# PharmPK Discussion - Relationship between AUC and Vd

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• On 24 Jan 2011 at 14:40:44, Ruby Mosher (mosherru.-at-.ksu.edu) sent the message
`I'm having trouble wrapping my head around this question from an online  pharmacology quiz.  The question is "The larger the volume of  distribution, the smaller the AUC of a given drug."  The answer is given  as "False." As I look at the equation AUC = D/KV, it would seem that if V is  larger, then AUC would be smaller.  Can anyone explain how the question is false instead of true? Thanks,Ruby`
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• On 24 Jan 2011 at 17:01:31, Anila Desai (adesai.-a-.ironwoodpharma.com) sent the message
`The following message was posted to: PharmPKIs it autoinduction?Regards,Anila`
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• On 25 Jan 2011 at 11:17:22, Nick Holford (n.holford.at.auckland.ac.nz) sent the message
`Ruby> I'm having trouble wrapping my head around this question from an  online pharmacology quiz.  The question is "The larger the volume of  distribution, the smaller the AUC of a given drug."  The answer is given  as "False.">> As I look at the equation AUC = D/KV, it would seem that if V is  larger, then AUC would be smaller.>> Can anyone explain how the question is false instead of true?This is a classic misunderstanding  by using maths and not biology to understand reality. The AUC is determined by dose and Clearance (CL).AUC = Dose/CL  Dose is real. Clearance is abstract but nevertheless directly linked to the ability to eliminate a drug which is real. Volume (V) is also abstract but directly related to how much drug is in the body and therefore real.On the other hand the so called 'rate constant' (K)  is a mathematical abstraction determined by the ratio of Clearance/Volume. i.e.K=CL/VThis biological relationship determining K can be re-arranged mathematically to come up withCL=K * VBut this does not make biological sense because K is determined by CL not the other way around. Unfortunately in some textbooks it may be written thatAUC=Dose/(K * V)but this is maths not biology and is quite misleading because it implies that a change in K will mean a change in AUC.Despite it's name, K is not constant. If either Clearance or Volume changes then K will change. It is possible to have a change in V which will change K but because CL does not change the AUC will not change.Perhaps you should contact the author of the source of your equation 'AUC=D/KV' and ask them to re-write the equation in a more sensible form. Feel free to use this email to support this view.Best wishes,Nick`
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• On 24 Jan 2011 at 14:21:26, Toufigh Gordi (tgordi.at.yahoo.com) sent the message
`Hi Ruby,The actual formula is AUC = Dose / CL. Now, since CL = k * Vd, you  can replace the CL and get the formula you refer to. However, you have  to remember that if CL is constant, when Vd changes so does the k. In  other words, in your formula, an increase/decrease in Vd results in a  corresponding decrease/increase in k so the CL, and thereby AUC is  unchanged.ToufighToufigh Gordi, PhDPresident, PK/PD and Clinical Pharmacology ServicesRosa & Co. LLC (www.rosaandco.com)E-mail: tgordi.at.rosaandco.com`
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• On 24 Jan 2011 at 17:34:13, Andrew Volosov (avolosov.-at-.gmail.com) sent the message
`Ruby,The important thing to remember is that V and CL are independent of each  other, and AUC = D/CL; therefore, AUC depends on CL, not on V. In a  (very) simplified case of a perfectly mono-exponential decline after IV  dosing, K = CL/V, which means that K depends both on CL and V.  Rearranging this equation to CL = KV gives the impression that CL  depends on V, hence the confusion. The increase in V will not increase  CL - rather, it will decrease K.Andrew`
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• On 24 Jan 2011 at 16:46:06, Sagar Agarwal (agar0077.-at-.umn.edu) sent the message
`The following message was posted to: PharmPKDear Ruby,The AUC is equal to Dose/Clearance and not Dose/KV. The reason I say thisis, is because even though clearance and volume are related to each other bythe equation K = CL/V, clearance is not dependent on volume. So even if thevolume changes, the clearance will not change and hence the AUC will remainthe same. Change in volume will only change K, not CL.Hope this helps.Sagar--Sagar AgarwalGraduate Student - Ph.D. Candidate3-140 WDH, 308 Harvard Street SEMinneapolis, MN 55455`
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• On 24 Jan 2011 at 22:56:04, Ronette Gehring (rgehring.-at-.vet.k-state.edu) sent the message
`The following message was posted to: PharmPKHi Ruby,Remember that K is also dependent on V. So if V goes up, K will decrease  proportionately and AUC stays the same!Ronette`
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• On 27 Jan 2011 at 03:27:48, wxmwqr (wxmwqr.aaa.163.com) sent the message
`I do not agree with that the k is dependent on the CL. And V can affect  the k. For the compartment model of iv. Bolus, the TBCL=kV (k is the  elimination rate constant, and the V is the apparent volume of  distribution). We can understand the K and V  is the parameter for physiological  condition of body. 1, K is the parameter that can express the physiological condition of  central compartment, such as the hepatic clearance, renal clearance et  al. So we can say the k is the physiological parameter of elimination. 2, And the V is another physiological parameter (if I can say). It is  co-related with the body weight et al. We can say the V is the  physiology parameter of distribution. And we assume that the V and k is independent to each other. In this equation the TBCL is the dependent parameter and the k and V are  the independent parameter. That means, the change of TBCL depend on (or  can say "because of") the change of k or V, or both of them.   In  the equation the AUC=X0/(kV), we can know that the AUC only  depends on the k and V., AUC  do not has the direct relationship with  the CL.So we can say the K and v is the primary parameter of PK, otherwise the  CL and AUC is the secondary parameter of PK.This question has two sides.1, if we assume that the k does not change (the physiological condition  of elimination does not change), V increase, the AUC should decrease.If we assume that the V does not change, the k increase, the AUC should  decrease also.2, If the k and V change at same time. The changing of  AUC will depend  on the (kV)xiaoming`
