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The following message was posted to: PharmPK
One member of this list recently asked about the meaning of the term
Cmax/AUC. This was a new term to me and I have not seen any replies to
date concerning this but I recently came across a statement, "The
average rate of absorption as determined by Cmax/AUC values was ...."
This term will have units of 1/Time and therefore cannot be considered
to be a rate. Also, one can presumably calculate this term for a
situation where there is no absorption (an iv bolus dose) using the
initial concentration as Cmax (for a one compartment model Cmax/AUC is
equal to the elimination rate constant).
However, if one considers two formulations that deliver identical doses
of the same drug at differing rates (i.e. one absorbed more rapidly than
the other) then the one with the faster rate of absorption will have the
higher value for Cmax but both will have identical values for AUC.
Therefore the one that is absorbed more rapidly will have the larger
value for Cmax/AUC.
Can anyone throw anymore light on this term and its origin?
Dr Paul S. Collier
School of Pharmacy
Queen's University, Belfast
97 Lisburn Road
Belfast BT9 7BL
N. Ireland, UK
[Didn't think about it until after I sent this but retrieved it first.
Let's try another ROA, i.e. IV Bolus (or a 'very' fast release oral dose
;-) Cmax/AUC = Cp(0)/AUC = (Dose/V)/AUC = CL/V = kel!!! (one compartment
model - sorry Nick). Maybe it is an indication of rate of elimination in
the form of a first order rate constant. Faster elimination means
earlier and higher Cmax as well as faster absorption - just a thought or
two - db]
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The following message was posted to: PharmPK
Hi Dr. Collier:
Accully the term Cmax/AUC is not new. It was proposed by Prof. Endrenyi
at the University of Toronto in 1991 (Endrenyi et al: J Clin Pharmacol
Ther Toxicol 29:394-). Basically, it's used to gain a clearer measure
than the Cmax for the evaluation of absorption rate. Usually it's used
for assessments of bioequivalence.
Regards,
Bang Qian Xu, Ph.D.
Apotex Inc.
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Dear Paul,
In "average" Bioequivalence testing, the problem inherent in Cmax as a
measure
(indirect) of absorption rate led Laszlo Endrenyi and his colleagues in
Toronto
to propose the use of Cmax/AUC as an alternative rate metric. Compared
with Cmax, Cmax/AUC better correlates with, and has the same units as,
the underlying
rate parameter, Ka.
{Ref. Endrenyi, L. et al., Int. J. Clin. Pharmacol. Ther. Tox., 29,
394-399, 1991; Tothfalusi, L & Endrenyi, L., Pharm. Res., 12, 937-942,
1995. See also
the comments of Tozer et al (Pharm. Sci., 13, 453-456, 1996) against the
use of
Cmax/AUC.}
Now for the commercial :-) for those who may be interested: This and
many other aspects of BA/BE testing are considered in the Kemic course
offering entitled: "Bioavailability and Bioequivalence: The Basics and
Beyond" (N.B. Actual course dates for 2002 have yet to be determined.
Also, our website is momentarily in some disarray while being moved to a
new server.)
Best regards,
Peter
Peter W. Mullen, Ph.D., FCSFS
Kemic Bioresearch
PO Box 878
Kentville
Nova Scotia, B4N 4H8
Canada
E-mail pmullen.at.kemic.com; URL
www.kemic.com
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The following message was posted to: PharmPK At 03:47 PM 3/14/02, Peter
Mullen wrote:
>In "average" Bioequivalence testing, the problem inherent in Cmax as a
measure
>(indirect) of absorption rate led Laszlo Endrenyi and his colleagues in
Toronto
>to propose the use of Cmax/AUC as an alternative rate metric. Compared
with Cmax, Cmax/AUC better correlates with, and has the same units as,
the underlying
>rate parameter, Ka.
>
This assumes Ka is a constant. It is not. Never was. Never will be.
Cannot be. (Sometimes it's almost constant, but not so often in our
experience.)
This is a very important point. Ka varies with time and position in the
gastrointestinal tract. It varies with time because concentrations vary
with time, and diffusion is dependent on a concentration gradient. It
varies with position because permeability varies with pH for ionized
compounds, with tight junction gap for paracellular compounds, and with
transporter density for carrier-mediated compounds. (If you have a low
molecular weight, ionized, carrier-mediated transported compound, you
may have to deal with all three!)
The pharmaceutical industry has long used constant Ka in a variety of
ways, ignoring Fick's first law for diffusion. Most drugs passively
diffuse across the apical membrane from high concentration to low
concentration in accordance with Fick's First law:
J = D (C1 - C2)
The difference between the concentration in the intestinal lumen and the
concentration in the enterocytes changes with time. As a result, the
absorption rate changes with time.
