Explore the Convergence of Drug Levels with Math, Physics, Biology, & Chemistry

[RossH]

Okay, this question straddles math, physics, biology, and chemistry, so I'm not sure if this is the correct forum to post it in, but I was using mostly chemistry knowledge to solve it, so I would guess that may be the correct method. Anyway, it's not homework, but it is similar to many homework problems, so mods, if you want to move the topic, no problem.

Here is the problem:
A person (me actually) takes a certain drug 3 times a day. He takes 10 mg of this drug at 8am, 10 mg at 2pm, and 10mg at 8pm. This drug has a half life of 4 hours. My question is, is it possible to solve for whether the quantity of the drug in the blood stream will approach a stable quantity, or at least fluctuate between certain boundaries?

So the doses look like this:

10mg -- 6 hrs --- 10 mg -- 6 hrs --10 mg -- 12 hrs, repeat

Here is the work I have done so far:
With a half life of 4 hours, the drug decays to 35.3 % during the two 6 hour breaks between doses.
It decays to 12.5% during the 12 hour break between doses.

I'm thinking that I may be able to solve this with a series, and then using an integral test to see if the quantity converges, but the problem is that the pattern is inconsistent. Then I considered trying to solve the problem with multiple series (one for each time break) but obviously since each series is dependent on all of the half-lives, this would also be problematic.

I also tried just working out a few times by hand to see what happens.
The first few quantities are:
10mg
3.53mg
4.78mg
1.81mg
4.17mg
5.00mg
1.84mg
4.18mg
5.00mg
1.84mg

So it looks like there might be a pattern, but obviously this is no mathematical proof. Any ideas?

 

[edgepflow]

Remember to add the 10 mg to the amount remaining from the last dose. For example, you have 3.53 mg remaining after the first six hours. So, apply your exponential decay law with an "initial" anount of:

3.53 mg + 10 mg = 13.53 mg.

And repeat this.

There is probably an exact solution to this, but I set it up in Excel and found that a pattern of three values (at each dose time) repeats !

If you took the medicine every 6 hours (or any equal interval), you would reach a single concentration in your system.

If the interval of time is too small, the dose take longer and longer to reach a steady level.

Try to set this up in Excel; it is interesting to see how it behaves for different half life, doses, intervals, and so on.

 

[Redbelly98]

The blood quantity will fluctuate, not approach a constant value. The fluctuation will not be a sinusoidal dependence.

Assuming the drug gets into the blood much much faster than the 6-hour time interval between doses, and by "half-life" we mean there is an exponential decay of the drug's level in the bloodstream, then this becomes a math problem. Without working out the exact solution, I can say that:

• You'll get a jump in level shortly after taking the each dose.
• The level will then drop exponentially, until the next dose.
• The level will be highest just after the 8 p.m. dose, and lowest just before the 8 a.m. dose.

It would be useful to compare the change in level ("high minus low") vs. the overall average level in the blood. This would essentially give you an idea of the variation, in a relative sense, of the level.

I'll see if I can work out the math in more detail, and post back if I can.

EDIT: I just noticed, the half life is 4 hours and there is a 12-hour interval -- 3 half-lives -- between the 8 pm and 8 am doses. So after reaching the maximum level just after 8 pm, the level will have dropped to 1/2³=1/8 of this before the 8 am dose.

 

[RossH]

The blood quantity will fluctuate, not approach a constant value. The fluctuation will not be a sinusoidal dependence.

Assuming the drug gets into the blood much much faster than the 6-hour time interval between doses, and by "half-life" we mean there is an exponential decay of the drug's level in the bloodstream, then this becomes a math problem. Without working out the exact solution, I can say that:

• You'll get a jump in level shortly after taking the each dose.
• The level will then drop exponentially, until the next dose.
• The level will be highest just after the 8 p.m. dose, and lowest just before the 8 a.m. dose.

It would be useful to compare the change in level ("high minus low") vs. the overall average level in the blood. This would essentially give you an idea of the variation, in a relative sense, of the level.

I'll see if I can work out the math in more detail, and post back if I can.

EDIT: I just noticed, the half life is 4 hours and there is a 12-hour interval -- 3 half-lives -- between the 8 pm and 8 am doses. So after reaching the maximum level just after 8 pm, the level will have dropped to 1/2³=1/8 of this before the 8 am dose.

I understand that there will be no constant value. I was just wondering if the limits- the 8 am level and the 8 pm level- would approach constant values. More if the range would be constant. I guess that I didn't really make that clear when I mentioned the sine function. I was thinking more that it would look like a noncontinuous sine wave.

EDIT: Oh, duh. It wouldn't even look like a discontinuous sine wave. Ha, I was thinking about that completely wrong. Oops.

And if that range is constant, at least as t->infinity, what the math would look like. I still feel like a might need a series, or a sum of multiple series, to properly describe the level of the drug, but I have no idea what the series would look like, given that this is a noncontinuous function and (I think) can't be integrated, so wouldn't have a corresponding series.

Am I right in these assumptions?

Thanks both of you for your comments!

 

[Redbelly98]

[EDIT: just saw your above post, after posting this. Yes, the level at a given time of day would approach a steady value. The variation during the day would consist of upward jumps of 10 mg each time a dose is taken (assuming the 10 mg enters the bloodstream rather quickly), with -- I assume for the purposes of working out the math -- exponential decay of the level in between doses. I'm not a biologist, so I don't know in actuality how accurate it is to model the level as an exponential decay. END OF EDIT]

Okay, I think I have worked things out.

