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Dear PharmPK group,
I know I am asking a question that has been discussed ad nauseam, but
nevertheless, I am interested in your opinions about the statistical
presentation of the tmax results of pk-studies. As we all know, concentration
measurements are taken only at pre-determined (fixed) time points and are thus
not continuous random variables in a proper sense. Obviously, the
data observed
are not normally distributed, so it is not particularly elegant to report means
and standard deviations. What do you suggest ? Medians, Modes,
other ? Thanks
for your responses.
David Tudor
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[Two replies - db]
Date: Wed, 15 Dec 1999 09:18:26 +0200
From: Andrew Volosov
X-Accept-Language: en,pdf
To: PharmPK.at.boomer.org
Subject: Re: PharmPK Statistical description of tmax
Dear David,
You might find useful the method we published a couple of years ago.
This method
deals with the case when tmax cannot be properly determined by simple
observation.
The strong side of the method is that it utilizes all data points you
have. The
drawback is that it can be used only when your data fits the Bateman "kit-kat"
equation. We have applied this method in several studies quite successfully.
The reference is:
"Use of Mean Residence Time to Determine the Magnitude of Difference
Between Rate
Constants and to Calculate tmax in the Bateman Equation". Biopharm.
Drug Dispos.
20: 3-9 (1999).
Andrew Volosov
The Hebrew University of Jerusalem
School of Pharmacy
volosov.-a-.md2.huji.ac.il
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Date: Wed, 15 Dec 1999 03:33:19 -0700 (MST)
X-Sender: ml11439.at.pop.goodnet.com
To: PharmPK.-at-.boomer.org
From: ml11439.aaa.goodnet.com (Michael J. Leibold)
Subject: Re: PharmPK Statistical description of tmax
David,
W.W.Daniel, the author of "Biostatistics", also has published a
statistics book: "Applied Nonparametric Statistics", PWS Kent, Boston 1990.
In nonparametric estimates of central tendency the median rather than the
mean is the location parameter for central tendency. Nonparametric
methods for estimating a population median include the Sign test and
the Wilcoxan Signed-ranks test. Other nonparametric methods are also
available for comparing test groups, similar to parametric analysis
of variance.(1)
A Chi-Sqaure goodness of fit test can be used to determine if the
distribution of a sample of variables is normal, log-normal or fits
another probability distribution(3). That is, to test for deviation from
a normal distibution and the need for nonparametric statistical analysis.
The nonparametric methods can also be used in when the samples fit a
normal distribution, and are proposed to have the same statistical power
as the parametric methods in this case(2).
Also:NPML
Also, there are advanced methods of nonparametric estimation such
as the the "Nonparametric Maximum Likelihood Method". For this method,
computer software has been developed which renders a nonparametric
maximum likelihood estimate of the distribution of parameters without
any prior assumptions of a normal distribution. The kinetic parameters
and distributions obtained with this method can then be used in Bayesian
regression programs for dose optimization.(4)
Mike Leibold, PharmD, RPh
ML11439.aaa.goodnet.com
References:
1) Daniel, W.W., Applied Nonparametic Statistics, PWS Kent, Boston 1990
2) Glantz, S.A., Primer of Biostatistics, McGraw Hill, New York 1997
3) Dowdy, S., Wearden, S., Statistic for Research, Wiley Interscience,
New York 1983
4) Taright, N., et al., Nonparamettic Estimation of Population Characteristics
in the kinetics of lithium......., Ther Drug Monitoring 1994;16(3):258-269
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[Two replies - db]
Reply-To: lesassays.at.ukneqasaa.win-uk.net (Les White)
To: PharmPK.-at-.boomer.org
Date: Thu, 16 Dec 1999 12:37:15
Subject: Re: PharmPK Statistical description of tmax
From: lesassays.aaa.ukneqasaa.win-uk.net (Les White)
both if they are very different
---
X-Sender: flerner.-at-.mail.intramed.net.ar
Date: Thu, 16 Dec 1999 17:57:21 -0300
To: PharmPK.aaa.boomer.org
From: Federico Ezequiel Lerner
Subject: Re: PharmPK Statistical description of tmax
Dear all:
A few month ago we proposed to analyses the normal distribution of Tmax an
other Pk parameters and describe those according to the results. See: J.
Clin Pharmacol 1999; 39: 1053-1061 and also Ther Drug monit 1999; 21: 200-207
regards
Federico Lerner, MD
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In nature, the observed data of Tmax better be described as from a discrete
uniform distribution. Even though, it may not be as that accurate. Consider
this: the likelihood of very large (i.e., towards the end portion of the
scheduled plasma sampling times) Tmax value usually is very small. I would
prefer using Median, Mode, Maximum, and Minimum measures. If there is a
need for confidence interval calculation, I would prefer using nonparametric
approach.
