Centering Covariates
Covariate Models Using CrCL
Predefined Models vs. "Context-Sensitive" Emperical Models

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Subject:Covariate Models Using Weight
Date: Tue, 16 Nov 1999 11:24:31 +0100

Dear Rebecca,

You should be more specific when posting your questions. The convergence behaviour of NONMEM depends very much on the METHOD you select for \$EST. It can be totally different for METHOD=0 (the default first order linearization method) and for METHOD=1 (first-order conditional method). It would be better if you attach the entire NM-TRAN control.

The way you implement the fixed effect of WT is not optimal. Firstly, it is preferable to center it using median WT (say, 70) as offset. Then, it worth to include an intercept in the fixed effect model, e.g.: TVCL=THETA(1)+THETA(2)*(WT-70). THETA(1) corresponds to the typical clearance at median WT. Lastly, I would strongly recommend to avoid using TRANS3. You will have much less troubles with TRANS4.

Hope this helps,

----------------------------------------------------------------------
Janssen Research Foundation
Clinical Pharmacokinetics
B-2340 Beerse
Belgium
Email: vpiotrov@janbe.jnj.com

*****

Date: Wed, 17 Nov 1999 14:09:55 +1300
From: Nick Holford <n.holford@auckland.ac.nz>
Subject: Covariate Models Using Weight

I agree with your comments about the value of centering to improve the estimation of parameters in the covariance model. There is no need to be obsessional about using the median. Any convenient value that is approximately in the middle of your data is fine. Remember that the final parameter estimate you obtain will be defined in terms of this centering value. I prefer to refer to this centering value as the standard covariate value e.g. a weight of 70 kg is a widely recognized standard for human adult weight.

However, I have argued (Holford 1996) on data driven and biological grounds that models for body size should be based on the allometric model:

CLi = CLstd * (Wi/Wstd)**3/4
Vi = Vstd * (Wi/Wstd)**1

where CLi, Vi are CL and V in an individual with weight Wi and Wstd is a standard weight e.g. 70 kg. CLstd and Vstd are the population parameters standardized to the size of an individual with weight Wstd.

The exponent value of 3/4 for CL can be justified on theoretical grounds (West et al. 1997) and is supported by allometric studies of a wide variety of body functions with an estimate of this exponent indistinguishable from 0.75 (Peters 1983). Justification for V and other body volumes having an allometric exponent of 1 has been reviewed by Anderson et al. 1997.

Note these models do not have an intercept parameter. I believe it is an a priori more reasonable model to expect that CL or V will be zero when WT is zero. I prefer to put my faith in biology and mechanism when choosing a model. I resort to statistical heuristics (e.g. change in log-likelihood) when the biology or mechanism is not obvious.

I suspect that empirical estimates of allometric exponents reported in the literature for PK parameters are most likely indistinguishable from the a priori value of 3/4 for CL and 1 for V. If the null hypothesis that the exponents are 3/4 and 1 is rejected then careful thought should be given to other confounding factors in the data rather than rejecting a priori well established biological principles.

Anderson BJ, McKee D, Holford NHG. Size, myths and the clinical pharmacokinetics of analgesia in paediatric patients. Clinical Pharmacokinetics 1997;33:313-327

Holford NHG. A size standard for pharmacokinetics. Clin. Pharmacokin. 1996: 30:329-332

Peters RH. The ecological implications of body size. Cambridge University Press.1983

West GB. Brown JH. Enquist BJ. A general model for the origin of allometric scaling laws in biology. Science. 1997; 276:122-6
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford@auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm

*****

Subject: RE: Covariate Models Using Weight
Date: Wed, 17 Nov 1999 09:08:04 +0100

I totally agree with you, Nick, when you highlight the importance of "biology and mechanism when choosing a model". And I also agree with the value of allometry in general. But only in general. And only in the context of interspecies correlations. I don't believe in allometry when it is applied to a single species like humans.

Specifically, the dependence of CL on body size is very complex and cannot be modelled in terms of a simple and universal function like CLi = CLstd * (Wi/Wstd)**3/4. First of all, we cannot apply the same function for drugs eliminated mainly via hepatic metabolism and via renal excretion controlled either by glomerular filtration, or by enzymes that govern active secretion and reabsorption, or both. Then, if a drug is extensively metabolised and the hepatic blood flow does not play any significant role (most typical case), the dependence of CL on body size is minor, and it is usually better correlated with LBM or IBW than with WT itself.

