In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. As such, the MPET equations represent a generalization of Biot's equations, and numerical discretizations of the MPET equations face similar challenges. Here, we focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly. Instead, we propose a new mixed finite element formulation based on introducing an additional total pressure variable. By presenting energy estimates for the continuous solutions and a priori error estimates for a family of compatible semi-discretizations, we show that this formulation is robust in the limits of incompressibility, vanishing storage coefficients, and vanishing transfer between networks. These theoretical results are corroborated by numerical experiments. Our primary interest in the MPET equations stems from the use of these equations in modelling interactions between biological fluids and tissues in physiological settings. So, we additionally present physiologically realistic numerical results for blood and tissue fluid flow interactions in the human brain.
The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multiporosity/multipermeability systems. These equations generalize Biot's equations, which describe the mechanics of the one network case. The efficient numerical solution of the MPET equations is challenging, in part due to the complexity of the system and in part due to the presence of interacting parameter regimes. In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically. In particular, we discuss an approach to formulating finite element methods and associated preconditioners for the MPET equations based on simultaneous diagonalization of the element matrices. We demonstrate the technique for the multicompartment Darcy equations, with large exchange variability, and the MPET equations for a nearly incompressible medium with large exchange variability. The approach is based on designing transformations of variables that simultaneously diagonalize (by congruence) the equations' key operators and subsequently constructing parameter-robust block diagonal preconditioners for the transformed system. The proposed approach is supported by theoretical considerations as well as by numerical results.
The intracranial pressure is implicated in many homeostatic processes in the brain and is a fundamental parameter in several diseases such as e.g.~idiopathic normal pressure hydrocephalus (iNPH). The presence of a small but persistent pulsatile intracranial pulsatile transmantle pressure gradient (on the order of a few mmHg/m at peak) has recently been demonstrated in iNPH subjects. A key question is whether pulsatile ICP and displacements can be induced by a small pressure gradient originating from the brain surface e.g. pial arteries alone. In this study, we model the brain parenchyma as either a linearly elastic or a poroelastic medium and impose a pulsatile pressure gradient acting between the ventricular and the pial surfaces. Using this high-resolution physics-based model, we compute the effect of the pulsatile pressure gradient on parenchyma displacement, volume change, fluid pressure, and fluid flux. The resulting displacement field is pulsatile and in qualitatively and quantitatively good agreement with the literature, both with elastic and poroelastic models. However, the pulsatile forces on the boundaries are not sufficient for pressure pulse propagation through the brain parenchyma. Our results suggest that pressure differences originating over the brain surface via e.g. pial artery pulsatility are not sufficient to drive interstitial fluid (ISF) flow within the brain parenchyma and that potential pressure gradients found within the parenchyma rather arise from local pressure pulsations of blood vessels within the brain parenchyma itself.
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