A general approach for modeling interacting flow through porous media under finite deformations

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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“…A three-field finite element using a continuous pressure approximation has been implemented in [52].…”

Section: Previous Results: Finite Strainmentioning

confidence: 99%

“…Relationships between the two theories are explored in [14,17]. As is most common in biological applications, we use the mixture theory for poroelasticity as outlined in [8] and recently summarized in [52].…”

Section: Poroelasticity Theorymentioning

confidence: 99%

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

Berger¹,

Bordas

Kay

et al. 2017

Comput Mech

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“…From our experience of similar porous medium systems [37,47] along with the numerical examples of Section 4, most of the computational time is spent in resolving the coupling between the structural deformation and the fluid flow or angiogenesis, respectively. Hence, in a first step, the nonlinear coupling between these two fields (42)- (43) can be resolved monolithically while for the coupling with species transport still a partitioned scheme is employed.…”

Section: Monolithic-partitioned Schemementioning

confidence: 99%

A monolithic multiphase porous medium framework for (a-)vascular tumor growth

Kremheller

Vuong

Yoshihara

et al. 2018

Computer Methods in Applied Mechanics and Engineering

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We present a dynamic vascular tumor model combining a multiphase porous medium framework for avascular tumor growth in a consistent Arbitrary Lagrangian Eulerian formulation and a novel approach to incorporate angiogenesis. The multiphase model is based on Thermodynamically Constrained Averaging Theory and comprises the extracellular matrix as a porous solid phase and three fluid phases: (living and necrotic) tumor cells, host cells and the interstitial fluid. Angiogenesis is modeled by treating the neovasculature as a proper additional phase with volume fraction or blood vessel density. This allows us to define consistent inter-phase exchange terms between the neovasculature and the interstitial fluid. As a consequence, transcapillary leakage and lymphatic drainage can be modeled. By including these important processes we are able to reproduce the increased interstitial pressure in tumors which is a crucial factor in drug delivery and, thus, therapeutic outcome. Different coupling schemes to solve the resulting five-phase problem are realized and compared with respect to robustness and computational efficiency. We find that a fully monolithic approach is superior to both the standard partitioned and a hybrid monolithic-partitioned scheme for a wide range of parameters. The flexible implementation of the novel model makes further extensions (e.g., inclusion of additional phases and species) straightforward.

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“…In particular, we consider large deformations, varying porosities and nonlinear material behaviour. A detailed derivation and further theory can be found in [25,26]. The governing equations are reviewed in the electronic supplementary material, section S1.…”

Section: Model and Governing Equations (A) Extracellular Matrix As Pomentioning

confidence: 99%

“…The numerical approach is based on schemes presented in [25,26,42]. Detailed explanations can be found in the electronic supplementary material, section S2, and a validation example of the numerical scheme for surface-volume reactions in the electronic supplementary material, section S3.…”

Section: Numerical Examplesmentioning

confidence: 99%

A biochemo-mechano coupled, computational model combining membrane transport and pericellular proteolysis in tissue mechanics

Vuong

Wall

2017

Proc. R. Soc. A.

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We present a computational model for the interaction of surface-and volume-bound scalar transport and reaction processes with a deformable porous medium. The application in mind is pericellular proteolysis, i.e. the dissolution of the solid phase of the extracellular matrix (ECM) as a response to the activation of certain chemical species at the cell membrane and in the vicinity of the cell. A poroelastic medium model represents the extra cellular scaffold and the interstitial fluid flow, while a surface-bound transport model accounts for the diffusion and reaction of membrane-bound chemical species. By further modelling the volumebound transport, we consider the advection, diffusion and reaction of sequestered chemical species within the extracellular scaffold. The chemo-mechanical coupling is established by introducing a continuum formulation for the interplay of reaction rates and the mechanical state of the ECM. It is based on known experimental insights and theoretical work on the thermodynamics of porous media and degradation kinetics of collagen fibres on the one hand and a damage-like effect of the fibre dissolution on the mechanical integrity of the ECM on the other hand. The resulting system of partial differential equations is solved via the finite-element method. To the best of our knowledge, it is the first computational model including contemporaneously the coupling between (i) advection-diffusion-reaction processes, (ii) interstitial flow and deformation of a porous medium, and (iii) the chemo-mechanical interaction

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A general approach for modeling interacting flow through porous media under finite deformations

Cited by 61 publications

References 29 publications

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

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

A monolithic multiphase porous medium framework for (a-)vascular tumor growth

A biochemo-mechano coupled, computational model combining membrane transport and pericellular proteolysis in tissue mechanics

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