- On 6 Mar 1998 at 10:05:05, Hans Proost (J.H.Proost.-at-.farm.rug.nl) sent the message

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To the PharmPK group:

During the last week several messages have been send recommending the

book 'Pharmacokinetic and Pharmacodynamic Data Analysis' by

Gabrielsson and Weiner, 2nd Ed. (1997).

About one year ago, I wrote to the PharmPK group that I was deeply

disappointed by the FIRST EDITION of that book (from 1994). I wrote

that the book contained errors on almost every page.

Dr. Gabrielsson wrote to me: 'Some people (....) interpreted your

letter (my E-mail message to the PharmPK group, JHP) as if there were

errors in the program, which there is not of course. PCNonlin and

WinNonlin are probably the most well validated software packages in

the area'.

However, the FIRST EDITION of the book (G&W-1) contained, apart from

a large number of typing errors, several erroneous results of

fitting procedures produced by PCNonlin.

My question is now: WHERE THESE ERRORS DUE TO THE PROGRAM PCNONLIN,

or where they due to inaccuracies by the authors?

To be concrete, I have selected 6 examples of erroneous results

presented in the first edition of the book G&W-1. For comparison,

the results of my program MultiFit are given in the right column:

(a) example PK 5, solution I (2-comp. model, constant variance)

G&W-1: MultiFit:

(Weighted) SS 0.130905 0.006486076

A 3.23130 1.057773

Alpha 0.376549 0.047824

B 1.22815 0.783247

Beta 0.006285 0.003297

(b) example PK 5, solution III (3-comp. model, constant variance)

G&W-1: MultiFit:

(Weighted) SS 0.00521384 0.003679095

A 0.887297 0.664345

Alpha 0.057217 0.087715

B 0.737811 0.626455

Beta 0.003081 0.021741

C 0.242603 0.638896

Gamma 0.021528 0.002530

(note: the G&W-1 solution presents values with gamma > beta, which is

against the usual convention)

(c) example PK 5, solution IV (3-comp. model, constant CV)

G&W-1: MultiFit:

Weighted SS 0.0135446 0.01094993

A 0.916094 0.589508

Alpha 0.054407 0.099807

B 0.493672 0.688816

Beta 0.009045 0.025186

C 0.432651 0.666444

Gamma 0.001632 0.002669

(note: in solution II, the difference between G&W-1 and MultiFit are

negligible)

(d) example PK 16 (multi-compartment modelling of single oral dose

data)

For solutions II and III, the printed values for standardized

residuals are clearly erroneous.

(e) examples PK 22 and PK 23 (Bayesian models for digoxin and

theophylline).

There are several errors in the equations (missing squares in 22:1,

22:5, 23:1, 23:5), the 95% confidence intervals do not match the

listed standard errors, and the results cannot be reproduced by other

clinical pharmacokinetic software (USC*PACK, MW\Pharm).

Finally, although I consider myself as reasonably well-known with

Bayesian fitting in clinical pharmacokinetics, I did not understand

the text of these chapters. I am afraid the text is not correct.

(f) example PD 11 (effect link-model II)

In the model definition file 11 the equation for the concentration in

the effect compartment is incorrect: the parameter ke0 in the

denominator is missing. As a result, the calculated value for EC50 is

incorrect. In solution II (sigmoid Emax model), EC50 should be 5.417,

not 4.233 mg/l.

My questions to the PharmPK group:

1. Are my comments correct (so, G&W-1 are wrong), or not?

2. If so, were the mistakes in G&W-1 due to errors of the authors or

to errors in PCNonlin?

3. does WinNonlin provides the correct results?

Please note that it is not my intention to set up a campaign against

the book, PCNonlin or WinNonlin. The purpose of my letter is to

notice flaws, simply from the scientific point of view that errors in

book and programs should be corrected.

I really would be glad if PCNonlin and WinNonlin would survive against

the aforementioned comments. You will understand that I have my

doubts, but I really hope that there is no reason for any doubt on

PCNonlin and WinNonlin.

