Question

Radioiodine treatment Roughly 12,000 Americans are diagnosed with thyroid cancer every year, which accounts for \(1 \$$ of all cancer cases. It occurs in women three times as frequently as in men. Fortunately, thyroid cancer can be treated successfully in many cases with radioactive iodine, or \)1-131 .\( This unstable form of iodine has a half-life of 8 days and is given in small doses measured in millicuries. a. Suppose a patient is given an initial dose of 100 millicuries. Find the function that gives the amount of \)1-131\( in the body after \)t \geq 0\( days. b. How long does it take the amount of \)1-131\( to reach \)10 \%\( of the initial dose? c. Finding the initial dose to give a particular patient is a critical calculation. How does the time to reach \)10 \%\( of the initial dose change if the initial dose is increased by \)5 \% ?$

Short answer

Answer: The time required to reach 10% of the initial dose is approximately 26.36 days. When the initial dose is increased by 5%, it takes approximately 28.32 days for the amount of iodine to reach 10% remaining, which is an increase of approximately 1.96 days.

Step-by-step solution

Recall the radioactive decay formula and plug in the given values

The radioactive decay formula is given by: \(A(t) = A_0 e^{-\lambda t}\) where - \(A(t)\) is the amount of substance remaining after time \(t\) - \(A_0\) is the initial amount of substance - \(e\) is the base of the natural logarithm - \(\lambda\) is the decay constant - \(t\) is the time The half-life of the radioactive iodine is given as 8 days. We can find the decay constant, \(\lambda\), using the formula: \(\lambda = \frac{\ln(2)}{T_{1/2}}\) where \(T_{1/2}\) is the half-life. Plugging in the values, we get \(\lambda = \frac{\ln(2)}{8}\) The initial amount, \(A_0\), is given as 100 millicuries. Now we can write our decay function: \(A(t) = 100 e^{-\frac{\ln(2)}{8} t}\)

Find the time when the amount of iodine is 10% of the initial dose

To find the time when the amount of iodine is 10% of the initial dose, we set \(A(t) = 10\% \times A_0\) which is \(10 e^{-\frac{\ln(2)}{8} t} = 100 e^{-\frac{\ln(2)}{8} t} \cdot 0.1\) Now divide by the initial dose: \(0.1 = e^{-\frac{\ln(2)}{8} t}\) Take the natural logarithm of both sides: \(-\ln(10) = -\frac{\ln(2)}{8} t\) Solve for \(t\): \(t = \frac{8 \ln(10)}{\ln(2)}\) Now plug those values into our time equation: \(t \approx 26.36\) days So it takes approximately 26.36 days for the amount of iodine to reach 10% of the initial dose.

Analyze how the time changes with changes in the initial dose

We are asked to find the change in time required to reach 10% remaining when the initial dose is increased by 5%. Let's denote this new initial dose as \(A_0'\). Then, \(1.05A_0 = A_0'\) Using the decay function with the new initial dose: \(A(t) = A_0' e^{-\frac{\ln(2)}{8} t} = 1.05A_0 e^{-\frac{\ln(2)}{8} t}\) Now, we want to find the time when the amount is 10% of this increased initial dose: \(0.1 A_0' = 1.05 A_0 e^{-\frac{\ln(2)}{8} t'}\) where \(t'\) represents the new time to reach 10% remaining. As before, divide by the increased initial dose and take the natural logarithm of both sides and isolate \(t'\): \(t' = \frac{8 \ln(10)}{\ln(2)} + \frac{8 \ln(1.05)}{\ln(2)} \approx 28.32\) days The time required to reach 10% remaining with the increased initial dose is approximately 28.32 days, which is an increase of approximately 1.96 days compared to the original scenario.

Key concepts

Thyroid Cancer Treatment

Thyroid cancer, although a serious health concern, is often treatable. The treatment frequently involves the use of radioactive iodine, particularly iodine-131. This isotope targets thyroid cells, enabling specialists to address the cancer effectively.

The thyroid gland absorbs iodine from the diet to produce essential hormones. Therefore, when iodine-131 is introduced into the body, it accumulates in the thyroid gland. There, it delivers radiation selectively to thyroid tissue, including cancerous cells, leading to their destruction.

This method of treatment is particularly advantageous because it is:

• Non-invasive.
• Highly targeted, sparing most normal cells elsewhere in the body.
• Effective in reducing the need for more aggressive treatments.

Patients undergoing this treatment must follow special precautions to limit radiation exposure to others until the iodine is sufficiently decayed.

Half-life

The concept of half-life is crucial in understanding radioactive substances, such as iodine-131. It describes the time required for half of the radioactive atoms in a sample to decay. In the case of iodine-131, this time period is 8 days.

Understanding half-life helps medical practitioners determine how long a radioactive substance remains active in the body and how often doses might need to be administered.

For students: think of half-life as the time it takes for half of your library books to disappear! If you start with a stack of 10 books and each day you return half of them, you will have 5 the next day, then 2.5 on the following day, and so on. The concept of half-life works similarly, but on an atomic scale.

In practical terms for thyroid treatment, the half-life helps determine how long a patient might remain a radiation concern to others and when a clinical response might be expected.

Exponential Decay Formula

Exponential decay is a mathematical concept that describes how a quantity decreases over time at a rate proportional to its current value. It's represented by the formula:

\( A(t) = A_0 e^{-\lambda t} \)

In this equation:

• \(A(t)\) is the amount remaining at time \(t\).
• \(A_0\) is the initial quantity.
• \(e\) is the base of natural logarithms (approximately 2.718).
• \(\lambda\) is the decay constant.

The decay constant \(\lambda\) is found using the half-life with the formula \(\lambda = \frac{\ln(2)}{T_{1/2}}\), where \(T_{1/2}\) is the half-life of the substance.

This formula is applied in medicine to predict how quickly a treatment such as radioactive iodine will lose activity in the body over time, aiding in dosage scheduling and patient care management.

Radioactive Iodine Therapy

Radioactive iodine therapy, a common treatment for thyroid cancer, employs iodine-131 because it is an effective isotope for targeting thyroid tissue.

After administration, iodine-131 is absorbed by the thyroid gland, leveraging its natural iodine uptake processes. Once there, the isotope emits beta particles and gamma rays which effectively damage and destroy the cancerous thyroid cells while sparing other organs.

Patients undergoing this therapy must be cautious about their exposure to others. Typically, within a few weeks, the radioactive potential of iodine-131 significantly diminishes due to its 8-day half-life. This decrease follows the predictable pattern outlined by the exponential decay formula.

• It is minimally invasive compared to surgical options.
• Usually administered in a controlled hospital setting.
• Allows normal activities to resume after a precautionary period.

Radioactive iodine therapy exemplifies how targeted therapies can effectively treat cancers with minimal overall impact on the body.