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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 $\tt {T}$-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 $\tt {T}$-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.

This paper is devoted to the mathematical analysis of the contact capabilities of the fluid-structure interaction (FSI) model with seepage reported in [Comput. Methods Appl. Mech., 392:114637, 2022]. 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. [ABSTRACT FROM AUTHOR]

In this work, we thoroughly analyze the linearized version of a poromechanics model developed to simulate soft tissues perfusion. This is a fully unsteady model in which the fluid and solid equations are strongly coupled through the interstitial pressure. As such, it generalizes Darcy, Brinkman and Biot equations of poroelasticity. The mathematical and numerical analysis of this model was initially performed for a compressible porous material. Here, we focus on the nearly incompressible case with a semigroup approach, which also allows us to prove the existence of weak solutions. We show the existence and uniqueness of strong and weak solutions in the incompressible limit case, for which a divergence constraint on the mixture velocity appears. Due to the special form of the coupling, the underlying problem is not coercive. Nevertheless, by using the notion of $ T $-coercivity, we obtain stability estimates and well-posedness results. Our study also provides guidelines to propose stable and robust approximations of the problem with mixed finite elements. In particular, we recover an inf-sup condition that is independent of the porosity. Finally, we numerically investigate the elliptic regularity of the associated steady-state problem and illustrate the sensitivity of the solution with respect to the different model parameters. [ABSTRACT FROM AUTHOR]

In this paper we analyse geometric multiscale models arising in the description of physiological flows such as blood flow in arteries or air flow in the bronchial tree. The geometrical complexity of the networks in which air/blood flows lead to a classical decomposition in two areas: a truncated 3D geometry corresponding to the largest contribution of the domain, and a 0D part connected to the 3D part, modelling air/blood flows in smaller airways/vessels. The fluid in the 3D part is described by the Stokes or the Navier–Stokes system which is coupled to 0D models or so-called Windkessel models. The resulting Navier–Stokes–Windkessel coupled system involves Neumann non-local boundary conditions that depends on the considered applications. We first show that the different types of Windkessel models share a similar formalism. Next we derive existence results and stability estimates for the continuous coupled Stokes–Windkessel or Navier–Stokes–Windkessel problem as well as stability estimates for the semi-discretized systems with either implicit or explicit treatment of the boundary conditions. In all the calculations, we pay a special attention to the dependency of the various constants and smallness conditions on the data with respect to the physical and numerical parameters. In particular we exhibit different kinds of behavior depending on the considered 0D model. Moreover even if no energy estimates can be derived in energy norms for the Navier–Stokes–Windkessel system, leading to possible and observed numerical instabilities for large applied pressures, we show that stability estimates for both the continuous and semi-discrete problems, can be obtained in appropriate norms for small enough data by introducing a new well chosen Stokes-like operator. These sufficient stability conditions on the data may give a hint on the order of magnitude of the data enabling stable computations without stabilization method for the problem. Numerical simulations illustrate some of the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we develop a lung ventilation model. The parenchyma is described as an elastic homogenized media. It is irrigated by a space-filling dyadic resistive pipe network, which represents the tracheobronchial tree. In this model, the tree and the parenchyma are strongly coupled. The tree induces an extra viscous term in the system constitutive relation, which leads, in the finite element framework, to a full matrix. We consider an efficient algorithm that takes advantage of the tree structure to enable a fast matrix-vector product computation. This framework can be used to model both free and mechanically induced respiration, in health and disease. Patient-specific lung geometries acquired from computed tomography scans are considered. Realistic Dirichlet boundary conditions can be deduced from surface registration on computed tomography images. The model is compared to a more classical exit compartment approach. Results illustrate the coupling between the tree and the parenchyma, at global and regional levels, and how conditions for the purely 0D model can be inferred. Different types of boundary conditions are tested, including a nonlinear Robin model of the surrounding lung structures. [ABSTRACT FROM AUTHOR]

We are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam-like elastic material containing millions of air-filled alveoli connected by a tree-shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We use the two-scale convergence theory to study the asymptotic behavior as ε goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges as ε goes to zero. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the mathematical modeling of the propagation of sound waves in the lung parenchyma, which is a foam-like elastic material containing millions of air-filled alveoli. In this study, the parenchyma is governed by the linearized elasticity equations, and the air by the acoustic wave equations. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We consider the time-harmonic regime forced by vibrations induced by volumic forces. We use the two-scale convergence theory to study the asymptotic behavior as e goes to zero and prove the convergence of the solutions of the coupled fluid-structure problem to the solution of a linear-elasticity boundary value problem. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we prove Hölder and logarithmic stability estimates associated with the unique continuation property for the Stokes system. The proof of these results is based on local Carleman inequalities. In the second part, these estimates on the fluid velocity and on the fluid pressure are applied to solve an inverse problem: we consider the Stokes system completed with mixed Neumann and Robin boundary conditions, and we want to recover the Robin coefficient (and obtain the stability estimate for it) from measurements available on a part of the boundary where the Neumann conditions are prescribed. For this identification parameter problem, we obtain a logarithmic stability estimate under the assumption that the velocity of a given reference solution stays far from zero on a part of the boundary where the Robin conditions are prescribed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coeffcient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coeffcient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim [11] and under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed. [ABSTRACT FROM AUTHOR]

We propose here a decomposition of the respiratory tree into three stages which correspond to different mechanical models. The resulting system is described by the Navier–Stokes equation coupled with an ODE (a simple spring model) representing the motion of the thoracic cage. We prove that this problem has at least one solution locally in time for any data and, in the special case where the spring stiffness is equal to zero, we obtain an existence result globally in time provided that the data are small enough. The behavior of the global model is illustrated by three-dimensional simulations. [ABSTRACT FROM AUTHOR]

Little is known about transport throughout the respiration cycle in the conducting airways. It is challenging to appropriately describe the time-dependent number of particles entering back into the model during exhalation. Modeling the entire lung is not feasible; therefore, multidomain methods must be used. Here, we present a new framework that is designed to simulate particles throughout the respiration cycle, incorporating realistic airway geometry and respiration. This framework is applied for a healthy rat lung exposed to ∼ 1 μm diameter particles, chosen to facilitate parameterization and validation. The flow field is calculated in the conducting airways (3D domain) by solving the incompressible Navier-Stokes equations with experimentally derived boundary conditions. Particles are tracked throughout inspiration by solving a modified Maxey-Riley equation. Next, we pass the time-dependent particle concentrations exiting the 3D model to the 1D volume conservation and advection-diffusion models (1D domain). Once the 1D models are solved, we prescribe the time-dependent number of particles entering back into the 3D airways to again solve for 3D transport. The coupled simulations highlight that about twice as many particles deposit during inhalation compared to exhalation for the entire lung. In contrast to inhalation, where most particles deposit at the bifurcation zones, particles deposit relatively uniformly on the gravitationally dependent side of the 3D airways during exhalation. Strong agreement to previously collected regional experimental data is shown, as the 1D models account for lobe-dependent morphology. This framework may be applied to investigate dosimetry in other species and pathological lungs. [ABSTRACT FROM AUTHOR]

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.