Radioactive Decay, Half-Life, and Medical Dosimetry Reasoning
A hospital uses a radioactive isotope with a half-life of 6.0 hours for a diagnostic scan. A vial is calibrated to contain 800 MBq at 08:00 on Monday. The isotope decays by beta emission. A patient scan requires an activity of 200 MBq at the moment of injection. The hospital must also dispose of residual activity; regulations allow disposal only when activity falls below 25 MBq. The radiopharmacy delivers a fresh vial every morning at 08:00. The isotope's biological half-life in the patient is 3.0 hours, and its physical half-life is 6.0 hours.
1. Determine the latest time on Monday that the vial can be used for a scan requiring 200 MBq, and the earliest time the residual vial can be legally disposed of.
Read the worked answer
Decay law: A = A₀ × (1/2)^(t/T½), with T½ = 6.0 h.
Time to reach 200 MBq from 800 MBq: 200 = 800 × (1/2)^(t/6) (1/2)^(t/6) = 0.25 = (1/2)² So t/6 = 2 → t = 12 h. Latest scan time = 08:00 + 12 h = 20:00 Monday.
Time to reach 25 MBq: 25 = 800 × (1/2)^(t/6) (1/2)^(t/6) = 25/800 = 0.03125 = (1/2)⁵ So t/6 = 5 → t = 30 h. Earliest legal disposal = 08:00 Monday + 30 h = 14:00 Tuesday.
Note: Since a fresh vial arrives at 08:00 Tuesday, the Monday vial could simply be held until 14:00 Tuesday for disposal, or the hospital could wait for the fresh delivery and dispose of the old vial later. The calculation shows the old vial cannot be disposed of on Monday at all (it would still be above 25 MBq at 20:00 Monday - indeed at 20:00 it is exactly 200 MBq, far above 25 MBq).
2. Calculate the effective half-life of the isotope in the patient and explain why this matters for the radiation dose received.
Read the worked answer
Effective half-life: 1/T_eff = 1/T_physical + 1/T_biological 1/T_eff = 1/6.0 + 1/3.0 = 0.1667 + 0.3333 = 0.5000 h⁻¹ T_eff = 1/0.5000 = 2.0 hours.
Why it matters: The radiation dose to the patient depends on how long the activity remains in the body, not just on the physical half-life. Because the body also clears the isotope biologically (T_biological = 3.0 h), the activity in the patient falls with an effective half-life of only 2.0 h - much faster than the 6.0 h physical half-life would suggest.
- A shorter effective half-life means the patient receives a lower total dose than a naive calculation using only the physical half-life would predict.
- It also means the isotope must be injected soon after calibration; by 12 hours after calibration the activity is only 200 MBq, and the dose delivered per MBq injected is the same, but the total residence time is short.
- For radiation protection, the effective half-life governs how long the patient remains a source of exposure to staff and family, and how long biological waste remains radioactive.
Practical implication: The scan should be performed as early as practical after calibration to maximise signal, and the short effective half-life is favourable for patient safety.