Of course, TVCL=THETA(1)+THETA(2)*(WT-70) is not ideal, but most suitable as the first approximation. TVCL=THETA(1)*(WT/70)**THETA(2) might be preferable if the range of WT is wide enough, however, THETA(2) certainly should not be fixed to 0.75.

The situation with V is not less complicated than with CL. Firstly, there is no V, but VC, VP, etc., and each volume may depend on body size differenly. Again, allometry is not something to be taken into account when building a model for one species.

Best regards,
----------------------------------------------------------------------
Janssen Research Foundation
Clinical Pharmacokinetics
B-2340 Beerse
Belgium
Email: vpiotrov@janbe.jnj.com

*****

Date: Wed, 17 Nov 1999 22:10:08 +1300
From: Nick Holford <n.holford@auckland.ac.nz>
Subject: Re: Covariate Models Using Weight

I also like to express some opinions as beliefs e.g. it is apparently our shared belief that biology and mechanism have a high priority in choosing a model. But the issue of the merits of allometric scaling within species e.g. humans, is not simply a belief but has hard data to support it. Please read the review paper by Brian Anderson which I cited (Anderson et al. 1997). I would be delighted to hear your comments about the merits of allometry for humans when you have done so.

> Specifically, the dependence of CL on body size is very complex and cannot
> be modelled in terms of a simple and universal function like CLi = CLstd *
> (Wi/Wstd)**3/4. First of all, we cannot apply the same function for drugs
> eliminated mainly via hepatic metabolism and via renal excretion controlled
> either by glomerular filtration, or by enzymes that govern active secretion
> and reabsorption, or both.

Why not? It is simple e.g. for a drug eliminated by a low extraction ratio mechanism (glomerular filtration) and also by a high extraction ratio process (hepatic metabolism) then I am sure you accept:

CLtotal = CLgfr + CLi*Q/(Q+CLi)

where CLi is intrinsic clearance of the organ with blood flow Q. The allometric expression for this is:

CLgrf=CLgfrstd*(W/Wstd)**3/4
CLi=CListd*(W/Wstd)**3/4
Q=Qstd*(W/Wstd)**3/4

What is the problem? Make the expression for clearance as complicated as you want -- the components can all be scaled allometrically. This has been done widely in cross-species scaling using PBPK (Physiologically Based Pharmaco Kinetic) models.

If allometric scaling is applied to predict the size covariate then it is possible to take a rational approach to unravelling the separate influences of other covariates such as age which reflect developmental maturation (Holford 1996).

> Then, if a drug is extensively metabolised and
> the hepatic blood flow does not play any significant role (most typical
> case), the dependence of CL on body size is minor

If I interpret this literally it would mean that you think clearance of such a drug is about the same in a 10 kg child and a 200 kg rugby player and that they both should get the same maintenance dose rate. You aren't serious are you?

>, and it is usually better
> correlated with LBM or IBW than with WT itself.

I have no idea why you make these remarks. LBM or IBW are no different from WT except they may be better predictors of body size when body composition differs from 'normal' e.g. in the obese person.

> Of course, TVCL=THETA(1)+THETA(2)*(WT-70) is not ideal, but most suitable as
> the first approximation. TVCL=THETA(1)*(WT/70)**THETA(2) might be preferable
> if the range of WT is wide enough, however, THETA(2) certainly should not be
> fixed to 0.75.

Please explain why "THETA(2) certainly should not be fixed to 0.75". I gave you references to support my statements that an allometric exponent of 3/4 has a strong theoretical basis (West et al. 1997) AND is supported by data (Peters 1983, Anderson et al. 1997). If you want to challenge my view then please give me chapter and verse not dogma. I ask you to pay particular attention to the dangers of rejecting the wrong null hypothesis (e.g. comparing a model for clearance with an exponent different from 3/4 to a model with an exponent of 0 (the 'no effect of weight' model) or with an exponent of 1 (the per kg model).

One also should be aware of the danger of concluding that weight is not a predictor of clearance when the range of weights actually observed is small and the between subject variability in clearance is large.

