You are here
Journal Club Theme of January 2012: Mechanics of Growth
"I can't understand how people are still working on growth. That stuff's all done." This was the beginning of the first lunch conversation at a recent Banff workshop on Mathematical Foundations on Mechanical Biology... somewhat frustrating for someone who is excited about growth. Fortunately, most of the presentations and discussions still focused on growth. Although "that stuff's all done".
Would anybody claim that plasticity was all done when Richard von Mises published his milestone work in 1913? Or was it all done when Geoffrey Ingram Taylor contributed his famous monograph on crystal plasticity in 1938? Or was it all done when Ekkehard Kröner introduced the concept of dislocations to explain the mechanistic origin of plastic slip in 1958? Or was it only all done when Juan Simo made it computationally manageable in 1985?
In a way, growth is like plasticity. It has its von Mises in Julius Wolff and D'Arcy Thompson, its Kröners in Steven Cowin and Dennis Carter, and its Simos in Rik Huiskes, Anne Hoger, Larry Taber, and Jay Humphrey. But... does that mean that "that stuff's all done"?
In the century of quantitative biology, mechanics has a lot to offer when it comes to exploring living systems. Continuum mechanics is a powerful tool, if not the only one, to bridge what a biologist sees in a dish and what a medical doctor diagnoses in a patient. As mechanics community we are very familiar with the tools to bring these two worlds together and characterize living systems across the scales, from the molecular to the subcellular, cellular, tissue, and organ levels. We are also familiar with the tools to characterize living systems across the fields, from mechanical, to biological, chemical, and sometimes even electrical.
We have to accept though that living systems are somewhat more complex than metal plasticity. Because living systems undergo a continuous turnover, they have the fascinating ability to adapt to their mechanical environment. From a continuum mechanics point of view, living systems are open systems interacting with their surroundings through exchanging mass. This allows them to grow in density and volume . Excellent overviews classify these adaptation phenomena  and summarize recent trends [2,11].
Figure 1. Growth of thin biological membranes: Skin growth in reconstructive surgery, see .
Key to most soft tissue growth models is the multiplicative decomposition of the deformation gradient into an elastic and a growth part . The specific format of the growth part, typically a second order tensor, depends on the particular type of tissue. It can be plain isotropic in the form of volume growth, e.g., for growing tumors , transversely isotropic in the form of area growth, e.g., for growing skin , transversely isotropic in the form of fiber growth, e.g., for growing muscle , or generally anisotropic . While some models focus on the plain kinematic characterization of growth  and illustrate its geometric interpretation , others combine the growth equation with mechanical equilibrium and introduce either stress  or strain  as the driving force for growth. Alternative approaches adopt density changes, initially introduced to model hard tissue growth, to avoid the introduction of internal variables when characterizing growing soft tissues .
Interest in modeling growth is currently shifting from hard to soft matter , from infinitesimal to finite deformation , from phenomenological to mechanistic , from single scale to multiscale , from single field to multifield , from generic to subject-specific , and from reproductive to predictive modeling . As such, growth of living biological tissues undoubtably remains one of the most challenging phenomena in continuum mechanics, and we hope that this Journal Club initiates interest amongst many of you to actively contribute to this rapidly evolving field.
If you don't think "that stuff's all done", please contribute additional references, comments, discussions, or figures.
 Ambrosi D, Mollica F. On the mechanics of a growing tumor. Int J Eng Sci. 2002;40:1297-1316. http://ezproxy.stanford.edu:2488/science/article/pii/S0020722502000149
 Ambrosi D, Ateshian GA, Arruda EM, Cowin SC, Dumais J, Goriely A, Holzapfel GA, Humphrey JD, Kemkemer R, Kuhl E, Olberding JE, Taber LA, Garikipati K. Perspectives on biological growth and remodeling. J Mech Phys Solids. 2011;59:863-883. http://www.sciencedirect.com/science/article/pii/S0022509610002516
 BenAmar M, Goriely A. Growth and instability in elastic tissues. J Mech Phys Solids. 2005;53:2284-2319. http://www.sciencedirect.com/science/article/pii/S0022509605000876
 Buganza Tepole A, Ploch CJ, Wong J, Gosain AK, Kuhl E. Growin skin: A computational model for skin expansion in reconstructive surgery. J Mech Phys Solids. 2011;59:2177-2190. http://www.sciencedirect.com/science/article/pii/S0022509611001074
 Epstein M, Maugin GA. Thermomechanics of volumetric growth in uniform bodies.
Int J Plast. 2000;16:951-978. http://www.sciencedirect.com/science/article/pii/S0749641999000819
 Garikipati K. The kinematics of biological growth. Appl Mech Rev.2009;62:030801.1-030801.7. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AMR...
 Göktepe S, Abilez OJ, Parker KK, Kuhl E. A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis. J Theor Bio. 2010;265:433-442. http://www.sciencedirect.com/science/article/pii/S0022519310002146
 Göktepe S, Abilez OJ, Kuhl E. A generic approach towards finite growth with examples of athlete's heart, cardiac dilation, and cardiac wall thickening. J Mech Phys Solids. 2010;58:1661-1680. http://www.sciencedirect.com/science/article/pii/S0022509610001328
 Humphrey JD, Rajagopal KR. A constrained mixture model for growth and remodeling of soft tissues. Math Mod Meth Appl Sci. 2002;12:407-430. http://ezproxy.stanford.edu:2062/ehost/detail?sid=c8b0e9be-6df6-4c2b-944...
 Jin L, Cai S, Suo Z. Creases in soft tissues generated by growth. EPL. 2011;95:64002,p1-p6. http://iopscience.iop.org/0295-5075/95/6/64002/
 Menzel A, Kuhl E. Frontiers in growth and remodeling. Mech Res Comm, 2012;42,1-14. http://www.sciencedirect.com/science/article/pii/S0093641312000225?v=s5
 Menzel A. Modelling of anisotropic growth in biological tissues - A new approach and computational aspects. Biomech Model Mechanobiol. 2005;3:147-171. http://www.springerlink.com/content/36hq4wwwufy4g983/
 Pang H, Shiwalkar AP, Madormo CM, Taylor RE, Andriacchi TP, Kuhl E. Computational modeling of bone density profiles in response to gait: A subject-specific approach. Biomech Model Mechanobiol. 2012;11:379-390. http://www.springerlink.com/content/fm26538l6720872u/
 Rodriguez EK, Hoger A, McCulloch AD. Stress-dependent finite growth in soft elastic tissues. J Biomech. 1994;27:455-467. http://ezproxy.stanford.edu:2488/science/article/pii/0021929094900213
 Taber LA. Biomechanics of growth, remodeling and morphogenesis. Appl Mech Rev. 1995;48:487-545. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AMR...
 Yavari A. A geometric theory of growth mechanics. J Nonlin Sci. 2010;20:781-830. http://ezproxy.stanford.edu:2062/ehost/pdfviewer/pdfviewer?sid=6fafa3dd-...
 Volokh KY. Stress in growing soft tissues. Acta Biomat. 2006;2:493-504. http://www.sciencedirect.com/science/article/pii/S1742706106000432
 Zöllner AM, Buganza Tepole A, Kuhl E. On the biomechanics and mechanobiology of growing skin. J Theor Bio. 2012;297:166-175. http://www.sciencedirect.com/science/article/pii/S0022519311006461
ME337 "Mechanics of Growth", Winter 2012, Stanford University. http://biomechanics.stanford.edu/Mechanics_of_growth_12