2023Activity reportProjectTeamCOMMEDIA
RNSR: 201923240H Research center Inria Paris Centre at Sorbonne University
 In partnership with:CNRS, Sorbonne Université
 Team name: Computational mathematics for biomedical applications
 In collaboration with:Laboratoire JacquesLouis Lions (LJLL)
 Domain:Digital Health, Biology and Earth
 Theme:Modeling and Control for Life Sciences
Keywords
Computer Science and Digital Science
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.4. Multiscale modeling
 A6.1.5. Multiphysics modeling
 A6.2.1. Numerical analysis of PDE and ODE
 A6.3.1. Inverse problems
 A6.3.2. Data assimilation
 A6.3.4. Model reduction
Other Research Topics and Application Domains
 B2.2.1. Cardiovascular and respiratory diseases
 B2.4.1. Pharmaco kinetics and dynamics
1 Team members, visitors, external collaborators
Research Scientists
 Miguel Angel Fernández Varela [Team leader, INRIA, Senior Researcher, HDR]
 Céline Grandmont [INRIA, Senior Researcher, HDR]
 Adrien Lefieux [INRIA, Advanced Research Position]
 Damiano Lombardi [INRIA, Researcher, HDR]
 Marina Vidrascu [INRIA, Emeritus]
Faculty Members
 Muriel Boulakia [USVQ, until Jun 2023, secondment , HDR]
 Guillaume Delay [SORBONNE UNIVERSITE, Associate Professor]
 Fabien Vergnet [SORBONNE UNIVERSITE, Associate Professor]
PostDoctoral Fellows
 Cyril Karamaoun [CNRS, PostDoctoral Fellow]
 Sébastien Riffaud [INRIA, PostDoctoral Fellow, until Nov 2023]
PhD Students
 Marguerite Champion [CNRS]
 Daniele Carlo Corti [INRIA]
 Sara Costa Faya [INRIA]
 Maria Fuente Ruiz [INRIA, until Feb 2023]
 Corrie James [UVSQ, from Oct 2023]
 Gael Le Ruz [SORBONNE UNIVERSITE]
 Fabien Lespagnol [POLITECNICO DI MILANO]
 Haibo Liu [NOTOCORD]
 Oscar Ruz [ANID (BECAS CHILE)]
Interns and Apprentices
 Riccardo Bianchi [INRIA, Intern, from Oct 2023]
 Marloes Coolen [INRIA, Intern, from Mar 2023 until Jun 2023]
 Corrie James [INRIA, Intern, from Apr 2023 until Aug 2023]
Administrative Assistant
 Julien Guieu [INRIA]
2 Overall objectives
COMMEDIA is a joint projectteam of the Inria Research Center of Paris and the JacquesLouis Lions Laboratory (LJLL) of Sorbonne Université and CNRS (UMR7598). The research activity of COMMEDIA focuses on the numerical simulation of biofluid flows in the human body, more specifically, blood flows in the cardiovascular system and air flows in the respiratory system. These simulations are intended to complement available clinical data with the following purpose: help clinicians or bioengineers to enhance the understanding of physiological phenomena, to improve diagnosis and therapy planning or to optimize medical devices. The main main objectives of COMMEDIA are:
 the development of appropriate mathematical models and efficient numerical methods for the simulations and for the interaction of simulations with measured data;
 the mathematical analysis of these models and numerical techniques;
 the development and validation of scientific computing software which implements these numerical techniques.
A distinctive feature of the mathematical models considered in COMMEDIA is that they often couple different types of partial differential equations (PDEs). This heterogeneous character in the models is a mathematical manifestation of the multiphysics nature of the considered problems.
3 Research program
3.1 Multiphysics modeling and simulation
The research activity in terms of modeling and simulation (i.e., the socalled forward problem) is driven by two application domains related to the cardiovascular and the respiratory systems.
3.1.1 Cardiovascular hemodynamics
We distinguish between cardiac hemodynamics (blood flow inside the four chambers of the heart) and vascular hemodynamics (blood flow in the vessels of the body).
Cardiac hemodynamics. The numerical simulation of cardiac hemodynamics presents many difficulties. We can mention, for instance, the large deformation of the cardiac chambers and the complex fluidstructure interaction (FSI) phenomena between blood, the valves and the myocardium. Blood flow can be described by the incompressible NavierStokes equations which have to be coupled with a biophysical model of the myocardium electromechanics and a mechanical model of the valves. The coupling between the fluid and the solid media is enforced by kinematic and dynamic coupling conditions, which guarantee the continuity of velocity and stresses across the interface. In spite of the significant advances achieved since the beginning of this century (see, e.g., 69, 70, 67, 72, 60), the simulation of all the fluidstructure interaction phenomena involved in the heart hemodynamics remains a complex and challenging problem.
Heart valves are definitely a bottleneck of the problem, particularly due to their fast dynamics and the contact phenomena at high pressuredrops. Computational cost is recognized as one of the key difficulties, related to the efficiency of the FSI coupling method and the robustness of the contact algorithm. Furthermore, the numerical discretization of these coupled systems requires to deal with unfitted fluid and solid meshes, which are known to complicate the accuracy and/or the robustness of the numerical approximations (see Section 3.3.2 below).
The ultimate goal of the proposed research activity is the simulation of the complete fluidstructurecontact interaction phenomena involved within the heart. Most of this work will be carried out in close collaboration with the M3DISIM projectteam, which has a wide expertise on the modeling, simulation and estimation of myocardium electromechanics. We will also consider simplified approaches for cardiac hemodynamics (see, e.g., 42, 55, 58). The objective is to develop mathematically sound models of reduced valve dynamics with the purpose of enhancing the description of the pressure dynamics right after the opening/closing of the valve (traditional models yield spurious pressure oscillations).
Vascular hemodynamics. The modeling and simulation of vascular hemodynamics in large vessels has been one of the core research topics of some members of COMMEDIA, notably as regards the fluidstructure interaction phenomena. Here we propose to investigate the modeling of pathological scenarios, such as the hemorrhage phenomena in smaller vessels. Modeling of hemorrhage is motivated by the medical constatation that, after a primary vessel wall rupture, secondary vessel wall ruptures are observed. Biologists postulate that the mechanical explanation of this phenomena might be in the change of applied stress due to blood bleeding. We propose to model and simulate the underlying coupled system, blood vessel flow through the external tissue, to estimate the effect of the subsequent stress variation.
3.1.2 Respiratory flows
The motivation of the proposed research activities is to develop a hierarchy of easily parametrizable models allowing to describe and efficiently simulate the physical, mechanical and biological phenomena related to human respiration, namely,
ventilation, particle deposition, gas diffusion and coupling with the circulatory system.
Ventilation. The current modeling approaches (either 3D–0D coupled models where the 3D NavierStokes equations are solved in truncated geometries of the bronchial tree with appropriate lumped boundary conditions, or 0D–3D coupled models where the lung parenchyma is described by a 3D elastic media irrigated by a simplified bronchial tree) provide satisfactory results in the case of mechanical ventilation or normal breathing. Realistic volumeflow phase portraits can also be simulated in the case of forced expiration (see 44, 52, 75), but the magnitude of the corresponding pressure is not physiological. The current models must be enriched since they do not yet correctly describe all the physiological phenomena at play. We hence propose to extend the 0D–3D (bronchial tree–parenchyma) model developed in the team, by considering a nonlinear, viscoelastic and possibly poroelastic description of the parenchyma with appropriate boundary conditions that describe ribs and adjacent organs and taking into account an appropriate resistive model.
So far, the motion of the trachea and proximal bronchi has been neglected in the ventilation models (see, e.g., 77). These features can be critical for the modeling of pathologic phenomena such as sleep apnea and occlusion of the airways. This would be a longterm goal where fluidstructure interaction and the possible contact phenomena will be taken into account, as in the simulation of cardiac hemodynamics (see Section 3.1.1).