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 - On 13 Mar 1998 at 09:51:42, "David Nix" (nix.-a-.Pharmacy.Arizona.EDU) sent the message

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In 1996, I attempted to use PCNONLIN for determining the elimination

rate constant of some oral PK data. The absorption phase was not

well characterized and modeling the entire profile was not

optimal. I was using weighted nonlinear regression in place

of linear regression following log transformation of concentrations

since peer review critisized the latter due to inadequate handling of

assay error. I fit the data which involved concentration

measurements over the period of 6-12 hours to a single exponentional

model and the rate constant was the only relevant parameter. I found

that variability was much greater for the PCNONLIN fit as

compared to the linear regression.

As followup, I used simulated data with random error for

concentrations over the same time period. PCNONLIN exhibited poor

precison in estimating the rate constant. Adapt II resulted in the

best accuracy and precison in estimating the rate constant. Linear

regression of log transformed values was only slightly less precise

than Adapt II. I then looked more closely at PCNONLIN. The default

algorhythm - (Method 2) I believe is a Gauss Newton method. I change

the Method to Method 1 (Nelder-Mead) and the results were very

similar to adapt. This finding warrants caution when using the

default method in PCNONLIN especially when the data is rather

limited. I am not sure if the same default is used for WIN NONLIN.

David Nix - On 13 Mar 1998 at 09:54:55, "Piotrovskij, Vladimir" (vpiotrov.at.janbe.jnj.com) sent the message

Back to the Top

Dear group members,

I checked a couple of examples where there are most significant

differences between PCnonlin results (as reported in Gabrielsson &

Weiner, 1st Edition, 1994) and MultiFit results (as reported in Dr.

Proost message, see balow). I used the well-validated `nls' function

of S-PLUS (version 4) package. Below the corresponding outputs of nls

summaries are shown:

(a) example PK 5, solution I (2-comp. model, constant variance)

Residual sum of squares : 0.00648608

Formula: y ~ A * exp( - alpha * x) + B * exp( - beta * x)

Parameters:

Value Std. Error t value

A 1.05777000 0.044946100 23.5343

alpha 0.04782730 0.004677480 10.2250

B 0.78327000 0.042544500 18.4106

beta 0.00329755 0.000305863 10.7812

Residual standard error: 0.0254678 on 10 degrees of freedom

Correlation of Parameter Estimates:

A alpha B

alpha -0.0555

B -0.5560 0.8140

beta -0.5530 0.6820 0.9100

(b) example PK 5, solution III (3-comp. model, constant variance)

Residual sum of squares : 0.0036791

Formula: y ~ A * exp( - alpha * x) + B * exp( - beta * x) + C * exp( -

gamma * x)

Parameters:

Value Std. Error t value

A 0.66443300 0.328345000 2.02358

alpha 0.08770400 0.044161700 1.98598

B 0.62639200 0.268073000 2.33664

beta 0.02173690 0.014778000 1.47090

C 0.63885900 0.146206000 4.36960

gamma 0.00252948 0.000766786 3.29881

Residual standard error: 0.021445 on 8 degrees of freedom

Correlation of Parameter Estimates:

A alpha B beta C

alpha -0.900

B -0.916 0.954

beta -0.965 0.868 0.827

C -0.828 0.694 0.582 0.931

gamma -0.766 0.634 0.504 0.878 0.983

As one can see, nls provides the same results as MultiFit.

BTW, with the latter sample (3-comp model) nls failed to converge

with the starting parameter values proposed in the book:

A=1.0, alpha=1.0, B=.7, beta=.1, C=.2, gamma=.01

The following initial values were used instead:

A=1.0, alpha=.5, B=.5, beta=.1, C=.2, gamma=.01

Apparently, those erroneous results presented in the book by

Gabrielsson & Weiner correspond to local minima of the objective

function, and would not be reproduced after running PCNonlin with

another sets of initial parameter values.

