# PharmPK Discussion - Weibull absorption models

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• On 22 Jun 2009 at 12:28:32, William Wolowich (wwolowic.-at-.nova.edu) sent the message
`Colleagues:I have bee struggling with describing the results of a deconvolutionanalysis in an efficient understandable manner. If I fit the %absorbed from the deconvolution to a weibull function. specificallythe 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 = rateconstant of absorption, and gamma is a shape parameter , as gammaincreases the abs curve becomes asymptotic instead of sigmoidal.What is the physiologic interpretation (related to in-vivo drugabsorption) of the parameters in this version of the Weibull function?What is the advantage of fitting a Weibull model Vs simply describingthe duration of absorption and perhaps the absorption t50%? Is there amore appropriate model to use? The data does not fit simple firstorder absorption.  I also note there are several versions of the Weibull function suchas 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 constantand gamma is a shaping factorWilliam R. Wolowich, Pharm.D., R.Ph.Chair and Assistant ProfessorDepartment of Pharmacy PracticeCollege of PharmacyNova Southeastern Universitywwolowic.-at-.nsu.nova.edu`
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• On 22 Jun 2009 at 19:50:58, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message
`William,The Weibull is a mathematical figment of the imagination. It has nophysiological interpretation when gamma is not 1. Even the first orderabsorption interpretation (gamma=1) is a gross simplification ofreality...However, like the sigmoidal Emax model, the empirical application of ashape parameter (gamma) may be be helpful for descriptive and perhapspredictive use of models.Note that bioavailability is not part of the Weibull function. It hasbeen added by authors who wish to show the relationship to standard PKmodels with first order oral absorption.Nick--Nick Holford, Dept Pharmacology & Clinical PharmacologyUniversity of Auckland, 85 Park Rd, Private Bag 92019, Auckland, NewZealandn.holford.at.auckland.ac.nzhttp://www.fmhs.auckland.ac.nz/sms/pharmacology/holford`
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• On 22 Jun 2009 at 14:45:16, "Davies, Brian" (brian.davies.-at-.roche.com) sent the message
`The following message was posted to: PharmPKDear WilliamThe Weibull function or the Rosin Ramler Sperling Bennett WeibullDistribution function, to give it its full name, was originally usedby Langenbucker to describe in vitro dissolution data.  I published amanuscript in 1988:On the absorption of clavulanic acid, Biopharmaceut Drug Dispo; 9:127, which may have been the first reference to the use of thisfunction for in vivo data - you may find this of interest.regardsbrian`
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• On 22 Jun 2009 at 16:46:34, JeeEun_Lee.at.vrtx.com sent the message
`Dear William,(1) Physiological interpretation of WeibullWhen Weibull distribution function is applied to drug absorptionkinetics, it is used to describe the probability distribution function(pdf) of "mean residence time (MRT) of drug molecules in the gut". Thecumulative density function (cdf) of it would be then "the probabilitythat MRT of drug molecules are shorter than time t". "MRT in the gutis shorter than time t" means that drug molecules are to be escapedfrom the gut before time t.Now then, the physiological application of this cdf can be:  Theprobability that the drug molecules have escaped from the gut beforetime t. This probability (cdf) is applied to be the fraction (orpercentage) of drug absorbed.(2) Applied equations in absorption kineticsThe cdf is expressed as 1-exp(-(t/alpha)^beta) in the Weibulldistribution, where alpha is a scale parameter and beta is a shapeparameter and these parameters determine the curve characteristics.Since 1-exp(-(t/alpha)^beta) is the fraction of drug absorbed asmentioned in (1),the amount of drug absorbed =F*D*(1-exp(-(t/alpha)^beta)).then the amount unabsorbed (remaining in the gut, A1) would beA1= F*D*exp(-(t/alpha)^beta)Differentiation of this leads todA1/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 coefficientof A1 also changes over time in Weibull, while the coefficient of A1is a constant (ka) for the first order absorption kinetics. This isthe power of Weibull for compounds with atypical absorptioncharacteristics.However, we often find different forms of Weibull equations fromliterature, like Zhou's. These modified equations have ka or someother variables empirically plugged into Weibull replacing 1/alpha orelse. You will be able to find quite different equations from thereferences appeared in Zhou's paper as well. However, I have not yetquite understood how they were derived and justified. If anyone canhelp 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 theoriginal paper where Weibull was first applied to in vivo kinetics byPiotrovskii (JPB 1987), and then compare it with other modifiedequations.I hope it helps.Thanks,Jee Eun`
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• On 22 Jun 2009 at 21:00:40, "Walt Woltosz" (walt.aaa.simulations-plus.com) sent the message
