BB9: Mathematical modelling of growth in physical and biological systems

Project completed March 31, 2013. Project report to follow

Background

Growth is an ubiquitous process in nature, common in some form to all living entities, and present in numerous physical and industrial applications as well.  Understanding how growth occurs has wide-ranging impact, from modelling tumour progression and wound healing to the building of smart materials to improving evolutionary understanding.  In this project, we are developing mathematical models of growth processes based on underlying mechanical forces.

Techniques and Challenges

We have developed a general framework to model surface growth in hollow structures by evolving a generating curve through space.  We utilise tools from differential geometry to obtain a description that is naturally linked to the growth process.  Kinematics must be driven by mechanics and the forces governing the way new material is added.  In many growth processes, this boils down to the interaction of a soft elastic tissue with a rigid body, for which we have developed several analytical and numerical techniques.  We have also developed an innovative computational approach for simulating growth of three-dimensional structures.

Results

We have applied our framework to the growth of seashells, in particular the formation of three-dimensional ornamentation on shells (e.g. spines, ribs, protrusions).  Here, the generating curve is the shell aperture, whose shape evolves due to the mechanical deformation of a soft tissue called the mantle that adheres to the shell edge and deposits new material.  By modelling the mantle edge as an elastic rod, we have successfully simulated a variety of realistic and complex shell structures.  Our results have shaped the basic understanding of how shells grow and also have important implications for theories of evolution.

The Future

We will continue to explore the numerous patterns and structures seen in seashells, with the aim of a unified theory of the morphogenesis of ornamentation.  We are also currently developing a general theory for the growth of complex filamentary structures.  A key goal is to uncover fundamental relationships between growth, curvature and stress in slender bodies.

References

[12/66] Moulton D. E., Lessinnes T., Goriely A.: Morphoelastic rods Part I: A single growing elastic rod, Journal of the Mechanics and Physics of Solids

[12/31] Chirat R., Moulton D.E., Goriely A.: Convergent evolution of spiny mollusk shells points to elastic energy minimum, PNAS

[12/01] Moulton D.E., Goriely A., Chirat R.: Mechanical growth and morphogenesis of seashells, Journal of Theoretical Biology

[11/53] Moulton D.E., Goriely A.: Surface growth kinematics via local curve evolution, Journal of Mathematical Biology

[10/48] Moulton D.E., Goriely A.: Possible role of differential growth in airway wall remodeling in asthma, Journal of Applied Physiology

[10/37] Moulton D.E., Goriely A.: Circumferential buckling instability of a growing cylindrical tube, Journal of the Mechanics and Physics of Solids

[10/19] Goriely A., Moulton D.E., Vandiver R.: Elastic cavitation, tube hollowing, and differential growth in plants and biological tissues, EuroPhysics Letters

[10/16] Goriely A., Moulton D.E.: The Physics and Mechanics of Biological Systems: Lecture Notes of the Les Houches Summer Schools (Book Chapter)

[10/09] Goriely A., Moulton D.E.: Anticavitation and differential growth in elastic shells, Journal of Elasticity

Skalak R, Farrow D, Hoger A Kinematics of surface growth, Journal of mathematical biology, 35(8): 869-907, 1997

Thompson D.: On growth and form, Cambridge, London,1942