- On 10 Dec 2000 at 21:36:21, "Henrik Kjer Petersen \(gmx\)" (henrik.kjer.-at-.gmx.net) sent the message

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Hi,

Could anybody please help me with the following question:

How many blood samples is needed in general to establish a

two-compartment model that describes the pharmacokinetics of a drug

(here: gentamicin) sufficiently? Is there a general rule, by which one

can calculate the number of demanded blood samples to establish a one-,

two, three- etc compartment model?

Any comment would be of great help.

Kirneh - On 11 Dec 2000 at 12:37:58, David_Bourne (david.at.boomer.org) sent the message

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[A few replies - db]

Sender: PharmPK.aaa.boomer.org

Reply-To: "Stephen Duffull"

MIME-Version: 1.0

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From: "Stephen Duffull"

Date: Mon, 11 Dec 2000 15:03:53 +1000

To: david.-at-.boomer.org

Subject: RE: PharmPK Number of samples for two compartment model

The following message was posted to: PharmPK

Kirneh

> (here: gentamicin) sufficiently? Is there a general rule, by which one

> can calculate the number of demanded blood samples to establish a one-,

> two, three- etc compartment model?

The answer to your question depends on whether you are undertaking an

individual or population study.

For the study of an individual the most parsimonious sampling strategy would

be one where you had 1 blood sample per parameter to estimate. In theory

using an optimal design technique this should be sufficient to fully

characterise the model - although this does depend on the accuracy of the

prior estimates of the parameters. These optimal design strategies are not

usually robust to misspecification of the model.

For the study of a population then the most parsimonious sampling strategy

where estimates of the fixed and random effects parameters are required

cannot be generalised easily. Many investigators rely on simulations and

more recently theoretic techniques for computing the "best" design. (see

http://www.uq.edu.au/pharmacy/pfim.htm for information on a theoretic

technique).

For so-called "Bayesian forecasting" using the MAP objective function it may

be possible to get reasonable parameter estimates when less blood samples

are taken than there are parameters to estimate. In these circumstances

however the values of the parameter estimates are greatly influenced by the

prior - and misspecification of the prior can result in biased estimates of

the parameters.

I hope this helps.

Regards

Steve

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

Stephen Duffull

School of Pharmacy

University of Queensland

Brisbane, QLD 4072

Australia

Ph +61 7 3365 8808

Fax +61 7 3365 1688

---

Sender: PharmPK.-at-.boomer.org

Reply-To: Nick Holford

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From: Nick Holford

Date: Tue, 12 Dec 2000 02:01:57 +1300

To: david.aaa.boomer.org

Subject: Re: PharmPK Number of samples for two compartment model

The following message was posted to: PharmPK

"Henrik Kjer Petersen (gmx) (by way of David_Bourne)" wrote:

>

> How many blood samples is needed in general to establish a

> two-compartment model that describes the pharmacokinetics of a drug

> (here: gentamicin) sufficiently? Is there a general rule, by which one

> can calculate the number of demanded blood samples to establish a one-,

> two, three- etc compartment model?

The general rule for the *minimum* number of samples is one per

parameter in your model.

eg. 1 cpt bolus input plus additive error = CL + V + SD = 3

2 cpt first order input plus additive error = Ka + CL + V1 + CLic

+ V2 + SD = 6

n cpt first order input + lag plus additive + proportional = Ka

+Lag + 2 * (CLn + Vn) + SD + CV = 4+2*n

But the more you can get the better your modelling will be!

Nick

--

Nick Holford, Divn Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand

email:n.holford.aaa.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556

http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm

---

Sender: PharmPK.at.boomer.org

Reply-To: "Robert Hunter"

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From: "Robert Hunter"

Date: Mon, 11 Dec 2000 08:16:27 -0600

To: david.-a-.boomer.org

Subject: Re: PharmPK Number of samples for two compartment model

The following message was posted to: PharmPK

Kirneh,

I have found that when using winnonlin, a minimum of 6 time points

per compartment are needed.

Rob Hunter, MS, PhD

Assistant Professor of Veterinary Pharmacology

Department of Anatomy & Physiology

1600 Denison Ave., 129 Coles Hall

College of Veterinary Medicine

Kansas State University

785-532-4524 (office)

785-532-4516 (lab)

785-532-4557 (fax)

rhunter.-at-.vet.ksu.edu

www.vet.ksu.edu/depts/ap/faculty/hunter.htm

---

Sender: PharmPK.at.boomer.org

Reply-To: Angusmdmclean.-at-.aol.com

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From: Angusmdmclean.aaa.aol.com

Date: Mon, 11 Dec 2000 09:43:14 EST

To: david.-at-.boomer.org

Subject: Re: PharmPK Number of samples for two compartment model

The following message was posted to: PharmPK

fit the experimental data you have to a 2 compartment model. Use the model

-predicted concentrations as a guide for the shape of the profile. Then

design timepoints you need to best characterize the shape of the 2

compartment model- calculated plasma concentrations. The timepoints selected

must characterize the shape you are trying to define.

same applies for all models.

