If a drug has a half-life of, say, 16 hours, how does the concentration in the body reach zero at some point?
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You already know that the concentration will get down to negligible because of the half-life. Then you must allow for the body’s mechanical elimination systems like urine/feces plus the constant tearing down and rebuilding of cells.
You can think of half life as the time it takes half of a substance to disappear, or you can think of it as the time over which one piece of that substance has a 50% chance of disappearing. In most cases these get you to the same place, since when you have enough of something (say trillions of trillions of molecules or atoms) those 50% chances per peice are going to reliably result in half of the pieces disapearing.
But when you have very few pieces left, like one, the first way of thinking of halflife doesn’t apply since you can’t have half a molecule of a drug. So instead you need to think of it as each molecule having a 50% chance of disapearing over one half life.
So with one left, half the time it will disapear after one half life, three quarters of the time it will be gone after two half lifes etc, which technically means there’s a miniscule chance that it never goes away, but long before that you likely have a concentration that’s too small to have a noticable impact anyway.
in short, it’s functionally zero after a few halflives (after 7 it’ll be under 1% of its initial concentration) but will probably be actually zero after a lot more (log base 2 of n) where n is the number of particles.
EDIT: Log base 2 of n, means what x do you need for 2^x = n. So in this case 2^6.6 = 100 which is how I know that after 7 halflives less than 1% will be left.
Half life is a bulk statistical property, it doesn’t really count when you’re down to just a tiny actual number of molecules.
The stragglers have some percentage chance of running into a metabolizing enzyme or decaying per hour, and eventually they’re all gonna get busted.
Some lucky stragglers may evade capture/degradation for many half lives though, just like you can theoretically flip ten coins and get ten heads.
So the idea of half-life is a bit simplified for biological processes. The way half-life is taught in school is usually with radioactive decay. Radioactive decay follows a first order rate of reaction. What this means is the rate of reaction is directly proportionally to the amount of substance. If you have half as much, it decays half as slowly.
Many other reactions have more complicated kinetic orders. There are second order reactions where reactions go four times faster if you double the reagent. Zero order reactions where the rate is not affected by the amount of something. You can have fractional order reactions that speed up as the concentration of a reagent decreases. A reaction can also have different orders for each reagent or even be catalyzed by some small amount of something to speed it up. Also, there can be a limited amount of certain reagents in the body meaning the order of the reaction can change as something is broken down.
So, for instance say you have a drug that is a zero order reagent. It is broken down at a rate of 1 g/hrs. You take two grams. Half life is 1 hour. After that hour, half life will be 30 minutes. 30 minutes from that, 15 minutes. You get the idea.
Typically, the half-lives are written at expected does. It can not necessarily be extrapolated to tell when it will all be out of the body, but it is a useful tool for healthcare providers.
Lay out 100 M&Ms on a table. Go through them one by one, and for each M&M, flip a coin. If heads, you eat it. If tails, you leave it and move on to the next.
After doing this for all 100 M&Ms, you’ll probably end up with somewhere close to 50 M&Ms left on the table. If our M&M half-life is an hour, you’ll wait an hour, and do that process again. You’ll have about 25, then about 17, then about 8, then about 4.
At this point, the number you have remaining becomes harder to predict. You’re only flipping the coin four times and it becomes far less likely that you’ll get the same number of heads and tails flips. The amount of M&Ms you eat every hour slows down a lot because you have fewer total M&Ms, but you’ll get to all of them eventually.
The basic rule of thumb that gets used is that you’re not considered to have a meaningful amount of drug in your blood after 4-5 half-lives (which will eliminate 95%+ of what’s in your system).
5% of the original concentration of a drug is usually not going to have much of a clinical effect, so in most cases this is a safe estimate.
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