability for models using richer recordings. Ultimately, the goal is to advance our physiological understanding of the processes to provide useful data to promote cardiovascular health. Accelerometer-based multichannel system is a contact method and laborious for use in practice, also even ultralight accelerometers can cause non-negligible loading effects. We propose a new contactless ultrasound imaging method to measure thoracic and abdominal surface motions, demonstrating that it is adequate for typical seismocardiogram use. The developed method extends non-contact surface-vibrometry to fast 2D mapping by originally combining multi-element airborne ultrasound arrays, a synthetic aperture implementation and pulsed-waves. Experimental results show the ability of the developed method to obtain 2D seismocardiographic maps of the body surface 30×40 cm 2 in dimension, with a temporal sampling rate of several hundred Hz, using ultrasound waves with the central frequency of 40 kHz. Our implementation was validated in-vivo on eight healthy human participants. The shape and position of the zone of maximal absolute acceleration and velocity during the cardiac cycle were also observed. This technology could potentially be used to obtain more complete cardio-vascular information than single-source SCG in and out of clinical environments, due to enhanced identifiability provided by distributed measurements, and observation of propagation phenomena.

Seismocardiography has been well studied in terms of analysis, applications, and methods of measurement. However, there remains a lack of research into the explanation and modelling of the involved phenomena. We propose a new contactless method to measure thoracic and sternal movements, demonstrate that it is adequate for typical seismocardiogram use. An ultrasonic diagnostic tool called ICARE (CArdio REspiratory Imager) was designed to perform non-contact ultrasonic waves imaging on the thorax and abdomen. In addition to ICARE measurements an accelerometer was placed above the xiphoid of 3 participants (male, age 39±11). Both ICARE and accelerometer measurements were performed concurrently. Experimental results show the ability of the ICARE system to obtain 3D seismocardiographic images with high frequency frame rate. Furthermore, this technology could potentially be used to obtain cardio-vascular information in and out of clinical environments, significantly lowering the required time and effort.

Proposed time discretization scheme for general two-phase poromechanical model.Discrete energy estimate with the total free energy of the mixture, in a general nonlinear framework.Numerical examples with representative test problems. We consider a general nonlinear poromechanical model, formulated based on fundamental thermodynamics principle, suitable for representing the coupling of rapid internal fluid flows with large deformations of the solid, and compatible with a wide class of constitutive behavior. The objective of the present work is to propose for this model a time discretization scheme of the partitioned type, to allow the use of existing time schemes and possibly separate solvers for each component of the model, i.e. for the fluid and the solid. To that purpose, we adapt and extend an earlier proposed approach devised for fluid-structure interaction in an Arbitrary Lagrangian-Eulerian framework. We then establish an energy estimate for the resulting time scheme, in a form that is consistent with the underlying energy principle in the poromechanical formulation, up to some numerical dissipation effects and some perturbations that we have carefully identified and assessed. In addition, we provide some numerical illustrations of our numerical strategy with test problems that present typical features of large strains and rapid fluid flows, and also a case of singular transition related to total drainage. An example of challenging application envisioned for this model and associated numerical coupling scheme concerns the perfusion of the heart.

In this work we introduce the modeling of atrioventricular valve regurgitation in a spatially reduced order biomechanical heart model. The model can be fast calibrated using non-invasive data of cardiac magnetic resonance imaging and provides an objective measure of contractile properties of the myocardium in the volume overloaded ventricle, for which the real systolic function may be masked by the significant level of the atrioventricular valve regurgitation. After demonstrating such diagnostic capabilities, we show the potential of modeling to address some clinical questions concerning possible therapeutic interventions for specific patients. The fast running of the model allows targeting specific questions of referring clinicians in a clinically acceptable time.

