Trying to solve this from a different approach now. started out here Calculate half life of esters
Is there a way to find a function that fits the graph below?
There are a few criterions. The area of the graph must be A The peak of the graph must be at X (7-8 in the image) The halflife after the peak must be H
The area in this case is the amount of testosterone, or rather the injected dose. The peak is when the compound is fully absorbed.
y = f(A,X,H, t)
Is this possible?
My math skills are way too limited here
A first idea is to try to fit a gamma function, in your case $f(x)=sx\mathrm e^{-x/m}$. The slope at the origin is $s$, the maximum is at $x=m$ and its value is $sm/\mathrm e$.
A caveat though: which parameters of the curve are important to you is not quite clear from your question but a general principle is that the more characteristics of the curve are imposed the more parameters are needed. In the present case, slope at the origin + location of the maximum + maximal height + area under the curve would a priori need four parameters (and the gamma family above is described by only two).