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Colleagues:
I have bee struggling with describing the results of a deconvolution
analysis in an efficient understandable manner. If I fit the %
absorbed from the deconvolution to a weibull function. specifically
the Weibull as parameterized by Zhou in J Clin Pharmacol 2003.
frac input = F*(1-f *e(-ka * (t^gamma)))
where F = bioavailability, 1-f is fraction of drug abs, ka = rate
constant of absorption, and gamma is a shape parameter , as gamma
increases the abs curve becomes asymptotic instead of sigmoidal.
What is the physiologic interpretation (related to in-vivo drug
absorption) of the parameters in this version of the Weibull function?
What is the advantage of fitting a Weibull model Vs simply describing
the duration of absorption and perhaps the absorption t50%? Is there a
more appropriate model to use? The data does not fit simple first
order absorption.
I also note there are several versions of the Weibull function such
as this one published by the same author in J Clin Pharmacol 1999
%absorbed = AB*(1-e(-ka*t)^gamma)
Where AB = bioavailability, ka is apparent absorption rate constant
and gamma is a shaping factor
William R. Wolowich, Pharm.D., R.Ph.
Chair and Assistant Professor
Department of Pharmacy Practice
College of Pharmacy
Nova Southeastern University
wwolowic.-at-.nsu.nova.edu
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William,
The Weibull is a mathematical figment of the imagination. It has no
physiological interpretation when gamma is not 1. Even the first order
absorption interpretation (gamma=1) is a gross simplification of
reality...
However, like the sigmoidal Emax model, the empirical application of a
shape parameter (gamma) may be be helpful for descriptive and perhaps
predictive use of models.
Note that bioavailability is not part of the Weibull function. It has
been added by authors who wish to show the relationship to standard PK
models with first order oral absorption.
Nick
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New
Zealand
n.holford.at.auckland.ac.nz
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
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The following message was posted to: PharmPK
Dear William
The Weibull function or the Rosin Ramler Sperling Bennett Weibull
Distribution function, to give it its full name, was originally used
by Langenbucker to describe in vitro dissolution data. I published a
manuscript in 1988:
On the absorption of clavulanic acid, Biopharmaceut Drug Dispo; 9:
127, which may have been the first reference to the use of this
function for in vivo data - you may find this of interest.
regards
brian
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Dear William,
(1) Physiological interpretation of Weibull
When Weibull distribution function is applied to drug absorption
kinetics, it is used to describe the probability distribution function
(pdf) of "mean residence time (MRT) of drug molecules in the gut". The
cumulative density function (cdf) of it would be then "the probability
that MRT of drug molecules are shorter than time t". "MRT in the gut
is shorter than time t" means that drug molecules are to be escaped
from the gut before time t.
Now then, the physiological application of this cdf can be: The
probability that the drug molecules have escaped from the gut before
time t. This probability (cdf) is applied to be the fraction (or
percentage) of drug absorbed.
(2) Applied equations in absorption kinetics
The cdf is expressed as 1-exp(-(t/alpha)^beta) in the Weibull
distribution, where alpha is a scale parameter and beta is a shape
parameter and these parameters determine the curve characteristics.
Since 1-exp(-(t/alpha)^beta) is the fraction of drug absorbed as
mentioned in (1),
the amount of drug absorbed =F*D*(1-exp(-(t/alpha)^beta)).
then the amount unabsorbed (remaining in the gut, A1) would be
A1= F*D*exp(-(t/alpha)^beta)
Differentiation of this leads to
dA1/dt = -F*D*(beta/alpha)*(t/alpha)^(beta-1)*exp(-(t/alpha)^beta)
If you rewrite the equation above using A1,
dA1/dt = -A1*(beta/alpha)*(t/alpha)^(beta-1)
The differential equation for A1 above indicates that the coefficient
of A1 also changes over time in Weibull, while the coefficient of A1
is a constant (ka) for the first order absorption kinetics. This is
the power of Weibull for compounds with atypical absorption
characteristics.
However, we often find different forms of Weibull equations from
literature, like Zhou's. These modified equations have ka or some
other variables empirically plugged into Weibull replacing 1/alpha or
else. You will be able to find quite different equations from the
references appeared in Zhou's paper as well. However, I have not yet
quite understood how they were derived and justified. If anyone can
help me find that out, it will be greatly appreciated.
If these equations confuse you even further, please accept my apology.
However, for better understanding, I would like you to go back to the
original paper where Weibull was first applied to in vivo kinetics by
Piotrovskii (JPB 1987), and then compare it with other modified
equations.
I hope it helps.
Thanks,
Jee Eun
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The following message was posted to: PharmPK
Dear Dr. Wolowich,
Your e-mail brings up a couple of important issues.
