- On 24 Nov 2005 at 19:51:49, hi2u.-a-.o2.co.uk sent the message

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hello

I have just been told that pharmacokinetic variables are usually log

normally

distributed and I am very surprised by this so !!..

Does anyone have any sources of evidence for this ?

Does log normal distribution mean that when the log of the values is

taken

the distribution becomes normal ??

Why are pharmacokinetic parameters log normally distributed instead

of normally

distributed ??

I have never heard of the log normal distribution before ! Please help

David Akers - On 25 Nov 2005 at 09:31:09, "Bruce Charles" (Bruce.-a-.pharmacy.uq.edu.au) sent the message

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This is well known and due to the complexities of variability in the

pharmacogenomics. My (basic) understanding is that complex/overlapping

gene arrays that code for expression of various proteins (enzymes,

receptors) summate to a merging of a number of smaller individual

"sub-populations" which is seen overall as a distribution which is

skewed away from the symmetrical normal (non-gaussian) distribution.

With drug metabolism, the distribution in a population is often skewed

towards poor/inefficient metabolic activities, thus PK parameters which

reflect this (e.g. CL, AUC) are skewed to the right (larger values) on a

frequency plot - Taking the logs of the PK values "transforms" the

larger values more than smaller values giving rise to a distribution

which is more "normal" looking, thus the term "log normal".

It also is seen with other PK parameters and also with PD parameters

(variability in expression of receptor protein complexes) to varying

extents - Others may have more specific/detailed/expert explanations but

this is the basic story more or less.

Hope this helps..

Cheers

BC

Bruce CHARLES, PhD

Reader

School of Pharmacy

The University of Queensland, 4072 Australia

[University Provider Number: 00025B]

TEL: +61 7 336 53194

FAX: +61 7 336 51688

B.Charles.-a-.pharmacy.uq.edu.au

http://www.uq.edu.au/pharmacy/brucecharles/charles.html - On 25 Nov 2005 at 09:12:50, Thierry Buclin (Thierry.Buclin.-at-.chuv.ch) sent the message

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David,

According to the central limit theorem, a variable influenced by the

addition/subtraction of a large number of random influences of

similar magnitude tends to reach a normal distribution. In

pharmacokinetics, random influences (variability in enzyme activity,

diffusion, permeation, flow rates etc.) affect clearance and volume

values in a multiplicative rather than additive way. A variable

influenced by the multiplication/division of random influences will

tend towards a log-normal distribution, and its log-transform will

follow a normal distribution. You have this in other areas of natural

sciences : e.g. pH measurement for acidity is a log-transform of H+

concentration, and noise measurement in decibel is a log-transform of

sound energy; the application of parametric statistical tests to such

results assumes in fact a log-normal distribution for H+

concentration or sound energy, respectively. The characteristic

parameters best describing a log-normally distributed variable are

the geometric mean and CV (see the thread about statistical terms in

2002). Hope this helps

Thierry Buclin - Lausanne (Switzerland) - On 25 Nov 2005 at 15:15:36, "sulagna" (sulagna.das.aaa.veedacr.com) sent the message

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A log normal distribution is one in which data that doesnot appear to be

normal appears normal after a log transformation.

Example : positively skewed- ie skewed to the right. On taking a log

transformation of such values you will see that the distribution becomes

almost normal as small values change more compared to large values

( ln1=0

ln10=1 , ln100=2 , ln 1000=3.....) . One of the greatest advantage of

log

normal distribution is that large skewed data can be transformed and

made

more uniform and compact .

Biological data is also positively skewed- it is time

dependent.

On taking a log transformation such data can be made normal. We can then

apply parametric tests like ANOVA and analyse it. Thus

pharmacokinetic data

( Cmax, AUC .) are log transformed and made normal. That's why they are

known as lognormally distributed

Sulagna

Statistician - On 25 Nov 2005 at 11:40:06, Hans Mielke (h.mielke.-a-.bfr.bund.de) sent the message

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David,

don't panic, the lognormal distribution is important and appears

frequently, but it is not too difficult to understand. As you

suspect "X is log-normally distributed" is another word for "log

X is normally distributed". Thus you need not learn new formulas

if you accept working with log-values.

