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

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Numerical Mathematics and Advanced Applications ENUMATH 2017 (ENUMATH 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 126))

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Abstract

We study perfusion by a multiscale model coupling diffusion in the tissue and diffusion along the one-dimensional segments representing the vasculature. We propose a block-diagonal preconditioner for the model equations and demonstrate its robustness by numerical experiments. We compare our model to a macroscale model by Tofts [Modelling in DCE MRI, 2012].

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Notes

  1. 1.

    Let x ∈ Γ and C R(x) be a circular crossection of the vessel surface with a plane defined by the tangent vector of Γ at x. The surface avarage Π R u of u is then defined by

    $$\displaystyle \begin{aligned} (\varPi_R u)(x) = \lvert C_R(x) \rvert ^{-1}\int_{C_R(x)} u(y)\,\mathrm{d}y. \end{aligned}$$
  2. 2.

    Note that in (2) and (1) the constraint/coupling is defined in terms of a surface averaging operator Π R.

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Author information

Authors and Affiliations

  1. University of Oslo, Department of Informatics, Oslo, Norway

    Karl Erik Holter

  2. Simula Research Laboratory, Fornebu, Norway

    Karl Erik Holter

  3. University of Oslo, Department of Mathematics, Oslo, Norway

    Miroslav Kuchta & Kent-André Mardal

  4. Simula Research Laboratory, Fornebu, Norway

    Kent-André Mardal

Authors
  1. Karl Erik Holter
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  2. Miroslav Kuchta
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  3. Kent-André Mardal
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Corresponding author

Correspondence to Kent-André Mardal .

Editor information

Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Florin Adrian Radu

  2. University of Bergen, Bergen, Norway

    Kundan Kumar

  3. University of Bergen, Bergen, Norway

    Inga Berre

  4. University of Bergen, Bergen, Norway

    Jan Martin Nordbotten

  5. Hasselt University, Diepenbeek, Belgium

    Iuliu Sorin Pop

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Holter, K.E., Kuchta, M., Mardal, KA. (2019). Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_2

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