NONMEM Users Network Archive

Hosted by Cognigen

Understanding allometry

From: Nick Holford <n.holford>
Date: Mon, 23 May 2016 09:17:00 +0200


Thanks for your comments and questions on theory based allometry. I have
cross-posted it to nmusers because this topic has been of interest there
over many years.

I am aware that you wrote an editorial on this topic with Denis Fisher
which you titled “Allometry, Shallometry!”. Having read your editorial
and your comments I don’t think the implication that allometry is
shallow is appropriate. On the contrary, I get the impression (see
below) that you and Denis do not really understand the biological
principles underlying allometry and seem to be unaware of the
substantial literature supporting theory based allometry and its
application in humans.

Would your journal be willing to receive a rejoinder to your editorial
with a deeper explanation of the science and the literature?

Best wishes,


Nick Holford, MBChB, FRACP

Professor of Clinical Pharmacology, University of Auckland

Adjunct Professor of Bioengineering and Therapeutic Sciences, UCSF

On 21-May-16 00:42, Steven L Shafer <sshafer
> Nick:
> You say below:
> "Theory based allometry predicts an exponent of 3/4 for many functional processes e.g. basal metabolism, cardiac output, lung volume flow (West et al 1997). It is not restricted to metabolism."
> The article you cite by West is an excellent treatise on the subject of scaling across species. There are numerous other papers that provide theoretical foundations for allometric scaling across species (e.g., Darveau, Nature, 2002; West, J Exp Biol. 2005). The allometric theory accounts for differences in rate-related functions across species that span many orders of magnitude in body mass. I am not aware of any theory that supports scaling by weight to the 3/4 power WITHIN A SPECIES.

NH: Unfortunately, you repeat a common misunderstanding of allometric
theory that it is somehow only applicable across species. Allometric
theory as originally proposed by West is based only on body mass (West,
Brown et al. 1997). Allometric theory does not require consideration of
species or any other covariate. This is the first commandment of
allometry (Holford 2008). If you read the work by West et al. you will
find that there is nothing in the theory that prevents its use for
within species scaling using mass. Therefore the theory of West supports
the use of the 3/4 exponent within species.

> Similarly, I am unaware of unambiguous data strongly supporting allometric scaling across the typical range of human weights.

NH: I recommend that you read the paper by McCune et al. that formally
tests the allometric theory prediction of an exponent of 3/4 for
clearance based on a large study of busulfan across the human size range
(McCune, Bemer et al. 2014). The theory was tested explicitly and no
evidence found to reject the value of 3/4. Work with a drug where your
expertise is renowned ( has
clearly demonstrated the benefit of allometric theory across humans from
infants to adults. Eleveld showed the fit was improved using theory
based allometric scaling (Eleveld, Proost et al. 2014)Schuttler also
demonstrated an improved fit with an estimated exponent for clearance
0.75 which is consistent with the theoretical value of 3/4 (Schuttler
and Ihmsen 2000).

In the spirit of modern scientific philosophy are you aware of
unambiguous data (and analysis) that falsifies the theory of allometry


> As applied to human pharmacokinetics, I do not believe any theory supports allometric scaling.

NH: As noted above there is nothing in the theory of allometry proposed
by West that would mean it is not applicable to humans. If you do not
want to believe this theory then that is your personal choice, as it
would be for any religious belief, and I will not attempt to change your


> You can see this if you consider the ends of the spectrum. Small size is associated with children. They are not a separate species, but are humans undergoing metabolic maturation.

NH: I have personally been a strong advocate on considering all humans,
regardless of age, as being a single species and have sought integrated
explanations of human clinical pharmacology. If you were aware of the
paediatric pharmacokinetic literature then you would know of many
publications supporting the use of a combination of theory based
allometry for size plus empirical maturation models for age (see this
review (Holford, Heo et al. 2013)).


> I am not aware of any allometric theory that accounts for metabolic maturation with age.

