- Physical, biological and effective half-life

## Physical, biological and effective half-life

**This technical mathematics article is written by Dr Chris Robbins of Grallator Limited**

Radioactive decay is based on the assumption that it is a random Poisson process and that over a given time interval there is a fixed probability that any given nuclide will decay. To obtain the total number from a collection that will decay we just multiply up by the population number. This gives a differential equation for the rate of change of population number due to radioactive decay as

where l is the removal rate per unit time, often described as the decay constant, and N is the population (N large). A negative sign is used as the decay reduces the population.

The solution to this equation is given by simple re-arrangement and direct integration with boundary conditions N(0) = No:

At the last stage we take exponentials of both sides to give

i.e. the standard exponential decay equation.

We now seek a value for λ, which is usually found from the last expression in (2) by timing how long a reduction in inventory takes so that the ratio N/No and t are known, from which

A useful measurement point is the time taken for half the initial amount of material to decay, better known as the half-life. At this time, t1/2, N = ½ No, so that (4) gives l in terms of the half life as:

The above relates to removal by physical decay. In biological systems there can be removal by excretion. As with decay there is a fixed probability that any one particle will be removed per unit time and so (1) can also be used to describe biological removal. When calculating a biological half-life the time taken for the body to remove half the initial material is considered and a biological removal rate is constructed:

where the b subscripts are used to differentiate from physical decay.

We can now ask “what happens when both physical decay and biological removal occur?” To consider this, we will re-visit (1) and write removal terms from both. Note they are independent removal mechanisms so the total probability of removal per unit time is obtained by summing the two individual probabilities to give

which we can write as

where λ_{eff} is the effective removal rate, given by λ_{eff}= λ+λ_{b}. Rearranging (5) we can calculate the effective half-life from the effective removal rate as

Using the relation λ_{eff}= λ+λ_{b} and (5) and (6) we can write the effective half life in terms of the physical and biological half life values as

Looking at the example of Po-210 in the body with t1/2 = 138 days and t1/2,b = 40 days. The effective half-life in the body is given by

This article was written by a good friend, and mathematics and physics wizard, Chris Collins. His website is well worth a visit if you need assistance in animation, software design, physical modeling or educational resource.