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• On 27 Jan 2011 at 08:22:48, Des Williams (Des.Williams.at.unisa.edu.au) sent the message
`It is unfortunate that misunderstanding continues for the relations that  involve clearance, volume of distribution, AUC, elimination rate  constant and elimination half-life.See Nick Holford's concise summary (25 January 2011) that has the  correct interpretation in response to earlier correspondence (see  attached). The elimination rate constant always depends on both  clearance and volume.Again to summarise, clearance and V are both primary pharmacokinetic  parameters that depend on the chemical properties of the drug and the  composition of the body in which the drug circulates. The elimination  rate constant, k, is not a primary pharmacokinetic function and is  derived from k = Cl/V. Clearly, from this relationship, k depends on  both Cl and V. We measure log(C) vs time to estimate k and also measure  AUC from plasma drug concentration vs time profiles, but that does not  make k a primary pharmacokinetic function. The observed plasma drug  profiles result from the specific values of Cl and V for a specific  drug. It is an excessively common mistake to call k a primary  pharmacokinetic function and it would be good for the scientific and  clinical community if this mistake could be at least minimised,  preferably eliminated. (The half-life for elimination of this mistake is  slow. Perhaps we have reached a steady-state situation.)Finally, if Cl goes down, often from reduced liver function, or  competition for metabolising enzymes, or from reduced renal clearance  then k will decrease if V remains constant. If Cl goes down and V  decreases to roughly the same proportion, then k, hence half-life, will  not change significantly. Other scenarios are easy to imagine.Des Williams--Des Williams, PhD, BPharm, FRACI, CChem Program Director, Pharmaceutical Sciences Sansom Institute for Health Research, and School of Pharmacy and Medical Sciences | CEA19University of South Australia North Terrace Adelaide`
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• On 28 Jan 2011 at 00:44:06, wxmwqr (wxmwqr.at.163.com) sent the message
`Thanks Des for the clearly explanation.I had processed a set of date for a friend. The result was interesting.He would like to study the bioavailability of old pharmaceutical form  (Oral Suspension) and new form (Granules). He did the experiment  according to the guidance of SFDA for the studying of bioavailability.The data of old form can be simulated by using one compartment model.   The data of the new form is interesting. Data of some animals is fitted  with one compartment model, and data of others can be fitted with the  two compartment model. Cmax, ka and half-life decrease but not so much;  K decrease and V increase.  Because of the k and V, the AUC and the CL  did not have the significantly change.  If we just see the CL and AUC,  the new form of the drug is not different from the old form. However,  the K and V changed.I thought that some of the excipients of the new form maybe affects the  clearance of drug. And some of the excipient can help the drug  distribute extensively, that can explain why some of the animal has the  distribution phase (two compartment model).  Consider the the drug can  be extensively metabolized by CYP, I suggested him to investigate the  inhibition between the excipient and the drug by using in-vitro method.For this case, I think that the k and V are more "sensitive" than the CL  and AUC when the drug property or body physiological condition changed.  CL and AUC are depending on the k and V.Xiaoming`
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• On 27 Jan 2011 at 11:59:15, "Pereira, Luis" (Luis.Pereira.-at-.childrens.harvard.edu) sent the message
`The following message was posted to: PharmPKDear AllIt's certainly very unfortunate to perpetuate this misunderstanding and  persist in arguing whether the egg or the chicken comes first. Mainly  for the elucidation of the new generations, I strongly advice reading  the original theoretical foundations of PK modeling. It's false to say  that k is derived from k=CL/V. This expression only applies to a  single compartment disposition. It's false to say that "the elimination  rate constant always depends on both clearance and volume". Truthfully,  every first order process has a rate proportional to its driving force,  with a proportionality constant denoted as rate constant, regardless of  the volume or clearance concepts (e.g. radioactive decay). In terms of  mass transfer, e.g. drug elimination, for a first order process, the  rate is proportional to the amount driving the process, regardless of  the APPARENT volume we choose to assess it. If we decide to  conceptualize that driving force in terms of concentration, which given  biological heterogeneity require`
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• On 28 Jan 2011 at 07:42:47, Stefan Soback (stefan.soback.at.gmail.com) sent the message
`First of all I think the original question was poorly worded and,  therefore, any answer can be accepted. In order to make things clear,  let's consider only Vss.We are obviously talking strictly about IV administration, as Vss cannot  be determined from extravascular administration data (and neither can  Vss/F, contrary to a number of papers claiming that). After IV  administration vast majority of drugs are best described by  multicompartment models (as Luis accurately pointed out).As we know, K = CL/Vss is correct only for one compartment model (Vss  = CL * 1/K) and the equation for any compartment model, which we  obviously have to use here, is Vss = CL * MRT. Maybe it is now easier  to understand that the MRT is not a constant determining the Vss or the  CL.It is also easy to see that if AUC changes, i.e. CL decreases or  increases for instance because of enzyme inhibition or induction, the  MRT will change, not the Vss. Regards, Stefan`
