- On 15 Jun 1998 at 11:26:19, "Zutshi, Anup" (zutshi.-a-.BATTELLE.ORG) sent the message

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Friends

Does anybody know of references, ICH/FDA guidelines, position papers etc.

that discuss the inappropriateness of using arbitrary estimates for

concentrations in blood/plasma/serum etc that are below the Limit of

Quantitation of your analytical method -- like setting the concentration of

one animal (in a group of 3 animals/group) to zero or 1/2 of the LOQ value

or some such arbitrary value in order to "complete the pharmacokinetic

profile" and get estimates of terminal rate constants etc.for

pharmacokinetic evaluation.

It appears to have been common practice before the advent of highly

sensitive analytical techniques, although I have never been able to find a

rationale for its use.

Thanks in advance

Anup Zutshi - On 17 Jun 1998 at 14:45:14, "Ette" (Ette.aaa.macnet.vpharm.com) sent the message

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I am not aware of any position paper from the regulatory agencies on this

subject. It would seem reasonable to explain your assumptions and rationale

for any imputation of concentrations you are willing to make. I believe there

is always room for good science at the regulatory agencies.

Ene Ette, Ph.D.

Vertex Pharmaceuticals, Inc.

ette.aaa.vpharm.com - On 18 Jun 1998 at 11:52:19, David Claghorn (dclaghorn.-at-.ilexonc.com) sent the message

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From: Larry Arthaud

Sent: Wednesday, June 17, 1998 1:57 PM

To: David Claghorn

Subject: RE: PharmPK Use of arbitrary values for plasma levels

below LOQ in toxicokine tic stuides

Dave,

I checked the ICH guidelines for toxicokinetic studies and did not see a

specific reference to the example mentioned. What they did say was that

if transformation is applied to the data a rationale has to be provided.

Jinee Rizzo and Sharon Baker would be good references for you to check.

I do know that they have applied particular rules for accepting or

rejecting data based upon an outlier.

Larry - On 23 Jun 1998 at 11:06:14, David_Bourne (david.-at-.pharm.cpb.uokhsc.edu) sent the message

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From: David Claghorn

To: "'zutshi.-a-.BATTELLE.ORG'",

"'PharmPK.-a-.pharm.cpb.uokhsc.edu'"

Subject: FW: PharmPK Use of arbitrary values for plasma levels below LOQ i

n toxicokine tic stuides

Date: Mon, 22 Jun 1998 13:37:08 -0500

MIME-Version: 1.0

From: Rizzo, Jinee [SMTP:jrizzo.at.saci.org]

Sent: Monday, June 22, 1998 12:30 PM

To: David Claghorn

Subject: RE: PharmPK Use of arbitrary values for plasma levels

below LOQ i n toxicokine tic stuides

Hello Dave:

I have an FDA paper briefly discussing the fact that NO samples should

be extrapolated below the LOQ. Although this paper applies to clinical

sample analysis, I use this guideline in my preclinical studies as well.

If you have any more analytical questions, I will try to help.

later....jinee

---

X-Sender: jelliffe.-a-.hsc.usc.edu

Date: Mon, 22 Jun 1998 17:47:21 -0700

To: PharmPK.at.pharm.cpb.uokhsc.edu

From: Roger Jelliffe

Subject: Re: PharmPK Use of arbitrary values for plasma levels below

LOQ in toxicokine tic stuides

Mime-Version: 1.0

Dear Anup:

The limit of detection is useful in toxicology, where there is no other

information as to whether or not there is any drug present. However, in PK

studies we know the drug is there, and usually decays exponentially,

"never" getting rid of the very last molecule. Because of this, we know the

drug is there, and our only question is how much.

It is useful to give weight to each data point by its Fisher

information,

the reciprocal of its variance. People have said that as the concentration

approaches zero, the coefficient of variation becomes infinite, and one

cannot use such data.

This view is incorrect. While the CV becomes infinite, the variance and

the SD are always finite. One can thus measure a blank sample in

quadruplicate, for example, as well as a low sample, a mid-range sample, a

high sample, and a very high one, and examine the empirical, usually

nonlinear, relationship between the measured concentration and the SD with

which it is measured.

This relationship can easily be captured with a polynomial

expression such as

SD = A0*C^0 + A1*C^1 + A2*C^2 + A3*C^3

where A is a coefficient of the polynomial

0,1,2,and 3 are the subscript for each coefficient, or the exponents to

which each coefficient is raised.

^ is this poor wordprocessor's way of indicating an exponent

and C is the observed concentration

This is a cheap and easy of capturing the relationship between the

measured concentration and the SD with which it is measured. It is a pretty

good way of getting the expected value of the SD for a new single measured

level.