First, please be careful about the definition of "absorption". The early
Wagner-Nelson and Loo-Riegelman papers use the word absorption to
refer to
the amount of drug reflected in the plasma concentration-time data,
not what
is referred to by the modern definition of absorption, i.e., the
amount of
drug that leaves the lumen and enters the enterocytes. Thus when early
methods deconvoluted what they called absorption vs time it was really
systemic availability vs time. Those methods have no way to deconvolute
absorption unless bioavailability is 100%, in which case they are equal.
Next, there is no such thing as constant absorption rate in reality.
Absorption rate is time-dependent, so Ka is not a constant, but is a
time-dependent absorption rate _coefficient_ that changes as the drug is
dissolved and absorbed from the lumen into the enterocytes.
From Fick's First Law, we know that for passive diffusion, the rate of
movement of molecules across a membrane is proportional to the
difference in
concentrations on either side. In the intestinal lumen, at time 0,
there is
no concentration gradient, so the absorption rate is zero. As the drug
goes
into solution (or drug in solution moves from stomach into small
intestine)
the absorption rate increases rapidly until sufficient drug is
absorbed to
cause the concentration within the enterocytes to rise, decreasing the
differential concentration across the membrane, and therefore
decreasing Ka.
If you think of the intestinal tract as divided into a series of
compartments as in the ACAT (Advanced Compartmental Absorption and
Transit)
model, you can imagine that the drug in solution as well as undissolved
particles gradually become distributed through these compartments, and
within each there is a different environment (pH, fluid volume, surface
area, etc.). So Ka that changes both with time and with location.
There are times when you can get away with using a constant Ka (for
example,
a high solubility drug completely in solution and rapidly absorbed).
There
are many more when it is a gross over-simplification. Mechanistic
modeling
with appropriate inputs can help you estimate how the complex
interplay of a
number of factors affects the absorption of more complex drugs/
formulations.
Best regards,
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (NASDAQ: SLP)
42505 10th Street West
Lancaster, CA 93534-7059
U.S.A.
http://www.simulations-plus.com
E-mail: walt.-at-.simulations-plus.com
[Just because the absorption rate changes with amount remaining to be
absorbed the value of the absorption rate constant, ka, the rate
constant can often be treated as (relatively) constant. However, as
with all pharmacokinetic parameters the ka may change with various
factors such as GI physiology, solubility, pH, etc. Not sure what Ka
is in this context - db]
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Dear Walt,
Vey nice information on absorption. Even I believe so. A step ahead of
the Wagner-Nelson and Loo-Riegelman papers definition of absorption,
there is rare report on the constant Ka even for the highly soluble
drugs and there will be huge proflies different in such kind of
compound as they will rather diversly disperse into the ACAT based
microenvironment localities and the difference in the absorption rates
comes there, while for the other compounds with the depot release
portfolio it is pure Fick's law mediated concentration gradient
dependent as the fraction of drugs get lesser diverse intragastrial
exposure.
So, there is not a rule of thumb for the conditions you can obtain for
almost constant Ka if you exclude the systematic availability as the
solo parameter.
Thanks and Best Regards,
Amrit
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Dear Walt,
In your explanation of absorption based on the Fick's law you are
confusing the rate of absorption to the rate constant:
"the absorption rate increases rapidly until sufficient drug is
absorbed to cause the concentration within the enterocytes to rise,
decreasing the differential concentration across the membrane, and
therefore decreasing Ka."
The correct end of the sentence is "and therefore decreasing the rate
of absorption". However the rate constant, Ka, remain constant.
radu
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Dear Radu,
I'm not confusing anything. Ka is never a constant. Never!
That is a simplification that you can get away with on occasion, but
is never actually true. It is often (mis)used as a number that
multiplies the total amount of drug in solution (regardless of where
it is located in the gut, and ignoring the resistance term of the
concentration on the other side of the membrane). So this use assumes
that the concentrations in all parts of the gut are the same, and the
surface areas are the same. If the drug is absorbed by the
paracellular route, it assumes the tight junctions are the same.
Clearly, this is not the case in any animal.
If you back out the value of Ka that would be needed to correctly
model absorption from the total amount of drug in solution in all the
regions of the gastrointestinal tract, and plot it vs time, you'll see
something that behaves the way I described - starting at zero,
increasing to a peak, and decreasing, with a shape similar to a
concentration-time curve.
This is done at each point in time by calculating the absorption rate
at any time using Fick's First Law in each compartment, summing them
to get the total absorption rate, and then dividing by the total mass
in solution in the traditional sense.
Ka = Sum(dAbs/dt)/M(solution)
GastroPlus provides this output to allow users to understand this
behavior.
Best regards,
Walt
[absorption rate IS NOT the same as absorption rate constant - db]
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