It worth to note that fitting the same models but with constant CV

residual model (using nls) gives results identical to those reported

in the above book.

Vladimir

--------

Vladimir Piotrovsky, Ph.D. Fax: +32-14-605834

Janssen Research Foundation Email: vpiotrov.aaa.janbe.jnj.com

Clinical Pharmacokinetics vpiotrov.aaa.janbelc1.ssw.jnj.com

B-2340 Beerse

Belgium - On 13 Mar 1998 at 09:58:14, "Cook, Jack" (COOKJ.-a-.wolf.research.aa.wl.com) sent the message

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Colleagues,

Recently Dr. Proost asked a series of questions regarding the 1st

edition of the book 'Pharmacokinetic and Pharmacodynamic Data Analysis'

by Gabrielsson and Weiner (referred to as G&W-1). Specifically, the

questions were:

1. Are my comments correct (so, G&W-1 are wrong), or not?

2. If so, were the mistakes in G&W-1 due to errors of the authors or

to errors in PCNonlin?

3. does WinNonlin provides the correct results?

I have chosen not to address #2 as PCNonlin is neither sold nor

supported at this time. In order to address #1 and 3 above, I obtained

the first edition of the text and repeated Dr. Proost's problems (a)-(c)

using WinNonlin Professional version 1.5. Results were essentially

identical to those provided by Dr. Proost's. Further, these items are

corrected in the second edition of Dr. Gabrielsson and Weiner's book

(Dr. Proost's problems (a) and (b) on pages 394-395 of the second

edition. Note I could not find problem (c) in the new book.)

Similarly for item (d), the standardized residuals in the 2nd edition

were in agreement with what I obtained using WinNonlin Professional, did

not appear erroneous and were different from the first edition. (Output

appears on pages 430-432 in the new text). Additionally equations

referred to in problem (3) (22:1, 22:5, 23:1 and 23:5) have been

corrected in the 2nd edition to include the "squares" (see equations

35:1 & 35:2).

The one possible uncorrected error that I found concerns item (f) and

the equations used to describe the effect compartment in the effect-link

model. Dr. Proost is correct in that keO is missing from the

coefficients of the exponential terms describing the effect compartment

model. However, I suspect that they are missing from the NUMERATORS

rather than the denominators as previously described (see Holford NHG,

Sheiner LB. Understanding the dose-effect relationship: Clinical

applications of pharmacokinetic-pharmacodynamic models. Clinical

Pharmacokinetics 6:429-453(1981)).

Thus I conclude that his comments were correct in that the results as

printed in the first edition are in error. Additionally while this does

not constitute a full validation of WinNonlin Professional's modeling

capabilities, these results and my previous experience lead me to

believe that WinNonlin "provides correct results".

Finally, even with the aforementioned errors, I found the first edition

helpful. I think the authors have been very responsive to errors found

in the first edition and have improved the book with additional topics

of interest. I find that the second addition a very good reference. I

will continue to recommend both the 2nd edition and WinNonlin

Professional.

Jack Cook

Parke-Davis

cookj.at.aa.wl.com - On 16 Mar 1998 at 13:47:29, David_Bourne (david.aaa.pharm.cpb.uokhsc.edu) sent the message

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[A few replies - db]

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

Date: Fri, 13 Mar 1998 16:41:37 -0800

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

From: Roger Jelliffe

Subject: Re: PharmPK Re: PCNonlin, WinNonLin, PK/PD book

Mime-Version: 1.0

Dear group members:

In the discussion about fitting procedures and results, as in the

recent

example of WinNonlin, one might suggest that fitting is not simply an art

in which one examines various possible weighting schemes, for example, to

see what one gives the best fit to the data. Instead of using common

weighting schemes such as these, one might consider that the actual error

pattern of the assay used to measure the response is easily determined, and

thus provides a good index of the relative credibility of each data point

being fitted. Instead of assuming a constant assay error or one of constant

CV, for example, there are many other options. Whatever weighting scheme is

used will significantly affect the resulting parameter estimates obtained.