The usual maximum aposteriori probability (MAP) Bayesian objective

function uses this data explicitly in finding the Bayesian posterior

parameter values. It balances the relative credibility of the population

means with their SD's against the likely SD with which each serum sample

was measured.

It is true that there are other sources of error than only measurement

error. However, it is useful to know the measurement error, and it can

easily be found as described. In this way, there is no lower detectable limit.

You might look at

Drug Monitoring and Toxicology No. DM 89-4 (DM-56) in Drug Monitoring and

Toxicology 10: 1-5, 1990

Therapeutic Drug Monitoring 15: 380-393, 1993

Therapeutic Drug Monitoring 16: 552-559, 1994

These discuss this all more fully. The first one is hard to find.

The last

one is a specific example of its use. Let me know if you have trouble.

Others may have comments.

Sincerely,

Roger Jelliffe - On 24 Jun 1998 at 10:16:40, Stephen Duffull (sduffull.-a-.fs1.pa.man.ac.uk) sent the message

Back to the Top

Dear Roger

Roger Jelliffe commented:

> The usual maximum aposteriori probability (MAP) Bayesian objective

function uses this data explicitly in finding the Bayesian posterior

parameter values. It balances the relative credibility of the population

means with their SD's against the likely SD with which each serum sample

was measured.

The part of the MAP objective function that addresses observations states that:

yi ~ N(f(.),var(yi))

where yi = ith observation, f(.)=model predicted concentration,

var(yi)=the variance of yi around f(.).

As you have commented the variance of the observations around the

expected concentration is not solely explained by assay error. While

many may employ MAP using assay error as a surrogate for var(yi) - it

should always be remembered that this is an approximation and may incur

significant error in estimating the posterior mode of the parameters if

the remaining sources of residual variability are considerable. In many

publications involving population analysis var(yi) is given explicitly and

therefore this value should be used rather than var(assay).

Regards

Steve

==========================

Stephen Duffull

School of Pharmacy

Manchester University M13 9PL, UK

Ph +44 161 275 2355

Fax +44 161 275 2396

Email: sduffull.-at-.fs1.pa.man.ac.uk - On 25 Jun 1998 at 08:51:44, David_Bourne (david.-a-.pharm.cpb.uokhsc.edu) sent the message

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[A couple of replies - db]

Date: Wed, 24 Jun 1998 13:18:36 -0400

From: "Zutshi, Anup"

Subject: RE: PharmPK Re: Use of arbitrary values for plasma levels below L OQ

To: "'PharmPK.-a-.pharm.cpb.uokhsc.edu'"

Dear Roger,

Your approach would be valid provided one assumes (from the standpoint of

errors) that the analytical detection system behaves in a similar manner at

levels BLOQ (Below the Limit of Quantitation) as it does in the quantifiable

range of the calibration curve. For example the photomultiplier in an UV-Vis

detector cannot detect subtle changes in transmitted light when the fraction

of the incident light that is absorbed is small (assuming Beer-Lambert law A

= E * l * [C]). This leads to an apparent deviation in the Beer's-Lambert's

Law. The same applies at the high end of the concentrations where other

physical effects such as Self-Association etc may affect the data. Compound

all of this with the fact that in liquid chromatography (the most widely

used analytical technique), solvents of various polarities are co-mixed with

water, which makes these physico-chemical interactions all the more

difficult to understand. Other analytical detectors also face similar

limitations. In these circumstances it would be prudent not to extrapolate

to regions beyond the calibration curve to determine plasma levels and

further rely on that information to make pharmacokinetic estimations.

Anup Zutshi

---

X-Sender: jelliffe.aaa.hsc.usc.edu

Date: Wed, 24 Jun 1998 12:33:02 -0700

To: PharmPK.at.pharm.cpb.uokhsc.edu

From: Roger Jelliffe

Subject: Re: PharmPK Re: Use of arbitrary values for plasma levels

below LOQ

Mime-Version: 1.0

Dear Steve:

Yes, you are right. There are significant other sources of error

than just

the assay error. However, it is nevertheless useful to know it explicitly.

Most of the other sources of variation are usually not measurement

noise,

but process noise. The model mis-specification, and the errors in dosage

preparation and administration are process noise, for example, and properly

belong in the differential equationsdescribing the bahevior of the drug

rather than in a measurement noise term. However, no one is doing this yet.

I think it is useful to know your assay error explicitly, and it is

easy

to do. Then, if you also want to include another term for the remaining

sources of noise, one can do so, either as a scaling factor for the assay

polynomial as we do in our iterative 2-stage Bayesian population modeling

software, or as an entire polynomial, with 4 coefficients, to define its

shape better still, if you wish, which our software also does.