NH: From the first commandment it necessarily follows that changes
associated with age are not predictable from the allometric theory of
West et al. There is no age related theory to predict quantitative
changes. However, plausible biological understanding of maturation means
that clearance will be zero (or at least very small) at conception and
will approach a maximum when it will be indistinguishable from the
mature adult value. So at least at the extremes there is a biological
and quantitative prediction of maturation. Joining these extremes
requires an empirical approach. A monotonic sigmoid emax function has
been suggested (Tod, Jullien et al. 2008)and widely applied (Holford,
Heo et al. 2013). A more complex function may be needed but this will
need to be driven first by data not by theory.

> Similarly, very large size is associated with morbid obesity. I am not aware of any allometric theory that suggests that clearance in morbid obesity is best estimated using allometric principles. Between these extremes, scaling by weight is not very different than scaling by weight to the three quarters power.

NH: Body composition contributes to body mass. Theory based allometry
does not specify how differences in body composition affect allometric
size. It is plausible however to propose that the size that is the
driving force behind allometric theory may not be determined simply by
total body weight. Application of theory based allometry in conjunction
with fat free mass can be used to determine a normal fat mass (NFM)
(Anderson and Holford 2009). The NFM concept has been used to account
for body composition differences determining allometric size. NFM is not
predicted from allometric theory but is a biologically plausible
extension of the theory of allometric size based on mass. NFM has been
used to show that total body mass rather than fat free mass provides a
better description of propofol pharmacokinetics in the obese (Cortinez,
Anderson et al. 2010). It has also be used to show that fat free mass is
a better predictor of dexmedetomidine but obesity is associated with
reduced clearance independently of allometric size based on fat free
mass (Cortinez, Anderson et al. 2015). This demonstrates how the
complexities of biology can be better understood based on a plausible
theory of allometry. The theory may not be perfect but it is compatible
with a very large number of observation studies in many domains.
Investigation of other phenomena such as maturation and obesity is aided
by building on allometric theory.


> Of course, I will defer to data. Can you point me to human PK examples where allometric scaling of weight to the three quarters power reliably provides substantially better fits to the data than scaling by weight alone?

NH: In addition to the large study of busulfan PK mentioned previously
(McCune, Bemer et al. 2014)I suggest you look at the prediction of
morphine clearance across the human size and age range using theory
based allometry with maturation. Prediction of clearance in a large
external data set was clearly better than other approaches including
empirical allometry (Holford, Ma et al. 2012). Other published examples
can be found in this review (Holford, Heo et al. 2013).


> I can point to many examples where it makes no difference. I can also point to many examples where investigators simply use allometric scaling without first seeing if allometric scaling was supported by the data.

NH: This is often the case when sample sizes are small, weight
distribution is narrow and power is small (Anderson and Holford 2008). A
pragmatic approach given the challenges of falsifying allometric theory
with small data sets is to assume it is useful. It is certainly better
than using empirical allometry or ignoring size altogether.

> However, I know of only one or two examples where models were estimated with and without allometric scaling, and the allometric scaling worked better than the simpler non-scaled model. If allometric scaling for human pharmacokinetics was “true” on first principles, as your comments imply, then the literature should abound with unequivocal examples.

NH: If you know of examples of suitably powered studies which can also
show they have accounted for other mass correlated factors that would
confound the estimation of a true allometric exponent then I would be
glad to know the details.

If you read the literature carefully and exclude those that are
underpowered to truly detect the difference between an exponent of 3/4
and say an exponent of 1 or an exponent of 2/3 and have accounted for
all other factors, such as maturation, that are necessarily correlated
with mass then you will not find many examples. I am not aware of any
that are inconsistent with allometric theory.


