# Calculations involving effective half-life

The biological half-life can be defined as the time taken for half of the available radioactive material to be removed from the body by natural processes. This assumes no new material is arriving. It is due to the metabolic turnover of the molecule the radioactive tracer atom has been attached to. This will vary from one person to another and from one organ to another, thereby making accurate estimates of biological half-life very difficult to make. It follows that effective half-life also has a larger than desirable error margin also.

Let lB be the fraction of the radioisotope removed per second by biological processes

Let lp be the fraction of the radioisotope removed per second by radioactive decay.

Let lE be the fraction of the radioisotope removed per second by both physical and biological processes.

Then lE = lB + lP

Let TB be the biological half-life

Let Tp be the physical half-life.

Let TE be the effective half-life.

Now

lE = ln2/TE

lP = ln2/TP

and lB = ln2/TB

So it follows that

ln2/TE = ln2/TP + ln2/TB

and

1/TE = 1/TP + 1/TB

(Note the similarity of this equation to the resistances in parallel one. This is because just as current flow has two routes through the potential drop when resistances are in parallel so the radioactive isotopes have 'two routes' out of the body… biological or by physical radioactive decay).

Try this calculation

The human serum, albumin can be labelled with atoms of the radioactive isotope iodine 131. This has a physical half life of 8 days and a biological half-life of 21 days. Find the effective half-life.