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Someone suggested to me that it is wrong to report a terminal elimination
rate constant (i.e. lambda-z) and half-life (i.e. ln2/lambda-z), obtained
from the terminal slope of a log-linear plot of concentrations vs. time,
when analyzing data from a drug that displays a multiexponential decline
according to non-compartmental theory?
I disagree.
Who's right?
Matthieu L. Kaltenbach, PharmD, PhD
Laboratoire de Pharmacologie et de Pharmacocin=E9tique
UFR de Pharmacie, 51 rue Cognacq-Jay
51100 Reims, France.
Phone: (33) 3 2689-8077
Fax: (33) 3 2605-3552
E-mail: matthieu.kaltenbach.at.univ-reims.fr
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[A few replies - db]
From: Nick Holford
Sender: nhol004.-a-.auckland.ac.nz
Reply-To: Nick Holford
To: PharmPK.-a-.pharm.cpb.uokhsc.edu
Subject: Re: PharmPK Non-compartmental analysis
Date: Fri, 4 Dec 1998 14:10:07 +0000 (GMT)
Priority: NORMAL
X-Authentication: none
MIME-Version: 1.0
Non-compartmental theory ignores the existence of
multiexponential decline so it isn't really relevant
(that's why it is called non-compartmental).
The terminal elimination rate constant is just a model
parameter. Its estimate can depend on the model you use. If
you are doing simple non-compartmental analysis you have to
make a decision when the terminal phase starts and then
estimate the rate constant. If you use a multi-compartment
model then this decision is taken out of your hands. If the
multi-compartment model fits the data well then the
terminal rate constant estimates will be similar. Neither
are "right".
"All models are wrong - some models are useful" George Box
--
Nick Holford, L226,Dept of Neurology,OHSU
3181 SW Sam Jackson Park Road,Portland,OR 97201-3098
n.holford.at.auckland.ac.nz,(503)494-4778,fax 494-7242
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm
---
Date: Fri, 04 Dec 1998 14:38:18 -0500
From: Mauricio Leal
To: PharmPK.-at-.pharm.cpb.uokhsc.edu
Subject: Re: PharmPK Non-compartmental analysis
Mime-Version: 1.0
I believe you are right. I see no reason why you can't report it as long
as it does describe the elimination phase.
M. Leal
WA-R
---
Date: Fri, 04 Dec 1998 13:19:51 -0700
From: "David Nix, Pharm D."
Organization: College of Pharmacy
MIME-Version: 1.0
To: PharmPK.at.pharm.cpb.uokhsc.edu
Subject: Non-compartmental analysis
It is not wrong to determine and report terminal elimination rate
constants for non-compartmental analyses. However there are some
problems that must be considered.
1. You must determine the points that will be used in the regression
of ln concentration vs time. This is usually subjective and some
individuals feel that there are better ways to analyse the data.
With modeling (especially with oral admin), there is less
subjectivity, but model mispecification and systematic bias is
common.
2. Typically linear regression is used and a weight of 1 is
assumed. This will place more emphasis on fitting the higher
concentrations
and may ignore the lowest concentrations. The log
transformation that is done will reduce this problem, but will not
eliminate it.
Despite this problem you will likely find almost identical slopes if
you use weighted nonlinear regression on the same points selected
for determination of terminal slope. Moreover, some nonlinear
fitting methods are unstable when only 3-4 points are available over
a narrow time range.
3. The most difficult problem occurs when the terminal slope does not
really look linear on a plot of ln(conc) vs time. There may be one
point that clearly deviates from the "line". This often occurs with
the last measured concentration.
Assumptions are always necessary regardless of the method used for
analyzing the data. Most pharmacokineticists do determine the terminal
slope of ln(conc) vs time using linear regression when non-compartmental
analysis is used.
David Nix,Pharm.D.
