The Heart - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 112

FIGURE 4.10 Schematic of the radial and circumferential stress in the
left ventricle. This assumes that the cardiac tissue is homogeneous in size
(and a perfect cylindrical shell) and homogeneous in mechanical proper-
ties, and that the pressure loading conditions are uniform. These formula-
tions also neglect the blood flow through the ventricles.

σ θ

b σ r

a

p i

p o

Section 4.3 ), but this is not necessarily accurate due to residual stresses within the cardiac
muscle fibers. However, it is not as simple as only equating the hydrostatic and normal
pressures. In fact, the systolic pressure is determined by the circumferential stress in the
ventricular wall, which is caused by muscle contraction. As the sarcomere length
increases, the maximal circumferential stress also increases, because with a longer sarco-
mere length, the sarcomere can perform more work (at least to a certain extent, which is
limited by the size of the sarcomere). A higher circumferential stress must be balanced by
the internal fluid pressure, so an increase in sarcomere length also increases the left ven-
tricular pressure waveform (the most critical component is the peak systolic pressure, i.e.,
120 mmHg). Some have shown that the number of sarcomeres in the heart remain fairly
constant with time. Therefore, with an increase in the internal radius of the heart, the mus-
cle tension and the blood pressure increases. This is seen in pathological conditions when
the heart enlarges but there is little increase in the number of cardiac myocytes. As a refer-
ence note, the external pressure is equal to the pleura pressure, which is slightly lower
than atmospheric pressure (negative 4 to negative 2 mmHg).

Example
Calculate the circumferential stress distribution in the left ventricle during peak systole
(assuming that the ventricle is a spherical shell). The inner radius and outer radius of the sphere
are 3.5 cm and 4 cm, respectively. The external pressure is

2 3 mmHg.

Solution

p o b 3

ð

2r 3 1 a 3

Þ

p i a 3

ð

2r 3 1 b 3

Þ

σ θ 5

Þ 2 2r 3 ð a 3 2 b 3 2r 3 ð a 3 2 b 3 Þ

3

2r 3

3 3

2r 3

3

σ θ 5 ð 2

3 mmHg Þð 4cm Þ ð

1 ð

3 5cm Þ Þ

2 ð

120 mmHg Þð 3 5cm Þ ð

1 ð

4cm Þ Þ

: :

:

3 3 3 3

2r 3

ðð Þ

2 ð

Þ Þ

2r 3

ðð Þ

2 ð

Þ Þ

3 : 5cm 4cm 3 : 5cm 4cm

1 065 J

r 3

:

σ θ 5 252 64 mmHg

: 1