03 Nuclear

Time between two activities

Elapsed time from A₀, A, and half-life.

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Simulation

Time between two activities — Change the numbers; the scene follows.

Where it works

Hot lab

Dose calibrator

In the dose calibrator well — assayed activity, decay between two times, Marinelli.

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Formula

t = \frac{T_{1/2}}{\ln 2}\ln\frac{A_0}{A}

Variables

A₀ · Initial activityMBqA · Current activityMBqT½ · Half-lifeh

Results

t
Elapsed time
6.02h
n
Number of half-lives
1

Explanation

t = \frac{T_{1/2}}{\ln 2}\ln\frac{A_0}{A}

What it means

Inverting the decay law gives the time between two assayed activities: t = (T½/ln2) ln(A₀/A). Useful when a labelled syringe was assayed earlier and you need the activity ‘now’ — or to date an unknown sample. This is a working relation in Nuclear medicine.

Where it is used

Clinically it sits on the Hot lab — Dose calibrator. In the dose calibrator well — assayed activity, decay between two times, Marinelli. Nuclear-medicine relations sit in the hot lab, on the camera, and in the voxel: decay, SUV, TOF, and counting statistics. They decide whether an uptake is real or a clock error.

Hot lab · Open this machine

How to use it

If current activity exceeds initial, the warning fires (growth, not decay, or swapped values). n = t/T½ is the number of elapsed half-lives. Change one input and watch the curve and the simulation follow.

Symbols

A₀Initial activity400 MBq
ACurrent activity200 MBq
T½Half-life6.02 h

Worked example

A typical case from the default values: A₀ = 400 MBq (Initial activity); A = 200 MBq (Current activity); T½ = 6.02 h (Half-life). Substituting into the relation gives t = 6.02 h; n = 1. These are teaching numbers — align them with your machine.

Typical values give

t = 6.02h
n = 1

Where it comes from

The displayed formula is the working relation. Elapsed time from A₀, A, and half-life. Usual reference: Cherry. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Cherry

Assumptions & limits

Same nuclide, no daughter in-growth. Assay geometry and dead-time must be consistent between the two measurements.

Pitfalls

Activity is not counts. SUV needs the true injected activity, the residual, and the correct decay time — a clock off by 10 min on ¹⁸F is a several-percent error. Do not compare SUVs across reconstructions. Same nuclide, no daughter in-growth. Assay geometry and dead-time must be consistent between the two measurements.

Keep this

Write the assay time next to every activity. Decay does the rest. Same nuclide, no daughter in-growth. Assay geometry and dead-time must be consistent between the two measurements.

In this specialty

Nuclear medicine