Using a concentration-gradient-based absorption model, a plot of the
"net effective Ka" calculated as
KaNet = SUM over i [(dMabs/dt)i / (Mdiss)i]
where
i = compartment number
(dMabs/dt)i = absorption rate in compartment i (Mdiss)i = total
mass in solution in compartment i
yields a curve that is shaped a bit like a plasma concentration-time
curve. KaNet starts at zero, because there is no lumen concentration at
time zero. It increases very rapidly as drug goes into solution, and the
lumen concentration is higher than the enterocyte concentration. After a
while, the unabsorbed portion of the dose moves downstream and the
concentration in the compartment decreases by both transit out of the
compartment and absorption. The difference between the lumen
concentration and the enterocyte concentration decreases, so the net Ka
decreases. This continues for all compartments. This is for the apical
membrane.
In the absence of carrier-mediated transport, the basolateral transfer
depends on the concentration different between enterocytes and blood,
which begins to equalize after a while as the blood concentration
increases, decreasing the basolateral transfer rate and causing the
concentration in the enterocytes to decrease more slowly than the
decreasing lumen concentration. This slows the apical transfer rate.
For a drug with long half-life and low protein binding, the
concentration in the blood can remain high long after absorption is
complete. This can result in higher concentration in blood than in
enterocytes, with subsequent transfer of drug from blood through the
basolateral membrane back into the enterocytes, and thence back into the
lumen, perhaps to be reabsorbed in compartments downstream. Thus an
entero-entero circulation is possible.
When the lumen concentration is lower than the enterocyte concentration,
then "Ka" becomes negative, and exsorption will occur. Clearly, if Ka
can go negative, it is not constant. Exsorption has been observed even
for IV doses.
I was at a pharmaceutical company in the past few weeks and an
experienced researcher made the comment that "pharmacokinetics does not
affect bioavailability". He justified this with an equation that had Ka
in the numerator (and which, of course, assumed a constant Ka). I
proceeded to demonstrate to him through simulation that pharmacokinetics
indeed can affect bioavailability - and absorption. We have always
treated absorption and pharmacokinetics as separate processes, but they
interact. Sometimes weakly, sometimes more strongly.
Treating Ka as a constant can lead to a variety of errors, some small
enough to ignore, many not so small. As with any other simplifying
mathematical assumption (including Fick's Law), it should be used with
caution.
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (SIMU)
1220 W. Avenue J
Lancaster, CA 93534-2902
U.S.A.
http://www.simulations-plus.com
E-mail: walt.aaa.simulations-plus.com
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[And a few more replies - db]
From: Dietrich.Tuerck.-at-.bc.boehringer-ingelheim.com Date: Fri, 15 Mar 2002
08:49:25 +0100
To: david.-at-.boomer.org
Subject: AW: PharmPK
The following message was posted to: PharmPK
Dear Paul,
I am certainly not the best person to respond on this item, but the term
Cmax/AUC was discussed extensively in the early nineties in the search
for a good parameter to determine the rate of absorption in
bioequivalence studies (see e.g. conference reports for BIO
International 92 by Shah, Blume et al. (Conferences with substantial
contributions by US and EC regulatory authorities). At this time, many
colleagues were not satisfied with the use of Cmax as a measure for rate
of absorption, because Cmax and AUC are correlated. I recall several
papers on the item, I think by Endrenyi and coworkers. There were also
some other derivations tested to find a better measure for the rate of
absorption.
Finally, the view was accepted that despite there is no good
noncompartmental measure for the rate of absorption, the maximum
concentration is valuable per se for safety reasons (I remember that
Geoff Tucker argued always in this direction).
Does this help?
Dietrich TŸrck, PhD
Boehringer Ingelheim Pharma KG
Clinical Pharmacoknetics - Group Registration Life Cycle Management
88397 Biberach an der Riss
Germany
---
From: "PERSIANI Stefano"Date: Fri, 15 Mar
2002 09:08:21 +0100
To: david.-at-.boomer.org
Subject: Re: Cmax/AUC
The following message was posted to: PharmPK
The term Cmax/AUC is the most sensitive and powerful indirect measure of
rate of drug absoprtion in comparatice PK studies involving
immediate-release dosage forms and someone think that should be used
instead of Cmax in bioequivalence studies.
A comparison of the different method for assessing the rate of
absorption after oral administartion or immediate relaese formulation
has been desrcibed by Lacey at al. Pharmaceutical Research 1994, 83(2):
212 "Evaluation of different indirect measures of rate of drug
absorption in comparative pharmacokinetic studies."
I have used this metric in the past and found it very usefull as it
describe the rate of absoprtion without any influence from extent.
I hope this help
Best regards.
Stefano
Stefano Persiani, PhD
Manager, Pharmacokinetics and Phase I
Rotta Research Laboratorium, S.p.A.
Via Valosa di Sopra, 7-9
20052 Monza (MI)
ITALY
e-mail stefano.persiani.-a-.rotta.com
---
From: "Delwar Hussain"Date: Fri, 15 Mar 2002
09:16:55 -0700
To: david.-a-.boomer.org
Subject: PharmPK Re: Cmax/AUC
The following message was posted to: PharmPK
Dear Walt,
I appreciate your discussion on the effect of position in the GI tract
on Ka. I am not clear when you explain
>>It varies with time because concentrations vary with time, and
diffusion is dependent on a
concentration gradient<<.
You have not defined Ka. If Ka is first order rate constant, how it can
vary with concentration?
