7.5 General Theory for Growth and Remodeling in 3D
365
Finally, the relative contributions of each constituent to the total stress-stretch
response are shown before (t = 0) and after (t = 10) remodeling (Fig. 7.8c, d).
These results are consistent with the idea that elastin dominates the behavior at
relatively small strains, while collagen dominates at large strains.
7.5 General Theory for Growth and Remodeling in 3D
In this section, we consider a tissue composed of 1D fibers embedded in a 3D
constrained mixture of cells and ground substance, which we call “cells” for
short. Consistent with the rest of this chapter, all constituents are assumed to
be incompressible, although it is straightforward to include compressibility in
the analysis. In general, the mixture may contain multiple fiber families, with
each family consisting of one type of fiber having a given orientation in the
reference configuration. A disk, for example, could have collagen fibers oriented
circumferentially and radially, as well as elastin aligned only radially. These fibers
are treated as three separate families: circumferential collagen, radial collagen, and
radial elastin.
Here, we assume that the fibers undergo both growth and remodeling, while
the cells only grow. Technically, the equations in the Humphrey-Rajagopal theory
can accommodate the growth of cells, as well as fibers. However, to be consistent
with our earlier growth formulation and to maintain the computational advantages
inherent in RHM theory, we treat cellular growth as before. Thus, our objective is
to combine the 1D theory for growth and remodeling of fibers with the 3D theory
for volumetric growth of cells presented in Sect. 6.6. For simplicity, we assume that
G&R is orthotropic relative to the local fiber direction.
In this section, cells are denoted by the index κ, 2 whereas fibers of the nth family
are indicated by n. All variables are functions of position, as well as time. For
convenience, however, position dependence is taken as implicit and not listed in
the arguments. Moreover, some of the previously derived equations are repeated
below.
7.5.1 Growth
Consider a rectangular element consisting of a mixture of cells and fibers. Because
the constituents are taken as incompressible, all changes in volume are attributed
to growth (J = J G ). During orthotropic growth, the lengths of the sides of the
2 We already have used c for collagen.
365
Finally, the relative contributions of each constituent to the total stress-stretch
response are shown before (t = 0) and after (t = 10) remodeling (Fig. 7.8c, d).
These results are consistent with the idea that elastin dominates the behavior at
relatively small strains, while collagen dominates at large strains.
7.5 General Theory for Growth and Remodeling in 3D
In this section, we consider a tissue composed of 1D fibers embedded in a 3D
constrained mixture of cells and ground substance, which we call “cells” for
short. Consistent with the rest of this chapter, all constituents are assumed to
be incompressible, although it is straightforward to include compressibility in
the analysis. In general, the mixture may contain multiple fiber families, with
each family consisting of one type of fiber having a given orientation in the
reference configuration. A disk, for example, could have collagen fibers oriented
circumferentially and radially, as well as elastin aligned only radially. These fibers
are treated as three separate families: circumferential collagen, radial collagen, and
radial elastin.
Here, we assume that the fibers undergo both growth and remodeling, while
the cells only grow. Technically, the equations in the Humphrey-Rajagopal theory
can accommodate the growth of cells, as well as fibers. However, to be consistent
with our earlier growth formulation and to maintain the computational advantages
inherent in RHM theory, we treat cellular growth as before. Thus, our objective is
to combine the 1D theory for growth and remodeling of fibers with the 3D theory
for volumetric growth of cells presented in Sect. 6.6. For simplicity, we assume that
G&R is orthotropic relative to the local fiber direction.
In this section, cells are denoted by the index κ, 2 whereas fibers of the nth family
are indicated by n. All variables are functions of position, as well as time. For
convenience, however, position dependence is taken as implicit and not listed in
the arguments. Moreover, some of the previously derived equations are repeated
below.
7.5.1 Growth
Consider a rectangular element consisting of a mixture of cells and fibers. Because
the constituents are taken as incompressible, all changes in volume are attributed
to growth (J = J G ). During orthotropic growth, the lengths of the sides of the
2 We already have used c for collagen.