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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
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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
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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)
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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
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The following message was posted to: PharmPK
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
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The following message was posted to: PharmPK
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
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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
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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
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The following message was posted to: PharmPK
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
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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
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The following message was posted to: PharmPK
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
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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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