A poroelastic model valid in large strains with applications to perfusion in cardiac modeling

Chapelle, Dominique; Gerbeau, Jean-Frédéric; Sainte-Marie, Jacques; Vignon-Clementel, Irene E.

doi:10.1007/s00466-009-0452-x

Cited by 109 publications

(161 citation statements)

References 37 publications

(43 reference statements)

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“…Historically, petroleum engineering has been the main applied field driving the theoretical development of porous media flow [3,17]. More recently, similar approaches have been applied to the modeling of fluid flow through biological tissues, with applications spanning from bio-engineering [16,32,38] to physiology [9,11,28].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models

Bociu

Guidoboni

Sacco

et al. 2016

Arch Rational Mech Anal

146

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We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid-solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory.

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Section: Introductionmentioning

confidence: 99%

Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models

Bociu

Guidoboni

Sacco

et al. 2016

Arch Rational Mech Anal

146

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“…The parameters chosen for this test problem are given in Table 5. This problem is similar to the one in [13] and highlights the method's ability to reliably capture steep gradients in the pressure solution due to rapid changes in material parameters.…”

Section: Swelling Testmentioning

confidence: 90%

“…However, using a continuous pressure element means that jumps in material coefficients may introduce large solution gradients across the interface, requiring severe mesh refinement or failing to reliably capture jumps in the pressure solution [54]. An operator splitting (iterative) approach for a near incompressible model is described by [13].…”

Section: Previous Results: Finite Strainmentioning

confidence: 99%

“…Fully saturated, incompressible poroelastic models have since been used to model a variety of biological tissues and processes. Biological examples include the coupling of flow in coronary vessels with the mechanical deformation of myocardial tissue to create a poroelastic model of coronary perfusion [13,15]. Other examples include modelling tissue deformation and the ventilation in the lungs [6], protein-based hydrogels embedded within cells [22], brain oedema and hydrocephalus [39,56], microcirculation of blood and interstitial fluid in the liver lobule [36], and interstitial fluid and tissue in articular cartilage and intervertebral discs [21,24,42].…”

Section: Introductionmentioning

confidence: 99%

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A stabilized finite element method for finite-strain three-field poroelasticity

Berger¹,

Bordas

Kay

et al. 2017

Comput Mech

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We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.

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“…In this regard we first list some of the straightforward extensions to the model compartments we have been already studied in previous contributions [185,216,219]. Some specific aspects of modelling cardiac function that have not been covered include the fast conduction system (Purkinje network [264] and Purkinje-muscle junctions), cardiac perfusion [46] and its coupling with coronary flow [66,169], autoregulation aspects of heart rate [139], myocardial tissue damage and remodelling [101], the modelling of the atria [47,64,252], the heart-torso coupling that is needed for the simulation of an ECG [33,55,75], fluid dynamics in idealized ventricles [194,248,249], and many others. As discussed in Sect.…”

Section: Discussionmentioning

confidence: 99%