Now let us face the difficult part of your question:

> Why are pharmacokinetic parameters log normally distributed

instead of

> normally distributed ??

To such a question, there cannot be a unique satisfying answer.

Not even talking about the philosophical question whether we

should state some real-world numbers *are* normally or

log-normally distributed. But let me make a few remarks which may

make the log-normality assumption a little plausible.

1. The central limit theorem (of stochatics) states (informally

speaking) the following: a parameter to which there are many

influences whose contributions add up is normally distributed.

A mere reformulation is: a parameter to which there are many

influences whose contributions multiply up is log-normally

distributed.

2. In the log-normal distribution, negative values always have

zero probability of occuring. This might be a theoretical issue

(when negative values do not make sense, usually their normal

distribution probability is neglegible small), but it is the

other side of the following.

3. Consider some pharmacokinetic parameter X with mean EX. The

value 2*EX = EX + EX is higher than EX. Which value below EX has

the same "distance" from EX? Is it 1/2 * EX or EX - EX ?

In case of the first answer (*2 vs. *1/2), you should assume X to

be log-normally distributed; in case of the second answer (+EX

vs. -EX), you should assume X to be normally distributed.

I hope these informal remarks are of some help.

Hans

--

Dr. Hans Mielke

Fed. Institute for Risk Assessment

Thielallee 88-92, D - 14195 Berlin

Tel. +49 1888 412 3969 Fax 3970 - On 28 Nov 2005 at 08:47:17, "Hans Proost" (j.h.proost.at.rug.nl) sent the message

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Dear David,

Many excellent remarks about the log-normal distribution have been

made by

others. In my opinion one should always prefer the log-normal

distribution

in pharmacokinetics, and usually in pharmacodynamics as well. Please

note

that one can always calculate an arithmetic mean value and a

geometric mean

value, irrespective of the distribution. But to interpret such a mean

value

as a meaningful 'measure of central tendency' or as a 'typical

value', one

should select a mean value that fits to the distribution of the data.

The

same holds of course for the measure of variability.

An example of the advantage of the log-normal distribution is the

calculation of half-life from clearance and volume of distribution,

or from

the elimination rate constant. In the past it has been proposed to

use the

harmonic mean for half-life if individual half-lives has been

obtained from

the elimination rate constant. This was based on the assumption that

elimination rate constant is normally distributed. But why? The

elimination

rate constant is clearance divided by volume of distribution. If

clearance

and volume of distribution are normally distributed, the elimination

rate

constant is not normally distributed. In particular in the case of wide

distributions, the differences between the (harmonic) mean of half-

life and

the half-life estimated from mean clearance and mean volume of

distribution

can be considerable. And what is the best answer?

As pointed out by others, there are sound reasons to assumed that

clearance

and volume of distribution are log-normally distributed. As a result,

elimination rate constant and half-life are also log-normally

distributed.

Therefore the geometric mean of clearance and volume of distribution

can be

used directly to calculate the (geometric) means of elimination rate

constant and half-life, and also their CV (similar for elimination rate

constant and half-life) can be easily calculated as the square root of

(CV_CL ^ 2 + CL_V ^ 2) (assuming that there is no correlation between

CL and

V). Even in case of wide distributions, this works perfectly.

Best regards,

Hans Proost

Johannes H. Proost

Dept. of Pharmacokinetics and Drug Delivery

University Centre for Pharmacy

Antonius Deusinglaan 1

9713 AV Groningen, The Netherlands

tel. 31-50 363 3292

fax 31-50 363 3247

Email: j.h.proost.-at-.rug.nl - On 28 Nov 2005 at 16:27:40, Roger Jelliffe (jelliffe.-at-.usc.edu) sent the message

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Dear Bruce and all:

About normal and lognormal distributions in population

parameters. You are quite correct that subpopulations within any

given population may well have quite different parameter values, and

that the parameter distributions may well not be normal, but skewed,

or more yet, multimodal. The question is how best to deal with this

problem.

