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The following message was posted to: PharmPK
Happy new year to all,
I have a question regarding dose linearity. In general, we examine, dose
linearity using a power model,
P = a * Dose^b
where P represents the dependent variable (Cmax, AUClast, AUCinf) and, a
and b are constants. A value of b~1 indicates linearity.
(To perform this simple regression in SAS, we use PROC REG on log
transformed data. The value of slope is 'b' and the exponentiation of
the intercept is 'a')
Do you have any comment or suggestion on this model?
Thanks in advance,
Shilpi
Shilpi Khan, M.S.
Staff Scientist/Biostatistician
CEDRA Corporation
Austin, TX 78754
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The following message was posted to: PharmPK
Shilpi:
Here is a reference for the method you mentioned. One can use SAS
Proc Mixed or the WinNonlin Linear Mixed Effects modelling feature.
Excel power curve plots give the same 'a' and 'b' parameter estimates
but does not compute confidence intervals, which are needed for the
assessment.
BP Smith, FR Vandenhende, KA DeSante, NA Farid, PA Welch, JT
Callaghan, ST Forgue. Confidence Interval Criteria for Assessment of
Dose Proportionality. Pharmaceutical Research, Vol. 17, No. 10, 2000.
Dan Combs
Combs Consulting Service
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Shilpi
You regession model is as you indicate and I reproduce it below:
P = a * Dose^b where P is a Pk exposure parameter e.g. Cmax.
My comment is as follows:
Can you tell me what your null hypothesis (Ho) is and how are you
evaluating the p value for this test.
Angus McLean Ph.D,
8125 Langport Terrace,
Suite 100,
Gaithersburg,
MD 20877
Tel 301-869-1009
fax 301-869-5737
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The following message was posted to: PharmPK
Hi Shilpi
Yes the equation of power model for testing Dose Linearity is
P = a * Dose^b
where P represents the dependent variable (Cmax, AUClast, AUCinf)
But i think apart from log-transformation data also needs to be dose-
normalized either to Higher Dose or Lower Dose.
On log transformation our equation becomes: Log(p) = a + b*log(dose)
Our null hypothesis is H0: b = 1.
We can use Proc Reg in SAS on the log-transformed data.
You can add following statement in Proc reg inorder to test for b=1:
SLOPE : test Log_DOSE = 1;
Hope this helps.....
Vikesh S
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The following message was posted to: PharmPK
Dear Shilpi,
The suggested model makes perfect sense. However, I would like to
challenge the use of log-transformation. In my humble opinion, you
may do a better job by running non-linear regression on untransformed
data. In that case, linear regression on log-transformed data would
still be useful in providing initial parameter estimates.
Hope this helps.
Henri
Henri MERDJAN, Pharm, AIHP
Head of Drug Metabolism and Pharmacokinetics
NOVEXEL S.A.
Parc Biocitech
102 Route de Noisy
F-93230 Romainville
France
Tel +33 (0)1 57 14 07 45
Fax +33 (0)1 48 46 39 26
Web www.novexel.com
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The following message was posted to: PharmPK
Dear Dr Merdjan, Dear All,
I agree that the power model (AUC = a * Dose^b) is a reasonable way
to statistically test dose linearity. There might be some drawbacks
for Cmax, because the rate of absorption is sometimes slower at
higher doses (e.g. if the drug has low solubility). In this case
fitting the full population PK model will be superior to NCA in order
to better understand the PK data.
It is important to adequately account for the between subject
variability (BSV) in Cmax and AUC, irrespective if one uses linear or
log-scale to fit the data. I would assume a log-normal distribution
for Cmax and AUC. If one fits the power model on linear scale, one
would have to account for a proportional error structure (BSV)
explicitly. Therefore, I would prefer ANCOVA on log-scale as a quick
way to analyze dose linearity data, because an additive error
structure on log-scale turns into a proportional error structure on
linear scale.
Besides a criterion for statistical significance of dose linearity,
an equivalence based criterion should also be considered. Even if a
drug shows a nonlinear curve of AUC vs. dose, this may have only a
limited effect on the pharmacodynamics at therapeutic doses. One way
to do so is to use equivalence statistics with dose-normalized AUC
(and Cmax) in addition to fitting the power model.
Hope this helps
Best regards
Juergen
--
Juergen Bulitta, M.Sc.
Research Scientist
IBMP - Institute for Biomedical and Pharmaceutical Research
Paul-Ehrlich-Strasse 19
90562 Nurnberg - Heroldsberg
Germany
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Henri:
What you suggest to evaluate nonlinear regression on untransformed
data makes sense but exactly how would you introduce the null
hypotheses and obtain a p factor. Exactly what computer program do
you have in mind.
can you comment?
Angus McLean Ph.D,
8125 Langport Terrace,
Suite 100,
Gaithersburg,
MD 20877
tel 301-869-1009
fax 301-869-5737
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The following message was posted to: PharmPK
Dear Angus,
I guess a sensible way of approaching the problem would be to state
the null hypothesis as: " exponent=1 ". I would expect a nonlinear
regression piece of software to provide confidence intervals around
parameters' estimates (by the way, I am not recommending any specific
soft). Accepting/rejecting H0 is then straightforward. Just answer
the question: does the CI include 1? So far, this is theory.
Real life may be a bit different. In my experience, the major
drawback of the hypothesis testing approach is that an exponent may
be statistically different from, however "close" to unity. Let me go
through a numerical example. Let's imagine the exponent is 1.08 with
a 95%CI of 1.03 to 1.13. In strict statistical terms, this is a
significant deviation from linearity. However, what is its practical
relevance? Such a power model would suggest that each time you double
the dose, the PK parameter of interest (eg Cmax or AUC) will increase
by a 2.1-fold factor instead of 2. Not a big deal!
In summary, you may find some interest in setting acceptance limits
for the exponent rather than, or in addition to, testing some
hypothesis.
Hope this helps,
Henri
Henri MERDJAN, Pharm, AIHP
Head of Drug Metabolism and Pharmacokinetics
NOVEXEL S.A.
France
Tel +33 (0)1 57 14 07 45
Fax +33 (0)1 48 46 39 26
Web www.novexel.com
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The following message was posted to: PharmPK
Dear Henri,
I fully agree with your statement about the relevance of confidence
intervals.
But is real life, there may be also a different problem. What should
one conclude if the exponent differs from 1 to a clinically relevant
extent, but with a confidence interval including 1? E.g. exponent 1.3
and confidence interval 0.9 to 1.7. What is the conclusion now?
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.at.rug.nl
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Johannes: what a physician want to know is if you double the dose
would you get double the maximum concentration (Cmax)? Given the
adverse event profile of the drug on a case by case basis there may
be concern about the probability of the drug (on dose doubling)
producing disproportionately higher adverse events on account of
higher Cmax values (than anticipated from linearity). Perhaps in
cases, where there is uncertainty about proportionality
considerations, then dose would be more carefully titrated with
clinical response in mind to a patient over a period of time with
intermediate dose strengths.
Best Regards,
Angus McLean Ph.D,
8125 Langport Terrace,
Suite 100,
Gaithersburg,
MD 20877
tel 301-869-1009
fax 301-869-5737
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The following message was posted to: PharmPK
Dear Johannes,
This is a fair point. The question you are asking illustrates the
level of uncertainty possibly associated with a parameter estimate.
In this particular example, the CI is so "wide" that you cannot
really decide whether the exponent value is actually closer to unity,
or to the central estimate of 1.3, or to some even higher value. In
such a scenario, I would simply recommend a careful dose escalation.
Best regards,
Henri
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