EditorialGrowing matter: A review of growth in living systems
Graphical abstract
Growth in living systems illustrated in a turtle trapped in a plastic six-pack ring, photo (left) and simulation (right).
Section snippets
Motivation
Growth is a distinguishing feature of all living things. Throughout the past century, the growth of living systems has fascinated plant physiologists, biologists, clinical scientists, mathematicians, physicists, computer scientists, and engineers alike (Taber, 1995). An intriguing feature of growth is the interplay of form and function, or, more specifically, the ability of the growing system to manipulate its microenvironment and, vice versa, the ability of the microenvironment to manipulate
Continuum modeling of growth
Before discussing the individual examples of growing matter, we briefly summarize the continuum modeling of finite growth. In general, growth of a living system is associated with an increase in mass (Epstein and Maugin, 2000, Kuhl and Steinmann, 2003). Most living systems are multiphase materials that consist of a solid and one or more fluid phases (Humphrey and Rajagopal, 2003). Within the context of continuum mechanics, growth of multiphase materials is nothing but the exchange of mass
Volume growth
Volume growth is the simplest type of finite growth, for which the amount of growth is identical in all directions (Chen and Hoger, 2000). Its growth tensor is simply the identity tensor scaled by a scalar-valued growth multiplier , In volume growth, the growth multiplier ϑ takes the interpretation of the grown volume . We can directly invert the growth tensor,and obtain an explicit representation of elastic tensor,From this explicit expression,
Area growth
Area growth is a type of finite growth, for which growth is isotropic in a plane characterized through the unit normal , while there is no growth in the out-of-plane direction (Buganza Tepole et al., 2011),In area growth, the growth multiplier ϑ takes the interpretation of the grown surface area . Since there is no growth in the normal direction, the amount of area growth is identical to the total amount of volume growth, . The growth tensor for area growth has
Fiber growth
Fiber growth is a type of finite growth, for which growth takes place exclusively along the fiber direction , while there is no growth in the cross-fiber direction (Zöllner et al., in press),In fiber growth, the growth multiplier ϑ takes the interpretation of the chronic fiber lengthening . Since there is no cross-fiber growth, this fiber lengthening is identical to the total amount of volume growth, . The growth tensor has a simple rank-one update structure and
Combined growth
Combined fiber and cross-fiber growth is a type of growth, for which growth may take place along the fiber direction and in the plane orthogonal to it (Göktepe et al., 2010),To model this combined type of growth, we introduce two independent growth multipliers, the cross-fiber growth as the change in area and the fiber growth as the change in length . This implies that the total amount of volume growth is equal to the product of the two, .
Discussion
Living systems can undergo a continuous turnover in response to microenvironmental cues. Alterations in these cues, in particular during development and disease, may cause the system to grow. Here we have illustrated the phenomenon of growth in arteries, tumors, lungs, plants, skin, muscle, and the heart. From a biological point of view, these types of growth are intrinsically different and entirely unrelated. From a mechanical point of view, however, they have a lot in common: They all fall
Acknowledgments
This study was supported by the National Science Foundation CAREER award CMMI 0952021, by the National Science Foundation INSPIRE grant 1233054, and by the National Institutes of Health grant U54 GM072970.
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