Kinetics of radioactive decay

Radioactive decay law

• Radioactive atoms decay at a rate proportional to the number of radioactive atoms present.

• N is the number of radioactive atoms, and λ is the decay constant of the radionuclide: the fraction of atoms that decay per unit time.
• The unit of λ is (time)⁻¹, for example s⁻¹ or h⁻¹.
• The disintegration rate (−dN/dt) is called the activity (A) of the radionuclide.
• Solving the equation gives the exponential decay law, where A₀ is the activity at time zero.

Half-life

• The physical half-life (t½) is the time taken for the activity of a radionuclide to fall to half its initial value.
• Each radionuclide has its own unique half-life.

• After n half-lives, the activity left is A₀/2ⁿ. After 10 half-lives, about 0.1 % remains.
• Example: 740 MBq of ⁹⁹ᵐTc (t½ 6 h) falls to 185 MBq after 12 h (two half-lives).

Mean life

• The mean life (τ) is the average lifetime of the atoms in a radioactive sample.

• After one mean life, the activity has fallen to 37 % (1/e) of its starting value.
• Example: for ⁹⁹ᵐTc, τ = 1.44 × 6.01 h ≈ 8.7 h.

Effective half-life

• The physical half-life (Tp) is the time for the activity to fall by half through radioactive decay alone. It is a property of the radionuclide and is the same inside or outside the body.
• In the body, a radiopharmaceutical is also cleared exponentially through urine, faeces, sweat and other routes. The time for half of it to be eliminated is its biological half-life (Tb).
• In a patient, activity is lost by both routes at once.

• The effective half-life (Te) is always shorter than the shorter of Tp and Tb.
• If Tp is very long and Tb short, Te ≈ Tb. If Tb is very long and Tp short, Te ≈ Tp.

Specific activity and concentration

• Specific activity is the activity per unit mass of a radionuclide or radioactive sample, for example MBq/mg.
• For a labelled compound, it is often given per mole, for example MBq/µmol.
• Concentration is the activity per unit volume of a sample, for example MBq/mL.
• A carrier-free sample contains only the radioactive atoms of that element, with no stable atoms of the same element mixed in.
• Its specific activity follows from A = λN. Per gram of the nuclide, N is Avogadro's number divided by the mass number A, and λ is 0.693/t½.

• So two things raise the specific activity: a short half-life (each atom decays sooner, so more decays per second) and a low mass number (lighter atoms, so more atoms in every gram).
• Example: ⁹⁹ᵐTc (t½ 6.0 h, A 99) has a specific activity of about 1.9 × 10¹⁷ Bq/g; ¹³¹I (t½ 8.0 d, A 131) about 4.6 × 10¹⁵ Bq/g, roughly 40 times lower, mainly because its half-life is 32 times longer.
• In practice, generator ⁹⁹ᵐTc is not quite carrier-free: it contains some ⁹⁹Tc from decay, so its real specific activity is lower.

Parent–daughter equilibrium

• When a radioactive parent decays to a radioactive daughter, the daughter's activity builds up and then settles into a fixed relationship with the parent. This is called equilibrium.

Transient equilibrium

• Occurs when the daughter's half-life is shorter than the parent's (λd > λp), with the two half-lives differing by a factor of about 10–50.
• The daughter activity first builds up, reaches a maximum, and then decays with the apparent half-life of the parent.
• At equilibrium, the ratio of daughter to parent activity stays constant.
• If all parent decays feed the daughter, the daughter activity exceeds the parent activity, because the ratio (t½)p / [(t½)p − (t½)d] is greater than 1.
• Example: ⁹⁹Mo (66 h) → ⁹⁹ᵐTc (6 h). ⁹⁹ᵐTc activity reaches its maximum about 23 h after elution.
• Only about 87.6 % of ⁹⁹Mo decays produce ⁹⁹ᵐTc, so in equilibrium the ⁹⁹ᵐTc activity is slightly lower than the ⁹⁹Mo activity, about 96 % of it.
• This is the basis of the daily elution of a ⁹⁹Mo/⁹⁹ᵐTc generator.

Secular equilibrium

• Occurs when the parent's half-life is much longer than the daughter's, by a factor of more than about 100 (λd ≫ λp).
• In secular equilibrium, parent and daughter activities become equal, and both decay with the parent's half-life.
• Equilibrium is reached after about 7 daughter half-lives.
• Examples: ⁶⁸Ge (271 d) → ⁶⁸Ga (68 min); ⁸²Sr (25 d) → ⁸²Rb (75 s).

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