So in the last video, we talked about the half-life and we covered all of the clinically important stuff there.
But what I said was for all those people who are interested in learning a little bit more,
I'll cover some of the math behind the half-life and dig in to what this first order elimination constant really is.
And we said that the half-life was inversely proportional to this first order elimination rate constant.
So the faster I eliminate drugs, the slower the - the shorter the half-life.
And so, I said yo don't have to memorize this equation. You should memorize an equation that is derived from it.
And here was the equation that was derived from it.
So what we want to do is really discuss how we get this.
So what we did is we drew a first order elimination rate graph using a linear plot and we said the half-life was equal to 1 hour.
So we started at 8. We went down to 4 after 1 hour, to 2, to 1, to 0.5 to 0.25.
and what I said to you is that as scientists, we want to know what the concentration is at any point in time
and the equation which would tell us that was that the concentration at a time (T) is equal to your initial concentration x E raised to the negative K x T
where K was your first order elimination rate constant.
So from this equation, we can derive the half-life equations.
So let's do that real quick on a blank sheet of paper.
So as I said the concentration at a time (T) is equal to your initial concentration x E raised to the negative k x T
and so, by definition, the concentration at T 1/2 right at the half life is 1/2 your initial concentration.
So I can just plug this into there and if I do that, I get 1/2 x your initial concentration is equal to the initial concentration
CNOT x e raised to the negative KT and this is not any T, this is T 1/2.
So what I'll see here is that the concentration actually cancels out and that's what I've said in the past
that the first order elimination is a constant proportion of drug is eliminated per time and that is independent of the concentration.
So if I was going to solve for something like this, what do I have to do?
I have to take the ln of both sides.
So I'll take the ln of 1/2 and I'll take the ln of e raised to the negative kt 1/2.
Now, the ln of 1/2 is going to be equal to -0.693 and the ln of e raised to anything is just the anything.
This is going to equal -kt and that t is the half-life.
So, if I solve for the half-life, I get that the half-life, the 1/2, the negatives cancel out
and this is where I get the 0.693 over that elimination rate constant.
So, here is where that equation is coming from.
So now we see where we get this first equation from where the half-life is 0.693 over this first order elimination rate constant.
And how could you solve for this first order of elimination rate constant by looking at this graph?
Well one way is that you can just figure out the half-life but the other way is that the first order elimination rate constant
is really equal to the ln of your initial concentration divided by your concentration at a time (T) over the amount of time that goes by between those 2 points.
So, if I was going to plug this in for any 2 numbers here, I get it would be - let's say I'm looking between 8 and 4.
So it would be the ln of 8 over 4 divided by the amount of time that went by which was 1 hour.
So the k here is the ln of 2 and the time that went by was 1 hour and so, the ln of 2 long and behold is 0.693
and that's over the units of 1 hour.
So the units of k are inverse hours or inverse amount of time that makes sense because it's in the denominator and we're solving for the half-life.
Now the last thing I want to do is show you how we get this equation here.
So we're going to relate the half-life to the clearance and the volume of distribution.
So we started with this equation here.
And what you should know is that this first order elimination rate constant is really determined by the clearance over the volume of distribution.
So what is the clearance?
We're going to dedicate a whole video to this but in short, the clearance is the volume of blood that gets of filtered of drug per unit time.
So this has a unit of liters per hour. It's the amount of blood that is getting filtered of drug in whatever organs that get rid of drugs.
So it includes your kidney, it includes your liver. You know if you're talking about inhaled drugs, it would include your lungs
and in the denominator, we have this volume of distribution which is the volume that the drug appears to be distributed in
and there's a whole video on volume of distribution.
So make sure to take a peek at that.
So, if I know that this is what determines the first order elimination rate constant, I can take this and plug it into there
but before we do that, let's just make sure we understand 2 points here.
First, with clearance. If I increase the clearance, that means more blood is being filtered of drug per unit of time.
That makes sense that it would increase the rate of drug elimination.
The one that doesn't make as much sense is this idea of volume of distribution.
Remember for a drug to be eliminated, it needs to be in the plasma.
So, if I have a low volume of distribution, what that means is that the drug likes to stay in the plasma.
Drug likes to stay in the plasma.
That's good because that means we can get rid of it.
And so, if you have a low volume of distribution, that's also going to increase the rate of drug elimination.
So, let's take this now and plug it in to this equation here and if I do that, I get the half-life is equal to 0.693
divided by the clearance over the volume of distribution and I just multiply by the reciprocal of that and I get this equation that you need to memorize
and that is the half-life is equal to 0.693 x the volume of distribution over the clearance.
And so, what we'll do in a future video is we'll discuss this term clearance and talk about how it relates to drug excretion.
So make sure to subscribe when new videos are posted.
Hope you enjoyed!