Re: [NMusers] Understanding allometry

From: Nick Holford <>
Date: Sun, 29 May 2016 21:55:57 +0200


I do not accept that my commentary and constructive criticism of the
public remarks made by Steve Shafer should be considered an insult.
Public discussion and criticism is essential to detect mistaken ideas
and fraud (

I am proud to be a co-author of the McCune et al (2014) article. I don't =

understand what you are implying with your "vested interest" remark. My
name was clearly written as a co-author in the reference list. Are you
suggesting I should not cite the science in this area that I have been
involved in for over 20 years?

Thank you for looking at the McCune article. However, you are mistaken
in suggesting that the "weight-normalized model" was not evaluated. This =

model (based on total body weight with an allometric exponent of 1) is a =

nested sub-model of size model described in the McCune article. When the =

allometric exponents are estimated the values provide no support for the =

"weight-normalized model" (see supplementary table 4). The objective
function improves by 532.12 when a non-linear allometric model with 4
additional parameters is compared to the linear model.

I find it strange that you offer anecdotally "Over the past two decades, =

I have analyzed data from > 200 studies" to support your
"weight-normalized" view yet insist I provide EVIDENCE. If you wish to
make a credible statement to support "weight-normalization" then I ask
you to publish the DATA with analysis details showing that you account
for factors such as maturation that are correlated with mass but are not =

part of theory based allometry, and apply allometric scaling to all
clearance and volume parameters.

With regard to SAFETY-- my work with allometric scaling applied to
prediction of clearance is not an ivory tower activity. You will find
details in the response I will send to Steve on PharmPK who raises a
similar point in his comments.

Best wishes,


McCune JS, Bemer MJ, Barrett JS, Scott Baker K, Gamis AS, Holford NHG.
Busulfan in Infant to Adult Hematopoietic Cell Transplant Recipients: A
Population Pharmacokinetic Model for Initial and Bayesian Dose
Personalization. Clin Cancer Res. 2014;20(3):754-63.