> Thanks,
> Steve
> --
> Steven L. Shafer, MD
> Professor of Anesthesiology, Perioperative and Pain Medicine, Stanford University
> Adjunct Associate Professor of Bioengineering and Therapeutic Sciences, UCSF
> ########################################################################
> PharmPK Home Page
> Information about PK/PD Jobs can be found at
> More information about my eBooks, including Basic Pharmacokinetics and Pharmacy Math, and now iOS and tvOS apps can be found at
> Thank you, David Bourne, PharmPK Moderator
> To unsubscribe from the PHARMPK list, click the following link:

Anderson, B. J. and N. H. Holford (2008). "Mechanism-based concepts of
size and maturity in pharmacokinetics." _Annu Rev Pharmacol Toxicol_
*48*: 303-332.

Anderson, B. J. and N. H. G. Holford (2009). "Mechanistic basis of using
body size and maturation to predict clearance in humans." _Drug Metab
Pharmacokinet_ *24*(1): 25-36.

Cortinez, L. I., B. J. Anderson, N. H. Holford, V. Puga, N. de la
Fuente, H. Auad, S. Solari, F. A. Allende and M. Ibacache (2015).
"Dexmedetomidine pharmacokinetics in the obese." _Eur J Clin Pharmacol_

Cortinez, L. I., B. J. Anderson, A. Penna, L. Olivares, H. R. Munoz, N.
H. Holford, M. M. Struys and P. Sepulveda (2010). "Influence of obesity
on propofol pharmacokinetics: derivation of a pharmacokinetic model."
_Br J Anaesth_ *105*(4): 448-456.

Eleveld, D. J., J. H. Proost, L. I. Cortinez, A. R. Absalom and M. M.
Struys (2014). "A general purpose pharmacokinetic model for propofol."
_Anesthesia and analgesia_ *118*(6): 1221-1237.

Holford, N. (2008). "Re: [NMusers] Scaling for pediatric study

Holford, N., Y. A. Heo and B. Anderson (2013). "A pharmacokinetic
standard for babies and adults." _J Pharm Sci_ *102*(9): 2941-2952.

Holford, N. H., S. C. Ma and B. J. Anderson (2012). "Prediction of
morphine dose in humans." _Paediatr Anaesth_ *22*(3): 209-222.

McCune, J. S., M. J. Bemer, J. S. Barrett, K. Scott Baker, A. S. Gamis
and N. H. G. Holford (2014). "Busulfan in Infant to Adult Hematopoietic
Cell Transplant Recipients: A Population Pharmacokinetic Model for
Initial and Bayesian Dose Personalization." _Clinical Cancer Research_
*20*(3): 754-763.

Schuttler, J. and H. Ihmsen (2000). "Population pharmacokinetics of
propofol: a multicenter study." _Anesthesiology_ *92*(3): 727-738.

Tod, M., V. Jullien and G. Pons (2008). "Facilitation of drug evaluation
in children by population methods and modelling." _Clin Pharmacokinet_
*47*(4): 231-243.

West, G. B., J. H. Brown and B. J. Enquist (1997). "A general model for
the origin of allometric scaling laws in biology." _Science_ *276*: 122-126.

Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology, Bldg 503 Room 302A
University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
office:+64(9)923-6730 mobile:NZ+64(21)46 23 53 FR+33(6)62 32 46 72

"Declarative languages are a form of dementia -- they have no memory of events"

Holford SD, Allegaert K, Anderson BJ, Kukanich B, Sousa AB, Steinman A, Pypendop, B., Mehvar, R., Giorgi, M., Holford,N.H.G. Parent-metabolite pharmacokinetic models - tests of assumptions and predictions. Journal of Pharmacology & Clinical Toxicology. 2014;2(2):1023-34.
Holford N. Clinical pharmacology = disease progression + drug action. Br J Clin Pharmacol. 2015;79(1):18-27.
Received on Mon May 23 2016 - 03:17:00 EDT

The NONMEM Users Network is maintained by ICON plc. Requests to subscribe to the network should be sent to:

Once subscribed, you may contribute to the discussion by emailing: