A finite element method for growth in biological development

Cornel M. Murea; H. G. E. Hentschel

doi:10.3934/mbe.2007.4.339

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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

Cornel M. Murea^1, and
H. G. E. Hentschel^2,

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: April 30, 2006

Revised: August 31, 2006

Published: January 2007

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Abstract

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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Mathematics Subject Classification: 65M60, 76Z05.

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