Aerosol and gas diffusion. The dynamics of aerosols in the lung have been widely studied from the mathematical modeling standpoint. They can be described by models at different scales: the microscopic one for which each particle is described individually, the mesoscopic (or kinetic) one for which a density of probability is considered, or the macroscopic one where reactiondiffusion equations describing the behavior of the constituant concentration are considered. The objective of COMMEDIA will mainly be to develop the kinetic approach that allows a precise description of the deposition area at controlled computational costs. Part of this study could be done in collaboration with colleagues from the Research Center for Respiratory Diseases at Inserm Tours (UMR1100).
The macroscopic description is also appropriate for the diffusion of gases (oxygen and carbon dioxide) in the bronchial tree (see 71). Regarding the influence of the carrier gas, if the patient inhales a different mixture of air such as a HeliumOxygen mixture, the diffusion mechanisms could be modified. In this context, the goal is to evaluate if the crossdiffusion (and thus the carrier gas) modifies the quantities of oxygen diffused. Part of this work will be carried out in collaboration with members of the LJLL and of the MAP5.
As a long term goal, we propose to investigate the coupling of these models to models of diffusion in the blood or to perfusion models of the parenchyma, and thus, have access thanks to numerical simulations to new indices of ventilation efficiency (such as dissolved oxygen levels), depending on the pathology considered or the resting or exercise condition of the patient.
3.2 Simulation with data interaction
The second research axis of COMMEDIA is devoted to the interaction of numerical simulations with measured data. Several research directions related to two specific applications are described below: blood flows and cardiac electrophysiology, for which the mathematical models have been validated against experimental data. This list is not exhaustive and additional problems (related to cardiac and respiratory flows) shall be considered depending on the degree of maturity of the developed models.
3.2.1 Fluid flow reconstruction from medical imaging
A first problem which is currently under study at COMMEDIA is the reconstruction of the flow state from Doppler ultrasound measurements. This is a cheap and largely available imaging modality where the measure can be interpreted as the average on a voxel of the velocity along the direction of the ultrasound beam. The goal is to perform a fullstate estimation in a time compatible with a realistic application.
A second problem which is relevant is the flow and wall dynamics reconstruction using 4Dflow MRI. This imaging modality is richer than Doppler ultrasound and provides directly a measure of the 3D velocity field in the voxels. This enables the use of direct estimation methods at a reduced computational cost with respect to the traditional variational data assimilation approaches. Yet, the sensitivity of the results to subsampling and noise is still not well understood.
We also propose to address the issues related to uncertainty quantification. Indeed, measurements are corrupted by noise and the parameters as well as the available data of the system are either hidden or not known exactly (see 66). This uncertainty makes the estimation difficult and has a large impact on the precision of the reconstruction, to be quantified in order to provide a reliable tool.
3.2.2 Safety pharmacology
One of the the most important problems in pharmacology is cardiotoxicity (see 65). The objective is to predict whether or not a molecule alters in a significant way the normal functioning of the cardiac cells. This problem can be formulated as inferring the impact of a drug on the ionic currents of each cell based on the measured electrical signal (e.g., electrograms from MicroElectrodes Arrays). The proposed approach in collaboration with two industrial partners (NOTOCORD and Ncardia) consists in combining available realistic data with virtual ones obtained by numerical simulations. These two datasets can be used to construct efficient classifiers and regressors using machine learning tools (see 49) and hence providing a rapid way to estimate the impact of a molecule on the electrical activity. The methodological aspects of this work are addressed in Section 3.3.3.
3.3 Methodological core
The work described in this section is aimed at investigating fundamental mathematical and numerical problems which arise in the first two research axes.
3.3.1 Mathematical analysis of PDEs
The mathematical analysis of the multiscale and multiphysics models are a fundamental tool of the simulation chain. Indeed, wellposedness results provide precious insights on the properties of solutions of the systems which can, for instance, guide the design of the numerical methods or help to discriminate between different modeling options.
Fluidstructure interaction. Most of the existing results concern the existence of solutions locally in time or away from contacts. One fundamental problem, related to the modeling and simulation of valve dynamics (see Sections 3.1.1 and 3.3.2), is the question of whether or not the model allows for contact (see 64, 62). The proposed research activity is aimed at investigating the case of both immersed rigid or elastic structures and explore if the considered model allows for contact and if existence can be proved beyond contact. The question of the choice of the model is crucial and considering different types of fluid (newtonian or non newtonian), structure (smooth or rough, elastic, viscoelastic, poroelastic), or various interface conditions has an influence on whether the model allows contact or not.
Fluid–structure mixture. The main motivation to study fluidsolid mixtures (i.e., porous media consisting of a skeleton and connecting pores filled with fluid) comes from the modeling of the lung parenchyma and cerebral hemorrhages (see Sections 3.1.1–3.1.2). The Biot model is the most widely used in the literature for the modeling of poroelastic effects in the arterial wall. Here, we propose to investigate the recent model proposed by the M3DISIM projectteam in 54, which allows for nonlinear constitutive behaviors and viscous effects, both in the fluid and the solid. Among the questions which will be addressed, some of them in collaboration with M3DISIM, we mention the justification of the model (or its linearized version) by means of homogenization techniques and its wellposedness.
Fluid–particle interaction. Mathematical analysis studies on the NavierStokesVlasov system for fluidparticle interaction in aerosols can be found in 46, 48. We propose to extend these studies to more realistic models which take into account, for instance, changes in the volume of the particles due to humidity.
3.3.2 Numerical methods for multiphysics problems
In this section we describe the main research directions that we propose to explore as regards the numerical approximation of multiphysics problems.
Fluidstructure interaction. The spatial discretization of fluidstructure interaction (FSI) problems generally depends on the amount of solid displacement within the fluid. Problems featuring moderate interface displacements can be successfully simulated using (moving) fitted meshes with an arbitrary LagrangianEulerian (ALE) description of the fluid. This facilitates, in particular, the accurate discretization of the interface conditions. Nevertheless, for problems involving large structural deflections, with solids that might come into contact or that might break up, the ALE formalism becomes cumbersome. A preferred approach in this case is to combine an Eulerian formalism in the fluid with an unfitted mesh discretization, in which the fluidstructure interface deforms independently of a background fluid mesh. In general, traditional unfitted mesh approaches (such as the immersed boundary and the fictitious domain methods 74, 45, 61, 43) are known to be inaccurate in space. These difficulties have been recently circumvented by a Nitschebased cutFEM methodolgy (see 40, 50). The superior accuracy properties of cutFEM approaches comes at a price: these methods demand a much more involved computer implementation and require a specific evaluation of the interface intersections.
As regards the time discretization, significant advances have been achieved over the last decade in the development and the analysis of timesplitting schemes that avoid strong coupling (fully implicit treatment of the interface coupling), without compromising stability and accuracy. In the vast majority these studies, the spatial discretization is based on body fitted fluid meshes and the problem of accuracy remains practically open for the coupling with thickwalled structures (see, e.g., 59). Within the unfitted mesh framework, splitting schemes which avoid strong coupling are much more rare in the literature.