> The situation with V is not less complicated than with CL. Firstly, there is
> no V, but VC, VP, etc., and each volume may depend on body size differenly.
> Again, allometry is not something to be taken into account when building a
> model for one species.

The allometric exponent of 1 applies to diverse volumes (e.g. blood volume, lung volumes, drug volumes of distribution). Please read these references as cited in Anderson et al. 1997. Then come back with your references which support your opinion. I would be delighted to debate this with you but lets use facts not ex cathedra statements.

Anderson BJ, McKee D, Holford NHG. Size, myths and the clinical pharmacokinetics of analgesia in paediatric patients. Clinical Pharmacokinetics 1997;33:313-327

Holford NHG. A size standard for pharmacokinetics. Clin. Pharmacokin. 1996: 30:329-332

Peters RH. The ecological implications of body size. Cambridge University Press.1983

West GB. Brown JH. Enquist BJ. A general model for the origin of allometric scaling laws in biology. Science. 1997; 276:122-6
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford@auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.html

*****

Subject: RE: Covariate Models Using Weight
Date: Wed, 17 Nov 1999 11:54:41 +0100

Sorry, Nick, I have no time to read your reviews. I've read a lot about allometry when I did PBPK modelling myself (more than 10 years ago).

You say CL is not the same in 10 kg child and in 200 kg rugby player. Sure, but it is another story. I this case the difference should better be described in terms of the age effect which may be different from the body size effect. Or do you really think WT is the only reason why CL in those two individuals differs?

Even for adults age and body size can independently affect CL. I give you just one example of the analysis I have done recently. The drug was eliminated via renal and hepatic routes in parallel, and the CL submodel was:

TVCL = 9.4+0.072*CLCR+0.034*(AGE-75)+0.050*(WT-67)

Note, this was mainly the elderly population (median age is 75), however, there were also a lot of young subjects (but no children) in the data set. Note also the model was developed using formal PK study data and sparse data, but the covariate model was based mainly on formal studies. I attach the scatter plot (MS Word Document) of CL random effects vs covariates. I don't know if you can open this attachment, but believe me, no trend can be seen even using a microscope.

*****JENNIFER, PLEASE MAKE THE WORD "MS Word Document" THE LINK TO THE

FILE 44.DOC

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I have a lot of similar examples (unfortunately, not published) where the linear model for CL vs. WT works perfectly. If you work with the industry you do not have opportunity to publish a lot.

Best regards,
----------------------------------------------------------------------
Janssen Research Foundation
Clinical Pharmacokinetics
B-2340 Beerse
Belgium
Email: vpiotrov@janbe.jnj.com

*****

Date: Thu, 18 Nov 1999 09:28:55 +0000
From: James <J.G.Wright@ncl.ac.uk>
Subject: Re: Covariate Models Using Weight

>
>I too agree that using prior information is essential in building a sensible model but I draw the line at extrapolating from points on a log-log plot representing different species to a clinical population of patients. Do we really believe that a human beings clearance changes in a predictable manner if they put on a few kilos? At least half of the western world is overweight these days. Comparisons using children and adults are necessarily confounded by many factors. I think in this day and age we should be looking for useful predictive covariates rather than claiming we can apply WT^3/4 to all possible drugs and patient populations on very limited evidence. In my personal experience, WT^3/4 and WT may not actually differ very much once other covariates have been used but I would need far more evidence before I could support Nicks view that WT^3/4 is the default choice.
>
>James

*****

Date: Thu, 18 Nov 1999 22:55:39 +1300
From: Nick Holford <n.holford@auckland.ac.nz>
Subject: Re: Covariate Models Using Weight

James Wright wrote:

> I too agree that using prior information is essential in building a
> sensible model but I draw the line at extrapolating from points on a
> log-log plot representing different species to a clinical population of
> patients.

Please read Anderson et al. (1997) for data using human clinical populations.

> Do we really believe that a human beings clearance changes in a
> predictable manner if they put on a few kilos?

If it is a matter of *belief* then YES I do believe that between subject differences in weight (even by a few kilos) are reflected in the typical individual by an increase in clearance. If I did not believe that then how would I explain in any meaningful biological way the extensive within and across species data showing that clearance and weight are correlated?

