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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
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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
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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
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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
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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
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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
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
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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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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Copyright 1995-2010 David W. A. Bourne (david@boomer.org)