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• On 28 Jan 2011 at 15:27:15, "J.H. Proost" (j.h.proost.-at-.rug.nl) sent the message
`The following message was posted to: PharmPKDear Luis,You seem to have missed the long discussion 'A question of clearance' onthis forum from December 2009 to February 2010 between the supporters of the'rate constant approach' and the 'clearance approach'. I suggest to readthat discussion carefully. You will read that all arguments in favor of therate constant approach were shown to be weak or incorrect, and that noconvincing arguments were given contradicting the clearance approach. Therecan be no doubt that the 'frivolous contest' was won by the clearanceapproach.In addition, you may read the recent excellent contributions of  NickHolford and Des Williams, who very clearly explained why k is dependent onV, and why CL is independent of V.You wrote:> Mainly for the elucidation of the new generations, I strongly advice> reading the original theoretical foundations of PK modeling. It's false to> say that k is derived from k=CL/V.The original theoretical foundations were very useful and very important,but not fully correct. The original PK started in the 1930s, and theclearance concept entered PK in the 1970s. This has led to a much betterunderstanding of PK, in particular with respect to clinicalpharmacokinetics. Keeping on original concepts does not seem to be fruitful.> This expression only applies to a single compartment disposition.Yes and no. It also applies to multicompartment models if written as k10 =CL/V1. Moreover, the problem is here in k (the 'toy' of the rate constantsupporters) and V, not in CL.> In terms of mass transfer, e.g. drug elimination, for a first order> process, the rate is proportional to the amount driving the process,> regardless of the APPARENT volume we choose to assess it.The driving force is not amount of drug, but concentration. To say itsimple: How does your liver and kidneys know how much drug is present inyour body? They only 'feel' the blood concentration.> If we decide to conceptualize that driving force in terms of> concentration, which given biological heterogeneity requires the> assumption of an APPARENT volume, then for a first order process, the> proportionality constant between rate and concentration is defined as> clearance.Here we agree. This is indeed the basis of the clearance concept.> It's the fraction of that APPARENT volume that gets cleared of drug per> unit time.No, cleareance is not the fraction of the apparent volume that gets cleared;The rate constant is the fraction of the apparent volume that gets cleared(and therefore it is dependent of that apparent volume).> I hope that the capitalizations above extol the fact that the blood> concentrations we work with daily, are really a surrogate measure of the> underlying amount of drug driving the elimination process.Yes, but again,  the driving force is not amount of drug, but concentration.See above. Blood concentration is not a 'surrogate measure'.> Consequently, AUC is just the integral of the concentrations profile over> time. Again, only under specific modeling assumptions, and data> availability, it may be estimated as D/CL, or Co/k, or D/kV, or> Co.t1/2/ln2, and on, and on.The only assumption for AUC = F*Dose/CL is that linear kinetics apply (as inthe whole discussion), irrespective of the model. It is CL=kV that is trueonly for a one-compartment model.> I will not participate in the frivolous contest for which parameters are> primary and which are secondary.You actually participated in the contest with your statement "It's false tosay that k is derived from k=CL/V". You have voted in favor of the 'rateconstant approach'!> That's so beyond the point that I just regret it being indoctrinated in> the minds of those starting to learn the field.Why would we regret to learn the basics of PK correctly?> All parameters (and models) are in fact secondary to reality.Of course, you are right here. But in this view, k is a tertiary parameter.Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics, Toxicology and TargetingUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlands`
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• On 28 Jan 2011 at 15:29:34, "J.H. Proost" (j.h.proost.at.rug.nl) sent the message
`The following message was posted to: PharmPKDear Xiaoming,You are making some wrong inferences, probably as a result of the incorrect'rate constant approach' (see other messages in this thread).> However, the K and V changed.It is not likely that V, and as a result k, will change by using a differentformulation. How can an excipient change the volume of distribution?> I thought that some of the excipients of the new form maybe affects the> clearance of drug.But you explained that CL was unaltered.> And some of the excipient can help the drug distribute extensively, that> can explain why some of the animal has the distribution phase (two> compartment model).Seems unlikely. What could be the mechanism? See above.> For this case, I think that the k and V are more "sensitive" than the CL> and AUC when the drug property or body physiological condition changed.Indeed, a change in V does not change CL and AUC; so CL and AUC are indeedinsensitive to changes in V. But you should not generalize this statement.For example, CL is 'sensitive' to changes in CYP activity, whereas V is notaffected.> CL and AUC are depending on the k and V.Again, CL and AUC are independent of V. See earlier messages of Nick Holfordand Des Williams.best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics, Toxicology and TargetingUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlands`
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• On 28 Jan 2011 at 14:31:07, "Pereira, Luis" (Luis.Pereira.at.childrens.harvard.edu) sent the message