In the examples given, if one assumes constant weighting and gives each

data point a weight of 1.0, when one looks at the objective function for

the maximum aposteriori probability (MAP) Bayesian objective function or

its equivalent for weighted nonlinear least squares, one sees that each

observation is instead actually weighted by the reciprocal of its variance.

Because of this, if one uses unity weighting, in which each data point is

given a weight of 1.0, this is equivalent to assuming that the assay

standard deviation (SD) is 1.0 assay units, whatever they are, for all

concentrations. One can also assume other constant assay SD's such as 0.1,

0.01, etc. They will often give different results in the parameter values

found, especially in the estimates of the precision of the parameter

values. This is easily seen in the NPEM population modeling software, where

one gets quite different results for the joint population parameter density

depending of the weighting scheme used for the assay data, and the 3D plots

of parameter joint density pairs are easily seen.

Similarly, if one assumes a constant assay CV, one must face the

fact that

this is equivalent to assuming that a concentration of 0.1 units has 100

times the weight of a level of 1.0, and 10,000 times the weight of a

concentration of 10.0. Is this realistic? I doubt it.

Instead, one might consider being as realistic as possible, and

weighting

each data point by its Fisher information, which is just what is

incorporated in the MAP Bayesian objective function, which weights each

point by the reciprocal of its variance. If the SD (or a reasonable

estimate of it) can be found for each data point, each point will then be

weighted optimally, and further attempts with other weighting schemes may

not be useful. This can be done by determining the SD of the assay with

quadruplicate measurements, for example, of a blank, a low level, a middle

level, a high level, and a very high level, and then fitting this data

relating concentration and SD with a polynomial of up to order 2 or 3 to

estimate the SD of each measured data point from the concentration. This is

easily done, and costs little.

This has been discussed in more detail in Therapeutic Drug

Monitoring, 15:

380-393, 1993. A number of other things come out of this, such as the fact

that for TDM, where we know when the last dose was given, there actually is

no lower detectable limit of an assay, in contrast to toxicology, where the

assay result is the sole source of information about the presence or

absence of the drug, and the time since the last dose is not kown.

It is true that as the concentrations approach zero, the assay CV

approaches infinity. However, the SD and variance are always quite finite,

and can provide useful and appropriate weighting of the data all the way

down to and including the blank.

Other sources of environmental noise, such as that produced by

errors in

preparation and administration of the doses, and model misspecification,

are more correctly part of the noise in the behavior of the system, and

constitute process noise which should optimally go in the differential

equations. If one lumps these together as intra-individual variability, it

still probably belongs in the differential equations rather than as

measurement noise. Whatever it is, it can also be used as a scaling factor

or multiplier (gamma, in our USC*PACK iterative Bayesian population

modeling software), for the assay error polynomial. This can also be used,

is one wishes, as a possible relative index of the quality of care, the

precision or amount of noise in the therapeutic environment for each

patient, or each population.

We tend to think that it is useful and appropriate to determine the

assay

error explicitly for each assay, and to describe it as a polynomial as done

in the TDM article. This is simply because it is easy to do and it actually

and specifically determines the precision of the assay itself. Thus this

source of noise in the therapeutic environnment can be spedifically

quantified. If one then wants to quantify the other sources of noise as

gamma, or as intra-individual variability, then it seems appropriate to do

this in a form which scales the assay error polynomial. The assay error is

easily found. The other sources can then also be found as a scaling factor

for the assay noise, or as some other term outside the assay error

polynomial itself.

We have favored this approach as it is easily and cheaply done, it

appears

to provide optimal weighting of the data, and also because it removes the

problem, largely a cultural one derived from toxicology, of the lower limit

of detection.

Sincerely,

Roger Jelliffe

---

From: "Bill "

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

Date: Sat, 14 Mar 1998 22:07:57 -0500

Reply-To: "Bill "

Priority: Normal

MIME-Version: 1.0

Subject: Re: PharmPK Re: PCNonlin, WinNonLin, PK/PD book

Yes, I just found out the same thing using WinNonlin. The default algorithm

is Gauss-Newton, however, Nelder-Mead is available, as in PCNonlin.

