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solving an integral for pharmacokinetics work

Hi,

 

Pharmacokinetics: mathematically characterizing how a drug  enters and leaves the body 

In pharmacokinetics experiments the influence of the stomach is ignored in most data analyses, which is not appropriate. Stomach emptying can be described with the equation (in which t=time):

Plateau×(1-e^(-Ks × t))^ß  (Ks=rate, beta =lag phase)

Subsequently, the absorption kinetics of a test component can be described with the equation:

Plateau ×(Ka× (Exp^(-Ke × t)-Exp^(-Ka × t)))/(Ka-Ke)  (Ka=absorption rate, Ke=elimination rate, Plateau is cummulative absorption)

 

Combining both gives:

∫ Plateau×(1-e^(-Ks × t))^ß * Plateau ×(Ka× (Exp^(-Ke × t)-Exp^(-Ka × t)))/(Ka-Ke)

Does anybody know how to solve such an equation in JMP, or is someone smart enough to integrate them (I am not)? I can only solve it in Excel, but then I don't get statistics.

 

Happy to include you as a co-author in a publication on the topic if we end with publishable outcomes.

 

thanks

 

Theo

Re: solving an integral for pharmacokinetics work

came across the wolfram site, and this one did yield an integral. Something to try tonight

Re: solving an integral for pharmacokinetics work

Wolfram Alpha...maybe?

http://www.wolframalpha.com/examples/Integrals.html

Re: solving an integral for pharmacokinetics work

Hi Steven,

 

Thanks for pointing this out to me; learning every day. I have tried the link you supplied, and this site can't find an answer, same answer I got from a couple of professors in mathematics.

 

Theo

Re: solving an integral for pharmacokinetics work

There are many sirtes on the internet that will solve derivatives and integrals for you.  See this link for one of my favorites:

 

http://www.integral-calculator.com/

Re: solving an integral for pharmacokinetics work

I think you could use JSL to solve the problem

 

pl=1;
ka=1;
ke=.5;
pk=expr(pl*(ka*(exp(-ke*t)-exp(-ka*t)))/(ka-ke));
integrate(pk, t, 0, 12);

Re: solving an integral for pharmacokinetics work

Solving an integral is a fundamental part of pharmacokinetics because it enables the quantification of drug concentration changes over time within the body. Integrals are used to calculate key pharmacokinetic parameters such as the area under the concentration–time curve (AUC), which reflects the extent of drug exposure and helps determine bioavailability, clearance, and dosing requirements. By integrating concentration data, researchers and clinicians can better understand drug absorption, distribution, metabolism, and elimination processes, supporting the development of safe and effective therapeutic regimens. https://mathssolverai.com/