Averages, Probability, and the Indivisible: Why you're morphine free in 1 week (probably)

Math makes everything less fun, right?

This morning I sat through a lecture discussing half-lives of drugs, and the lecturer declared that "If a drug follows first-order kinetics [meaning its concentration is cut in half every few hours] it will never be entirely eliminated from your body.  So, you still have some of the formula you drank as a baby!" Okay, so Aristotle thought of this one too.  But zealots beware: the world does not have an infinite resolution for everything.

The basic principle here is simple: if you have three baseballs, and you divide them into two groups, one group gets two baseballs and the other gets one.  You can't have 1.5 baseballs.  Sure, a half cup of water is a half cup of water, but a half baseball is a useless lump of string and leather, and at that point it has ceased to be a baseball.  Thus, you cannot have half a baseball!  If you say you have 1.5, it really means you have either one or two on average.  You cannot always divide by two, and as numbers get small, the remainder gets to be a significant part of what you have.  No matter how big of a number you start with, if you keep dividing you eventually get to a number that is small enough that it matters if you can cut a baseball in half.  Basically, we're going to have a deathmatch between Achilles and the Tortoise and the Wheat and Chessboard problem, but we're going to let the chessboard have as many squares as it needs.

Let's say you take a 20mg dose of morphine (285.34 g/mol, half life of about 2 hours).  20mg does not represent an infinite number of molecules.  Cue Avagadro's number and a bit of multiplication that I'll omit (if you'd understand it, you probably already know it), and we see that 20mg of morphine is about 4.22 * 10^19 molecules of morphine.  After one half life, that number is cut in half (2.11 * 10^19).  After two, you divide by 4.  After three, divide by 8.  After 10 half lives, 4.12 * 10^16 molecules left.  Still a lot, right?  After ten half lives, you are dividing the original number by 1024.  That's a lot, but not compared to the original 42,200,000,000,000,000,000 molecules.  Well, because we're dividing by powers of two, just one more half life and we're not dividing by 2048, then 4096, etc.  In the case of morphine, at 65 half lives you have 1.1 molecules of morphine still around.  Of course, this is impossible,  we can only have zero, 1, 2, etc.  So, even assuming that each division was a perfect half, we can only have 66 half lives before the morphine is gone entirely.  Morphine has a 2-hour half life, so the bottom line is that after about 132 hours (about 5.5 days, if you're into counting "half days"), your body will not have a single molecule of morphine in it.

Now, of course what really happens is that each half life does not eliminate exactly half of what's in you. Heck, it might be off by trillions of molecules each half life, but when we're working with a trillion-trillion-trillion-etc... molecules, we don't care too much.  But what does that mean for any given molecule of morphine in you?  We might think of a half life as just another way of saying "a length of time during which any given molecule has a 50% chance of being eliminated." For big numbers, that expression behaves like a fraction, because on average 50% end up being eliminated each half life.  However, as things get small, you have to go back to the probability definition.

In conclusion, all of this really illustrates is two things:
1)You can only divide things by two infinitely if it is some quantity that it makes sense to have a half of.
2)Averages and statistics work fine as fractions if you are working with a big population (like saying 25% of people die of cancer), but when you get down to single items, it's more useful to think of them as probabilities (I have a 25% chance that I will die in an accident).

Of course, that 25% could mean that 75% of people have 0% chance of dying from cancer, and the other 25% are genetically doomed to cancer with 100% certainty, but that's a discussion for another day.

[Note: I'd love to give proper attribution for the art above, but I can't find the original source.  Let me know if you know it.]