---

Sender: PharmPK.-a-.boomer.org

Reply-To: "Dr. Sander Vinks"

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From: "Dr. Sander Vinks"

Date: Mon, 11 Dec 2000 10:31:16 -0500

To: david.at.boomer.org

Subject: Re: PharmPK Number of samples for two compartment model

The following message was posted to: PharmPK

Dear Kirneh,

An attractive approach is to employ optimal sampling strategy

to determine

information-rich sampling times for obtaining data. Given the intended

dosage regimen and assuming that structural pharmacokinetic and

observational model are known, along with reasonable estimates of the

expected pharmacokinetic parameters and observation error, D-optimal times

can be determined to maximize the precision of pharmacokinetic parameter

value estimation for an individual subject . Typically for a 2-compartment

model one would need 4 samples; one time point for every parameter in the

model. For this the sample module in the ADAPT software can be used:

See also http://www.usc.edu/dept/biomed/BMSR/Software/1999.html:

D'Argenio DZ. Optimal sampling times for pharmacokinetic experiments. J

Pharmacokinet Biopharm 1981;9:739-56.

and examples for 2-compt PK:

Drusano GL, Forrest A, Snyder MJ, Reed MD, Blumer JL. An evaluation of

optimal sampling strategy and adaptive study design. Clin Pharmacol Ther

1988;44:232-8.

Forrest A, Ballow CH, Nix DE, Birmingham MC, Schentag JJ. Development of a

population pharmacokinetic model and optimal sampling strategies for

intravenous ciprofloxacin. Antimicrob Agents Chemother 1993;37:1065-72.

Vinks AATMM, Mouton JW, Touw DJ, Heijerman HGM, Danhof M, Bakker W.

Population pharmacokinetics of ceftazidime in cystic fibrosis patients

using a nonparametric algorithm and optimal sampling strategy. Antimicrob

Agents Chemother 1996;40:1091-7.

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

Alexander A. Vinks, PharmD, PhD,

Research Professor of Pediatrics

Pharmacology Research Center & Clinical Trials Office

Children's Hospital Medical Center

3333 Burnet Avenue

Cincinnati, Ohio 45229-3039

Tel: (513) 636-0159; (513) 636-0160 (secretary).

Fax: (513) 636-0168.

CHMCC email: sander.vinks.aaa.chmcc.org

**************************************************************************** - On 12 Dec 2000 at 13:47:22, "Dr. Sander Vinks" (sander.vinks.aaa.chmcc.org) sent the message

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Dear Kirneh,

An attractive approach is to employ optimal sampling strategy

to determine

information-rich sampling times for obtaining data. Given the intended

dosage regimen and assuming that structural pharmacokinetic and

observational model are known, along with reasonable estimates of the

expected pharmacokinetic parameters and observation error, D-optimal times

can be determined to maximize the precision of pharmacokinetic parameter

value estimation for an individual subject . Typically for a 2-compartment

model one would need 4 samples; one time point for every parameter in the

model. For this the sample module in the ADAPT software can be used:

See also http://www.usc.edu/dept/biomed/BMSR/Software/1999.html:

D'Argenio DZ. Optimal sampling times for pharmacokinetic experiments. J

Pharmacokinet Biopharm 1981;9:739-56.

and examples for 2-compt PK:

Drusano GL, Forrest A, Snyder MJ, Reed MD, Blumer JL. An evaluation of

optimal sampling strategy and adaptive study design. Clin Pharmacol Ther

1988;44:232-8.

Forrest A, Ballow CH, Nix DE, Birmingham MC, Schentag JJ. Development of a

population pharmacokinetic model and optimal sampling strategies for

intravenous ciprofloxacin. Antimicrob Agents Chemother 1993;37:1065-72.

Vinks AATMM, Mouton JW, Touw DJ, Heijerman HGM, Danhof M, Bakker W.

Population pharmacokinetics of ceftazidime in cystic fibrosis patients

using a nonparametric algorithm and optimal sampling strategy. Antimicrob

Agents Chemother 1996;40:1091-7.

