Posted by Adam on April 10, 2002, at 23:03:12
In reply to Re: WHY is 2^-5 used to compute max # half-lives? » JohnX2, posted by 2sense on April 8, 2002, at 19:56:30
I always think that any discussion of this kind benefits from some understanding of some simple math and physical principles.
You can, if you know the half-life, calculate the amount of drug you are going to have around at any point in the future. You can also determine the half life of something by watching it dissapear over any span of time.
It all can be explained in the law of exponential decay, expressed thusly:
At=Ai * e^kt
(Ai = initial amount, At equals amount at time t, k is the "decay constant")If I don't know what k was, I can find out: Say at the beginning of the experiment I had 50g of stuff in a box, and when I looked 100 min later, I had only 30g of stuff left. Using the formula above...
30g = 50g * e^(k*100 min)
Now solve for k...
ln(30g/50g) / 100min = k = -0.005108
(k is in units min^-1 in this case)Now I can calculate the half life of stuff. Since the half life is the time it takes for half of the stuff to disappear...
0.5 = e^-0.005108t
Solve for t...
ln0.5/-0.005108 = t = 135.7 minutes
This is kind of cool: If I know what k is, I can calculate the half life (t(1/2)) of anything just by using the formula ln0.5 / k = t(1/2)
Conversely, if I know the half life of something, I can always find k:
k = ln0.5 / t(1/2)
And if I know k, I can calculate how much of something there will be at any time in the future if I know how much is around now.
There's a neat physical concept in all of this: The statitistical nature of exponential decay. When you think about it, molecules of a drug aren't like, say, guppies. It doesn't make much sense, under normal circumstances, to talk about the half-life of guppies. Say you had a bowl of ten guppies, all the same age. If you watched the bowl of guppies day after day for months, the number probably wouldn't change much. Then one day there'd be one dead guppy. Maybe the next day, there'd be three. By the end of the week, maybe, they'd all be dead. What happened? They got old and died. Guppies have a "life span". That's a lot different than a "half life". The guppies swim along happily for however long their life span allows, and then, all of the sudden, they die.
Now let's imagine a rather perverse scenerio, to illustrate the concept of half life. Say I have a tank full of 100 guppies. Let's just ignore life span, since the time-frame of the experiment is so much shorter than the life of a guppy...for our puroposes, guppies live forever, undisturbed. Now, into that tank, I throw a blind pirhanna. The voracious pirhanna swims about randomly, and when he encounters a guppy, he eats it.
Now imagine the scene in the tank. The pirhanna begins in an environment dense with guppies. He can't see them, but by random chance, he's bound to bump into one eventually, and he eats them as soon as he does. In the beginning, finding guppies is easy, since there are so many around to bump into. As time goes by, and the guppies become more sparse, the rate that the pirhanna eats them slows, as he is less likely with each guppy he eats to bump into another. Say you counted guppies in the tank each half hour after you introduced the pirhanna. To make it easy, in 30 min. there are 50 guppies (the half life of guppies with the pirhanna around is thus 30 min.) After an hour, there are 25, in two hours there are only six or seven, and so on. You could plot out the disappearance of guppies over time on a graph, and what you would see is an exponential decay curve.
You might notice that the life of the poor guppy in this experiment is determined only by chance. If you watch any individual guppy, it might live only a few minutes, or it might evade the pirhanna for hours if it was lucky. On average, though, you might say in half an hour, any individual guppy has a 50-50 chance of survival. Given those odds, in half an our, you would expect 50% of the guppies to be gone.
In a way, the same goes for molecules. If you took a tab of sertraline, the sertraline doesn't "get old" in your body. Molecules of sertraline float about essentially randomly until something happens. It's theoretically possible that a molecule of sertraline that entered your body could still be there a year from now. It would be exactly the same as it was when you ingested it. The elimination half life of sertraline is about 26 hours. That means that on average, half of the sertraline molecules you ingested will have been excreted, or reacted with something, or been converted by an enzyme, etc. You could also say that any given molecule of sertraline has a 50% chance of being excreted by or chemically altered in your body after 26 hours. It's purely statistical, and exponential decay is just the cumulative result of the chance of something happening once to a single representative entity in a collection of such entities over time.
A funny thing about exponential decay: You never reach zero. When somebody says five or ten half-lives is enough to eliminate something, they just mean for practical purposes. In reality, some portion of that tab of sertraline you took last tuesday could be with you for the rest of your life...even if it's just a molecule. Something only has a chance of disappearing in a given number of half-lives...never a certainty!
(Of course, the mathematical expressions for drug clearance in standard pharmicokinetics can look different even when equivalent to the simple decay law given above, and they becomes a bit more complicated with distribution time, number of compartments, etc. This is the basic idea, though.)
poster:Adam
thread:101843
URL: http://www.dr-bob.org/babble/20020408/msgs/102698.html