Computational efficiency is a major bottleneck in the numerical simulation of fluidstructure interaction problems with unfitted meshes. The proposed research activity is aimed at addressing these issues. Another fundamental problem that we propose to face is the case of topology changes in the fluid, due to contact or fracture of immersed solids. This challenging problem (fluidstructurecontactfracture interaction) has major role in many applications (e.g., heart valves repair or replacement, breakup of drugloaded microcapsules) but most of the available studies are still merely illustrative. Indeed, besides the numerical issues discussed above, the stability and the accuracy properties of the numerical approximations in such a singular setting are not known.
Fluid–particle interaction and gas diffusion.
Aerosols can be described through mesoscopic equations of kinetic type, which provide a tradeoff between model complexity and accuracy. The strongly coupled fluidparticle system involves the incompressible NavierStokes equations and the Vlasov equation. The proposed research activity is aimed at investigating the theoretical stability of timesplitting schemes for this system. We also propose to extend these studies to more complex models that take into account the radius growth of the particles due to humidity, and for which stable, accurate and mass conservative schemes have to be developed.
As regards gas diffusion, the mathematical models are generally highly nonlinear (see, e.g., 71, 73, 47). Numerical difficulties arise from these strong non linearities and we propose to develop numerical schemes able to deal with the stiff geometrical terms and that guarantee mass conservation. Moreover, numerical diffusion must be limited in order to correctly capture the time scales and the crossdiffusion effects.
3.3.3 Statistical learning and mathematical modeling interactions
Machine learning and in general statistical learning methods (currently intensively developed and used, see 41) build a relationship between the system observations and the predictions of the QoI based on the a posteriori knowledge of a large amount of data. When dealing with biomedical applications, the available observations are signals (think for instance to images or electrocardiograms, pressure and Doppler measurements). These data are high dimensional and the number of available individuals to set up precise classification/regression tools could be prohibitively large. To overcome this major problem and still try to exploit the advantages of statistical learning approaches, we try to add, to the a posteriori knowledge of the available data an a priori knowledge, based on the mathematical modeling of the system. A large number of numerical simulations is performed in order to explore a set of meaningful scenarios, potentially missing in the dataset. This in silico database of virtual experiments is added to the real dataset: the number of individuals is increased and, moreover, this larger dataset can be used to compute semiempirical functions to reduce the dimension of the observed signals.
Several investigations have to be carried out to systematically set up this framework. First, often there is not a single mathematical model describing a physiological phenomenon, but hierarchies of model of different complexity. Every model is characterized by a model error. How can this be accounted for? Moreover, several statistical estimators can be set up and eventually combined together in order to improve the estimations (see 68). Other issues have an actual impact and has to be investigated: what is the optimal number of in silico experiments to be added? What are the most relevant scenarios to be simulated in relation to the statistical learning approach considered in order to obtain reliable results? In order to answer to these questions, discussions and collaborations with statistics and machine learning groups have to be developed.
3.3.4 Tensor approximation and HPC
Tensor methods have a recent significant development because of their pertinence in providing a compact representation of large, highdimensional data. Their applications range from applied mathematics and numerical analysis to machine learning and computational physics. Several tensor decompositions and methods are currently available (see 63). Contrary to matrices, for tensors of order higher or equal to three, there does not exist, in general, a best low rank approximation, the problem being ill posed (see 76). Two main points will be addressed: (i) The tensor construction and the multilinear algebra operations involved when solving highdimensional problems are still sequential in most of the cases. The objective is to design efficient parallel methods for tensor construction and computations; (ii) When solving highdimensional problems, the tensor is not assigned; instead, it is specified through a set of equations and tensor data. Our goal is to devise numerical methods able to (dynamically) adapt the rank and the discretization (possibly even the tensor format) to respect the chosen error criterion. This could, in turn, improve the efficiency and reduce the computational burden.
These sought improvements could make the definition of parsimonious discretizations for kinetic theory and uncertainty quantification problems (see Section 3.2.1) more efficient and suitable for a HPC paradigm. This work will be carried out in collaboration with Olga Mula (Université ParisDauphine) and the ALPINES and MATHERIALS projectteams.
4 Application domains
4.1 Cardiovascular hemodynamics
The heart is a double pump whose purpose is to deliver blood to the tissue and organs of the body. This function is made possible through the opening and closing of the heart valves. Cardiac diseases generally manifest by affecting the pumping function of the heart. Numerical simulations of cardiac hemodynamics, in normal and pathological conditions, are recognized as a tool of paramount importance for improving the understanding, diagnosis and treatment of cardiac pathologies, and also for the development of implantable devices (see, e.g., 72, 53). As an example, we can mention the case of cardiac mitral valve regurgitation, one of the most common heart valve diseases. For this pathology, clinical data are known to be insufficient for determining the optimal timing for surgery, the best surgical strategy and the longterm outcome of a surgical repair. Contrary to imaging techniques, numerical simulations provide local information, such as pressure and stresses, which are of fundamental importance for the prediction of the mechanical behavior of native valves and of implantable devices.
4.2 Respiratory flows
Respiration involves the transport of air through the airways from the mouth to the alveoli of the lungs. These units where diffusion of oxygen and carbon dioxide take place, are surrounded by a viscoelastic medium (the parenchyma) consisting of blood vessels and collagen fibers. Air flows due to the displacement of the diaphragm, which drives the pulmonary parenchyma. Accidental inhalations of foreign bodies or pathologies such as asthma, emphysema and fibrosis might prevent the lung of fulfilling its function. Therapies mostly use aerosols (set of small particles, solid or liquid), which must reach the specific areas of the lung targeted for treatment. Understanding the airflow mechanisms within the respiratory network is a fundamental ingredient for predicting the particles motion and their deposition (see, e.g., 51). Moreover, understanding of the gas diffusion in the lung is also of major importance since the main fonction of this organ is to deliver oxygen to the blood.
4.3 Safety pharmacology
The problem of safety pharmacology can be summarized as follows: given a molecule which is a candidate to become a drug, is its use dangerous due to side effects? Among all the different problems to be addressed, one of the most relevant questions in pharmacology is cardiotoxicity (see 65). More precisely, the objective is to determine whether or not a molecule alters in a significant way the normal functioning of the cardiac cells. To answer these questions, the CiPA initiative promotes the introduction of novel techniques and their standardisation (see 57). One of the proposed tests of the CiPA panel is to measure the the electrical activity using MicroElectrodes Array: these are microchips that record the electrical activity of an ensemble of cells. The task is to infer the impact of a drug on the ionic currents of each cell based on the electrical signal measured (electrograms) and, in perspective, to be able to assess whether a molecule can induce arrhythmia (see 56).
5 New software, platforms, open data
5.1 New software
5.1.1 FELiScE

