# Looking for function of bell-like curve that peaks quickly.

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 "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 )
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).

-

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:
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.
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