Editorial
Growing matter: A review of growth in living systems

https://doi.org/10.1016/j.jmbbm.2013.10.009Get rights and content

Highlights

  • Growth is a distinguishing feature of all living things.

  • We review and categorize growth by means of microstructure and microenvironment.

  • The microstructural appearance of growth can be either isotropic or anisotropic.

  • The microenvironmental cues of growth can be either biochemical or mechanical.

  • Exploring the underlying mechanisms helps understand growth in health and disease.

Abstract

Living systems can grow, develop, adapt, and evolve. These phenomena are non-intuitive to traditional engineers and often difficult to understand. Yet, classical engineering tools can provide valuable insight into the mechanisms of growth in health and disease. Within the past decade, the concept of incompatible configurations has evolved as a powerful tool to model growing systems within the framework of nonlinear continuum mechanics. However, there is still a substantial disconnect between the individual disciplines, which explore the phenomenon of growth from different angles. Here we show that the nonlinear field theories of mechanics provide a unified concept to model finite growth by means of a single tensorial internal variable, the second order growth tensor. We review the literature and categorize existing growth models by means of two criteria: the microstructural appearance of growth, either isotropic or anisotropic; and the microenvironmental cues that drive the growth process, either chemical or mechanical. We demonstrate that this generic concept is applicable to a broad range of phenomena such as growing arteries, growing tumors, growing skin, growing airway walls, growing heart valve leaflets, growing skeletal muscle, growing plant stems, growing heart valve annuli, and growing cardiac muscle. The proposed approach has important biological and clinical applications in atherosclerosis, in-stent restenosis, tumor invasion, tissue expansion, chronic bronchitis, mitral regurgitation, limb lengthening, tendon tear, plant physiology, dilated and hypertrophic cardiomyopathy, and heart failure. Understanding the mechanisms of growth in these chronic conditions may open new avenues in medical device design and personalized medicine to surgically or pharmacologically manipulate development and alter, control, or revert disease progression.

Graphical abstract

Growth in living systems illustrated in a turtle trapped in a plastic six-pack ring, photo (left) and simulation (right).

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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 I scaled by a scalar-valued growth multiplier ϑ1/3, Fg=ϑ1/3Iwhereϑ=Jg.In volume growth, the growth multiplier ϑ takes the interpretation of the grown volume Jg. We can directly invert the growth tensor,Fg1=ϑ1/3I,and obtain an explicit representation of elastic tensor,Fe=ϑ1/3F.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 n0, while there is no growth in the out-of-plane direction (Buganza Tepole et al., 2011),Fg=ϑI+[1ϑ]n0n0whereϑ=ηg.In area growth, the growth multiplier ϑ takes the interpretation of the grown surface area ηg. Since there is no growth in the normal direction, the amount of area growth is identical to the total amount of volume growth, Jg=ϑ. 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 n0, while there is no growth in the cross-fiber direction (Zöllner et al., in press),Fg=I+[ϑ1]n0n0whereϑ=λg.In fiber growth, the growth multiplier ϑ takes the interpretation of the chronic fiber lengthening λg. Since there is no cross-fiber growth, this fiber lengthening is identical to the total amount of volume growth, Jg=ϑ. 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 n0 and in the plane orthogonal to it (Göktepe et al., 2010),Fg=ϑI+[ϑϑ]n0n0whereϑ=ηgϑ=λg.To model this combined type of growth, we introduce two independent growth multipliers, the cross-fiber growth as the change in area ϑ=ηg and the fiber growth as the change in length ϑ=λg. This implies that the total amount of volume growth is equal to the product of the two, Jg=ϑϑ.

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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