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Mathematical Biosciences and Engineering

2007, Volume 4, Issue 2: 339-353. doi: 10.3934/mbe.2007.4.339
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A finite element method for growth in biological development

  • 1.
    Laboratoire de Mathématiques, Informatique et Applications, Université de Haute-Alsace, 4, rue des Frères Lumière, 68093 MULHOUSE Cedex
  • 2.
    Department of Physics, Emory University, Maths/Science Center, 400 Dowman Drive, Atlanta, GA 30322
  • Received: 01 May 2006 Accepted: 29 June 2018 Published: 01 February 2007

  • MSC : 65M60, 76Z05.

  • We describe finite element simulations of limb growth based on Stokes flow models with a nonzero divergence representing growth due to nutrients in the early stages of limb bud development. We introduce a ''tissue pressure'' whose spatial derivatives yield the growth velocity in the limb and our explicit time advancing algorithm for such tissue flows is described in detail. The limb boundary is approached by spline functions to compute the curvature and the unit outward normal vector. At each time step, a mixed-hybrid finite element problem is solved, where the condition that the velocity is strictly normal to the limb boundary is treated by a Lagrange multiplier technique. Numerical results are presented.

    Citation: Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 339-353. doi: 10.3934/mbe.2007.4.339

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  • We describe finite element simulations of limb growth based on Stokes flow models with a nonzero divergence representing growth due to nutrients in the early stages of limb bud development. We introduce a ''tissue pressure'' whose spatial derivatives yield the growth velocity in the limb and our explicit time advancing algorithm for such tissue flows is described in detail. The limb boundary is approached by spline functions to compute the curvature and the unit outward normal vector. At each time step, a mixed-hybrid finite element problem is solved, where the condition that the velocity is strictly normal to the limb boundary is treated by a Lagrange multiplier technique. Numerical results are presented.


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    2. Ricardo Ruiz-Baier, Primal-mixed formulations for reaction–diffusion systems on deforming domains, 2015, 299, 00219991, 320, 10.1016/j.jcp.2015.07.018
    3. Bernd Boehm, Henrik Westerberg, Gaja Lesnicar-Pucko, Sahdia Raja, Michael Rautschka, James Cotterell, Jim Swoger, James Sharpe, Alfonso Martinez Arias, The Role of Spatially Controlled Cell Proliferation in Limb Bud Morphogenesis, 2010, 8, 1545-7885, e1000420, 10.1371/journal.pbio.1000420
    4. C. M. Murea, Arbitrary Lagrangian Eulerian approximation with remeshing for Navier-Stokes equations, 2010, 26, 20407939, 1435, 10.1002/cnm.1223
    5. Paramita Chatterjee, Tilmann Glimm, Bogdan Kaźmierczak, Mathematical modeling of chondrogenic pattern formation during limb development: Recent advances in continuous models, 2020, 322, 00255564, 108319, 10.1016/j.mbs.2020.108319
    6. Stuart A. Newman, Scott Christley, Tilmann Glimm, H.G.E. Hentschel, Bogdan Kazmierczak, Yong-Tao Zhang, Jianfeng Zhu, Mark Alber, 2008, 81, 9780123742537, 311, 10.1016/S0070-2153(07)81011-8
    7. Yipei Guo, Mor Nitzan, Michael P. Brenner, Alexandre V. Morozov, Programming cell growth into different cluster shapes using diffusible signals, 2021, 17, 1553-7358, e1009576, 10.1371/journal.pcbi.1009576
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  • © 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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