328
6 Growth
c
med
p
= 450 kPa
c
adv
p = 225 kPa
α 1 = 0.04
α 2 = 0.8
α 3 = 0.4
α 4 = 0.004
α 5 = 0.08
α 6 = 0.04.
(6.152)
With φ p = φ a = 0.5 in the media and φ p = 1, φ a = 0 in the adventitia, the
layers have the same equivalent passive modulus φ p c p = 225 kPa. The parameters
in W ∗ med
a
are
c a , max = 225 kPa,
K min = 0.6.
The parameters in the growth and contraction laws have the values
a r = 0.5
a θ = 0.25
a τ = 0.04
(σ θ0 ) m = 160 kPa
(τ 0 ) m = 1.5 Pa
b τ = 0.5
b K = 0.05
K 0 = 0.9.
Since contraction generally occurs much faster than shear-induced growth, b τ is
taken considerably larger than a τ . Finally, the blood viscosity is μ = 0.03 Pa s
(poise). Unless stated otherwise, all results are based on these values. Moreover,
because the aorta undergoes relatively little contraction, the effects of contraction
are ignored (c a = 0) until the end of this section.
Normal Development As pressure and flow increase, the radius a and wall
thickness h increase and approach a homeostatic state by t = 20 (Fig. 6.27a).
The ratio a/ h ≈ 10 is reasonable for the pressurized aorta. The stresses τ and σ θ
rise rapidly at first and approach their target values by t = 20 ( ˆ
σ θ = ˆ
τ 0 = 1;
Fig. 6.27b). Stress distributions plotted at selected times show how σ θ evolves
toward its homeostatic value of 160 kPa uniformly across the wall (Fig. 6.27c). 15
According to the model, τ first jumps to more than 100 times its value of 1.5 Pa
in the mature rat (not shown). This high shear stress, which is short-lived and likely
unrealistic, is caused by the circumferential growth rate being too slow to keep up
with the assumed increase in flow rate near t = 0. The model yields more than a
15-fold increase in the loaded lumen radius during development (Fig. 6.27a).
A key assumption in this model is that the target stresses increase during
development. If they do not evolve and always have the same values as in the
mature artery, then the stresses during early development would be far below their
homeostatic values, and the vessel would initially atrophy, which is not realistic
behavior. By remaining above the increasing target stresses, the stresses caused
by the increasing hemodynamic loads drive growth and expansion of the vascular
15 To facilitate comparisons between vessels with large differences in size during development,
stresses are plotted as functions of the undeformed coordinate R.
6 Growth
c
med
p
= 450 kPa
c
adv
p = 225 kPa
α 1 = 0.04
α 2 = 0.8
α 3 = 0.4
α 4 = 0.004
α 5 = 0.08
α 6 = 0.04.
(6.152)
With φ p = φ a = 0.5 in the media and φ p = 1, φ a = 0 in the adventitia, the
layers have the same equivalent passive modulus φ p c p = 225 kPa. The parameters
in W ∗ med
a
are
c a , max = 225 kPa,
K min = 0.6.
The parameters in the growth and contraction laws have the values
a r = 0.5
a θ = 0.25
a τ = 0.04
(σ θ0 ) m = 160 kPa
(τ 0 ) m = 1.5 Pa
b τ = 0.5
b K = 0.05
K 0 = 0.9.
Since contraction generally occurs much faster than shear-induced growth, b τ is
taken considerably larger than a τ . Finally, the blood viscosity is μ = 0.03 Pa s
(poise). Unless stated otherwise, all results are based on these values. Moreover,
because the aorta undergoes relatively little contraction, the effects of contraction
are ignored (c a = 0) until the end of this section.
Normal Development As pressure and flow increase, the radius a and wall
thickness h increase and approach a homeostatic state by t = 20 (Fig. 6.27a).
The ratio a/ h ≈ 10 is reasonable for the pressurized aorta. The stresses τ and σ θ
rise rapidly at first and approach their target values by t = 20 ( ˆ
σ θ = ˆ
τ 0 = 1;
Fig. 6.27b). Stress distributions plotted at selected times show how σ θ evolves
toward its homeostatic value of 160 kPa uniformly across the wall (Fig. 6.27c). 15
According to the model, τ first jumps to more than 100 times its value of 1.5 Pa
in the mature rat (not shown). This high shear stress, which is short-lived and likely
unrealistic, is caused by the circumferential growth rate being too slow to keep up
with the assumed increase in flow rate near t = 0. The model yields more than a
15-fold increase in the loaded lumen radius during development (Fig. 6.27a).
A key assumption in this model is that the target stresses increase during
development. If they do not evolve and always have the same values as in the
mature artery, then the stresses during early development would be far below their
homeostatic values, and the vessel would initially atrophy, which is not realistic
behavior. By remaining above the increasing target stresses, the stresses caused
by the increasing hemodynamic loads drive growth and expansion of the vascular
15 To facilitate comparisons between vessels with large differences in size during development,
stresses are plotted as functions of the undeformed coordinate R.