Problem 20 - Chemistry for Allied Health

Question

Lead-212 is one of the radioisotopes used in the treatment of breast cancer. A patient needs a \(15 \mu \mathrm{Ci}\) dose and it is supplied as a solution with a concentration of \(2.5 \mu \mathrm{Ci} / \mathrm{mL}\). What volume does the patient need? Given the half-life of lead is \(10.6\) hours, what will be the radioactivity of the sample after approximately four days?

Step-by-Step Solution

Answer

The required volume is 6 mL, and the radioactivity after four days is approximately 0.0282 bcCi.

Step 1: Identify the Required Volume

To find the volume of the solution required, we apply the formula for concentration: \( V = \frac{C}{c} \), where \( V \) is the volume needed in mL, \( C \) is the desired concentration in \( \mu \text{Ci} \), and \( c \) is the given concentration in \( \mu \text{Ci/mL} \). Substituting the given values, we have \( V = \frac{15}{2.5} = 6 \text{ mL} \). Therefore, the volume needed is 6 mL.

Step 2: Calculate the Decay Constant

The decay constant, \( \lambda \), is calculated using the formula \( \lambda = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} \) is the half-life. The half-life of lead-212 is 10.6 hours. So, \( \lambda = \frac{\ln(2)}{10.6} \approx 0.0654 \text{ hour}^{-1} \).

Step 3: Determine Time in a Single Unit

Convert the time from days to hours, since the decay rate is given in hours. Four days is equivalent to \( 4 \times 24 = 96 \text{ hours} \).

Step 4: Calculate Remaining Activity

The remaining radioactivity after a period of time can be found using the formula \( A = A_0 e^{-\lambda t} \), where \( A_0 \) is the initial activity and \( t \) is the time. The initial activity is 15 \( \mu \text{Ci} \), \( \lambda \approx 0.0654 \text{ hour}^{-1} \), and \( t = 96 \text{ hours} \). Substitute into the equation: \( A = 15 e^{-0.0654 \times 96} \approx 15 e^{-6.2784} \approx 15 \times 0.00188 \approx 0.0282 \text{ \( \mu \text{Ci} \)} \).

Key Concepts

Lead-212 Decay

Lead-212 is a radioactive isotope, which means it naturally undergoes a breakdown process that emits radiation. This process is scientifically known as radioactive decay. Each atom of lead-212 decays at a unique rate to become a more stable form. During this decay, lead-212 primarily transforms into other elements until it reaches a stable lead isotope. This chain reaction is what makes lead-212 significant in medical treatments, such as fighting cancer cells by targeting them with radiation.

Understanding the decay of lead-212 is crucial, particularly because the emitted radiation energy can be used to destroy harmful cells while limiting damage to healthy ones. This property makes lead-212 an effective agent in specific medical radioisotope therapies, like those for breast cancer.

• It decays by emitting particles and energy.
• Radioactive decay is a random process.
• Lead-212 eventually stabilizes into a non-radioactive isotope.

Half-Life Calculation

The half-life of a radioactive isotope is a measure of the time required for half of the radioactive atoms in a sample to decay. For lead-212, this half-life is approximately 10.6 hours. This brief period is critical in medical scenarios as it determines how long the radioisotope remains active and effective. To calculate the decay over time, we use the decay constant formula: \[ \lambda = \frac{\ln(2)}{t_{1/2}}\]where \( \lambda \) is the decay constant and \( t_{1/2} \) is the half-life.

The half-life assists in planning and optimizing the dosage schedule for patients, ensuring that the radioisotope provides maximum effectiveness during treatment. This is important in balancing the radiative exposure to both maximize efficacy and minimize potential side effects.

• Helps to predict how long the isotope remains potent.
• Key in determining dosing schedules.
• Influences the effectiveness and safety of treatment.

Concentration Formula

In medical treatments involving radioisotopes, knowing the concentration of the solution is crucial for accuracy. The concentration formula \( V = \frac{C}{c} \) enables the calculation of the required volume of a radioactive solution needed to deliver a specific dose. Here, \( V \) represents volume, \( C \) is the dosage needed, and \( c \) is the concentration of the solution.

For example, if a patient requires a dose of \( 15 \, \mu \text{Ci} \) and the solution concentration is \( 2.5 \, \mu \text{Ci/mL} \), the volume needed can be calculated as:
\[ V = \frac{15}{2.5} = 6 \, \text{mL} \]
This formula ensures that patients receive the correct amount of radioactivity, which is crucial in achieving the desired therapeutic effect with precision.

• Allows precise dosage calculation.
• Ensures safe and effective treatment.
• Reduces potential for dosing errors.

Radioactivity Measurement

Measuring radioactivity helps in determining how much ionizing radiation is present in a sample. This measurement is crucial because it helps to understand how much radiation is being administered in treatments and how it decreases over time due to decay. This is done using the formula:\[ A = A_0 e^{-\lambda t} \]Where \( A \) is the remaining activity, \( A_0 \) is the initial radioactivity, \( \lambda \) is the decay constant, and \( t \) is the time.

For the given problem, the initial radioactivity was \( 15 \, \mu \text{Ci} \), and after four days (96 hours), the radioactivity decreased to about \( 0.0282 \, \mu \text{Ci} \). This significant drop highlights the need to monitor and calculate the radioactivity, ensuring treatments are timely and effective. Regular measurement and calculation of radioactivity are essential to maintaining safe levels while treating patients efficiently.

• Crucial for monitoring treatment effectiveness.
• Helps adjust treatment plans accordingly.
• Ensures accurate radiation dose is given.