`The following message was posted to: PharmPKDear Hans, Xiaoming and All,I won't give up assuming that we are all genuinely convinced of our  rightfulness and this is above all a scientific forum for the spreading  knowledge.  But as much as I respect everybody's opinion, whoever  engages on a clan feud between "supporters of this" and "supporters of  that", or a "contest was won" state of mind, or "that's your 'toy' " and  therefore you're wrong, may only get my commiseration. There's no place  in science for this line of argumentation. Science is not about being  'fruitful' and retiring what is not. The Jan10 thread was not the only  one, nor apparently will be the last, on this topic. But if I recall  correctly, it ended with a clear cut and scientific argumentation by  Roger Jelliffe, as he always does, showing that ranking parameters by  level of importance is nonsense. Believes and preferences are not facts.  I just intervened this time because the recent contributions to this  tread were again very misleading. To say that CL=K.V and CL=k10.V1,  therefore CL is one and the s`
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• On 28 Jan 2011 at 14:53:21, Roger Jelliffe (jelliffe.-at-.usc.edu) sent the message
`The following message was posted to: PharmPKDear Hans:  I think you yourself should read that discussion more carefully. Youare simply wrong when you say that " all arguments in favor of the rateconstant approach were shown to be weak or incorrect, and that no convincingarguments were given contradicting the clearance approach. There can be nodoubt that the 'frivolous contest' was won by the clearance". Where is yourscience? Where is your good sense? The plain fact is that kel and clearanceare simply interconvertible. You have many words but few make sense.I enclose once again our contribution (with references) to the subject.Please pay more attention this time. I would greatly appreciate knowing justwhat is "weak or incorrect" about our discussion. You seem to be stucksomewhere.Sincerely,Roger JelliffeDear All: from 1/26/10 in PharmPK:           Let us return once again to the question of kel and clearance. Iwould like to respond to several recent comments.It is disappointing to hear some of the discussants trying to "get away fromtheoretical arguments and deal with real data". Really now, if we don't havetheory, what do we have, after all? Isn't that what we, as scientists, arereally trying to develop - a firm theory that helps us explain, predict, andcontrol drug behavior? What about other theories, such as those of quantummechanics and Einstein? Once we can construct them, we have a firmfoundation and rationale for our actions. We have had so many words aboutthis issue that some theory might, after all, be helpful. It was Kurt Lewin,I think, who said, "there is nothing as practical as a good theory".           Why shouldn't we base our discussion on theoretical arguments?If we don't do this, what basis do we have for what we actually do, and whatis our  scientific rationale for our actions?  Let's look at the data, andthe theory, that will help us evaluate the utility of K and Cl. In my view,K and Cl are both equally useful. The beauty is in the eye of the beholder.No one is more "biologically relevant" than the other, unless someone thinksso. Then that is a belief, not a fact.No one is more "orthogonal" than the other. That depends upon the particularmodel parameter values. The correlation between Cl and V simply depends onthe model parameter values one is dealing with. By itself, Cl is no moreorthogonal than K with respect to V.The result presented,  Corr(CL&V)=0 gets one's attention. Is this real or anaccident? Answer - an accident of the values for V, CL and K that were used.In an earlier discussion, the model used wasln(y)= ln(1/V) - Kt + noiseAlan says it is straightforward to calculate analytically the formula forthe asymptotic covariance matrix between CL and V.  The off-diagonal term,Corr(CL&V), is exactlyCorr(CL&V)= -((1/4)*V-CL)*VThe stated parameter values were V=1, K=0.25, which implies CL=V*K=0.25 andCorr(CL=0.25&V=1)=0But what if K=0.35?  Fixing V=1, Alan shows a short table for CLvs.Corr(CL&V)     CL         Corr(CL&V)  0.0500   -0.2000   0.1500   -0.1000   0.2500         0   0.3500    0.1000   0.4500    0.2000   0.5500    0.3000   0.6500    0.4000   0.7500    0.5000   0.8500    0.6000   0.9500    0.7000So the correlation between Cl and V simply depends on the model parametervalues one is dealing with. By itself, Cl is no more orthogonal than K withrespect to V.In addition, it is clear, as described by David Bayard, that "a fundamentalproperty of the maximum likelihood estimator is the result that the maximumlikelihood estimate of a function of the parameters can be computed bytaking the function of the maximum likelihood estimates of the parameters.This property is known as the principle of invariance" [1]. Because of this,the MLE estimation of  = (K, V) gives exactly the same result as the MLEestimation of g() = (Cl, V). If you estimate it in clearance form (Cl,V) andthen convert it to elimination form (K,V), you get the same estimate as ifyou had originally estimated it in elimination form. And vice versa.           The examples of losing a kidney or losing 2 legs are interestingexamples. I am not sure what they settle. If you lose 2 legs you losevolume. Renal perfusion probably stays the same, though. Lose a kidney andyou lose renal elimination - that is intuitive. For a drug eliminated fromthe body only by glomerular filtration, for example, then ke is roughlyequivalent to creatinine clearance, as a relatively crude but easilyobtained estimate of GFR.  Oral ganciclovir and the aminoglycosideantibiotics are good examples of this.In addition, there has been recent discussion about K and T1/2.  In ourMM-USCPACK software, we often model ke as the product of knr + ke-slopetimes CLCr.  Therefore, if you lose a kidney and your CLCr halves (at leastuntil the other kidney hypertrophies to compensate), then ke will alsohalve.  By the way, there has been more recent discussion of themisperception of ke as the percentage of drug eliminated over a period oftime.  The precise way to view ke is the instantaneous rate of drugelimination, i.e. when the slice of time is infinitely small.Further, from another perspective, if the kidneys are responsible forgetting rid of the drug, then removing one will halve that function, andwill halve the fractional elimination of the drug.  