Decay Correction

Radioactive decay is first-order with decay constant λ = ln(2) / t½. The activity at time t is:

• A(t) = A₀ · e^(−λ · Δt) where Δt = t − t₀ (positive when t is after t₀, negative when t is before t₀).
• Equivalent forms: A(t) = A₀ · 2^(−Δt / t½) — convenient when Δt is naturally expressed as a multiple of half-lives.
• Decay-correction factor (DCF) = e^(−λ · Δt) = 2^(−Δt / t½). Multiply the reference activity by the DCF to get the activity at t.

Every unit-dose label carries a calibration time (the timestamp at which the labelled activity is stated to be true) and an activity at that time. The technologist administering the dose computes the activity at the actual administration time. The calibration time is the single anchor that makes every downstream computation reconcilable to the same number.

• Calibration time before administration → DCF < 1 → activity at administration is lower than the label. Draw a larger volume.
• Calibration time after administration → DCF > 1 → activity at administration is higher than the label. Draw a smaller volume.
• Calibration time = administration time → DCF = 1 → draw the labelled volume.

For typical morning FDG batches, calibration time is set to the first scan slot of the day. Doses administered earlier (rare) or later (the entire afternoon) are decay-corrected from that anchor.

F-18, t½ = 109.77 min. Unit-dose label reads 370 MBq @ 09:30. Patient injection scheduled for 11:15.

• Δt = 11:15 − 09:30 = 1:45 = 105 min.
• DCF = 2^(−105 / 109.77) = 2^(−0.9566) = 0.5159.
• A(11:15) = 370 × 0.5159 = 190.9 MBq.
• If the prescribed dose is 350 MBq, draw a larger volume from a higher-concentration vial — or accept the smaller dose if within clinical tolerance.

The same maths runs in reverse when calibration time is in the future: a unit dose drawn at 07:30 carrying a 09:30 calibration time of 370 MBq currently contains 370 / 0.5159 = 717 MBq.

End-of-bombardment (EOB) activity is decay-corrected to end-of-synthesis (EOS), to dispense calibration time, to unit-dose calibration time, to administration time. The reconciliation chain is the audit trail:

• EOB activity (Bq, measured on the synthesis hot-cell ion chamber).
• EOS activity (after radiochemical synthesis losses + decay during synthesis).
• Bulk vial activity at dispense calibration time (after QC sampling losses).
• Sum of unit-dose activities at unit-dose calibration time (after dispense pipetting losses).
• Patient-administered activity (after residual in syringe + line losses).

All five steps reconcile decay-corrected back to the same EOS reference. A reconciliation gap > a few percent is a deviation under 21 CFR 212.71.

Parent–daughter generator systems (Mo-99 / Tc-99m, Ge-68 / Ga-68, Sr-82 / Rb-82) do not follow simple A = A₀·e^(−λt). The daughter activity follows the Bateman equation, reaching transient equilibrium when the parent t½ is long compared to the daughter t½. Practical implications:

• Tc-99m elution yield depends on time since last elution, parent activity at calibration, and elution efficiency.
• Mo-99 breakthrough must be measured at each elution and compared against the USP limit (≤ 0.15 µCi Mo-99 / mCi Tc-99m at administration).
• Ga-68 generator elution recovers the daughter that built up since the last elution (typically every 4 hours).

The MES decay model must distinguish simple-decay isotopes from generator-system daughters.

USP <821> / <825> and EANM procedure guidelines accept ±5 % (some texts ±10 %) on activity assay against the labelled value, after decay correction. National dispense rules often impose tighter tolerances for therapy doses (e.g. Lu-177 PSMA at ±5 %) than diagnostic doses (FDG at ±10 % is typical clinical practice). Tolerance limits should be documented in the dispense SOP and enforced in the MES at calibration time.

• Decay correction using λ = 1 / t½ instead of λ = ln(2) / t½ — gives an activity ~1.44× too high.
• Mixed units (minutes and hours) in the same calculation — silent error.
• Calibration time recorded only on the label, not the batch record — auditor cannot reconstruct the dose.
• Time-zone confusion across multi-site networks (synthesis site UTC, dispense site local time, hospital local time, daylight-saving switchover days).
• Daughter activity calculated using parent half-life — Bateman ignored.
• Tolerance check done against pre-decay (labelled) activity, not against decay-corrected activity at administration time.
• Operator overrides the system-computed activity without two-person e-signature — Part 11 violation.
• Reconciliation done only EOS → administration, skipping intermediate steps — losses cannot be attributed.