On 23-May-16 15:51, Dennis Fisher wrote:
> Nick
> Your condescending tone is not appropriate — Steve did not insu=
lt you nor should you insult him. An apology is due.
> You, Steve, and I had the same mentor — Lewis Sheiner. His mos=
t important teaching was LET THE DATA SPEAK. When theory and evidence cl=
ash, Lewis would not blindly stick to theory. In this instance, Steve as=
ked you to provide EVIDENCE to support your claim that allometric scaling=
 would yield markedly better fits than weight-normalization. You offered=
 the McCune article to support your argument (without mentioning your ves=
ted interest as an author). In that article, you wrote:
> all clearance (CL,Q) and volume (V1,V2) parameters were scaled for body=
 size and composition using allometric theory and predicted fat free mass=
> It does not appear that you evaluated a weight-normalized model. If yo=
u don’t look, you never see!
> I also note that the table reports the following:
> CL Clearance L/h/62kg NFM CL 11.4 (1.1)
> This brings up the issue of SAFETY. I was a clinician for several deca=
des and Steve continues to be an active clinician. I don’t know =
if you see patients. However, many participants in this mailing list hav=
e never selected a dose of a drug, then administered it to a patient. I =
venture to say that most clinicians when faced with a dosing regimen that=
 requires raising weight to the 3/4 power would run in the opposite direc=
tion (or, make a dosing error). The entry in the table is even more prob=
lematic. A clinician first needs to calculate NFM, then they need to rea=
lize that the dose is not proportional to 11.4 by a factor of weight/62.
> If the goal of PK is publishing journal articles about pure science, yo=
ur approach might be OK. But, as far as I know, the goal is to improve p=
atient safety. If there were a strong (or even moderate) preference for =
allometrically-scaled models, I would support their use. But, Steve aske=
d you to provide EVIDENCE for this and you failed to do you. You also ci=
ted Eleveld’s article. Although his manuscript did include one w=
eight-normalized model, the allometric model required multiple additional=
 terms to fit the data. In particular, that article (as yours) required =
a “maturation” term to describe younger patients. In oth=
er words, neither of these articles demonstrates that allometric models a=
re sufficient to describe the range of sizes in humans. Had Eleveld adde=
d those extra terms to the weight-normalized model, it might have perform=
ed as well as the model he published.
> Over the past two decades, I have analyzed data from > 200 studies incl=
uding many in infants and children. In many of these, I have compared we=
ight-normalized and allometric (and, in adults, unscaled) approaches. In=
 virtually all cases, the difference in fit between the weight-normalized=
 and allometric approaches was trivial and often favored the weight-norma=
lized approach. Can you cite cases in which the allometric approach fare=
s much better?
> I also have had the opportunity to be an editor for a journal and to re=
view articles for many journals (and I have reviewed submissions by many =
people who participate in this mailing list). In many instances, authors=
 refuse to evaluate a weight-normalized model, citing you. In many of th=
ese instances, I have insisted that the authors conduct that analysis and=
 (as far as I can recall) there has never been strong evidence to support=
 the allometric model.
> I repeat — if you don’t look, you don’t see.
> I suspect that Steve will have more to add.
> Dennis
> Dennis Fisher MD
> P < (The "P Less Than" Company)
> Phone / Fax: 1-866-PLessThan (1-866-753-7784)
>> On May 23, 2016, at 12:17 AM, Nick Holford<> =
>> Steve,
>> Thanks for your comments and questions on theory based allometry. I ha=
ve cross-posted it to nmusers because this topic has been of interest the=
re 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 th=
at allometry is shallow is appropriate. On the contrary, I get the impres=
sion (see below) that you and Denis do not really understand the biologic=
al principles underlying allometry and seem to be unaware of the substant=
ial literature supporting theory based allometry and its application in h=
>> 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
>> 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<> wrote:
>>> Nick:
>>> You say below:
>>> "Theory based allometry predicts an exponent of 3/4 for many functi=
onal processes e.g. basal metabolism, cardiac output, lung volume flow (W=
est et al 1997). It is not restricted to metabolism."
>>> The article you cite by West is an excellent treatise on the subjec=
t of scaling across species. There are numerous other papers that provide=
 theoretical foundations for allometric scaling across species (e.g., Dar=
veau, Nature, 2002; West, J Exp Biol. 2005). The allometric theory accoun=
ts for differences in rate-related functions across species that span man=
y orders of magnitude in body mass. I am not aware of any theory that su=
pports 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 theo=
ry as originally proposed by West is based only on body mass (West, Brown=
 et al. 1997). Allometric theory does not require consideration of specie=
s or any other covariate. This is the first commandment of allometry (Hol=
ford 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 exp=
onent within species.
>> SS:
>>> Similarly, I am unaware of unambiguous data strongly supporting allom=
etric 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 clearan=
ce 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 fo=
und to reject the value of 3/4. Work with a drug where your expertise is =
renowned ( has clearly demo=
nstrated the benefit of allometric theory across humans from infants to a=
dults. Eleveld showed the fit was improved using theory based allometric =
scaling (Eleveld, Proost et al. 2014)Schuttler also demonstrated an impro=
ved 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 unambig=
uous data (and analysis) that falsifies the theory of allometry (https://=
>> SS:
>>> As applied to human pharmacokinetics, I do not believe any theory s=
upports allometric scaling.