Xinwei Jia
Biostatistics
Janssen Research Foundation
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Hi colleagues
in BIO-EQUIVALENCE STUDIES
I'll appreciate if anyone explain to me the significance and the
limit of tmax ,in case of comparing two oral products e.g tablets
for AUC I know that the two products are considered bioequivalant if
the AUC as well as C-max are from 80% -120% from the reference
product , this is according to FDA
Please correct me if this not right.
Also what can be happen if the new product absorption is improved to
be far better than the original reference product. i.e more than 120%
,should this considered NON equivalent ??
thanks
Osama Ziada
Julphar
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To: Osama Ziada
Your inquiry concerning bioavailability/bioequivalence indicates that you have
not yet reviewed the FDA's guidances on the subject which can be found at
http://www.fda.gov/cder/guidance/index/htm. In contradiction to what you stated
you knew, the FDA does not consider two products bioequivalent if the Cmax and
AUC are within 80-120%. What they do require is that the 90% confidence
interval for the ln transformed metrics must be within 80-125%. This is very
different than the average AUC and Cmax (I assume you meant average) of the
test formulation being within 80-120% of the reference. This is a perception
that many people have and it must be made clear that if a formulation passes
confidence intervals it is highly unlikely that the mean differences are more
than 10%, and in most cases they will be within 5%. Next, if a test
formulation is "better" than the reference (for example: CI's for AUC are 110%
to 140%, it is not bioequivalent and that's that. This does not mean that a
firm cannot market this "better" formulation, but it will not be approvable as
therapeutically equivalent. It must undergo further clinical testing for safety
and efficacy. Finally, although the FDA is not currently requiring Tmax's to
pass some specified criteria, they must be reported, and the draft guidances do
address some of the issues associated with how they will deal with Tmax in the
future. Go to the web.
Art Straughn
Memphis
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Hello,
You can find relevant guidance on BE studies at :
http://www.fda.gov/cder/guidance/index.htm
These are my personal views and not necessarily of the agency:
I think the pharmacodynamic implications of the test/reference ratio
being greater than 125% (or < 80%) are more meaningful before deeming
some test formulation inequivalent to a reference. A concentration -
effect model should help you in translating the PK differences into PD
differences.
Regards,
Joga
Joga Gobburu,
Pharmacometrics,
CDER, FDA
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[A few replies - db]
Date: Tue, 28 Dec 1999 00:44:56 -0700 (MST)
X-Sender: ml11439.-a-.pop.goodnet.com
To: PharmPK.-at-.boomer.org
From: ml11439.-a-.goodnet.com (Michael J. Leibold)
Subject: Re: PharmPK Re: Statistical description of tmax
Hello Osama,
Tmax is related to Ka according as follows:
Tmax= [1/(Ka-Ke)][Ln(Ka/Ke)]
A decrease in Ka or rate of absorption would not change the AUC, which
is determined by F, Xo, Vd and Ke. Bioeqivalence studies standardize the
other parameters in this equation so that AUC is effected principally by
F or bioavailability.
Cmax should decrease as the Tmax increases according to the following
equation:
Cmax= [FXo/V]e-KeTmax
The Tmax or Cmax should be useful in comparing the rate of absorption of
different oral products. But, I think that F is the most important factor
in comparing bioequivalence, and I think that this is what the FDA uses.
Mike Leibold, PharmD, RPH
ML11439.-a-.goodnet.com
---
From: HARRY.MAGER.HM.aaa.bayer-ag.de
To: " - *PharmPK.aaa.boomer.org"
Subject: Antwort: PharmPK Re: Statistical description of tmax
Date: Tue, 28 Dec 1999 08:57:44 +0100
Due to the logarithmic transformation, the confidence intervals should be
within 80% to about 125%.
Testing for equivalence does not involve inferiority/superiority. Accordingly,
when one out of two formulations turns out to be superior in terms of a
pharmacokinetic parameter, the two formulations are not equivalent. However,
you can always test the hypothesis equivalent or superior versus inferior.
Best regards and have a great 2000.
---
X-Sender: denucci.-at-.mailhost.dglnet.com.br
Date: Tue, 28 Dec 1999 10:02:21 -0300
To: PharmPK.at.boomer.org
From: Gilberto de Nucci
Subject: Re: PharmPK Re: Statistical description of tmax
Dear Osamar
Please note that 80-125% is the range accepted by the FDA for both AUC and
Cmax
Regards
Gilberto De Nucci
---
From: Thomas.Senderovitz.aaa.ferring.com
To: PharmPK.aaa.boomer.org
Subject: RE: PharmPK Re: Statistical description of tmax
Date: Tue, 28 Dec 1999 14:56:18 +0100
Dear Osama,
To answer your last question first: Yes, if the AUC of the test
product is outside the 90% CI for the ratio of the means, then the
two products are NOT bioequivalent. If you are not able to establish
bioequivalence, then therapeutic equivalence might be established,
although this is considerable more difficult - you need to show
equivalence of clinical efficacy and safety!
tmax should preferably be described my nonparametric methods (in
accordance to guidelines), and hence median and range as descriptive
statistics are appropriate. For comparison between test and reference
product with respect to tmax, a nonparametric test can be used, and
guidelines do not state level of significance (but i guess 5% would
be the usual).
