Question

To treat a brain tumor with gamma knife radiosurgery, the patient's head is positioned within a hemispherical dome covered by 201 individual \({ }^{60}\) Co sources, whose narrow beams are directed inward toward the tumor target point. The tumor receives a very high dose of radiation because all the beams converge on it, while any irradiated healthy tissue receives only the radiation of a single beam. For a prescribed dose of \(1800 \mathrm{rad}\), how long should the radiation treatment go if \(2.2 \times 10^{11}\) disintegrations of \({ }^{60} \mathrm{Co}\) are required to give a dose of \(1.0\) rad? Assume that all the sources are directed at the tumor with an activity of 30 Ci each.

Short answer

The treatment should last approximately 1.775 seconds.

Step-by-step solution

Understanding the Problem

We need to determine the time for which gamma knife radiosurgery should be administered to achieve a dose of 1800 rad on the tumor.

Find Disintegrations for 1800 rad

First, calculate the total number of disintegrations required for 1800 rad. Since 1 rad requires \(2.2 \times 10^{11}\) disintegrations, 1800 rad requires:\[ 1800 \times 2.2 \times 10^{11} = 3.96 \times 10^{14} \text{ disintegrations} \]

Calculate Total Activity

Each of the 201 sources has an activity of 30 Ci. Therefore, the total activity is:\[ 201 \times 30 = 6030 \text{ Ci} \]

Convert Curie to Disintegrations per Second

1 Curie (Ci) is equivalent to \(3.7 \times 10^{10}\) disintegrations per second. Thus, the total disintegrations per second (activity in disintegrations per second) is:\[ 6030 \times 3.7 \times 10^{10} = 2.2311 \times 10^{14} \text{ disintegrations/second} \]

Determine Time Duration for Treatment

Solve for the time \(t\) using the formula:\[ \text{Total Disintegrations} = \text{Rate of Disintegrations} \times t \]Plug in values:\[ t = \frac{3.96 \times 10^{14}}{2.2311 \times 10^{14}} = 1.775 \text{ seconds} \]

Key concepts

Radiation Dose Calculation

When treating a tumor with gamma knife radiosurgery, understanding radiation dose calculation is crucial. A radiation dose is the amount of radiation energy absorbed by the target tissue. It is measured in units called rads, where 1 rad is equivalent to the absorption of 0.01 joules of energy per kilogram of tissue. For controlled radiation treatments, like the one described in the exercise, calculating the exact dose absorbed by the tumor is essential. This ensures the maximum effect on cancerous cells while sparing the surrounding healthy tissue.

To achieve the desired therapeutic effect, the beams from multiple sources converge on the tumor. In this exercise, we compared the radiation dose needed, 1800 rads, against a specified known standard, stating that you need roughly \(2.2 \times 10^{11}\) disintegrations of cobalt-60 to receive 1 rad. By multiplying the number of disintegrations for 1 rad by 1800, we determine how many total disintegrations are needed to achieve the dose sufficient for the medical treatment.

Cobalt-60 Disintegration

Cobalt-60, known as \(^{60}Co\), plays a significant role in gamma knife radiosurgery due to its ability to emit high-energy gamma rays. These emissions are the reason cobalt-60 is preferred for such precise treatments. The disintegration process refers to how cobalt-60 atoms emit radiation as they decay. Each disintegration releases gamma rays, which then converge as narrow beams on the tumor, maximizing the radiation dose in a specific area.

In our exercise, each cobalt-60 disintegration contributes to achieving the necessary radiation dose. The process of determining the required number of these disintegrations involves calculating the expected number of disintegrations needed for each rad and then scaling this up to the total required dose, such as 1800 rads. This requires understanding the decay rate and calculating the total disintegrations utilizing the entire setup, where 201 sources of cobalt-60 converge their beams on the target. Understanding the decay process of cobalt-60 ensures precise dose delivery.

Activity Conversion

Radioactive activity refers to the rate at which a radioactive substance disintegrates, releasing ionizing radiation. Activity is commonly measured in curies (Ci), with one curie representing \(3.7 \times 10^{10}\) disintegrations per second. In treatment planning, converting this activity into a measurable rate of disintegration per second is necessary to determine treatment time.

In our exercise, each of the 201 sources assigned contributes 30 Ci, collectively reaching a total activity of 6030 Ci. By converting these curies into disintegrations per second, practitioners calculate how long those sources need to remain active to achieve a specific radiation dose, which in this scenario involved matching the needed 1800 rads dose with the converged beams.

• Determine total activity from sources in curies.
• Convert curies to disintegrations per second.
• Calculate time by dividing required disintegrations by the total activity rate.

By mastering activity conversion, precise treatment durations are set, ensuring a consistent dose delivery to the tumor.