Let L₀ be the level in the blood just before taking the 8 a.m. dose.

24 hours later -- just before 8 a.m. the following day -- the level must be

L₀·2^-6 (due to the decay of the L₀ that was present 24 hours, or 6 half-lives, earlier)
+ 10·2^-6 (due to the decay of the 10 mg taken at 8 a.m., 24 hours earlier)
+ 10·2^-4.5 (due to the decay of the 10 mg taken at 2 p.m., 4.5 half-lives earlier)
+ 10·2^-3 (due to the decay of the 10 mg taken at 8 p.m., 3 half-lives earlier)
​

But that sum must itself be equal to L₀, since that is always the level present just before 8 a.m. every day. So that gives us an equation to solve:

[tex]L_0 \cdot 2^{-6}
+ 10 \cdot 2^{-6}
+ 10 \cdot 2^{-4.5}
+ 10 \cdot 2^{-3}
= L_0[/tex]
[tex]10 \cdot 2^{-6}
+ 10 \cdot 2^{-4.5}
+ 10 \cdot 2^{-3}
= L_0 \cdot (1 - 2^{-6})[/tex]
[tex]10 \cdot 0.185 = L_0 \cdot 0.984[/tex]
And so L₀ = 1.88 mg, the lowest level present in the blood, just before 8 a.m.

We can work out the levels present at various times:

1.88 mg just before 8 a.m.
11.9 mg just after 8 a.m. (added 10 mg to 1.88 mg)
This decays to 4.20 mg in 1.5 half-lives later, just before 2 p.m.
Then 14.2 mg just after 2 p.m. (added another 10 mg)
This decays to 5.02 mg in 1.5 half-lives later, just before 8 p.m.
Then 15.0 mg just after 8 p.m. (added another 10 mg)
This decays to 1.88 mg in 3 half-lives, just before 8 a.m. the following morning​

 

[RossH]

Wow! Thank you very much. I'll have to look at it a little more to completely get it, but I think I see where you're going. Now I'm off to try the same thing with all my other drugs. Haha. Thanks!

 

[Mike H]

Disclaimer - pharmacokinetics can get fairly messy. While the drug itself might have a half-life in the body of four hours, the tablet that you take might have a particular rate of dissolution that is particular to that formulation so as to ensure a more steady concentration in the blood. Also, the necessary therapeutic concentration can vary for different pharmaceuticals, so that if it fluctuates, as long as it's above a certain concentration you should be fine. But having said that, I presumed that one could use a simple exponential decay, although I went at it via basic chemical kinetics. I did this after seeing the original post but before seeing the other comments, so consider it an independent verification of Redbelly98's approach through another method. :)

Since you know the half-life for the drug, you can apparently go the very simple route and calculate the rate constant for the drug’s metabolism through the first-order rate equation

ln 0.5 = –kt_1/2

which is 0.17325 hr^-1 by my math. You can then estimate the amount of drug in your system as [A]_eff prior to the 8 am dose as a sum of previous dosages, with [A]₀ = 10 mg, where

[A]_eff = [A]_eff*e^-kt₂₄ + [A]₀*e^-kt₂₄ + [A]₀*e^-kt₁₈ + [A]₀*e^-kt₁₂

Collect similar terms together on each side, and you have

[A]_eff - [A]_eff*e^-kt₂₄ = [A]₀*e^-kt₂₄ + [A]₀*e^-kt₁₈ + [A]₀*e^-kt₁₂

[A]_eff (1- e^-kt₂₄) = [A]₀ (e^-kt₂₄ + e^-kt₁₈ + e^-kt₁₂)


Solve for [A]_eff, and you have the following expression:

[A]_eff = {[A]₀ (e^-kt₂₄ + e^-kt₁₈ + e^-kt₁₂)}/(1- e^-kt₂₄)

Plug in the calculated rate constant, and just switch in 24 hours/18 hours/12 hours for t₂₄, t₁₈, and t₁₂, and I got 1.87 mg. You can pretty much apply this sort of analysis at any time point, you’ll just need to tweak things here and there. You could just rewrite the equation for [A]_t at some time t, and generalize t₂₄, t₁₈, and t₁₂ as t₃, t₂, and t₁ as the time of your last three dosages.

It makes for a fun little mathematical exercise, even if we're not sure about the actual kinetics.

 

[RossH]

Disclaimer - pharmacokinetics can get fairly messy. While the drug itself might have a half-life in the body of four hours, the tablet that you take might have a particular rate of dissolution that is particular to that formulation so as to ensure a more steady concentration in the blood. Also, the necessary therapeutic concentration can vary for different pharmaceuticals, so that if it fluctuates, as long as it's above a certain concentration you should be fine. But having said that, I presumed that one could use a simple exponential decay, although I went at it via basic chemical kinetics. I did this after seeing the original post but before seeing the other comments, so consider it an independent verification of Redbelly98's approach through another method. :)

[snipped]

It makes for a fun little mathematical exercise, even if we're not sure about the actual kinetics.

Thanks! Very interesting. I was more interested in the math behind this than the results anyway, so it's neat to see a completely separate method come out with almost identical results. Don't worry, I'm not planning on using this data for any real pharmacological application.