Associate Professor
The University of Arizona
---
X-Sender: dfarrier.at.mail.bright.net
Date: Fri, 04 Dec 1998 18:17:10 -0800
To: PharmPK.aaa.pharm.cpb.uokhsc.edu
From: "David S. Farrier"
Subject: Re: PharmPK Non-compartmental analysis
Mime-Version: 1.0
Matthieu,
You are right. How much did you bet?
Each of the exponential terms comprising a multiphasic semi-log plot will
have its own intercept and slope which can be extracted by the method of
residuals (curve stripping). The half-life calculated from the terminal
portion is an appropriate measure regardless of the number of other terms
or the multiphasic nature of the curve. The usual limitation is, of course,
that you have sufficiently good data over the terminal time points.
David
David S. Farrier, Ph.D.
Summit Research Services
Pharmacokinetics and Metabolism Software
1374 Hillcrest Drive
Ashland, OH 44805 USA
Tel: (419) 289-9207
Email: dfarrier.-a-.bright.net
Web Site: http://www.bright.net/~dfarrier
---
Reply-To:
From: "Steven L. Shafer"
To:
Subject: RE: PharmPK Non-compartmental analysis
Date: Sat, 5 Dec 1998 20:00:03 -0800
MIME-Version: 1.0
X-Priority: 3 (Normal)
Importance: Normal
Dear Dr. Kaltenbach:
It's not wrong to calculate and report terminal half life, but it may not
mean very much clinically. See, for example, Anesthesiology. 1992 Mar;
76(3):334-41.
Best regards,
Steve Shafer
---
X-Originating-IP: [194.95.249.29]
From: "Krishna Devarakonda"
To: PharmPK.-a-.pharm.cpb.uokhsc.edu
Subject: Re: PharmPK Non-compartmental analysis
MIME-Version: 1.0
Date: Mon, 07 Dec 1998 03:27:21 PST
Dear Dr.Kaletbach,
Normally you determine what is called MRT (mean residence time) when you
apply Statistical Moments Theory (SMT) in noncompartmental analysis and
it is comparable to t1/2. Lambda(z) and terminal half life are
calculated from the terminal data by the method you used even though you
don't call it Lamda(z) when you employ noncompartmental analysis. You
would get the same value for this parameter even if you use
compartmental analysis. You were right.
D.R.Krishna, Ph.D
---
Reply-To: Dr Les White
To: PharmPK.aaa.pharm.cpb.uokhsc.edu
Date: Mon, 07 Dec 1998 09:35:37
Subject: Re: PharmPK Non-compartmental analysis
From: lesassays.-at-.ukneqasaa.win-uk.net (Dr Les White)
THE FIGURE YOU GET WILL DEPEND ON THE LENGTH OF TIME AFTER THE
DO YOU TAKE SAMPLES.
THIS MAY NOT BE TERRIBLY USEFUL FOR DOSING PREDICTIONS
==================
The UK NEQAS for Antibiotic Assays is provided by:
Department of Microbiology, Southmead Health Services NHS Trust
Bristol BS10 5NB, UK.
Tel INT+UK+117 9595653 Fax INT+UK+117 9593217
http://www.ukneqasaa.win-uk.net
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Dear Matthieu,
Assuming that your terminal phase has been analytically detected with the
appropriate statistical and acceptable confidence, and you have good
regression parameters for this phase, you need to consider the following
aspects of the disposition before you conclude whether your terminal phase
is a `meaningful' elimination phase.
If the area contribution (AUC) of your `terminal' phase to your overall
disposition area (total AUC measured by linear/log-linear trapezoidal
methods) is significant (>= 10%--since less than this value usually (one
exception I can think of is THC) does not significantly affect predictions
of steady state levels or time to steady state), the `terminal' phase you
have defined becomes meaningful and the rate constant estimate and
corresponding half-life are significant.
If on the other hand the area contribution of the `terminal' phase is less
than 10% of the total AUC then the defined `terminal' phase does not play a
significant role in the disposition of the drug. Obviously,
Non-Compartmental analysis cannot address this issue as you cannot directly
determine the rate constant of the pre-terminal phase which has now become
the significant `elimination' phase for the overall disposition. Modeling
the data or compartmental analysis becomes useful at this stage.
