Effective Half Life

What is half-life?

An atom with an unstable nucleus undergoes radioactive decay to achieve stability by releasing either alpha, beta, or gamma radiation. A nuclide with an unstable nucleus is referred to as a radionuclide or radioisotope. The timing of the decay cannot be accurately predicted as it is a random process, and so it is impossible to predict when any single unstable nucleus will decay. There are several radioisotopes that exist naturally in nature, but the others are artificially created in nuclear reactors.

However, if we have a large number of atoms, we can predict that decay will occur somewhere within the sample. So, if we can somehow time how long it takes for half of the number of atoms in the sample to decay, we have a measure of its half-life.

Plotting the time it takes for our sample to halve each time, we get a graph similar to the one below:

N is the number of nuclei left in the sample, while T is the number of halves. In the graph, there are a total of 5 half-lives, T_1/2 being the symbol for half-life, and nT being the number of half-lives passed. 2T, therefore, means after two half-lives.

It is also important to keep in mind that the rate of radioactive decay has no relationship with the radionuclide’s physical state, its chemical composition in which the nucleus is bonded, and exterior factors like temperature or pressure. The graph below indicates the amount of a radioactive sample in pink remaining after several half-lives have passed.

Why is half-life important?

The half-life can be used to calculate the time it would take for half of the atoms that have not yet changed to decay. Consider the equation below:

\[N = N_0 (\frac{1}{2})^{\frac{t}{t_{1/2}}}\]

• N = new amount of radioactive substance after decomposition.
• N₀ = initial amount.
• t = time elapsed for decomposition.
• t_1/2 = half-life.

The effective half-life of radiopharmaceuticals in medical uses

The choice of radionuclides for medical imaging and therapy primarily depends on their chemical, biological, and physical properties.

Radionuclides need to be chemically suitable in order for them to bond with the molecules as a pharmaceutical product. From a biological standpoint, the radiopharmaceutical must be acceptable so that it does not hinder the body's everyday functions while building up in the organ to be assessed. Finally, the radionuclide needs to have the correct physical properties, such as the type of radiation, the duration of half-life, and its energy.

Physical half-life

In medicine, the half-life of a radionuclide T_1/2 is referred to as its physical half-life T_p. The half-life values for some commonly used medical radionuclides that can be used are listed in the table below.

Biological half-life

The radionuclide is merged into a chemical compound that results in a radiopharmaceutical. Radiopharmaceuticals are radioisotopes that combine with biological molecules that target specific organs, tissues, or cells within the body. Via the everyday processes of respiration, urination, or defecation, a radiopharmaceutical’s concentration in the body will naturally decrease exponentially with time, which is known as the biological half-life T_B.

Effective-half life formula

The effective half-life is the biological excretion along with the radioactive decay after a radiopharmaceutical enters a body. The effective half-life can be represented as:

\[\frac{1}{T_E} = \frac{1}{T_B} + \frac{1}{T_P}\]

• T_E = Effective Half-life.
• T_B = Biological Half-life.
• T_P = Physical Half-life.

Rearranging the above yields:

\[T_E = \frac{T_P \cdot T_B}{T_P + T_B}\]

When comparing the effective half-life and physical half-life, keep in mind that the effective half-life consists of both the physical half-life and the biological half-life.

Factors affecting the half-life of radionuclides in medical uses

The effective half-life can change depending on the biological half-life as it depends on the person’s health and overall metabolism. With that being said, the effective half-life will always be less than the physical half-life because the nuclide is removed from the body quickly due to the biological processes of respiration, urination, and defecation.

The physical half-life of a radionuclide, however, remains the same and is unaffected.