- On 3 May 2000 at 22:47:11, Todd Herbst (THerbst.-a-.CortexPharm.com) sent the message

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Hi,

I have two data sets analyzed in WinNonLin as a first order in/first

order elimination with lag(model 4) for animals given a PO dose of

drug. The two data sets appear quite similar in plasma concentration

values at respective time points and in the line fit produced by

Winnonlin. One sample set has reasonable (5-20%) %CV for K01-HL,

K10-HL, K01, K10, volume, etc., while the other set has %CV's that are

very high(>700). Convergence was reached with 150 iterations under a

criteria of 0.0001 by a Gauss-Newton minimization method. How can two

very similar data sets have such a large degree of variability in %CV?

I would very much like to understand this.

Thanks

Todd Herbst - On 9 May 2000 at 23:02:47, David_Bourne (david.aaa.boomer.org) sent the message

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[A few replies - db]

From: harry.mager.hm.-at-.bayer-ag.de

To: " - *PharmPK.-a-.boomer.org"

Subject: Antwort: PharmPK Statistical Problem?

Date: Fri, 5 May 2000 07:43:02 +0200

Hi,

it might be that the estimates are highly correlated, thus causing the

(asymptotic) variance-covariance being nearly ill-conditioned. I experienced

the same problem several times and I got the impression that CV's and variances

based on the Fisher information matrix should be judged with due caution. This

is for sure not a problem with WinNonlin, but with the asymptotic estimation

procedure and the additional approximations for getting the estimate.

Harry Mager

---

Date: Fri, 05 May 2000 09:01:00 -0400

From: "Ed O'Connor"

Reply-To: efoconnor.-a-.snet.net

Organization: PM PHARMA

X-Accept-Language: en

To: PharmPK.aaa.boomer.org

Subject: Re: PharmPK Statistical Problem?

Has the raw data and the data you entered been through QA? Are all decimal

points where the ought to be? Are the data entered with the same number of

significant figures? Are there equal replicates in each data set? Have

the data been screened appropriately? Are there any values above or below

the the dynamic curve of the analytical method? If the data is acceptable

then pre-analytical errors may be suspected-wrong tube used, amount

delivered too variable, timing of dose/sampling too variable, timing of

dose delivery variable (eg too close to eating). Are the sets from the

same sex? same formulation lot? Analyzed at the same time?

The data appear to be quite similar? What does that mean? Same order of

magnitude? WinNonLin (or any other software) shouldn't generate

differences if there are none. Without further info there may have been an

error in dosing, in sampling, in analysis or in data entry.

---

From: "Stephen Duffull"

To:

Subject: RE: PharmPK Statistical Problem?

Date: Wed, 10 May 2000 08:12:06 +1000

X-Priority: 3 (Normal)

Importance: Normal

Todd,

This is a good question. In theory the SEs of parameter

estimates are dependent on the model, model parameter values

and experimental design. A change in any of these factors

will affect the estimation variance-covariance matrix. It

could be that a good experimental design for one subject may

not be good for another if their parameter values

significantly diverge or their data supports a more complex

model thus leading to model misspecification.

It could, however, be a problem with the minimisation

technique that somehow found a local minimum that

represented a flat part of the surface of the objective

function. In this case try using another method for

minimisation, eg Simplex (I think WinNonLin supports various

methods).

Regards

Steve

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

Stephen Duffull

School of Pharmacy

University of Queensland

Brisbane, QLD 4072

Australia

Ph +61 7 3365 8808

Fax +61 7 3365 1688 - On 10 May 2000 at 22:31:19, "Hans Proost" (J.H.Proost.-at-.farm.rug.nl) sent the message

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Dear colleagues,

With respect to the data analysis problem of Todd Herbst:

The high %CV in one data set might be related to the situation that

the absorption rate constant (K01) and the elimination rate

constant (K10) have become almost equal. This is a common

phenomenon in fitting oral data.

As a result, %CV and correlations are high (as mentioned earlier

by Harry Mager), indicating that there is something 'unusual' , or,

possibly, wrong.

For Monte Carlo simulations it can be concluded that this occurs

quite often, even if the true values of K01 and K10 are really

different (e.g. a factor 2). From these Monte Carlo simulations I

learned that this problem has nothing to do with the initial

estimates, 'flip-flop' situation, numerical problems if K01 and K10

approach (in that case there are serious numerical problems, but

these can be solved by appropriate programming techniques), or

weighting.

