Drug Half-Life Calculator

Here we go again with some baffling stuff. I wanted to understand the implications of drug (i.e., medicine or medication) half-lives, in particular for drugs taken daily. The half-life calculators that I found were not useful at all, so I created my own (including an interactive graph), for use on a desktop or laptop, with a keyboard and biggish screen:

http://com.hemiola.com/half-life/

This page does not explain what a half-life is. There are plenty of other sites that do that.

For drugs with a short half-life (e.g., a few hours), I can see how if taken daily, there is no buildup because the daily residual is negligible. It was intuitively obvious to me that with a long half-life (e.g., a half-day or more), taking the drug daily would cause an overlap and buildup—convergent, but still, you would have more drugs in your system than you take daily, and I wanted to know that number.

The basics

Wikipedia recently instituted a format for its drug entries that includes the drug’s half-life. That makes it easy and convenient to look up the half-life for all the drugs I’ve checked.

The basic idea is that if doses overlap (i.e., taken again before the earlier dose[s] leave the system), they stabilize in the system at a higher dose than what you take each time. Also, on a different point, there seems to be an assumption that drugs with a long half-life are slower acting (i.e., weaker). I find that interesting.

The math

There is the Wikipedia page on biological half-life, but the math there is way beyond me. Here is what was obvious to me:

After x hours with half-life H (in hours) and dose D1, the fractional amount D2 leftover is:

When you take drugs at regular intervals, there might be some nonnegligible amount left over from previous doses. Here is essentially what my calculator is doing, where p is the hours between doses:

The disclaimers

Yes, I realize the real-world implications of drugs and their half-lives are way more complicated than a simple power-of-two equation. Still, I wanted a quick and easy way to compute the oversimplified numbers.

A few notes about the user interface

  • Units don’t matter. Quantities are just numbers. Units need to be consistent; that is all.
    • Dosage: the UI uses mg, but you can mentally substitute a different unit.
    • Durations: the UI uses hours, but you can mentally substitute a different unit. Yes, the UI divides by 24 to get (assumed) days; that’s because the half-lives I’ve been interested in could be expressed in hours.
  • The arrow keys work as a convenience for incrementing and decrementing values, but you can edit the values by typing or pasting, too.
  • The time for a logarithmic function to level off is infinite (asymptotic). The percentage elimination sets the point (near 100%) that you want to consider close enough to 100% (of the elimination) to be considered eliminated. For example, it might take 6 days for elimination to reach 97%, which to me is more useful than saying 100% elimination is always (mathematically) infinite.

At what point does a drug stop being effective and get flushed out in your urine, leaving no trace? Different drugs are different. I want to emphasize that I created the UI to show the sum of residuals, not to take other real-life things into account.

I’m now learning that there are drugs that are effective for about four hours or less, with half-lives of 15 – 30 hours. If you take them based on effectiveness (and why wouldn’t you?), there’s massive build-up in your system that no one seems to acknowledge.

An example

Aimovig was approved by the FDA on 2018-05-17. It’s taken once per month, and this really got me wondering, because I’m used to thinking about things taken about once per day. Aimovig’s half-life turns out to be a whopping 28 days (672 hours)! Being that you take it once per half-life, the swing in your system is 70 – 140 mg, even though you take 70 mg per month. Once stabilized after six doses, it takes about 170 days to leave your system, which is a serious commitment!

See also

Some valuable links:


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Comments

36 responses to “Drug Half-Life Calculator”

  1. Ben Wolfenden Avatar

    This was exactly what I needed! I am titrating up on a drug with a half-life of 26hrs – the drug has side effects initially – so I wanted to know where roughly the peak was for each movement upwards

  2. Aleksander Avatar
    Aleksander

    Thank you so much! I knew what i was looking for and was so happy when i found your calculator. In am on Intuniv for ADHD and know it’s building up in the body over time. T 1/2 is 18 hours and i wanted to know what blood levels i had of the drug when i was feeling the best. This is an easy formula like you say yourself, not including bio availability (which should not be hard to add) although it is a very useful one.

  3. versenumber2 Avatar
    versenumber2

    Don’t see the need for factoring in bioavailability Aleksander. Especially when you are looking at how it affects you or as you state “when I was feeling the best”.

    It doesn’t matter if the bioavailability is 10% or 90%. Sure it affects plasma concentration but unless your changing ROA, the actual noticeable effects will remain constant. Since you are used to this constant you want to know what percentage of a drug is in your system over time whether you are titrating, tapering, or taking a consistent dose. Once you’re used to taking a drug you know it’s effects on your system…and that’s all we really care about right?
    I mean if your doing a study on blood plasma concentration levels that’s a different story…and even that is going to vary from person to person.

    1. Kyle Avatar
      Kyle

      If I took a 30MG XR adderall how long is it in system for? I had a urine test about 100 hours after taking it. Never took one before. So what’s the odds I fail ha.

