If a drug has a half life of, let say 72 hours, and you take it every day, how do the levels not continuously climb?
In: Biology
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It’s hard to answer without more detail but it could be the case that rate of elimination is dependent on the blood-concentration of the substance.
Is there a specific example you have for this?
A “half life” is the time it takes for the amount to drop by half. This is an exponential decay, meaning the more there is the faster it’s eliminated. After a few half lives, the amount left is relatively small compared to the next dose.
At first, the amount in your system *does* rise pretty fast. But it will eventually reach an equilibrium where the amount eliminated in a day is the same as the daily dose.
Figuring out exactly *when* you’ll get within, say, 90% of equilibrium takes some serious calculation, but figuring out *how much* the equilibrium amount is is much easier. All you have to do is figure out what percentage is eliminated each day and set that equal to your daily dose.
For example, if a drug has a half life of one day, then 50% = 0.5 of it is eliminated each day. So set “0.5 * equilibrium = daily_dose” and solve for equilibrium to find that equilibrium = 2 * daily_dose. So if you take 125 mg per day for several days, then you’ll have 2 * 125 mg = 250 mg in your system right after taking your daily dose.
As another example, you mentioned a drug with a half life of 72 hours (3 days). First we have to figure out how much is eliminated each day. The amount left after a day is cube_root(1/2) = (1/2)^(1/3) ≈ 0.8 = 80%, so the amount eliminated each day is 100% – 80% = 20%. (Sorry, I can’t explain exponents like you’re 5.)
… Now that you know 20% is eliminated each day, you know that your daily dose will be 20% of the equilibrium amount that your system stabilizes at. Thus you can calculate the equilibrium as “0.2 * equilibrium = daily_dose”, yielding “equilibrium = daily_dose / 0.2 = daily_dose * 5”. So with the 125 mg dose example above, your body would eventually have 5*125 mg = 625 mg of the drug right after your daily dose. This is a lot more than if the drug had a shorter 1-day half life, but it still stops at a point.
This is a good question. Let’s look at a basic case. You take 10 mg of a drug every 24 hours. Just for easy math, let’s say this drug has a 24 hour half – life.
Think of taking a drug as having peaks and valleys. When you take the drug, the amount of drug in your body is at a high point, or “peak,” and slowly declines until you take the next dose. The drug concentration right before you take your next dose is at a relative low point, or “trough.”
Going back to our example, let’s say you take your first dose on day 1. The peak is 10 mg, and then the amount of drug in your body will start to decline. After 24 hours, there will be 5 mg left (since the half-life is 24 hours and 5 is 50% of 10).
Now it’s day 2 and you still have 5 mg left from day 1. You take the 10 mg so your peak for day 2 is 15 mg. After 24 hours, you have 7.5 mg left in your body.
Let’s see how this pattern progresses over the next couple days.
Day 3: Peak = 17.5, trough = 8.75
Day 4: Peak = 18.75, trough ~ 9.4
Day 5: Peak = 19.4, trough = 9.7
Day 6: Peak = 19.7, trough = 9.85
Day 7: Peak = 19.85, trough ~9.9
Day 8: Peak = 19.9, trough ~10
Day 9: Peak = 20, trough = 10
Day 10: Peak = 20, trough = 10
Do you see how after awhile, the peaks and troughs stay the same day after day? This is known as “steady state” since the peaks and troughs don’t change over time.
If we look at the values, the steady state peak concentration is 20 and the steady state trough is 10. Let’s look at each day’s concentration as a percentage of steady state (we will just use peaks here).
Day 1: 10/20 = 50%
Day 2: 15/20 = 75%
Day 3: 17.5/20 = 87.5%
Day 4: 18.75/20 = 93.75%
Day 5: 19.4/20 = 97%
Day 6: 19.7/20 = 98.5%
And so on and so forth.
What we see is that after about 5 days, we’re at greater than 95% of steady state concentration. In this case, since one day is also one half – life, we can say that we’re at greater than 95% steady state after 5 half-lives. And this turns out to be the case for many, many drugs. You can approximate the time at which a person will reach steady state by multiplying the half-life by 5. So in your example, if a drug has a half – life of 72 hours, steady state will be reached in 72 hours x 5 = 360 hours, or 15 days.
This type of question falls into a field called pharmacokinetics, and there are many other variables that can affect drug disposition (and then equations can get fairly complex). However, if you need to come up with how long it takes for someone to reach steady state, 5 times the half-life is a good rule of thumb.
Because you’re adding a fixed amount of the drug every day, but progressively larger *amounts* are decaying (since a constant *fraction* goes away at a fixed interval). Eventually you hit an equilibrium.
Imagine you have a very large leaky bucket. Every hour, half of *whatever amount* is in the bucket will leak out. If you fill it 1/4, 1/8 will be gone in an hour. If you fill it to 1/2, 1/4 will be gone.
So let’s say every hour you add 1/4 to the empty bucket.
Hour 1: You add 1/4, bucket goes from empty to 1/4 full.
Hour 2: You add 1/4, 1/8 leaked out, you now have 3/8.
Hour 3: You add 1/4, 3/16 leaked out, so you now have 7/16.
Hour 4: You add 1/4, 7/32 leaked out, you have now 15/32.
Every hour you have seen larger amounts leak out, 1/8 < 3/16 < 7/32 … getting closer and closer to 1/4. At the same time, you’re seeing the height of the water in the bucket slow down, going from 0 -> 1/4 -> 3/8 -> 7/16 -> … -> 1/2.
Eventually, when the bucket is 1/2 full, you walk away and come back and find 1/4. You add another 1/4, walk away, come back and find 1/4 again. That’s your equilibrium.
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