******

Alexander A. Vinks, PharmD, PhD,

Research Professor of Pediatrics

Pharmacology Research Center & Clinical Trials Office

Children's Hospital Medical Center

3333 Burnet Avenue

Cincinnati, Ohio 45229-3039

Tel: (513) 636-0159; (513) 636-0160 (secretary).

Fax: (513) 636-0168.

CHMCC email: sander.vinks.-a-.chmcc.org

****** - On 12 Dec 2000 at 20:20:16, GLDrusano.at.aol.com sent the message

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Hi!

There is a lot written on this subject.

The literature notes that if one assumes one "true" mean parameter vector,

that there will be EXACTLY four optimal points to sample under a D optimality

criterion (the factor being minimized or maximized is a scalar and is related

to the determinant of the Fisher Information Matrix). This has the particular

property of replication. That is, if one asks for the 5th through 8th points,

it will say to replicate the first four points, as this will be the minimum

variance solution. David D'Argenio and JJ Di Stefano have written extensively

on this topic.

HOWEVER, this approach misses the fact that there IS NO ONE "TRUE" MEAN

PARAMETER VECTOR. Patients have true between-patient variability. To approach

this in a stochastic framework, D'Argenio has developed a new optimality

criterion that is related to the log of the D-optimal criterion and has an

expectation taken over a population to make it explicitly stochastic. This

does not have the property of replication and, when asked for the 5th point,

provides a time that is different from the first four and provides more

information about patients removed from the mean parameter values.

Tod and Rocchisani have also written a few papers on this topic.

Hope this helps,

George Drusano --part1_55.e93fef4.2768094c_boundary-- - On 15 Dec 2000 at 13:18:52, Thorsten Mueller (muellert.-a-.physik.uni-freiburg.de) sent the message

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The following message was posted to: PharmPK

Dear Sir,

I was quite puzzled by the answers to the question how

many samples are 'needed in general to establish a

two-compartment model that describes the pharmacokinetics of a drug

(here: gentamicin) sufficiently?'

In my view, the mathematical questions that arise are not

yet precise; i.e. what do we want?

1) Model selection in order to show that a two compartent

model is sufficient (best) to describe the data.

2) Parameter estimation with help of data in order to

determine a quantitative description for the dynamic

behaviour of the drug.

3) Experimental design in order to optimize data

acquisition with respect to number of samples,.....

Not for all of these problems, in my view, there is a

best answer.

In case 1, one could rely on one of the

many model selection approaches (AIC, BIC, LR-tests, ....).

Unfortunately there are still many open questions and I do

not agree with Nick Holford that 'the more you can get

the better your modelling will be!' since the complexity of

the model will certainly depend of the number of data

points.

In case 2, assuming that the two compartment model is

sufficient, there should be no problems. Just write down

the error model for the measurement error, determine which

level of accuracy is needed/desired for the estimated

parameters and calculate the number of data points needed to

establish this level (e.g. for equally spaced sampling

times).

Case 3 was already commented and many experimental design

techniques are readily available.

I hope that my comments are of any value and I would

appreciate any reply.

Sincerely yours,

Thorsten M=FCller

--

Thorsten Mueller FDM, Freiburg Centre for

muellert.aaa.fdm.uni-freiburg.de Data Analysis and Modelling

Tel. ++49 761 / 203 5764 Eckerstr.1 79104 Freiburg (Germany)

=46ax. ++49 761 / 203 7700 http://www.fdm.uni-freiburg.de - On 15 Dec 2000 at 21:43:44, ml11439.at.goodnet.com sent the message

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The following message was posted to: PharmPK

Kirneh,

Although there is literature regarding the theoretical

aspects of how many levels to obtain to adeqautely determine

the parameters of a two compartment model, practical considerations

would demand at least two levels for each phase of the plasma

concentration curve. Two levels in the alpha phase, and two levels

in the elimination phase, would allow some estimation of alpha

and beta macroconstants. The goodness of fit the of the two

compartment model could be assessed by such tests as the Akaike

Information Criterion and F-test.

Mike Leibold, PharmD, RPH

ML11439.-a-.goodnet.com - On 15 Dec 2000 at 21:44:27, Nick Holford (n.holford.-a-.auckland.ac.nz) sent the message

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The following message was posted to: PharmPK

"Thorsten Mueller (by way of David_Bourne)" wrote:

Thorsten,

> In my view, the mathematical questions that arise are not

> yet precise; i.e. what do we want?

I quite agree that the design of an experiment intended to build a

model and estimate its parameters is dependent on the question being

asked.

> Unfortunately there are still many open questions and I do

> not agree with Nick Holford that 'the more you can get

> the better your modelling will be!' since the complexity of

> the model will certainly depend of the number of data

> points.

