James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 8.4 Applications to Economics and Biology 571
The resulting equation
2
F − PR4
8l
vein
pulmonary
arteries
right
atrium
pulmonary
veins
vein
FIGURE 6
aorta
pulmonary
arteries
pulmonary
veins
left
atrium
is called Poiseuille’s Law; it shows that the flux is proportional to the fourth power of
the radius of the blood vessel.
Cardiac Output
Figure 6 shows the human cardiovascular system. Blood returns from the body through
the veins, enters the right atrium of the heart, and is pumped to the lungs through the
pulmonary arteries for oxygenation. It then flows back into the left atrium through the
pulmo nary veins and then out to the rest of the body through the aorta. The cardiac
output of the heart is the volume of blood pumped by the heart per unit time, that is, the
rate of flow into the aorta.
The dye dilution method is used to measure the cardiac output. Dye is injected into the
right atrium and flows through the heart into the aorta. A probe inserted into the aorta measures
the concentration of the dye leaving the heart at equally spaced times over a time
interval f0, T g until the dye has cleared. Let cstd be the concentration of the dye at time t.
If we divide f0, T g into subintervals of equal length Dt, then the amount of dye that flows
past the measuring point during the subinterval from t − t i21 to t − t i is approximately
sconcentrationdsvolumed − cst i dsF Dtd
where F is the rate of flow that we are trying to determine. Thus the total amount of dye
is approximately
o n
cst i dF Dt − F o n
cst i d Dt
i−1
i−1
and, letting n l `, we find that the amount of dye is
Thus the cardiac output is given by
A − F y T
cstd dt
0
3
F −
A
y T
cstd dt
0
where the amount of dye A is known and the integral can be approximated from the
concentration readings.
t cstd t cstd
0 0 6 6.1
1 0.4 7 4.0
2 2.8 8 2.3
3 6.5 9 1.1
4 9.8 10 0
5 8.9
Example 2 A 5-mg bolus of dye is injected into a right atrium. The concentration
of the dye (in milligrams per liter) is measured in the aorta at one-second intervals as
shown in the table. Estimate the cardiac output.
SOLUTION Here A − 5, Dt − 1, and T − 10. We use Simpson’s Rule to approximate
the integral of the concentration:
y 10
0
cstd dt < 1 3 f0 1 4s0.4d 1 2s2.8d 1 4s6.5d 1 2s9.8d 1 4s8.9d
< 41.87
1 2s6.1d 1 4s4.0d 1 2s2.3d 1 4s1.1d 1 0g
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