>> NH: As noted above there is nothing in the theory of allometry propose=
d by West that would mean it is not applicable to humans. If you do not w=
ant 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 relig=
>> SS:
>>> 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 hu=
mans 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 pa=
ediatric pharmacokinetic literature then you would know of many publicati=
ons supporting the use of a combination of theory based allometry for siz=
e plus empirical maturation models for age (see this review (Holford, Heo=
 et al. 2013)).
>> SS:
>>> I am not aware of any allometric theory that accounts for metabolic m=
aturation with age.
>> NH: From the first commandment it necessarily follows that changes ass=
ociated with age are not predictable from the allometric theory of West e=
t al. There is no age related theory to predict quantitative changes. How=
ever, plausible biological understanding of maturation means that clearan=
ce 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 predic=
tion 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 f=
unction may be needed but this will need to be driven first by data not b=
y theory.
>> SS:
>>> Similarly, very large size is associated with morbid obesity. I am no=
t aware of any allometric theory that suggests that clearance in morbid o=
besity is best estimated using allometric principles. Between these extre=
mes, scaling by weight is not very different than scaling by weight to th=
e three quarters power.
>> NH: Body composition contributes to body mass. Theory based allometry =
does not specify how differences in body composition affect allometric si=
ze. 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 fr=
ee mass can be used to determine a normal fat mass (NFM) (Anderson and Ho=
lford 2009). The NFM concept has been used to account for body compositio=
n differences determining allometric size. NFM is not predicted from allo=
metric 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 al=
so be used to show that fat free mass is a better predictor of dexmedetom=
idine but obesity is associated with reduced clearance independently of a=
llometric size based on fat free mass (Cortinez, Anderson et al. 2015). T=
his 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 m=
any domains. Investigation of other phenomena such as maturation and obes=
ity is aided by building on allometric theory.
>> SS:
>>> Of course, I will defer to data. Can you point me to human PK examp=
les where allometric scaling of weight to the three quarters power reliab=
ly provides substantially better fits to the data than scaling by weight =
>> 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 morph=
ine clearance across the human size and age range using theory based allo=
metry with maturation. Prediction of clearance in a large external data s=
et 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).
>> SS:
>>> 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 distrib=
ution is narrow and power is small (Anderson and Holford 2008). A pragmat=
ic approach given the challenges of falsifying allometric theory with sma=
ll data sets is to assume it is useful. It is certainly better than using=
 empirical allometry or ignoring size altogether.
>> SS:
>>> However, I know of only one or two examples where models were estimat=
ed with and without allometric scaling, and the allometric scaling worked=
 better than the simpler non-scaled model. If allometric scaling for huma=
n pharmacokinetics was “true” on first principles, as you=
r comments imply, then the literature should abound with unequivocal exam=
>> NH: If you know of examples of suitably powered studies which can also=
 show they have accounted for other mass correlated factors that would co=
nfound 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 underp=
owered 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 f=
actors, such as maturation, that are necessarily correlated with mass the=
n you will not find many examples. I am not aware of any that are inconsi=
stent with allometric theory.
>> SS:
>>> Thanks,
>>> Steve
>>> --
>>> Steven L. Shafer, MD
>>> Professor of Anesthesiology, Perioperative and Pain Medicine, Stanfor=
d University
>>> Adjunct Associate Professor of Bioengineering and Therapeutic Science=
>>> #####################################################################=
>>> PharmPK Home Page
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d Pharmacy Math, and now iOS and tvOS apps can be found athttps://www.pha=
>>> 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 usi=
ng 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 Fuen=
te, H. Auad, S. Solari, F. A. Allende and M. Ibacache (2015). "Dexmedetom=
idine pharmacokinetics in the obese." _Eur J Clin Pharmacol_ *doi:10.1007=
>> 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 obesi=
ty 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." _A=
nesthesia and analgesia_ *118*(6): 1221-1237.
>> Holford, N. (2008). "Re: [NMusers] Scaling for pediatric study plannin=
>> Holford, N., Y. A. Heo and B. Anderson (2013). "A pharmacokinetic stan=
dard for babies and adults." _J Pharm Sci_ *102*(9): 2941-2952.
>> Holford, N. H., S. C. Ma and B. J. Anderson (2012). "Prediction of mor=
phine 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 Initia=
l and Bayesian Dose Personalization." _Clinical Cancer Research_ *20*(3):=
>> Schuttler, J. and H. Ihmsen (2000). "Population pharmacokinetics of pr=
opofol: a multicenter study." _Anesthesiology_ *92*(3): 727-738.
>> Tod, M., V. Jullien and G. Pons (2008). "Facilitation of drug evaluati=
on 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 fo=
r the origin of allometric scaling laws in biology." _Science_ *276*: 122=

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

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ypendop, B., Mehvar, R., Giorgi, M., Holford,N.H.G. Parent-metabolite pha=
rmacokinetic models - tests of assumptions and predictions. Journal of Ph=
armacology & Clinical Toxicology. 2014;2(2):1023-34.
Holford N. Clinical pharmacology = disease progression + drug action. B=
r J Clin Pharmacol. 2015;79(1):18-27.

Received on Sun May 29 2016 - 15:55:57 EDT

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