      1. Kyle Avatar
        Kyle

        Also, if it matters, (I know it’s still assuming), but I’m 5’9 sitting at 211. I rarely work out but I am fit. 28 years old

        1. Jens-Ingo Farley Avatar
          Jens-Ingo Farley

          I have no inside knowledge of or experience with drug testing. I created my calculator to compute the sum-of-residuals of drugs taken regularly. You can certainly use it to get an idea of how long it takes for a single dose to be eliminated. I’d recommend, first, to make sure you understand half-lives, which I’ve made no attempt to explain here.

  4. versenumber2 Avatar
    versenumber2

    I spoke too soon (before I tried it). This won’t help if your titrating or tapering as it assumes a consistent dose.

  5. anon Avatar
    anon

    Thanks exactly what i was searching for. You would think other people who built calculators would understand that what people are actually interested in is knowing the residual accumulation but you are the only one who figured it out. Well done.

  6. Matt Avatar
    Matt

    This is great. Searching for something similar and haven’t been able to find it.

    Appreciate your work.

  7. C stone Avatar
    C stone

    It would be useful to have a multistage calculator: one that can calculate for tapering regimens, such as, 40mg for 3 days, followed by 32mg for 3 days, for example.

    I’m trying to understand the concept of percentage of interest as well.

    Overall, thank you very much for this.

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      Tapering regimens are not something I’ve investigated. I suspect there isn’t a universal one.

      The time for a logarithmic function to level off is infinite. The ‘percentage of interest’ sets the point (near 100%) that you want to consider close enough to 100% (of the elimination) to be eliminated. For example, it might take 6 days for elimination to reach 97%, which to me is more useful than saying 100% elimination is always (theoretically) infinite.

  8. Mary Avatar
    Mary

    I’m sorry I don’t understand this stuff. In the span of 2-3 days I took 20 milligrams of Xanax. Yesterday, oct 5, 2018 was the last couple of doses. I have a urine test on oct 26, 2018. I’m 45, in very good shape and drugs usually leave my system quickly. Please please help me! I just don’t understand your mathematics here. Thanks, worried in philly

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      You can first look up the half-life on the generic drug’s Wikipedia page:

      https://en.wikipedia.org/wiki/Alprazolam

      The half-life is either 4 – 6 hours (immediate release) or about 11 – 16 (extended release).

      The half-life is short enough in either case that 21 days is way, way more than enough to eliminate it. If you use my calculator, you can see that with a half-life of 4 hours, it should be gone in less than a day. Even with a half-life of 16 hours, it should be gone in less than four days.

  9. Jay Avatar
    Jay

    What about mutiple doses within the half life. For example ampodipine and a 35 hour half life with a daily dosage of 10 mg? What about the same drug at 5 mg twice daily?

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      I might not understand the question, but this calculator visually and interactively answers your exact questions, and is pretty much exactly why I found other calculators useless.

      1. Jay Avatar
        Jay

        Oh sorry only read the blog and didn’t use the calculator. Looks good thanks!

  10. Brian Avatar
    Brian

    Hi,
    I sent you an email but in case you did not get it I wanted to bounce this off of you.

    I am trying to come up with a logic based approach to titration of vasopressors/vasoactive medications on critically ill patients. I am trying to emphasize reaching a steady state before adjusting the dose. For example Epinephrine has a half life of 3 minutes and 5x half lives at a continuous infusion is 15 min.

    In my effort to explain this to providers I am trying to calculate how much epinephrine is floating around in the plasma (forgetting individual variation and proteins and such) – just shooting for simple.

    I am using 8mcg/min as a dose with a 3 minute half life.
    If I understand the math correctly this is roughly equivalent to building up to ~35mcg/min in the plsama when it reaches a steady state…?

    I tried to fudge it on your calculator but the time interval would not go less than 2.

  11. Russell Jurney Avatar

    Thanks for building this and putting it up! This is exactly what I was about to research and build…

  12. Aly W. Avatar
    Aly W.

    This is great, thank you for your work!. Could you add an option for days for the half-life though? Some drugs like bicalutamide and tamoxifen have long half-lives of like 5 to 10 days. And intramuscularly or subcutaneously injected drug esters like estradiol valerate can have durations of weeks or even months. It’d be nice to have a calculator that accommodates these as well!

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      I attempted to address your concerns already via explanation.

      #1: I gave an example of a 28-day half-life.

      Example

      #2: I tried to explain that units don’t matter as long as they are consistent. If you want to think of half-lives in seconds, hours, days, weeks, or months, that’s fine.

      Units

  13. David H Avatar
    David H

    This is great!
    I’m missing the option to add a loading dose, though.

    Would it be difficult to add?
    Thanks!

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      Thank you for the term “loading dose.”

      Depicting this graphically might be interesting, but I don’t think this tool is the place for it. A loading dose has no effect on the long-term convergence levels. And, if, say, the loading dose is an injection and the maintenance doses are oral, that’s already getting into more real-world differences than this tool can claim to consider. As it is, this tool could possibly help compute a loading dose based on what the eventual convergence of the maintenance doses will be.