Your equation shows Ka is a net summation of different first order rate
constants at different segments or compartments:
>>Using a concentration-gradient-based absorption model, a plot of the
"net effective Ka" calculated as
KaNet = SUM over i [(dMabs/dt)i / (Mdiss)i]
where
i = compartment number
(dMabs/dt)i = absorption rate in compartment i (Mdiss)i = total
mass in solution in compartment i
yields a curve that is shaped a bit like a plasma concentration-time
curve. KaNet starts at zero, because there is no lumen concentration at
time zero. It increases very rapidly as drug goes into solution, and the
lumen concentration is higher than the enterocyte concentration. After a
while, the unabsorbed portion of the dose moves downstream and the
concentration in the compartment decreases by both transit out of the
compartment and absorption. The difference between the lumen
concentration and the enterocyte concentration decreases, so the net Ka
decreases.<<
>>When the lumen concentration is lower than the enterocyte
concentration, then "Ka" becomes negative, and exsorption will occur.
Clearly, if Ka can go negative, it is not constant. Exsorption has been
observed even for IV doses.<<
If Ka is associated with the input process (i.e. absorption), why you
associate it with exsorption? The equation you used above is for KaNet,
both absorption and exsorption. I think you really need to define Ka
first.
Thanks,
M. Delwar Hussain, Ph.D.
Atrix Laboratories
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The following message was posted to: PharmPK
I am not happy with Ka varying wildly and even going negative. Wouldn't
it be simpler to accept that passive absorption across a membrane is
indeed made up of constant factors but leading to a two-way dynamic
process with forward and backward rate constants determining transfer
rates. In most cases the blood concentration remains close to zero and
the forward rate constant only applies to give a constant 'ka'. If the
drug is not cleared from the blood, then an equilibrium will be reached
depending on Kforward/kbackward. For an iv dose and slow clearance, drug
will appear in the oral absorption site due to kbackward (Walt's
negative ka).
Joe Chamberlain
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David
I'd like to come back on this one in relation to absorption and exsorption.
My view of things is that drugs are small molecules that dash about here and
there, going into solution (in lipid layers, in aqueous microenvironments,
etc) even going into the vapour phase if necessary all according to the
forces of nature (physical laws if you like). I don't believe my molecules,
on meeting a membrane barrier, pop their heads over the top to assess how
many spaces are available on the other side before deciding to head across.
I don't see any problem in considering that the drug on either side of a
membrane is in dynamic equilibrium leading to classical absorption if the
blood side acts as a sink if dug is continually removed, and to exsorption if
the blood concentration is higher than the gut concentration (I'm using
concentration for want of a better word just to argue the concept).
While I'm up and running:
I don't think we should invoke time-dependent kinetics when I think we all
agree that it is space-dependent; that is drug absorbing at different sites
will be absorbed at different rates as of course there is no reason why the
absorption at different sites should be the same. The so-called
time-dependent absorption is merely a consequence of the drug being moved to
the other sites by other physical processes and not time-dependent per se.
A more valid time-dependent process would be say diurnal variation in some
physiological process such as something that may effect the nature of the
absorption site.
Joe Chamberlain
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Just a few clarifications about the Cmax/AUC metric. It was recommended to be
used as a secondary metric, in addition to the primary metric of AUC, in
determinations of bioequivalence. It was perceived to be a better, more
informative tool than Cmax for the quality control of pharmaceutical
preparations
since it is generally not correlated with AUC. However, Cmax is a more direct
measure of peak exposure. The metrics have differing roles and, conceivably,
both have places in evaluations of bioequivalence.
Both Cmax/AUC and Cmax are indirect and not very good measures for the average
rate of absorption. Still, the ratio metric does a somewhat cleaner
job for the
purpose. Neither metric measures the absorption rate constant (ka).
This brings us to the more general question whether there is such a quantity as
ka. Well, strictly speaking, there is none. But strictly speaking,
none of the
models and model parameters are correct. Models are conceptual simplifications
that are convenient to describe natural phenomena. They have obvious
limitations. Still, they are very often useful and serve us very well.
Laszlo Endrenyi, Ph.D.
Department of Pharmacology
University of Toronto
Toronto, Ont. M5S 1A8
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The following message was posted to: PharmPK At 08:20 AM 3/18/02, Joe
Chamberlain wrote:
>My view of things is that drugs are small molecules that dash about here and
>there, going into solution (in lipid layers, in aqueous microenvironments,
>etc) even going into the vapour phase if necessary all according to the
>forces of nature (physical laws if you like). I don't believe my molecules,
>on meeting a membrane barrier, pop their heads over the top to assess how
>many spaces are available on the other side before deciding to head across.
>I don't see any problem in considering that the drug on either side of a
>membrane is in dynamic equilibrium leading to classical absorption if the
>blood side acts as a sink if drug is continually removed, and to exsorption if
>the blood concentration is higher than the gut concentration (I'm using
>concentration for want of a better word just to argue the concept).
