Question 10.3.Q2: The molybdenum-99 (Mo-99) → technetium-99m (Tc-99m) → techne......
The molybdenum-99 (Mo-99) → technetium-99m (Tc-99m) → technetium99 (Tc-99) decay series plays an important role in nuclear medicine, since it serves as the source of Tc-99m, the most widely used radionuclide for nuclear imaging tests. The series parent radionuclide Mo-99 decays through β^− decay with a \text { half-life }\left(t_{1 / 2}\right)_{\mathrm{Mo}-99}=66.0 hours into daughter radionuclide Tc-99m.
Subsequently, the daughter Tc-99m decays through gamma emission with a \text { half-life }\left(t_{1 / 2}\right)_{\mathrm{Tc}-99 \mathrm{~m}}=6.02 hours to the granddaughter radionuclide Tc-99. The Tc-99 radionuclide has a much longer \text { half-life }\left[\left(t_{1 / 2}\right)_{\mathrm{Tc}-99}=2.1 \times 10^5 \mathrm{y}\right] in comparison with Mo-99 and Tc-99m and decays through β^− decay to ruthenium-99 (Ru-99).
Starting with a pure 10 mCi (0.37 GBq) Mo-99 source:
(a) State or derive equations for activities of the Mo-99 parent and Tc-99m daughter as a function of time.
(b) Calculate the characteristic \text { time }\left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}} at which the Tc-99m daughter radionuclide attains its maximum activity.
(c) Determine the maximum \text { activity } \mathcal{A}_{\mathrm{D}}\left[\left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}}\right] of the Tc-99m radionuclide.
(d) Show that activities of the \text { parent } \mathcal{A}_{\mathrm{p}}\left[\left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}}\right] and the daughter \mathcal{A}_{\mathrm{D}}\left[\left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}}\right] are equal at the characteristic time \left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}}.
(e) Sketch the activities of the Mo-99 parent and Tc-99m daughter as a function of time and highlight the salient features of the two radioactive decay curves.
(a) Activities of the parent \mathcal{A}_{\mathrm{P}}(t)=\mathcal{A}_{\mathrm{Mo}-99}(t) \text { and of the daughter } \mathcal{A}_{\mathrm{D}}(t)=\mathcal{A}_{\mathrm{Tc}-99 \mathrm{~m}}(t) as a function of time t are, respectively, given by [see (T10.10) and (T10.35), respectively]
\mathcal{A}_{\mathrm{P}}(t)=\mathcal{A}_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t} (10.59)
or
\mathcal{A}_{\text {Mo-99 }}(t)=\mathcal{A}_{\text {Mo-99 }}(0) e^{-\lambda_{\text {Mo-99 }} t} (10.60)
and
\mathcal{A}_{\mathrm{D}}(t)=\mathcal{A}_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[e^{-\lambda_{\mathrm{P}} t}-e^{-\lambda_{\mathrm{D}} t}\right] (10.61)
or
\mathcal{A}_{\mathrm{Tc}-99 \mathrm{~m}}(t)=\mathcal{A}_{\mathrm{Mo}}(0) \frac{\lambda_{\mathrm{Tc}-99 \mathrm{~m}}}{\lambda_{\mathrm{Tc}-99 \mathrm{~m}}-\lambda_{\mathrm{Mo}-99}}\left[e^{-\lambda_{\mathrm{Mo}-99} t}-e^{-\lambda_{\mathrm{Tc}-99 \mathrm{~m}} t}\right], (10.62)
where
\mathcal{A}_P(0) is the activity of the parent P at time t = 0.
λ_P is the decay constant for the parent P radionuclide.
λ_D is the decay constant for the daughter D radionuclide.
Decay constants λ for molybdenum-99 and technetium-99m are obtained from their known half-lives t_{1/2} using the standard relationship \lambda=(\ln 2) / t_{1 / 2}. Thus, for Mo99 we have
\lambda_{\mathrm{P}}=\lambda_{\mathrm{Mo}-99}=\frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Mo}-99}}=\frac{\ln 2}{66.0 \mathrm{~h}}=1.05 \times 10^{-2} \mathrm{~h}^{-1} (10.63)
and for Tc-99m
\lambda_{\mathrm{D}}=\lambda_{\mathrm{Tc}-99 \mathrm{~m}}=\frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Tc}-99 \mathrm{~m}}}=\frac{\ln 2}{6.02 \mathrm{~h}}=0.115 \mathrm{~h}^{-1} (10.64)
Inserting (10.63) and (10.64) into (10.59) and (10.61) and using the initial activity \mathcal{A}_P(0) = 10 mCi of the parent (Mo-99) radionuclide we get
\mathcal{A}_{\mathrm{P}}(t)=\mathcal{A}_{\mathrm{Mo}-99}(t)=(10 \mathrm{mCi}) \times e^{-\left(1.05 \times 10^{-2} \mathrm{~h}^{-1}\right) \times t} (10.65)
and
(b) Activity of the parent \mathcal{A}_P(t) falls exponentially with increasing time t; however, activity of the daughter \mathcal{A}_D(t) initially increases from zero to reach a maximum at a specific characteristic time \left(t_{max}\right)_D and then decreases with increasing time t to return to zero at t → ∞.