I would suggest that the best way to deal with the problem

is not go go about making transformations of normal distributions. It

doesn't seem useful to me to make such transformations unless you

first know what the actual shape of the distribution is, with no

preconceived assumptions about what its shape might be. Taking logs

of an unknown but assumed lognormal parameter distribution may well

not be whet you want to do. The nonparametric approach to population

modeling does this - either the original Mallet NPML approach, or the

newer NPEM and NPAG approaches. They make no assumptions at all about

the shape of the parameter distributions. They simply get the most

likely distribution based on the population raw data and the error

model used. The methods are also consistent, as the likelihoods are

computed exactly. In general, both parametric (PEM, Lavelle, and

others now, and the nonparametric approaches above have the property

that if you study more subjects, the results get closer to the true

ones. That is statistical consistency. Other approaches that use

approximate methods to compute the likelihoods such as FO or FOCE,

for example, as in the USC*PACK iterative 2 stage Bayesian (IT2B) or

NONMEM, for example, do not have this desirable property, and the

results may actually get worse with more subjects. In addition,

NONMEM and IT2B are considerably less precise in their parameter

estimates. More information is available on our web site

www.lapk.org. A manuscript describing this is in press in Clinical

Pharmacokinetics. In any event, if you want to capture and quantify

the prevalence of subpopulations, and dealing rigorously with your

verygood statement of the problem, I would really suggest that

instead of using various empirical parametric transformations, that

you seriously consider using nonparametric population modeling.

Now, what to do about the population parameter shape that

you find? With or without clearly visible subpopulations, how do you

propose to develop the initial dosage regimen based on this

population raw data? Using population mean parameter values can be

quite dangerous. Median values are better. However, the best is the

regimen that hits the desired target with the greatest precision.

Parametric population models have only one single point

estimator for each parameter distribution. One can only assume that

the regimen developed will hit the target exactly, and yet we all

know this is not so. On the other hand, the multiple support points

provided by a nonparametric population distribution permit one to

make many predictions of future serum concentrations, for example,

and to find the weighted squared error with which they fail to hit

the target. It is only a short step from that to finding the regimen

that specifically minimizes the weighted least square error with with

the target is hit. For the first time, we now have a maximally

precise regimen, based on all the data we know up to now. This is

multiple model dosage design. Again, more information is available on

our web site.

Very best regards,

Roger Jelliffe

Roger W. Jelliffe, M.D. Professor of Medicine,

Division of Geriatric Medicine,

Laboratory of Applied Pharmacokinetics,

USC Keck School of Medicine

2250 Alcazar St, Los Angeles CA 90033, USA

Phone (323)442-1300, fax (323)442-1302, email= jelliffe.aaa.usc.edu

Our web site= http://www.lapk.org - On 28 Nov 2005 at 23:33:03, "Walt Woltosz" (walt.at.simulations-plus.com) sent the message

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Dear Roger,

I'll admit to begin with that I have not yet taken the time to

understand

the nonparametric methods that you so eloquently have described. In

fact, at

a PopPK session at AAPS a couple weeks ago it seemed that everyone

was using

everything but. Yet your frequent posts here are quite compelling,

and when

I've heard you speak in person, I wished I could absorb your

knowledge about

the subject.

My (perhaps naive) question is regarding your statement that included

" . .

..the multiple support points provided by a nonparametric population

distribution . . .". It would seem like this means you're using extra

fitted

parameters to create the model compared to the other methods. If so,

then

how do you get a fair comparison of methods, and how do you guard

against

overfitting?

Best regards,

Walt

Walt Woltosz

Chairman & CEO

Simulations Plus, Inc. (AMEX: SLP)

1220 W. Avenue J

Lancaster, CA 93534-2902

U.S.A.

http://www.simulations-plus.com

Phone: (661) 723-7723

FAX: (661) 723-5524

E-mail: walt.-at-.simulations-plus.com - On 30 Nov 2005 at 10:24:24, "Frederik B. Pruijn" (f.pruijn.-at-.auckland.ac.nz) sent the message

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Dear Dr Jelliffe,

I also have a question about nonparametric population modeling (and

please forgive my utter ignorance on this subject), which is about the

so-called "target".

In your most recent post, but also in many others, you mention "hitting

the target with the greatest precision" and "hitting the target

exactly". Could you please elaborate on what is known about the

distribution of the target (value)? How is this determined? Etc. When I

read your posts I always think of the target as a "bull's eye" but

surely that's due to my lack of imagination & understanding.

TIA

Frederik Pruijn

Frederik B. Pruijn PhD MSc (Senior Research Fellow)

Experimental Oncology Group

Auckland Cancer Society Research Centre

Faculty of Medical and Health Sciences

The University of Auckland

Private Bag 92019

Auckland

New Zealand

Phone: +64-9-3737 599 x86939 or x86090

Fax: +64-9-3737 571

E-mail: f.pruijn.-at-.auckland.ac.nz - On 29 Nov 2005 at 16:37:53, Roger Jelliffe (jelliffe.-at-.usc.edu) sent the message

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Dear Walt and Frederick:

More about nonparametric modeling. First, about the number

of parameters. With parametric modeling, you assume either normal or

lognormal, or multimodal distributions. The parameters for this are

the means and covariances. The whole distribution in not estimated -

only the above parameters, as they define the shape of the normal curve.

With nonparametric (NP) modeling, you are right, Walt -

there are many more parameters - up to one set of model parameters

for each subject studied, each set with an estimate of the

probability of that set of parameters. That is not overfitting - it

is simply the way the NP methods work. NP methods have been called

the ultimate mixture model, as there is one density (with an implied

mean and covariance) for each subject as well. The population

parameter joint densith is simply the sum of all the individual

subject Bayesian posterior support points. For 100 subjects, with a 5

parameter model, you will have aup to (I think) 10**5 support points

(parameters) along with the means, medians, covariances. It

approaches the impossible ideal of being able to directly observe

each of the parameter values in each of the 100 subjects. Since

parameters cannot be directly observed, but must be inferred or

estimated from the data of the doses, serum concentrations (and other

responses) and the error pattern used, the NP method provides the

best approach to the impossible ideal. If there is significant error,

the data will be resolved into fewer that 1 support point per

subject. The point is that instead of estimating the parameters of an

assumed distribution, the NP methods estimate the entire

distribution, whatever may be its shape. Because of this, it stands

the best chance of discovering unsuspected subpopulations. Look at

our web site, under teaching topics, and click on nonparametric

population modeling.

Now, about targets. What do we want to do with our model? We

like to hit specific desired therapeutic target values, for serum

concentrations, for example. Not just some window, but a specific

target - a gentamicin peak of 12, for example. The problem is this -

how do you do this most precisely? With parametric models, there is

only 1 model, and onlly 1 regimen to hit the target, which is

assumed to be hit exactly. One reason for this is that the parameter

distributions are usually neither normal or lognormal, but simply are

what they are. Sometimes, for example, the mean Vd for Vancomycin is

at about the 73rd percentile of the distribution. Because of this,

and other things being equal, about 73 % of the time the Vd is less

than this, and so the serum concentrations are greater. Only about

27% of the levels will be less than predicted. This is a good

illustration of the DANGERS of using mean population parameter values

in developing dosage regimens. Medians are better. But the best

regimen is that which specifically hits the target with the greatest

precision, such as the minimum weighted squared error. That is what

"multiple model" dosage design does, and that is a very good reason,

we think, for using NP pop models are useful for. They generate

multiple predictions of future serum concentrations (or other

responses) and it is then easy to find the regimen that minimizes

that weighted squared error. This approach is widely used in the

aerospace industry in fixed wing and helicopter flight control

systems, and spacecraft guidance systems.

Further, what about the behavior of these approaches? Would

you like to use a method that has or does not have the property that

the more subjects you study, the closer the parameter values get to

the true ones? The FOCE or FO approximations in calculation the

likelihood in IT2B or NONMEM, for example, do NOT have that proven

property. Methods, both parametric (Lavelle, Leary's PEM), that

compute the likelihood exactly DO have that property. Also the NP

methods such as NPML, NPEM, and NPAG do have this property. It is

interesting to me how few people are ever concerned with examining

the behavior of the methods they use. Again, a paper describing this

is in press in Clinical Pharmacokinetics.

