Question

A typical chest x-ray radiation dose is $\mathbf{250}\mathit{\mu }\mathit{s}\mathit{v}$, delivered by x rays with an RBE factor of 0.85. Assuming that the mass of the exposed tissue is one-half the patient’s mass of 88 kg, calculate the energy absorbed in joules.

Short answer

The energy absorbed by the chest in joules is $1.3\times {10}^{-2}\mathrm{J}$.

Step-by-step solution

Write the given data

1. Equivalent dose by the chest, $Doseequivalent=250\mu Sv$
2. RBE factor is 0.85.
3. Mass of the patient, $m_{patient}=88\mathrm{kg}$

Determine the concept of dose equivalent  

The equivalent dose of radiation is a measure of the biological damage to the human body due to the ionizing radiation in the radioactive decay processes. The new international system (SI) unit of radiation dose, expressed as absorbed energy per unit mass of tissue is the SI unit "gray". 1 Gy = 1 Joule/kilogram or 1 Gy =1 Sv. RBE (relative biological effectiveness) is a relative measure of the damage done by a given type of radiation per unit of energy deposited in biological tissues.

The dose absorbed of a radiation source:

$Absorbeddose=\frac{\mathrm{Total}\mathrm{absorbed}\mathrm{energy}}{\mathrm{Mass}\mathrm{of}\mathrm{the}\mathrm{sample}}$ …… (i)

The dose equivalent of a radiation source is:

$D.E=RBE\times Absorbeddose$ ….. (ii)

Determine the absorbed energy

Now, using the given data in equation (ii), the absorbed dose of the radiation source can be given as follows:

$Absorbeddose=\frac{Doseequivalent}{RBE}$

Substitute the values and solve as:

$$\begin{gathered} Absorbeddose=\frac{250\times {10}^{-6}\mathrm{Sv}}{0.85} \\ =2.94\times {10}^{-4}\mathrm{Gy} \\ =2.94\times {10}^{-4}\frac{\mathrm{J}}{\mathrm{kg}} \end{gathered}$$

But the mass of the exposed tissue is half the mass of patient, thus, the required mass is given as:

$m=\frac{1}{2}\left(\mathrm{mass}\mathrm{of}\mathrm{patient}\right)$

Substitute the values and solve as:

$$\begin{gathered} m=\frac{88\mathrm{kg}}{2} \\ =44\mathrm{kg} \end{gathered}$$

Thus, using these above values in equation (i), the absorbed energy by the chest in joules can be calculated as follows:

$$\begin{gathered} Absorbedenergy=\left(2.94\times {10}^{-4}\frac{\mathrm{J}}{\mathrm{kg}}\right)\left(44\mathrm{kg}\right) \\ =1.29\times {10}^{-2}\mathrm{J} \\ =1.3\times {10}^{-2}\mathrm{J} \end{gathered}$$

Hence, the value of energy is $1.3\times {10}^{-2}\mathrm{J}$.