Problem 53 - College Physics

Question

A cancer treatment called the gamma knife (see Insight 29.1, Biological and Medical Application of Radiation) uses focused \({ }^{60}\) Co sources to treat tumors. Each \({ }^{60}\) Co nucleus emits two gamma rays, of energy \(1.33 \mathrm{MeV}\) and \(1.17 \mathrm{MeV}\), in quick succession. Assume that \(50.0 \%\) of the total gamma-ray energy is absorbed by a tumor. Further assume that the total activity of the \({ }^{60}\) Co sources is \(1.00 \mathrm{mCi}\), the tumor's mass is \(0.100 \mathrm{~kg}\), and the patient is exposed to the gamma radiation for an hour. Determine the effective radiation dose received by the tumor. (Since the \({ }^{60}\) Co half-life is 5.3 years, changes in its activity during treatment are negligible.)

Step-by-Step Solution

Answer

The effective radiation dose received by the tumor is 133.4 Grays.

Step 1: Calculate the Total Energy Emitted

First, we need to determine the energy of gamma rays emitted per unit time. Each \(^{60}\)Co nucleus emits two gamma rays with energies of 1.33 MeV and 1.17 MeV. Thus, the total energy per disintegration is \(1.33 \mathrm{MeV} + 1.17 \mathrm{MeV} = 2.50 \mathrm{MeV}\).

Step 2: Convert Energy to Joules

Convert the total gamma ray energy from MeV to Joules. \(1 \mathrm{MeV} = 1.602 \times 10^{-13} \mathrm{J}\). Hence, \(2.50 \mathrm{MeV} = 2.50 \times 1.602 \times 10^{-13} \mathrm{J} = 4.005 \times 10^{-13} \mathrm{J}\).

Step 3: Calculate the Number of Disintegrations

Given the activity \(A = 1.00 \mathrm{mCi}\), convert it to disintegrations per second. \(1 \mathrm{mCi} = 3.7 \times 10^7 \) decays/second, so the total number of disintegrations in one hour (3600 seconds) is \(3.7 \times 10^7 \times 3600\).

Step 4: Calculate Total Energy Absorbed by the Tumor

Only 50% of the gamma-ray energy is absorbed. Therefore:\[ E_{\text{absorbed}} = \frac{1}{2} \times 4.005 \times 10^{-13} \times (3.7 \times 10^7 \times 3600) \]

Step 5: Calculate the Dose in Joules per Kilogram

The dose absorbed (in Greys) is the absorbed energy divided by the mass of the tumor:\[ \text{Dose} = \frac{E_{\text{absorbed}}}{\text{mass}} \]Substitute \( \text{mass} = 0.100 \mathrm{kg} \).

Step 6: Solve the Equation for the Dose

Substituting the calculated values:\[ \text{Dose} = \frac{1}{2} \times 4.005 \times 10^{-13} \times 3.7 \times 10^7 \times 3600 \div 0.100 \]After computing, you find \(\text{Dose} = 133.4 \mathrm{Gr} \).

Key Concepts

Radiation Dosage

Radiation dosage is a critical concept when talking about treatments like the gamma knife, which is used for targeting tumors with precision. The amount of radiation absorbed by tissue is measured in Grays (Gy), where one Gray is equivalent to one Joule of radiation energy absorbed per kilogram of tissue. In practical scenarios, it becomes essential to calculate how much radiation dose a tumor receives. The energy from gamma rays emitted by a source, like cobalt-60, is only partially absorbed by the target tissue. The absorption rate might be different depending on the medium it travels through before reaching the tumor.
For treatments, it is vital to compute the expected dose accurately, which involves knowing the energy released by the radioactive decay, the percentage of energy absorbed, and the duration of exposure. All of these are factored into determining the radiation dosage that can effectively treat the tumor while minimizing damage to surrounding healthy tissues.

Gamma Knife

The Gamma Knife is a non-invasive medical tool designed for treating brain tumors and other lesions. Despite its name, it isn't a traditional surgical knife; rather, it uses highly focused gamma radiation to target and treat specific areas within the brain. A gamma knife system employs numerous beams of gamma rays, typically from cobalt-60 sources, directed to converge at a specific point within the brain. This technique ensures high-dose delivery to the tumor while sparing the surrounding healthy tissue.
Patients undergoing gamma knife treatment benefit from:

• Minimal impact on surrounding healthy brain tissue due to precise targeting.
• No need for physical incisions, reducing risks associated with conventional surgery.
• Often shorter recovery times and lower complication rates.

Gamma Knife Procedures are commonly used for:

• Brain tumors, including both malignant and benign types.
• Arteriovenous malformations (AVM).
• Trigeminal neuralgia and other nerve disorders.

Despite its benefits, careful planning by skilled specialists is necessary to ensure the maximum effectiveness and safety of the treatment.

Cobalt-60

Cobalt-60 ( ^{60} Co) is a radioactive isotope of cobalt and plays a significant role in medical applications, particularly in the Gamma Knife technology. It is known for emitting two high-energy gamma rays quickly one after the other, making it extremely effective for use in radiotherapy for cancer treatments. This isotope is produced by neutron activation of cobalt-59. It has a half-life of 5.3 years, which is relatively long, allowing it to be used in medical applications for extended periods before needing replacement. Key features to understand about cobalt-60 include:

• It emits powerful gamma rays used to target cancerous tissues precisely.
• Its decay process involves emitting two gamma rays, with energies of 1.33 MeV and 1.17 MeV, which are designed to maximize the dose to the tumor.
• The efficiency of cobalt-60 in delivering radiation to specific areas makes it essential in devices like linear accelerators and the gamma knife.

These properties, coupled with advanced medical engineering, allow cobalt-60 to be a powerful tool in the fight against cancer, providing targeted and effective treatment options with precision.