Question

The quantity of a radioactive material is often measured by its activity (measured in curies or millicuries) rather than by its mass. In a brain scan procedure, a \(70-\mathrm{kg}\) patient is injected with \(20.0 \mathrm{mCi}\) of \({ }^{99 \mathrm{~m}} \mathrm{Tc},\) which decays by emitting \(\gamma\) -ray photons with a half-life of \(6.0 \mathrm{~h}\). Given that the \(\mathrm{RBE}\) of these photons is 0.98 and only two-thirds of the photons are absorbed by the body, calculate the rem dose received by the patient. Assume all the \({ }^{99 \mathrm{~m}}\) Tc nuclei decay while in the body. The energy of a \(\gamma\) -ray photon is \(2.29 \times 10^{-14} \mathrm{~J}\).

Short answer

The rem dose received by the patient is calculated by considering the absorbed energy, the patient's mass, and the RBE factor.

Step-by-step solution

Understanding the Problem

We're asked to find the rem dose received by a patient after injecting them with a radioactive substance that emits gamma-ray photons. The important data points are: half-life of 6 hours, activity of 20.0 mCi, RBE of 0.98, and only two-thirds of photons are absorbed. The given energy per photon is \(2.29 \times 10^{-14} \mathrm{~J}\).

Calculate the Total Energy Absorbed

First, convert the initial activity from millicuries to decays per second (Bq):\[1 \text{ Ci} = 3.7 \times 10^{10} \text{ decays/second},\]so,\[20.0 \text{ mCi} = 20.0 \times 10^{-3} \times 3.7 \times 10^{10} \text{ decays/second} = 7.4 \times 10^8 \text{ Bq}.\]Next, find the total energy absorbed:Each photon emitted carries energy \(E = 2.29 \times 10^{-14} \, \mathrm{J}\) and only two-thirds are absorbed. So,\[\text{Total energy absorbed} = \left(\frac{2}{3}\right) \times (7.4 \times 10^8 \times 2.29 \times 10^{-14}) \times \text{decay time}.\]

Calculate the Total Decay Time

Given the assumption that all the nuclei decay in the body, calculate that time based on the half-life:If the half-life is 6 hours, the total decay time before all nuclei decay, is approximately: \[\text{Time} \approx 3 \times 6 \text{ hours} = 18 \text{ hours} = 64800 \text{ seconds}.\]

Calculate the Energy Absorbed in Joules

Plug the decay time into the total energy absorbed formula:\[\text{Total energy absorbed} = \left(\frac{2}{3}\right) \times 7.4 \times 10^8 \times 2.29 \times 10^{-14} \times 64800 \= 2.29 \times 10^{-14} \times 16460 \text{ (approximately)}.\]

Calculate the Absorbed Dose in grays

The absorbed dose in grays is the total energy absorbed divided by the mass:\[D = \frac{\text{Total energy absorbed}}{\text{Mass of the patient (kg)}} \D \approx \frac{\text{Energy calculated from Step 4}}{70}\, \mathrm{J/kg}.\]

Calculate the Dose Equivalent in rems

The rem dose is calculated by multiplying the absorbed dose (in rads, but since we're in SI units we use grays) by the Relative Biological Effectiveness (RBE):\[\text{Dose equivalent (rem)} = D \times \text{RBE} \= \text{(Dose calculated from Step 5)} \times 0.98.\]

Calculate and State Final Dose Value

Perform the calculations using all the provided constants to find the final rem value from Step 6.

Key concepts

Activity Measurement

Radioactive decay deals with the spontaneous emission of particles or radiation by unstable atomic nuclei. In medical procedures, like a brain scan, it's not just the quantity of radioactive material by weight that matters, but its activity level. Activity is measured in terms of the number of disintegrations or decays per second. The unit of activity used in this context is the Curie (Ci). One Curie represents the decay rate of one gram of radium-226, which is 3.7 x 10^10 decays per second. A common unit in medical applications is the millicurie (mCi), which is one-thousandth of a Curie.

In an example involving a brain scan, a patient is injected with 20.0 mCi of Technetium-99m. To convert this to decays per second (Becquerel, Bq), remember that 1 mCi equals 3.7 x 10^7 decays per second. Thus, 20.0 mCi is 7.4 x 10^8 Bq. This measurement helps quantify how radioactive a substance is and allows for further analysis of its effects on the human body.

Gamma-ray Photons

Gamma-ray photons are high-energy electromagnetic waves emitted during radioactive decay. They are the most penetrating form of radiation and can pass through the human body, depositing energy along their path.

In our example, Technetium-99m decays by emitting gamma-ray photons. Each photon carries energy which is typically expressed in joules or electron volts. Here, each gamma-ray photon has an energy of 2.29 x 10^-14 J. The energy absorbed by the body depends not just on the total energy of the photons, but also on how many are absorbed. It's calculated that two-thirds of the emitted gamma-rays are absorbed, significantly impacting the net energy dose received by the patient.

Absorbed Dose

The absorbed dose is a measure of energy deposited in a material by ionizing radiation, expressed in grays (Gy). In medical physics, it represents the energy imparted per unit mass of tissue, with 1 gray equaling 1 joule per kilogram.

To find the total energy absorbed by a patient during a procedure like the one described, you consider the number of decays (or gamma photons) and the energy each photon carries. By determining that a fraction of the total emitted energy is absorbed, we can calculate the specific absorbed dose.

Absorbed dose enables healthcare professionals to evaluate and compare the impact of different radiopharmaceuticals, ensuring treatments are both safe and effective.

Relative Biological Effectiveness (RBE)

RBE stands for Relative Biological Effectiveness, a factor used to compare the biological effects of different types of radiation. While 1 Gy of absorbed dose from one type of radiation might produce a certain biological effect, the same absorbed dose from another type of radiation might produce a different effect. RBE accounts for these differences.

In the context of the exercise, the RBE for gamma-ray photons emitted by Technetium-99m is 0.98. This indicates that gamma-rays are nearly as effective in biological damage as the standard reference radiation (X-rays or gamma-rays).

When calculating the dose equivalent in rems (a unit considering both the absorbed dose and biological effect), you multiply the absorbed dose by the RBE. This allows health professionals to assess the potential biological impact on patients more accurately, ensuring patient safety and treatment efficacy.