Question 5 of 7: Internal Dosimetry of Iodine-131 in the Thyroid

Question 5: Internal Dosimetry of Iodine-131 in the Thyroid (18 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Administered activity $A_0=100$ MBq; physical half-life $T_{\text{phys}}=8$ d; biological half-life $T_{\text{bio}}=2$ d; thyroid uptake $f=60\%$, immediate; mean beta energy 192 keV, mean gamma energy 370 keV; infinite-medium dose-rate constants $S_\beta=27.72\times10^{-15}$ and $S_\gamma=48.55\times10^{-15}$ kg·Sv/(Bq·s); thyroid mass $m=16$ g; per-gram absorbed-fraction table above.

Find. (a) the rationale for iodine as the therapeutic radionuclide; (b) time to reach $1/8$ activity; (c) cumulated activity in the thyroid; (d) the total absorbed fraction of 370 keV gammas in the 16 g thyroid; (e)–(f) cumulative gamma and beta absorbed dose to the thyroid; (g) which emission is therapeutically dominant.

Approach. Combine physical and biological clearance into an effective half-life, use it to get both the $1/8$-activity time and the time-integrated (cumulated) activity in the thyroid, interpolate the absorbed-fraction table to 370 keV and scale it by thyroid mass, then apply the given infinite-medium dose-rate constants — used directly for beta (whose short range makes the thyroid effectively "infinite" for local absorption) and scaled down by the finite absorbed fraction for gamma (which is not locally absorbed in a 16 g organ).

1. Part (a) — why iodine. The thyroid gland actively and selectively concentrates iodine from the bloodstream because iodine is the essential raw material for synthesizing thyroid hormones (T3, T4). Radioactive iodine administered systemically is therefore preferentially taken up and concentrated in thyroid tissue by the gland's own normal physiology, delivering a targeted internal radiation dose to (over)active thyroid tissue with comparatively little uptake elsewhere in the body.
2. Part (b) — time to $1/8$ activity. Physical decay and biological clearance act in parallel, so their rate constants add, giving an effective half-life $$\frac{1}{T_{\text{eff}}} = \frac{1}{T_{\text{phys}}}+\frac{1}{T_{\text{bio}}} = \frac{1}{8}+\frac{1}{2} = \frac{5}{8}\text{ d}^{-1} \quad\Rightarrow\quad T_{\text{eff}} = 1.6\text{ d}$$ Since $1/8=(1/2)^3$, reaching one-eighth of the initial activity takes exactly three effective half-lives: $$\boxed{t_{1/8} = 3\,T_{\text{eff}} = 3(1.6) = 4.8\text{ days}}$$
3. Part (c) — cumulated activity. With immediate 60% thyroid uptake, the thyroid's initial activity is $A_{\text{thy},0}=0.60\times100=60$ MBq $=6.0\times10^7$ Bq. Clearance from the thyroid follows the same effective half-life found in (b), so the time-integral of activity (the "cumulated activity," $\tilde A=\int_0^\infty A(t)\,dt$) is $$\tilde A = \frac{A_{\text{thy},0}}{\lambda_{\text{eff}}} = A_{\text{thy},0}\,\frac{T_{\text{eff}}}{\ln2}, \qquad T_{\text{eff}}=1.6\text{ d} = 1.6\times86{,}400\text{ s} = 1.3824\times10^5\text{ s}$$ $$\boxed{\tilde A = 6.0\times10^7 \times \frac{1.3824\times10^5}{0.6931} = 1.197\times10^{13}\text{ Bq}\!\cdot\!\text{s}}$$
4. Part (d) — absorbed fraction at 370 keV. The table gives absorbed fraction per gram at 100/200/500 keV; interpolating linearly between the 200 keV and 500 keV points for the given 370 keV mean gamma energy: $$\phi(370\text{ keV}) = 1.55\times10^{-3} + \frac{370-200}{500-200}\big(1.66\times10^{-3}-1.55\times10^{-3}\big) = 1.612\times10^{-3}\text{ g}^{-1}$$ Scaling by the 16 g thyroid mass gives the total (dimensionless) absorbed fraction of the 370 keV gammas actually stopped in the thyroid itself: $$\boxed{\text{AF}_\gamma = 1.612\times10^{-3}\times16 = 0.0258\ \ (2.6\%)}$$ i.e. roughly 97.4% of the thyroid's own gamma emissions escape the 16 g gland without depositing energy in it.
5. Part (e) — cumulative gamma dose. The given $S_\gamma$ is defined for an infinite medium (absorbed fraction $=1$ by construction); for the real, small thyroid the dose must be scaled down by the actual absorbed fraction found in (d). Using the cumulated activity concentration $\tilde A/m$ ($m=16$ g $=0.016$ kg): $$\frac{\tilde A}{m} = \frac{1.197\times10^{13}}{0.016} = 7.479\times10^{14}\text{ Bq}\!\cdot\!\text{s/kg}$$ $$D_{\gamma,\infty} = S_\gamma\,\frac{\tilde A}{m} = 48.55\times10^{-15}\times7.479\times10^{14} = 36.31\text{ Sv (infinite-medium value)}$$ $$\boxed{D_\gamma = D_{\gamma,\infty}\times\text{AF}_\gamma = 36.31\times0.0258 = 0.937\text{ Sv}}$$
6. Part (f) — cumulative beta dose. Beta particles from ${}^{131}$I have a range of only a few millimetres in tissue — far smaller than the thyroid's own dimensions (a compact 16 g gland spans several centimetres) — so essentially all beta energy is absorbed locally and the infinite-medium constant applies directly, with no absorbed-fraction correction: $$\boxed{D_\beta = S_\beta\,\frac{\tilde A}{m} = 27.72\times10^{-15}\times7.479\times10^{14} = 20.73\text{ Sv}}$$
7. Part (g) — beta vs. gamma for therapy. Beta delivers roughly 22 times the local thyroid dose that gamma does (20.7 Sv vs. 0.94 Sv) precisely because its short range confines essentially all its energy to the gland itself, while gamma's much longer range lets 97% of its energy escape and irradiate the rest of the body instead. For treating Graves' disease the goal is to destroy overactive thyroid tissue while minimizing dose to everything else, so the therapeutically effective (and dose-efficient) emission is beta. Gamma still has clinical value — it is exactly what makes the uptake and biodistribution externally imageable/measurable — but from a pure dose-delivery standpoint it is mostly "wasted" dose to the rest of the body rather than useful therapeutic dose to the target.