7.6 Case Study: Growth and Remodeling of Arteries
373
representing deposition time and current time, respectively. In addition, subscript i
represents cylindrical coordinates (r, θ, z), while superscript n denotes c (collagen)
and e (elastin).
Kinematic Relations: The total stretch ratios are
λ r =
∂r
∂R
,
λ θ =
r
R
,
λ z = λ.
(7.73)
With smooth muscle assumed to exist without stress at t = 0, λ m∗
i is obtained by
substituting λ m
i0 = 1 and λ i (τ ) = λ i (0) = 1 into Eq. (7.20). Thus, the elastic stretch
ratios are
λ
m∗
i (t) = λ(t)
G m
i (0)
G m
i (t)
λ
n∗
θ (t, τ ) = λ
n
0
λ θ (t)
λ θ (τ )
G n
θ (τ )
G n
θ (t)
,
(7.74)
where λ n
0 is the deposition stretch of the circumferential collagen or elastin fibers.
With the assumption that the fiber volume does not change, the volume ratios are
given by Eqs. (7.54)–(7.56) in the form
J = λ r λ θ λ z = J
m
+ J
c
+ J
e
J
m (t) = G
m
r (t)G
m
θ (t)G
m
z (t)
J
n (t) = J
n (0) = φ
n (0).
(7.75)
The initial conditions for the growth laws (7.72) are
G
m
r (0) = G
m
θ (0) = [φ
m (0)]
1/3 ,
(7.76)
and we set G n
θ (t) = G n
θ (0) = [φ n (0)] 1/3 for elastin and collagen, which do not
grow in volume.
Equilibrium:
∂σ r
∂r
+
σ r − σ θ
r
= 0
(7.77)
Constitutive Relations: For circumferential fibers, the total stresses are given by
σ θ = σ
m
θ + σ
c
θ + σ
e
θ
σ r = σ
m
r ,
σ z = σ
m
z .
(7.78)
The Lagrange multiplier p needed to enforce incompressibility is found using
equilibrium and the boundary conditions on the curved surfaces. These conditions