COMMEDIA is a joint project-team of the Inria Research Center of Paris and the Jacques-Louis Lions Laboratory (LJLL) of Sorbonne Université and CNRS (UMR7598). The research activity of COMMEDIA focuses on the numerical simulation of bio-fluid 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 bio-engineers 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:

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 multi-physics nature of the considered problems.

The research activity in terms of modeling and simulation (i.e., the so-called forward problem) is driven by two application domains related to the cardiovascular and the respiratory systems.

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
fluid-structure interaction (FSI) phenomena between blood, the valves and the myocardium. Blood flow can be described by the incompressible Navier-Stokes equations which
have to be coupled with a bio-physical model of the myocardium electro-mechanics 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 fluid-structure 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 pressure-drops. 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 fluid-structure-contact interaction phenomena involved within the heart. Most of this work will be carried out in close collaboration with the M3DISIM project-team, which has a wide expertise on the modeling, simulation and estimation of myocardium electro-mechanics. 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 fluid-structure 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.

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 Navier-Stokes 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 volume-flow 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 non-linear, viscoelastic and possibly poro-elastic 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 long-term goal where fluid-structure 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 reaction-diffusion 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 Helium-Oxygen mixture, the diffusion mechanisms could be modified. In this context, the goal is to evaluate if the cross-diffusion (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.

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.

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 full-state estimation in a time compatible with a realistic application.

A second problem which is relevant is the flow and wall dynamics reconstruction using 4D-flow 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.

One of the the most important problems in pharmacology is cardio-toxicity (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 Micro-Electrodes 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.

The work described in this section is aimed at investigating fundamental mathematical and numerical problems which arise in the first two research axes.

The mathematical analysis of the multi-scale and multi-physics models are a fundamental tool of the simulation chain. Indeed, well-posedness 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.

Fluid-structure 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, poro-elastic),
or various interface conditions has an influence on whether the model allows contact or not.

Fluid–structure mixture.
The main motivation to study fluid-solid 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 poro-elastic effects in the arterial wall.
Here, we propose to investigate the recent model proposed by the M3DISIM project-team 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 well-posedness.

Fluid–particle interaction.
Mathematical analysis studies on the Navier-Stokes-Vlasov system for fluid-particle 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.

In this section we describe the main research directions that we propose to explore as regards the numerical approximation of multi-physics problems.

Fluid-structure interaction. The spatial discretization of fluid-structure 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 Lagrangian-Eulerian (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 fluid-structure 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 Nitsche-based cut-FEM methodolgy (see 40, 50). The superior accuracy properties of cut-FEM 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 time-splitting 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 thick-walled 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 fluid-structure 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 (fluid-structure-contact-fracture interaction) has major role in many applications (e.g., heart valves repair or replacement, break-up of drug-loaded micro-capsules) 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 trade-off between model complexity and accuracy. The strongly coupled fluid-particle system involves the incompressible Navier-Stokes equations and the Vlasov equation. The proposed research activity is aimed at investigating the theoretical stability of time-splitting 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 non-linear (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 cross-diffusion effects.

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 electro-cardiograms, 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 semi-empirical 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.

Tensor methods have a recent significant development because of their pertinence in providing a compact representation of large, high-dimensional 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 multi-linear algebra operations involved when solving high-dimensional 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 high-dimensional 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 Paris-Dauphine) and the ALPINES and MATHERIALS project-teams.

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 long-term 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.

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.

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 cardio-toxicity (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 Micro-Electrodes 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).

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 high-fidelity model and a reduced-order model to speedup numerical simulations while maintaining accurate approximations. In particular, a residual-based error indicator is developed to determine when the reduced-order model is not sufficiently accurate and the high-fidelity model needs to be solved. Then, we propose an adaptive-extended version of the hybrid approach to update the reduced-order model with the solution snapshots generated by the high-fidelity model when the reduced-order model was not sufficiently accurate. In this way, we expect the reduced-order model to become more robust for predicting new out-of-sample solutions. The performance of the proposed method is evaluated on parametrized, time-dependent, 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 high-fidelity model.

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 encoder-decoder pair, we are able to introduce a Lipschitz stable non-linear 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

In 16, we propose and study a new continuum mechanics model for the fluid-structure 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 Piola-Kirchoff tensor , which mimics the action of internal biological motors, inducing the motility of the structures. The well-posedness 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 fluid-structure 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.

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 fluid-structure 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 thick-walled solids the error analyses reported in the literature are limited to 1/2-order accuracy in time. In 34 we introduce two important extensions of the analysis of the Robin-Robin loosely coupled scheme recently reported in [Numer. Math., 151(4):807–840, 2022]. First, we give a formulation of the scheme in a general non-linear setting and prove its unconditional energy stability. Then we show that nearly-optimal 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 energy-based estimates and T-coercivity methods, so that the numerical analysis benefits from the essential tools used in the existence analysis of the continuous-time and continuous-space formulation.

37 We consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixed-dimensional 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 mixed-dimensional discrete formulation as compared to a fully resolved one. We establish the continuous formulation using the Navier-Stokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixed-dimensional version of the fluid-structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional 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.

This research project has the objective of developing mathematical models of photoplethysmography (PPG) measurements in the wrist and their connection to blood pressure estimation.

IMFIBIO: Innovative Methods for Forward and Inverse problems in BIO-medical applications

INSPIRE: INnovation in Safety Pharmacology for Integrated cardiovascular safety assessment to REduce adverse events and late stage drug attrition

ADAPT: Adaptive Dynamical Approximations by Parallel Tensor methods

SIMR: Simulation and Imaging for Mitral Regurgitation