From: Nick Holford <*n.holford*>

Date: Sat, 20 Sep 2008 08:40:17 +1200

Hi Everyone,

Joachim, Thanks for starting this popular thread and for your clear

statements of two allometric commandments as applied to pharmacokinetics.

However, I feel I must emphasize that the first commandment is

"allometry predictions ONLY explain the effects of size".

Many times discussions of allometric predictions conclude with something

like 'allometry does not work' (e.g. see post in this thread from Masoud

in relation to children under 2 years). These conclusions are based on

ignoring the first commandment. Allometry ONLY explains the effect of

size and not the multitude of other effects such as maturation in young

children, body composition, renal function, species differences,

pharmacogenetic differences, disease effects (e.g. receptor density as

pointed out by Diane). Empirical tests of allometry are therefore very

difficult because they need to account for all other important non-size

related covariate that may be correlated with size e.g. age. See Savage

et al. (2004) to appreciate the scale of this task and Anderson &

Holford (2008) for the impracticality of common study designs in humans

to obtain precise estimates of allometric coefficients.

I've rewritten the next two commandments slightly to emphasise the

relationship to the quarter power 'law' from which they are derived (See

Savage et al. 2004 for references). Note that WTstd is a convenience for

standardising human PK parameters to a 70 kg value and is not really a

part of allometry itself.

Volume: Vstd*(WT/WTstd)** (4/4)

CL: CLstd*(WT/WTtd)**(3/4)

Joachim's statement about scaling of distribution and elimination rate

constants does not really reflect a separate commandment because it is

simply an algebraic consequence of applying the volume and clearance

commandments.

Absorption rate constants used to describe the rate of oral drug

absorption or absorption from a depot such as muscle are harder to

relate to allometric scaling principles. Under the simplest of

assumptions the absorption rate constant is just a diffusion

coefficient reflecting the local membrane structure. Given that cell

membrane structure is essentially the same for all sizes or organisms I

would not expect it to scale with weight. Oral drug absorption is of

course more complicated than just diffusion (cue for Walt W to appear).

One of the major determinants of the rate of absorption of drugs such as

BCS Type 1 (rapid dissolution, high permeability) is the rate of gastric

emptying. This rate can be understood in terms of the flow of gastric

contents into the duodenum which may well scale with size to the 3/4

power just like other flow like processes. The absorption process is

then more like a zero-order process with a duration inversely

proportional to the flow rate. Thus the duration might scale with a

power 4/3.

My own preference is not to try to scale absorption rate processes using

weight.

The Km of a mixed-order elimination process is of course a receptor

affinity property and not expected to scale with size. If you

empirically find a relationship with size then you should be looking

elsewhere for the real cause because it is certainly not based on allometry.

I derive the allometric scaling of mixed order elimination from the

clearance commandment. We know that CL at any given concentration for a

mixed order process is:

CL=Vmax/(Km+C)

As C tends to zero we get the first-order CL:

CL=Vmax/Km

As explained above there is no expectation that Km will scale with size

so in order to make first-order and mixed-order elimination consistent

it is necessary that Vmax scales to the 3/4 power just like CL. Note

that I prefer to parameterise Vmax as mass/time and not conc/time. The

conc/time parameterisation is confounded with volume of distribution and

thus the allometric power for that is -1/4. Leonid used the symbol VM'

for the mass/time parameter and VM for the conc/time parameter but

suggests the same conclusion.

Nick

Anderson BJ, Holford NH. Mechanism-based concepts of size and maturity

in pharmacokinetics. Annu Rev Pharmacol Toxicol. 2008;48:303-32.

Savage VM, Gillooly JF, Woodruff WH, West GB, Allen AP, Enquist BJ, et

al. The predominance of quarter-power scaling in biology. Functional

Ecology. 2004;18(2):257-82.