With heart and cardiovascular diseases continually challenging healthcare systems worldwide, translating basic research on cardiac (patho)physiology into clinical care is essential. Exacerbating this already extensive challenge is the complexity of the heart, relying on its hierarchical structure and function to maintain cardiovascular flow. Computational modelling has been proposed and actively pursued as a tool for accelerating research and translation. Allowing exploration of the relationships between physics, multiscale mechanisms and function, computational modelling provides a platform for improving our understanding of the heart. Further integration of experimental and clinical data through data assimilation and parameter estimation techniques is bringing computational models closer to use in routine clinical practice. This article reviews developments in computational cardiac modelling and how their integration with medical imaging data is providing new pathways for translational cardiac modelling.

Cardiac resynchronisation therapy (CRT) is an effective treatment for patients with congestive heart failure and a wide QRS complex. However, up to 30% of patients are non-responders to therapy in terms of exercise capacity or left ventricular reverse remodelling. A number of controversies still remain surrounding patient selection, targeted lead implantation and optimisation of this important treatment. The development of biophysical models to predict the response to CRT represents a potential strategy to address these issues. In this article, we present how the personalisation of an electromechanical model of the myocardium can predict the acute haemodynamic changes associated with CRT. In order to introduce such an approach as a clinical application, we needed to design models that can be individualised from images and electrophysiological mapping of the left ventricle. In this paper the personalisation of the anatomy, the electrophysiology, the kinematics and the mechanics are described. The acute effects of pacing on pressure development were predicted with the in silico model for several pacing conditions on two patients, achieving good agreement with invasive haemodynamic measurements: the mean error on dP/dtmax is 47.5 ± 35 mm Hg s−1, less than 5% error. These promising results demonstrate the potential of physiological models personalised from images and electrophysiology signals to improve patient selection and plan CRT.

International audience; In this paper we propose a muscle tissue model -- valid for striated muscles in general, and for the myocardium in particular -- based on a multi-scale physiological description. This model extends and refines an earlier-proposed formulation by allowing to account for all major energy exchanges and balances, from the chemical activity coupled with oxygen supply to the production of actual mechanical work, namely, the biological function of the tissue. We thus perform a thorough analysis of the energy mechanisms prevailing at the various scales, and we proceed to propose a complete discretization strategy -- in time and space -- respecting the same balance laws. This will be crucial in future works to adequately model the many important physiological -- normal and pathological -- phenomena associated with these energy considerations.

International audience; Parameter estimation from non-invasive measurements is a crucial step in patient-specific cardiac modeling. It also has the potential to provide sig- nificant assistance in the clinical diagnosis of cardiac diseases through the quantification of myocardial material heterogeneity. In this paper, we for- mulate a novel Reduced-order Unscented Kalman Filter (rUKF) applied to the left ventricular (LV) nonlinear mechanical model based on cubic-Hermite finite elements. Material parameters in the widely-employed transversely isotropic Guccione's constitutive law are successfully identified for both ho- mogeneous and heterogeneous cases. We conclude that the four parameters in Guccione's law can be uniquely and correctly determined in silico from noisy displacement measurements of material points located on the myocar- dial surfaces. The future application of this novel and effective approach to real clinical measurements is thus promising.

International audience; The loss of cardiac pump function accounts for a significant increase in both mortality and morbidity in Western society, where there is currently a one in four lifetime risk, and costs associated with acute and long-term hospital treatments are accelerating. The significance of cardiac disease has motivated the application of state-of-the-art clinical imaging techniques and functional signal analysis to aid diagnosis and clinical planning. Measurements of cardiac function currently provide high-resolution datasets for characterizing cardiac patients. However, the clinical practice of using population-based metrics derived from separate image or signal-based datasets often indicates contradictory treatments plans owing to inter-individual variability in pathophysiology. To address this issue, the goal of our work, demonstrated in this study through four specific clinical applications, is to integrate multiple types of functional data into a consistent framework using multi-scale computational modelling.

