03 Nuclear
Cumulated activity
Time-integral of activity to infinity: Ã = A₀ / λ_e.
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Simulation
Cumulated activity — Change the numbers; the scene follows.
Where it works
PET/CT
Patient / uptake
In the patient on the PET couch — activity concentration, SUV, and internal dose.
Open this machineFormula
Variables
Results
Ã
Cumulated activity
1,154.156MBq·h
Ã
Cumulated activity
4.155e+6MBq·s
Explanation
What it means
Cumulated activity à is the time integral of A(t). For a single exponential to infinity, à = A₀/λ_e = 1.443 A₀ T_e. Multiply by the S-value to get organ dose (MIRD). This is a working relation in Nuclear medicine.
Where it is used
Clinically it sits on the PET/CT — Patient / uptake. In the patient on the PET couch — activity concentration, SUV, and internal dose. 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.
PET/CT · Open this machineHow to use it
Use T_e (effective), not T_physical, when biology clears the tracer. Output is in MBq·h and MBq·s to match either S-value convention. Change one input and watch the curve and the simulation follow.
Symbols
- A₀Initial activity200 MBq
- T_eEffective half-life4 h
Worked example
A typical case from the default values: A₀ = 200 MBq (Initial activity); T_e = 4 h (Effective half-life). Substituting into the relation gives à = 1,154.156 MBq·h; à = 4.155e+6 MBq·s. These are teaching numbers — align them with your machine.
Typical values give
- Ã = 1,154.156MBq·h
- Ã = 4.155e+6MBq·s
Where it comes from
The displayed formula is the working relation. Time-integral of activity to infinity: Ã = A₀ / λ_e. Usual reference: MIRD / Cherry. Derive it in the specialty lesson, then return here to pin the numbers.
Reference: MIRD / Cherry
Assumptions & limits
Integration from 0 to ∞ of one exponential. Imaging-derived time-activity curves with several phases need a sum of Ã_i.
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. Integration from 0 to ∞ of one exponential. Imaging-derived time-activity curves with several phases need a sum of Ã_i.
Keep this
Write the assay time next to every activity. Decay does the rest. Integration from 0 to ∞ of one exponential. Imaging-derived time-activity curves with several phases need a sum of Ã_i.
In this specialty