Hope this helps.
Thomas Senderovitz
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[A few replies - db --- Happy New Year everyone!]
Date: Wed, 29 Dec 1999 01:45:49 -0700 (MST)
X-Sender: ml11439.aaa.pop.goodnet.com
To: PharmPK.-at-.boomer.org
From: ml11439.-a-.goodnet.com (Michael J. Leibold)
Subject: Re: PharmPK Statistical description of tmax
Hello David,
I further considered your question regarding statistical treatment
of pharmacokinetic data:
I checked the FDA CDER site and found that the FDA uses parametric
analysis of log-transformed data for AUC and Cmax. This is in reference to
your question regarding the statistical analysis of Cmax data (or Tmax data)
using nonparametric methods. That is, a log-normal distribution is
assumed for AUC and Cmax data for a parametric analysis of the data. So, if
your data does not appear normally distributed, try looking at the log-
transformed data (log10 or ln).
http://www.fda.gov/cder/guidance/index.htm
Average, Population, and Individual Approaches to Establishing
Bioequivalence [HTML] or [Acrobat] or [Word97] (Issued 8/1999, Posted
8/27/1999, Word and Acrobat version reposted 9/8/99)
From Web Site:
1. Average Bioequivalence
a. Overview
"Parametric (normal-theory) methods are recommended for the analysis of
log-transformed BE measures. For average BE using the criterion stated
in equations 2 or 3 above, the general approach is to construct a 90%
confidence interval for the quantity :T-:R and to reach a conclusion
of average BE if this confidence interval is contained in the interval
[-2A , 2A]. Due to the nature of normal-theory confidence intervals,
this is equivalent to carrying out two one-sided tests of hypothesis
at the 5% level of significance (Schuirmann 1987).
The 90% confidence interval for the difference in the means of the
log-transformed data should be calculated using methods appropriate to
the experimental design. The antilogs of the confidence limits obtained
constitute the 90% confidence interval for the ratio of the geometric
means between T and R product."
Mike Leibold, PharmD, RPH
ML111439.-a-.goodnet.com
---
Date: Wed, 29 Dec 1999 21:56:07 +1300
From: Nick Holford
X-Accept-Language: en
To: PharmPK.at.boomer.org
Subject: Re: PharmPK Re: Statistical description of tmax
> A decrease in Ka or rate of absorption would not change the AUC, which
> is determined by F, Xo, Vd and Ke.
AUC is determined by F, Dose and Clearance. Volume of distribution
does not influence AUC 0-inf.
CL = Vd * Ke is algebra not pharmacokinetics.
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm
---
X-Sender: walt.at.mail.simulations-plus.com
Date: Wed, 29 Dec 1999 12:25:46 -0800
To: PharmPK.aaa.boomer.org
From: Walt Woltosz
Subject: Re: PharmPK Re: Statistical description of tmax
I have always been impressed with the responses provided by certain members
of the PharmPK list server - especially those from Mike Leibold and Nick
Holford, but several others as well, so I do not mean to be critical here.
There are some hidden assumptions in some of the replies to this inquiry,
and they have been incorporated in other posts to this list server as well.
These assumptions may hold for some compounds, but could get you into
trouble for others.
I think it is very important to recognize that any analytical approach that
uses a single, constant value for Ka is a simplification that may work for
many compounds that have high solubility and permeability, but it can be
very misleading for those that are either low in permeability or
solubility, or both.
As a relative newcomer to this field, but with over 25 years' experience in
modeling and simulation, it has struck me as a bit strange that many
discussions seem to dwell on handling pharmacokinetics with higher order
models, while maintaining a constant Ka. If you're trying to
predict/estimate Cmax and Tmax, which are affected by the balance between
absorption and clearance pulling plasma concentration in opposite
directions, then you need to consider that for many compounds, absorption
rate is not a constant, but can vary with pH, local concentration, transit
times, saturation of transport mechanisms, dissolution rates, and other
factors. For such compounds, only an analysis that incorporates the
interactive effects of both pharmacokinetics and varying absorption rates
can hope to provide reasonable estimates.
Last year I visited a generic house to demonstrate GastroPlus(TM). Within
an hour, we showed by simulation that one of the company's generic
formulations would significantly overshoot Cmax compare to the original
formulation. (I found out later they had already learned this in clinical
trials, but they wanted to see what the prediction would show.) If the
company had used the simulation prior to going to clinical trials, they'd
have saved themselves considerable time and money. A simple constant Ka
model did not provide such a prediction.
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (SIMU)
1220 W. Avenue J
Lancaster, CA 93534-2902
U.S.A.
http://www.simulations-plus.com
Phone: (661) 723-7723
FAX: (661) 723-5524
walt.-a-.simulations-plus.com
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