396
7 Remodeling
(a) Write the equations for computing the total volume ratio J (t), as well as
the volume fractions φ p (t) and φ a (t).
(b) Determine an equation to be solved for the total stretch ratio λ x (t).
(c) Compute and plot λ x , the total stress σ x , the volume fractions, and the ratio
of the current to initial cross-sectional area as functions of time (0 ≤ t ≤
100) for the following baseline parameter values:
c p = c a = 1,
λ
a
0 = 1.2,
˙
J
a +
0 = 0.5,
β = 0.
(d) Using appropriate plots, show the effects of changing the values of c p , ˙
J a +
0 ,
and β. Explain the trends in your results and discuss how they may apply
to tissue engineering of heart muscle.
7.7 The problem in Sect. 7.8 deals with changes in cell and stress-fiber orientation
in response to cyclic stretch. Write a computer program to implement the
analysis presented in that section, and check your results against those from the
baseline model in Figs. 7.18 and 7.19. (Also, be sure the code yields no changes
in fiber orientation for equibiaxial stretch.) Next, use the code to simulate
classic uniaxial extension and explore the following perturbations relative to
the baseline model:
(a) Effects of stretch amplitude: Compute the solution for a = 0.05 and 0.2.
(b) Effects of fiber off-loading: Studies have shown that stress fibers stabilize
under increased stress and may disassemble under reduced stress. Simulate
this behavior by setting K = 0 in Eq. (7.101) (minimum k n − ) for fibers with
λ n∗ > λ 0 , while keeping K at its baseline value for fibers with λ n∗ < λ 0 .
(c) Effects of monotonically increasing stretch: Some studies have shown that
fibers exposed to slow increases in stretch orient parallel to, rather than
away from, the direction of stretch. Examine the behavior of the model for
λ x (t) = 1 + ae −t .
Plot and explain selected results for each case.
7.8 Consider growth and remodeling of an artery modeled as a thin-walled
cylindrical membrane composed of a single layer of smooth muscle and
circumferentially aligned collagen fibers. At t = 0, the membrane has a radius
a 0 and wall thickness h 0 . Assume the following:
• For a pressurized cylindrical membrane, the radial stress is approximately
zero, and equilibrium is governed by Laplace’s law (4.108). The fluid shear
stress τ f is given by Eq. (7.81), and the ends of the cylinder are fixed at the
longitudinal stretch ratio λ 0 .
• The total stress is
σ i = σ
m
i + σ
a
i + σ
c
i
(i = r, θ, z),
7 Remodeling
(a) Write the equations for computing the total volume ratio J (t), as well as
the volume fractions φ p (t) and φ a (t).
(b) Determine an equation to be solved for the total stretch ratio λ x (t).
(c) Compute and plot λ x , the total stress σ x , the volume fractions, and the ratio
of the current to initial cross-sectional area as functions of time (0 ≤ t ≤
100) for the following baseline parameter values:
c p = c a = 1,
λ
a
0 = 1.2,
˙
J
a +
0 = 0.5,
β = 0.
(d) Using appropriate plots, show the effects of changing the values of c p , ˙
J a +
0 ,
and β. Explain the trends in your results and discuss how they may apply
to tissue engineering of heart muscle.
7.7 The problem in Sect. 7.8 deals with changes in cell and stress-fiber orientation
in response to cyclic stretch. Write a computer program to implement the
analysis presented in that section, and check your results against those from the
baseline model in Figs. 7.18 and 7.19. (Also, be sure the code yields no changes
in fiber orientation for equibiaxial stretch.) Next, use the code to simulate
classic uniaxial extension and explore the following perturbations relative to
the baseline model:
(a) Effects of stretch amplitude: Compute the solution for a = 0.05 and 0.2.
(b) Effects of fiber off-loading: Studies have shown that stress fibers stabilize
under increased stress and may disassemble under reduced stress. Simulate
this behavior by setting K = 0 in Eq. (7.101) (minimum k n − ) for fibers with
λ n∗ > λ 0 , while keeping K at its baseline value for fibers with λ n∗ < λ 0 .
(c) Effects of monotonically increasing stretch: Some studies have shown that
fibers exposed to slow increases in stretch orient parallel to, rather than
away from, the direction of stretch. Examine the behavior of the model for
λ x (t) = 1 + ae −t .
Plot and explain selected results for each case.
7.8 Consider growth and remodeling of an artery modeled as a thin-walled
cylindrical membrane composed of a single layer of smooth muscle and
circumferentially aligned collagen fibers. At t = 0, the membrane has a radius
a 0 and wall thickness h 0 . Assume the following:
• For a pressurized cylindrical membrane, the radial stress is approximately
zero, and equilibrium is governed by Laplace’s law (4.108). The fluid shear
stress τ f is given by Eq. (7.81), and the ends of the cylinder are fixed at the
longitudinal stretch ratio λ 0 .
• The total stress is
σ i = σ
m
i + σ
a
i + σ
c
i
(i = r, θ, z),