Also, statistical efficiency (precision of parameter

estimation). This is also an important property of a method. In that

paper in press, the relative efficiency and parameter precision were

examined. They were as follows. NPOD is another variant on NPAG. They

all have about the same precision and efficiency.

Estimator Relative efficiency

Relative error

DIRECT OBSERVATION 100.0

% 1.00

PEM

75.4% 1.33

NPOD

61.4% 1.63

NONMEM FOCE

29.0% 3.45

IT2B FOCE

25.3% 3.95

NONMEM FO

0.9% 111.11

Notice the difference between the efficiency of the two

methods (PEM and NPOD) that use exact computations of the likelihood,

versus those below it that use the FOCE or FO approximations. The

difference is quite apparent. How to compare them? Compare the

likelihood value found. It is very important to report these

likelihoods, and yet, very few parametric studies do this, using

instead values of "goodness of fit" instead. The likelihoods also can

be directly compared between IT2B and the NP methods, as the IT2B

program takes the collection of Bayesian posterior support points at

the end of the computation, and then computes the likelihood

directly, just as if it came from an NP program. Go to our web site

www.lapk.org, and click on new advances in population modeling.

Very best regards,

Roger Jelliffe

--

Roger W. Jelliffe, M.D. Professor of Medicine,

Division of Geriatric Medicine,

Laboratory of Applied Pharmacokinetics,

USC Keck School of Medicine

2250 Alcazar St, Los Angeles CA 90033, USA

Phone (323)442-1300, fax (323)442-1302, email= jelliffe.-a-.usc.edu

Our web site= http://www.lapk.org - On 30 Nov 2005 at 17:35:48, "Frederik B. Pruijn" (f.pruijn.-a-.auckland.ac.nz) sent the message

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Dear Dr Jelliffe,

Thank you for your reply. I went to your Website and got a 'bit

overwhelmed' and I guess it will take me some time to get my head around

it. However, in your reply you give the example of gentamicin peak of

12. It is still not clear to me where this comes from; for example, is

11 way off target or just a little and how is this best determined? Is

12 the target for each patient or is this the individualised target? In

other words, how important is it to hit the target exactly & precisely

if the target itself is a rather diffuse parameter (value)?

Perhaps you could point me (and others?) to the appropriate PDF on your

Website and I'll do some homework.

Many thanks for your help.

Frederik Pruijn

Frederik B. Pruijn PhD MSc (Senior Research Fellow)

Experimental Oncology Group

Auckland Cancer Society Research Centre

Faculty of Medical and Health Sciences

The University of Auckland

Private Bag 92019

Auckland

New Zealand

Phone: +64-9-3737 599 x86939 or x86090

Fax: +64-9-3737 571

E-mail: f.pruijn.-at-.auckland.ac.nz - On 30 Nov 2005 at 12:54:54, Roger Jelliffe (jelliffe.at.usc.edu) sent the message

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Dear Frederick:

Thanks for your reply. The idea is not to limit your therapy

to being in some overall "therapeutic range" where most (but not all)

patients do well. Instead, the idea is to select a specific

individualized target goal for that particular individual patient,

based on your assessment of his/her meeds and the acceptable risks of

toxicity, again for that patent, and then to hit that target with

maximum precision. There is no zone of indifference. 11 instead of 12

is not bad, but it all depends on each patient's particular

situation, and his/her (or the bug's) clinical sensitivity to the

drug. We also use a Zhi-Nightingale model to describe the Hill model

effect relationship between the serum concentration profile and the

kill.

You might go to our web site again, click on teaching

topics, and click on section 9, and click on multiple model dosage

design. It specifically discusses setting individual target goals for

each patient. See what you think, and please let me know.

All the best,

Roger Jelliffe

Roger W. Jelliffe, M.D. Professor of Medicine,

Division of Geriatric Medicine,

Laboratory of Applied Pharmacokinetics,

USC Keck School of Medicine

2250 Alcazar St, Los Angeles CA 90033, USA

Phone (323)442-1300, fax (323)442-1302, email= jelliffe.at.usc.edu

Our web site= http://www.lapk.org

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