Name:
Finite Elements for Life SCiences and Engineering problems

Keywords:
Finite element modelling, Cardiac Electrophysiology, Cardiovascular and respiratory systems

Functional Description:
FELiScE is a finite element code which the M3DISIM and REO projectteams initially jointly develop in order to build up on their respective experiences concerning finite element simulations. One specific objective of this code is to provide in a unified software environment all the stateoftheart tools needed to perform simulations of the complex respiratory and cardiovascular models considered in the two teams – namely involving fluid and solid mechanics, electrophysiology, and the various associated coupling phenomena. FELISCE is written in C++ and open source, and may be later released as an opensource library. FELiScE was registered in July 2014 at the Agence pour la Protection des Programmes under the Inter Deposit Digital Number IDDN.FR.001.350015.000.S.P.2014.000.10000.
 URL:

Contact:
Miguel Angel Fernandez Varela

Participants:
Daniele Carlo Corti, Miguel Angel Fernandez Varela, Marina Vidrascu, Sara Costa Faya, Mocia Agbalessi, Mihaisimion Nechita, Oscar Ruz, Fabien Lespagnol, Vicente Mataix Ferrandiz
5.1.2 FELiScENS

Functional Description:
FELiScENS is a set finite elements solvers for incompressible fluids (fractionalstep schemes) and nonlinear thinwalled structures (3D shells, and 2D curved beams) developed in the framework of the FELiScE library. FELiSCeNS was registered in 2018 at the Agence pour la Protection des Programmes Inter Deposit Digital Number IDDN.FR.001.270015.000.S.A.2018.000.31200.

Contact:
Miguel Angel Fernandez Varela

Participants:
Miguel Angel Fernandez Varela, Marina Vidrascu, Daniele Carlo Corti, Mocia Agbalessi, Sara Costa Faya, Vicente Mataix Ferrandiz, Mocia Agbalessi
5.1.3 DCIMaL

Functional Description:
DCIMaL is a Python and C++ software for safety pharmacology studies and particularly field potentials signals measured with microelectrode array (MEA). The software includes a solver for field potential simulations and a dictionary of entries corresponding to features which can be extracted from real or simulated potential signals. It also includes an algorithm for drug classification (channel blockade or torsadogenic risk) and a tool for estimating ion channel activity (based on the CMAES library). DCIMaL was registered in 2018 at the Agence pour la Protection des Programmes Inter Deposit Digital Number IDDN.FR.001.270003.000.S.P.2018.000.31230

Contact:
Damiano Lombardi

Participants:
Fabien Raphel, Damiano Lombardi
5.1.4 ADAPT

Name:
Adaptive Dynamical Approximation via Parallel Tensor methods

Keywords:
Scientific computing, Tensor decomposition, Partial differential equation

Functional Description:
ADAPT is a library containing methods for scientific computing based on tensors. In many fields of science and engineering we need to approximate the solution of highdimensional problems. In this library we propose a collection of methods to parsimoniously discretise highdimensional problems. These methods are mainly based on tensors.