Alternatively, since wehave already agreed that CL and k are algebraically related by V, and theargument has been made that if CL halves, then for a constant V, or forconstant renal perfusion, ke must halve as well.           It was also said that "no one makes any dose adjustments basedon volume of distribution changes". This actually contradicts much clinicalexperience with acutely ill and unstable patents who have significantchanges in several parameters, including volume of distribution of thecentral compartment, during their therapy. Most fitting methods do notpermit parameter values to change during data analysis.  However, MarcusHaug and Peter Slugg at the Cleveland Clinic in the 1980's fitted individualclusters of TDM data, and found, for example, significant changes ingentamicin volume of distribution. They would say that a patient had "VDcollapse", and that this decreasing gentamicin volume of distribution meantthat the patient would get well.And they were right. This was about the time that it was becoming known thatICU patients had larger volumes of distribution than general medicalpatients. At about the same time we became aware that each patient wouldoften increase his/her aminoglycoside central compartment VD upon becomingsicker, and later on decrease it as they recovered, perhaps due to changesin capillary permeability. Clearly while they were sicker they needed largerdoses to achieve effective serum concentrations, and lower doses later. Sowe, and quite a few others, I think, have quite often made dose adjustmentsbased on volume of distribution. We have even developed an interactingmultiple model (IMM) sequential Bayesian procedure to deal specifically withthis problem [2]. This method tracks the behavior of gentamicin andvancomycin in post cardiac surgical patients better that any other method[3]. The method comes from the aerospace community where it is widely usedto track and hit hostile targets trying to take evasive action.           "The half time following multiple dosing is only calculated whendosing has ended and the kinetics are in the terminal phase". Clearly notso. Clinically, using target oriented, model based TDM, one computes allrelevant parameter distributions during therapy, and uses that patient'sBayesian posterior model to develop the next, adjusted, dosage regimen.           About digoxin. We use the equations described by Reuning and hiscolleagues back in 1973 [4]. They clearly showed that the inotropic effectof digoxin correlates not with the serum concentrations, but withconcentrations in the peripheral, nonserum, compartment. We have used hismodel to develop a clinical model for the use of digoxin. It has worked verywell for us. You can see relationships between dose, serum concentration,and clinical effect that you cannot see otherwise. And you can select targettherapeutic goals in either the central, but better in the peripheralcompartment, (especially in acute situations) and develop dosage regimens toachieve them, not for a steady state, but NOW. You can control patientsclinically in acute situations when the simple raw data of serumconcentrations are not at all helpful. Good models and good software are thething. Further, using this model, it is clear that quinidine does not reducethe clearance of digoxin - it simply reduces the uptake in the peripheralcompartment, raising the serum concentrations, and resulting in a smallerapparent volume of distribution in the central compartment. If you add upthe total amount in digoxin in the central and peripheral compartments whena patient is on quinidine in the steady state, and then off quinidine a weeklater, there is surprisingly little difference in the total amount ofdigoxin in the patient.           We have a number of materials in our web site www.lapk.org thatbear on these subjects. If you like, go there and click around. You can goto Teaching Topics and to New Advances and download and print a bunch ofstuff.Best regards to you all,Roger JelliffeReferences1.         Goodwin and Payne: Dynamic System Identification - ExperimentDesign and Data Analysis, Academic Press, NY, 1977,p.50].2.         Bayard D, and Jelliffe R: A Bayesian Approach to TrackingPatients having Changing Pharmacokinetic Parameters. J. Pharmacokin.Pharmacodyn. 31 (1): 75-107, 2004.3.         Macdonald I,  Staatz C, Jelliffe R, and Thomson A: Evaluation andComparison of Simple Multiple Model, Richer Data Multiple Model, andSequential Interacting Multiple Model (IMM) Bayesian Analyses of Gentamicinand Vancomycin Data Collected From Patients Undergoing CardiothoracicSurgery. Ther. Drug Monit. 30:67-74, 2008.4.         Reuning R, Sams R, and Notari R: Role of Pharmacokinetic s inDrug Dosage Adjustment. I. Pharmacologic Effect Kinetics and Apparent Volumeof Distribution of Digoxin. J. Clin. Pharmacol. 13:127-141, 1973.`
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• On 29 Jan 2011 at 10:30:47, Stefan Soback (stefan.soback.at.gmail.com) sent the message
`> Luis wrote:>> "if the consideration of the bioavailability factor in the interpretation> of V/F and CL/F. Again, the extent of absorption does not affect the> elimination kinetics. Once drug molecules reach the systemic circulation> they bear no recollection about whether they came from a suspension, a> tablet or a syringe, and the body will not process ones differently from> others."Dear Luis,I may have missed the point, but was this remark for me? I personally thinkthat Vss is the only volume term of interest. The original PK quiz question(title of the thread) was about the connection between the apparent volumeof distribution and the AUC. What I wanted to say in my previous message wasthat there is no such thing as calculating Vss/F unless we *a priori* assumeone compartment model (of that drug after IV administration). OtherwiseVss/F cannot be calculated because its value is not dependent only on F butalso on MAT. Therefore, I considered the original quiz question (AUC vs.volume of distribution) certainly inappropriate for extravascularadministration data.Vss = CL * MRT. In other words the ratio of Vss and CL is MRT. Clearly wecan determine CL/F and it is often a useful tool for PK data analysis. If wecould also determine Vss/F, the ratio of Vss/F and CL/F would be identicalto the ratio of Vss and CL, i.e. the MRT after intravenous administration.To