5.5 Case Study: Cardiac Mechanics
251
5.8 The bladder is modeled as a thick-walled spherical shell with undeformed inner
radius a 0 = 2.5 cm and outer radius b 0 = 5 cm. The wall is composed of
incompressible smooth muscle, with passive and active strain-energy density
functions given by
W p = c p (I 1 − 3)
3
W a = c a (I a − 2),
where
I 1 = λ
2
r + λ
2
θ + λ
2
φ
I a = λ
∗2
θ + λ
∗2
φ
for isotropic contraction parallel to the surface of the wall. Volume fractions are
absorbed into the moduli, so the total stress tensor is given by σ = ¯
σ p + σ a −
p I. Write a computer program to examine the following cases:
(a) The bladder remains passive while urine inflow increases the inner radius
from a 0 = 2.5 cm to a = 5 cm. Following the analysis in Sect. 4.6, compute
and plot the fluid pressure p i as a function of a/a 0 . Neglect external loads
and take c p = 100 Pa.
(b) The bladder expels urine by contracting equally in the θ - and φ-directions.
If the inner radius decreases linearly in time from 5 cm at t = 0 to 2.5 cm
at t = T /2, compute and plot the fluid pressure as a function of ¯
t = 2t/T .
In addition, plot the distribution of σ θ across the wall at ¯
t = 0 and ¯
t = 1.
Neglect the effects of contraction velocity, and assume that K(t) is given
by (5.4) with K min = 0.3, while (5.23) provides c a (t) with c a,max = 250
Pa.
(c) Is it possible for the fluid pressure to become negative during the contraction phase? If so, how could this happen and what would it mean?
5.9 Write a computer program to solve the artery problem in Example 5.3 (page
227).
(a) For the same parameter values, check your results against those in Fig. 5.12.
Also, check that Eq. (5.43) is satisfied.
(b) Reverse the properties of the layers, i.e., the inner layer is passive with a
strain-energy density function given by (5.37), while the outer layer is muscle with properties defined by (5.36). Taking a 0 = 50 µm, b 0 = 100 µm,
and c 0 = 80 µm (Fig. 5.11a), use this model to simulate contraction of the
tubular heart in the early embryo (see Fig. 1.3a). In this case, the inner layer
(“adventitia”) represents cardiac jelly (φ adv = 1; c adv = 3 Pa; β = 0.4),
and the outer layer (“media”) is myocardium (φ med
p
= 0.4; φ med
a
= 0.6;
c p = 20 Pa).
251
5.8 The bladder is modeled as a thick-walled spherical shell with undeformed inner
radius a 0 = 2.5 cm and outer radius b 0 = 5 cm. The wall is composed of
incompressible smooth muscle, with passive and active strain-energy density
functions given by
W p = c p (I 1 − 3)
3
W a = c a (I a − 2),
where
I 1 = λ
2
r + λ
2
θ + λ
2
φ
I a = λ
∗2
θ + λ
∗2
φ
for isotropic contraction parallel to the surface of the wall. Volume fractions are
absorbed into the moduli, so the total stress tensor is given by σ = ¯
σ p + σ a −
p I. Write a computer program to examine the following cases:
(a) The bladder remains passive while urine inflow increases the inner radius
from a 0 = 2.5 cm to a = 5 cm. Following the analysis in Sect. 4.6, compute
and plot the fluid pressure p i as a function of a/a 0 . Neglect external loads
and take c p = 100 Pa.
(b) The bladder expels urine by contracting equally in the θ - and φ-directions.
If the inner radius decreases linearly in time from 5 cm at t = 0 to 2.5 cm
at t = T /2, compute and plot the fluid pressure as a function of ¯
t = 2t/T .
In addition, plot the distribution of σ θ across the wall at ¯
t = 0 and ¯
t = 1.
Neglect the effects of contraction velocity, and assume that K(t) is given
by (5.4) with K min = 0.3, while (5.23) provides c a (t) with c a,max = 250
Pa.
(c) Is it possible for the fluid pressure to become negative during the contraction phase? If so, how could this happen and what would it mean?
5.9 Write a computer program to solve the artery problem in Example 5.3 (page
227).
(a) For the same parameter values, check your results against those in Fig. 5.12.
Also, check that Eq. (5.43) is satisfied.
(b) Reverse the properties of the layers, i.e., the inner layer is passive with a
strain-energy density function given by (5.37), while the outer layer is muscle with properties defined by (5.36). Taking a 0 = 50 µm, b 0 = 100 µm,
and c 0 = 80 µm (Fig. 5.11a), use this model to simulate contraction of the
tubular heart in the early embryo (see Fig. 1.3a). In this case, the inner layer
(“adventitia”) represents cardiac jelly (φ adv = 1; c adv = 3 Pa; β = 0.4),
and the outer layer (“media”) is myocardium (φ med
p
= 0.4; φ med
a
= 0.6;
c p = 20 Pa).