Contact:
Damiano Lombardi

Participants:
Virginie Ehrlacher, Maria Fuente Ruiz, Damiano Lombardi, Sebastien Riffaud
6 New results
6.1 Reduced Order modelling
Participants: Muriel Boulakia, Céline Grandmont, Fabien Lespagnol, Sébastien Riffaud.
In 32, we propose a reduced model associated to the Poisson problem in a domain with small holes. The reduction method is based on a fictitious domain formulation combined with a projection of Dirichlet boudary constraints on a finite dimensional approximation space. After analyzing the existence of a solution for the reduced problem and its convergence towards the full problem, we propose a numerical discretization which relies on augmented finite elements and allows to achieve optimal convergence properties that are illustrated through numerical illustrations.
In 38, we introduce a hybrid approach that alternates between a highfidelity model and a reducedorder model to speedup numerical simulations while maintaining accurate approximations. In particular, a residualbased error indicator is developed to determine when the reducedorder model is not sufficiently accurate and the highfidelity model needs to be solved. Then, we propose an adaptiveextended version of the hybrid approach to update the reducedorder model with the solution snapshots generated by the highfidelity model when the reducedorder model was not sufficiently accurate. In this way, we expect the reducedorder model to become more robust for predicting new outofsample solutions. The performance of the proposed method is evaluated on parametrized, timedependent, nonlinear problems governed by the 1D Burgers' equation and 2D compressible Euler equations. The results demonstrate the accuracy and computational efficiency of the adaptive hybrid approach with respect to the highfidelity model.
6.2 Safety pharmacology
Participants: Muriel Boulakia, Haibo Liu, Damiano Lombardi.
In 33, we study how to regularise, by means of a dataset of existing observations, parameter estimation problems in dynamical systems. Thanks to a representation of the dataset in the form of an encoderdecoder pair, we are able to introduce a Lipschitz stable nonlinear change of variables such that the parameter estimation problem can be cast as an optimisation of a convex function. We prove that, given a Hölder regularity ${C}^{1,\beta}$ of the composition of the encoder and the dynamical system equation, we can have some theoretical guarantees on the convergence. Some encouraging numerical results are shown on the Van der Pol and the FizHughNagumo dynamical systems.
6.3 Mathematical analysis of PDEs
Participants: Marguerite Champion, Miguel Angel Fernández Varela, Céline Grandmont, Fabien Vergnet, Marina Vidrascu.
In 16, we propose and study a new continuum mechanics model for the fluidstructure interaction problem involving active thin structures embedded in a Stokes flow. In particular, this model is able to reproduce the behavior of cilia or flagella betting in a viscous flow. In the context of linear or nonlinear elasticity, the model is based upon the definition of a suitable internal PiolaKirchoff tensor , which mimics the action of internal biological motors, inducing the motility of the structures. The wellposedness of this coupled system is studied and, for the numerical resolution, an equivalent formulation using Lagrange multipliers is introduced, allowing for the use of standard (fluid and structure) solvers, up to an iterative procedure. Numerical simulations are presented, which illustrate the potential of the proposed active elasticity model.
In 36 we analyse of the contact capabilities of the fluidstructure interaction (FSI) model with seepage reported in 4. In the case of a rigid disk moving over a fixed horizontal plane, we show that this model encompasses contact and hence removes the non collision paradox of traditional FSI models which rely on Dirichlet or Dirichlet/Navier boundary conditions. Numerical evidence on the theoretical results is also provided.
6.4 Numerical methods for multiphysics problems
Participants: Daniele Corti, Guillaume Delay, Miguel Angel Fernández Varela, Céline Grandmont, Fabien Lespagnol, Oscar Ruz, Fabien Vergnet, Marina Vidrascu.
One of the main difficulties that has to be faced with fictitious domain approximation of incompressible flows with immersed interfaces is related to the potential lack of mass conservation across the interface. In 15, we propose and analyze a low order fictitious domain stabilized finite element method which mitigates this issue with the addition of a single velocity constraint. We provide a complete a priori numerical analysis of the method under minimal regularity assumptions. A comprehensive numerical study illustrates the capabilities of the proposed method, including comparisons with alternative fitted and unfitted mesh methods.
The numerical simulation of incompressible fluidstructure interaction systems with loosely coupled schemes is a delicate problem. Indeed, the splitting method must both be stable for the full nonlinear system and have sufficient accuracy to be of use in practice. In the case of the coupling of an incompressible fluid with thickwalled solids the error analyses reported in the literature are limited to 1/2order accuracy in time. In 34 we introduce two important extensions of the analysis of the RobinRobin loosely coupled scheme recently reported in [Numer. Math., 151(4):807–840, 2022]. First, we give a formulation of the scheme in a general nonlinear setting and prove its unconditional energy stability. Then we show that nearlyoptimal accuracy can be achieved in the linear case. These theoretical findings are illustrated in a series of numerical examples.
In 11 we analyse the convergence of the full discretization of a generalized poromechanical model resulting from the linearization of an initial model fitted to soft tissue perfusion. Our strategy here is based on the use of energybased estimates and Tcoercivity methods, so that the numerical analysis benefits from the essential tools used in the existence analysis of the continuoustime and continuousspace formulation.
37 We consider the simulation of slender structures immersed in a threedimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixeddimensional coupled equations (3D for the fluid and 1D for the solid). Several challenges must be faced when dealing with this type of problems. From a mathematical point of view, these include defining wellposed trace operators of codimension two. On the computational standpoint, the nonstandard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixeddimensional discrete formulation as compared to a fully resolved one. We establish the continuous formulation using the NavierStokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixeddimensional version of the fluidstructure interface conditions, based on the projection of kinematic coupling conditions on a finitedimensional Fourier space. Furthermore, we develop a discrete formulation within the framework of the finite element method, establish the energy stability of the scheme, provide extensive numerical evidence of the accuracy of the discrete formulation, notably with respect to a fully resolved (ALE based) model and a standard reduced modeling approach.
7 Bilateral contracts and grants with industry
7.1 Bilateral contracts with industry
Withings
Participants: Miguel Ángel Fernández Varela [coordinator], Adrien Lefieux, Damiano Lombardi, Marina Vidrascu, Fabien Vergent.
This research project has the objective of developing mathematical models of photoplethysmography (PPG) measurements in the wrist and their connection to blood pressure estimation.
8 Partnerships and cooperations
8.1 International initiatives
8.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria International Program
IMFIBIO: Innovative Methods for Forward and Inverse problems in BIOmedical applications
Participants: Muriel Boulakia, Daniele Carlo Corti, Guillaume Delay, Miguel Ángel Fernández Varela [coordinator].

Title:
Innovative Methods for Forward and Inverse problems in BIOmedical applications

Duration:
20222025

Coordinator:
Miguel Angel Fernández Varela

Partner:
University College London London (RoyaumeUni)

UCL contact:
Erik Burman

Summary:
The purpose of the IMFIBIO Associate Team is to exploit the complementary expertise of both partners in mathematical analysis, numerical analysis, scientific computing and data assimilation in order to develop innovative methods for the study of forward and inverse problems in the context of biomedical applications.
 Web site:
8.1.2 Visits of international scientists
 Erik Burman (UCL), March and October 2023.
 Mihai Nechita (Tiberiu Popoviciu Institute of Numerical Analysis), May 2023.
 Cecilia Pagliantini (Università di Pisa), February and October 2023.
 Gianluca Ceruti (University of Innsbruck), February and October 2023.
8.2 European initiatives
8.2.1 Horizon Europe
INSPIRE: INnovation in Safety Pharmacology for Integrated cardiovascular safety assessment to REduce adverse events and late stage drug attrition
Participants: Sara Costa, Miguel Ángel Fernández Varela, Haibo Liu, Damiano Lombardi [coordinator].

Funding:
Horizon 2020  MSCAITN

Duration:
20202024

Coordinator:
University of Antwerp

Local coordinator:
Damiano Lombardi

Summary:
INSPIRE is an European Training Network (ETN) projet funding 15 Early Stage Researchers (ESRs) aimed to exploit innovative techniques for better assessment and prediction of cardiovascular safety liabilities.
 Web site:
8.3 National initiatives
8.3.1 ANR
Participants: Maria FuenteRuiz, Damiano Lombardi [coordinator], Sébastien Riffaud.
ADAPT: Adaptive Dynamical Approximations by Parallel Tensor methods

Funding:
ANR JCJC

Duration:
20182023

Coordinator:
Damiano Lombardi

Summary:
The main goal of the ANR is to investigate the numerical approximation of the solution of highdimensional problems. In particular, the applications that motivate this study are the Uncertainty Quantification and the Kinetic theory. The main objective is to construct in an adaptive way parsimonious discretisations starting from arbitrarily chosen separated discretisations.
 Web site:
SIMR: Simulation and Imaging for Mitral Regurgitation
Participants: Daniele Carlo Corti, Miguel Ángel Fernández Varela [coordinator], Céline Grandmont, Marina Vidrascu.