>
>While I'm up and running:
>
>I don't think we should invoke time-dependent kinetics when I think we all
>agree that it is space-dependent; that is drug absorbing at different sites
>will be absorbed at different rates as of course there is no reason why the
>absorption at different sites should be the same. The so-called
>time-dependent absorption is merely a consequence of the drug being moved to
>the other sites by other physical processes and not time-dependent per se.
>A more valid time-dependent process would be say diurnal variation in some
>physiological process such as something that may effect the nature of the
>absorption site.
>
Sorry Joe, but I could not disagree more. It is not only
space-dependent. It is both time and space-dependent. It is not
"merely a consequence of drug being moved to other sites by other
physical processes" - it is that passive diffusion must follow Fick's
Law.
Even in a sink condition, Fick's Law should apply (in the absence of
carrier-mediated transport):
J = D (C1 - C2)
In a perfect sink, C2 = 0, but C1 still varies with time.
Of course in reality, C2 is not zero, and the time-dependent
concentration difference can be even more dramatic than just the C1
variation. In a multi-compartment model where different compartments
have different volumes, the same amount of drug in solution will be
absorbed at different rates in each compartment, because the
concentrations will be different - even if the Peff is identical in
the compartments. Failure to recognize this will lead to significant
errors as I have described earlier.
As I understand it, passive diffusion is a result of Brownian motion.
Molecules move randomly in both directions with equal ease across the
membrane. It is not a matter of a molecule "popping its head over to
see how many spaces there are on the other side" - it will go through
the membrane if is moving in the right direction with sufficient
momentum and finds a "soft spot". The side that has more molecules
hitting the membrane because of higher concentration will have more
successful permeations, and the side with lower concentration will
have fewer. That's why the net transfer is concentration dependent.
If the transfer rate across the apical membrane has a certain
time-dependence (i.e., it is not instantaneous) then so must the
transfer across the basolateral membrane. Literature says it's a bit
faster across the basolateral membrane - not instantaneous. So a
concentration builds up in the enterocytes.
We don't look at transfer across a membrane as two processes. The net
transfer is what we want to predict. The net movement of molecules
will be from high concentration to low concentration. The rate will
change with concentration, which changes with time. So the rate will
be time-dependent. If the concentrations become equal, as they will
when the lumen concentration is low enough to become equal to that in
the enterocytes (because of absorption and transit), the net transfer
will be zero. I cannot accept that the drug will be pumped in by a
constant Ka in defiance of the laws of physics.
I agree with G. Box that all models (including compartmental models)
are wrong, but some are useful. In the case of oral drug absorption,
a constant Ka model may prove useful for some drugs for some
purposes. But to use it as a general model is like saying my car got
25 miles to the gallon on the 200 mile trip I completed this weekend.
It did. But it got about 28 mpg on the way to my destination, and
about 22 on the way back. (I went from an elevation of 2600 feet to
sea level on the way there, and had to climb back to 2600 feet on the
way back.) If all I care about is the average, then 25 mpg is OK. If
I care about a more accurate description of the rate of fuel
consumption vs time, then it is not.
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (SIMU)
1220 W. Avenue J
Lancaster, CA 93534-2902
U.S.A.
http://www.simulations-plus.com
E-mail: walt.aaa.simulations-plus.com
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Dear Forum;
I would be interested to know if the absorption rate constant, Ka,
mentioned by Laszlo Endrenyi and others recently has any relevance
for efficacy because there is some evidence from pharmacodynamic
studies of analgesics using Lasers to monitor pain thresholds that
rate of change of concentration could be an important determinant of
efficacy (see Neilsen JC et al Eur J Clin Phamacol 1992 42:261-
264.) Does anyone have any more evidence or know whether this
could apply to other types of drug? Anaesthetics might be one example
and possibly alcohol, since it is a common belief that drinking with
food reduces the chances of getting drunk. With antibiotics I have often
wondered if injected forms are more effective due to faster rates of
increase in concentration as well as the higher Cmaxs obtained.
Thanks
Andrew Sutton
Guildford Clinical Pharmacology
asutton.-a-.gcpl.co.uk
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This was a very interesting discussion, I've learned a lot!
I agree 100% with Laszlo Endrenyi and Meyer Katzper. Ka is a
parameter obtained from models which are mathematical representations
of the average behavior of extremely complex factors such as
dissolution, active/passive transport, difussion, pH, flow-rates,
etc. Could Ka be function of time, pH, etc? Yes, it depends of how we
build the model. For instance we could build a model with no Ka at
all, if we considered the dissolution of a oral MR formulation rather
than the absorption as the rate limiting step of the overall process
for a drug having high water-solubility and permeabilty, meaning that
molecules are released so slowly that are absorbed immediately after
being transported to the absorption site.
Cmax would be a measure not only of the rate but also of the externt
of the absorption process. In this sense Cmax/AUC would seem to be a
better measure of the absorption rate. However, Cmax is obtained
directly from experimental data, thus it's a one-point parameter (AUC
is an average, we use all data for its calculation). Cmax depends on
the sampling schedule, the amount of points that define the peak.
Therefore, we would expect a bigger associated error of estimation.
Cmax/AUC would have the same drawback.
