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Sum of exponentials: non-iterative approximate solution?

  1. Mar 24, 2015 #1
    Hi everyone,
    I am trying to solve a problem (related to pharmacokinetics) that requires finding t for an equation like the following:

    [itex]A e^{-a t} - B e ^{-b t} + C e^{-c t} = 0[/itex]

    where A, a, B, b, C and c are all real positive numbers (they are constants related to the absorption, uptake, efflux and elimination of a substance in the body) and t is time.
    a is always different from b.
    When c is equal to either a or b, I know how to solve the equation analytically, but that is an exception, it almost never happens.
    In the most general case, it appears that there is no analytic solution for this equation (is it true, BTW?).

    In such cases I would normally use an iterative method.
    However, I have an Excel file where the constants A,a...C can be varied by the user, and I would like t to be recalculated automatically without having to run Solver each time.

    So the question is, can an approximate solution (for t) to the above equation be written as a function of the constants A,a...c?

    I tried a few things already, e.g. writing y = f(t), Taylor and then inverting, but I always get zero as the solution.
    I also searched this forum and the web for similar problems, but I couldn't find anything close (in most cases it seems iterative methods are used).
    I suspect the Newton-Puiseux method could be applicable, but I can only find very technical/theoretical descriptions of it, I'm not sure how I could use it in this case.

    Any suggestions/links?
  2. jcsd
  3. Mar 24, 2015 #2


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    I would be surprised by such a formula.

    Instead of running the solver, you can write a macro, or even implement the steps as normal formulas somewhere.
  4. Mar 25, 2015 #3
    Thanks! I hadn't thought of writing down the steps. That may indeed work.
    What numerical method would you advise for this type of function?
    I had considered the macro, but I thought it would slow down an optimisation routine that I have in another sheet and depends on a, b etc.

    BTW, in the meantime I looked again at the Taylor method (I use Maxima for this), and I discovered that the reversion command wasn't working as I thought.
    Now I think I understood how to use it properly. I tested it on finding x for 1+x+exp(x)=0, and it worked pretty well.
    So there may be a way to write a single formula for the approximate solution (huge though, probably not worth it if the steps you suggested do the job).
  5. Mar 27, 2015 #4


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    Newton should be fine. I guess you have some rough estimate for the time that can be used a starting value. If your patient needs years to get the drugs, something went wrong.
  6. Mar 29, 2015 #5
    Thanks mfb, I'll give it a try.

    In fact the above equation comes from the solution of a linear system of 2 differential equations describing the mass balance in a 2-compartment (plasma and tissue) after oral administration of a drug. I solved it by Laplace, obtaining concentration_in_plasma(t) and concentration_in_tissue(t), which I re-differentiated w.r.t. t. What I got was a sum of 3 exponentials (with different A, B and C depending if it's plasma or tissue), like the one above.
    So the solution for t tells you the time for which the maximal concentration is reached after administration ('Tmax').
    The literature says that this Tmax can't be calculated exactly, and must be found graphically or by iteration. That's why I tried to calculate it of course :smile: - I never take no for an answer.

    It's possible that a more elegant solution exists, which uses the original differential equations (as in that case you already have dC/dt), but I could not find it.
    Pharmacokinetics is really one of those fields where you need a good understanding of maths to grasp certain concepts. But they're stuck with me. :biggrin:
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