Funding:
ANR PRC

Duration:
20202024

Coordinator:
Miguel Ángel Fernández Varela

Partners:
CREATIS, HCL, LGEF, M3DISIM, TIMC

Summary:
The SIMR project aims at evaluating the physical consequences of mitral repair using efficient numerical simulations, advanced imaging techniques and an innovative measurement tools in a clinical study.
 Web site:
9 Dissemination
9.1 Promoting scientific activities
9.1.1 Scientific events: organisation
 Guillaume Delay
 Coorganiser of the scientific computing seminar, joint event between Inria and Laboratoire JacquesLouis Lions.
 Organizer of the Workshop on data assimilation, Sorbonne Université, November 2023.

Miguel Ángel Fernández Varela
 Coorganizer of an FIMH sateltite workshop, Simulation and Imaging for Mitral Regurgitation, Lyon, June 2023.
 Damiano Lombardi
 Coorganiser of the scientific computing seminar, joint event between Inria and Laboratoire JacquesLouis Lions.
 Minisymposium organiser, HighDimensional Approximation and ReducedOrder Models, in SIAM CSE, Amsterdam, March 2023.
9.1.2 Journal
Member of the editorial boards
 Céline Grandmont
 Mathematical Modelling of Natural Phenomena
 Journal of Mathematical Fluid Mechanics
 ESAIM: Mathematical Modelling and Numerical Analysis
9.1.3 Invited talks
 Muriel Boulakia
 Invited speaker, Workshop ANR Trecos, ENS Rennes, June 2023
 Invited speaker, Conference Optimization and Control in Burgundy, Dijon, May 2023
 Minisymposium keynote speaker, CFC 2023, Cannes, April 2023
 Miguel Angel Fernández Varela
 Invited talk in MS, ICIAM, Tokyo, August, 2023
 Céline Grandmont
 Conference in honnor pf Grigory Panasenko, Saint Etienne University, Octobre 2023.
 Séminaire EDP, Grenoble Rhone Alpes Univ., Sept. 2023.
 Colloquium Nantes, April 2023
 Damiano Lombardi
 Keynote speaker, Biophysicsbased modeling and data assimilation in medical imaging, Berlin, August 2023.
 Invited talk in MS, ICIAM, Tokyo, August, 2023
 Fabien Vergnet
 European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), September 2023, Lisbon, Portugal
9.1.4 Research administration
 Muriel Boulakia
 Member of the committee of the doctoral school FMJH/LMH ParisSaclay for the thematics Mathematics of Scientific Computing and Engineering
 Miguel Fernández Varela
 Head of Science,Inria Paris
 Member of the Inria Evaluation Committee
 Céline Grandmont
 Member of the Inria Evaluation Commitee
 Member of the Inria Parity
 Member of the scientific commitee of the doctoral school EDMH, ParisSaclay.
 Member of the scientific commitee of GDR MathSAv: Mathématiques, Santé, Sciences de la Vie. Commitee
 Damiano lombardi
 Copresident of CES (Commission Emploi Scientifique), Inria Paris.
9.2 Teaching  Supervision  Juries
9.2.1 Teaching
 Licence:
 Marguerite Champion
 Numerical analysis, 24h, L3, Sorbonne University
 Python, 32h, L2, Sorbonne University
 Guillaume Delay
 Analyse Numérique L3, (48h)
 Remise à Niveau en Math à PolyTech Sorbonne L3, (12h)
 Corrie James
 Math Tutoring, 9h, L1, Université de Versailles SaintQuentinenYvelines
 Gaël le Ruz
 Linear algebra and ODE, 30h, L3, Polytech Sorbonne, Sorbonne Université.
 Fourier Analysis, 24h, L3, Polytech Sorbonne, Sorbonne Université.
 Numerical Analysis for PDE, 12h, L3, Polytech Sorbonne, Sorbonne Université.
 Fabien Vergnet
 Numerical analysis and ODE, 58h, L3, Polytech Sorbonne, Sorbonne Université.
 Nonlinear systems and optimization, 30h, L3, Polytech Sorbonne, Sorbonne Université.
 Marguerite Champion
 Master:
 Guillaume Delay
 Preparation to Agrégation, 34h, M2, Sorbonne Université
 Numerical analysis for PDE, 18h, M1,PolyTech Sorbonne
 Initiation to C++, 27h, M2, Sorbonne Université.
 C++ projet, 21h, M2, Sorbonne Université.
 Miguel Ángel Fernández Varela
 Mathematical models and numerical methods for hemodynamics simulations, 20h, M2, Sorbonne Université.
 Damiano Lombardi
 Lecture, 1.5h, Modelling the heart, June 2023, ENPC.
 Lecture, 1.5h, Modeling the electrophysiology of heart, November 2023, Ecole des Mines Paristech.
 Guillaume Delay
9.2.2 Supervision
 PhD Mathieu Barré, Mathematical and numerical study of a poroelastic model.Supevisors: C. Grandmont & P. Moireau (M3DISIM, Inria Saclay defended in October 2023).
 PhD in progress: Marguerite Champion, Modeling, analysis and simulation of fluidstructure contact interaction. Supervisors: M.A. Fernández Varela, C. Grandmont, F. Vergnet & M. Vidrascu.
 PhD in progress: Daniele Corti, Modeling and numerical simulation of the mitral apparatus.Since October 2020. Supervisors: M.A. Fernández Varela, G. Delay , F. Vergnet & M. Vidrascu.
 PhD in progress: Sara Costa Faya, An in silico approach to monitor and predict haemodynamics during safety pharmacology studies. Since September 2020. Supervisors: M.A. Fernández Varela, D.Lombardi.
 PhD in progress: Fabien Lespagnol, A new computational approach for fluidstructure interaction of slender bodies immersed in threedimensional flow. Since September 2020. Supervisors: M. Boulakia, M.A. Fernández Varela, C. Grandmont & Paolo Zunino (MOX, Politecnico de Milano).
 PhD in progress: Haibo Liu, Data assimilation for highthroughputs creening in safety pharmacology. Since September 2020. Supervisors: D. Lombardi & M. Boulakia.
 PhD in progress: Gaël le Ruz, Observer theory in general constrained spaces – from formulations to applications. Since September 2020. Supervisors: D. Lombardi & P. Moireau.
 PhD defended (March 2023): Maria Fuente Ruiz, Adaptive tensor methods for scientific computing. Supervisors: D. Lombardi & V. Ehrlacher.
 PostDoc: Cyril Karamaoun (CNRS, Sorbonne Université), Until dec 2023, Modelling of gas transport and exchange in the lung. Supervisors: L. Boudin (Sorbonne Univ.) & C. Grandmont.
 PostDoc: Sebastien Riffaud. Tensor methods for parametric fluidstructure interaction and data assimilation. Supervisors: D. Lombardi & M.A. Fernández Varela.
 PostDoc in progress: Lamis Sabbagh (ARCULB, hosted by ULB), Mathematical analysis of fluidstructure interactions problems. Supervisor: C. Grandmont.
 Internship Marloes Coolen. Supervisor: M.A. Fernández Varela.
 Internship Corrie James. Supervisors: M. Boulakia & D. Lombardi.
 Internship Riccardo Bianchi. Supervisors: D. Lombardi & Stefano Pagani (Politecnico di Milano)
9.2.3 Juries
 Muriel Boulakia
 Hiring committees: MdC at GEMaC laboratory, UVSQ ; Professor at Laboratoire de Mathématiques, Université du Littoral Côte d'Opale
 Céline Grandmont
 Member of the PhD thesis prize SMAIGamni 2023
