NONMEM Users Network Archive

Hosted by Cognigen

[NMusers] Understanding allometry

From: Nick Holford <>
Date: Wed, 1 Jun 2016 10:27:04 +0200

Dear Steve,

Thanks for your responses of 22 May and 24 May PDT. Due to some
unexplained technical problem I have not been receiving posts from
PharmPK since 23 May and only just became aware of your 24 May post. I
have tried twice to post a response to PharmPK but it has not appeared
on the list as far as I can tell. I am therefore transferring this
discussion to nmusers.

On 22 May you wrote “I am not aware of any theory that supports scaling
by weight to the 3/4 power WITHIN A SPECIES”. Having read the 22 May
post from Douglas Eleveld and my own reply do you now accept that the
theory of West et al. supports 3/4 power weight scaling within a species
including humans?

My comments below are in response to your PharmPK post of 24 May.

Best wishes,



Dear Nick:

You stated in your comments to me, both on NMUSERS and on PHARMPK, that
a paper of yours with Dr. McCune offered a validation of the use of
allometric scaling. Dennis Fisher has already commented on this paper.
Dennis said I might have additional comments. As usual, Dennis is right.

The paper is McCune et al, Busulfan in infant to adult hematopoietic
cell transplant recipients: a population pharmacokinetic model for
initial and Bayesian dose personalization.. Clin Cancer Res.
2014;20:754-63. Based on my reading of your paper, and the supplementary
material, I would like to offer several observations.

1. Quoting from page 755: “To characterize busulfan pharmacokinetics
over the entire age continuum, all clearance (CL,Q) and volume (V1, V2)
parameters were scaled for body size and composition using allometric
theory and predicted fat-free mass.” In other words, you assumed
allometric theory at the outset of your analysis.

NH: It is correct that theory based allometry was used at the outset to
guide the analysis. This was based not only the biological plausibility
of this theory but also the consistent use of non-linear weight scaling
in all previous reports of IV busulfan PK (see McCune Supplementary
Table 1). The results of non-linear regression using NONMEM are well
known to be better when starting from a priori more reasonable models
and parameters. A deeper understanding of the problem is always helpful.

2. I think your testing of this assumption is described on page 759:
“we estimated the allometric exponents for each of the 4 main
pharmacokinetic parameters (Supplementary Table S4). Initial estimates
of 2/3 and 1.25 were used for the clearance and volume exponents.”
However, you tested this with bootstrap, not with log-likelihood
profiles. Why?


A recognized limitation of log-likelihood profiles is the focus on just
one parameter at a time plus interpretation of the confidence interval
is dependent on the assumption that the difference in -2LL is
Chi-squared distributed. As I am sure you know that is a questionable
assumption when using the NONMEM approximation to the likelihood.

While the bootstrap cannot be considered perfect it has useful
properties compared with log-likelihood profiles because it allows the
uncertainty for all exponents to be estimated simultaneously. The
log-likelihood profile method provides a rather superficial view because
it only allows the uncertainty for one parameter at time while fixing
other parameters at some other values.

3. If allometry makes little difference, then it is an expected
result that your final estimates would be close to the starting
parameters. This might especially be the case where there are 10
parameters in the calculation of clearance (see item 10 below),
compromising the “navigability” of the model away from the starting
estimate of PWR when the other 9 parameters start at the value
determined by assuming that PWR=0.75.

NH: The initial estimates for the bootstrap were deliberately set to
values that were quite distant from the theory based values (page 759,
para 1). This would be expected to allow the exponent estimates to
“navigate” more freely. As you point out it is possible that there would
be some kind of “memory” in the other parameters of the model determined
using theory based exponents.This is a good idea. I have tried to test
it by fitting the data with initial estimates assuming that total body
weight is the allometric mass with an allometric exponent for CL and V
parameters equal to 1 (linear weight model). The results of such a fit
should have no “memory” of theory based values. I then refit the data
starting from the final estimates of this linear weight model and
estimated the allometric exponents for CL, V, Q and V2 (empirical
allometric model). The bootstrap confidence intervals for these
exponents estimated from the empirical allometric model with total body
weight are shown here:





Bootstrap Average


2.5% ile


97.5% ile


Bootstrap RSE



Allometric exponent for CL











Allometric exponent for V1











Allometric exponent for Q











Allometric exponent for V2









It is clear that the exponent estimates for CL and V consistent with
theory based allometry and inconsistent with a linear weight model. Not
accounting for body composition probably explains why the confidence
interval for V2 just misses the theory based prediction. This is not due
to a “memory” problem because very similar results were obtained
starting from theory based initial estimates and total body weight as
shown in McCune supplementary table 4.