International audience; Effective shell finite elements are of utmost interest in engineering practice, and more particularly triangular shell elements are often needed to handle complex geometries. The quadrilateral elements of the MITC family have been shown to be effective for general asymptotic behaviors, but a triangular MITC shell finite element of comparable performance remains to be found. The main difficulty is to reduce the locking phenomenon arising in bending dominated situations without destroying the ability of the procedure to accurately capture membrane dominated and mixed behaviors. In this context, we have designed a simple numerical rank test aimed at assessing the reliability of MITC triangular elements. This test led to selecting some specific MITC6 tying schemes, which have been further assessed using some meaningful test problems. The assessment results showed that the MITC6a shell element is a most promising candidate, although it may feature some non-physical membrane spurious modes in some specific shell problems.

We present a novel strategy to perform estimation for a dynamical mechanical system in standard operating conditions, namely, without ad hoc experimental testing. We adopt a sequential approach, and the joint state-parameter estimation procedure is based on a state estimator inspired from collocated feedback control. This type of state estimator is chosen due to its particular effectiveness and robustness, but the methodology proposed to adequately extend state estimation to joint state-parameter estimation is general, and - indeed -applicable with any other choice of state feedback observer. The convergence of the resulting joint estimator is mathematically established. In addition, we demonstrate its effectiveness with a biomechanical test problem defined to feature the same essential characteristics as a heart model, in which we identify localized contractility and stiffness parameters using measurements of a type that is available in medical imaging.

This paper presents a numerical study in which several partitioned solution procedures for incompressible fluid-structure interaction are compared and validated against the results of an experimental fluid-structure interaction benchmark. The numerical methods discussed cover the three main families of coupling schemes: strongly coupled, semi-implicit, and loosely coupled. Very good agreement is observed between the numerical and experimental results. The comparisons confirm that strong coupling can be efficiently avoided, via semi-implicit and loosely coupled schemes, without compromising stability and accuracy. Copyright © 2016 John WileySons, Ltd.

Seismocardiography (SCG) is measurement of sternal acceleration caused by heart beats. Although fiducial cardiac events have been associated with seismocardiogram extrema, the forces that cause the vibrations are unknown. The goal of this study was to create a 3D model of the thorax capable of modelling its vibrations under heart-like forces.

International audience; We propose a Luenberger observer for reaction-diffusion models with propagating front features, and for data associated with the location of the front over time. Such models are considered in various application fields, such as electrophysiology, wild-land fire propagation and tumor growth modeling. Drawing our inspiration from image processing methods, we start by proposing an observer for the eikonal-curvature equation that can be derived from the reaction-diffusion model by an asymptotic expansion. We then carry over this observer to the underlying reaction-diffusion equation by an "inverse asymptotic analysis", and we show that the associated correction in the dynamics has a stabilizing effect for the linearized estimation error. We also discuss the extension to joint state-parameter estimation by using the earlier-proposed ROUKF strategy. We then illustrate and assess our proposed observer method with test problems pertaining to electrophysiology modeling, including with a realistic model of cardiac atria. Our numerical trials show that state estimation is directly very effective with the proposed Luenberger observer, while specific strategies are needed to accurately perform parameter estimation – as is usual with Kalman filtering used in a nonlinear setting – and we demonstrate two such successful strategies.

International audience; We present a patient-specific biomechanical modeling framework and an initial case study for investigating cardiac amyloidosis (CA). Our patient-specific heartbeat simulations are in good agreement with the data, and our model calibration indicates that the major effect of CA in the biophysical behavior lies in a dramatic increase of the passive stiffness. We also conducted a preliminary trial for predicting the effects of pharmacological treatments – which is an important clinical challenge – based on the model combined with a simple venous return representation. This requires further investigation and validation, albeit provides some valuable preliminary insight.

