# Thread: A problem of half lives

1. ## A problem of half lives

Hey guys, I'm stuck on a problem:

The concentration C of a drug in the bloodstream decays exponentially according to the equation

C(t) = (D/V)(e^-kt)

where t is time, D is initial dosage in mg, V is the volume in litres and k is a rate constant.

A drug is said to be active if the concentration in the bloodstream is above some minimum concentration C(min). The duration of response t_duration is the amount of time for which a drug is active in the bloodstream.

I need to prove that the duration of response increases by one half life when the initial dosage is doubled.

Anybody know how I would approach this problem?

2. ## Re: A problem of half lives

If $\displaystyle t_h$ is half-life, then $\displaystyle \frac{D}{V}e^{-kt_h}=\frac{1}{2}\frac{D}{V}$.

If $\displaystyle t_d$ is the duration of response for the initial concentration D / V and $\displaystyle t$ is the increase of the duration of response for the initial concentration 2D / V, then $\displaystyle \frac{2D}{V}e^{-k(t_d+t)}=C_{\text{min}}=\frac{D}{V}e^{-kt_d}$. From here, compare $\displaystyle t$ with $\displaystyle t_h$.

3. ## Re: A problem of half lives

Thanks a lot! I was struggling on this problem for a long time