Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm writing a little Sage/Python script that would graph the cumulative effects of taking a particular medication at different time intervals / doses.

Right now, I'm using the following equation:

$$ q\cdot e^{- \displaystyle \frac{(x-(1+t))^2}{d}} $$


$q =$ dose

$t = $ time of ingestion

$d =$ overall duration of the effects

$p =$ time it takes to peak (missing from eq. 1)

While the curve should be a rough approximation, I need more control over its shape. In particular, right now the graph peaks in the middle of the bell curve, but I need a curve that is near $0$ at time $t$ and then quickly peaks at time $t+p$ (say, an equation that quickly peaks in one hour, then slowly declines for the rest of the duration period).

How do I create a "left-heavy" curve like that?

Here is the Sage/Python code, with a sample graph below, so you get an idea of what it looks like vs. what it should look like:

(In this example, the person takes his medication at 1:00, 3:00, 5:00, and 8:00; and effects last him 2.5 hours.)

duration = 2.5
times = [1, 3, 5, 8]
dose = 5
totalDuration = 0
graphs = []
all = []
plotSum = 0

def gaussian():
    i = 0
    while i < len(times):    
        time = times[i]
        gaussian = (dose)*e^-( (  x-(1+time)  )^2/duration )
        i = i+1

def plotSumFunction():
    global plotSum
    i = 0
    while i < len(graphs):
        plotSum = plotSum + graphs[i]
        i = i+1


allPlot = plot(all, (x, 0, times[len(times)-1]+3))

multiPlot = plot(graphs, (x, 0, times[len(times)-1]+3))


You can see that the graph is far from realistic (he has medicine in his system before he even takes the first dose!):

Four doses and their cumulative effect.

The top line is the sum of all four (the cumulative effect).

share|improve this question
add comment

2 Answers

up vote 5 down vote accepted

You could imagine that the medicine gets absorbed by the digestive system at a fast rate $\alpha$ and then consumed by the body at a slower rate $\beta$. Then you have the following system of ordinary differential equations, $$\begin{align} x' &= -\alpha x, \\ y' &= \alpha x - \beta y, \end{align}$$ where $x$ and $y$ are the amounts of unabsorbed medicine in the stomach and absorbed but unconsumed medicine in the bloodstream respectively. For initial conditions $x(0) = q$ and $y(0) = 0$, the solution is simply $$\begin{align} x(t) &= e^{-\alpha t}q, \\ y(t) &= -\frac{\alpha}{\alpha-\beta}e^{-(\alpha+\beta)t}\left(e^{\beta t}-e^{\alpha t}\right)q. \end{align}$$ For $q = 1$, $\alpha = 1$, $\beta = 0.1$, the curve looks like this:

enter image description here

share|improve this answer
Thanks. I don't know why it didn't even cross my mind to switch to parametric equations... head in the clouds. –  iDontKnowBetter Mar 22 '12 at 17:21
add comment

How about $ate^{(-t/t_0)}$? If that doesn't fall fast enough, $ate^{-(t/t_0)^2}$. These have the advantage of being zero (you have to truncate) for $t \le 0$ and are left heavy.

share|improve this answer
unless I misunderstand, the first one doesn't follow a bell pattern (ascend -> descend) because it's of the form $ab^{-x}$, and the second is similar to the one I used and also has the problem of peaking in the middle. -- If you look at the final graph that results from my equation, you'll see what I need to fix. Truncating the graphs will help somewhat. Still, I hope there's a way to fix the math. –  iDontKnowBetter Mar 22 '12 at 5:28
@fakaff: Sorry, I meant to have a factor of $t$ on both so they start at $0$. Fixed. –  Ross Millikan Mar 22 '12 at 15:54
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.