The objective function value (OFV) difference between the linear model
and the empirical allometric model is 532.12 with 4 estimated
parameters. This difference in OFV would usually be interpreted to show
that the non-linear allometric model is superior to the linear function
of weight. This is letting the data speak and it is shouting out that
the linear model is inconsistent with the data.


4. I stated in my comments that allometric theory did not account for
the upper extreme of obesity. You agree, since you found it necessary to
corrected allometric scaling with an additional parameter to account for
the effects of obesity.
NH: I am happy to see that we agree that “obesity” is not part of theory
based allometry. However, please recognise that the mass associated with
“obesity” would be expected under theory based allometry to have an
influence. A deeper, biological question is what is this mass? The West
et al.theory does not specify how to account for mass, such as excess
fat, that probably has metabolic properties different from mass with
“normal” body composition. The NFM method estimates the combination of
fat free mass and fat mass that best agrees with allometric theory. The
assumption of allometric theory then allows insights into the
contributions of these components to “normal” allometric size.


5. I stated in my comments that allometric theory did not account for
maturation. You agree, since you found it necessary to add an additional
parameter for maturation.

NH: I am very happy to see that we agree that allometric theory does not
account for maturation. That is why a separate model component (with 2
parameters TM50CL and HIllCL) is included to account for maturation of


6. You had actual body weight for only 133 subjects in this study, of
which only 24 subjects were less than 18 years of age (supplementary
table 2). Although your model has 1610 individuals, you only estimated
the allometric portion of your model from 24 children. This allometric
scaling parameter was assumed to be true for all 1407 subjects
(calculated from supplementary table 2). Since your allometric
parameter, Ffat, was derived from just 24 children, and applied to all
1407 children, your testing (supplementary table S4) may be a tautology.

NH: It is indeed a limitation of this data analysis that most of the
hospitals who contributed data did not supply actual body weight but a
“dosing weight” (see below). Ffat contributes to the allometric part of
the model but recall that allometric theory does not include maturation.
The Ffat parameters were estimated from all 133 subjects , irrespective
of age, with actual body weight recorded. Therefore I do not agree with
your assertion that I “only estimated the allometric portion of your
model from 24 children”.


7. You state on page 755 that you used the dosing weight in reference
18. Reference 18 is Gibbs, et al, The Impact of Obesity and Disease on
Busulfan Oral Clearance in Adults, Blood 1999;93:4436-40. Reference 18
discusses actual body weight, body surface area, adjusted ideal body
weight, and ideal body weight, all calculated from standard formulae.
There is no reference to Dosing Weight in this publication.

NH: You are correct. There is no reference describing DWT for patients
outside Seattle. This is because the method used to calculate DWT was
calculated using each institution’s own practice (see page 755, col 2,
end of 2nd paragraph describing the study population). The exact method
was not supplied with the data provided for each patient by the
institution. The dosing weight in Seattle was not used for model
development because actual body weight was recorded.


8. Dennis pointed out the potential safety concerns of allometric
scaling. I suggest that interested readers look at page 756 of your
paper. If that does not scare clinicians, then the exact math for dose
calculation appears in supplementary table 7. Would you be comfortable
if the oncologist treating your child had to calculate dose based on the
complex, interlocking equations required to estimate body size? What
theoretical advantage in dose calculation justifies the potential for
computational error inherent in supplementary table 7? The risk vs.
benefit of allometric scaling cannot be determined from the data in the

NH: You bring up an interesting issue that was never the subject of this
paper.Unfortunately, the literature abounds with evidence that doctors
e.g. anesthetists (Nanji, Patel et al. 2016), often make dosing errors.
My work with allometric scaling applied to prediction of clearance and
dosing is not an ivory tower activity.Safe dosing of dangerous medicines
requires good science and validated methods to ensure the correct dose
is administered.

