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Mathematical Models and Methods for Living Systems pp 211–274Cite as

Mathematical Modeling of Morphogenesis in Living Materials

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2167))

Abstract

From a mathematical viewpoint, the study of morphogenesis focuses on the description of all geometric and structural changes which locally orchestrate the underlying biological processes directing the formation of a macroscopic shape in living matter. In this chapter, we introduce a continuous chemo-mechanical approach of morphogenesis, deriving the balance principles and evolution laws for both volumetric and interfacial processes. The proposed theory is applied to the study of pattern formation for either a fluid-like or a solid-like biological system model, using both theoretical methods and simulation tools.

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Notes

  1. 1.

    In mathematical terms, this latin expression can be roughly translated as the set of the strictly necessary causes (from Cicero, Rhetorica, Tusculanae Disputationes, Liber Quintus, 70).

  2. 2.

    This idea was enforced by Luca Pacioli, a student of Leonardo da Vinci, in his book De divina proportione, published in 1509.

  3. 3.

    Since S is related to \(\boldsymbol{\sigma }\) through a Piola transformation, it is often called first Piola-Kirchhoff stress tensor [85]. Nonetheless, some authors consider the latter being the transpose of S [90]. Accordingly we prefer to call it nominal stress in order to avoid misunderstanding to the readers.

  4. 4.

    We remark that for Galilean invariance they should be somehow dependent on a relative velocity field.

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Acknowledgements

We are thankful to Chiara Giverso for performing the numerical simulations shown in Fig. 4.8, and to Ellen Kuhl for sharing the numerical FE code used in Sect. 4.6.2.4. Partial funding by the European Community grant ERG-256605, FP7 program, and by the “Start-up Packages and PhD Program” project, co-funded by Regione Lombardia through the “Fondo per lo sviluppo e la coesione 2007-2013”, formerly FAS program, is gratefully acknowledged by the corresponding author.

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Authors and Affiliations

  1. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Valentina Balbi

  2. Institut Jean le Rond d’Alembert, CNRS and Sorbonne Universités, Université Paris 6, UMR 7190, 4 place Jussieu case 162, 75005, Paris, France

    Pasquale Ciarletta

  3. Mox, Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 33, 20133, Milano, Italy

    Pasquale Ciarletta

Authors
  1. Valentina Balbi
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  2. Pasquale Ciarletta
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Corresponding author

Correspondence to Pasquale Ciarletta .

Editor information

Editors and Affiliations

  1. Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Luigi Preziosi

  2. School of Mathematics and Statistics, University of St. Andrews, St. Andrews, United Kingdom

    Mark Chaplain

  3. Department of Mathematics, Università di Trento, Povo, Italy

    Andrea Pugliese

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© 2016 Springer International Publishing Switzerland

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Balbi, V., Ciarletta, P. (2016). Mathematical Modeling of Morphogenesis in Living Materials. In: Preziosi, L., Chaplain, M., Pugliese, A. (eds) Mathematical Models and Methods for Living Systems. Lecture Notes in Mathematics(), vol 2167. Springer, Cham. https://doi.org/10.1007/978-3-319-42679-2_4

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