Biomedical Engineering Reference
In-Depth Information
EXAMPLE 5.5
The aorta of a male patient had an inner radius of 13 mm and was 2.2 mm thick in the dia-
stolic state. It was 50 cm long and expanded due to the pumping of the heart. When the
heart valve opened in the systolic phase, 70 mL of blood was discharged. Half of this blood
was initially stored in the aorta, expanding its wall to some inner radius. Assume the dia-
stolic pressure and systolic pressure to be 80 mmHg and 130 mmHg, respectively, and the
heart rate to be 72 beats per minute. Calculate the systolic radius and the wall thickness.
What is the stress in the wall in the systolic state? How much energy is stored in the elastic
wall? What is the average Reynolds number in the aorta? Is the flow laminar or turbulent?
Solution:
(a) Volume of the lumen in diastole
=
π
* (13 * 10
−
3
)
2
*
=
0.5
=
0.265*10
−
3
m
3
Increase in volume due to storage of blood
=
70/2
=
35 mL
=
0.035 m
3
Volume of the lumen in systole
=
(0.265
+
0.035) * 10
−
3
=
0.3 * 10
−
3
m
3
=
π
* (
r
s
)
2
* 0.5
Systolic radius
=
13.8 mm
(b) Cross-sectional area in diastole
=
π
* [((13
+
2.2) * 10
−
3
)
2
−
(13 * 10
−
3
)
2
]
=
194.9 *
−
3
m
2
Cross-sectional area in systole
=
194.9 * 10
−
3
m
2
=
π
* [((13.8
+
t
s
) * 10
−
3
)
2
−
(13.8 *
10
−
3
)
2
]
t
s
=
2 mm
Hoop stress,
σ
s
=
P
s
R
s
/
t
s
=
130 mmHg * (133 Pa/mmHg) * 0.0138m/0.002
=
0.113 MPa
(c) Energy stored in the elastic wall =
P
mean
Δ
V
P
mean
=
(
P
systolic
+
2
P
diastolic
)/3
=
(130
+
2 * 80)/3
=
96.67 mmHg
=
96.96 * 133 Pa
=
12.9
Δ
V
=
35 mL
=
35 * 10
−
6
m
3
P
mean
Δ
V
=
12.9 * 10
3
* 35 * 10
−
6
=
0.45J
(d) Volumetric flow rate
=
70*10
−
6
(m
3
/beat)/(60 sec/72 beats)
=
84.1*10
−
6
(m
3
/s)
Velocity
=
volumetric flow rate/average cross-sectional area
=
84.1 * 10
−
6
/(
π
* ((13
+
13.8)/2 * 10
−
3
)
2
]
=
0.149 m/s
Density of blood (assume),
ρ
= 1.1 gm/cc = 1,100 kg/m
3
Viscosity (assume),
μ =
3 cP
=
3 * 10
−
3
kg/m.s
From (5.2),
N
Re
=
ρΔ
v
/
μ
=
1,100 * (2 * 0.0134) * 0.149/3 * 10
−
3
=
1,465
Hence, the flow is laminar.
Compliance
is an expression used in biological materials to indicate their elas-
tic nature. This concept is applied to blood vessels, the heart, lungs, and practi-
cally any other situation involving a closed structure. Compliance is defined as the
change in volume for a given change in pressure. The reciprocal of compliance is
elastance. For example, the lungs expand and contract during each breath. The
pressure within and around the lungs changes equally at the same time. If a pres-
sure-volume curve of the tissue is drawn, then the slope of the line is the compli-
ance of the tissue. In the majority of the tissues, the pressure-volume relation is not
linear due to the heterogeneous nature of tissues. For example, in a normal healthy
lung at a low volume, relatively little negative pressure outside (or positive pressure
inside) is applied to the lung expansion. However, lung compliance decreases with
increasing volume. Therefore, as the lung increases in size, more pressure must be
applied to get the same increase in volume. Hence, local compliance is calculated.
Compliance can also change in various disease states; compliance decreases in scar-
ring and increases in fibrous tissues. For example, fibrotic lungs are stiffer and
require a large pressure to maintain a moderate volume. Such lungs would have a
