Goriely - The Mathematics and Mechanics of Biological Growth

Growth. The term growth by itself refers to a change in mass. It is colloquially understood as an increase in mass, but the concept extends naturally to describe mass reduction or shrinking. Change in mass can be created either by addition of mass at constant density, as found in the development of soft tissues, a change in density at constant volume, as in the case of bone densification, or both as found in a developing bone. Mathematically, a theory of growth must allow for changes in mass, volume, and density and must be flexible to account for mass permeating through the boundary of the body, accumulating at the boundary, or occurring within the body itself.

Remodeling. It is well known that in the process of aging, tissues may become stiffer or softer. The term remodeling refers to an evolution of material properties in a system without change of mass such as stiffness, fiber orientation, fiber strength, and so forth. These remodeling processes are due to a change in the microstructure that determines the overall behavior of the tissue. For instance, the typical composition of soft tissues in many animals is a mixture of collagen fibers within an elastin matrix. Whereas elastin content remains mostly unchanged over many years, there is a continuous turnover of collagen that depends on the local biochemical and mechanical stimuli acting on the cells. The relative content of different types of collagen fibers and elastin determines the overall response of the tissue [635]. This process can occur without a change of mass, but it is crucial to understand the response of a tissue under mechanical loads. From a mathematical perspective, the variation of material properties can either be modeled by considering a separate evolution of the material parameters of a system or, at a lower scale, by taking into account the evolution of separate tissue components.

Morphogenesis. Early in embryonic life, new tissues and organs are formed. In this process, major reorganization and differentiation of cells take place after cell division, and, importantly, there is a restructuring of material elements. This reorganization process can only happen if the adhesion between different components is weak enough so that they can separate and reattach. This simple observation has important consequences for modeling as tissues undergoing morphogenesis exhibit rapid elastic stress relaxation and plastic-like flow. Mathematically, this evolution is often described by modeling tissues as fluid or viscoelastic rather than elastic, even though these two points of views are equivalent, as we will show.