In this article, we present a framework to estimate cardiac function parameters like local myocardium contractility using clinical MRI, a heart model and data assimilation. First, we build a generic anatomical model of the ventricles including muscle fibre orientations and anatomical subdivi- sions. Then, this model is deformed to fit a segmented MRI, using an affine registration method and a local deformable biomechanical model. An electromechanical model of the heart can be simulated on this mesh. Data assimilation makes it possible to estimate local contractility from given displacements. Presented results on simulated data and adjustment to clinical data are very promising. Current work on model calibration and estimation of patient parameters open up possibilities for clinical application of this framework. Resume. Dans cet article, nous presentons un cadre de travail pour estimer certains parametres de la fonction cardiaque (comme la contractilite locale) en utilisant l'IRM, un modele du myocarde et l'assimilation de donnees. Tout d'abord, nous detaillons la construction d'un modele generique du my- ocarde incluant l'orientation des fibres musculaires et les differents segments anatomiques. Ce modele peutdeforme pour s'ajuster aux images cliniques. Un modeleecanique du myocarde est ensuite presentee t simule. Enfin, l'assimilation de donnees permet d'estimer la contractilitel ocalea partir de donnees telles que les deplacements. Les resultats sur l'ajustement du modelea geometrie du patient et l'assimilation sur des donnees simulees sont tres encourageants. Les travaux en cours sur la calibration du modele et l'estimation de parametres du patient ouvrent de nouvelles possibilites pour l'application de ce cadre de travail dans un environnement clinique.

Les progrès réalisés en mathématiques appliquées permettent aujourd’hui d’envisager la simulation sur ordinateur de certains compartiments du système cardiovasculaire. Nous proposons de faire un point sur quelques modèles, en nous focalisant sur la simulation de l’écoulement du sang dans des artères déformables et sur la simulation de la contraction du myocarde sous l’effet de la propagation d’un signal électrique. Nous tentons également de présenter des applications possibles de ce type de travaux., In this article, we aim at giving a non-technical overview of some mathematical models currently used in the numerical simulation of the cardiovascular system. A hierarchy of models for blood flows in large arteries is briefly described as well as an electromechanical model for the heart. We discuss some possible applications of the numerical simulations of such models, for example the optimization of prostheses. We also address the issue of the data assimilation for the calibration of the models.

International audience; We propose a variational formulation suitable for coupling a shell and a surrounding softer 3D medium. This is intended to model the behavior of a reinforcing sheet as frequently occurs in applications. The crucial issue of the kinematical coupling conditions is addressed by taking into account the rotation degrees of freedom of the shell. We then perform an asymptotic analysis of the coupled model in which the distinction between bending-dominated and membrane-dominated shells arises. For the limit solutions we identify coupling kinematical conditions that do not involve rotations. Finally we present some numerical results that illustrate our discussions.

Resume La plupart des methodes de simulation numerique du tombe de tissus sont basees sur un modele mecanique s'appuyant, en general, sur un maillage surfacique uniforme. Le tissu etant un materiau tres souple, de nombreux plis peuvent se former sur sa surface lorsque celui-ci est en mouvement libre ou contraint (collisions, points d'arrimage, etc.). Le probleme de la simulation est de faire evoluer le systeme mecanique tout en ayant une representation geometrique realiste de la surface. Il se pose alors un probleme de discretisation adequate de la surface. Pour remedier, a ce probleme, nous proposons une methode basee sur un maillage adaptatif permettant au modele mecanique d'evoluer sans la contrainte d'un maillage uniforme. Nous decrivons aussi un nouveau modele mecanique adapte ainsi que notre methode de raffinement local du maillage. Un exemple numerique vient illustrer l'interet de notre methode. Pour citer cet article : J. Villard et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 561–566.