> At least half of the western world is overweight these days.
...
> Comparisons using children and adults are necessarily confounded
...

Body composition and stage of maturation are separate covariates that can be helpful in describing between subject variability in clearance. These factors are correlated with weight and therefore a systematic approach to separating them can be based initially on the use of an allometric model using weight (Holford 1996).

> I think in this day and
> age we should be looking for useful predictive covariates rather than
> claiming we can apply WT^3/4 to all possible drugs and patient populations
> on very limited evidence.

I am perfectly willing to listen to your suggestions for how to disentangle the separate influences of weight, body composition and maturation. My proposal is firstly to account for the influence of size using allometric scaling (perhaps using lean body weight instead of total body weight for obese patients), then introduce other covariates such as body mass index and age. How would *you* suggest it be done "in this day and age"?

The allometric scaling exponent of 3/4 for functional properties (e.g. metabolic rate) and 1 for structural properties (e.g. blood volume) is based on extensive evidence (see Peters 1983 for a book on the subject and West et al. 1997 Table 1). I *believe* it is reasonable to extend the principle that if an exponent of 3/4 is reasonable for metabolic rate then it should also work for clearance, similarly an exponent of 1 for blood volume is reasonable for apparent volume of distribution. Of course, it has not been evaluated for "all possible drugs and patient populations". That would be absurd. But the model does have a theoretical and biological basis (West et al. 1997). It is not a black box empirical model. If a drug is eliminated renally then I feel comfortable using a marker of renal function e.g. predicted creatinine clearance, as a covariate for the renal component of drug clearance. This is the basic application of biology to modelling. The allometric relationship is being applied in the same way.

Anderson BJ, McKee D, Holford NHG. Size, myths and the clinical pharmacokinetics of analgesia in paediatric patients. Clin. Pharmacokin. 1997;33:313-327

Holford NHG. A size standard for pharmacokinetics. Clin. Pharmacokin. 1996; 30:329-332

Peters RH. The ecological implications of body size. Cambridge University Press. 1983

West GB. Brown JH. Enquist BJ. A general model for the origin of allometric scaling laws in biology. Science. 1997; 276:122-6
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford@auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm

*****

Date: Thu, 18 Nov 1999 10:04:51 +0000
From: James <J.G.Wright@ncl.ac.uk>
Subject: Re: Covariate Models Using Weight

Dear Nick,

Just to make this clear - if I put on weight ie fat tissue, my clearance increases? How? Does my liver get bigger? I shall read your selected references (again, as I have read two of them in the past, although asking me to read a book before considering my views seems a little demanding), but given that only two of them appear to be about clinical populations, I can hardly consider this an overwhelming body of evidence. Can I suggest that you take a look at?

Nawaratne S. Brien JE. Seeman E. Fabiny R. Zalcberg J. Cosolo W. Angus P. Morgan DJ. Relationships among liver and kidney volumes, lean body mass and drug clearance. British Journal of Clinical Pharmacology. 46(5):447-52, 1998 Nov

James

*****

Date: Fri, 19 Nov 1999 12:01:50 +1300
From: Nick Holford <n.holford@auckland.ac.nz>
Subject: Re: Covariate Models Using Weight

James Wright wrote:

> Just to make this clear - if I put on weight ie fat tissue, my clearance
> increases? How? Does my liver get bigger?

I do not expect an increase in weight due entirely to an increase in fat to increase clearance as predicted by the allometric model using weight. However, it is possible that there are extra metabolic demands put on the body by the extra fat that could lead to increased liver size and perhaps enhanced drug clearance.

The allometric scaling model has to be underststood as a model for ONE covariate (weight) among many that might be associated with changes in clearance. Some of these covariates (age, obesity) will be correlated with weight so it will require some thought about a model and appropriate data with sufficient variability in the covariates to test the model.

> I shall read your selected references... but given that only two
> of them appear to be about clinical populations, I can hardly
> consider this an overwhelming body of evidence.

Would you like to offer some guidelines for what *you* consider "an overwhelming body of evidence". How much evidence does I have to provide to persuade you that an allometric exponent of 3/4 is reasonable when you provide no plausible counter evidence (see below)?

May I suggest, you add to the evidence I wish you to consider the recent generalization of the theoretical basis for the 3/4 exponent model by West et al (1999)?

