Question 28.10: Estimating blood volume When a small amount of radioactive m......
Estimating blood volume
When a small amount of radioactive material is placed in 1.0 cm³ of water, it has an activity of 75,000 decays/min. Imagine that an identical amount of radioactive material is injected into an individual’s bloodstream. Later, after the material has spread throughout the blood, a 1.0-cm³ sample of blood taken from the individual has an activity of 16 decays/min. Determine the individual’s total blood volume.
Sketch and translate The total blood volume can be thought of as a large container of unknown volume and the 1.0-cm³ sample as a small portion of that. We know that a direct relationship exists between the decays/min in the sample and the decays/min in the total volume of blood. Therefore, we can use the value of one to determine the value of the other.
Simplify and diagram Assume that during this process the person does not drink any water (so that their blood volume doesn’t change), that all the radioactive material is absorbed by the blood, and that none is filtered out by the kidneys. Assume also that the material disperses evenly throughout the blood. Lastly, assume that the half-life of that type of material is much longer than the time interval needed to complete the experiment so that the activity of the radioactive material can be considered constant.
Represent mathematically We use a ratio method to determine the blood volume by comparing the activity A and volume V of the two amounts of blood (the whole body and the sample):
\frac{V_{\text {blood }}}{V_{\text {sample }}}=\frac{A_{\text {blood }}}{A_{\text {sample }}}Solve and evaluate Solve for Vblood and substitute the known quantities into the above to determine the blood volume:
V_{\text {blood }}=\frac{A_{\text {blood }}}{A_{\text {sample }}} V_{\text {sample }}=\left(\frac{75,000 \text { decays } / \mathrm{min}}{16 \text { decays } / \mathrm{min}}\right)\left(1.0 \mathrm{~cm}^3\right)
= 4.7× 10³ cm³
where (4.7 × 10³ cm³)(1 L/10³ cm³) = 4.7 L. This value is within the 4- to 6-L range that is considered a normal human blood volume.
Try it yourself: Suppose the radioactive sample had a half life of 1.0 h and you waited 2 h for the injected sample to distribute uniformly before measuring the activity of 1 cm³ of blood. What would the activity of that blood sample be?
Answer: In two half-lives, the activity would be reduced
to (1/2²)(16 decays/min) = 4 decays/min.