Regards
Dr. Marcelo Befumo
Osmotica Argentina
mbefumo.-at-.osmotica.com.ar
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The following message was posted to: PharmPK At 08:11 AM 3/19/02,
Andrew Sutton wrote:
>I would be interested to know if the absorption rate constant, Ka,
>mentioned by Laszlo Endrenyi and others recently has any relevance
>for efficacy . . .
>
This is a great discussion - and one that is long overdue.
The absorption rate coefficient certainly does have relevance to
efficacy - and safety.
The use of Ka as an average constant rather than as a time-varying
coefficient results in significantly different predicted effect for
most drugs. Consider a drug with pharmacodynamic effect that behaves
according to a simple sigmoid Emax effect model:
Effect = Emax * C^g / (EC50^g + C^g)
where C is the effect compartment concentration and g is a fitted
exponent. Let's assume the central compartment is the effect
compartment (so C=Cp).
Effect = Emax * Cp / (EC50 + Cp)
Variations in Cp (including, of course, Cmax) are directly affected by Ka.
Suppose you are doing research on a particular compound, you have IV
PK parameters and you have values for Emax and EC50. Now you'd like
to predict what dose to use in oral dosage forms. To simplify, assume
F=100%.
If you use a simple constant Ka at the average value for 24 hours,
you will underpredict Cp (and, of course, Cmax) for about 3-5 hours,
and then you will overpredict it after that. So your calculation of
effect during the first 5-6 hours will be too low. Your estimate of
Cmax will be also too low. You may decide to use a higher dose than
you should and possibly run into safety problems when the real Cmax
turns out to be much higher than you expected.
If your effect model is more complex, the problem gets more complex.
IMHO, constant Ka is an artifact of the days when computing power was
minimal and work had to be done with calculators and simpler methods.
The notebook computer I'm using is about 3,000 times faster than the
original IBM PC. On this machine, a typical 24-hour simulation for a
complex drug like midazolam, including all the effects of
dissolution, transit, absorption, nonlinear gut and hepatic
metabolism, and multi-compartment pharmacokinetics takes about two
seconds. It would have taken over 90 minutes on the original IBM PC -
the power available when constant Ka models were in their heyday.
The industry has recognized that today's tools and computing power
mean we don't need to oversimplify oral absorption any more. It is
complex - but it is manageable.
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (SIMU)
1220 W. Avenue J
Lancaster, CA 93534-2902
U.S.A.
http://www.simulations-plus.com
E-mail: walt.-at-.simulations-plus.com
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I think that you are right when you are speaking about simulation
mode. However when we deal with observed data from which we want to
retrieve the parameters of a model, we are limited about how complex
we can go in terms of modeling. We must find a compromise between the
potential bias inherent to model misspecification and the reliability
of the parameters we try to estimate. In other words you can always
simulate whatever you want but you cannot do the same once you fit
data to a model
Serge Guzy
Head of Pharmacometrics
Xoma
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[Three more replies - db]
From: Spiresgate.at.aol.com
Date: Wed, 20 Mar 2002 05:30:33 EST
To: david.aaa.boomer.org
Subject: PharmPK Re: Cmax/AUC - Diffusion etc
Dear Walt
Thanks for your reply; I do hope others will join in this discussion.
We obviously have a disagreement, but I am not sure I can see what the
disagreement really is.
If one does a simulation assuming the transfer is proportional to the
concentration gradient, the effect is exactly the same as a simulation which
assumes forward and backward transfers operating simultaneously, so no
disagreement there.
You use the expression net transfer from which I assume you recognise that
there are forward and backward processes resulting in a knet value, yet you
suggest movement ceases once the concentration gradient equals zero. Perhaps
someone would like to do the following experiments (I'm an armchair
pharmacokineticist these days):
Set up a membrane with drug in buffer on one side, buffer only on the other.
Monitor the change in concentration on each side. The result will be curves
tending towards equal concentration on each side; do we all agree?
Repeat the experiment with equal concentrations on each side but with
radiolabelled drug on one side. Monitor concentrations of drug and label.
The concentration of cold drug will be unchanged on either side. If
diffusion stops when the concentrations are equal, then the radiolabel will
stay where it is. If there is a dynamic equilibrium, then the radiolabel
curves will resemble those for cold drug in the first experiment.
Now to the problem of supposedly variable Ka.
You state that the rate of absorption is dependent on the concentration. No
argument there, this is the basis of most pharmacokinetic theory, that is the
rate of absorption is proportional to the concentration at the absorbing
site, or put another way:
dc/dt = K * c
so K is constant by definition.
You may well argue that absorption is more complex than just a single
process, but this isn't the point. You can make the simulation as complex as
you like by invoking multi-comparments with many k values (or other
parameters determining transfer rates), all the while maintaining constants
as constants.
Serge Guzy makes a similar point; pharmacokinetics has mainly concentrated on
the analytical side. Usually the problem is to tease from a limited amount
of data some parameters related to a description of the drug's behaviour and
the quality and quantity of data will not justify fitting the data to
anything more complex than a two or three-compartment model. Pharmacokinetic
synthesis on the other hand (and I think we are totally together on this)
seeks to assemble all the factors affecting (and effecting) drug transfer or
transformation with a view to prediction. It does seem to me though that
invoking "variable constants" is unnecessarily complicating the issue.
