How do i derive an equation from a pharmacokinetic differential equation to model first-order absorption/elimination

Question

I watched this video, https://youtu.be/9W_kwn7ZHtI where they gave a differential equation to model a first-order absorption/elimination drug

I understood each of the equations separately, but I’m trying to derive the equation for X by solving the differential equation.

I looked at this page, https://www.pharmpress.com/files/docs/clinical_pharmacokinetics_samplechapter.pdf, where they get the equation as so:

M from the 1st equation is a constant that is probably equal to the (X0*ka)/(ka-k) term from the 2nd equation.

I tried solving by the integrating factor like this:

I'm not getting the correct term though. Any guidance on where I might’ve gone wrong or how to derive the equation for X?

Answer 1

Hey Sam!

This is definitely outside of my scope, but my initial thought would be that you can't treat X_a as a constant. It is likely also a function of time. That doesn't get you all the way there, but it might get you started.