# Finding the best Equation for concentration substance

1. Mar 18, 2013

### jackscholar

1. The problem statement, all variables and given/known data
I need to find the best equation possible for administration of a drug. I have found two types of equations; the first is by starting with a dose (d) and adding that same dose so that it builds up slowly over time. This is shown as the equation y(T)=de^(-kT) where T is the time interval between the dosages, e is the exponential constant and k is a proportionality constant.

Or there is the second option where the first dose is the maximum effective dose followed by smaller doses to 'top up' the concentration to the maximum. The maximum effective dosage is denoted by
Ys=d/1-e^-kT
where Ys is the saturation level
Therefore the equation for this method would be Ys=Yse^-kT+Yd where the small dose is d. Are there any equations more effective than the latter?

2. Mar 18, 2013

### epenguin

You cannot arrive at the best drug dosage by pure mathematics!
You have to have a model of at least some rough assumptions about how the drug works, and what happens to the drug. E.g. is it excreted or metabolised, and do the rates of those processes depend on its concentration in the bloodstream or what, are there any thresholds, in what way is it administered and if e.g. orally do you know anything about how much and and what rate it goes into the bloodstream. Are there any experimental data that cam be incorporated in a model. Are there any risks or side effects that can be incorporated? Then a model or simulation could be attempted. Though common sense could probably take you most of the way.

That said, medicine can profit from, does need, mathematics and has always needed it.

Last edited: Mar 18, 2013
3. Mar 18, 2013

### Ray Vickson

I second the remarks of 'epenguin'.

You can easily do a Google search on 'drug concentration equation' and come up with loads of useful material. For example, see
http://www.mhprofessional.com/downloads/products/0071476288/BauerCh2.pdf (which has numerous formulas at the end, depending on scenarios and model details)
or
http://www.intmath.com/blog/math-of-drugs-and-bodies-pharmacokinetics/4098
(which looks at the effects of the five steps in the body: liberation, absorption, distribution, metabolism and secretion) and obtains/solves a differential equation for concentration vs. time.