Problem 42 - 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 palladium-103, which has a half-life of 16.99 days. a) Find the decay rate, \(k\), of palladium-103. 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) Decay rate \(k\) is approximately 14.88 year⁻¹. b) 0.64 rems transmitted in the first month. c) Total energy transmitted is 0.67 rems.

Step 1: Calculate Decay Rate

The decay constant \(k\) is related to the half-life \(T_{1/2}\) of the material. The formula to find \(k\) is: \[ k = \frac{\ln(2)}{T_{1/2}} \]Given the half-life of palladium-103 is 16.99 days, convert this into years:\[ T_{1/2} = \frac{16.99}{365} \text{ years} \]Calculate \(k\):\[ k = \frac{\ln(2)}{16.99/365} \approx 14.88 \text{ year}^{-1} \]

Step 2: Calculate Energy Transmitted in First Month

First, convert one month to years:\[ a = \frac{30}{365} \approx 0.0822 \text{ years} \]The formula for energy \(E\) is:\[ E = \int_{0}^{a} 10 e^{-14.88 t} dt \]Integrate:\[ E = \left[ -\frac{10}{14.88} e^{-14.88 t} \right]_{0}^{0.0822} \]\[ E = -\frac{10}{14.88} \left( e^{-14.88 \times 0.0822} - 1 \right) \]Calculate:\[ E \approx -\frac{10}{14.88} (e^{-1.2238} - 1) \approx 0.64 \text{ rems} \]

Step 3: Determine Total Energy Transmitted

For the total energy transmission, set \(a\) to infinity. The formula for \(E\):\[ E = \int_{0}^{\infty} 10 e^{-14.88 t} dt \]Integrate:\[ E = \left[ -\frac{10}{14.88} e^{-14.88 t} \right]_{0}^{\infty} \]\[ E = -\frac{10}{14.88} (0 - 1) = \frac{10}{14.88} \approx 0.67 \text{ rems} \]

Key Concepts

Calculus Applications

In the realm of calculus applications, understanding how calculus can be used in real-world scenarios is crucial. One such application is in the field of radioactive decay, particularly in medical contexts like cancer treatment, where it's used to calculate the energy transmission of radioactive implants. Calculus provides tools to model how quantities change, using derivatives and integrals. Derivatives help us understand rates of change, whereas integrals allow us to accumulate quantities over a certain period.

For example, when we want to calculate the total energy emitted by a radioactive implant over time, we use the integral of an exponential decay function. By setting up the integral with an exponential decay formula, we can determine the accumulated energy transmitted to the body from the implant. This is especially useful in medical physics to ensure that patients receive the correct dosage of radiation therapy.

• Integrals are used to accumulate quantities over intervals, making them ideal for calculating total emissions.
• Radioactive decay follows an exponential model, showcasing calculus's ability to handle complex, variable processes over time.

Integrals

Integrals play a key role in determining the energy emitted by radioactive materials. In our example, the energy transmission is modeled with an integral of the form:\[E=\int_{0}^{a} P_{0} e^{-k t} d t\]This integral calculates the total energy, in rems, imparted to the body over a time period \(a\). The function \(P_{0} e^{-k t}\) represents the decay of initial energy rate \(P_{0}\) over time \(t\) with decay constant \(k\).

Calculating this integral involves evaluating how much energy is transmitted from \(t = 0\) to \(t = a\). Here are some steps to work through such integrals:

• Identify the variables and constants in the expression \(P_{0} e^{-k t}\).
• Determine the time period \(a\) over which the energy is calculated (expressed in years in this case).
• Use integration techniques to solve the integral, considering appropriate bounds.

Such integrals allow healthcare professionals to predict and measure radiation dosage effectively, keeping treatments within safe limits.

Half-Life Calculations

Half-life calculations are fundamental in understanding radioactive decay. The half-life is the time taken for half of a radioactive substance to decay. This concept can be mathematically expressed and directly influences the rate at which energy is released in radioactive implants.

The relationship between the decay constant \(k\) and the half-life \(T_{1/2}\) is given by:\[k = \frac{\ln(2)}{T_{1/2}}\]Where \(\ln(2)\) is the natural logarithm of 2, approximately 0.693. For palladium-103, with a given half-life of 16.99 days, we calculate \(k\) by converting the half-life into years and then applying the formula.

• Convert days into years for consistency in units when using \(k\).
• Use the formula \(k = \frac{\ln(2)}{T_{1/2}}\) to find the rate of decay.
• Applying \(k\) helps in solving integrals that predict the radioactive material's behavior over time.

Understanding half-life calculations is pivotal, allowing us to assess how long a radioactive substance remains active, which directly impacts both the safety and the effectiveness of treatments involving radiation.