`The following message was posted to: PharmPKDear Dr. Wolowich,Your e-mail brings up a couple of important issues.First, please be careful about the definition of "absorption". The earlyWagner-Nelson and Loo-Riegelman papers use the word absorption torefer tothe amount of drug reflected in the plasma concentration-time data,not whatis referred to by the modern definition of absorption, i.e., theamount ofdrug that leaves the lumen and enters the enterocytes. Thus when earlymethods deconvoluted what they called absorption vs time it was reallysystemic availability vs time. Those methods have no way to deconvoluteabsorption 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 atime-dependent absorption rate _coefficient_ that changes as the drug isdissolved and absorbed from the lumen into the enterocytes. From Fick's First Law, we know that for passive diffusion, the rate ofmovement of molecules across a membrane is proportional to thedifference inconcentrations on either side. In the intestinal lumen, at time 0,there isno concentration gradient, so the absorption rate is zero. As the druggoesinto solution (or drug in solution moves from stomach into smallintestine)the absorption rate increases rapidly until sufficient drug isabsorbed tocause the concentration within the enterocytes to rise, decreasing thedifferential concentration across the membrane, and thereforedecreasing Ka.If you think of the intestinal tract as divided into a series ofcompartments as in the ACAT (Advanced Compartmental Absorption andTransit)model, you can imagine that the drug in solution as well as undissolvedparticles gradually become distributed through these compartments, andwithin each there is a different environment (pH, fluid volume, surfacearea, etc.). So Ka that changes both with time and with location.There are times when you can get away with using a constant Ka (forexample,a high solubility drug completely in solution and rapidly absorbed).Thereare many more when it is a gross over-simplification. Mechanisticmodelingwith appropriate inputs can help you estimate how the complexinterplay of anumber of factors affects the absorption of more complex drugs/formulations.Best regards,Walt WoltoszChairman & CEOSimulations Plus, Inc. (NASDAQ: SLP)42505 10th Street WestLancaster, CA  93534-7059U.S.A.http://www.simulations-plus.comE-mail: walt.-at-.simulations-plus.com[Just because the absorption rate changes with amount remaining to beabsorbed the value of the absorption rate constant, ka, the rateconstant can often be treated as (relatively) constant. However, aswith all pharmacokinetic parameters the ka may change with variousfactors such as GI physiology, solubility, pH, etc. Not sure what Kais in this context - db]`
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• On 23 Jun 2009 at 07:25:51, "amrit paudel.at.gmail.com" (amrit.niper.-at-.gmail.com) sent the message
`Dear Walt,Vey nice information on absorption. Even I believe so. A step ahead ofthe Wagner-Nelson and Loo-Riegelman papers definition of absorption,there is rare report on the constant Ka even for the highly solubledrugs and  there will be huge proflies different in such kind ofcompound as they will rather diversly disperse into the ACAT basedmicroenvironment localities and the difference in the absorption ratescomes there, while for the other compounds with the depot releaseportfolio it is pure Fick's law mediated concentration gradientdependent as the fraction of drugs get lesser diverse intragastrialexposure.So, there is not a rule of thumb for the conditions you can obtain foralmost constant Ka if you exclude the systematic availability as thesolo parameter.Thanks and Best Regards,Amrit`
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• On 23 Jun 2009 at 10:56:55, "Radu Pop" (rpop.aaa.pharmamedica.com) sent the message
`Dear Walt,In your explanation of absorption based on the Fick's law you areconfusing the rate of absorption to the rate constant:"the absorption rate increases rapidly until sufficient drug isabsorbed to cause the concentration within the enterocytes to rise,decreasing the differential concentration across the membrane, andtherefore decreasing Ka."The correct end of the sentence is "and therefore decreasing the rateof absorption". However the rate constant, Ka, remain constant.radu`
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• On 23 Jun 2009 at 09:30:05, "Walt Woltosz" (walt.at.simulations-plus.com) sent the message
`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, butis never actually true. It is often (mis)used as a number thatmultiplies the total amount of drug in solution (regardless of whereit is located in the gut, and ignoring the resistance term of theconcentration on the other side of the membrane). So this use assumesthat the concentrations in all parts of the gut are the same, and thesurface areas are the same. If the drug is absorbed by theparacellular 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 correctlymodel absorption from the total amount of drug in solution in all theregions of the gastrointestinal tract, and plot it vs time, you'll seesomething that behaves the way I described - starting at zero,increasing to a peak, and decreasing, with a shape similar to aconcentration-time curve.This is done at each point in time by calculating the absorption rateat any time using Fick's First Law in each compartment, summing themto get the total absorption rate, and then dividing by the total massin solution in the traditional sense.             Ka = Sum(dAbs/dt)/M(solution)GastroPlus provides this output to allow users to understand thisbehavior.Best regards,Walt[absorption rate IS NOT the same as absorption rate constant - db]`
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