## SHAM(e) math [NCA / SHAM]

❝ 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 t_{½} 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 AUC_{0–∞} (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 t

^{2}. If you want to have a constant proportion you need a vessel with a form of parabola x

^{4}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 C

_{c}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 C

^{2}in it, it's variability was always lower than that of C

_{max}, 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 C

_{max}?

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-AUC

_{0-t}/AUC

_{0-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 T

_{1/2,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 AUC

_{resid}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 AUC

_{resid}=1% the duration should be n=7.64 T

_{1/2}, for AUC

_{resid}=20% the duration should be n=3.32 T

_{1/2}(the exact value is 3.24.)

In particular, $$AUC_{resid}(T_{1/2},x)=\frac{x-2^{1-x}}{2(x-1)};\quad AUC_{resid}(T_{max},x)=\frac{x^{\frac{2-x}{1-x}}-x^{\frac{x}{1-x}}}{(x-1)};\quad AUC_{resid}(2T_{max},x)=\frac{x^{\frac{3-x}{1-x}}-x^{\frac{2x}{1-x}}}{(x-1)}. $$

AUC

_{resid}(T

_{max},x) is a monotone function of x limited from 2/e (0.736) to 1;

AUC

_{resid}(2T

_{max},x) is a monotone function of x limited from 3/e

^{3}(0.406) to 1.

"Being in minority, even a minority of one, did not make you mad"

### Complete thread:

- Fantastic PK parameters and where to find them Astea 2020-06-12 14:22 [NCA / SHAM]
- Fantastic post ?? Helmut 2020-06-12 16:23
- Rattleback Astea 2020-06-13 10:17
- Chatter Helmut 2020-06-13 12:28
- Reinventing the Hula-Hoop Astea 2020-06-16 01:33
- Hula-Hoop references Helmut 2020-06-16 13:07
- Prof. Keller vs. Yamaoka mittyri 2020-06-17 14:28
- SHAM(e) mathAstea 2020-06-23 14:41
- SHAM(e) math Helmut 2020-06-23 15:55
- OT: Möbius strip Astea 2020-06-23 21:41
- OT: Möbius strip Helmut 2020-06-24 11:29

- OT: Möbius strip Astea 2020-06-23 21:41

- SHAM(e) math Helmut 2020-06-23 15:55

- SHAM(e) mathAstea 2020-06-23 14:41

- Prof. Keller vs. Yamaoka mittyri 2020-06-17 14:28

- Hula-Hoop references Helmut 2020-06-16 13:07

- Reinventing the Hula-Hoop Astea 2020-06-16 01:33

- Chatter Helmut 2020-06-13 12:28

- Rattleback Astea 2020-06-13 10:17
- More stuff Helmut 2020-06-13 15:28
- MRT and Gravity duration mittyri 2020-06-14 22:24
- Fantastic PK parameters and where to find them ElMaestro 2020-06-16 09:46
- Cauchy distribution mittyri 2020-06-16 10:54
- Cauchy distribution Helmut 2020-06-16 13:14
- Cauchy distribution ElMaestro 2020-06-20 10:33
- noncentral normal ratio mittyri 2020-06-20 23:04

- Cauchy distribution mittyri 2020-06-16 10:54

- Fantastic post ?? Helmut 2020-06-12 16:23