International audience; Complex 3D beating heart models are now available, but their complexity makes calibration and validation very difficult tasks. We thus propose a systematic approach of deriving simplified reduced-dimen\-sional models, in ''0D'' --~typically, to represent a cardiac cavity, or several coupled cavities --~and in ''1D'' --~to model elongated structures such as muscle samples or myocytes. We apply this approach with an earlier-proposed 3D cardiac model designed to capture length-dependence effects in contraction, which we here complement by an additional modeling component devised to represent length-dependent relaxation. We then pre\-sent experimental data produced with rat papillary muscles samples when varying preload and afterload conditions, and we achieve some detailed validations of the 1D model with these data, including for the length-dependence effects that are accurately captured. Finally, when running simulations of the 0D model pre-calibrated with the 1D model parameters, we obtain pressure-volume indicators of the left ventricle in good agreement with some important features of cardiac physiology, including the so-called Frank-Starling mechanism, the End-Systolic Pres\-sure-Volume Relationship (ESPVR), as well as varying elastance properties. This integrated multi-dimensional modeling approach thus sheds new light on the relations between the phenomena observed at different scales and at the local vs. organ levels.

We obtain estimates for the consistency errors arising in general shell element procedures, which are widely used in engineering practice. These estimates improve a previous result by the same authors. Moreover, numerical experiments indicate that these new estimates are optimal. Further, we introduce a modified procedure for which nominal convergence is recovered. These results are of much practical significance.

International audience; We present an inf-sup test for general mixed shell finite element discretizations. The test is useful in the thorough evaluation of a shell finite element discretization scheme. We apply the test to the MITC shell elements and find that these elements pass the test.

We propose a simple modification of a recently introduced locking-free finite element method for the Reissner–Mindlin plate model. By this modification, we are able to obtain optimal convergence rates on numerical benchmarks. These results are substantiated by a complete mathematical analysis which provides optimal a priori error estimates.

The objective in this paper is to present fundamental considerations regarding the finite element analysis of shell structures. First, we review some well-known results regarding the asymptotic behaviour of a shell mathematical model. When the thickness becomes small, the shell behaviour falls into one of two dramatically different categories; namely, the membrane-dominated and bending-dotninated cases. The shell geometry and boundary conditions decide into which category the shell structure falls, and a seemingly small change in these conditions can result into a change of category and hence into a dramatically different shell behaviour. An effective finite element scheme should be applicable to both categories of shell behaviour and the rate of convergence in either case should be optimal and independent of the shell thickness. Such a finite element scheme is difficult to achieve but it is important that existing procedures be analysed and measured with due regard to these considerations. To this end, we present theoretical considerations and we propose appropriate shell analysis test cases for numerical evaluations.

International audience; We propose a direct method for computing modal coupling coefficients - due to geometrically nonlinear effects - for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the finite element formulation, with specificities pertaining to the shell elements considered, namely, here elements of the ''Mixed Interpolation of Tensorial Components'' family (MITC). Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing - with a continuation procedure - the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes (NNMs), and we find that the most important part of the dynamics can be predicted with a single oscillator equation.

Updated version of previously published research report; International audience; We formulate a general poromechanics model -- within the framework of a two-phase mixture theory -- compatible with large strains and without any simplification in the momentum expressions, in particular concerning the fluid flows. The only specific assumptions made are fluid incompressibility and isothermal conditions. Our formulation is based on fundamental physical principles -- namely, essential conservation and thermodynamics laws -- and we thus obtain a Clausius-Duhem inequality which is crucial for devising compatible constitutive laws. We then propose to model the solid behavior based on a generalized hyperelastic free energy potential -- with additional viscous effects -- which allows to represent a wide range of mechanical behaviors. The resulting formulation takes the form of a coupled system similar to a fluid-structure interaction problem written in an Arbitrary Lagrangian-Eulerian formalism, with additional volume-distributed interaction forces. We achieve another important objective by identifying the essential energy balance prevailing in the model, and this paves the way for further works on mathematical analyses, and time and space discretizations of the formulation.; Nous présentons, dans le cadre de la théorie des mélanges, un model poromécanique général valide en grands déplacements et sans simplification sur le bilan de conservation des moments, en particulier pour le système fluide. Les seules hypothèses faites sont l'incompressibilité du fluide et des conditions isothermes. Notre formulation s'appuie sur les principes de la thermodynamiques et nous obtenons une inégalité de Clausius-Duhem fondamentale pour l'obtention de lois de comportement adaptées. Nous proposons alors de modéliser le solide avec un potentiel d'énergie libre hyperélastique généralisé auquel s'ajoute un potentiel visqueux, permettant ainsi de représenter une large gamme de comportements mécaniques. La formulation résultante prend la forme d'un système couplé similaire à ceux rencontrés en interaction fluide-structure de type ALE comprenant un couplage volumique supplémentaire. Nous sommes alors en mesure d'écrire sur le modèle complet des estimations d'énergie qui seront à l'origine de travaux futurs, que ce soit pour l'analyse mathématique du système ou la formulation de discrétisations en temps et en espace adaptées.