 Member of the scientific commitee of SMAI 2023.
 Hiring committees: Inria CR/IFSP Nancy, "Repyramidage PR" La Rochelle University, Inria DR2, and "Chaire de Professeur Junior", Sorbonne Université.
 Miguel Ángel Fernández Varela
 Hiring committe: Inria CR/IFSP Saclay.
9.3 Popularization
 Marguerite Champion
 Coorganizer of "Rencontres Jeunes Mathématiciennes et Informaticiennes", Inria Paris, October 2023
10 Scientific production
10.1 Major publications
 1 articleAnalysis of a linearized poromechanics model for incompressible and nearly incompressible materials.Evolution Equations and Control Theory2022HAL
 2 articleFully discrete loosely coupled RobinRobin scheme for incompressible fluidstructure interaction: stability and error analysis.Numerische MathematikJuly 2022HALDOI
 3 articleSoTT: greedy approximation of a tensor as a sum of Tensor Trains.SIAM Journal on Scientific Computing2021HAL
 4 articleA mechanically consistent model for fluidstructure interactions with contact including seepage.Computer Methods in Applied Mechanics and Engineering2022HALDOIback to text
 5 articleExistence and uniqueness for a quasistatic interaction problem between a viscous fluid and an active structure.Journal of Mathematical Fluid Mechanics2345March 2021HALDOI
 6 articleA method to enrich experimental datasets by means of numerical simulations in view of classification tasks.ESAIM: Mathematical Modelling and Numerical Analysis555September 2021, 22592291HALDOI
 7 articleState estimation in nonlinear parametric time dependent systems using Tensor Train.International Journal for Numerical Methods in Engineering2022HALDOI
 8 articleAugmented Resistive Immersed Surfaces valve model for the simulation of cardiac hemodynamics with isovolumetric phases.International Journal for Numerical Methods in Biomedical Engineering363February 2020, e3223HALDOI
10.2 Publications of the year
International journals
 9 articleTracking of blood vessels motion from 4Dflow MRI data.Cardiovascular Engineering and Technology14August 2023, 577604HALDOI
 10 articleSplitting schemes for a Lagrange multiplier formulation of FSI with immersed thinwalled structure: stability and convergence analysis.IMA Journal of Numerical Analysis4322023, 881–919HALDOI
 11 articleNumerical analysis of an incompressible soft material poromechanics model using Tcoercivity.Comptes Rendus. Mécanique351S12023, 136HALDOIback to text
 12 articleA coupled model for the dynamics of gas exchanges in the human lung with Haldane and Bohr's effects.Journal of Theoretical Biology2023, 111590HALDOI
 13 articleLoosely coupled, noniterative timesplitting scheme based on RobinRobin coupling: Unified analysis for Parabolic/Parabolic and Parabolic/Hyperbolic problems.Journal of Numerical Mathematics3112023, 5977HALDOI
 14 articleConvergence analysis of an unfitted mesh semiimplicit coupling scheme for incompressible fluidstructure interaction.Vietnam Journal of Mathematics512023, 37–69HALDOI
 15 articleLoworder fictitious domain method with enhanced mass conservation for an interface Stokes problem.ESAIM: Mathematical Modelling and Numerical AnalysisDecember 2023HALDOIback to text
 16 articleA continuum active structure model for the interaction of cilia with a viscous fluid.Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikOctober 2023HALDOIback to text
 17 articlePropagation of an idealized infection in an airway tree, consequences of the inflammation on the oxygen transfer to blood.Journal of Theoretical BiologyJanuary 2023HALDOI
Invited conferences
 18 inproceedingsData assimilation problems in haemodynamics.Workshop on Biophysicsbased modeling and data assimilation in medical imagingBerlin, GermanyAugust 2023HAL
 19 inproceedingsSoTT: a greedy construction of a sum of Tensor Trains.ICOSAHOM 2023  International Conference on Spectral and High Order MethodsSeoul, South KoreaAugust 2023HAL
International peerreviewed conferences
 20 inproceedingsParametric solvers for simulation of blood flows.CFC 2023  IACM Computational Fluids ConferenceCannes, FranceApril 2023HAL
Conferences without proceedings
 21 inproceedingsFluidstructure interaction calibration from 4Dflow MRI data.CFC 2023  IACM Computational Fluids ConferenceCannes, FranceApril 2023HAL
 22 inproceedingsLipschitz Stabilised Autoencoders in Parameter Identification of Dynamical Systems.10th International Congress on Industrial and Applied Mathematics (ICIAM 2023)Tokyo, JapanAugust 2023HAL
 23 inproceedingsLipschitz Stabilised Autoencoders to Study the Intrinsic Dimensionality of Dynamical Systems and Build Datadriven Models.Math 2 Product (M2P): Emerging Technologies in Computational Science for Industry, Sustainability and InnovationTaormina, ItalyMay 2023HAL
 24 inproceedingsAn optimal recovery formulation of data assimilation in haemodynamics.Data AssimilationParis, FranceNovember 2023HAL
 25 inproceedingsParametric FSI solvers for haemodynamics.ICIAM 2023  10th International Congress on Industrial and Applied MathematicsTokyo, JapanAugust 2023HAL
 26 inproceedingsParametric PDE solvers for parameter estimation and Uncertainty Quantification.ENUMATH 2023  European Conference on Numerical Mathematics and Advanced ApplicationsLisboa, PortugalSeptember 2023HAL
 27 inproceedingsParametric PDEs solvers, uncertainty quantification and parameter estimation.First International Conference Math 2 Product  M2P 2023Taormina, ItalyMay 2023HAL
 28 inproceedingsSolving parametric PDEs via parsimonious discretizations: application to Uncertainty Quantification and Data Assimilation.5th International Conference on Uncertainty Quantification in Computational Science and Engineering  UNCECOMP 2023Athens, GreeceJune 2023HAL
Edition (books, proceedings, special issue of a journal)
 29 proceedingsB.Beatrice BattistiT.Tobias BlickhanG.Guillaume EncheryV.Virginie EhrlacherD.Damiano LombardiO.Olga MulaWasserstein model reduction approach for parametrized flow problems in porous media.CEMRACS 2021  Data Assimilation and Reduced Modeling for High Dimensional Problems73EDP SciencesAugust 2023, 2847HALDOI
Doctoral dissertations and habilitation theses
Reports & preprints
 32 miscNumerical approximation of the Poisson problem with small holes, using augmented finite elements and defective boundary conditions..January 2023HALback to text
 33 miscParameter identification through gradient flow on latent variables.December 2023HALback to text