The calculation of the dose for busulfan is complex. This is the only
drug I know of where the FDA has recommended detailed dose
individualization including calculation of an AUC in order to use the
drug (FDA 2015). At the request of my clinical colleagues in Auckland, I
was involved in the implementation and testing of a tool to guide
busulfan dosing in children and adults. The tool was initially based on
an FDA study (Booth, Rahman et al. 2007) describing an allometrically
scaled model for clearance and was subsequently updated using the
results in the McCune paper when an audit showed the predictions were
better. The web based dosing tool ( has been in use for
over 4 years to guide dosing of all patients in New Zealand receiving
high dose busulfan for bone marrow ablation. The tool is available for
use by anybody who has access to the internet. I understand that the
model will be used to guide dosing of all patients in the USA who have
samples submitted to the national laboratory in Seattle for measurement
of busulfan concentrations.

It is my personal hope that dosing decisions will be taken out of the
hands of doctors who rarely recognize the principles of rational dosing
and continue to use ad hoc empirical methods.

To quote from your “Allometry, Shallometry!” editorial, with an example
based on what you claim is a simple approach using the linear weight
model: “It is OK if you skipped the math. As a clinician, all you need
to know is the punchline”. You have been a pioneer in this area with
target controlled infusions so I don’t think I have to convince you that
this is the way of the future. Doctors should be responsible for
providing the data to decide on an appropriate dose and after that a
science based computation tool should work out the dose.


9. You have no data showing how well your model predicts individual
patients. The closest you come are the visual predictive checks (figure
1) and the prediction corrected visual predictive check (supplement 2).
This tells me that the cloud of points is about right. That’s fine, but
the average patient does not die. It is patient at the extremes of
prediction accuracy who are at increased risk. The data, as presented,
does not provide this information.

NH: This paper is based on pharmacokinetic data. There is no
effectiveness or safety data to judge risk. However, we have described
the expected fraction of patients that would be expected to be within an
acceptable range of concentrations using a predicted initial dose. Our
model performs better than other methods in nearly all the scenarios we
tested and is never worse to a clinically important degree. As noted
above a web based dosing system using this approach has been used by
clinicians in Auckland for over 4 years.


10. Clearance (page 757) is calculated 10 parameters:
a population estimate, which is adjusted for F(size), F(maturation), and
F(sex). F(size) is based on dosing weight (not explained, see 7 above),
height, WHS(50), WHS(max), F(fat), FFEM(DW), and PWR (your allometric
parameter, fixed at ¾). F(maturation) is based on PMA, TM(50), and the
Hill coefficient. F(sex) is a further adjustment for sex. When clearance
is a function of 10 parameters, I do not see how this tests allometric
scaling. Indeed, if allometric scaling were hurting your fit (unlikely –
more likely it makes no difference, see below), other parameters might
compensate to fit the data.

NH: Please look at Table 2 to count the parameters in the fixed effect
model. There are 12 estimated parameters. The number of parameters is
not a “test of allometric scaling”. It is a measure of the complexity of
variability of busulfan PK. These parameters identify predictable
sources of variability that can be used to aid initial dosing.


11. You compare this model to models by Trame, Paci,
and Bartelink, noting that your model performs much better than these
models. You are comparing your model with 12 structural parameters to
models with 2 (Trame), 4 (Paci), and 5 (Bartelink) structural
parameters. Your 12 parameter model better described your data than
these simpler structural models fit to your data. Did you expect
anything else?

NH: I certainly expected to find our model would do better because it
has a stronger mechanistic and biological basis. It is more complex than
others because it goes more deeply into biological understanding and
does better over a wide range of human size and age.


