COMMEDIA: Mathématiques et calcul scientifique pour les applications bio-médicales

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

Abstract In this paper, we consider a system modeling the interaction between a viscous incompressible fluid and an elastic structure. The fluid motion is represented by the classical Navier–Stokes equations while the elastic displacement is described by the linearized elasticity equation. The elastic structure is immersed in the fluid and the whole system is confined into a general bounded smooth domain of . Our main result is the local in time existence and uniqueness of a strong solution of the corresponding system.

In order to reduce the complexity of heart hemodynamics simulations, uncoupling approaches are often considered for the modeling of the immersed valves as an alternative to complex fluid-structure interaction (FSI) models. A possible shortcoming of these simplified approaches is the difficulty to correctly capture the pressure dynamics during the isovolumetric phases. In this work, we propose an enhanced resistive immersed surfaces (RIS) model of cardiac valves, which overcomes this issue. The benefits of the model are investigated and tested in blood flow simulations of the left heart where the physiological behavior of the intracavity pressure during the isovolumetric phases is recovered without using fully coupled fluid-structure models and without important alteration of the associated velocity field.

The present contribution deals with the estimation of haemodynamics Quantities of Interest by exploiting Ultrasound Doppler measurements. A fast method is proposed, based on the PBDW method. Several methodological contributions are described: a sub-manifold partitioning is introduced to improve the reduced-order approximation, two different ways to estimate the pressure drop are compared, and an error estimation is derived. A test-case on a realistic common carotid geometry is presented, showing that the proposed approach is promising in view of realistic applications.

Novel studies conducting cardiac safety assessment using human-induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) are promising but might be limited by their specificity and predictivity. It is often challenging to correctly classify ion channel blockers or to sufficiently predict the risk for Torsade de Pointes (TdP). In this study, we developed a method combining in vitro and in silico experiments to improve machine learning approaches in delivering fast and reliable prediction of drug-induced ion-channel blockade and proarrhythmic behaviour. The algorithm is based on the construction of a dictionary and a greedy optimization, leading to the definition of optimal classifiers. Finally, we present a numerical tool that can accurately predict compound-induced pro-arrhythmic risk and involvement of sodium, calcium and potassium channels, based on hiPSC-CM field potential data.

The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces , ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space , SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantly challenged, and can potentially spoil the final quality of the approximated solution due to the convergence to bad local minima, and bad generalization capabilities. In this paper, we present an exhaustive numerical study of the merits and limitations of these schemes when solutions exhibit low-regularity, and compare performance with respect to more benign cases when solutions are very smooth. As a support for our study, we consider singularly perturbed convection-diffusion problems where the regularity of solutions typically degrades as certain multiscale parameters go to zero.

In this article, we consider a reaction–diffusion equation where the reaction term is given by a cubic function and we are interested in the numerical reconstruction of the time-independent part of the source term from measurements of the solution. For this identification problem, we present an iterative algorithm based on Carleman estimates which consists of minimizing at each iteration cost functionals which are strongly convex on bounded sets. Despite the nonlinear nature of the problem, we prove that our method globally converges and the convergence speed evaluated in weighted norm is linear. In the last part of the paper, we illustrate the effectiveness of our method with several numerical reconstructions in dimension one or two.

A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information and their combination is proposed. The framework is tested on the analysis of protocols—a combination of angles along which strain measurements can be acquired—in a biaxial experiment of soft tissues for the estimation of hyperelastic constitutive model parameters. The proposed framework considers the information gain about the parameters from the experiment as the key criterion to be maximised, which can be directly used for optimal design. Information gain is computed through k-nearest neighbour algorithms applied to the joint samples of the parameters and measurements produced by the forward and observation models. For biaxial experiments, the results show that low angles have a relatively low information content compared to high angles. The results also show that a smaller number of angles with suitably chosen combinations can result in higher information gains when compared to a larger number of angles which are poorly combined. Finally, it is shown that the proposed framework is consistent with classical approaches, particularly D-optimal design.

We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. Since the notion of linear vectorial spaces does not exist in general metric spaces, both approaches result in nonlinear approximation methods. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be theL2-Wasserstein space.

Abstract We present a loosely coupled, non-iterative time-splitting scheme based on Robin–Robin coupling conditions. We apply a novel unified analysis for this scheme applied to both a parabolic/parabolic coupled system and a parabolic/hyperbolic coupled system. We show for both systems that the scheme is stable, and the error converges as $\mathcal{O}\big({\Delta t} \sqrt{T +\log(\frac{1}{{\Delta t}})}\big),$ where Δt is the time step.

In this paper, we propose a coupled fluid-kinetic model taking into account the radius growth of aerosol particles due to humidity in the respiratory system. We aim to numerically investigate the impact of hygroscopic effects on the particle behaviour. The air flow is described by the incompressible Navier-Stokes equations, and the aerosol by a Vlasov-type equation involving the air humidity and temperature, both quantities satisfying a convection-diffusion equation with a source term. Conservations properties are checked and an explicit time-marching scheme is proposed. Twodimensional numerical simulations in a branched structure show the influence of the particle size variations on the aerosol dynamics.

