2.2 Decay Calculations and Counting Statistics

Decay Calculations and Counting Statistics

Quick Answer: Use A = A₀ e^(−λt) with λ = 0.693/T½. For Tc-99m (T½ ≈ 6.02 h) activity halves every ~6 h; for F-18 (T½ ≈ 110 min) it halves every ~1.8 h. Counting error is approximately 100/√N percent — more counts (higher activity or longer time) improve precision.

The Decay Law

Radioactive decay is a first-order process. If A₀ is activity at a reference (calibration) time t = 0, activity at later time t is:

A = A₀ e^(−λt)

where the decay constant λ is related to half-life by:

λ = ln(2) / T½ ≈ 0.693 / T½

Units of λ are inverse time (h⁻¹, min⁻¹). Keep t and T½ in the same units. You may also write remaining fraction as (1/2)^(t/T½) — often faster for mental half-life multiples:

• t = 1 × T½ → 50% remains
• t = 2 × T½ → 25% remains
• t = 3 × T½ → 12.5% remains
• t = 0.5 × T½ → √0.5 ≈ 70.7% remains

Pre-calibration (future calibration time): if the dose is calibrated for a later time, activity now is higher: A_now = A_cal / e^(−λt) = A_cal × e^(+λt), where t is time until calibration.

Worked Example 1 — Tc-99m Dose Decay

A Tc-99m MDP kit is calibrated as 25.0 mCi at 08:00. The patient is injected at 11:00. T½ = 6.02 h.

1. Elapsed time t = 3.0 h
2. λ = 0.693 / 6.02 h ≈ 0.1151 h⁻¹
3. λt = 0.1151 × 3.0 ≈ 0.345
4. e^(−λt) ≈ e^(−0.345) ≈ 0.708
5. A = 25.0 × 0.708 ≈ 17.7 mCi at 11:00

Sanity check: 3 h is half of one half-life, so remaining fraction should be near √0.5 ≈ 0.707 — matches.

Another Tc-99m check: at 08:00 next day (24 h later), t/T½ ≈ 24/6.02 ≈ 4.0 half-lives → remaining ≈ (1/2)⁴ = 6.25% → 25 × 0.0625 ≈ 1.6 mCi. Overnight residual in a syringe shield is non-zero; treat waste and surveys accordingly.

Worked Example 2 — Multi-Step F-18 Decay

An FDG unit dose is 10.0 mCi at 10:00 (calibration). T½ ≈ 110 min.

Part A — Activity at 11:50 (110 min later):
t = 110 min = 1 half-life → A = 10.0 × 0.5 = 5.0 mCi.

Part B — Activity at 12:45 (165 min after calibration):
t = 165 min; t/T½ = 165/110 = 1.5
Remaining = (1/2)^1.5 = 0.5 × √0.5 ≈ 0.5 × 0.707 = 0.354
A ≈ 10.0 × 0.354 = 3.54 mCi.

Part C — How early must you assay if you need ≥8 mCi at injection and you will inject at 10:40?
Required at 10:40: 8 mCi; t from calibration to injection = 40 min if calibrated at 10:00.
λ = 0.693/110 min⁻¹ ≈ 0.0063 min⁻¹; e^(−λ·40) ≈ e^(−0.252) ≈ 0.777
A₀ needed at 10:00 = 8 / 0.777 ≈ 10.3 mCi. If the delivered dose is only 10.0 mCi at 10:00, at 10:40 you have ~7.8 mCi — slightly short; reschedule or request a higher calibrated amount.

Remaining Activity Tables (Quick Reference)

Tc-99m (T½ = 6.0 h used for rounded table)

F-18 (T½ = 110 min)

Exam tip: when times are exact multiples of T½, prefer the half-life power method over calculator work — faster and less error-prone under timed conditions.

Counting Statistics (Poisson)

Nuclear decay and detector counts are random. For a large number of independent decays, counts follow a Poisson distribution. For mean count N, the standard deviation σ ≈ √N, and the percent error (coefficient of variation × 100) is:

% error ≈ 100 / √N

Effect of count time: If the true count rate is constant (neglecting decay during the count and dead time), counts N = rate × time. Doubling count time doubles N and multiplies % error by 1/√2 ≈ 0.71 (about a 29% relative reduction in percent error). Quadrupling counts halves percent error.

Clinical implications:

• Well-counter assays of wipe tests or blood samples: choose count time so expected net counts give acceptable % error (often aiming for a few percent or better for quantitative work).
• Imaging: longer acquisition or more administered activity increases counts and improves signal-to-noise, balanced against patient dose, motion, and throughput.
• Background: net counts = gross − background; low-level contamination surveys need adequate time or you risk false negatives from large % error.
• Decay during counting: for very short T½ (Rb-82) or long counts, apply decay correction to the count interval — beyond intro CNMT math but know the concept.

Putting It Together

A technologist who can (1) convert mCi ↔ MBq, (2) decay-correct Tc-99m and F-18 across clinic schedules, and (3) estimate whether a 1-minute wipe count with 100 net counts (~10% error) is good enough for a release decision is operating at Domain I competence. Practice both formula and half-life-table paths until either is automatic.