We do think, though, that since the assay error can be easily

known, it is

useful to do that, and then to see what you wish to do after that. The MAP

Bayesian objective function appears to describe the variance of each

measured level itself, to balance this properly against the variance of

each population parameter value in the fitting process. The other sources

of variation mentioned above are likely to be in the parameter values, as

they are mostly process noise rather than measurement noise.

************************************************

Roger W. Jelliffe, M.D.

USC Lab of Applied Pharmacokinetics

CSC 134-B, 2250 Alcazar St, Los Angeles CA 90033

**Note our new area codes below, since 6/15/98!**

Phone (323)342-1300, Fax (323)342-1302

email=jelliffe.at.hsc.usc.edu

************************************************

You might also look at our Web page for announcements of

new software and upcoming workshops and events. It is

http://www.usc.edu/hsc/lab_apk/

************************************************ - On 26 Jun 1998 at 11:21:53, "Hans Proost" (HANSP.at.farm.rug.nl) sent the message

Back to the Top

Dear colleagues,

I fully agree with the previous responders; in particular the answers

of Roger Jelliffe and Steve Dufull reflect the difficulties and

possible solutions.

I would like to make one additional statement about the question WHY

are we bothering about concentrations below LOQ?

Simply, because they contain important information!

In the case of PK modelling and Bayesian forecasting, any knowledge

about the concentration at any time is useful information, and can be

used if weighted appropriately.

The alternative, i.e., ignoring this information, may lead to false

conclusions!

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 29 Jun 1998 at 10:45:12, Roger Jelliffe (jelliffe.-a-.hsc.usc.edu) sent the message

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Dear Anup:

I am not sure that the assay must behave the same way even at such low

levels. I think the important thing is to know your assay well, and to know

the relationship between what you measure (compared with standerds to deal

with your question) and the precision of that measure as shown by replicate

determinations. These do not have to be standards, and are probably best as

regular samples. One can nevertheless quantify the relationship between a

measured level and its SD, and can do so all the way down to a blank. The

important thing is to know, if possible (and it usually is) the Fisher

information of the assay throughout its range. The whole question of a

lower detectable limit arises only from a toxicological point of view, when

the assay info is the only soure of info as to whether the drug is present

or not. Then you must have a lower detectable limit.

What is this limit? Usually about 2 SD above the blank. Why is that? So

you can be confident that something is present if you get a result above

that. But what should this be? 2SD? 3SD? It is a statistical thing that one

decides to buy in on, depending on the probability that the result is not a

blank.

In TDM, though, we KNOW the sample is not a blank, because we know how

long it was since the last dose. One can report these levels as a gent

level of 0.2 ug/ml, for example, below our usual detection limits of 0.5

ug/ml. Then both the toxicologists and the TDM people and the pop modeling

people have what they need to make their pop or patient-specific model, and

the toxicologists also have what they heec if there was no other info about

the time since the last dose. What do you do now with such data, and why?

Do you simply withold such a result? If so, Why?

Does this help?

Roger Jelliffe

************************************************

Roger W. Jelliffe, M.D.

USC Lab of Applied Pharmacokinetics

CSC 134-B, 2250 Alcazar St, Los Angeles CA 90033

**Note our new area codes below, since 6/15/98!**

Phone (323)342-1300, Fax (323)342-1302

email=jelliffe.at.hsc.usc.edu

************************************************

You might also look at our Web page for announcements of

new software and upcoming workshops and events. It is

http://www.usc.edu/hsc/lab_apk/

************************************************ - On 6 Jul 1998 at 13:15:35, ahmadmir.-a-.nrcgeb.ac.ir (Ahmad Mirfazaelian) sent the message

Back to the Top

Dear Anup,

There is sth that i sometimes do when the computer programme that is

going to solve my data has a problem with 0 and i have a blank or a conc.

under LOD( that i regard it as 0 ). i input 0.001 in the programme. & i

think that doesn't make any problem.

yours

A. Mirfazaelian Pharm.D.

Dept. of Pharmaceutical Sciences

College of Pharmacy

Teheran Univ. Med. Sci.

Teheran,Iran

Phone: (+98)-021-6026452,844162

Fax: (+98)-021-6419539

e.mail: ahmadmir.-a-.nrcgeb.ac.ir - On 27 Jul 1998 at 12:40:47, Roger Jelliffe (jelliffe.-a-.hsc.usc.edu) sent the message

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Dear Anup:

Thanks for your comments. The problem is that you are going back to the

assumption that you know nothing about whether the drug is present or not

except from the assay results. Once again, if that is the case, then very

clearly therer is a lower detectable limit that is usually about 2SD above

the blank inmost cases.