I don't know the reason why this situation occurs so often!

If anyone has the same experience, and/or has suggestions for the

reason for this apparent weakness of fitting oral data, please let me

know.

Johannes H. Proost

Dept. of Pharmacokinetics and Drug Delivery

University Centre for Pharmacy

Antonius Deusinglaan 1

9713 AV Groningen, The Netherlands

tel. 31-50 363 3292

fax 31-50 363 3247

Email: j.h.proost.at.farm.rug.nl - On 11 May 2000 at 20:52:07, Nick Holford (n.holford.-at-.auckland.ac.nz) sent the message

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I am not sure if the original question was referring to a population PK

analysis or an individual analysis. It is important to distinguish TWO

kinds of correlation between parameters. One of them can be thought of a

biological e.g. clearance and volume tend to get bigger as body size

increases so they are correlated when one looks across a group of

individuals. This kind of correlation can be detected and modelled in a

variety of ways using population analysis. The second kind of

correlation arises from the estimation error and is critically dependent

on the design (sample times), parameterization and model

misspecification. In an individual analysis this is the only kind of

correlation that can be detected.

It seems unlikely that there would be a biological reason to expect the

absorption rate constant (which Hans calls K01) and the elimination rate

constant (K10) to be correlated so an estimation related factor must be

considered. Hans does not explicitly mention that he explored the

sampling times in his simulations. If the sampling times are not well

chosen then there may indeed be a high correlation because the

parameters are poorly a posteriori identifiable.

One way to try and understand more about the source of correlation is to

use a more physiological parameterization ie. use clearance and volume

instead of volume and an elimination rate constant. In this case you may

well find that the correlation is primarily between the absorption rate

constant and volume because information about these 2 parameters is

critically dependent on sampling during the rapid phase of drug

absorption.

--

Nick Holford, Dept Pharmacology & Clinical Pharmacology

University of Auckland, 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 - On 14 May 2000 at 22:30:05, exfamadu.aaa.savba.sk sent the message

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> The high %CV in one data set might be related to the situation that

> the absorption rate constant (K01) and the elimination rate

> constant (K10) have become almost equal. This is a common

> phenomenon in fitting oral data.

In such situations, the model equation

C(t)=A*t*exp(-K*t)

can be used, where

K=K01=K10.

This model equation is the solution of the differential equations

dX1(t)/dt=-K01*X1(t)+D*delta(t) X1(0)=0

dX2(t)/dt=K01*X1(t)-K10*X2(t) X2(0)=0

for K10=K01.

D>0 is the drug dose, delta(t) is the Dirac delta function, the product

D*delta(t) is the drug input, C(t)=X2(t)/V, and A=D*K01/V.

With best regards,

Maria Durisova

Dipl. Engineer Maria Durisova D.Sc.

Senior Research Worker

Scientific Secretary

Institute of Experimental Pharmacology

Slovak Academy of Sciences

SK-842 16 Bratislava

Slovak Republic

http://nic.savba.sk/sav/inst/exfa/advanced.htm - On 25 May 2000 at 20:50:31, "Hans Proost" (J.H.Proost.aaa.farm.rug.nl) sent the message

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Dear Nick Holford,

Thank you for your clever comments to my posting, which in turn was

a comment to a question raised by Todd Herbst.

The original question, and my comments, refer to an individual

analysis. I fully agree with your comments with respect to the two

types of correlations.

To make my point more clear, the following example could be used.

The two data sets were derived by Monte Carlo simulation,

similar to the procedure described by Metzler (J Pharm Sci 1987;76:565-571),

with V = 10 liter, k10 = 0.5 1/h (or clearance 5 liter/h),

k01 = 2 1/h (first-order absorption rate constant), Dose = 1500 mg,

and 15% data noise.

plasmaconc. (mg/l)

time (h) set #1 set #2

0.12 25 27

0.2 63 35

0.28 72 48

0.37 64 60

0.49 86 73

0.57 91 89

0.69 91 75

0.81 115 90

1.02 87 86

1.5 105 89

2 84 78

2.5 60 67

3 50 49

4 29 23

5 15 12

Although set #1 raises somewhat faster than set #2, they do not differ

with respect to the time to peak. Also their AUCs are broadly similar.

Set #1 can be fitted satisfactorily (there are of course two

solutions, ie k01 and k10 can be interchanged, yielding exactly the

same sum-of-squares).

In set #2 the rate constants k10 and k01 become (almost) equal.

Their standard errors are large and their correlation is high. A

parametrization using V, CL and k01 does not change anything,

and the same solution is found. Now, V and k01 are highly

correlated. Irrespective of the parametrization, CL can be obtained

precisely, as expected since the AUC is well-defined. Various

weighting schemes (OLS, 1/C, 1/C^2, log-transformation, ELS)

yield similar results. Also, the time schedule does not seem to be

inappropriate. There is quite some information during the absorption

phase. Finally, changing the initial estimates does not help either.

My particular point is: this situation is not uncommon in data sets

generated by the aforementioned procedure. However, I hardly ever found a

set where k10 and k01 differed by, say, 0.1% to 30%. Either they are

really different, or they become equal.

This makes me suspicious about the procedure, or at least, I would be

interested to know whether or not this phenomenon affects the accuracy

of the parameter estimates in such cases.

I would appreciate any suggestion to this topic.

Hans Proost

Johannes H. Proost

Dept. of Pharmacokinetics and Drug Delivery

University Centre for Pharmacy

Antonius Deusinglaan 1

9713 AV Groningen, The Netherlands

tel. 31-50 363 3292

fax 31-50 363 3247

Email: j.h.proost.aaa.farm.rug.nl

--

From: "Hans Proost"

Organization: Pharmacy Dept Groningen University

To: PharmPK.-a-.boomer.org

Date: Wed, 24 May 2000 10:16:33 MET

Subject: Re: Statistical Problem?

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

X-pmrqc: 1

Priority: normal

Dear Maria Durisova,

Thank you for your comments to my message on equal values of

k10 and k01. I am well aware of the fact that the 'normal' equation

cannot be used if k10 and k01 are equal. 'Your' equations are

implemented in my program.

I have two further comments:

1) In cases where k10 and k01 are close to each other, there are

numerical problems. In that case, 'your' equation does not provide

exact results (the differential equation remain valid, of course).

The 'normal' equation gives a problem in dividing the difference of

two almost similar numbers (exponentials of k10 and k01) by the

difference of two almost similar numbers (k10 and k01).

Even in double precision calculations, the result may be very

inaccurate if k10 and k01 become close.

To overcome this problem, the equations may be replaced by

Taylor approximations, which I have implemented in my programs

in cases where their approximation is more accurate than the

numerically hampered 'exact' calculation. I know that this

procedure may cause problems in fitting due to discontinuity, in

particular in optimization algorithms using derivatives.

However, my 'problem of equal k10 and k01' occurs both without

and with these procedure, both in the simplex (not using

derivatives) and Marquardt algorithm, both in single and double

precision, as well as with Borland Pascal's 'extended' numbers, et

cetera. Therefore I think that there must be a different reason for

the 'problem of equal k10 and k01'.

2) Please see my comments to the message of Nick Holford.

Equal values of k10 and k01 are relatively frequently encountered.

However, in the real world, it is extremely unlikely that k10 and k01

are equal (say, within a relative difference of 0.001).

Therefore I do not understand the relative extensive literature on the

pharmacokinetic analysis for the special case that k10 and k01 are

equal. In my opinion, this situation is mainly an artifact of the fitting

procedures. Therefore I would suggest not to focus on equations for

this special case, but on procedures to avoid, or reduce, the

occurence of equal k10 and k01.

Any suggestion is welcomed.

Best regards,

Hans Proost

Johannes H. Proost

Dept. of Pharmacokinetics and Drug Delivery

University Centre for Pharmacy

Antonius Deusinglaan 1

9713 AV Groningen, The Netherlands

tel. 31-50 363 3292

fax 31-50 363 3247

Email: j.h.proost.-a-.farm.rug.nl - On 29 May 2000 at 13:45:30, David_Bourne (david.at.boomer.org) sent the message

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[Two replies - db]

X-Sender: st005899.-a-.brandywine.otago.ac.nz

Date: Sat, 27 May 2000 09:51:28 +1200

To: PharmPK.at.boomer.org

From: Robert Purves

Subject: Re: PharmPK Re: Statistical Problem?

There's a small paper discussing exactly this phenomenon:

Purves RD (1993) Anomalous parameter estimates in the one-compartment model

with first order absorption. J Pharm Pharmacol 45 934-936.

And a paper on this and related pathologies in the 2 compartment model:

Purves RD (1996) Multiple solutions, illegal parameter values, local minima

of the sum of squares, and anomalous parameter estimates in least-squares

fitting of the two-compartment model with absorption. J Pharmacokin

Biopharm 24 79-101.

Robert Purves

Department of Pharmacology

University of Otago

New Zealand

---

From: exfamadu.at.savba.sk

To: PharmPK.-a-.boomer.org

Date: Mon, 29 May 2000 10:29:35 +0200

Subject: Re: PharmPK Re: Statistical Problem?

X-Confirm-Reading-To: exfamadu.at.savba.savba.sk

X-pmrqc: 1

Priority: normal

To the PharmPK list,

My suggestion to explain the problems Dr. Proost had observed is as follows:

I have analyzed his data using the CXT software (a version of it

is available

form the www page given below). This software is based on modeling

transfer functions in the complex domain. The functions of that type are

called the disposition functions in pharmacokinetic literature.

To model the transfer functions corresponding to set#1 and set#2, I have

used the second-order model in the form of Eq. 1

a0

H(s) = G --------------, (1)

1+b1.s+b2.s^2

where G is the gain (in the given examples it is the

reciprocal value of

clearance), a0, b1, b2 are the model parameters, and s is the

Laplace variable.

For the drug input in the form of the single dose (what is the

case in the

given examples), the time-domain solution of this model is the Bateman

function under the condition that the roots of this model are real

and distinct.

For set#1 I have obtained the following model estimates (+-SD):

G=0.215 h/l

a0=1.000+-0.0184

b1=2.3615+-0.1611 h

b2=1.1482+-0.1006 h^2

The model has two different real roots (-0.5964, -1.4603) and thus its

solution in the time domain has the form of the Bateman function. This time

domain solution is

C(t)=324.93*(exp(-0.5964*t)-exp(-1.4603*t)).

(2)

It yields the value of the Akaike criterion about 110.

For set#2 I have obtained the following parameter estimates:

G=0.189 h/l

a0=1.000+-0.0117

b1=2.2007+-0.1038 h

b2=1.3572+-0.0692 h^2.

The model has one complex conjugated root (-0.8108+-0.2819i), where i

is the imaginary unit, and thus the solution of this model in the time domain

does not have the form of the Bateman function. This solution is

C(t)= 742.07*exp(-0.8108*t)*sin(0.2819*t).

(3)

It yields the value of the Akaike criterion about 90.

If I neglect the imaginary part of the model root (in spite that it

is not small in

comparison with the real part), I would obtain the time-domain solution

C(t)=209.21*t*exp(-0.8108*t).

(4)

This solution has the same form as that I mentioned in my previous reply. It

yields the value of the Akaike criterion about 100 and it does not approximate

correctly the terminal phase.

The difference in the models of set#1 and set#2 may be the consequence of

the use the Monte Carlo method with rather low number of data and rather high

data noise, what significantly modified the original profile.

Closing this reply, I would like to add that the model transfer

function given by

Eq.1 is a simple form of the general m-order model transfer function given by

Eq. 5

a0+a1.s+...+an.s^n

H(s)= G ----------------------------.

(5)

1+b1.s+b2.s^2+...+bm.s^m

All the compartment models and many other models (including models with

time delays, shunts, etc.) can be rewritten in the form of the model given by

Eq.5. This model can be used to model the drug behavior, drug effect,

etc., as it

is mentioned at our www page. For example, this model allows to determine

MRT and VRT using the simple formulae:

a1

MRT= b1 - -----

a0

a2 a1^2

VRT= b1^2 -2.b2 + 2------ - -------- .

a0 a0^2

Since the model transfer functions corresponding to set#1 and set#2 have:

a1=a2=0

here

MRT=b1

VRT= b1^2 -2.b2.

With best regards,

Maria Durisova

Dipl. Engineer Maria Durisova D.Sc.

Senior Research Worker

Scientific Secretary

Institute of Experimental Pharmacology

Slovak Academy of Sciences

SK-842 16 Bratislava

Slovak Republic

http://nic.savba.sk/sav/inst/exfa/advanced.htm

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