> Can I suggest that you take a look at:
>
> Nawaratne S. Brien JE. Seeman E. Fabiny R. Zalcberg J. Cosolo W. Angus P.
> Morgan DJ.
> Relationships among liver and
> kidney volumes, lean body mass and drug clearance. British Journal of
> Clinical Pharmacology. 46(5):447-52, 1998 Nov

I do not consider this small study of 21 healthy volunteers, with less than a 2 fold range in any of the covariates and more than a 4 fold range in clearance, relevant to testing the hypothesis about the 3/4 power model. I would like to refer you to Beuchat (1997) with an accompanying comment from Brown et al. which deals with the problems of considering empirical alternative exponents based on sparse data and the lack of any theory.

I note that the authors of this study make the classical error of accepting the null hypothesis that LBM was not a predictor of anitipyrine clearance via a relationship to liver volume without any consideration of the power they had for detecting such a prediction. I would think the signal to noise ratio is just too small for this kind of study to offer any insights.

Beuchat CA. Allometric scaling laws in biology [letter; comment]. Science 1997;278(5337):371; discussion 372-3.

West GB, Brown JH, Enquist BJ. The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science 1999;284(5420):1677-9.
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford@auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm

*****

Date: Fri, 19 Nov 1999 12:23:36 +0000
From: James <J.G.Wright@ncl.ac.uk>
Subject: Re: Covariate Models Using Weight

Dear Nick,

The point where you and I differ, I think, is in what we consider relevant evidence. I don't consider between species scaling to have any relevance to the use of weight in a human population. Nor am I impressed by elaborate fractal models. As you have provided only two references that I consider relevant, (and you are an author on both of them), I remain totally underwhelmed.

Steve raises several interesting issues, and the truth is that unless you have a truly massive sample there is always a certain amount of subjectivity in how you construct a covariate model (A Miller, Subset Selection in Regression, is a very scary book). If you require predictive accuracy, then you should be very wary of overfitting, so I guess I am not that keen on including covariates that are not justified by improvements in fit. Steve's example of creatinine clearance and the Cockcroft & Gault formula is an interesting one - the Cockcroft & Gault formula contains someone else's prior knowledge, but if your population isn't the same as theirs (or you are using a modern creatinine assay) maybe this will do more harm than good. Personally, I would use the raw covariate (and weight etc separately (if justified), although this will use more parameters) in this particular case.

My view is we should use prior information, but evaluate its relevance critically.

James

*****

From: "Stephen Duffull" <sduffull@fs1.pa.man.ac.uk>
Subject: Re: Covariate Models Using Weight
Date: Fri, 19 Nov 1999 09:15:41 -0000

There has been considerable discussion about covariate models etc. I perceive that the discussion has slightly wider implications. Are covariates that have a mechanistic flavour better than those that do not (or none at all)?

1) If you have a choice of two covariates (eg creatinine clearance or serum creatinine). eg. If you have a renally cleared drug and you can use either estimated creatinine clearance (eg Cockcroft and Gault or Jelliffe & Jelliffe) as a covariate or some empirical model based on serum creatinine and both models yield the same objective function with the same number of parameters then which do you use? I personally prefer the generality of the creatinine clearance equation. The same is true for Nick's argument - if an allometric scaling factor has been developed that has generality outside of the particular experiment that is being considered currently and still predicts as well as an empirical choice then I personally would prefer using it (even if I could not substantiate the value of the power (3/4) in this particular experiment).

2) If a covariate does not improve the fit of your model do you include it? eg If adding WT^(?) as a covariate for Vd or CL does not alter the objective function significantly - then do you include it as a descriptor? This depends on why you are modelling (eg descriptive or predictive). If "predictive" then it would be difficult not to believe that larger people don't have larger Vd or CL - even if you couldn't show this based on the current experimental design.

I don't think it is bad to include prior beliefs, as long as they have some basis in reality, in the model building exercise and indeed if the process needs to be formalised then a Bayesian solution may be appropriate.

In both examples assumptions in model building need to be transparent to the user.

Just a thought
Steve
=====================
Stephen Duffull
School of Pharmacy
University of Manchester
Manchester, M13 9PL, UK
Ph +44 161 275 2355
Fax +44 161 275 2396

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