## AUMC ⇒ MRT [PK / PD]

BE-proff,

» […] I can't catch the idea why AUMC is required?

In the BE-context you generally don’t need it at all. It’s only an intermediate step in calculating MRT. Who asked for it?

» Can anybody explain me?

We can see plasma profiles as the result of a stochastic process (drug molecules are absorbed, distributed, and metabolized/excreted). At any given time point a certain fraction is in the cir­cu­la­tion. Hey, that’s a sta­tis­tical distribution!

Distributions can be described by ‘statistical moments’

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

where $$\small{S_0}$$ is the AUC and $$\small{S_1}$$ the AUMC (Area Under the Moment Curve). In PK $$\small{x=t}$$ and $$\small{f(x)=C}$$. We need the AUMC to calculate the MRT (Mean of Residence Times), which is for
• iv bolus
MRT = AUMC / AUC
• and for infusion (of length tinf)
MRT = AUMC / AUC – ½tinf
Rule of thumb: After MRT ~⅔ of the drug is eliminated.

S2 is rarely used (for the calculation of the ‘Variance of Residence Times’): VRT = S2/S0 – (S1/S0)2. MRT and VRT are the coordinates (x|y) of the ‘center of gravity’ of the profile. Try it: Plot a PK-curve on cardboard and stick a pin through x|y. Makes a nice children’s wind wheel.

Dif-tor heh smusma 🖖
Helmut Schütz

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