372
7 Remodeling
W
c∗
=
c c
α c
e
α c [(λ c∗
θ ) 2 −1] 2 − 1
W
e∗
= c e [(λ
e∗
θ )
2
− 1]
2 ,
(7.71)
where the elastic strain components for the muscle are given by
E
m∗
i =
1
2
λ
m∗
i
2 − 1
in cylindrical polar coordinates (i = r, θ, z).
Blood pressure P and flow rate Q are specified functions of time, consistent with
the curves in Fig. 6.25a (page 320). But because the focus here is on mature arteries,
the boundary conditions at the ends of the tube are different. In the previous model,
the ends were closed but free to move axially to allow for the longitudinal expansion
that occurs during development. Here, to simulate mechanics in a mature artery, the
ends are fixed at a specified axial stretch ratio λ.
To focus on the effects of fiber remodeling, we assume that the matrix does not
grow, i.e., J c (t) = J c (0) and J e (t) = J e (0) for all t ≥ 0. Thus, all changes in
volume are caused by smooth muscle growth, which, for simplicity, is assumed to
depend on the total wall stress σ θ and fluid shear stress τ f . In the mature artery, the
growth laws for smooth muscle are given by modifying Eqs. (6.126) to obtain
˙
G
m
θ = [a θ ( ˆ
σ θ − 1) + a τ ( ˆ
τ f − 1)]G
m
θ
˙
G
m
r = a r ( ˆ
σ θ − 1)G
m
r ,
(7.72)
where 3
ˆ
σ θ =
σ θ
σ 0
,
ˆ
τ f =
τ f
τ f 0
.
Axial growth is ignored (G m
z = 1), and the target stresses σ 0 and τ f 0 are assumed
to remain constant in the mature vessel.
The results presented below begin with the homeostatic state at P = 16 kPa
and Q = 1400 mm 3 /s for the cardiovascular system of the adult rat. To establish
a homeostatic state in the mature vessel, we follow the same procedure used in
the growth model of Chap. 6, including the pressure-dependent target stresses given
by (6.128).
Analysis The analysis follows the basic steps outlined in Sect. 6.11.3. The governing equations from that section are listed below, modified and supplemented
as necessary to include matrix remodeling. All variables generally depend on both
radial position and time, but explicit arguments are listed only for time, with τ and t
3 Subscript f is used here to distinguish the fluid shear stress τ f from deposition time τ .
7 Remodeling
W
c∗
=
c c
α c
e
α c [(λ c∗
θ ) 2 −1] 2 − 1
W
e∗
= c e [(λ
e∗
θ )
2
− 1]
2 ,
(7.71)
where the elastic strain components for the muscle are given by
E
m∗
i =
1
2
λ
m∗
i
2 − 1
in cylindrical polar coordinates (i = r, θ, z).
Blood pressure P and flow rate Q are specified functions of time, consistent with
the curves in Fig. 6.25a (page 320). But because the focus here is on mature arteries,
the boundary conditions at the ends of the tube are different. In the previous model,
the ends were closed but free to move axially to allow for the longitudinal expansion
that occurs during development. Here, to simulate mechanics in a mature artery, the
ends are fixed at a specified axial stretch ratio λ.
To focus on the effects of fiber remodeling, we assume that the matrix does not
grow, i.e., J c (t) = J c (0) and J e (t) = J e (0) for all t ≥ 0. Thus, all changes in
volume are caused by smooth muscle growth, which, for simplicity, is assumed to
depend on the total wall stress σ θ and fluid shear stress τ f . In the mature artery, the
growth laws for smooth muscle are given by modifying Eqs. (6.126) to obtain
˙
G
m
θ = [a θ ( ˆ
σ θ − 1) + a τ ( ˆ
τ f − 1)]G
m
θ
˙
G
m
r = a r ( ˆ
σ θ − 1)G
m
r ,
(7.72)
where 3
ˆ
σ θ =
σ θ
σ 0
,
ˆ
τ f =
τ f
τ f 0
.
Axial growth is ignored (G m
z = 1), and the target stresses σ 0 and τ f 0 are assumed
to remain constant in the mature vessel.
The results presented below begin with the homeostatic state at P = 16 kPa
and Q = 1400 mm 3 /s for the cardiovascular system of the adult rat. To establish
a homeostatic state in the mature vessel, we follow the same procedure used in
the growth model of Chap. 6, including the pressure-dependent target stresses given
by (6.128).
Analysis The analysis follows the basic steps outlined in Sect. 6.11.3. The governing equations from that section are listed below, modified and supplemented
as necessary to include matrix remodeling. All variables generally depend on both
radial position and time, but explicit arguments are listed only for time, with τ and t
3 Subscript f is used here to distinguish the fluid shear stress τ f from deposition time τ .