---

From: Hans Proost

Organization: Pharmacy Dept Groningen University

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

Date: Mon, 16 Mar 1998 10:12:02 CET

Subject: Re: PharmPK Re: PCNonlin, WinNonLin, PK/PD book

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

X-pmrqc: 1

Priority: normal

Dear colleagues,

Several of you responded to my message to the PharmPk group regarding

PCNonlin, WinNonlin, and the PK/PD book by Gabrielsson and Weiner.

I would like to thank these colleagues, and to make the following

comments:

To David Nix:

The main cause of the differences seems to be the inability of the

Gauss-Newton method to find the global minimum. This may be due to

the method itself, or to the implementation in PCNonlin; in my

experience, small differences in the implementation may have

important consequences with respect to robustness.

Anyhow, it seems incredible that the makers of PCNonlin used such a

method as the default.

Note: Did anyone try these examples in WinNonlin?

To Vladimir Piotrovskij:

The data on standard error and correlation matrix of examples (a) and

(b) are practically identical to that obtained with MultiFit.

You wrote that nls gave the same results as G&W-1, and thus, results

different from my outcomes. Since the latter gave a lower weighted SS,

they must be 'better'. Did you try to test nls with my results as

starting parameters?

To Jack Cook:

You are fully correct with respect to item (f): ke0 is missing from

the NUMERATORS, not from the denominators. This was indeed meant, as

can be concluded from my numerical example.

My question #2 seems to be less relevant since PCNonlin is neither

sold nor supported at this time. However, the claimed validation of

PCNonlin seem to be highly questionable, and so, results obtained

with earlier versions of PCNonlin are questionable.

Problem (c) does not seem to be in the new version of the book.

However, the question is still: Do PCNonlin and WinNonlin provide the

correct answer (see also comment to Dr. Piotrovskij)?

I hope that the answers to these questions could give a clear light

on the discussion.

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

---

From: "Thomas Senderovitz"

To:

Subject: Sv: PharmPK Re: PCNonlin, WinNonLin, PK/PD book

Date: Fri, 8 Aug 1997 22:52:49 +0200

MIME-Version: 1.0

X-Priority: 3

X-MSMail-Priority: Normal

X-MimeOLE: Produced By Microsoft MimeOLE V4.71.1712.3

Nelder-Mead is not the default in WinNonlin, but this really shows the

essence of PK modelling: No software package should be a black box. Every

data set has to be looked at very carefully, and the methods must be chosen

with care. This applies to selection of the weighting scheme as well.

The WinNonlin software has just been upgraded to a version 1.5, and it is

still a very good programme to work with and easy to use.

Regarding the 2nd ed. of the PK/PD Data Analysis (Gabrielsson &

Weiner) book, there are still some minor typographic errors, but I still

find it very usefull, and as far as I know there aren't many alternatives

with this practical approach.

Thomas Senderovitz, MD.

Unit of Clinical Pharmacology

Bispebjerg Hospital, University of Copenhagen, Denmark

E-mail: senderovitz.-at-.dadlnet.dk - On 17 Mar 1998 at 16:10:16, "Piotrovskij, Vladimir" (vpiotrov.-at-.janbe.jnj.com) sent the message

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Dear Dr. Proost,

As a matter of fact, there were no differences between the results

provided by the nls function and those by your program MultiFit.

Besides the examples you mentioned I also checked another examples not

mentioned in your mail and found that nls and PCNonlin results (as

presented in the book by Gabrielsson and Weiner) coincided.

In the course of the present discussion people addressed the question

of fitting algorithm (Gauss-Newton vs. Nelder-Mead). Indead, a simplex

algorithm like Nelder-Mead is almost safe w/r to lokal minima, but it

is much slower as compared to Gauss-Newton and other gradient-based

algorithms which may take advantage of using second partial

derivative.

BTW, which algorithm is used in MultiFit?

Best regards,

Vladimir

--------

Vladimir Piotrovsky, Ph.D. Fax: +32-14-605834

Janssen Research Foundation Email: vpiotrov.-at-.janbe.jnj.com

Clinical Pharmacokinetics vpiotrov.-at-.janbelc1.ssw.jnj.com

B-2340 Beerse

Belgium - On 17 Mar 1998 at 16:09:27, "CHARLES" (bruce.-a-.pharmacy.uq.edu.au) sent the message

Back to the Top

We routinely use the Simplex algorithm (Nelder-Mead) because of its

robustness. The relatively slower convergance on some problems is not

a practical limitation any more with pentium PCs.

Cheers,

BC

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Bruce CHARLES, PhD

School of Pharmacy

The University of Queensland

Brisbane, Qld, Australia 4072

Telephone : +61 7 336 53194

Facsimile : +61 7 336 51688

Email : Bruce.Charles.-a-.pharmacy.uq.edu.au

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + - On 25 Mar 1998 at 09:53:44, Maria Durisova (exfamadu.-at-.savba.savba.sk) sent the message

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To members of PharmPK,

I have analyzed the data set of example PK5, item a, in Dr.

Proost's message of March 7, (kindly provided by Dr.Proost),

Time Concentration

(min) (mg/l)

5 1.625

10 1.384

15 1.280

20 1.105

30 0.973

45 0.806

60 0.740

90 0.582

120 0.530

150 0.458

180 0.416

240 0.342

300 0.321

360 0.246

using the linear dynamic system approach and the software

described in studies (1,2). The data set corresponds to an

i.v. bolus dose of 100 mg, given to a human volunteer. On the

basis of the given data, I defined the system describing the

fate of the drug in the volunteer, in such a way that the

product of the dose and the Dirac delta function (an

explanation is given at the bottom) was considered the system

input, and the measured concentration profile was considered

the system output. I selected, an optimal model of this

system, assuming a constant variance, i.e. equal weight of

1 for each measurement. The output of this model in the time

domain is given by Eq.1

Eq.1:

C(t)= 269.8 (3.115E-03 e^(-3.322E-03 t)+3.492E-03 e^(-0.046 t)+

+ e^(-0.461 t) (-6.608E^-03 cos(0.495t) - 5.811E-03 sin(0.495 t))).

This model has the following values, e.g.:

Time Model value

(min) (mg/l)

0.00000 0.00000

2.05000 0.83069

4.15000 1.52744

5.00000 1.61332

10.00000 1.41323

15.00000 1.26617

20.05000 1.15685

30.05000 0.99335

45.05000 0.83960

60.05000 0.74626

90.00000 0.63770

120.00000 0.56779

150.00000 0.51160

180.00000 0.46249

240.00000 0.37874

300.00000 0.31029

360.05000 0.25417

The corresponding values of SS and AIC are 0.0165 and

-45.42, respectively. The estimated value of the system gain

(1,2) is G=2.2996 min/l. For the system describing the fate

of the drug after a single bolus dose this value represents

the reciprocal value of clearance. The presence of the sine

and cosine functions in Eq.1 indicates the time delay in the

studied system (3,4).

As seen, the output of this non-compartment model has the

zero value at time zero, (please note that the parameters in

Eq.1 are truncated) even for the data after a bolus i.v.

dose. This is in agreement with the physically obvious fact

that the initial concentration of any drug is zero in all the

body pools at time zero, whatever the route of

administration. Except for linearity, the linear dynamic

system approach is totally independent. It does not employ

any concepts concerning the drug distribution and

elimination. Furthermore, it does not impose the abstract

constrains typical for the deterministically represented

compartment models, i.e. the assumption of homogeneous

instantaneously well mixed pools in the body.

In our study (5) we proposed and applied a criterion for

testing similarity of two dynamic systems, on the basis of

the normalized model weighting functions of these systems.

