# RE: calculation of AUC

From: Wang, Yaning <Yaning.Wang>
Date: Wed, 25 Mar 2009 23:43:19 -0400

I think Mats' description about NCA based AUC can be simplified to the
following(imagine only 2 time points):
Given two log-normally distributed random variable,
C1~log-N(log(F1),sigma**2) and C2~log-N(log(F2),sigma**2), what is the
relationship between C1+C2 (model simulated) and F1+F2(model predicted)?
Mats is saying average (mean) of C1+C2 is systematically greater than
F1+F2 (but a specific C1+C2 can be either less or greater than F1+F2).
This can be easily proved mathematically.
The distribution of the sum of two log-normally distributed random
variables is approximately another log-normal distribution(Fenton L).
Say Y=C1+C2, then Y's distribution can be approximately by
log-N(log(median_Y),sigmasum**2), where
median_Y=(F1+F2)*exp(sigma**2/2-sigmasum**2/2). And
mean_Y=(F1+F2)*exp(sigma**2/2). Since exp(sigma**2/2) is always =
greater
than 1, mean_Y will be greater than F1+F2. I believe Mats' table is
based on simulation. The exact ratio (difference) is exp(sigma**2/2).

Notice F1 and F2 are the medians. Therefore I believe a more reasonable
statistic to be compared with F1+F2 should be median_Y. Then the ratio
will be exp(sigma**2/2-sigmasum**2/2), which will be much closer to 1
but still greater than 1 (sigmasum**2 can be proved to be less than
sigma**2). If there is only 1 time point (C1), median_Y will be equal
to F1 while mean_Y will still be greater than F1 by exp(sigma**2/2).

Fenton L., The Sum of Log-Normal Probability Distributions in Scatter
Transmission Systems, IRE Trans. Commun. Syst., vol. CS-8, pp. 57-67

Yaning Wang, Ph.D.
Office of Clinical Pharmacology
Office of Translational Science
Center for Drug Evaluation and Research
Phone: 301-796-1624
Email: yaning.wang

"The contents of this message are mine personally and do not necessarily
reflect any position of the Government or the Food and Drug

-----Original Message-----
From: owner-nmusers
On Behalf Of Mats Karlsson
Sent: Wednesday, March 25, 2009 5:16 PM
To: 'Nick Holford'; 'nmusers'
Subject: RE: [NMusers] calculation of AUC

Nick,

Mats

Mats Karlsson, PhD
Professor of Pharmacometrics
Dept of Pharmaceutical Biosciences
Uppsala University
Box 591
751 24 Uppsala Sweden
phone: +46 18 4714105
fax: +46 18 471 4003

-----Original Message-----
From: owner-nmusers
On Behalf Of Nick Holford
Sent: Tuesday, March 24, 2009 8:07 PM
To: nmusers
Subject: Re: [NMusers] calculation of AUC

Mats,

Thanks for trying to explain things further but I am still confused. I
agree for the moment we can forget about shrinkage and about computing
the central tendency over a group of subjects.

Lets just fit one individual with a model and estimate clearance then
calculate AUC from Dose/CL. If we take the same concentration
observations and compute an AUC using a trapezoidal rule then I still

M> AUCs from a model are usually calculated either to drive PD or to
compare with NCA AUCs of observed data, as an internal validation. I was
making a point regarding the second use of model based AUCs. Of course
there are many situations where AUC is not dose/CL, whenever AUC is
calculated by trapezoidal rule is one of them.
If you want to compare like with like - model-based trapezoidal rule
AUCs with real data trapezoidal rule AUCs - you should also take into
the error generation structure. If you get data that have reported
negative concentration (as you discuss below) it is appropriate to use a
simulation that mimics that. I never see that type of data and try to
mimic the error generation process of more common structure. Naturally
you should treat your simulated data just as the real data.

You say:

"The exponentiation of (LOG(F)+EPS(1)) will not give the expected mean
of F, but something higher. [this is what you can calculate
model-simulated NCA AUCs from]"

but I dont understand what you mean by "this is what you calculate
model-simulate NCA AUCs from".

I am not thinking of calculating NCA AUCs from simulated concentrations.

I expect to use real measured concentrations.

Simulating concentrations which force all concentrations to be
non-negative is a biased simulation of reality. If there is any
possibility of an additive error then there is a possibility of a
negative measured concentration. Real assays can have additive errors so

real assays must be capable of measuring values that appear to be
negative. Note the difference between the true concentration which must
be non-negative and the measured concentration, i.e. the truth plus
error, which can be negative if the error is additive.

