So now, we finally get to talk about the half-life or the T 1/2 and this is an incredibly important topic so I’m going to put some stars next to it. So what is the elimination half life? Well, the elimination half is the time it takes to eliminate 50% of that drug from your body or from your plasma and the reason we care about this is two-fold. One, we want to know how long it takes to get rid of all of that drug from our body and two, by knowing how quickly we eliminate that drug, it gives us a sense of how frequently we should be dosing that drug. Now what I want to do is kind of front load this lecture and give you the 3 most important points to remember right off the bat and then we’ll develop each of these points. So, the first point we’ve already talked about and that is the definition of half-life. This is the time it takes to go from some initial concentration CNOT to 1/2 of that initial concentration.
The second point I want you to remember is that it doesn’t matter what your initial dosage is or your initial concentration 95% of that drug will always be eliminated after 4.5 half lives. So, if we’re at any sort of steady state, after 4.5 half-lives, 95% of that drug will be eliminated regardless of the initial concentration. The third point I want you to remember is that the half-life is inversely proportional to the rate of drug metabolism. So, if I metabolize the drug faster, we expect the half life to go down Now, we have a term which represents this rate of drug elimination and this is KE or KEL. It’s the first order elimination rate constant. first order elimination rate constant Now you don’t actually have to memorize this equation but you need to understand what this term is here. So this term represents how quickly we are eliminating the drug and if this increases, we expect the half-life to decrease. Now if you’re unfamiliar with this term first order elimination, I’d recommend going back to the last video and watching what that means.
In a nutshell, it says that the first order elimination which is happening for most drugs at most dosages, the proportion of drug metabolized over time is constant. So while you don’t have to memorize this equation, you need to memorize an equation that is derived from this and that is the half-life is equal to 0.693 x the volume of distribution divided by a term that we haven’t heard before called the clearance. This guy is called the clearance. And we’ll talk about clearance in a future video but real quick, the clearance is the volume of blood that gets eliminated of drug per unit time. So these are the 3 most important points to remember and if you wanted to stop this video right here and you just learned these 3 points, you would probably be okay but that’s not how we roll.
We want to really understand things. So, let’s work through an example with the table here. So the example that we’re going to use is we’re going to say that the half life of the drug is equal to 1 hour and that the initial concentration is equal to 8 mg of drug per liter of plasma. So we’ll start off by looking at the left hand side of this table. Notice that the time increment is 1 hour and this 1 hour is the half-life of the drug. So every hour that goes by, we expect the concentration of the drug to decrease by 1/2. So let’s look at the plasma concentration. If I started at 8 mg/L, after 1 hour I expect to be at 4 and after another hour, I expect to be at 2 and then at 1 and then at 0.5 and then at 0.25. Now, moving on over to the right here. This part is a little more interesting. We have the amount of drug that is remaining and I listed it as fractions and also, as a percentage. So after 1 half-life, we expect 1/2 of the drug to be remaining.
After 2 half-lives, I have gone down by 1/2 and by another 1/2. So another way of writing this is I’ve gone down by 1/2 squared or I have 1/4 of the drug remaining or 25%. and now that means that I’ve metabolized 75% of the drug. If I go by, if I let another hour go by, another half-life has gone by. So I’ve gone 1/2 x 1/2 x 1/2 or 1/2 to the 3rd and I have 1/8 remaining. So, notice this trend here. If you want to figure out what fraction of the drug is remaining after a certain number of half-lives, you just take oh 3 half-lives have gone by, it’s 1/2 raised to the 3rd or 1/8 of the drug is remaining.
Now, let’s get to that second point that we’ve talked about that. That 95% of the drug is always eliminated after around 4.5 half lives. So, at 4 half-lives, I have 6.25% remaining. That means I’ve metabolized around 94% of the drug. And so, as more time goes by, I’m going to metabolize another half-life right. So if I had 1/16 then I have 1/32 left at 5 hours or 5 half-lives and that’s around 97% of that drug has been eliminated. So somewhere between 4 and 5 half-lives, we expect 95% of that drug to be eliminated and this does not depend on the initial concentration. If I would’ve started at 32 then I would’ve been at 16, then at 8, then at 4, then at 2, then at 1 and somewhere between here, I would’ve also eliminated 95% of the drug. So that is 0.2. And the 3rd point has to do with the first order elimination rate constant and how it relates to the half-life and to do that, we need to graph a first order elimination rate graph using a linear plot and then using a semi-log plot. So, using a linear plot, remember that the half life is 1 hour and so if I started at 8 mg/L, after an hour, I’d be around 4 and then I’d be around 2 and then I’d be around 1 and then I’d be around 0.5 and if I wanted to graph this, I get this exponential graph.
Now, as scientists right, we want to be able to describe this graph and if I wanted to know the plasma concentration at any point, it can be given by this equation. and that is that the concentration at any time (T) is equal to your initial concentration x E raised to the -KT where K is that first order elimination rate constant. Now, you don’t have to memorize this equation but from this equation, I can solve this for the half-life and if I do that, that’s where I get the T 1/2 is equal to 0.693 over that first order elimination rate constant. Now, maybe what I’ll do in a future video is just talk about how this is derived but let’s be on the scope of what we’re talking about now.
The bigger point is to understand this inverse relationship between the half-life and the elimination rate constant. Now, the best way to do that is by looking at a semi-log graph. And so, what is a semi-log graph? It’s probably been awhile since people have used this. So, when I say semi-log, half of the graph is a logarithmic graph or half of the scale and that’s the Y axis is logarithmic where the other half is linear. and so notice the units here. I start at 0.25 and then I double 0.5 and then I double to 1 and then a 2, 4, 8 or another way of thinking of this is that the concentration is decreasing by 1/2 each time. So, this is the perfect graph to graph the half-life. So, if I started at 8 after a certain period of time, 1 half-life I would be at 4 and then at 2 and then at 1 and then at 0.5 and if I went to another one, I would be at 0.25.
Now there is no way this graph is going to be exponential. It is a linear graph and that is because I’m using a logarithmic scale. Now, this graph helps us in a couple of ways. First off, I want you to note that this slope correlates very well. Let’s just write correlates with that first order elimination rate constant. And maybe in that future video were I talk about how half-life is derived, I’ll tell you how you can solve for K but just know that it correlates with it. And for your purposes, you can just say, the slope is K. And so, the point here is that the steeper the slope, this increases the first order elimination rate constant K and as a result, I’m eliminating faster and I decrease the half-life. Remember, there is an inverse relationship between the 2. So, what I’ll do here is I’m just going to graph a new plot. So this new plot, I’m still starting at 8 but after 3 hours, I’ve gotten rid of most of that drug.
So using this graph, if I wanted to solve for the half-life, I just take 2 points that I have half the concentration of 1 another. So, let’s just say here at you know at 2. What I meant is value of 2 for the plasma concentration. 2 mg/L, I was at 1 hour and half of that would be when I’m at 1 and that looks like I’m at about 1 1/2 hours. 1.5. So the time it takes to go from an initial concentration to 1/2 your initial concentration is the half-life. So, the T 1/2 here is equal to 0.5 hours. And so, as this graph is becoming steeper, as I’m increasing the rate of elimination, the half-life is going down.
It went from 1 to 0.5. And so, the point is that you need to remember this inverse relationship between the rate of metabolism and the drug half-life. Now, you should walk away from this with these 3 main points that we’ve listed up here. Maybe what I’ll do in the next video is just quickly cover some of the math for all of us nerds who like this very much but the next big topic we’re going to talk about is drug clearance and really, we’ll get into excretion.
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