However, that is not the point with PK studies or with TDM. In

contrast to

the toxicology situation, in TDM and in PK studies, one usually knows

pretty well when the doses were given and the levels drawn. Because of

this, and because usually you never get rid of the last molecule of an

exponentially decaying drug, you KNOW that the drug IS PRESENT, and you are

only asking HOW MUCH. It is true that you are down in the noise region, but

you can quantify the noise as well as the result.

Because of this, and because you can determine the SD of an assay right

down and to the blank, there is no lower detectable limit of an assay in

the PK or TDM situation.

************************************************

Roger W. Jelliffe, M.D.

USC Lab of Applied Pharmacokinetics

CSC 134-B, 2250 Alcazar St, Los Angeles CA 90033

**Note our new area codes below, since 6/15/98!**

Phone **(NOTE NEW AREA CODE AND PREFIX)** (323)442-1300, Fax (323)442-1302

email=jelliffe.-at-.hsc.usc.edu

************************************************

You might also look at our Web page for announcements of

new software and upcoming workshops and events. It is

http://www.usc.edu/hsc/lab_apk/

************************************************ - On 3 Aug 1998 at 10:20:14, Brennan (brennan.-a-.ids.net) sent the message

Back to the Top

Once again we return to the practical application of pharmacokinetics. The

question must revolve around the patient. Are we better off piggybacking drug

levels with routine blood work and accepting the errors implied but reaching a

pratical and effective dose or are we more concerned with the "numbers" as

an end

point in them selves?

Certainly the best model we can obtain is what we should work with.

Still we

must not forget that these models are just one more tool to use in the care

of our

patients

Bob B. - On 4 Aug 1998 at 13:53:47, "Zutshi, Anup" (zutshi.-a-.BATTELLE.ORG) sent the message

Back to the Top

Dear Roger

I agree with you about the fact that an exponentially decaying drug is still

present in the body and once an `exponential component' in a sum of

exponentials model is known one can predict the levels BLOQ. However in

situations where a kinetic (exponential) phase is incompletely defined (i.e.

you end up with one or two data points--at the low end of your

assay--defining a slower eliminating phase) it is `hazardous' (I would

think) to extrapolate your calibration curve to define this phase and make

categorical statements about the pharmacokinetic parameters derived from

this phase. I guess in the absence of any data, information derived from

extrapolations may be the best we have-- but I am sure you would agree that

it would be pointless to spend money on improving the sensitivities of our

analytical methods if one can extrapolate the data BLOQ from any convenient

and easily achievable levels. Infact the conference on Analytical Methods

Validation held in Washington DC in December 1990 debated this issue and

suggested that extrapolations above and below the standard curve are not

recommended (Shah VP et.al; Int. J. of Pharmaceutics, 82, 1-7, 1992).

Regards

Anup Zutshi - On 5 Aug 1998 at 12:48:41, David_Bourne (david.aaa.pharm.cpb.uokhsc.edu) sent the message

Back to the Top

[A few replies - db]

From: "Hans Proost"

Organization: Pharmacy Dept Groningen University

To: PharmPK.aaa.pharm.cpb.uokhsc.edu

Date: Wed, 5 Aug 1998 08:58:57 CET

Subject: Re: PharmPK Re: Use of arbitrary values for plasma levels

X-Confirm-Reading-To: "Hans Proost"

X-pmrqc: 1

Priority: normal

Dear Anup,

With respect to your arguments against the use of plasma

concentration values below the LOQ, it might be useful to make a

clear distinction between PK research (e.g. assessment of PK

parameters) and clinical PK (e.g. treating patients with a drug).

In PK research, the use of concentration values below the LOQ may

give rise to the problems you mentioned. However, it should be

remembered that simply leaving out this information may also lead to

serious errors (as I stated in my earlier message of 26 June).

Personally, I think that such values should be used since they

contain essential information. In practice, it may be difficult to

use these data properly, however.

In clinical PK, ALL INFORMATION on the patient should be taken into

account to maximise the chance that the patient is treated optimally,

as has been pointed out in several messages by Roger Jelliffe.

Of course, appropriate weighting is essential. Although some

arguments against the method of Roger Jelliffe can be found (as

mentioned by Stephen Duffull and others), I think that, at present,

his approach is still 'state-of-the-art'.

