Back to the Top
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
Back to the Top
[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
Back to the Top
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
Back to the Top
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
Back to the Top
> 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
Back to the Top
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
Back to the Top
[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
PharmPK Discussion List Archive Index page
Copyright 1995-2010 David W. A. Bourne (david@boomer.org)