## Fantastic post 👍🏽 [NCA / SHAM]

Hi Nastia,

limited time (bloody bread-and-butter job). Some desultory thoughts about PK metrics (more to come).

» Cmax/AUC is referred sometimes to be more appropriate in BE studies than Cmax cause it has much lower variability…

Not only that (it’s a side-effect). What are we interested in? Extent of absoption (cleary AUC) and rate of absorption (ka and possibly tlag). ka is not easily accessible in NCA. Cmax is a composite surrogate (because influenced by AUC). Easy to show: Define any PK model and vary ƒ whilst keeping ka and tlag constant. Cmax will change… Cmax/AUC is an attempt to deal with that.

» The distribution type of the function is also questionable (ratio of log-normal - paranormal ?).

Do you know ’Pataphysics? Seriously, László (The Younger) asked me the same question years ago, which I could not answer. Martin helped us out. It doesn’t matter: The sum/difference of two normal distributions will be normal, the same here: It will be log-normal.

» pAUC (partial AUC, truncated AUC), index of early exposure, as was discussed in the parallel thread could be more sensitive than AUClast ('pAUC was always more sensitive than Cmax'2). The question is what time should be the end time of the calculation? […] The time should be related to clinically relevant pharmacodynamic measure. What are their advantages?

The jury is out. E.g., for biphasic methylphenidate the cut-off time (FDA: 3 h fasting, 4 h fed) is based on PD indeed (at that time ~90% of patients show the maximum effect). Makes sense. The EMA it its eternal wisdom asks to set the cut-off based on PK (a trough between the two parts). Splendid. Some subjects show just a shoulder (see there) and mean curves of the innovator are completely useless.

» AUMC (first order momentum, the area under the curve 'Concentration*time-time'). If we'll look at the physics analogy - the first-order momentum for the plain figure is the static momentum which defines the center of mass coordinates, while the second-order momentum is moment of inertia. The definition could be generalised to the n-th
» order momentum (why to limit ourselves by first order? There could be second or third order as well...).

Not only physics. Statistical distributions have also moments and we can interpret the behavior of drug molecules as a stochastic process. I love moments. I general

$$S_i=\int x^i\cdot f(x)dx$$

and in PK $$\small{i=0\ldots2}$$. Hence,

$$S_0=\int x^0\cdot f(x)dx = \int f(x)dx$$,
$$S_1=\int x^1\cdot f(x)dx = \int x\cdot f(x)dx$$,
$$S_2=\int x^2\cdot f(x)dx$$,

where in PK $$\small{x=t}$$ and $$\small{f(x)=C}$$. Then $$\small{AUC=S_0}$$, $$\small{AUMC=S_1}$$ and $$\small{MRT=AUMC/AUC}$$. $$\small{S_2}$$ is practically useless. OK, some people calculated $$\small{VRT = S_2/S_0-(S_1/S_0)^2}$$, the “Variance of Residence Times” or “Gravity Duration” (stop searching; out of fashion for decades). The coordinates $$\small{\{MRT\:|\:VRT\}}$$ define the “Center of Gravity” of the curve. Only nice to print a profile, cut it a out, push a pin through it, and make a weird whizz wheel for kids.

» MRT is a most common PK parameters that compels to look at the AUMC. There existed approaches4 to use MRT instead of T1/2 in order to choose the appropriate washout period.

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.

More to come.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes