292
6 Growth
the inner perimeter causes compression in the inner part. The opposite effects occur
when G r < 1. These qualitative observations agree with the quantitative results in
Fig. 6.11b.
The stress distributions for an artery plotted in Fig. 6.8d show that a positive
opening angle corresponds to circumferential residual stress that is tensile outside
and compressive inside. Therefore, in a tube undergoing uniform growth, negative
circumferential growth and positive radial growth tend to make opening angles
positive, and vice versa (Fig. 6.11a, b).
In general, arteries undergo various combinations of radial, circumferential, and
longitudinal growth. Moreover, the growth need not be uniform. Suppose an artery
grows only in the circumferential direction, but with a negative gradient across the
wall. If the average value of G θ remains unchanged, the growth gradient can change
the sign of the residual stress gradient (Fig. 6.11c) and the corresponding opening
angle.
6.8 Case Study: Growth of a Spherical Brain Tumor
It is now widely accepted that microenvironmental factors, including mechanical
forces, play an important role in the development and progression of cancer
(Northey et al. 2017). Studies have shown, for example, that stress and tissue
stiffness affect tumor growth and metastasis (Jain et al. 2014; Northey et al. 2017).
Growth, in turn, can have important biomechanical effects both inside and outside
the tumor.
To help understand these effects, we consider two models for a growing brain
tumor in the shape of a sphere. The models, which consist of two spherical regions
(Fig. 6.13, upper left), are used to examine the following questions:
1. How does a growing tumor alter stress in surrounding brain tissue? Elevated
stresses may have detrimental effects on brain function. For this problem, the
inner region represents the growing tumor and the outer region nongrowing brain
tissue.
2. How does a necrotic core affect stress in a tumor? Necrosis (cell death) develops
inside tumors that become too large for diffusion or blood flow to deliver
sufficient nutrients to the interior (Jain et al. 2014). Here, the inner region
represents the nongrowing core while the outer region represents a growing shell
of proliferating cancer cells.
Both models remain unconstrained and free of external loads; thus, all stresses are
residual stresses generated by differential growth.
To gain qualitative insight into these problems, consider the sequence shown
in Fig. 6.13. Beginning with the initial stress-free configuration (upper left), 8 we
8 The tumor is assumed to be small enough at t = 0 for any initial stresses to be either very small
or relieved by internal remodeling or relaxation of the surrounding brain tissue.
6 Growth
the inner perimeter causes compression in the inner part. The opposite effects occur
when G r < 1. These qualitative observations agree with the quantitative results in
Fig. 6.11b.
The stress distributions for an artery plotted in Fig. 6.8d show that a positive
opening angle corresponds to circumferential residual stress that is tensile outside
and compressive inside. Therefore, in a tube undergoing uniform growth, negative
circumferential growth and positive radial growth tend to make opening angles
positive, and vice versa (Fig. 6.11a, b).
In general, arteries undergo various combinations of radial, circumferential, and
longitudinal growth. Moreover, the growth need not be uniform. Suppose an artery
grows only in the circumferential direction, but with a negative gradient across the
wall. If the average value of G θ remains unchanged, the growth gradient can change
the sign of the residual stress gradient (Fig. 6.11c) and the corresponding opening
angle.
6.8 Case Study: Growth of a Spherical Brain Tumor
It is now widely accepted that microenvironmental factors, including mechanical
forces, play an important role in the development and progression of cancer
(Northey et al. 2017). Studies have shown, for example, that stress and tissue
stiffness affect tumor growth and metastasis (Jain et al. 2014; Northey et al. 2017).
Growth, in turn, can have important biomechanical effects both inside and outside
the tumor.
To help understand these effects, we consider two models for a growing brain
tumor in the shape of a sphere. The models, which consist of two spherical regions
(Fig. 6.13, upper left), are used to examine the following questions:
1. How does a growing tumor alter stress in surrounding brain tissue? Elevated
stresses may have detrimental effects on brain function. For this problem, the
inner region represents the growing tumor and the outer region nongrowing brain
tissue.
2. How does a necrotic core affect stress in a tumor? Necrosis (cell death) develops
inside tumors that become too large for diffusion or blood flow to deliver
sufficient nutrients to the interior (Jain et al. 2014). Here, the inner region
represents the nongrowing core while the outer region represents a growing shell
of proliferating cancer cells.
Both models remain unconstrained and free of external loads; thus, all stresses are
residual stresses generated by differential growth.
To gain qualitative insight into these problems, consider the sequence shown
in Fig. 6.13. Beginning with the initial stress-free configuration (upper left), 8 we
8 The tumor is assumed to be small enough at t = 0 for any initial stresses to be either very small
or relieved by internal remodeling or relaxation of the surrounding brain tissue.