Question 7 of 7: Radioiodine ( 131 I) Thyroid Dosimetry for Graves' Disease

Question 7: Radioiodine (¹³¹I) Thyroid Dosimetry for Graves' Disease (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; $T_p=8$ d, $T_b=2$ d; thyroid uptake $60\%$, immediate; mean $\bar E_\beta=192$ keV, $\bar E_\gamma=370$ keV per decay; infinite-medium dose-rate constants $k_\beta=27.72\times10^{-15}$ Sv·kg/(Bq·s), $k_\gamma=48.55\times10^{-15}$ Sv·kg/(Bq·s); thyroid mass $m=16$ g; gamma absorbed-fraction table above.

Find. (a) rationale for iodine; (b) time to $A_0/8$; (c) cumulative (time-integrated) thyroid activity; (d) gamma absorbed fraction at 370 keV; (e) cumulative gamma dose; (f) cumulative beta dose; (g) which radiation best suits treatment.

Approach. Combine physical and biological clearance into an effective half-life, use it to answer the elapsed-time question (b) and to time-integrate the thyroid's activity concentration (c). Interpolate the absorbed-fraction table to the actual 370 keV gamma energy (d); since beta particles have a range of only fractions of a millimetre they are essentially fully absorbed locally in a 16 g organ (absorbed fraction $\approx1$), while penetrating gammas mostly escape, so only the gamma dose needs the absorbed-fraction correction (e)–(f). Compare the two cumulative doses to answer (g).

1. Part (a) — why iodine. The thyroid gland actively and selectively concentrates iodine from the bloodstream as the essential raw material for synthesizing thyroid hormones (T3/T4); radioactive $^{131}$I is chemically identical to stable iodine, so the gland takes it up by the same natural physiological pathway. This gives a highly selective, targeted radiation dose to thyroid tissue itself, with comparatively little uptake or dose to other organs — exactly the selectivity a systemic (whole-body-administered) radiotherapy needs.
2. Part (b) — time to decay to one-eighth. Physical and biological clearance act in parallel, combining into an effective half-life: $$\frac{1}{T_{\text{eff}}} = \frac{1}{T_p}+\frac{1}{T_b} = \frac{1}{8}+\frac{1}{2} = \frac{5}{8}\ \text{d}^{-1}$$ $$T_{\text{eff}} = \frac{8}{5} = 1.6\ \text{days}$$ A decrease to one-eighth is exactly three half-lives ($\left(\tfrac12\right)^3=\tfrac18$), so: $$\boxed{t_{1/8} = 3\,T_{\text{eff}} = 3(1.6) = 4.8\ \text{days}}$$
3. Part (c) — cumulative activity in the thyroid. With immediate uptake, the activity resident in the thyroid at $t=0$ is $A_{0,\text{thy}}=u\,A_0=0.60(100\ \text{MBq}) =60$ MBq, decaying with the effective (physical + biological clearance) rate. The time-integral of an exponentially decaying activity is $A_0\times\tau_{\text{eff}}$, where $\tau_{\text{eff}}=T_{\text{eff}}/\ln2$ is the mean life: $$\tau_{\text{eff}} = \frac{T_{\text{eff}}}{\ln2} = \frac{1.6\times86{,}400\ \text{s}}{0.6931} \approx 1.994\times10^{5}\ \text{s}$$ $$\tilde A = A_{0,\text{thy}}\times\tau_{\text{eff}} = (60\times10^{6}\ \text{Bq})(1.994\times10^{5}\ \text{s})$$ $$\boxed{\tilde A \approx 1.197\times10^{13}\ \text{Bq}\cdot\text{s}}$$
4. Part (d) — gamma absorbed fraction at 370 keV. The table gives the per-gram absorbed fraction at 200 keV and 500 keV, bracketing the actual 370 keV emission; interpolating linearly: $$\phi_{370}(\text{per g}) = 1.55\times10^{-3} + \frac{370-200}{500-200} \left(1.66\times10^{-3}-1.55\times10^{-3}\right) \approx 1.612\times10^{-3}\ \text{g}^{-1}$$ Scaling by the thyroid's 16 g mass gives the total fraction of emitted gamma energy actually absorbed in the whole gland: $$\boxed{\phi_{370,\text{total}} = 1.612\times10^{-3}\times16 \approx 0.0258\ \ (2.58\%)}$$ — i.e. only about 2.6% of the emitted gamma energy is absorbed within the small thyroid; the rest escapes the organ entirely, which is exactly what "absorbed fraction" quantifies for a penetrating, weakly-interacting photon in a small target.
5. Part (e) — cumulative gamma dose to the thyroid. The given dose-rate constant $k_\gamma$ already assumes full local absorption (the "infinitely large" tissue premise), so it must be scaled down by the actual absorbed fraction from part (d) to reflect the finite 16 g thyroid. Using the activity concentration $c_0=A_{0,\text{thy}}/m=60\times10^6/ 0.016=3.75\times10^9$ Bq/kg and the same $\tau_{\text{eff}}$ from part (c): $$D_{\gamma,\infty} = k_\gamma\, c_0\, \tau_{\text{eff}} = (48.55\times10^{-15})(3.75\times10^{9})(1.994\times10^{5}) \approx 36.3\ \text{Sv}$$ $$\boxed{D_\gamma = D_{\gamma,\infty}\times\phi_{370,\text{total}} \approx 36.3\times0.0258 \approx 0.937\ \text{Sv}}$$
6. Part (f) — cumulative beta dose to the thyroid. Beta particles of this energy have a range in tissue of only fractions of a millimetre, far smaller than the thyroid itself, so essentially all emitted beta energy is absorbed locally — no absorbed-fraction correction is needed (absorbed fraction $\approx1$): $$\boxed{D_\beta = k_\beta\, c_0\, \tau_{\text{eff}} = (27.72\times10^{-15})(3.75\times10^{9})(1.994\times10^{5}) \approx 20.7\ \text{Sv}}$$
7. Part (g) — which radiation is best suited to treatment. Beta radiation delivers roughly 22 times the local thyroid dose of gamma radiation for the same administered activity ($20.7$ Sv vs. $0.937$ Sv), precisely because its short range confines essentially all of its energy to the target organ itself, while the penetrating gamma rays mostly escape the small thyroid and are largely "wasted" — delivering unwanted dose to surrounding tissue, other organs, and people nearby, rather than doing therapeutic work. Both findings point the same direction: $$\boxed{\text{Beta radiation is the therapeutically effective component}}$$ — it is what actually ablates the overactive thyroid tissue, consistent with why $^{131}$I is chosen precisely for its beta emission for Graves' disease therapy, with its gamma emission being useful mainly for imaging/dosimetry confirmation rather than treatment itself.