The characteristic time \left(t_{max}\right)_D is determined by setting \mathrm{d} \mathcal{A}_{\mathrm{D}}(t) / \mathrm{d} t=0 at t=\left(t_{\max }\right) \mathrm{D}. Differentiating (10.61) results in
\frac{\mathrm{d} \mathcal{A}_{\mathrm{D}}(t)}{\mathrm{d} t}=\mathcal{A }_{P}(0) \frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[\lambda_{\mathrm{D}} e^{-\lambda_{\mathrm{D}} t}-\lambda_{\mathrm{P}} e^{-\lambda_{\mathrm{p}} t}\right] (10.67)
and setting d\mathcal{A}_D(t)/dt = 0\ at\ t = \left(t_{max}\right)_D we get
\lambda_{\mathrm{D}} e^{-\lambda_{\mathrm{D}}\left(t_{\max }\right)_{\mathrm{D}}}=\lambda_{\mathrm{P}} e^{-\lambda_{\mathrm{P}}\left(t_{\max }\right)_{\mathrm{D}}} (10.68)
Solving (10.68) for \left(t_{max}\right)_D finally yields the following result for characteristic time \left(t_{max}\right)_D.
\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}} . (10.69)
For the Mo-99 → Tc-99m → Tc-99 decay series the characteristic time \left(t_{max}\right)_D at which \mathcal{A}_{Tc-99m}(t) attains its maximum value is calculated as follows
\left(t_{\max }\right)_{\mathrm{D}}=\left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}}=\frac{\ln \frac{1.05 \times 10^{-2}}{0.115}}{\left(1.05 \times 10^{-2} \mathrm{~h}^{-1}-0.115 \mathrm{~h}^{-1}\right)}=22.88 \mathrm{~h} \approx 23 \mathrm{~h} . (10.70)
(c) Maximum activity \mathcal{A}_{\mathrm{D}}\left[\left(t_{\max }\right)_{\mathrm{D}}\right] of the daughter Tc-99m at \left(t_{\max }\right)_{\mathrm{D}} is obtained by inserting into (10.66) the characteristic time t=\left(t_{\max }\right)_{\mathrm{D}} \approx 22.88 \mathrm{hr} that was calculated in (10.70) to get
(d) Daughter activity \mathcal{A}_D(t) in (10.61) can be expressed as a function of parent activity \mathcal{A}_P(t) as follows
\mathcal{A}_{\mathrm{D}}(t)=\mathcal{A}_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t} \frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[1-e^{-\left(\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}\right) t}\right]=\mathcal{A}_{\mathrm{P}}(t) \frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[1-e^{-\left(\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}\right) t}\right], (10.72)
and at characteristic time \left(t_{max}\right)_D (10.72) becomes
\mathcal{A}_{\mathrm{D}}\left[\left(t_{\max }\right)_{\mathrm{D}}\right]=\mathcal{A}_{\mathrm{P}}\left[\left(t_{\max }\right)_{\mathrm{D}}\right] \frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[1-e^{-\left(\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}\right)\left(t_{\max }\right)_{\mathrm{D}}}\right] (10.73)
Inserting (10.69) for \left(t_{max}\right)_D into (10.73) gives the following expression for \mathcal{A}_{\mathrm{D}}\left[t=\left(t_{\max }\right)_{\mathrm{D}}\right].
showing explicitly that at t = \left(t_{max}\right)_D the parent and daughter activities are equal.
We now determine the maximum activity of the daughter Tc-99m in \mathcal{A}_{\mathrm{D}}\left[\left(t_{\max }\right)_\mathrm{D}\right] using (10.65) with t = \left(t_{max}\right)_D ≈ 22.88 hr
\mathcal{A}_{\mathrm{D}}\left[\left(t_{\max }\right)_{\mathrm{D}}\right]=\mathcal{A}_{\mathrm{P}}\left[\left(t_{\max }\right)_{\mathrm{D}}\right]=(10 \mathrm{mCi}) \times e^{-1.05 \times 10^{-2} \times 22.88}=7.86 \mathrm{mCi} (10.75)
and obtain the same result as we did with (10.66) for the daughter activity \mathcal{A}_{\mathrm{D}}\left[\left(t_{\max }\right)_{\mathrm{D}}\right] \text {. Thus, at } t=\left(t_{\max }\right)_{\mathrm{D}} activities of the parent and the daughter are equal in general and, in our case with 10 mCi parent at t = 0, the activity of parent and daughter are 7.86 mCi and \left(t_{max}\right)_D ≈ 23 h.
(e) The parent (Mo-99) and daughter (Tc-99m) activities \mathcal{A}_P(t)\ and\ \mathcal{A}_D(t), respectively, are shown in Fig. 10.3 plotted against time t using (10.65) and (10.66), respectively. A sketch of the two activity curves can be drawn based on a few important features or anchor points shown on the curves. The following features of P → D → G radioactive decay series should be considered:
(1) Parent activity \mathcal{A}_p(t)=\mathcal{A}_{\mathrm{Mo}-99}(t) follows exponential decay starting at \mathcal{A}_{\mathrm{Mo}-99}(0)=10 \mathrm{mCi} (see Point 1 in Fig. 10.3).
(2) Since \left(t_{1 / 2}\right)_{\mathrm{Mo}}=66 \mathrm{~h} \text {, we know that } \mathcal{A}_{\mathrm{Mo}-99}(66 \mathrm{~h})=5 \mathrm{mCi} (see Point 4 in Fig. 10.3), \mathcal{A}_{\mathrm{Mo}-99}(132 \mathrm{~h})=2.5 \mathrm{mCi}, etc.
(3) Daughter activity \mathcal{A}_{\mathrm{D}}(t)=\mathcal{A}_{\mathrm{Tc}-99 \mathrm{~m}}(t) \text { is zero at } t=0 (initial condition: see Point 2 in Fig. 10.3). With increase in time, \mathcal{A}_{\mathrm{Tc}-99 \mathrm{~m}}(t) first increases, reaches a peak of 7.86 mCi (see Point 3 in Fig. 10.3) as determined in (c), at a characteristic time \left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}} \approx 23 \mathrm{~h} as determined in (b).
