6.11 Case Study: Functional Adaptation in Arteries
327
The first term vanishes because of the boundary conditions σ R = 0 at R = ˆ
a c and
R = ˆ
b c . After substituting Eq. (6.146) for ∂σ R /∂R, combining the above equations
with (6.144) gives
M = 0 →
ˆ
b c
ˆ
a c
( ¯
σ − ¯
σ R ) R dR = 0.
(6.151)
Solution Procedure To obtain results for the unloaded passive state B 0 , the
simulation for the loaded artery is paused at one or more time points, and the
solution for P = 0, K θ = 1 is computed. The values of λ ∗
rp , λ ∗
θp , and λ ∗
zp
provide λ ∗
ρ , λ ∗
ϑ , and λ ∗
ζ , respectively. The opening angle is computed by solving
Eqs. (6.148), (6.149), and (6.151) simultaneously for ˆ
a c , , and φ 0 . Each iteration
requires the following steps:
1. Use (6.143) to compute R corresponding to each point ρ as a function of ˆ
a c , ,
and φ 0 .
2. Compute λ and λ Z using (6.142), and then λ R = (λ λ Z ) −1 .
3. Compute λ ∗
R , λ ∗
, λ ∗
Z , and the E ∗
I using (6.141).
4. Compute ¯
σ R , ¯
σ , and ¯
σ Z using (6.145).
With the values of ˆ
a c , , and φ 0 , as well as ˆ
b c = R( ˆ
b), stress distributions can be
computed using Eqs. (6.144) and (6.147). As defined by Fung and Liu (1989), the
opening angle is = φ/2 = π − φ 0 (Fig. 6.26b).
6.11.5 Illustrative Results
Results are shown for the rat aorta. In the rat, significant blood flow begins about
10 days after fertilization, birth occurs about 11 days later, and the animal nears
maturity (growth slows) about 3 months after birth. For convenience, time begins
at the onset of flow (t = 0) and is normalized to the time of birth (t = 1). Thus,
maturity occurs at about t = 10.
Representative parameter values are based on published experimental data for
the rat, except for the growth parameters, which were chosen to give reasonable
qualitative behavior. (In terms of dimensionless time, the growth-rate coefficients
are unitless.) In the initial configuration, the cross-sectional geometry is defined by
(a 0 , b 0 , c 0 ) = (0.20, 0.25, 0.24) mm (Fig. 6.24). During normal development, the
specified pressure and flow rate increase according to Eqs. (6.121) with P max =
16 kPa (peak systolic pressure), Q max = 1400 mm 3 /s, β P = 0.6, and β Q = 0.3.
With these parameters, the pressure reaches its peak before the flow rate (Fig. 6.25).
To examine fundamental behavior, we establish a baseline model with both layers
having equivalent passive properties. For the rat aorta, the material constants in
W med
p
and W adv
p , as given by Eq. (6.123), are taken as
327
The first term vanishes because of the boundary conditions σ R = 0 at R = ˆ
a c and
R = ˆ
b c . After substituting Eq. (6.146) for ∂σ R /∂R, combining the above equations
with (6.144) gives
M = 0 →
ˆ
b c
ˆ
a c
( ¯
σ − ¯
σ R ) R dR = 0.
(6.151)
Solution Procedure To obtain results for the unloaded passive state B 0 , the
simulation for the loaded artery is paused at one or more time points, and the
solution for P = 0, K θ = 1 is computed. The values of λ ∗
rp , λ ∗
θp , and λ ∗
zp
provide λ ∗
ρ , λ ∗
ϑ , and λ ∗
ζ , respectively. The opening angle is computed by solving
Eqs. (6.148), (6.149), and (6.151) simultaneously for ˆ
a c , , and φ 0 . Each iteration
requires the following steps:
1. Use (6.143) to compute R corresponding to each point ρ as a function of ˆ
a c , ,
and φ 0 .
2. Compute λ and λ Z using (6.142), and then λ R = (λ λ Z ) −1 .
3. Compute λ ∗
R , λ ∗
, λ ∗
Z , and the E ∗
I using (6.141).
4. Compute ¯
σ R , ¯
σ , and ¯
σ Z using (6.145).
With the values of ˆ
a c , , and φ 0 , as well as ˆ
b c = R( ˆ
b), stress distributions can be
computed using Eqs. (6.144) and (6.147). As defined by Fung and Liu (1989), the
opening angle is = φ/2 = π − φ 0 (Fig. 6.26b).
6.11.5 Illustrative Results
Results are shown for the rat aorta. In the rat, significant blood flow begins about
10 days after fertilization, birth occurs about 11 days later, and the animal nears
maturity (growth slows) about 3 months after birth. For convenience, time begins
at the onset of flow (t = 0) and is normalized to the time of birth (t = 1). Thus,
maturity occurs at about t = 10.
Representative parameter values are based on published experimental data for
the rat, except for the growth parameters, which were chosen to give reasonable
qualitative behavior. (In terms of dimensionless time, the growth-rate coefficients
are unitless.) In the initial configuration, the cross-sectional geometry is defined by
(a 0 , b 0 , c 0 ) = (0.20, 0.25, 0.24) mm (Fig. 6.24). During normal development, the
specified pressure and flow rate increase according to Eqs. (6.121) with P max =
16 kPa (peak systolic pressure), Q max = 1400 mm 3 /s, β P = 0.6, and β Q = 0.3.
With these parameters, the pressure reaches its peak before the flow rate (Fig. 6.25).
To examine fundamental behavior, we establish a baseline model with both layers
having equivalent passive properties. For the rat aorta, the material constants in
W med
p
and W adv
p , as given by Eq. (6.123), are taken as