Looking for function of belllike curve that peaks quickly.
Solution 1
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:
Solution 2
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.
Comments

iDontKnowBetter over 3 years
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:
$$ q\cdot e^{ \displaystyle \frac{(x(1+t))^2}{d}} $$
where
$q =$ 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 "leftheavy" 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 ) graphs.append(gaussian) i = i+1 def plotSumFunction(): global plotSum i = 0 while i < len(graphs): plotSum = plotSum + graphs[i] i = i+1 gaussian() plotSumFunction() all.append(graphs) all.append(plotSum) allPlot = plot(all, (x, 0, times[len(times)1]+3)) multiPlot = plot(graphs, (x, 0, times[len(times)1]+3)) allPlot.show()
You can see that the graph is far from realistic (he has medicine in his system before he even takes the first dose!):
The top line is the sum of all four (the cumulative effect).

iDontKnowBetter over 11 yearsunless 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.

Ross Millikan over 11 years@fakaff: Sorry, I meant to have a factor of $t$ on both so they start at $0$. Fixed.

iDontKnowBetter over 11 yearsThanks. I don't know why it didn't even cross my mind to switch to parametric equations... head in the clouds.