382
7 Remodeling
In these relations, b θ , b a , and b r are positive constants and
ˆ
σ θ =
σ θ
σ h
,
ˆ
a =
a
a h
,
where σ h and a h are the (specified) wall stress and inner radius, respectively, in
the homeostatic artery. In (7.72), circumferential growth depends on the fluid shear
stress τ f , which is inversely proportional to a 3 . With the plus sign changed to minus,
the modified term in (7.87) 1 controls radius in a similar manner, i.e., if a > a h , the
circumference grows smaller and vice versa.
After a homeostatic state is achieved (at P = 16 kPa), the second phase of
aneurysm development is initiated by degrading the muscle and elastin. To simulate
this process, the moduli c m and c e are decreased by 80% exponentially with time. In
addition, growth of the degrading muscle is turned off during this phase, but collagen
remodeling continues. Finally, because researchers have used models to show that
increased collagen formation can slow or arrest expansion of an aneurysm (Watton
et al. 2004; Baek et al. 2005, 2006), we assume that the collagen deposition rate is
governed by the remodeling law
˙
J
c +
θ (t) = ˙
J
c +
φ (t) = ˙
J
c +
0 [1 + K c ( ˆ
σ θ − 1)],
(7.88)
which is a slightly modified form of Eq. (7.68), with K c being a constant. If we set
˙
J
c +
0 = k
c + ((
c
θ ) 0 = k
c + ((
c
φ ) 0
with k c + = k c − , then no net collagen growth occurs as long as the total wall stress
remains at its homeostatic value. Collagen volume increases, however, if the total
wall stress exceeds σ h ( ˆ
σ θ > 1).
To summarize, the initial homeostatic state is created by a combination of
smooth-muscle growth and collagen remodeling. This state is then perturbed by
degrading the muscle and elastin, inducing further development of the aneurysm
through collagen growth and remodeling.
Illustrative Results As in the previous subsection, we focus here on the behavior
in the mature artery, with t = 0 moved to the beginning of the second phase of the
simulation. During this phase, the baseline parameters are the following:
c m = 20 kPa
c c = 10 kPa
c e = 10 kPa
α m = 2
α c = 0.2
γ = 1
k
c + = k
c − = 5 yr
−1
k
e + = k
e − = 0
K c = 0
σ h = 70 kPa
a h = 12 mm
λ
c
0 = 1.1
λ
e
0 = 1.05.
7 Remodeling
In these relations, b θ , b a , and b r are positive constants and
ˆ
σ θ =
σ θ
σ h
,
ˆ
a =
a
a h
,
where σ h and a h are the (specified) wall stress and inner radius, respectively, in
the homeostatic artery. In (7.72), circumferential growth depends on the fluid shear
stress τ f , which is inversely proportional to a 3 . With the plus sign changed to minus,
the modified term in (7.87) 1 controls radius in a similar manner, i.e., if a > a h , the
circumference grows smaller and vice versa.
After a homeostatic state is achieved (at P = 16 kPa), the second phase of
aneurysm development is initiated by degrading the muscle and elastin. To simulate
this process, the moduli c m and c e are decreased by 80% exponentially with time. In
addition, growth of the degrading muscle is turned off during this phase, but collagen
remodeling continues. Finally, because researchers have used models to show that
increased collagen formation can slow or arrest expansion of an aneurysm (Watton
et al. 2004; Baek et al. 2005, 2006), we assume that the collagen deposition rate is
governed by the remodeling law
˙
J
c +
θ (t) = ˙
J
c +
φ (t) = ˙
J
c +
0 [1 + K c ( ˆ
σ θ − 1)],
(7.88)
which is a slightly modified form of Eq. (7.68), with K c being a constant. If we set
˙
J
c +
0 = k
c + ((
c
θ ) 0 = k
c + ((
c
φ ) 0
with k c + = k c − , then no net collagen growth occurs as long as the total wall stress
remains at its homeostatic value. Collagen volume increases, however, if the total
wall stress exceeds σ h ( ˆ
σ θ > 1).
To summarize, the initial homeostatic state is created by a combination of
smooth-muscle growth and collagen remodeling. This state is then perturbed by
degrading the muscle and elastin, inducing further development of the aneurysm
through collagen growth and remodeling.
Illustrative Results As in the previous subsection, we focus here on the behavior
in the mature artery, with t = 0 moved to the beginning of the second phase of the
simulation. During this phase, the baseline parameters are the following:
c m = 20 kPa
c c = 10 kPa
c e = 10 kPa
α m = 2
α c = 0.2
γ = 1
k
c + = k
c − = 5 yr
−1
k
e + = k
e − = 0
K c = 0
σ h = 70 kPa
a h = 12 mm
λ
c
0 = 1.1
λ
e
0 = 1.05.