Continuum Modeling of Biological Network Formation

Giacomo Albi, Martin Burger, Jan Haskovec, Peter Markowich, Matthias Schlottbom

Research output: Chapter in Book/Report/Conference proceedingChapterAcademicpeer-review

3 Citations (Scopus)

Abstract

We present an overview of recent analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transportation networks. The model describes the pressure field using a Darcy type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. We first introduce micro- and mesoscopic models and show how they are connected to the macroscopic PDE system. Then, we provide an overview of analytical results for the PDE model, focusing mainly on the existence of weak and mild solutions and analysis of the steady states. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on finite elements and study the qualitative properties of network structures for various parameter values.
Original languageEnglish
Title of host publicationActive Particles, Volume 1
Subtitle of host publicationAdvances in Theory, Models, and Applications
EditorsNicola Bellomo, Pierre Degond, Eitan Tadmor
PublisherSpringer
Pages1-48
Volume1
ISBN (Electronic)978-3-319-49996-3
ISBN (Print)978-3-319-49994-9
DOIs
Publication statusPublished - 2017

Publication series

NameModeling and Simulation in Science, Engineering and Technology

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continuum modeling
pulse detonation engines
transportation networks
pressure distribution
partial differential equations
decay
simulation

Cite this

Albi, G., Burger, M., Haskovec, J., Markowich, P., & Schlottbom, M. (2017). Continuum Modeling of Biological Network Formation. In N. Bellomo, P. Degond, & E. Tadmor (Eds.), Active Particles, Volume 1: Advances in Theory, Models, and Applications (Vol. 1, pp. 1-48). (Modeling and Simulation in Science, Engineering and Technology ). Springer. https://doi.org/10.1007/978-3-319-49996-3
Albi, Giacomo ; Burger, Martin ; Haskovec, Jan ; Markowich, Peter ; Schlottbom, Matthias . / Continuum Modeling of Biological Network Formation. Active Particles, Volume 1: Advances in Theory, Models, and Applications. editor / Nicola Bellomo ; Pierre Degond ; Eitan Tadmor. Vol. 1 Springer, 2017. pp. 1-48 (Modeling and Simulation in Science, Engineering and Technology ).
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Albi, G, Burger, M, Haskovec, J, Markowich, P & Schlottbom, M 2017, Continuum Modeling of Biological Network Formation. in N Bellomo, P Degond & E Tadmor (eds), Active Particles, Volume 1: Advances in Theory, Models, and Applications. vol. 1, Modeling and Simulation in Science, Engineering and Technology , Springer, pp. 1-48. https://doi.org/10.1007/978-3-319-49996-3

Continuum Modeling of Biological Network Formation. / Albi, Giacomo; Burger, Martin; Haskovec, Jan; Markowich, Peter; Schlottbom, Matthias .

Active Particles, Volume 1: Advances in Theory, Models, and Applications. ed. / Nicola Bellomo; Pierre Degond; Eitan Tadmor. Vol. 1 Springer, 2017. p. 1-48 (Modeling and Simulation in Science, Engineering and Technology ).

Research output: Chapter in Book/Report/Conference proceedingChapterAcademicpeer-review

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AU - Albi, Giacomo

AU - Burger, Martin

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AU - Markowich, Peter

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AB - We present an overview of recent analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transportation networks. The model describes the pressure field using a Darcy type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. We first introduce micro- and mesoscopic models and show how they are connected to the macroscopic PDE system. Then, we provide an overview of analytical results for the PDE model, focusing mainly on the existence of weak and mild solutions and analysis of the steady states. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on finite elements and study the qualitative properties of network structures for various parameter values.

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Albi G, Burger M, Haskovec J, Markowich P, Schlottbom M. Continuum Modeling of Biological Network Formation. In Bellomo N, Degond P, Tadmor E, editors, Active Particles, Volume 1: Advances in Theory, Models, and Applications. Vol. 1. Springer. 2017. p. 1-48. (Modeling and Simulation in Science, Engineering and Technology ). https://doi.org/10.1007/978-3-319-49996-3