6.9 Mechanical Feedback
297
geared toward fighting the disease (Jain et al. 2014). In addition, the relatively large
tension just outside the tumor could damage axons in the brain.
For the incompressible solution, because the tumor does not deform under
isotropic compression, the size of the tumor is determined entirely by growth with
a = G 1 a 0 = 1.5a 0 , independently of its stiffness. When compressibility is included,
the size of the tumor and the associated stresses become smaller, but only slightly
for ν = 0.2 (Fig. 6.14a, dashed curves). If the tumor stiffens by more than the
specified factor of two, these effects would be even smaller (not shown). Therefore,
incompressibility may be a reasonable approximation in this problem.
In the second problem, where the core is softer and does not grow, results are
shown for b 0 /a 0 = 1.7 and c 2 /c 1 = 2, with G 1 = 1 and G 2 = 1.5. The
growing outer shell pulls the necrotic inner region outward, putting the latter under
uniform isotropic tension. For incompressible material, the size of the core does
not change (r = a 0 ). In contrast, for compressible tissue with ν = 0.2, the
radius increases by about 40% and the stresses drop dramatically (Fig. 6.14b). As
in the first problem, the proliferating cancer cells (outer region in this model) are
subjected to circumferential compression, although the radial stress is tensile. If the
proliferation rate depends on stress, these results suggest that treatments that alter
the compressibility (or compliance) of tumor tissue may affect the rate of growth.
As a check, prescribing the same growth for both regions (G 1 = G 2 ) yields no
residual stress (not shown). This result is consistent with the expected behavior, as
well as Eq. (6.82), which gives λ ∗
i = 1 in both regions.
6.9 Mechanical Feedback
In the problems considered thus far, growth is a specified function of position
and time, with feedback playing no role. In general, however, tissues also grow
in response to changing functional demands or other environmental conditions.
For example, increased oxygen demand induces the left ventricular cavity to grow
larger, allowing the heart to pump more blood with each beat. High blood pressure,
on the other hand, causes the wall of the heart to grow thicker, presumably to
improve efficiency by returning wall stress toward normal levels. These responses
are examples of functional adaptation (see Chap. 1). Changes in oxygen demand
and blood pressure provide biochemical and mechanical stimuli, respectively, that
trigger and regulate growth in the heart.
This book considers only mechanical feedback, which plays a major role in the
growth of load-bearing structures of the cardiovascular and musculoskeletal systems. While we have already seen that growth influences tissue stress, experiments
have shown that stress affects growth. In other words, biology (growth) affects
mechanics and vice versa. This feedback loop is a hallmark of mechanobiology.
The fundamental principles that underlie mechanical regulation of growth are
poorly understood, and many of the ideas presented in this section remain the subject
297
geared toward fighting the disease (Jain et al. 2014). In addition, the relatively large
tension just outside the tumor could damage axons in the brain.
For the incompressible solution, because the tumor does not deform under
isotropic compression, the size of the tumor is determined entirely by growth with
a = G 1 a 0 = 1.5a 0 , independently of its stiffness. When compressibility is included,
the size of the tumor and the associated stresses become smaller, but only slightly
for ν = 0.2 (Fig. 6.14a, dashed curves). If the tumor stiffens by more than the
specified factor of two, these effects would be even smaller (not shown). Therefore,
incompressibility may be a reasonable approximation in this problem.
In the second problem, where the core is softer and does not grow, results are
shown for b 0 /a 0 = 1.7 and c 2 /c 1 = 2, with G 1 = 1 and G 2 = 1.5. The
growing outer shell pulls the necrotic inner region outward, putting the latter under
uniform isotropic tension. For incompressible material, the size of the core does
not change (r = a 0 ). In contrast, for compressible tissue with ν = 0.2, the
radius increases by about 40% and the stresses drop dramatically (Fig. 6.14b). As
in the first problem, the proliferating cancer cells (outer region in this model) are
subjected to circumferential compression, although the radial stress is tensile. If the
proliferation rate depends on stress, these results suggest that treatments that alter
the compressibility (or compliance) of tumor tissue may affect the rate of growth.
As a check, prescribing the same growth for both regions (G 1 = G 2 ) yields no
residual stress (not shown). This result is consistent with the expected behavior, as
well as Eq. (6.82), which gives λ ∗
i = 1 in both regions.
6.9 Mechanical Feedback
In the problems considered thus far, growth is a specified function of position
and time, with feedback playing no role. In general, however, tissues also grow
in response to changing functional demands or other environmental conditions.
For example, increased oxygen demand induces the left ventricular cavity to grow
larger, allowing the heart to pump more blood with each beat. High blood pressure,
on the other hand, causes the wall of the heart to grow thicker, presumably to
improve efficiency by returning wall stress toward normal levels. These responses
are examples of functional adaptation (see Chap. 1). Changes in oxygen demand
and blood pressure provide biochemical and mechanical stimuli, respectively, that
trigger and regulate growth in the heart.
This book considers only mechanical feedback, which plays a major role in the
growth of load-bearing structures of the cardiovascular and musculoskeletal systems. While we have already seen that growth influences tissue stress, experiments
have shown that stress affects growth. In other words, biology (growth) affects
mechanics and vice versa. This feedback loop is a hallmark of mechanobiology.
The fundamental principles that underlie mechanical regulation of growth are
poorly understood, and many of the ideas presented in this section remain the subject