Alternatively one can `feather' the pre-terminal phase using simple linear
regression and determine the true elimination rate constant of this phase.
Hope this has been helpful
Anup Zutshi
Battelle Pulmonary Therapeutics
(614) 424-5997 Tel.
(614) 424-3268 Fax.
zutshi.-at-.battelle.org
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Dear Matthieu,
Assuming that your terminal phase has been analytically detected with the
appropriate statistical and acceptable confidence, and you have good
regression parameters for this phase, you need to consider the following
aspects of the disposition before you conclude whether your terminal phase
is a `meaningful' elimination phase.
If the area contribution (AUC) of your `terminal' phase to your overall
disposition area (total AUC measured by linear/log-linear trapezoidal
methods) is significant (>= 10%--since less than this value usually (one
exception I can think of is THC) does not significantly affect predictions
of steady state levels or time to steady state), the `terminal' phase you
have defined becomes meaningful and the rate constant estimate and
corresponding half-life are significant.
If on the other hand the area contribution of the `terminal' phase is less
than 10% of the total AUC then the defined `terminal' phase does not play a
significant role in the disposition of the drug. Obviously,
Non-Compartmental analysis cannot address this issue as you cannot directly
determine the rate constant of the pre-terminal phase which has now become
the significant `elimination' phase for the overall disposition. Modeling
the data or compartmental analysis becomes useful at this stage.
Alternatively one can `feather' the pre-terminal phase using simple linear
regression and determine the true elimination rate constant of this phase.
Hope this has been helpful
Anup Zutshi
Battelle Pulmonary Therapeutics
(614) 424-5997 Tel.
(614) 424-3268 Fax.
zutshi.at.battelle.org
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[Two more replies - db]
Date: Tue, 08 Dec 1998 10:36:18 +0000
From: Richard Knapp
Reply-To: rknapp.-at-.cyberannex.com
MIME-Version: 1.0
To: PharmPK.-at-.pharm.cpb.uokhsc.edu
Subject: Re: PharmPK Re: Non-compartmental analysis
Dear Colleagues:
We have an observation that illustrates a problem associated with defining
the terminal elimination rate. Consider the following rate constants obtained
for a small molecule drug using two or three exponential model fits:
10 mg/kg i.v. data
k(1) = 0.440/min (t = 1.58 min)
k(2) = 0.0336/min (t = 20.6 min)
15 mg/kg p.o. data
k(1) = 0.397/min (t = 1.75 min)
k(2) = 0.0257/min (t = 27.0 min)
15 mg/kg p.o. data
k(1) = 0.251/min (t = 2.76 min)
k(2) = 0.0797/min (t = 8.70 min)
k(3) = 0.0146/min (t = 47.5 min)
If we perform a conventional p.o. administration of a drug in solution and
measure its plasma concentration at 10 or more time points out to about 6 hr
we often find a best fit to a 3 exponential non-compartmental model. This is
not true for i.v. administration data that are well fitted by the
biexponential model.
The biexponential fits for the i.v. and p.o. data are in reasonable agreement
with respect to the terminal (elimination) phase. The last (elimination) rate
constant for the p.o. data is slower than for the i.v. data (0.0257 vs.
0.0336) which might suggest continued absorption from the gut. Fitting the
p.o. data to the triexponential model gives a significantly better (F-ratio
test) fit but now the rate constants are in disagreement with the i.v. data.
This is most evident for the apparent elimination rate constants (0.0336 vs.
0.0146).
We hypothesize that while there is an input to the central (plasma)
compartment late in the sampling process, it comes from the liver. This
hypothesis is supported by studies showing that 30 min after p.o.
administration the drug plasma to liver ratio is about 1:10 consistent with
drug accumulation in the hepatic tissue.
I present these data to show how confusing multiexponential p.o. data can be
and to invite comments. It is likely that the first rate constant (0.251/min)
accurately measures absorption but the second rate constant (0.0797/min) is
likely to be a hybrid of four processes; absorption, distribution, hepatic
release and elimination. The final rate constant (0.0146/min) is probably a
hybrid of hepatic release and elimination. I would be interested in knowing of
other drugs showing similar properties.
Richard Knapp
Director of Pharmacology
Cortex Pharmaceuticals, Inc.
---
From: "Hans Proost"
Organization: Pharmacy Dept Groningen University
To: PharmPK.at.pharm.cpb.uokhsc.edu
Date: Wed, 9 Dec 1998 10:21:30 MET
Subject: Re: PharmPK Re: Non-compartmental analysis
X-Confirm-Reading-To: "Hans Proost"
X-pmrqc: 1
Priority: normal
Many good answers to this question have been given already.
I would add a comment to the following statement by David Nix:
> 2. Typically linear regression is used and a weight of 1 is
> assumed. This will place more emphasis on fitting the higher
> concentrations
> and may ignore the lowest concentrations. The log
> transformation that is done will reduce this problem, but will not
> eliminate it.
The log transformation completely eliminates the problem of weighting
if the errors are log-normally distributed, which is, in general, a
reasonable assumption. In my view, it is even (slightly) better, and
easier, than the use of a weighting factor 1/Y^2.
Both methods are based on the assumption of a constant coefficient of
variation.
The real problem with both log-transformation and 1/Y^2 weighting is
that it places too much emphasis on the LOWER concentrations, if the
assumption of constant coefficient of variation is violated, eg. at
concentrations close to the limit of quantification.
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University Centre for Pharmacy
Groningen, The Netherlands
tel. 31-50 363 3292
fax 31-50 363 3247
Email: j.h.proost.at.farm.rug.nl
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If I recall it well, the original query was: are the elimination rate
constant and elimination half-life noncompartmental parameters or not.
People can have doubts since the exponential terminal tail is usually
related to (multi) compartmental models. As a matter of fact, however, it is
a general property of any linear time-invariant pharmacokinetic system, as
it has been proved by M. Weiss (see Journal of Pharmacokinetics &
Biopharmaceutics. 14(6):635-57, 1986; 15(1):57-74, 1987).
Best regards,
Vladimir
----------------
Vladimir Piotrovsky
Clinical Pharmacokinetics, ext 5463
Janssen Research Foundation
2340 Beerse, Belgium
E-mail: vpiotrov.-at-.janbe.jnj.com
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>People can have doubts since the exponential terminal tail is
>usually related to (multi) compartmental models. As a matter
>of fact, however, it is a general property of any linear
>time-invariant pharmacokinetic system, as it has been proved
>by M. Weiss (see Journal of Pharmacokinetics &
>Biopharmaceutics. 14(6):635-57, 1986; 15(1):57-74, 1987).
To support this, the following comments could be added:
Arguments presented in these papers by Weiss have the
background in the probability theory. If h(t) is
a probability density function, H(t) is the corresponding
cumulative function, H1(t)=1-H(t), and k(t)=h(t)/H1(t), then
for k(t) being constant (at least asymptotically) the density
h(t) is asymptotically exponential. A special class of the
probability distributions with non-increasing k(t) are
so-called completely monotone distributions. A probability
distribution function is completely monotone on
(0,+infinity) if and only if it is a linear combination of
exponentials (1). If h(t) is the probability density function
of the residence time of the drug in the body, k(t)
represents the probability of elimination at a given time
point under the condition that the drug molecule has not been
eliminated up to this time (2,3).
1. Feller, W. 1966. An Introduction to Probability Theory and
Its Applications, Vol.8, New York; Wiley.
2. Smith, Ch. E., Lansky, P., Lung, T.H., Bull. Math. Biol.,
1997, 59, 1-22.
3. Wimmer, G., Dedik, L., Durisova, M., et al., Bull. Math.
Biol., 1999, 61, in press.
Dr. Maria Durisova PhD
Institute of Experimental Pharmacology
Slovak Academy of Sciences
Bratislava, Slovak Republic
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