Problem 55 - College Physics

Question

\(\mathrm{A}^{60} \mathrm{Co}\) source with activity 15.0 Ci is imbedded in a tumor that has a mass of 0.500 \(\mathrm{kg}\) . The Co source emits gamma-ray photons with average energy of 1.25 MeV. Half the photons are absorbed in the tumor, and half escape. (a) What energy is delivered to the tumor per second? (b) What absorbed dose (in rad) is delivered per second? (c) What equivalent dose (in rem) is delivered per second if the RBE for these gamma rays is 0.70\(?(\mathrm{d})\) What exposure time is required for an equivalent dose of 200 \(\mathrm{rem} ?\)

Step-by-Step Solution

Answer

The energy delivered to the tumor is approximately 4.635×10³ J/s. The absorbed dose is 9.27 rad/s, the equivalent dose is 6.49 rem/s, and the exposure time required is about 30.8 s.

Step 1: Determine Energy Delivered to the Tumor per Second

The activity, \( A \), of the \( ^{60}\mathrm{Co} \) source is given as 15.0 Ci. First, convert this to disintegrations per second using the conversion factor: 1 Ci = \( 3.7 \times 10^{10} \) disintegrations/s. Thus, \( A = 15.0 \times 3.7 \times 10^{10} \) disintegrations/s. The energy per disintegration is given as 1.25 MeV, which is equal to \( 1.25 \times 1.602 \times 10^{-13} \) J (since 1 MeV = \( 1.602 \times 10^{-13} \) J). Only half of these photons are absorbed, so the energy absorbed by the tumor per second is \( 0.5 \times A \times 1.25 \times 1.602 \times 10^{-13} \). Calculate this to find the energy delivered per second.

Step 2: Calculate Absorbed Dose in rad

The absorbed dose in rad is calculated as the energy absorbed per unit mass. The energy absorbed per second is calculated in Step 1. The mass of the tumor is 0.500 kg, which is equivalent to 500 g since 1 kg = 1000 g. The absorbed dose \( D \) in rad is given by \( D = \frac{\text{Energy absorbed (J)}}{\text{mass (g)}} \times 100 \). Complete this calculation to find the dose in rad.

Step 3: Determine Equivalent Dose in rem

The equivalent dose in rem is the absorbed dose in rad multiplied by the Relative Biological Effectiveness (RBE factor). For gamma rays, the given RBE factor is 0.70. The equivalent dose \( H \) is given by \( H = D \times 0.70 \). Use the absorbed dose found in Step 2 to calculate the equivalent dose in rem.

Step 4: Calculate Required Exposure Time for 200 rem

To find the exposure time required to deliver a total equivalent dose of 200 rem, use the equivalent dose per second found in Step 3. The exposure time \( t \) in seconds is calculated by dividing the total desired dose (200 rem) by the equivalent dose per second. Use the formula \( t = \frac{200}{H} \), where \( H \) is the equivalent dose per second from Step 3.

Key Concepts

Absorbed Dose

The concept of absorbed dose is fundamental in radiation dosimetry. It is the amount of energy deposited by radiation per unit mass of a substance. This is typically measured in rads (radiation absorbed doses). To find the absorbed dose for the exercise, we need to calculate how much energy from gamma rays is absorbed by the tumor. The important steps include:

• Converting the activity of the source, given in curies (Ci), to disintegrations per second. This is done because the energy delivered is closely related to how frequently gamma photons are emitted.
• Determining the energy of each gamma photon, which is given in mega-electron volts (MeV), and converting this into joules so it's compatible with other physical units in the calculation.
• Considering the fraction of energy that actually gets absorbed by the tumor.

By performing these steps, you determine the total energy absorbed per second, which, when divided by the tumor mass, gives the absorbed dose in rads.

Equivalent Dose

Equivalent dose takes into account not only the amount of radiation absorbed but also the type of radiation and its potential biological effect. Different types of radiation can cause different levels of harm even with the same absorbed dose. This is where Relative Biological Effectiveness (RBE) comes into play. The RBE is a factor used to measure the biological effectiveness of a specific type of radiation. In this exercise:

• The absorbed dose calculated in rads is multiplied by the RBE for gamma rays, provided as 0.70, to compute the equivalent dose in rems (roentgen equivalent man).
• This step helps quantify the potential biological impact the radiation can have on the human body, tailored to the type of radiation.

By using the RBE, we can understand that not all radiation is created equal in terms of its effects on living tissues.

Gamma Rays

Gamma rays are high-energy photons emitted from the nucleus of radioactive atoms, such as cobalt-60 used in the exercise. They have no mass and no charge. Instead, they possess very high penetration power, capable of passing through many kinds of materials and tissues. Some essential characteristics:

• High frequency and energy compared to other types of electromagnetic radiation, like X-rays or UV light.
• Used in medical applications for treatment and diagnostic purposes due to their penetrating abilities.

In radiation dosimetry, understanding these properties is crucial for accurately calculating how much energy gamma rays transfer to tissues, as well as for determining safety measures needed during their application.

Relative Biological Effectiveness

Relative Biological Effectiveness (RBE) reflects the efficiency of different types of radiation in causing biological damage. The RBE allows comparison of the biological effects of radiation at a similar absorbed dose. Key points about RBE include:

• It's used to modify the absorbed dose to better estimate the actual biological damage.
• RBE varies depending on factors like the type of radiation, the energy it carries, and the biological tissue affected.

In the context of gamma rays emitted by cobalt-60, the RBE value provided is 0.70. This indicates that, relative to standard radiation (often X-rays or beta radiation), gamma rays are less effective biologically at the same absorbed dose. Understanding RBE is crucial for accurately assessing risk and planning radiation treatments, making it a necessary consideration in medical physics and health physics.