BIO A ${}^{\mathbf{60}}\mathit{C}\mathit{o}$ source with activity $\mathbf{2}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$Ci is embedded in a tumor that has mass 0.200 kg. The source emits $\mathit{\gamma }$photons with average energy 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$\mathit{\gamma }$rays is 0.70? (d) What exposure time is required for an equivalent dose of 200 rem?

Absorbed dose $\mathbf{=}\frac{\mathbf{E}}{\mathbf{M}}$ .............(i)

Equivalent dose = Absorbed dose * RBE .............(ii)

First, convert the activity to decay/s,

$$\begin{gathered} \mathit{R}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{)}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{70}\mathbf{\times }{\mathbf{10}}^{\mathbf{10}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{d}\mathit{e}\mathit{c}\mathit{a}\mathit{y}\mathbf{/}\mathit{s} \end{gathered}$$

So, $\mathbf{9}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}$photons are produced each second; Since only half of them are absorbed, so the number of absorbed photons each second by the tumor is:

$$\begin{gathered} \mathit{N}\mathbf{=}\frac{\mathbf{9}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}}{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{81}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathit{p}\mathit{h}\mathit{o}\mathit{t}\mathit{o}\mathit{n}\mathbf{/}\mathit{s} \end{gathered}$$

Multiply by the average energy of a photon, the total energy absorbed by the tumor each second is given as:

role="math" localid="1664108395449" $$\begin{gathered} \mathit{E}\mathbf{=}\mathit{N}{\mathit{E}}_{\mathbf{\gamma }} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{81}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{25}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{01}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{M}\mathit{e}\mathit{V}\mathbf{/}\mathit{s} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{}\mathit{J}\mathbf{/}\mathit{s} \end{gathered}$$

From equation (i) and (ii),

Absorbed dose is:$$\begin{gathered} \frac{\mathbf{E}}{\mathbf{m}}\mathbf{=}\frac{\mathbf{9}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}}{\mathbf{0}\mathbf{.}\mathbf{200}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{81}\mathbf{}\mathit{\mu }\mathit{G}\mathit{y}\mathbf{/}\mathit{s} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{481}\mathbf{}\mathit{\mu }\mathit{r}\mathit{a}\mathit{d}\mathbf{/}\mathit{s} \end{gathered}$$

Equivalent dose is:$\mathbf{0}\mathbf{.}\mathbf{70}\mathbf{\times }\mathbf{481}\mathbf{=}\mathbf{337}\mathbf{}\mathit{\mu }\mathit{r}\mathit{e}\mathit{m}\mathbf{/}\mathit{s}$

$$\begin{gathered} \mathit{t}\mathbf{=}\frac{\mathbf{200}}{\mathbf{481}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}} \\ \mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{594}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{s} \\ \mathbf{}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{88}\mathbf{}\mathit{d}\mathit{a}\mathit{y} \end{gathered}$$

Thus, the energy delivered to the tumor per second is $\mathbf{9}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{}\mathit{J}\mathbf{/}\mathit{s}$. The absorbed dose delivered per second is role="math" localid="1664108480796" $\mathbf{481}\mathbf{}\mathit{\mu }\mathit{r}\mathit{a}\mathit{d}\mathbf{/}\mathit{s}$. The equivalent dose delivered per second is $\mathbf{337}\mathbf{}\mathit{\mu }\mathit{r}\mathit{e}\mathit{m}\mathbf{/}\mathit{s}$. The exposure time required for an equivalent dose of 200 rem is 6.88 day.