Is it possible to reparametrise this non-linear equation?

Question

I am currently working on fitting data to this pharmacokinetics model (also called "one compartment first order absorption model"):

$y(T) = \frac{Q K_a K_e \left(e^{-K_e T} - e^{-K_a T}\right) }{K_a - K_e}$,

where $T$ is time, $K_a$, $K_e$, and $Q$ are all parameters (constants); $y(T)$ is always positive.

I was wondering if it would be possible to re-parametrise this equation above such that the following conditions are met:

$0 < K_a$

$0 < K_e < 1$

$0 < Q$

$0 < Q K_e < 1$

Thanks!

Answer 1

Your model being $$y(T) = \frac{Q K_a K_e \left(e^{-K_e T} - e^{-K_a T}\right) }{K_a - K_e}$$ you could rewrite it as $$y(T) = \frac{A K_a \left(e^{-K_e T} - e^{-K_a T}\right) }{K_a - K_e}\qquad \text{where} \qquad A=Q K_e$$

Now, to take into account the constraints, let $K_a=e^B$, $K_e=\frac 1 {1+e^C}$ and $A=\frac 1 {1+e^D}$.

For sure, this introduces a lot of nonlinearities.