Nick

Mats Karlsson wrote:
> Dear Nick,
>
> I did not discuss shrinkage because it didn't concern the point I was
trying (and maybe failing) to make. [However, I don't think that if one
wants to compare with NCA AUCs, data are likely to be rich with
reasonably small shrinkage]
>
> I used proportional residual error as an example. Doesn't really
matter which residual error you use - going from the normality
assumption on the log scale to normal scale would always make the mean
of a simulated observation higher than the mean. Mean(exp(epsilon)) is
going to be higher than 1 regardless of residual error model.
>
> The point I'm trying to make is not how you calculate the central
tendency of several AUCs, it concerns the calculation of individual
AUCs.
>
> The problem I point out is relevant when you compare NCA AUCs from
observed data with NCA AUCs from model predictions, regardless if you
use linear or log-linear trapezoidal rules. Observed NCA AUCs are
expected to be higher than NCA AUCs from model-predicted (but not higher
than model simulated) AUCs calculated by NCA (from the same sampling
schedule).
> For a model:
> Y=LOG(F)+EPS(1)
> The exponentiation of LOG(F) will give the expected mean of F [from
which model-predicted NCA AUC will be calculated]
> The exponentiation of (LOG(F)+EPS(1)) will not give the expected mean
of F, but something higher. [this is what you can calculate
model-simulated NCA AUCs from]
>
> Thus model-predicted and model-simulated NCA AUCs will be
systematically different if they are calculated in this way. I expect
that if the model is correct, the observed NCA AUCs will be more similar
to the simulated NCA AUCs.
>
> Hope this makes it clearer.
>
> Best regards,
> Mats
>
>
> Mats Karlsson, PhD
> Professor of Pharmacometrics
> Dept of Pharmaceutical Biosciences
> Uppsala University
> Box 591
> 751 24 Uppsala Sweden
> phone: +46 18 4714105
> fax: +46 18 471 4003
>
>
> -----Original Message-----
> From: owner-nmusers
[mailto:owner-nmusers
> Sent: Sunday, March 22, 2009 7:09 AM
> To: nmusers
> Subject: Re: [NMusers] calculation of AUC
>
> Mats,
>
> This is an interesting idea but it seems to be more complicated than
> just a consideration of the residual variability (RV%) when using log
> transformation with transform both sides (TBS) estimation.
>
> First of all you appear to assume that the RV% is only a proportional
> residual error but if could also include an additive component when
> using TBS so that there is not a single RV% that would describe a
> particular situation because it would change with concentration.
>
> A model based estimate of AUC would typically be based on an empirical

> Bayes estimate (EBE) of CL. This estimate is of course a shrinkage
> estimate which will typically be biased towards the population CL but
I
> have realized that there is also EBE bias from the choice of
> transformation used in parameter estimation. Thus I would not expect
the
> model based estimate to be additionally biased because of using EBEs
> with TBS. This is probably something you have thought about so please
> inform me.
>
> Turning to the NCA method - I dont know if a bias is expected from the

> NCA calculated AUC but I would naively assume that the trapezoidal
part
> would not be biased. I am ready to learn if there is a bias expected
> with trapezoidal NCA. I expect this has been investigated and reported

> but I am not familiar with it. The extrapolated portion typically
relies
> on a log linear transformation to estimate the elimination rate
constant
> which so in this respect the log transformed model based and NCA based

> methods would seem to be similar.
>
> Another source of difference between model and NCA based AUCs might
> arise from the use of different statistics to describe the central
> tendency of the indidual estimates. NCA estimates could be based on
the
> arithmetic mean of the individual AUC sor on the geometric mean (most
> commonly used for bioequivalence analysis). The model based estimates
> based on the arithmetic mean of the EBE predicted AUCs would be biased

> towards the geometric mean because the population value would
typically
> be estimated with an exponential ETA.
>
> If you have the time would you expand on the details of your assertion

> so that I and others can understand the basis more clearly? It seems
to
> me that comparison of model based AUCs with NCA based AUCs is more
> complicated than just a consideration of the typical value of the
> residual error.
>
> Nick
>
>
> Mats Karlsson wrote:
>
>> Dear Ethan,
>>
>>
>>
>> Just a caution when comparing model-based AUCs with NCA calculated
>> AUCs. If you have done your modeling using log-transformation of
>> observations and model predictions and then compared AUCs on the
>> linear scale, you should not expect a perfect agreement between the
>> two. The reason is that the mean of an exponentiated distribution of
>> epsilons is not the same as the median, but higher. Thus, the AUCs of