12. You state on page 762: “The model is based on
principles that have already been shown to be robust for predictions
with other small molecule agents from neonates to adults.” I don’t see
that. If “robust” means that it allometric helps describe PK at the
extremes of weight, then the allometric model was not robust. It
required adjustments for both maturation and for obesity. Between these
extremes, say 30-100 kg, any optimal coefficient times weight to the ¾
power will differ by less than 10% from an optimal coefficient times
weight alone. This will be invisible given the order of magnitude
variability in clearance (your figure 2).

NH: Robust refers to principles which recognize the major role of size
and maturation (the key components of our model) in explaining
variability in PK for many drugs (see (Holford, Heo et al. 2013)).The
influence of body composition as a predictor of allometric size has
fewer examples but it is only by digging below the surface that we can
discover new things and evaluate their importance.

It is no surprised that over a narrow range (30-100 kg) a linear model
is a reasonable approximation to theory based allometry. But this is not
true over the range of TBW (3 to 140 kg; see Figure 1) in the patients
in this study (see below).

I see little to no evidence that your paper with Dr. McCune demonstrated
superiority of allometry. Rather, your paper demonstrated that even a
model with 12 parameters could not reduce the variability of busulfan
estimated clearance beyond an order of magnitude.

NH: If the model can predict how to reduce variability by an order of
magnitude for a very toxic drug such as busulfan then I think this is a
major advance. It makes no difference how many parameters are needed.
The important thing is to be able to predict differences in PK which can
then be applied to achieve safe and effective dosing. If this was my
child faced with a bone marrow transplant I would want to use every
means possible to improve the chances of a successful graft and reduce
the substantial risk of serious toxicity and death.

The simulations you provide in your “Allometry, Shallometry!” editorial
replicate what I demonstrated 20 years ago (Holford 1996). I pointed out
at that time the underestimation of doses predicted from adults if a
linear weight model was assumed. You appear to propose using the same
mg/kg dose in a children as in adults for computational convenience. But
clinically recommended dosing regimens for busulfan use a higher mg/kg
dose in younger and lighter children with lower mg/kg doses for older
and heavier children. The allometric and maturation model we have
developed predicts and explains this pattern of mg/kg dosing
recommendations for busulfan and all other drugs used in humans
(Holford, Heo et al. 2013).


You also demonstrated that allometric models require specific
adjustments for maturation and dosing. You will recall this was one of
the points that I made in my comments, which are also discussed in the
Allometry Shallomatry! editorial.

NH: I think we are in agreement that theory based allometry can only
explain variability due to differences in body mass. Other factors also
explain variability such as maturation, organ function, drug
interactions, genotypes, etc. These factors have no influence on the
allometric component of the model. I do not agree with you when you say
that allometric models requires “specific adjustments” using factors
such as maturation. If you tried to understand more deeply the
allometric model you would realize it is not based on these other factors.


Perhaps there are other analyses of these data that would demonstrate a
significant benefit of allometric scaling of data. If you are willing to
share with me your data on the 133 subjects for whom you have actual
body weights, I would be happy to address the question directly.

NH: I understand that Jeannine McCune has contacted you and offered to
work with you to obtain permission to use the data.



Steven L. Shafer, MD
Professor of Anesthesiology, Perioperative and Pain Medicine, Stanford
Adjunct Associate Professor of Bioengineering and Therapeutic Sciences, UCSF

NH: References

Booth, B. P., A. Rahman, R. Dagher, D. Griebel, S. Lennon, D. Fuller, C.
Sahajwalla, M. Mehta and J. V. Gobburu (2007). "Population
pharmacokinetic-based dosing of intravenous busulfan in pediatric
patients." _J Clin Pharmacol_ *47*(1): 101-111.

FDA (2015). "Busulfex Product Label"

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. (1996). "A size standard for pharmacokinetics." _Clin
Pharmacokinet_ *30*(5): 329-332.

Nanji, K. C., A. Patel, S. Shaikh, D. L. Seger and D. W. Bates (2016).
"Evaluation of Perioperative Medication Errors and Adverse Drug Events."
_Anesthesiology_ *124*(1): 25-34.

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 Wed Jun 01 2016 - 04:27:04 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: