- On 4 Dec 1998 at 11:14:32, "Matthieu L. Kaltenbach" (matthieu.kaltenbach.aaa.univ-reims.fr) sent the message

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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 - On 7 Dec 1998 at 16:48:41, David_Bourne (david.at.pharm.cpb.uokhsc.edu) sent the message

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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

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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

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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 - On 8 Dec 1998 at 12:08:39, "Zutshi, Anup" (zutshi.-at-.BATTELLE.ORG) sent the message

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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 - On 8 Dec 1998 at 12:08:39, "Zutshi, Anup" (zutshi.aaa.BATTELLE.ORG) sent the message

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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 - On 9 Dec 1998 at 15:52:47, David_Bourne (david.-a-.pharm.cpb.uokhsc.edu) sent the message

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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 - On 10 Dec 1998 at 11:25:21, "Piotrovskij, Vladimir [JanBe]" (VPIOTROV.-at-.janbe.jnj.com) sent the message

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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 - On 22 Dec 1998 at 13:49:27, Maria Durisova (exfamadu.aaa.savba.savba.sk) sent the message

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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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