the best of my understanding we cannot determine the MRT afterintravenous administration from extravascular administration data. Hence,Vss/F can be determined only if IV data is available (and who needs Vss/F insuch case?).Roger, convincingly, pointed out the clinical relevance of changes in thecentral compartment volume. In the case of gentamicin, I presume that wasbased on IV data. As indicated CL = Vc * k10 and any change in Vc would alsochange the k10 (at least if CL is unaffected). After extravascularadministration CL/F = Vc/F * k10, but only if the k10 here is identical tothe k10 obtained after IV administration. Obviously we need IV data heretoo. So every time I see a volume term based on extravascular administrationdata, I wonder how they were calculated mathematically (in the case of Vbetayou only have to have a strong faith that the terminal slope is identical tothe IV administration data). We actually wrote a short communication on thisin the J. Vet. Pharmacol. Therap. (presently available as an onlinepre-publication).Best regards,StefanStefan Soback, DVM, PhD`
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• On 1 Feb 2011 at 13:53:17, "J.H.Proost" (j.h.proost.-a-.rug.nl) sent the message
`The following message was posted to: PharmPKDear Roger,Thank you for your comments. You wrote in reply to my message to Luis  Pereira:> You are simply wrong when you say that " all arguments in favor of the  rate constant approach were shown to be weak or incorrect, and that no  convincing arguments were given contradicting the clearance approach.I have not seen the arguments in favor of the rate constant approach  that were not refuted by me or others, and I have not seen any  convincing argument refuting the clearance approach.> Where is your science?My science is in my arguments.> Where is your good sense?My good sense is in discussing ALL arguments mentioned by others,  instead of ignoring them. Indeed, in this particular case I did not  mention all my arguments in my message, since that would have taken many  pages. Therefore I simply referred to the discussion.> The plain fact is that kel and clearance are simply interconvertible.No, this is not a plain fact. For me, and I have explained this in  numerous messages in this group, this statement is simply false, and  this is certainly not trivial. Please see my message today to Luis  Pereira.> I enclose once again our contribution (with references) to the  subject. Please pay more attention this time. I would greatly appreciate  knowing just what is "weak or incorrect" about our discussion. You seem  to be stuck somewhere.Please note I replied to your message of from 1/26/10 in PharmPK on  1/28/10, which you seem to have missed. I will repeat my message, so you  can see that I carefully read your contribution:--The following message was posted to: PharmPKDear Roger,Thank you for your thoughful comments. I fully agree with your plea for  theory and theoretical arguments. But I have still a few comments.> In my view, K and Cl are both equally useful. The beauty is in the> eye of the beholder. No one is more "biologically relevant" than> the other, unless someone thinks so. Then that is a belief, not a> fact.Over the last two months I have given several arguments why clearance is  more "biologically relevant" than elimination rate constant. This is not  a belief, and it is not a fact. It follows from sound theory. All  arguments in favor of K by Peter Mullen and Yaning Wang were refuted by  Nick Holford or by me. Do you have an argument why K is 'as biologically  relevant' as clearance?> No one is more "orthogonal" than the other.I agree. Please note that I stated this explicitly in my message of  December 9 to David. Your numerical example is a nice illustration of my  theoretical view in that message.ADDED COMMENT: Please note that orthogonality is a mathematical  property. We are not dealing with mathematics (although we use  mathematics), but with pharmacokinetics. The mechanistic basis of  pharmacokinetics is physiology, not mathematics.> If you lose 2 legs you lose volume. Renal perfusion probably> stays the same, though.OK, but the question was also: what changes, CL, K, CL and K, or none? I  would like to hear your opinion.ADDED COMMENT: I repeated this question in my message today to Luis  Pereira.Your examples on TDM of antiobiotics and digoxin are important 'golden  classics', and are highly appreciated. I have one comment to:> Further, using this model, it is clear that quinidine does not> reduce the clearance of digoxin - it simply reduces the uptake in> the peripheral compartment, raising the serum concentrations, and> resulting in a smaller apparent volume of distribution in the> central compartment.In many handbooks it is stated that quinidine reduces both clearance (by  competition of active secretion in the proximal tubules) and volume of  distribution of digoxin (by displacement from tissue binding sites). I  don't have primary references for this statement, which may be a wrong  interpretation of data. But your reasoning does not seem to be correct:  if the steady state serum concentration of digoxin raises after  administration of quinidine, clearance is reduced, irrespective of any  change in distribution.best regards,Hans Proost--So far the status at 28/1/10. For further discussion, please read my  message today to Luis Pereira.best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics, Toxicology and TargetingUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlands--The following message was posted to: PharmPKDear Luis.Thank you for your comments. You wrote in reply to me:> But as much as I respect everybody's opinion, whoever engages on a  clan feud between "supporters of this" and "supporters of that", or a  "contest was won" state of mind, or "that's your 'toy' " and therefore  you're wrong, may only get my commiseration. There's no place in science  for this line of argumentation.(Sorry, you started with the words 'the frivolous contest').> But if I recall correctly, it ended with a clear cut and scientific  argumentation by Roger Jelliffe, as he always does, showing that ranking  parameters by level of importance is nonsense.You have missed my reply to that message by Roger Jelliffe. And I have  repeated this reply in my message today to Roger Jelliffe.