Continuum Modeling of Biological Network Formation

Handle URI:
http://hdl.handle.net/10754/623812
Title:
Continuum Modeling of Biological Network Formation
Authors:
Albi, Giacomo; Burger, Martin; Haskovec, Jan; Markowich, Peter A. ( 0000-0002-3704-1821 ) ; Schlottbom, Matthias
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.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Albi G, Burger M, Haskovec J, Markowich P, Schlottbom M (2017) Continuum Modeling of Biological Network Formation. Modeling and Simulation in Science, Engineering and Technology: 1–48. Available: http://dx.doi.org/10.1007/978-3-319-49996-3_1.
Publisher:
Springer International Publishing
Journal:
Modeling and Simulation in Science, Engineering and Technology
Issue Date:
10-Apr-2017
DOI:
10.1007/978-3-319-49996-3_1
Type:
Book Chapter
ISSN:
2164-3679; 2164-3725
Sponsors:
MB and MS acknowledge support by ERC via Grant EU FP 7 - ERC Con- solidator Grant 615216 LifeInverse. MB acknowledges support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster, Germany. GA acknowledges the ERC-Starting Grant project High-Dimensional Sparse Optimal Control (HDSPCONTR).
Additional Links:
http://link.springer.com/chapter/10.1007/978-3-319-49996-3_1
Appears in Collections:
Book Chapters; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorAlbi, Giacomoen
dc.contributor.authorBurger, Martinen
dc.contributor.authorHaskovec, Janen
dc.contributor.authorMarkowich, Peter A.en
dc.contributor.authorSchlottbom, Matthiasen
dc.date.accessioned2017-05-31T11:23:07Z-
dc.date.available2017-05-31T11:23:07Z-
dc.date.issued2017-04-10en
dc.identifier.citationAlbi G, Burger M, Haskovec J, Markowich P, Schlottbom M (2017) Continuum Modeling of Biological Network Formation. Modeling and Simulation in Science, Engineering and Technology: 1–48. Available: http://dx.doi.org/10.1007/978-3-319-49996-3_1.en
dc.identifier.issn2164-3679en
dc.identifier.issn2164-3725en
dc.identifier.doi10.1007/978-3-319-49996-3_1en
dc.identifier.urihttp://hdl.handle.net/10754/623812-
dc.description.abstractWe 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.en
dc.description.sponsorshipMB and MS acknowledge support by ERC via Grant EU FP 7 - ERC Con- solidator Grant 615216 LifeInverse. MB acknowledges support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster, Germany. GA acknowledges the ERC-Starting Grant project High-Dimensional Sparse Optimal Control (HDSPCONTR).en
dc.publisherSpringer International Publishingen
dc.relation.urlhttp://link.springer.com/chapter/10.1007/978-3-319-49996-3_1en
dc.titleContinuum Modeling of Biological Network Formationen
dc.typeBook Chapteren
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalModeling and Simulation in Science, Engineering and Technologyen
dc.contributor.institutionApplied Numerical Analysis, Technical University of Munich, Boltzmannstr. 3, 85478, Garching bei München, Germanyen
dc.contributor.institutionInstitute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149, Münster, Germanyen
dc.contributor.institutionMultiscale Modeling and Simulation, University of Twente, P.O. Box 217, NL-7500, AE Enschede, The Netherlandsen
kaust.authorHaskovec, Janen
kaust.authorMarkowich, Peter A.en
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