Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem

Holter, Karl Erik; Kuchta, Miroslav; Mardal, Kent‐André

doi:10.1007/978-3-319-96415-7_2

Cited by 6 publications

(7 citation statements)

References 19 publications

(31 reference statements)

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“…Motivated by the observation in [29], that in a discrete finite element setting the trace operator is stable as a map L 2 (Ω) → L 2 (∂Ω), we propose an alternative approach to construct the preconditioners. We start off by outlining the construction of an operator ∂ n, : H 1 (Ω p ) → L 2 (Γ) which will be an approximation to ∂ n .…”

Section: Approximating the Trace Normal Gradient Operatormentioning

confidence: 99%

“…We remark that when using the finite element method, the identity or the mass matrix has eigenvalues such that both the smallest and the largest eigenvalues scale as h d on uniform mesh. First we consider S = I, with eigenvalues ≈ h. Then, following [29], we let S = h −1 I, i.e., a matrix with eigenvalues ≈ 1. Finally, the choice of S = (−∆ + I) −1/2 is included to show that the relevant trace space in (12) is not (by viewing the trace as an order 1/2 operator) H 1/2 so that dual variable would reside in H −1/2 .…”

Section: Approximating the Trace Normal Gradient Operatormentioning

confidence: 99%

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Robust preconditioning for coupled Stokes-Darcy problems with the Darcy problem in primal form

Holter¹,

Kuchta²,

Mardal³

2020

Preprint

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The coupled Darcy-Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolitic solution algorithms of the coupled Darcy-Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters, i.e. the permeability, viscosity and Beavers-Joseph-Saffman condition. Our approach is similar to that of [1], but since the Darcy problem is in primal form, the mass conservation at the interface introduces some challenges. These challenges will be specifically addressed in this paper. Numerical experiments illustrating the performance are provided. The preconditioner is posed in non-standard Sobolev spaces which may be perceived as an obstacle for its use in applications. However, we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice.

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Section: Approximating the Trace Normal Gradient Operatormentioning

confidence: 99%

Section: Approximating the Trace Normal Gradient Operatormentioning

confidence: 99%

Robust preconditioning for coupled Stokes-Darcy problems with the Darcy problem in primal form

Holter¹,

Kuchta²,

Mardal³

2020

Preprint

Self Cite

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“…Related models have been used to study the proliferation of cancer drugs, [27][28][29] the transport of oxygen, [30][31][32][33][34] and nanoparticle transport for hypothermia therapy. 35 A recent study 36 describes contrast agent perfusion based on diffusive transport with a mixed-dimension model. The herein presented fluid-mechanical model is similar to the drug proliferation model described in Cattaneo and Zunino 27,37 and introduced in Possenti et al 28 It is derived here for the specific application of contrast agent perfusion in brain tissue.…”

Section: Introductionmentioning

confidence: 99%

A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data

Koch

Flemisch

Helmig

et al. 2020

Numer Methods Biomed Eng

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We propose a new mathematical model to learn capillary leakage coefficients from dynamic susceptibility contrast MRI data. To this end, we derive an embedded mixed‐dimension flow and transport model for brain tissue perfusion on a subvoxel scale. This model is used to obtain the contrast agent concentration distribution in a single MRI voxel during a perfusion MRI sequence. We further present a magnetic resonance signal model for the considered sequence including a model for local susceptibility effects. This allows modeling MR signal‐time curves that can be compared with clinical MRI data. The proposed model can be used as a forward model in the inverse modeling problem of inferring model parameters such as the diffusive capillary wall conductivity. Acute multiple sclerosis lesions are associated with a breach in the integrity of the blood‐brain barrier. Applying the model to perfusion MR data of a patient with acute multiple sclerosis lesions, we conclude that diffusive capillary wall conductivity is a good indicator for characterizing activity of lesions, even if other patient‐specific model parameters are not well‐known.

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“…Different strategies have been proposed to remedy this, e.g., Kuchta et al studied suitable preconditioners in [27]. Holter et al then applied this preconditioner to simulate flow through the microcirculature found in a mouse brain [23]. An alternative coupling scheme was intro-duced by Köppl et al in [25], where the source term was taken to live on the boundary of the inclusions.…”

Section: Introductionmentioning

confidence: 99%

A singularity removal method for coupled 1D–3D flow models

2019

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In reservoir simulations, the radius of a well is inevitably going to be small compared to the horizontal length scale of the reservoir. For this reason, wells are typically modelled as lower-dimensional sources. In this work, we consider a coupled 1D-3D flow model, in which the well is modelled as a line source in the reservoir domain and endowed with its own 1D flow equation. The flow between well and reservoir can then be modelled in a fully coupled manner by applying a linear filtration law.The line source induces a logarithmic type singularity in the reservoir pressure that is difficult to resolve numerically. We present here a singularity removal method for the model equations, resulting in a reformulated coupled 1D-3D flow model in which all variables are smooth. The singularity removal is based on a solution splitting of the reservoir pressure, where it is decomposed into two terms: an explicitly given, lower regularity term capturing the solution singularity and some smooth background pressure. The singularities can then be removed from the system by subtracting them from the governing equations. Finally, the coupled 1D-3D flow equations can be reformulated so they are given in terms of the well pressure and the background reservoir pressure. As these variables are both smooth -singular), the reformulated model has the advantage that it can be approximated using any standard numerical method. The reformulation itself resembles a Peaceman well correction performed at the continuous level.

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Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem

Cited by 6 publications

References 19 publications

Robust preconditioning for coupled Stokes-Darcy problems with the Darcy problem in primal form

Robust preconditioning for coupled Stokes-Darcy problems with the Darcy problem in primal form

A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data

A singularity removal method for coupled 1D–3D flow models

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