We consider an elastic model for a curved rod with arbitrary three-dimensional geometry, incorporating shear and membrane as well as bending and torsion effects. We define an approximation procedure based on a discretization by linear Timoshenko beam elements. Introducing an equivalent mixed problem, we establish optimal error estimates independent of the thickness, thereby proving that shear and membrane locking is avoided. The approximation scheme is tested on specific examples and the numerical results confirm the estimates obtained by theory.

To be published as proceedings in: Functional Imaging and Modeling of the Heart 7th International Conference, FIMH 2013, London, UK, Lecture Notes in Computer Science 7945; International audience; Complex 3D beating heart models are now available, but their complexity makes calibration and validation very difficult tasks. We thus propose a systematic approach of deriving simplified reduced- dimensional models, in "0D" - typically, to represent a cardiac cavity, or several coupled cavities - and in "1D" - to model elongated structures such as fibers or myocytes. As illustrations of our approach, we demon- strate model validation based on experiments performed with papillary muscles, and calibration using patient-specific pressure-volume loops.

International audience; The objective of this paper is to assess a previously-proposed surface-based electrophysiology model with detailed atrial simulations. This model - derived and substantiated by mathematical arguments - is specifically designed to address thin structures such as atria, and to take into account strong anisotropy effects related to fiber directions with possibly rapid variations across the wall thickness. The simulation results are in excellent adequacy with previous studies, and confirm the importance of anisotropy effects and variations thereof. Furthermore, this surface-based model provides dramatic computational benefits over 3D models with preserved accuracy.

International audience; Patient-specific cardiac modelling can help in understanding pathophysiology and therapy planning. However it requires to combine functional and anatomical data in order to build accurate models and to personalize the model geometry, kinematics, electrophysiology and mechanics. Personalizing the electromechanical coupling from medical images is a challenging task. We use the Bestel-Clément-Sorine (BCS) electromechanical model of the heart, which provides reasonable accuracy with a reasonable number of parameters (14 for each ventricle) compared to the available clinical data at the organ level. We propose a personalization strategy from cine MRI data in two steps. We first estimate global parameters with an automatic calibration algorithm based on the Unscented Transform which allows to initialize the parameters while matching the volume and pressure curves. In a second step we locally personalize the contractilities of all AHA (American Heart Association) zones of the left ventricle using the Reduced Order Unscented Kalman Filtering on Regional Volumes. This personalization strategy was validated synthetically and tested successfully on eight healthy and three pathological cases.

International audience; We analyze the performances of two types of Luenberger observers -- namely, the so-called Direct Velocity Feedback and Schur Displacement Feedback procedures, originally devised for elasto-dynamics -- to estimate the state of a fluid-structure interaction model for hemodynamics, when the measurements are assumed to be restricted to displacements or velocities in the solid. We first assess the observers using hemodynamics-inspired test problems with the complete model, including the Navier-Stokes equations in Arbitrary Lagrangian-Eulerian formulation, in particular. Then, in order to obtain more detailed insight we consider several well-chosen simplified models, each of which allowing a thorough analysis -- emphasizing spectral considerations -- while illustrating a major phenomenon of interest for the observer performance, namely, the added mass effect for the structure, the coupling with a lumped-parameter boundary condition model for the fluid flow, and the fluid dynamics effect per se. Whereas improvements can be sought when additional measurements are available in the fluid domain in order to more effectively deal with strong uncertainties in the fluid state, in the present framework this establishes Luenberger observer methods as very attractive strategies -- compared, e.g., to classical variational techniques -- to perform state estimation, and more generally for uncertainty estimation since other observer procedures can be conveniently combined to estimate uncertain parameters.