Joachim.Grevel

*>
*

*> Dear NM_Users,
*

*>
*

*> we have all been good students and listened to Nick when he told us
*

*> again and again the rock-solid truths of allometry:
*

*>
*

*> Volume: *(WT/70)
*

*>
*

*> CL: *(WT/70)**0.75
*

*>
*

*> any rate constant related to distribution or elimination:
*

*> *(WT/70)**(-0.25)
*

*>
*

*> Here my questions:
*

*> a) how do we allometrically scale a first-order rate constant of
*

*> absorption after oral dosing?
*

*>
*

*> b) how do we allometrically scale a first-order rate constant of
*

*> absorption from a subcutaneous injection site?
*

*>
*

*> Thank you for your thoughts,
*

*>
*

*> Joachim
*

*>
*

*> __________________________________________
*

*> Joachim GREVEL, Ph.D.
*

*> MERCK SERONO International S.A.
*

*> Exploratory Medicine
*

*> 1202 Geneva
*

*> Tel: +41.22.414.4751
*

*> Fax: +41.22.414.3059
*

*> Email: joachim.grevel *

--

Nick Holford, Dept Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand

n.holford

http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford

Received on Fri Sep 19 2008 - 16:40:17 EDT

Date: Sat, 20 Sep 2008 08:40:17 +1200

Hi Everyone,

Joachim, Thanks for starting this popular thread and for your clear

statements of two allometric commandments as applied to pharmacokinetics.

However, I feel I must emphasize that the first commandment is

"allometry predictions ONLY explain the effects of size".

Many times discussions of allometric predictions conclude with something

like 'allometry does not work' (e.g. see post in this thread from Masoud

in relation to children under 2 years). These conclusions are based on

ignoring the first commandment. Allometry ONLY explains the effect of

size and not the multitude of other effects such as maturation in young

children, body composition, renal function, species differences,

pharmacogenetic differences, disease effects (e.g. receptor density as

pointed out by Diane). Empirical tests of allometry are therefore very

difficult because they need to account for all other important non-size

related covariate that may be correlated with size e.g. age. See Savage

et al. (2004) to appreciate the scale of this task and Anderson &

Holford (2008) for the impracticality of common study designs in humans

to obtain precise estimates of allometric coefficients.

I've rewritten the next two commandments slightly to emphasise the

relationship to the quarter power 'law' from which they are derived (See

Savage et al. 2004 for references). Note that WTstd is a convenience for

standardising human PK parameters to a 70 kg value and is not really a

part of allometry itself.

Volume: Vstd*(WT/WTstd)** (4/4)

CL: CLstd*(WT/WTtd)**(3/4)

Joachim's statement about scaling of distribution and elimination rate

constants does not really reflect a separate commandment because it is

simply an algebraic consequence of applying the volume and clearance

commandments.

Absorption rate constants used to describe the rate of oral drug

absorption or absorption from a depot such as muscle are harder to

relate to allometric scaling principles. Under the simplest of

assumptions the absorption rate constant is just a diffusion

coefficient reflecting the local membrane structure. Given that cell

membrane structure is essentially the same for all sizes or organisms I

would not expect it to scale with weight. Oral drug absorption is of

course more complicated than just diffusion (cue for Walt W to appear).

One of the major determinants of the rate of absorption of drugs such as

BCS Type 1 (rapid dissolution, high permeability) is the rate of gastric

emptying. This rate can be understood in terms of the flow of gastric

contents into the duodenum which may well scale with size to the 3/4

power just like other flow like processes. The absorption process is

then more like a zero-order process with a duration inversely

proportional to the flow rate. Thus the duration might scale with a

power 4/3.

My own preference is not to try to scale absorption rate processes using

weight.

The Km of a mixed-order elimination process is of course a receptor

affinity property and not expected to scale with size. If you

empirically find a relationship with size then you should be looking

elsewhere for the real cause because it is certainly not based on allometry.

I derive the allometric scaling of mixed order elimination from the

clearance commandment. We know that CL at any given concentration for a

mixed order process is:

CL=Vmax/(Km+C)

As C tends to zero we get the first-order CL:

CL=Vmax/Km

As explained above there is no expectation that Km will scale with size

so in order to make first-order and mixed-order elimination consistent

it is necessary that Vmax scales to the 3/4 power just like CL. Note

that I prefer to parameterise Vmax as mass/time and not conc/time. The

conc/time parameterisation is confounded with volume of distribution and

thus the allometric power for that is -1/4. Leonid used the symbol VM'

for the mass/time parameter and VM for the conc/time parameter but

suggests the same conclusion.

Nick

Anderson BJ, Holford NH. Mechanism-based concepts of size and maturity

in pharmacokinetics. Annu Rev Pharmacol Toxicol. 2008;48:303-32.

Savage VM, Gillooly JF, Woodruff WH, West GB, Allen AP, Enquist BJ, et

al. The predominance of quarter-power scaling in biology. Functional

Ecology. 2004;18(2):257-82.

Joachim.Grevel

--

Nick Holford, Dept Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand

n.holford

http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford

Received on Fri Sep 19 2008 - 16:40:17 EDT