Best regards,

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.aaa.farm.rug.nl

---

From: "Leon Aarons"

To: pharmpk.at.pharm.cpb.uokhsc.edu

Date: Wed, 5 Aug 1998 09:21:25 GMT

Subject: Re: Concentration values below assay limits

Reply-to: l.aarons.aaa.man.ac.uk

Priority: normal

There is a 'formal' way to handle this problem which is to use

integral contributions to the likelihood, i.e.

INTEGRAL{from y = 0 to QL} Likelihood dy

But this is difficult, obviously, because one needs to integrate

numerically and the support of the likelihood over this region is

unknown because it depends on theta, etc. From an MCMC perspective

this is alleviated somewhat because you one can integrate at each

iteration, but the shape of the likelihood is still unknown - i.e.

you have to make sure that your integration routine is very robust.

Also, I don't imagine any existing software (except BUGS) will let

this be done.

See D.J. Lunn, L. Aarons, 'Markov Chain Monte Carlo techniques for

studying interoccasion and intersubject variability: application to

pharmacokinetic data', Appl.Statist. 46, 73-91 (1997).

The logic behind this approach is that data below the quantification

limit carries information and that information is greatest the closer

the sample is to the LOQ.

_______________

Leon Aarons

School of Pharmacy and Pharmaceutical Sciences

University of Manchester

Manchester, M13 9PL, U.K.

tel +44-161-275-2357

fax +44-161-275-2396

email l.aarons.-a-.man.ac.uk

---

From: Nick Holford

Sender: nhol004.-a-.auckland.ac.nz

Reply-To: Nick Holford

To: PharmPK.aaa.pharm.cpb.uokhsc.edu

Subject: PharmPK Re: Use of arbitrary values for plasma levels below LOQ

Date: Wed, 5 Aug 1998 09:47:01 +0100 (British Summer Time)

Priority: NORMAL

X-Authentication: none

MIME-Version: 1.0

I suspect that this decision is a reflection of a

reasonable regulatory viewpoint (given Vinod works for FDA

and the conference was in DC). Science, however, is another

world which is forever trying to discover things at the

limits of resolution. Roger's recommendation to extend what

we are trying to learn by using a reasonable error model is

exactly what science is about. Regulators have a different

role. Consider traffic cops and astronauts going to the

moon. We need the cops to control the crowds at the launch

site but they have no legitimate (in the sense of

Congressional mandate) role for recommending on how to get

to the moon. The FDA and other agencies have a traffic cop

role. Their policies may not be relevant to any science

based investigation.

--

Nick Holford, L226,Dept of Neurology,OHSU

3181 SW Sam Jackson Park Road,Portland,OR 97201-3098

n.holford.-a-.auckland.ac.nz,(503)494-4778,fax 494-7242

http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm - On 21 Aug 1998 at 14:30:33, Roger Jelliffe (jelliffe.-at-.hsc.usc.edu) sent the message

Back to the Top

Dear Anup:

Thanks for your note. I think we must keep in mind that one first must

define the structural model for the drug. The situation you describe, where

one wishes to extrapolate to infinity, I believe, leaves the "tail" of the

problem undefined. We have avoided this situation whenever possible.

Excluding this situation, we would advocate weighting all observations by

the reciprocal of their variance, and this, using our polynomial to

describe the relationship between the concentration and the SD with which

it is measured, appears to us to be the optimal thing to do.

Now, if you wish to extrapolate out to infinity, what should one do?

First, be aware that we are not making "categorical" statements, as you

suggest. What we are suggesting is that each data point be given weight

according to its credibility. Actually, I think you have said it - that it

is certainly hazardous to make categorical statements about parameter

values, period.

You may, or may not, choose, for various reasons, to include or exclude

various data points. Each choice carries risks. We suggest that it may well

be at least as good to include such points if you correctly weight them by

the carefully determined reciprocal of their variance rather than simply to

exclude them and without them from analysis. The point is that their

credibility can be reasonably determined by a careful examination of the

assay error pattern over its entire working range. Most of the time, people

have simply not done this, It strikes some as a new idea that while the

coefficient of variation becomes infinite as the concentration approaches a

blank, the SD and variance are in fact finite, and the Fisher information

of such a point can, in fact, be explicitly determined.

Sincerely,

Roger Jelliffe

************************************************

Roger W. Jelliffe, M.D.

USC Lab of Applied Pharmacokinetics

CSC 134-B, 2250 Alcazar St, Los Angeles CA 90033

**Note our new area codes below, since 6/15/98!**

Phone **(NOTE NEW AREA CODE AND PREFIX)** (323)442-1300, Fax (323)442-1302

email=jelliffe.aaa.hsc.usc.edu

************************************************

You might also look at our Web page for announcements of

new software and upcoming workshops and events. It is

http://www.usc.edu/hsc/lab_apk/

************************************************

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