## SHAM(e) math [NCA / SHAM]

Dear Helmut!

❝ Reach for the stars, even if you have to stand on a cactus. Susan Longacre

Uno puede estar mirando las estrellas y al mismo tiempo verse la punta de las pestañas (Julio Cortázar)

I am very grateful for your answers and the references provided! I've found some of them and will search for others although for some reason I have strong doubts that our local library has books on pharmacokinetics on german printed in 50th ❝ Not only that. As a rule of thumb at $$\small{MRT}$$ ~⅔ of the drug is eliminated. It is very useful comparing PK models with different compartments. The slowest t½ might be misleading (see there, slides 24–28). There is a big problem with it. To get a reliable estimate of AUC one has to cover 95% (!) of AUC0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.

I was wondering from where such a rule of thumb was going and integrated the area for simple exponential elimination. It turns out that at MRT (1-exp(-1))~0,632 of the drug is eliminated for IV and slightly lower for EV (so the rule of pinky is 0,632 versus the rule of thumb (2/3=0,(6))

As for physics there exists inaccuracy in the considerations on the slide "Excursion to Hydrodynamics" . "Same proportions is emptied in the same time interval" is true only when you are solving school problems with a pool. Exactly the unexpired volume leaked depends on the form of the vessel. For the cylindric vessel for example water height and thus the volume is proportional to t2. If you want to have a constant proportion you need a vessel with a form of parabola x4 that is clepsydra or consider Mariotte’s bottle.

❝ What I learned: The variability of VRT sucks. Not surprising cause we have $$\small{t^2}$$ and $$\small{C^2}$$ in it.

I've calculated Cc for several real studies according to simple linear trapezoidal rule:
$$C_c=\frac{1}{3}\frac{\sum\limits_{i}(t_{i+1}-t_i)(C^{2}_{i}+C^{2}_{i+1}+C_{i}\cdot C_{i+1})}{\sum\limits_{i}(t_{i+1}-t_{i})(C_{i+1}+C_i)}\tag{7}.$$ Although it has C2 in it, it's variability was always lower than that of Cmax, but I should've check it more carefully.

Dear ElMaestro!

❝ I think F may be in its own right also included on your list of crackpot ideas from the odd sock drawer? PMDA have a sentence about it in their guidance. "If F can be calculated by deconvolution, F may be used instead of AUC"

Thank you! I will definitely add it to my collection of weird PK parameters! Need to know more about deconvolution... Dear mittyri!

❝ By the way I couldn't follow Yamaoka's logic regarding that magic cut-off errors. How did they find it?

I am puzzled with the same question. How did they calculated the time to reach 5% of Cmax?
I slightly modified the Helmut's considerations on the article of Scheerans et al. (2008)
Let us consider a one-compartment model with first-order absorbtion of the form:
$$C=\frac{A}{k_a-k_e}(\textrm{e}^{-k_et}-\textrm{e}^{-k_at}) \tag{6}$$
then residual area (1-AUC0-t/AUC0-inf) should be as follows:
$$AUC_{resid}(x,t)=\frac{x\textrm{e}^{-t\cdot k_e}-\textrm{e}^{-x\cdot k_e*t}}{x-1},\quad \textrm{where}\qquad x=\frac{k_a}{k_e}.$$ Let n define the ratio of t to T1/2,e, then
$$AUC_{resid}(x,n)=\frac{x\cdot2^{-n}-2^{-nx}}{x-1}\sim \frac{x\cdot 2^{-n}}{x-1}\qquad for\qquad nx>>1. \tag{8}$$
In order to estimate the duration of sampling to achieve specific AUCresid we can use the simplifyed formula
$$n=\textrm{log}_2\left(\frac{x}{(x-1)AUC_{resid}}\right) \tag{9},$$
for example for x=2 and AUCresid=1% the duration should be n=7.64 T1/2, for AUCresid=20% the duration should be n=3.32 T1/2 (the exact value is 3.24.)
In particular, $$AUC_{resid}(T_{1/2},x)=\frac{x-2^{1-x}}{2(x-1)};\quad AUC_{resid}(T_{max},x)=\frac{x^{\frac{2-x}{1-x}}-x^{\frac{x}{1-x}}}{(x-1)};\quad AUC_{resid}(2T_{max},x)=\frac{x^{\frac{3-x}{1-x}}-x^{\frac{2x}{1-x}}}{(x-1)}.$$
AUCresid(Tmax,x) is a monotone function of x limited from 2/e (0.736) to 1;
AUCresid(2Tmax,x) is a monotone function of x limited from 3/e3 (0.406) to 1.

"Being in minority, even a minority of one, did not make you mad"  Ing. Helmut Schütz 