> To say that CL=K.V and CL=k10.V1, therefore CL is one and the same  thing is flat out wrong.I didn't say that. It's just the other way around. k = CL/V and also  k10 = CL/V1. It is not a conclusion that CL is the same in both cases,  it is the definition of k and k10 (k and k10 are dependent on one and  the same CL).> To say that liver and kidneys 'feel' blood concentrations is  tremendously naive and misses the true understanding of what blood  concentrations are all about.Good to hear that I am tremendously naive. Please explain me the truth  of what blood concentrations are all about. In my tremendously naive  understanding the blood (or plasma) concentration of a drug is the  concentration of that drug in blood (or plasma). That blood passes  organs like liver and kidneys. The diffusion into the liver, metabolism  in the liver, filtration in the kidneys and active secretion in the  kidneys are all dependent on the blood (or plasma) concentration. Of  course, the driving force is the unbound concentration, but in most  cases the unbound concentration is proportional to the total  concentration. So my tremendously naive understanding is that blood (or  plasma) concentration is the driving force of drug elimination. Where am  I wrong?> To say that "rate constant is the fraction of the apparent volume" is  even dimensionally incongruent.The rate constant k is a fraction of the apparent volume cleared per  unit of time. Nothing dimensionally incongruent here.> So, the sheer number of molecules that enter a clearing organ is the  driving force for a first order elimination, if that's the underlying  kinetics, no matter which parameterization we choose.To my opinion this is not true. Consider the situation of hepatic  elimination with low extraction ratio (Eh).Case A: blood concentration 10 mg/l, hepatic blood flow (Qh) 1 l/minCase B: blood concentration 10 mg/l, hepatic blood flow (Qh) 0.5 l/minIn case A the number of molecules entering the clearing organ is two  fold that in case B.Now the question is: what is the elimination rate and what is the  clearance in both cases?Since the extraction ratio is low, hepatic clearance is not limited by  hepatic blood flow, and so the clearance will be the same in both cases  (the lower hepatic blood flow might result in a reduced metabolic  clearance, e.g. due to limited oxygen supply, but that aspect is not  considered here). As a result, the elimination rate (amount/time) and  the elimination rate constant are the same in both cases. However, the  number of molecules entering the clearing organ is different. So, that  number cannot be the driving force.> But it's still a free world, and anybody may decide to do differently  at one's own peril.I have indicated in earlier message that this question is not trivial in  clinical situations, and that it would be strange or even dangerous if  anybody may decide what to do. Two examples from earlier messages:A) From my message to Yaning Wang, 22-12-2009:What happens with the pharmacokinetic variables in a patient after  connection to an extracorporeal circulation? In your view, K would not  be affected, and the increase in V will lead to an increase in  clearance. Does this imply that the renal and/or hepatic function of the  patient is increased? (the original posting mentioned here erroneously:  decreased).In the clearance approach, one would assume that renal and/or hepatic  function of the patient are not altered, so clearance remains the same.  The increased volume results in a decrease of k, and an increase of  half-life.What do you expect to happen?B) From my message to Peter Mullen, 28-12-2009:Let me add another example why this topic is not trivial, and why the  clearance approach should be preferred.Consider an interaction of drug A by drug B, which displaces drug A from  tissue binding sites. Should the dose of drug A be adjusted in the case  of coadministration of drug B?The displacement from tissue binding sites causes a decrease of the  volume of distribution of drug A.The 'rate constant fetishists' might think that the elimination rate  constant does not change (drug elimination is not affected, isn't it?).  Would this imply that the dose need not to be changed? Or does a lower  volume of distribution implies that the dose should be lowered? If it is  known, either by reasoning or by measurement, that half-life is  decreased, one might think that the dose should be increased. I really  can't guess the answer in the rate constant approach, and I'm worried  about the patient.On the contrary, the 'clearance absolutists' have an easy job. Since  clearance is not affected, the dosing rate (daily dose) should not be  changed. Because of the decreased volume of distribution, half-life will  be decreased, so a more frequent dosing (shorter dosing interval with  proportionally lower dose) would be advisable. No puzzling here.best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics, Toxicology and TargetingUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlands`
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• On 1 Feb 2011 at 20:46:22, Stefan Soback (stefan.soback.aaa.gmail.com) sent the message
`> Hans Proost wrote:> I didn't say that. It's just the other way around. k = CL/V and also k10  > CL/V1. It is not a conclusion that CL is the same in both cases, it is the> definition of k and k10 (k and k10 are dependent on one and the same CL).This was not intended for me, but I didn't quite understand the messagehere. The equation K = CL/V is valid only for one-compartment model. Theequation k10 = CL/V1 can be used only for multi-compartment models. AlthoughCL is model independent, these equations would not exist simultaneously fora single data set.If you need to fit a (compartment) model for instance for dosingpredictions, you have to determine the rate constants. If you are onlyinterested in clearance, the non-compartmental approach works fine.Best regards,StefanStefan Soback, DVM, PhD`
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• On 1 Feb 2011 at 13:54:42, Roger Jelliffe (jelliffe.-at-.usc.edu) sent the message
