# Medicine half life

A short challenge concerning the decay of medicines in the body.

## Problem

A certain medicine is broken down by the body so that half of the remaining dose in the system is broken down every 12 hours.

If I regularly take 400mg of medicine in the morning and 400mg of medicine in the evening what is the long term effective level in my system after I take each tablet?

Extension: What will be the long term stable level after I take each tablet if the body breaks down half of the remaining dose every 6 hours?

If I regularly take 400mg of medicine in the morning and 400mg of medicine in the evening what is the long term effective level in my system after I take each tablet?

Extension: What will be the long term stable level after I take each tablet if the body breaks down half of the remaining dose every 6 hours?

Did you know ... ?

The 'half-life' of medicines is an important concept in medicine, particularly in the area of anti-depressants which take some time to reach active levels in the body.

The 'half-life' of medicines is an important concept in medicine, particularly in the area of anti-depressants which take some time to reach active levels in the body.

## Student Solutions

As the question indicates a long-term level I might imagine having taken the tablets for several days and will assume that the tablets are taken in 12-hourly intervals. When I take a tablet, the amount remaining of the previous tablet will have halved, the tablet before that quartered an so on.

If I think carefully and clearly about this then I can write down an expression for the total mass $M(n)$ remaining in the system upon taking the $n$th tablet:

$$

M(n) = 400\left(1 +\frac{1}{2}+\frac{1}{4}+\dots +\frac{1}{2^{n-1}}\right)mg

$$

(With such an expression I should check that the limits are correct: My formula gives$M(1) = 400(1)$ and $M(2) = 400( 1+\frac{1}{2^1})$, which are correct).

$M(n)$ is a geometric series, which sums to

$$

M(n) = 400\left(2-\frac{1}{2^{n-1}}\right)

$$

The long term level is found by taking the large $n$ limit, giving a level of $800$mg in the system, as one might intuitively guess.

Extension:

If the body breaks down half of the dose every $6$ hours then the levels of the previous tablets will have quartered each time. Thus the relevent geometric series is

$$

M(n) = 400\left(1 +\frac{1}{4}+\frac{1}{16}+\dots +\frac{1}{4^{n-1}}\right)

$$

The sum of this series is

$$

M(n) = 400\left(\frac{1-0.25^n}{1-0.25}\right)

$$

The long term level will be $533\frac{1}{3}$mg