Here we go again with some baffling stuff. I wanted to understand the implications of drug (i.e., medicine or medication) 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 (including an interactive graph), for use on a desktop or laptop, with a keyboard and biggish screen:

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.

## 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 D_{1}, the fractional amount D_{2} 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 are way more complicated than a simple power-of-two equation. Still, I wanted a quick and easy way to compute the oversimplified numbers.

## An example

Aimovig was approved by the FDA on 2018-05-17. It’s taken once per month, and this really got me wondering, because I’m used to thinking about things taken about once per day. Its half-life turns out to be a whopping 28 days (672 hours)! Being that you take it once per half-life, the swing in your system is 70 – 140 mg, even though you take 70 mg per month. It takes about 142 days for one dose to (97%) leave the system, which is a serious commitment!