Question : exponential growth

Hi

The equation for exponential growth of a tumour is

V = Voe^(kt)
Where V is volume of tumour, Vo = initial volume, k is the costant and t is time

If a tumour is fatal when its volume is 500cm3 and the size of someone's tumour is Vd (for volume at diagnosis) how do you predict how long they have to live presuming k is known for lung cancer tumours.

My book says ts = t2 * ln(500/Vd) /  ln(2)
where ts = survival time and t2 is the doubling time which = Ln(2)/k

Please could you show me where they get this from? A derivation without t2 would be nice as I can substitute in t2 at the end.

Thanks

Answer : exponential growth

Well if the initial volume Vo is Vd, and you want to find the time ts when V=500, then you have the equation:

   500 = Vd * e^(k*ts)

So divide both sides by Vd to get:

   500/Vd = e^(k*ts)

Take the natural log of both sides:

   ln(500/Vd) = k*ts

Divide out the k:

  1/k * ln(500/Vd) = ts

which is what you want.