Problem 41 - Calculus and its applications

Question

In the treatment of prostate cancer, radioactive implants are often used. The implants are left in the patient and never removed. The amount of energy that is transmitted to the body from the implant is measured in rem units and is given by $$ E=\int_{0}^{a} P_{0} e^{-k t} d t $$ where \(k\) is the decay constant for the radioactive material, \(t\) is the number of years since the implant, \(a\) is the time (in years) until the rem measurement is made, and \(P_{0}\) is the initial rate at which energy is transmitted. (Source: www.cancer.gov.) Use this information. Suppose the treatment uses iodine-125, which has a halflife of 60.1 days. a) Find the decay rate, \(k,\) of iodine- 125 . b) How much energy (measured in rems) is transmitted in the first month if the initial rate of transmission is 10 rems per year? c) What is the total amount of energy that the implant will transmit to the body?

Step-by-Step Solution

Answer

a) \(k \approx 4.21 \text{ per year}\); b) 0.773 rems; c) 2.38 rems in total.

Step 1: Find the Decay Constant, k, of Iodine-125

The decay constant \(k\) is calculated using the formula \(k = \frac{\ln(2)}{T_{1/2}}\), where \(T_{1/2}\) is the half-life. For iodine-125, the half-life is 60.1 days, so first convert it into years: \(T_{1/2} = \frac{60.1}{365}\) years. Then calculate \(k = \frac{\ln(2)}{60.1/365} \approx 4.21 \text{ per year}\).

Step 2: Calculate Energy Transmitted in the First Month

In this step, calculate \(E\) for the first month. Here, \(a = \frac{1}{12}\) of a year, and \(P_0 = 10\) rems/year. The integral becomes \(E = \int_0^{1/12} 10 e^{-4.21 t} \, dt\). Solve this integral to get \(E = \left[-\frac{10}{4.21} e^{-4.21 t} \right]_0^{1/12}\), which evaluates to approximately 0.773 rems.

Step 3: Calculate the Total Energy Transmitted

Since the radioactive material decays, the total energy transmitted is when \(a \to \infty\). Calculate \(E = \int_0^{\infty} 10 e^{-4.21 t} \ dt\), which simplifies to \(E = \left[-\frac{10}{4.21} e^{-4.21 t} \right]_0^{\infty}\). Evaluating gives \(E = \frac{10}{4.21} \approx 2.38\text{ rem in total}\).

Key Concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics. It describes how unstable atomic nuclei lose energy by emitting radiation. This decay process is random but can be characterized statistically, and it impacts many fields, including medicine, geology, and archaeology. Each radioactive isotope has its own decay rate, defined by its half-life.

The rate at which a radioactive material decays is often measured by a decay constant, denoted as \(k\). The decay constant relates to the probability of decay per unit of time, for a given nucleus.

In the context of cancer treatment with radioactive implants, understanding the decay process helps determine how much radiation is emitted over time and thus enables precise treatment planning.

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms in a sample to decay. It is a critical measurement because it determines how long the material will remain active.

To calculate the half-life of a substance, you can use the formula \(k = \frac{\ln(2)}{T_{1/2}}\), where \(T_{1/2}\) is the half-life period and \(k\) is the decay constant. In the case of iodine-125, with a half-life of around 60.1 days, we first convert days into years to align with other time measurements, before solving for \(k\).

This calculation is essential in medical applications. It allows healthcare professionals to predict the duration and intensity of the radiation exposure from a radioactive implant, ensuring safe and effective use.

Definite Integral

In integral calculus, a definite integral calculates the area under a curve. It is represented as \(\int_{a}^{b} f(x) \, dx\), spanning from a specific start point \(a\) to an endpoint \(b\).

When calculating how much energy is emitted from a radioactive source over time, the definite integral is used to sum the continuous radiation emission rate over a given time period. This is crucial for knowing the total radiation a patient receives from their treatment.

For example, if the initial emission rate of energy \(P_0\) is known, and the decay constant \(k\) is given, setting up the integral \(E = \int_{0}^{a} P_{0} e^{-k t} \, dt\) allows us to evaluate the energy transmitted over time \(a\).

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. This results in a curve that falls steeply at first and then levels off over time, never quite reaching zero.

In equations, exponential decay is expressed as \(N(t) = N_0 e^{-kt}\), where \(N(t)\) is the quantity at time \(t\), \(N_0\) is the initial quantity, and \(k\) is the decay constant.

This form of decay is typical in natural processes, including radioactive decay, where the rate of decay diminishes over time. It's integral in calculating how much radioactive material remains effective over a specific interval, helping determine treatment duration and dosage in therapies involving radioactive implants.