The combination of reduced models of cardiac valve dynamics with a one-way kinematic uncoupling of blood flow and electromechanics is a widespread approach for reducing the complexity of cardiac hemodynamics simulations. This comes, however, with a number of shortcomings: artificial pressure oscillations, missing isovolumetric phases, and valve laws without precise continuous formulation. This paper is aimed at overcoming these three difficulties while still mitigating computational cost. A novel reduced model of valve dynamics is proposed in which unidirectional flow is enforced in a mathematically sound fashion. Artificial pressure oscillations are overcome by considering a fluid-structure interaction model, which couples bi-ventricular electromechanics and blood flow in the left cavities. The interface coupling is solved in a partitioned fashion via an unconditionally stable loosely coupled scheme. A priori energy estimates are derived for both the continuous coupled problem and its numerical approximation. The benefits and limitations of the proposed approaches are illustrated in a comprehensive numerical study.

We address the question of the modelling of the fluid–structure interactions for a microcapsule enclosed by a finite-thickness wall, and of the prediction of the buckling behaviour when it is subjected to large displacements and deformations. Specifically, we model the strong coupling between the solid (the wall dynamics) and fluid (the flow inside and outside the capsule) mechanics, for a wall material that can be strain-hardening or softening, while accounting for the bending resistance due to thickness. The fluid flow is assumed to be inertialess on the capsule scale, which allows the use of the boundary integral formulation for the fluid velocity. We discuss the different simplifications that are made when designing a fluid–shell interaction model for large deformations, and present a shear-membrane-bending (SMB) shell model that allows for a non-linear wall stretching law. The performance of the model, as compared to a simple membrane model where bending resistance is neglected, is illustrated on a generic example: we consider an initially ellipsoidal capsule, freely suspended in a plane hyperbolic flow, that is subjected to such stringent deformation, that its short axis becomes the long one. We show that the simple membrane model predicts reasonably well the overall shape of the capsule, but cannot capture the detailed post buckling behaviour, for which a robust shell model is necessary. The SMB shell model complies with dominant membrane effects, remains stable even under large deformation and avoids numerical locking. It allows predicting post-buckling behaviour, which depends on the material constitutive law.

Abstract In the present work, we propose a reduced‐order method to solve the state estimation problem when nonlinear parametric time‐dependent systems are at hand. The method is based on the approximation of the set of system solutions by means of a tensor train format. The particular structure of tensor train makes it possible to set up both a variational and a sequential method. Several numerical experiments are proposed to assess the behavior of the method.

This paper studies the unique continuation problem for the Stokes equations given a database of population measurements. The problem is set up as a minimization problem under a PDE constraint and discretized using the finite element method. It is then regularized by the population data, by imposing that the solution lives near a finite-dimensional subspace generated by the database. This study examines how the inclusion of population data in the resolution improves both theoretical and numerical results. Using the proposed method, global error estimates for the velocity and pressure are obtained at an improved rate of convergence. The inclusion of the population data also has a positive impact on the numerical test cases, in both 2D and 3D, and especially when the measurements are scarce.

This article is devoted to the numerical analysis 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. In particular, our T -coercivity strategy allows us to obtain the necessary inf-sup condition for the global system from the inf-sup condition restricted to a subsystem having the same structure as the Stokes problem. This allows us to prove that any finite element pair adapted to the Stokes problem is also suitable for this global poromechanical model regardless of porosity and permeability, generalizing previous results from the literature studying this model.

The aim of this work is to provide a preliminary comparison of different classes of methods to automatically detect the effect of ageing from in vivo data. The application which motivated this work is related to safety pharmacology, whose major goal is to determine, in a pre-clinical phase, whether a drug is potentially dangerous for the health. In particular, we are going to compare statistical, machine learning and mathematical modelling methods.

We propose a two-dimensional benchmark for fluid-structure-contact interaction consisting of a deformable elastic disk falling under gravity within a viscous incompressible fluid and rebounding in the vicinity of the bottom wall. Solid deformability is essential, as rigid solids do not rebound in this framework. Besides this, the setting is deliberately kept simple to facilitate reproduction. The configuration is particularly challenging due to the well-known no-contact paradox, which can lead to a contactless rebound and forces numerical methods to resolve a vanishingly thin fluid layer in the near-contact region, making the dynamics highly sensitive to the spatial and temporal discretizations. In addition to no-slip boundary and interface conditions, a reduced porous modeling of surface roughness, either on the disk boundary or on the bottom wall, is also considered; this circumvents the no-contact paradox and enables genuine contact. An energy balance law is derived theoretically for all three cases. Eight numerical methodologies, developed by five research groups and spanning different model formulations, numerical methods, and codes (including both fitted and unfitted discretizations), are applied to the benchmark at several levels of spatial and temporal refinement. Quantities of interest of varying complexity are collected and compared, showing close agreement during the falling phase and increased sensitivity in the near-contact and rebound regimes. The setting and the results provide a suitable reference for the systematic assessment of fluid-structure-contact interaction solvers. The time histories of all quantities of interest for every approach and refinement level are provided as supplementary material.

Abstract We revisit the direct method for reconstructing coefficients in second-order elliptic partial differential equations. Two model problems are considered: first, the reconstruction of the diffusion coefficient in a scalar elliptic problem, and second, the reconstruction of the shear modulus in the elastography problem. To highlight the versatility of the framework, different notions of stability are exploited in the two situations. In the scalar case, the system is interpreted as a hyperbolic transport equation and an inf-sup condition on the discrete level is leveraged for the analysis of the numerical method. We obtain error estimates on the reconstruction coefficient that are suboptimal by half an order, which is known to be sharp on general meshes. In the vector case, the minimization of the residual in dual norm and a stability result on the continuous problem lead to error estimates that are optimal compared to the approximation. For both problems, the theoretical results are illustrated by some numerical examples.