 34 miscRobinRobin loose coupling for incompressible fluidstructure interaction: nonlinear setting and nearlyoptimal error analysis.October 2023HALback to text
 35 reportBilan de la mandature 20192023 de la Commission d'Évaluation Inria.InriaAugust 2023HAL
 36 miscOn the analysis of a mechanically consistent model of fluidstructurecontact interaction.October 2023HALback to text
 37 miscA mixeddimensional formulation for the simulation of slender structures immersed in an incompressible flow.December 2023HALback to text
 38 miscAccurate and robust predictions for model order reduction via an adaptive, hybrid FOM/ROM approach.December 2023HALback to text
Other scientific publications
 39 inproceedingsAn in silico approach to monitor and predict haemodynamics during safety pharmacology studies.SPS Annual MeetingBrussels, BelgiumSeptember 2023HAL
10.3 Cited publications
 40 articleNitscheXFEM for the coupling of an incompressible fluid with immersed thinwalled structures.Comput. Methods Appl. Mech. Engrg.3012016, 300335back to text
 41 bookIntroduction to machine learning.MIT press2009back to text
 42 articleA robust and efficient valve model based on resistive immersed surfaces.Int. J. Numer. Meth. Biomed. Engng.2892012, 937959back to text
 43 articleA fictitious domain/mortar element method for fluidstructure interaction.Int. Jour. Num. Meth. Fluids352001, 743761back to text
 44 articleMultiscale modeling of the respiratory tract.Math. Models Methods Appl. Sci.2012010, 5993back to text
 45 articleFinite element approach to immersed boundary method with different fluid and solid densities.Math. Models Methods Appl. Sci.21122011, 25232550back to text
 46 articleGlobal existence of solutions for the coupled Vlasov and NavierStokes equations.Differential Integral Equations2211122009, 12471271back to text
 47 incollectionDiffusion models of multicomponent mixtures in the lung.CEMRACS 2009: Mathematical modelling in medicine30ESAIM Proc.EDP Sci., Les Ulis2010, 90103back to text
 48 articleGlobal existence of solutions to the incompressible NavierStokesVlasov equations in a timedependent domain.J. Differential Equations26232017, 13171340back to text
 49 bookClassification and regression trees.Routledge2017back to text
 50 articleAn unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes.Comput. Methods Appl. Mech. Engrg.2792014, 497514back to text
 51 articleNumerical simulation of respiratory flow patterns within human lung.Respir. Physiol. Neurobiol.13022002, 201221back to text
 52 articleHomogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs.Math. Models Methods Appl. Sci.2562015, 11251177back to text
 53 articleRole of Computational Simulations in Heart Valve Dynamics and Design of Valvular Prostheses.Cardiovasc. Eng. Technol.112010, 1838back to text
 54 articleGeneral coupling of porous flows and hyperelastic formulations—From thermodynamics principles to energy balance and compatible time schemes.Eur. J. Mech. B Fluids.462014, 8296back to text
 55 articleImageBased Simulations Show Important Flow Fluctuations in a Normal Left Ventricle: What Could be the Implications?Ann. Biomed. Eng.44112016, 33463358back to text
 56 articleThe comprehensive in vitro proarrhythmia assay (CiPA) initiative—update on progress.J. Pharmacol. Toxicol. Methods812016, 1520back to text
 57 articleAn evaluation of 30 clinical drugs against the comprehensive in vitro proarrhythmia assay (CiPA) proposed ion channel panel.J. Pharmacol. Toxicol. Methods812016, 251262back to text
 58 articleA patientspecific aortic valve model based on moving resistive immersed implicit surfaces.Biomech. Model. Mechanobiol.1652017, 17791803back to text
 59 articleConvergence and error analysis for a class of splitting schemes in incompressible fluidstructure interaction.IMA J. Numer. Anal.3642016, 17481782back to text
 60 articleA coupled mitral valveleft ventricle model with fluidstructure interaction.Med. Eng. Phys.4709 2017, 128136back to text
 61 articleA distributed Lagrange mutiplier/fictitious domain method for particulate flows.Int. J. of Multiphase Flow251999, 755794back to text
 62 articleExistence of global strong solutions to a beamfluid interaction system.Arch. Ration. Mech. Anal.22032016, 12831333back to text
 63 articleA literature survey of lowrank tensor approximation techniques.GAMMMitt.3612013, 5378back to text
 64 articleLack of collision between solid bodies in a 2D incompressible viscous flow.Comm. Partial Differential Equations32792007, 13451371back to text
 65 articleDruginduced functional cardiotoxicity screening in stem cellderived human and mouse cardiomyocytes: effects of reference compounds.J. Pharmacol. Toxicol. Methods6812013, 97111back to textback to text
 66 bookStatistical and computational inverse problems.160Applied Mathematical SciencesSpringerVerlag, New York2005back to text
 67 articleImmersogeometric cardiovascular fluidstructure interaction analysis with divergenceconforming Bsplines.Comput. Methods Appl. Mech. Engrg.3142017, 408472back to text
 68 bookTargeted learning.Springer Series in StatisticsSpringer, New York2011back to text
 69 articleAn immersed boundary method with formal secondorder accuracy and reduced numerical viscosity.J. Comp. Phys.16022000, 705719back to text
 70 articleA combined fictitious domain/adaptive meshing method for fluid–structure interaction in heart valves.International Journal for Numerical Methods in Fluids4652004, 533544back to text
 71 articleModeling of the oxygen transfer in the respiratory process.ESAIM Math. Model. Numer. Anal.4742013, 935960back to textback to text
 72 articleComputational modeling of cardiac hemodynamics: current status and future outlook.J. Comput. Phys.3052016, 10651082back to textback to text
 73 articleAerosol transport throughout inspiration and expiration in the pulmonary airways.Int. J. Numer. Methods Biomed. Eng.3392017back to text
 74 articleThe immersed boundary method.Acta Numer.112002, 479517back to text
 75 articleA comprehensive computational human lung model incorporating interacinar dependencies: Application to spontaneous breathing and mechanical ventilation.Int. J. Numer. Method. Biomed. Eng.331e027872016back to text
 76 articleTensor rank and the illposedness of the best lowrank approximation problem.SIAM J. Matrix Anal. Appl.3032008, 10841127back to text
 77 articleFluidstructure interaction including volumetric coupling with homogenised subdomains for modeling respiratory mechanics.Int. J. Numer. Method. Biomed. Eng.334e28122016back to text