We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the FitzHugh − Nagumo 1D model. Finally, we give detailed illustrations of the approach in two parameter estimation applications, the first in a variational estimation framework with the FitzHugh − Nagumo model, and the second with a beating heart mechanical model for which we employ a sequential estimation method to characterize model parameters using real image data in a clinical case.

International audience; Computational electrophysiology is a very active field with tremendous potential in medical applications, albeit leads to highly intensive simulations. We here propose a surface-based electrophysiology formulation, motivated by the modeling of thin structures such as cardiac atria, which greatly reduces the size of the computational models. Moreover, our model is specifically devised to retain the key features associated with the anisotropy in the diffusion effects induced by the fiber architecture, with rapid variations across the thickness which cannot be adequately represented by naive averaging strategies. Our proposed model relies on a detailed asymptotic analysis in which we identify a limit model and establish strong convergence results. We also provide detailed numerical assessments which confirm an excellent accuracy of the surface-based model -- compared with the reference 3D model -- including in the representation of a complex phenomenon, namely, spiral waves.

This paper presents a numerical study in which several partitioned solution procedures for incompressible fluid-structure interaction are compared and validated against the results of an experimental FSI benchmark. The numerical methods discussed cover the three main families of coupling schemes: strongly coupled, semi-implicit and loosely coupled. Very good agreement is observed between the numerical and experimental results. The comparisons confirm that strong coupling can be efficiently avoided, via semi-implicit and loosely coupled schemes, without compromising stability and accuracy.

International audience; We analyze an observer strategy based on partial --~i.e.~in a subdomain --~measurements of the solution of a wave equation, in order to compensate for unknown initial conditions. We prove the exponential convergence of this observer under a non-standard observability condition, whereas using measurements of the time-derivative of the solution would lead to a standard observability condition arising in stabilization and exact controlability. Nevertheless, we directly relate our specific condition to the classical geometric control condition. Finally, we provide some numerical illustrations of the effectiveness of the approach.; Nous analysons un observateur formulé pour l'équation des ondes et utilisant des mesures de la solution dans un sous-domaine afin de compenser des conditions initiales inconnues. Nous démontrons la convergence exponentielle de cet observateur moyennant une condition d'observabilité non-standard, alors qu'une mesure de la dérivée en temps de la solution conduirait à la forme standard de la condition d'observabilité apparaissant en stabilisation et contrôla\-bilité exacte. Néanmoins, nous établissons un lien direct entre notre condition spécifique et la condition de contrôle géométrique classique.

International audience; Viscoelastic support has been previously established as a valuable modeling ingredient to represent the effect of surrounding tissues and organs in a fluid-structure vascular model. In this paper, we propose a complete methodological chain for the identification of the corresponding boundary support parameters, using patient image data. We consider distance maps of model to image contours as the discrepancy driving the data assimilation approach, which then relies on a combination of (1) state estimation based on the so-called SDF filtering method, designed within the realm of Luenberger observers and well adapted to handling measurements provided by image sequences, and (2) parameter estimation based on a reduced-order UKF filtering method which has no need for tangent operator computations and features natural parallelism to a high degree. Implementation issues are discussed, and we show that the resulting computational effectiveness of the complete estimation chain is comparable to that of a direct simulation. Furthermore, we demonstrate the use of this framework in a realistic application case involving hemodynamics in the thoracic aorta. The estimation of the boundary support parameters proves successful, in particular in that direct modeling simulations based on the estimated parameters are more accurate than with a previous manual expert calibration. This paves the way for complete patient-specific fluid-structure vascular modeling in which all types of available measurements could be used to estimate additional uncertain parameters of biophysical and clinical relevance.