      1. David H Avatar
        David H

        True, the convergence level would be the same in the end.
        The loading dose just helps you quickly get as close as possible to that level.
        But it’s true, there are many permutations one could take into consideration!
        Thank you for a neat tool 🙂

  14. Chad Summers Avatar
    Chad Summers

    Can you please publish the code or create an excel spreadsheet that I can download? There used to be a site like this many years ago but it’s since offline.

    I will even pay you for your intellectual property so I can always have a working copy on my laptop if I don’t have Internet.

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      The code is JavaScript and is completely unobfuscated, meaning you can view it / save it using a browser. I’m not sure what you mean about Excel. Offline usage is a little tricky; I don’t think there is a stable and straightforward way to achieve that.

  15. Lisa Robinson Avatar
    Lisa Robinson

    I breastfeed and take methylphinadate 40mg 3x day. How much gets in my milk at the end of the day? Taken 4 hours apart? What is the RID and maternal adjusted weight dose? Estimated. Assuming I have a half life of 1.2hours.

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      I have no idea.

  16. Phil W Avatar
    Phil W

    This is very helpful and relevant to me. I am on Cabozantonib (stage IV renal cancer) and the long half life of 110 hours means after a couple of weeks I started to get side effects and by the 4th on my first 30 day cycle, I was suffering side effects.

    So I created a simple Google sheet, added in my log data that I’d kept and so looking back I can anticipate what level I need to reign back. It means I can try and stay on the drug (20, 40 and 60mg pills) rather than take a break for a few days.

    The advice I had was to see how I go and it’s trial and error. When I asked i was told it was crude and a daily blood test was the only reliable method. Hardly helpful when I don’t want more hospital visits in Covid times and we know the average half life really!

    I wonder if a Web app would be helpful for patients like me as that is what I used to do before my health got to a point I had to give up?

  17. Anonymous Avatar
    Anonymous

    Excellent calculator! Most online calculators simply express the half-life of initial dose and when it is cleared from your system… These are just useless because they don’t give any practical effect of repeat dosing which yours does…

    Well done and quite pragmatic!

  18. D Monroe Avatar
    D Monroe

    Very helpful calculator. I was doing multiple iterations on a basic half life calculator to guestimate the rough asymptotic convergences of the min and max levels in my system of daily pain meds. Had been on same monthly pain med prescription after spinal cord injury 17 years ago. As of January 1st this year, insurance cut daily/monthly dosage coverage down to 75% of what I’d been taking all these years (nothing to do with my personal situation or doctor recommendation). Totally messed with my delicate balancing act of daily living. Very informative to see the difference between old dosage and new dosage, varying the amount and interval. I can take x amount every 4 hours or 2x amount every 8 hours for my insurance daily max. At 2x, I get about a 35% increase at the max but a 34% decrease at the min compared to the smaller amount taken more often. This helps the discussion with my doctor on how to deal with this.

  19. Ed Avatar
    Ed

    I’m looking at pain medication Gabapentin for
    Nerve pain (diabetes).
    300mg dose at breakfast, lunch,dinner.
    900mg dose at bedtime.
    7 hrs half-life
    But it takes 2-3 hrs to get into my system.
    (Every dose)
    So how does this model work in this case?

    1. Jens-Ingo Farley Avatar
      Jens-Ingo Farley

      It doesn’t.

  20. Anon Avatar
    Anon

    Thank you very much for this, it’s exactly something I was looking for in order to speed up calculating some possible dosages for a medication I need with some weird practical requirements. I wasn’t prepared for how useful the actual visualization was, though.

    On that note, one change that may be very useful to add to this could be allowing one to tweak what the 2nd, 3rd, etc. doses are and see the results.

    That way you could quickly calculate and visualize the effects of e.g. taking 2mg for the first dose but 1mg for every subsequent dose.

    1. Ruth Avatar
      Ruth

      I agree! Trying to work out the ideal dose of levothyroxine and I know roughly what cumulative dose is too much now trying to work back from that after an accidental loading dose – having different doses for different days would be super helpful

      1. Janice Avatar
        Janice

        I was in a similar situation and ended up using http://half-life-calculator.com for it, so I’m sharing here. Hope it’ll serve you as well!

  21. Cheryl Avatar
    Cheryl

    Hi, I’m trying to take a 200mg medication every 24 hours, but the half-life is 36 hours.

    I’m trying to use your equation to calculate the residuals (aka nonnegligible amounts leftover). I don’t know how to input it into a Casio calculator or where to go to input online. I understand this uses limits but I was never the smartest at math, but I want to know amount left in my system for 1 month period or approx. 28 days to be exact. Your calculator only goes till 8 days.

    What is the ‘i’ in that second equation of yours? Does this look remotely correct? See below (I used https://www.symbolab.com/solver/limit-calculator).

    https://imgur.com/Mkw9rNf

    Any help would be appreciated for a dunce like me! 🙂

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