I cannot understand why you disagree with my general statement. The

more data one has then generally speaking the more one will learn

about the reality underlying the observations. In that sense the more

data points (with suitable design) then the more complex the model

may become but in practice there is some user specified perspective

of the use of the model that says the model is good enough and the

focus will turn perhaps to the precision of the estimates. In this

case the more data points (with suitable design) the better the

precision.

Can you please explain more clearly why you disagree with my assertion?

> In case 2, assuming that the two compartment model is

> sufficient, there should be no problems. Just write down

> the error model for the measurement error, determine which

> level of accuracy is needed/desired for the estimated

> parameters and calculate the number of data points needed to

> establish this level (e.g. for equally spaced sampling

> times).

You do not say how you would do the calculation of the number of

points nor do you mention how you would decide on the spacing (design

times). There is a strong interaction between number and timing of

points and there is no simple calculation that I know of that can

solve this problem. A solution can be obtained depending on what you

want to know. D-optimality methods will attempt to improve precision

of parameter estimates but pay no attention to accuracy as far as I

understand the methods. A better optimization metric may be one that

include both imprecision and bias (aka precision and accuracy) e.g.

based on the root mean square error of the parameter estimate

compared with its true value.

--

Nick Holford, Divn Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand

email:n.holford.at.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556

http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm - On 16 Dec 2000 at 23:22:58, "Tata, Prasad N" (Prasad.Tata.at.MKG.COM) sent the message

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The following message was posted to: PharmPK

I guess it is the practicality that dictates number of samples in a clinical

study. While there are no hard and fixed rules that guides how many samples

to be drawn in a clinical study following guidelines should give you some

clue.

1. At least 2 plasma draws before the Tmax for the purpose of

calculating the early exposure (which is a recommendation in the recently

issued BA/BE guidance).

2. We need at least three plasma draws for calculating terminal phase.

3. Same criteria as item # 2 if you want to calculate distribution

half-life.

4. While items # 1 and 2 are a must my personal preference is to have

two/three blood draws before and after the Tmax.

5. Maximum blood one can draw from a subject is pretty much limited to

a pint in a study of one month duration.

6. Finally in the current scenario where majority of the studies are

conducted in CROs. Convenience of the CROs and subject of the volunteer

dictating the blood draw times and number of draws.

These are my thoughts may be I am not be totally accurate.

Prasad Tata - On 22 Dec 2000 at 11:40:39, Thorsten Mueller (muellert.-a-.physik.uni-freiburg.de) sent the message

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The following message was posted to: PharmPK

Dear Nick Holford,

thank you for your reply, I hope I am able to clarify my

point:

> > Unfortunately [...] I do

> > not agree with Nick Holford that 'the more you can get

> > the better your modelling will be!' since the complexity of

> > the model will certainly depend of the number of data

> > points.

> I cannot understand why you disagree with my general statement. The

> more data one has then generally speaking the more one will learn

> about the reality underlying the observations. In that sense the more

> data points (with suitable design) then the more complex the model

> may become but in practice there is some user specified perspective

> of the use of the model that says the model is good enough and the

> focus will turn perhaps to the precision of the estimates. In this

> case the more data points (with suitable design) the better the

> precision.

In my view, biological and medical dynamical systems

possess an infinite number of degrees of freedom.

Although, for the researcher, more data points

may mean more insight into the experiment and the

underlying dynamics, current model selection procedures will

select the most complex model which is not always the one

desired. Instead the researcher will use "some user

specified perspective of the use of the model that says

the model is good enough" which, in my view, is a rather

subjective method.

Although I am no expert in model selection procedures, in

my view the current alternatives are rather unsatisfactory,

especially in bio/med dynamical systems.

I would be happy to receive any comment on this topic.

> You do not say how you would do the calculation of the number of

> points nor do you mention how you would decide on the spacing (design

> times). There is a strong interaction between number and

> timing of

> points and there is no simple calculation

> that I know of that can

> solve this problem.

This is a question for an expert for experimental

design, but I would use D-optimality methods to compute the

time points and then do a Monte Carlo simulation to

approximate the confidence intervals of the estimates. For a

specified desired level of precision the number of time

points can then be calculated.

Merry Christmas and a happy new year 2001!

Best wishes,

Thorsten M=FCller

--

Thorsten Mueller FDM, Freiburg Centre for

muellert.at.fdm.uni-freiburg.de Data Analysis and Modelling

Tel. ++49 761 / 203 5764 Eckerstr.1 79104 Freiburg (Germany)

=46ax. ++49 761 / 203 7700 http://www.fdm.uni-freiburg.de

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