(This criterion can be used to advantage also in

bioequivalence trials.) The closer the value of this

criterion to 100%, the higher the probability that the

dynamic properties of the two systems are identical. The

closer the value of the criterion to 0%, the higher the

probability that these properties fail to be identical.

The 2- exponential function estimated by G&W and MultiFit

have the form of Eq.2 and Eq.3, respectively

Eq.2

C(t)=3.2313 e^(-0.376549 t)+1.22815 e^(-0.006285 t)

Eq.3

C(t)=1.057773 e^(-0.047824 t)+0.783247 e^(-0.003297 t).

The corresponding weighting function can be obtained (by

dividing them by the dose of 100 mg) in the form of Eq.4 and

Eq.5, respectively

Eq.4

WF(t)=0.032313 e^(-0.376549 t)+0.0122815 e^(-0.006285 t)

Eq.5

WF(t)=0.01057773 e^(-0.047824 t)+0.00783247 e^(-0.003297 t).

The corresponding normalized weighting functions can be

obtained by dividing of the function given by Eq.4 and Eq.5

by AUC under these functions, i.e. by the values of 2.0399

and 2.5966, respectively. The normalized weighting functions

are given by Eq.6 and Eq.7

Eq.6

WF(t)_{N}=0.0158404 e^(-0.376549 t)+0.006020638 e^(-0.006285 t)

Eq.7

WF(t)_{N}=0.004073519 e^(-0.047824 t)+0.0030162 e^(-0.003297 t).

The application of the criterion presented in our study (3)

to the normalized model weighting functions given by Eq.6 and

Eq.7 yields the value of 80.66%, indicating that 80.66% of

the dynamic properties of the models given by Eq.2 and Eq.3

are equivalent. The difference between the dynamic properties

of the two models compared can be said to be significant when

the criterion value obtained is less than a preset threshold

value, e.g. the value of 95 or 90%. With respect to this, the

difference between the 2-exponential G&W and MultiFit models

can be considered significant. Consequently, MultiFit model

(with smaller SS) can be considered a better approximation of

the data by the 2-exponential function than G&W model.

The criterion used in our study (5) and in this example is

formally similar to the bioequivalence index introduced in

study (6). However, since this criterion is based on the

weighting functions of the systems compared, i.e. on the

inherent functions of these systems, it is a general

criterion for testing dynamic similarity of two linear

dynamic time invariant systems.

1. L. Dedik, M. Durisova, Int. J. Bio-Med. Comput., 39, 1995,

231-241.

2. L. Dedik, M. Durisova, Comput. Methods Programs Biomed.,

51, 1996, 183-192.

3. M. Durisova, L. Dedik, M. Balan, Bull. Math. Biol., 57,

1995, 407-412.

4. G. A. Baker, P. G. Morris, P. A. Carruthers, Pade

approximants, In: Encyclopedia of Mathematics and Its

Applications, Volume 13, ed.: G. C. Rota, Addison-Wesley

Publishing Company, Massachusetts, 1992.

5. M. Durisova, L. Dedik, Pharm. Res., 14, 1997, 860-864.

6. A. Rescigno, Pharm. Res., 9, 1992, 925-928.

Maria Durisova

Institute of Experimental Pharmacology

Slovak Academy of Sciences

Dubravska cesta 9, 842 16 Bratislava

Slovak Republic

exfamadu.at.savba.sk

Phone/fax: 004217375928

P.S.

The Dirac delta function is not a function in the classical

sense, since a function represents a way of associating

unique objects to each points in a given set. However, it can

be treated as a function in the generalized sense, and in

fact, it is called "a generalized function." It is

a theoretical not a real function, having the value 0 except

at 0, the value infinity at 0, and an integral from minus

infinity to plus infinity of 1. It is named after a British

physicist P. A. M. Dirac (1902-1984). In fact, Dirac said:

"All electrical engineers are familiar with the idea of

a pulse, and this function is just a way of expressing

a pulse mathematically".

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