Time-dependency? I know you like analogies. Presumably your car is often
somewhere else some weekends but is safely back in Lancaster in time for work
on Monday mornings; is it's appearance in Lancaster time-dependent (that is,
is it in Lancaster because it's Monday) or is it because it was subject to a
number of forces and controls that conveyed it to Lancaster? (like models, no
analogy is perfect).
One genuine time-dependent phenomenon I would accept would be something like
gut motility; you could introduce a modifying factor to transfer of drug down
the gastrointestinal tract which is dependent on the time of day to reflect
the different phases of gut motility. Perhaps your excellent simulations
already do this?
best regards
Joe Chamberlain
---
From: phil.lowe.-at-.pharma.novartis.com
Date: Wed, 20 Mar 2002 12:30:27 +0100
To: david.-at-.boomer.org
Subject: Re: PharmPK Re: Cmax/AUC
Walt is correct in saying that the rate of absorption may affect an effect
parameter (toxic or efficacious), but there again it may not.
Consider an indirect response system where a drug affects a process which
is turning over slowly. Then, any peakiness in the plasma PK profile will
have limited effect on the PD process. In fact, the majority of the drug
could have left the system, but the effect "persists" due to the length of
time it takes to return to baseline. An extreme example would be a drug
which affects erythropoiesis, with the average 100 day life of an
erythrocyte. This apparent disconnect between presence of the drug and the
time course of effects confuses many people, who then say there is no link
between exposure and response.
Another train of thought is to consider the partition of drug into a tissue
effect site. If the kinetics of exchange are slow, then KA (or any further
parameterisation of KA) will have little sensitivity on tissue levels. The
system will appear to be well damped. This can occur for poorly perfused
tissue, or if there is some barrier to drug exchange (low
permeability*surface area coefficient).
It all depends on the PK and PD model, which should be a mathematical
representation of the mechanism of action of the drug through all its
stages from dose to effect.
Philip Lowe
Head of Modelling and Simulation
Preclinical Safety
Novartis Pharma AG
CH-4002 Basel
Switzerland
---
From: "Durisova Maria"
Date: Wed, 20 Mar 2002 13:11:59 +0100
To: david.aaa.boomer.org
Subject: Re: PharmPK Re: Cmax/AUC
The ratio Cmax/AUC is a number which determined very easily.
However this number cannot fully characterize such a complex
process as is drug bioavailability. This process can be fully
characterized
by the rate of bioavailability which is a function of time.
The integral of the rate of bioavailability from time zero to infinity
is the extent of bioavailability. Dividing the rate of bioavailability
by the extent of bioavailability yields the function called
the normalized rate of bioavailability. This function is not
influenced by the extent of bioavailability.
The criterion introduced in our study
Durisova M., Dedik L. Pharm. Res., 14, 1997, 860-864 can
be used to test similarity of two normalized rates of bioavailability of
a drug form two drug formulations in a bioequivalence study.
Regards,
Maria Durisova
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Dear Walt,
Your contribution with respect to the concept of Ka were quite
useful, and I have no comments on that. However, it remains unclear
what this has to do with bioequivalence and Cmax/AUC.
You wrote:
> The absorption rate coefficient certainly does have relevance to
> efficacy - and safety.
> The use of Ka as an average constant rather than as a time-varying
> coefficient results in significantly different predicted effect for
> most drugs.
Although this is fully correct, IMHO, you are looking into the wrong
direction with respect to the discussion on bioequivalence and
Cmax/AUC.
The interest in Cmax and Cmax/AUC is twofold: (#1) efficacy and
safety may be expected to be directly related to Cmax (Cmax/AUC
has a limited meaning, if any, in this case), and (#2) Cmax and/or
Cmax/AUC may be informative about the concentration profile, for a
given AUC (eg peak - trough difference). Cmax/AUC is preferred
here since it is not confounded by the extent of absorption (which is
fully covered by AUC).
In my opinion, #1 is the more important issue. This implies that Ka is
not relevant by itself; the absorption process may be very
complicated, but the only thing that counts is the peak concentration
(and of course AUC, but that is beyond discussion). So there is no
need to discuss about whether or not there is a single, constant Ka
or not. The only reason to talk about Ka is that rate of absorption is
usually expressed in a Ka value, just for clarity and simplicity. A fast
absorption implies a high Ka, and results in a high peak level. That's
all.
Your reasoning is about as follows: We need a measure for the rate
of absorption in bioequivalence, so 'others' are asking for a measure
for Ka. You say: 'Ka' does not exist since it is not a single constant.
Then somebody asks what is the relevance of Ka in bioequivalence,
and then you say that Ka has relevance for efficacy and safety
since it affects the plasma concentration profile (and thus Cmax),
but that the concept of Ka may introduce errors. This makes really
no sense in the discussion about Cmax/AUC in bioequivalence.
Please note that I do not say that your opinion about Ka itself makes
no sense. But it is simply a different (and equally interesting)
discussion.
