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Continuum Modeling of Biological Network Formation

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Active Particles, Volume 1

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.

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Acknowledgements

MB and MS acknowledge support by ERC via Grant EU FP 7 - ERC Consolidator 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).

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Authors and Affiliations

  1. Applied Numerical Analysis, Technical University of Munich, Boltzmannstr. 3, 85478, Garching bei München, Germany

    Giacomo Albi

  2. Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149, Münster, Germany

    Martin Burger

  3. Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, Kingdom of Saudi Arabia

    Jan Haskovec & Peter Markowich

  4. Multiscale Modeling and Simulation, University of Twente, P.O. Box 217, NL-7500, AE Enschede, The Netherlands

    Matthias Schlottbom

Authors
  1. Giacomo Albi
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  2. Martin Burger
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  3. Jan Haskovec
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  4. Peter Markowich
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  5. Matthias Schlottbom
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Corresponding author

Correspondence to Jan Haskovec .

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Editors and Affiliations

  1. Department of Mathematical Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Degond

  3. CSCAMM, University of Maryland, College Park, USA

    Eitan Tadmor

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Albi, G., Burger, M., Haskovec, J., Markowich, P., Schlottbom, M. (2017). Continuum Modeling of Biological Network Formation. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49996-3_1

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