Drug Half-Life Calculator

Here we go again with some baffling stuff. I wanted to understand the implications of drug half-lives, in particular for drugs taken daily. The half-life calculators that I found were not useful at all, so I created my own, here:

http://com.hemiola.com/half-life/

This page does not explain the basics of half-lives. There are plenty of other sites that do that.

For drugs with a short half-life (e.g., a few hours), I can see how if taken daily, there is no buildup because the daily residual is negligible. It was intuitively obvious to me that with a long half-life (e.g., a half-day or more), taking the drug daily would cause an overlap and buildup—convergent, but still, you would have more drugs in your system than you take daily, and I wanted to know that number.

I guess no one cares, which is the nature of this blog—finding things that I care a lot about that others don’t seem to care about but maybe they do. One can argue that the sum of the daily drug residuals is meaningless to know because it changes nothing, but that makes as much sense to me as using unlabeled bottles of analgesics. I want to know simply because I want to be informed and not be piddling around in the dark.

The basics

Wikipedia recently instituted a format for its drug entries that includes the drug’s half-life. That makes it easy and convenient to look up the half-life for all the drugs I’ve checked.

There seems to be an assumption that drugs with a long half-life are slower acting. Mathematically, they stabilize in the system at a higher dose than what you take. I find that interesting.

The math

There is the Wikipedia page on biological half-life, but the math there is way beyond me. Here is what was obvious to me:

After x hours with half-life H (in hours) and dose D1, the fractional amount D2 leftover is:

When you take drugs at regular intervals, there might be some nonnegligible amount left over from previous doses. Here is essentially what my calculator is doing, where p is the hours between doses:

The disclaimers

Yes, I realize the real-world implications of drugs and their half-lives is way more complicated than a simple power-of-two equation. Still, I wanted a quick and easy way to compute the oversimplified numbers.