International audience; The objective of this paper is to propose and assess an estimation procedure - based on data assimilation principles - well-suited to obtain some regional values of key biophysical parameters in a beating heart model, using actual Cine-MR images. The motivation is twofold: (1) to provide an automatic tool for personalizing the characteristics of a cardiac model in order to achieve predictivity in patient-specific modeling, and (2) to obtain some useful information for diagnosis purposes in the estimated quantities themselves. In order to assess the global methodology we specifically devised an animal experiment in which a controlled infarct was produced and data acquired before and after infarction, with an estimation of regional tissue contractility - a key parameter directly affected by the pathology - performed for every measured stage. After performing a preliminary assessment of our proposed methodology using synthetic data, we then demonstrate a full-scale application by first estimating contractility values associated with 6 regions based on the AHA subdivision, before running a more detailed estimation using the actual AHA segments. The estimation results are assessed by comparison with the medical knowledge of the specific infarct, and with late enhancement MR images. We discuss their accuracy at the various subdivision levels, in the light of the inherent modeling limitations and of the intrinsic information contents featured in the data.

International audience; We propose a numerical analysis of Proper Orthogonal Decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and -- whereas a general uniform continuity bound seems out of reach -- we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound -- and the effectiveness of the POD approach altogether -- for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

International audience; We propose an observer-based approach to circumvent the issue of unbounded approximation errors -- with respect to the length of the time window considered -- in the discretization of wave-like equations in bounded domains, which covers the cases of the wave equation per se and of linear elasticity as well as beam, plate and shell formulations, and so on. Namely, taking advantage of some measurements available on the system over time, we adopt a strategy inspired from sequential data assimilation and by which the discrete system is dynamically corrected using the discrepancy between the solution and the measurements. In addition to the classical cornerstones of numerical analysis made up by stability and consistency, we are thus led to incorporating a third crucial requirement pertaining to observability -- to be preserved through discretization. The latter property warrants exponential stability for the corrected dynamics, hence provides bounded approximation errors over time. Special care is needed to establish the required observability at the discrete level, in particular due to the fact that we focus on an original observer method adapted to measurements of the main variable, whereas measurements of the time-derivative -- admissible, of course, albeit less frequent in practical systems -- lead to a stability analysis in which existing results can be more directly applied. We also provide some detailed application examples with several such wave-like equations, and the corresponding numerical assessments illustrate the performance of our approach.

International audience; The objective of this work is to address the formulation of an adequate model of the external tissue environment when studying a portion of the arterial tree with fluid-structure interaction. Whereas much work has already been accomplished concerning flow and pressure boundary conditions associated with truncations in the fluid domain, very few studies take into account the tissues surrounding the region of interest to derive adequate boundary conditions for the solid domain. In this paper, we propose to model the effect of external tissues by introducing viscoelastic support conditions along the artery wall, with two--possibly distributed--parameters that can be adjusted to mimic the response of various physiological tissues. In order to illustrate the versatility and effectiveness of our approach, we apply this strategy to perform patient-specific modeling of thoracic aortae based on clinical data, in two different cases and using a distinct fluid-structure interaction methodology for each, namely an Arbitrary Lagrangian-Eulerian (ALE) approach with prescribed inlet motion in the first case and the coupled momentum method in the second case. In both cases, the resulting simulations are quantitatively assessed by detailed comparisons with dynamic image sequences, and the model results are shown to be in very good adequacy with the data.

We propose a methodology for performing the estimation of a key constitutive parameter in a biomechanical heart model - namely, the tissue contractility - using tagged-MRI data. We adopt a sequential data assimilation strategy, and the image data is assumed to be processed in the form of deforming tag planes, which we employ to obtain a discrepancy between the model and the data by computing distances to these surfaces. We assess our procedure using synthetic measurements produced with a model representing an infarcted heart as observed in an animal experiment, and the estimation results are found to be of superior accuracy compared to assimilation based on segmented endo- and epicardium surfaces.