Basic Physics of Nuclear Medicine/Print version

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Operation of a 99m-Tc Generator

Suppose we have a sample of ⁹⁹Mo and suppose that at time t = 0 there are N₀ nuclei in our sample and nothing else. The number N(t) of ⁹⁹Mo nuclei decreases with time according to radioactive decay law as discussed in Chapter 3:

$N(t)=N_\mathrm{0}e^{-\lambda_\mathrm{Mo}t}$

where ?_Mo is the decay constant for ⁹⁹Mo.

Thus the number of ⁹⁹Mo nuclei that decay during a small time interval dt is given by

$\mathrm{d}N(t)=-\lambda_\mathrm{Mo} N_\mathrm{0} e^{-\lambda_\mathrm{Mo}t}\mathrm{d}t$

Since ⁹⁹Mo decays into ^99mTc, the same number of ^99mTc nuclei are formed during the time period dt. At a time t', only a fraction dn(t') of these nuclei will still be around since the ^99mTc is also decaying. The time for ^99mTc to decay is given by t' ? t. Plugging this into radioactive the decay law we arrive at:

$\mathrm{d}n(t')=-\mathrm{d}N(t)e^{-\lambda_\mathrm{Tc}\left(t'-t\right)}=\lambda_\mathrm{Mo} N_\mathrm{0}e^{-\lambda_\mathrm{Mo}t}e^{-\lambda_\mathrm{Tc}\left(t'-t\right)}\mathrm{d}t$

Now we sum up the little contributions dn(t'). In other words we integrate over t in order to find the number n(t'), that is the number of all ^99mTc nuclei present at the time t':

$n(t')=\int_0^{t'}-\mathrm{d}N(t')e^{-\lambda_\mathrm{Tc}\left(t'-t\right)}=\lambda_\mathrm{Mo}N_\mathrm{0}e^{-\lambda_\mathrm{Tc}t'} \int_0^{t'}e^{\left( \lambda_\mathrm{Tc}-\lambda_\mathrm{Mo}\right)t} \mathrm{d}t$

Finally solving this intergral we find:

$\begin{matrix} \Rightarrow n(t')&=& \frac{\lambda_\mathrm{Mo}}{\lambda_\mathrm{Tc}-\lambda_\mathrm{Mo}}N_\mathrm{0} e^{-\lambda_\mathrm{Tc}t'}\left(e^{\left( \lambda_\mathrm{Tc}-\lambda_\mathrm{Mo}\right)t'} -1\right) \\ \end{matrix}$

The figure below ilustrates the outcome of this calculation. The horizontal axis represents time (in days), while the vertical one represents the number of nulcei present (in arbitrary units). The green curve illustrates the exponential decay of a sample of pure ^99mTc. The red curve shows the number of ^99mTc nuclei present in a ^99mTc generator that is never eluted. Finally, the blue curve shows the situation for a ^99mTc generator that is eluted every 12 hours.

Comparison of the physical decay of ^99mTc with its activity arising from ⁹⁹Mo decay in a radioisotope generator with and without elution at 12 hour intervals.