Finally, we really do not need complicated PK modelling in
bioequivalence. And I would raise the statement that we cannot
obtain reliable parameters of more complicated absorption models
and a disposition model (which drug really obeys single-
compartment kinetics?) from, say, 15 plasma levels (including the
elimination phase), without an intravenous reference administration.
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
Email: j.h.proost.at.farm.rug.nl
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Dear Proost,
Ka is important here in the discussion since the FDA definition of
bioequivalence clearly mentions that the rate and extent of
absorption must be equivalent. extent of absorption can be explained
by AUC and partially by Cmax but what about rate?
with regards
Sanjay
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The following message was posted to: PharmPK Replies to several postings:
I hope everyone else is enjoying this discussion as much as I am -
it's one of the most lively discussions I've seen on PharmPK in a
while! The fopllowing responds to postings by Joe Chamberlain, Phil
Rowe, and Hans Proost:
At 08:14 AM 3/20/02, Joe Robinson wrote:
>You use the expression net transfer from which I assume you recognise that
>there are forward and backward processes resulting in a knet value, yet you
>suggest movement ceases once the concentration gradient equals zero.
>
Not really - I suggested that the *net* movement ceases, because the
rate of molecules moving in each direction are "equal". It seems to
me that it is only practical to consider net movement as absorption.
If there is no net transfer of drug, who cares if some molecules are
more or less exchanging positions with others on the other side of
the membrane?
>Set up a membrane with drug in buffer on one side, buffer only on the other.
>Monitor the change in concentration on each side. The result will be curves
>tending towards equal concentration on each side; do we all agree?
>
Yes - and the rate of change in concentration on both sides will
decrease with time as the concentrations become more and more equal.
When they are equal, the rate will be zero, but there will still be
molecules crossing in both directions.
>Repeat the experiment with equal concentrations on each side but with
>radiolabelled drug on one side. Monitor concentrations of drug and label.
>The concentration of cold drug will be unchanged on either side. If
>diffusion stops when the concentrations are equal, then the radiolabel will
>stay where it is. If there is a dynamic equilibrium, then the radiolabel
>curves will resemble those for cold drug in the first experiment.
>
I think there would be an "equal" amount of cold and radiolabeled
drug after a long enough time - similar to partial pressures in
gasses, each component should seek its own "equilibrium" distribution
(which is really a steady state transfer in both directions).
>You state that the rate of absorption is dependent on the concentration. No
>argument there, this is the basis of most pharmacokinetic theory, that is the
>rate of absorption is proportional to the concentration at the absorbing
>site, or put another way:
>
>dc/dt = K * c
>
>so K is constant by definition.
>
I don't see this as a definition - just an equation. The dissolution
rate equation I mention below does not "define" Kd as a constant, yet
we call it a rate constant.
Your equation is not Fick's Law, but is instead a variation that
assumes a perfect sink. It is equivalent to the way I've seen Ka
used, which is:
dMdiss/dt = - Ka * Mdiss
or
dMabs/dt = Ka * Mdiss
In a lumen compartment, the first equation can be divided by the
volume of the compartment to produce your equation:
dC/dt = Ka * C
But this applies only when there is no second concentration (i.e.,
perfect sink). When there is a second concentration, it should be
dC1/dt = -Ka * (C1 - C2)
which is
d(Mdiss1/V1)/dt = -Ka * (Mdiss1/V1 - Mdiss2/V2)
and if you multiply through by V1 you get
dMdiss1/dt = -Ka * (Mdiss1 - Mdiss2*V1/V2)
This Ka is what we like to call Ka' (Peff * absorption scale factor),
because it is not applied in the same way that Ka has traditionally
been applied (which is equivalent to your equation above):
dMdiss/dt = - Ka * Mdiss
The absorption scale factor is theoretically surface/volume, or 2/R,
where R is the effective volumetric radius of the compartment (i.e.,
the radius that produces the average cross-sectional area as pi*R^2).
NOTE: We also use the well-known and often-used Nernst-Brunner
equation (variation of Noyes-Whitney) for the dissolution rate
"constant" to calculate the rate of dissolution of undissolved drug
mass in a compartment:
Kd = 3 D (Cs - C) / (rho * r0 * T)
where D= diffusion coefficient, Cs=solubility (at local pH),
C=current concentration, rho=particle density, r0=initial particle
radius, and T= diffusion layer thickness.
Clearly, this "rate constant" is also time-varying, because C varies
with time. (In some methods, T also varies with time.) So when we say
"rate constant" it is, I think, a term that indicates a number with
units of 1/time that indicates the instantaneous fractional rate of
change of some variable. But I think "rate constant" does not
automatically imply constant value for all time.
>You may well argue that absorption is more complex than just a single
>process, but this isn't the point. You can make the simulation as complex as
>you like by invoking multi-comparments with many k values (or other
>parameters determining transfer rates), all the while maintaining constants
>as constants.
