3.10.6.2 - Half-life

3.10.6.2 - Half-life

A tracer can leave a region by biological clearance or stop being radioactive by nuclear decay. Distinguish these processes, combine their rates and calculate how activity changes inside a patient.

Physical half-life and nuclear decay

The physical half-life, TPT_{\mathrm P}, is the time for half the undecayed nuclei in a sample to decay. If no nuclei are added or removed by other processes, activity also halves in that time because A=λPNA=\lambda_{\mathrm P}N.

For the medical model used here, physical half-life is a property of the radionuclide and is unchanged by ordinary temperature, chemical labelling or bodily clearance. The decay of any one nucleus is unpredictable, but a large population follows an exponential law:

N=N0 2−t/TP.N=N_0\,2^{-t/T_{\mathrm P}}. λP=ln⁡2TP.\lambda_{\mathrm P}=\frac{\ln2}{T_{\mathrm P}}.

Technetium-99m has TP≈6.0 hT_{\mathrm P}\approx6.0\,\mathrm{h}. In a closed sample, about one quarter of the initially undecayed technetium-99m nuclei remain after 12 h12\,\mathrm{h}. The decayed nuclei have changed nuclear state; they have not necessarily left the container.

Biological clearance of the substance

The biological half-life, TBT_{\mathrm B}, is the time for biological processes alone to remove half the substance from the specified body region. It is defined without counting radioactive decay. A cleared radioactive atom can still be radioactive elsewhere; clearance from an organ does not mean its nucleus has decayed.

The region must be clear: removal from blood, an organ and the whole body need not occur at the same rate. Chemical form, organ function and the patient's metabolism affect clearance. A radionuclide attached to a different compound can therefore have a different biological half-life while retaining the same physical half-life.

A constant biological half-life assumes that the same fraction is removed in equal time intervals, with no continuing intake. Real uptake and clearance may have several phases. In calculations, use the stated single exponential model and the specified region rather than treating a quoted biological half-life as universal for every patient.

Combining decay and clearance rates

The effective half-life, TET_{\mathrm E}, is the time for activity retained in the specified region to halve when physical decay and biological removal both act. Assume independent exponential processes, constant rates and no further supply.

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