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Mathematical Medicine and Biology Advance Access published online on November 28, 2006
Mathematical Medicine and Biology, doi:10.1093/imammb/dql029
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Received May 22, 2005
Revised September 1, 2006

Article

Mathematical analysis of a free-boundary model for lung branching morphogenesis

Dirk Hartmann 1 * and Takashi Miura 2

1 Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany
2 Department of Human Anatomy and Genetics, University of Oxford, Oxford, UK and Department of Anatomy and Developmental Biology, Kyoto University Graduate School of Medicine, Yoshida Konoe-chou, Japan

* To whom correspondence should be addressed.
Dirk Hartmann, E-mail: dirk.hartmann{at}iwr.uni-heidelberg.de


  Abstract

Lung branching morphogenesis has been widely studied in the field of developmental biology. Lung airway trees consist of relatively regular-sized distal branches, but how this regular branched pattern is formed is not well understood. In the present study, we undertake a detailed mathematical analysis of the model proposed in Hartmann & Miura (2006), which numerically captures branching morphogenesis of the simplest possible experimental system in vitro. We investigate analytically the stability of 1D travelling waves with respect to periodic perturbations in two dimensions. This linear stability analysis leads to the so-called dispersion relations, predicting that a certain representative length dominates in this model. As the analytical analysis is restricted to travelling waves, we generalize the linear analysis to any 1D solution by numerical simulations. Both results predict how the representative lengths will change by experimentally changing specific parameters. Finally, we discuss the importance of the analytical results from a biological point of view and propose an experimental scheme for a quantitative comparison between experiments and theory.

Keywords: free-boundary problem; travelling waves; diffusive instability; branching morphogenesis.
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