Question

Radioactive isotopes used in cancer therapy have a "shelf-life," like pharmaceuticals used in chemotherapy. Just after it has been manufactured in a nuclear reactor, the activity of a sample of \(^6$$^0\)Co is 5000 Ci. When its activity falls below 3500 Ci, it is considered too weak a source to use in treatment. You work in the radiology department of a large hospital. One of these \(^6$$^0\)Co sources in your inventory was manufactured on October 6, 2011. It is now April 6, 2014. Is the source still usable? The half-life of \(^6$$^0\)Co is 5.271 years.

Short answer

Yes, the source is still usable as its activity is approximately 3795 Ci, which is above the 3500 Ci threshold.

Step-by-step solution

Determine the Time Elapsed

Calculate the total time that has elapsed from the manufacturing date to the evaluation date. The source was manufactured on October 6, 2011, and we are evaluating it on April 6, 2014. This gives a total time of 2 years and 6 months, which converts into years as follows: \[2.5 \text{ years}\]

Use the Radioactive Decay Formula

Apply the radioactive decay formula to calculate the remaining activity after 2.5 years. The formula is:\[A = A_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}\]Where:- \(A\) is the final activity,- \(A_0\) is the initial activity (5000 Ci),- \(t\) is the time elapsed (2.5 years),- \(T_{1/2}\) is the half-life of \(^6$$^0\)Co (5.271 years).Substituting the values:\[A = 5000 \times \left(\frac{1}{2}\right)^{2.5/5.271}\]

Solve for Remaining Activity

Calculate the remaining activity using the formula from Step 2:\[A \approx 5000 \times 0.759\]This approximately evaluates to:\[A \approx 3795\text{ Ci}\]

Compare Remaining Activity to Usability Threshold

Since the remaining activity (approximately 3795 Ci) is greater than the threshold of 3500 Ci, the source is still considered usable for treatment purposes.

Key concepts

Half-life Calculation

Half-life is a fundamental concept in understanding radioactive decay. It describes the time required for half of a radioactive substance to decay into a more stable form. This concept is crucial when working with isotopes like cobalt-60 in medical applications. The half-life formula used in such calculations is: \[A = A_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}\]

• \(A\): remaining activity
• \(A_0\): original activity
• \(t\): time elapsed
• \(T_{1/2}\): half-life of the substance

To determine if a radioactive source is still viable, like in the exercise with cobalt-60, one needs to calculate how much of it has decayed over a certain period. By inputting the values, you can compute the remaining activity and compare it to the necessary threshold. Understanding this formula helps ensure isotopes are used effectively and safely.

Medical Physics

Medical physics is a branch of physics that applies physical principles to medicine. It is instrumental in developing technologies and treatments for various medical conditions, including cancer. Radioactive isotopes, such as cobalt-60, are often used in this field.
Medical physicists work closely with other healthcare professionals to optimize the use of radiation therapy, ensuring it is both effective and safe. This involves calculating the correct dosage and managing the isotopic decay to guarantee that treatments remain within safe and effective limits.
By understanding both the physics and biology involved, medical physics can enhance patient outcomes, making treatments less harmful and more successful in targeting diseases.

Cobalt-60

Cobalt-60 is a radioactive isotope with a variety of applications, particularly in medicine and industry. Known for its gamma radiation, it is frequently used in radiotherapy to treat cancer. This isotope has a half-life of approximately 5.271 years, which makes it suitable for medical use for a limited time.

• Cobalt-60 decays by beta and gamma emission, helping to eliminate cancerous cells by damaging their DNA.
• The isotopes are carefully managed to prevent exposure to healthy tissues and the environment.
• It is also used in sterilizing medical equipment, food irradiation, and industrial radiography.

For medical purposes, it is important to monitor the decay of cobalt-60 to ensure that it remains effective without posing harm to patients or medical staff.

Radiotherapy

Radiotherapy is a treatment method that uses high-energy radiation to kill cancer cells or shrink tumors. It is usually employed when cancerous cells need localized treatment. Cobalt-60, a potent source of gamma radiation, is often utilized in this process.
During radiotherapy:

• Accurate dose calculations are performed to maximize damage to cancer cells while minimizing harm to healthy cells.
• Patient treatment plans consider factors such as the type of cancer, its stage, and location.
• The delivery of radiation is adjusted as the isotopic activity decreases, according to its half-life.

Radiotherapy not only halts the growth of cancer cells but can also make them more susceptible to surgery or chemotherapy. Understanding the decay of isotopes like cobalt-60 is vital for maintaining the effectiveness of this critical cancer treatment tool.