Question

A blood-volume determination was carried out on a patient by injection with \(20.0 \mathrm{~mL}\) of blood that had been radioactively labeled with Cr- 51 to an activity of \(4.10 \mu \mathrm{Ci} / \mathrm{mL}\). After a brief period to allow for mixing in the body, blood was drawn from the patient for analysis. Unfortunately, a mix-up in the laboratory prevented an immediate analysis, and it was not until \(17.0\) days later that a measurement on the blood was made. The radiation level was then determined to be \(0.00935 \mu \mathrm{Ci} / \mathrm{mL}\). If \({ }^{51} \mathrm{Cr}\) has \(t_{1 / 2}=27.7\) days, what is the volume of blood in the patient?

Short answer

The blood volume in the patient is approximately 5733 mL.

Step-by-step solution

Calculate the decay constant

The decay constant (\(\lambda\)) can be found using the half-life formula: \( \lambda = \frac{0.693}{t_{1/2}} \). For \( {}^{51}\text{Cr} \) with a half-life of \( 27.7 \) days, we have:\[ \lambda = \frac{0.693}{27.7 \text{ days}} \approx 0.025 \text{ day}^{-1} \]

Determine the initial total activity

Find the initial total activity \(A_0\) of the injected blood. Given that the activity is \(4.10 \mu\text{Ci}/\text{mL}\) and the volume injected is \(20.0 \text{ mL}\), we calculate:\[ A_0 = 4.10 \mu\text{Ci/mL} \times 20.0 \text{ mL} = 82.0 \mu\text{Ci} \]

Calculate the activity after 17 days

Use the decay formula \( A = A_0 e^{-\lambda t} \) to find the activity after \(17\) days. With \( A_0 = 82.0 \mu\text{Ci} \), \( \lambda = 0.025 \text{ day}^{-1} \), and \( t = 17 \text{ days} \):\[ A = 82.0 \mu\text{Ci} \times e^{-0.025 \times 17} \approx 82.0 \mu\text{Ci} \times e^{-0.425} \approx 53.6 \mu\text{Ci} \]

Compare with measured activity

We know that the measured activity \(A_m\) per \(\text{mL}\) was \(0.00935 \mu\text{Ci/mL}\). Thus, the total activity \(A_m\) in the full blood volume \(V\) in the patient is:\[ A_m = 0.00935 \mu\text{Ci/mL} \times V \]

Solve for the blood volume

Set the calculated activity equal to the measured activity equation to find the blood volume \(V\):\[ 53.6 \mu\text{Ci} = 0.00935 \mu\text{Ci/mL} \times V \]Solving for \(V\), we get:\[ V = \frac{53.6 \mu\text{Ci}}{0.00935 \mu\text{Ci/mL}} \approx 5733 \text{ mL} \]

Key concepts

Half-life Calculation

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. A key concept in understanding radioactive decay is the half-life, which is defined as the time required for half of the radioactive substance to decay. In mathematical terms, the half-life equation can be used to calculate the decay constant, which helps to predict the activity of a radioactive substance over time.

The decay constant, denoted as \( \lambda \), can be calculated using the formula: \( \lambda = \frac{0.693}{t_{1/2}} \), where \( t_{1/2} \) represents the half-life of the substance. This formula is derived from the natural logarithm of 2 (approximately 0.693), which reflects the rate at which half of the substance undergoes decay.

The decay constant allows us to then apply the exponential decay equation: \( A = A_0 e^{-\lambda t} \). Here \( A \) is the activity at time \( t \), \( A_0 \) is the initial activity, and \( e \) is the base of natural logarithms, approximately equal to 2.718. This equation gives us a powerful tool to calculate the remaining activity of a radioactive sample at any given time after it began decaying.

Chromium-51

Chromium-51 is a radioactive isotope of the element chromium that is often used in medical and scientific applications, particularly those involving tracing blood flow and determining blood volume. This isotope emits gamma radiation, which can be detected and measured to provide valuable data on bodily functions.

One of the key benefits of using chromium-51 is its relatively long half-life, which is about 27.7 days. This duration is long enough to conduct extensive studies or medical evaluations without the isotope decaying too quickly, yet not so long as to result in prolonged radiation exposure. Additionally, chromium-51's properties make it ideal for labeling red blood cells, which can then be used to track blood circulation or measure total blood volume in a patient.

In clinical settings, a known quantity of chromium-51 is injected into the bloodstream. By measuring the initial and subsequent concentrations of the isotope, practitioners can assess various hematological parameters, such as blood volume. Chromium-51's utilization in these contexts exemplifies how modern medicine leverages the principles of radioactive decay to enhance diagnostic capabilities.

Blood Volume Determination

Finding out a person's blood volume is essential in both clinical and research settings. By understanding how much blood circulates within the body, medical professionals can better diagnose and treat various medical conditions. One technique to determine blood volume is called radiotracer dilution, which often utilizes chromium-51 for its effectiveness and accuracy.

The process begins with injecting a known volume of blood, labeled with an isotope like chromium-51, into the patient's bloodstream. By waiting a short period for the isotope to adequately mix within the circulatory system, a blood sample can then be taken for analysis. The activity of the isotope is measured, and, using the known decay rate, the concentration of chromium-51 in the blood can be evaluated at any later time.

The volume of blood can be calculated by measuring the ratio of the initial isotope activity to the measured activity at a specific time. Using the equation \( V = \frac{A_{\text{initial}}}{C_{\text{measured}}} \), where \( A_{\text{initial}} \) is the total initial activity and \( C_{\text{measured}} \) is the concentration of activity per milliliter in the blood, one can calculate the blood volume. This method not only provides an accurate measurement but also aids in the diagnosis of conditions that alter normal blood volume, such as hemorrhage or dehydration.