>> model-predicted individual profiles will be expected to be lower than

>> either simulated or observed. The magnitude of the difference will
>> depend on the residual error magnitude and will typically be:
>>
>>
>>
>> %RV expected AUC difference
>>
>> 10 0.50%
>>
>> 20 2%
>>
>> 30 5%
>>
>> 40 9%
>>
>> 50 14%
>>
>> 70 29%
>>
>>
>>
>> Best regards,
>>
>> Mats
>>
>>
>>
>> Mats Karlsson, PhD
>>
>> Professor of Pharmacometrics
>>
>> Dept of Pharmaceutical Biosciences
>>
>> Uppsala University
>>
>> Box 591
>>
>> 751 24 Uppsala Sweden
>>
>> phone: +46 18 4714105
>>
>> fax: +46 18 471 4003
>>
>>
>>
>> *From:* owner-nmusers
>> [mailto:owner-nmusers
>> *Sent:* Friday, March 20, 2009 6:52 PM
>> *To:* Michael.J.Fossler
>> *Subject:* Re: [NMusers] calculation of AUC
>>
>>
>>
>> sorry for being lazy this morning and wish relying on others
knowledge
>>
>> just to share, I used DADT=C method, and it didn't depend on =
sampling

>> after I tried with my model (which took quite a while to get results)
>>
>> -- I could do as Bill suggested setting up some small dataset and
>> simple model to check first, then would share with the group ealier
:-)
>>
>>
>>
>>
>>
>>
>>
------------------------------------------------------------------------
>>
>> *From:* "Michael.J.Fossler
>> *To:* nmusers
>> *Sent:* Friday, March 20, 2009 9:42:59 AM
>> *Subject:* Fw: [NMusers] calculation of AUC
>>
>>
>> I second Bill's suggestion to work this out on your own for your
>> specific problem. This forum can help you with general questions and
>> overall approaches, but very specific queries like this are for you
>> and your colleagues to hash out.
>>
>> *Error! Filename not specified.*
>> ----- Forwarded by Michael J Fossler/PharmRD/GSK on 03/20/2009 09:40
>> AM -----
>>
>> *"Bill Bachman" <bachmanw
>> Sent by: owner-nmusers
>>
>> 20-Mar-2009 09:17
>>
>>
>>
>>
>>
>> To
>>
>>
>>
>> "'Martin Bergstrand'" <martin.bergstrand

>> <ethan.wu75
>>
>> cc
>>
>>
>>
>> Subject
>>
>>
>>
>> RE: [NMusers] calculation of AUC
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> The easiest answer is to work it out. Do some simulations (without
>> variability) with multiple subjects with identical PK parameters BUT
>> different sampling times. Tabulate your AUCs and compare the results

>> for different sampling times!
>>
>>
>>
>>
>>
------------------------------------------------------------------------
>>
>>
>> *From:* owner-nmusers
>> [mailto:owner-nmusers
Bergstrand*
>> Sent:* Friday, March 20, 2009 8:45 AM*
>> To:* 'Ethan Wu'; nmusers
>> Subject:* RE: [NMusers] calculation of AUC
>>
>> Dear Ethan,
>>
>> You need to provide more information on how you plan to calculate AUC

>> otherwise the question can't be answered. It is of course possible to

>> calculate the AUC without any influence of the sampling frequency.
You
>> should be able to find examples of how to do this in the NMusers
>> archive. See for example the answer from Mats Karlsson in this thread

>> (http://nonmem..org/nonmem/nm/98apr032002.html
>> <http://nonmem.org/nonmem/nm/98apr032002.html>).
>>
>> Kind regards,
>>
>> Martin Bergstrand, MSc, PhD student
>> -----------------------------------------------
>> Department of Pharmaceutical Biosciences,
>> Uppsala University
>> -----------------------------------------------
>> P.O. Box 591
>> SE-751 24 Uppsala
>> Sweden
>> -----------------------------------------------
>> martin.bergstrand
<mailto:martin.bergstrand
>> -----------------------------------------------
>> Work: +46 18 471 4639
>> Mobile: +46 709 994 396
>> Fax: +46 18 471 4003
>>
>>
>> *From:* owner-nmusers
>> [mailto:owner-nmusers
>> Sent:* den 20 mars 2009 13:05*
>> To:* nmusers
>> Subject:* [NMusers] calculation of AUC
>>
>> Hi all, to calculate AUC of one of the compartments using ADVAN6, if
>> it is a fixed time interval, will the AUC be influenced by the
>> frequncy of sampling of the dataset within this interval or not?
>> thanks
>>
>>
>>
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>>
>>
>
>

--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New
Zealand
n.holford
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
Received on Wed Mar 25 2009 - 23:43:19 EDT

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