>
We would say the part that is constant is the Ka' term (Peff *
absorption scale factor) - although an argument can be made that some
drugs exhibit concentration-dependent Peff when there is saturable
carrier-mediated transport - but let's leave that out for now. My
point was and is that if you want to define Ka in the traditional
sense so that you can use the equation
dMdiss/dt = - Ka * Mdiss
rather than the equation
dMdiss/dt = - Ka' * V * (CLumen - CEnterocyte)
then in order to use the first equation, you would need a
time-dependent Ka to get the same results as the second equation.
That is what I have called the net Ka - the value you get when you
solve the equation for Ka
KaNet = - (dMdiss/dt) / (Mdiss)
or
KaNet = (dMabs/dt) / (Mdiss)
KaNet can be for one compartment, or summed over all compartments in
a multiple compartment gastrointestinal model. It is time-dependent
and is not the same as the traditional Ka.
[By the way, and this is another important (and possibly
controversial) point - we define absorption as crossing the apical
membrane - not as reaching the portal vein. I believe Gordon Amidon
defines it the same way. This becomes important when comparing
various methods of "predicting Fa" - different numbers may not
disagree if the definitions were different.]
>Serge Guzy makes a similar point; pharmacokinetics has mainly concentrated on
>the analytical side. Usually the problem is to tease from a limited amount
>of data some parameters related to a description of the drug's behaviour and
>the quality and quantity of data will not justify fitting the data to
>anything more complex than a two or three-compartment model. Pharmacokinetic
>synthesis on the other hand (and I think we are totally together on this)
>seeks to assemble all the factors affecting (and effecting) drug transfer or
>transformation with a view to prediction. It does seem to me though that
>invoking "variable constants" is unnecessarily complicating the issue.
>
This has not been our experience after working with a large number of
companies with a wide variety of compounds over the past several
years.
We have been able to fit reasonable models using the 9-compartment
ACAT (Advanced Compartmental Absorption and Transit) model to
relatively limited amounts of data - including cases where there were
no IV data (of course, it's much more tricky, and we don't always
succeed in developing a single model, but you'd be amazed [as I was]
at how well it works most of the time). With rare exception, we use
all of the default values for transit times, compartment volumes,
enterocyte volumes, compartment pH's, blood flow rates, etc. that
define the physiological model (i.e., we do not adjust any of these
many parameters, so although a number of compartments are involved,
there is only one model).
When we also have IV data for PK parameters, we've been very
successful in predicting oral dose behavior -- including saturable
gut and hepatic metabolism using in vitro Vmax and Km values along
with all the other complications of low solubility, dissolution rate,
concentration-gradient based absorption, etc. This is not even
possible with a simple constant Ka absorption model (because you need
enterocyte concentrations to calculate gut metabolism rate).
>One genuine time-dependent phenomenon I would accept would be something like
>gut motility; you could introduce a modifying factor to transfer of drug down
>the gastrointestinal tract which is dependent on the time of day to reflect
>the different phases of gut motility. Perhaps your excellent simulations
>already do this?
>
We have not found the need to do this (yet?). The literature seems to
indicate that although gastric emptying time has significant
within-individual variation, small intestine and colon transit times
are not much affected by food, etc. Perhaps by disease-, food- or
drug-induced motility, but for "normal" foods, not much effect. So we
have just used our default values, and haven't found the need to
modify them in the vast majority of cases.
>From: phil.lowe.aaa.pharma.novartis.com
>Walt is correct in saying that the rate of absorption may affect an effect
>parameter (toxic or efficacious), but there again it may not.
>
I totally agree with Phil, and the further examples he provided and
that was why I couched my statement with "The use of Ka as an average
constant rather than as a time-varying coefficient results in
significantly different predicted effect for most drugs." I used the
phrase "most drugs" in order to allow for exceptions like the ones
Phil mentioned.
>Hans Proost wrote:
>Your contribution with respect to the concept of Ka were quite
>useful, and I have no comments on that. However, it remains unclear
>what this has to do with bioequivalence and Cmax/AUC.
I've been thinking the same thing, and I almost suggested to David
that he start a new thread with a different subject line, but the
momentum was already established.
My original posting was in response to Peter Mullen's original
comment that: "Compared with Cmax, Cmax/AUC better correlates with,
and has the same units as, the underlying rate parameter, Ka."
I took that to be an implication that there was such a thing as a
single, constant-valued Ka. So the thread sort of took of on a
tangent.
Sure has been fun, though, eh?
Walt
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (SIMU)
1220 W. Avenue J
Lancaster, CA 93534-2902
U.S.A.
http://www.simulations-plus.com
E-mail: walt.-a-.simulations-plus.com
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Dear Dr. Patel,
You wrote:
> Ka is important here in the discussion since the FDA definition of
> bioequivalence clearly mentions that the rate and extent of
> absorption must be equivalent. extent of absorption can be explained
> by AUC and partially by Cmax but what about rate?
You are right that the rate of absorption is important. However, my
comment referred to the question that we do not need a Ka. We
need a reliable measure of the rate of absorption (and Cmax/AUC is
probably the best we have), but we do not need to know whether or
not the process behaves as a first-order process or not, nor we
need to know a parameter describing this process directly such as
Ka. That was my point.
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
Email: j.h.proost.at.farm.rug.nl
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