`The following message was posted to: PharmPKDear Hans:  Please read the references. You really seem to have a blind spot.Just read. It is all there. You haven't seen the facts that Kel and Cl areinterconvertible because I don't think you have read the relevant material.Look at the references. Please. Your arguments are simply false because youhaven't incorporated the relevant references, which are there for you, andeveryone else, to see. Expand your horizons a bit. Please,Hans, use a 2 compartment model of digoxin. Quinidine reduces the uptake ofdigoxin on Na-K APTase and in most tissues except, perhaps, the CNS, whichis why the apparent central volume of distribution is reduced and the serumconcentrations increase. But the tissue amounts are less. It is interestingthat if you add up the total amount of drug in both compartments, on and offquinidine, there is not a great deal of difference.Sorry, Hans, but K and CL are in fact both equally useful andinterconvertible. Read the refs and you will see. I bet you haven't readthem, as they may not be in the usual PK culture, but there they arenevertheless. If you prefer Cl for some reason, that is your choice. I don'tadvocate anything except that they are equally useful and areinterconvertible, and are supported by good and well known work. Read therefs please. They are good math, not arguments.Sincerely,Roger`
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• On 1 Feb 2011 at 17:25:16, "Pereira, Luis" (Luis.Pereira.-a-.childrens.harvard.edu) sent the message
`The following message was posted to: PharmPKDear Hans and All,Naive reasoning is as old as human nature. We may all genuinely say that  it just takes to step out the door and see the sun coming up on one side  of the horizon in the morning, and going down on the other at the end of  the day, to absolutely and conclusively say that the sun is going around  the earth. It's such an obvious evidence. Not too long ago, whoever said  the contrary would be imprisoned, tortured or even killed. But as much  as there's no shame in any kind of naivete, every time we manage to leap  ahead and reach understandings that essentially do not disprove our  previous beliefs, but rather allow us to explain why we evolved beyond  them, that's what scientific advancement is all about. Then it's just  for each one to decide what to see.Science is not in anyone's arguments, but rather in self-contained  facts. Although I understand why Hans keeps going in circles, and by  personal choice decides to be naive, as anyone is entitled to, I'll just  try once more to point out where the fallacy is, as explained in so many  textbook references in much more detail. What I cannot explain are  statements like "It is not a conclusion that CL is the same ... k and  k10 are dependent on one and the same CL".When mass transfer occurs by passive diffusion, according to Fick's  first law, the rate (mass per time) is proportional to the concentration  gradient in a perfectly homogeneous, or well stirred, system. It's truly  a probabilistic phenomenon which allows the derivation of this law from  counting a finite number of molecules (with a binomial distribution) all  the way to a macroscopic concentration (approximated by either a  Gaussian function or an error function). Transferring this reasoning to  a biological system one ought to remember that it's no longer about a  beaker full of a solvent where a certain amount of drug is dissolved  resulting in a concentration. The mere fact that organs and tissues and  cells and all kinds of heterogeneous dwellings exist, must always remind  us about where we started and what concentration now means. So, one  possibility is to rationalize in terms of mass, or flux, (as with heat  transfer or energy transfer), which eliminates the notion of volume.  Then, according to the`
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• On 25 Feb 2011 at 13:27:16, "J.H. Proost" (j.h.proost.at.rug.nl) sent the message
`Dear Roger,Thank you for your comments. You wrote:>  Please read the references. You really seem to have a blind spot. Just> read. It is all there.I don't have the book of Goodwin and Payne, but the title (Dynamic SystemIdentification - Experiment Design and Data Analysis) and year (1977) makeit unlikely that this book contains the modern pharmacokinetic concepts. Inrefs. 2 and 3 there is nothing about the present discussion. Ref. 4 is aninteresting paper, but it is based on the rate constant approach, and theword 'clearance' only appears twice as 'creatinine clearance'. Did you everrealize why kidney function is expressed as creatinine clearance?Physiologists know for long that renal function should be expressed asclearance.> Please, Hans, use a 2 compartment model of digoxin. Quinidine reduces the> uptake of digoxin on Na-K APTase and in most tissues except, perhaps, the> CNS, which is why the apparent central volume of distribution is reduced> and the seum concentrations increase.This is not the complete interpretation (see the contribution of LuisPereira about 'the sun is going around the earth'). If the volume ofdistribution decreases, the serum concentration would rise indeed. However,if clearance and dosing rate would remain the same, the increased serumconcentration would result in an increased rate of elimination, and theserum concentration would decrease until the serum concentration is again atthe original average level; the average steady-state serum concentrationwould not be altered.However, it is observed that the serum concentration increases. This impliesthat clearance decreases. This is not a result of the decrease of the volumeof distribution (probably your view), but a result of the reduced renalexcretion. This mechanism of interaction has been described in numerouspapers (e.g. Woodland C, Ito S, Koren G. A model for the prediction ofdigoxin-drug interactions at the renal tubular cell level. Ther Drug Monit.1998).> But the tissue amounts are less. It is interesting that if you add up the> total amount of drug in both compartments, on and off quinidine, there is> not a great deal of difference.Indeed, this is what is expected. The reduced uptake of digoxin results in adecreased volume of distribution, AND in a decreased renal excretion (seeabove). As a result of the decreased renal excretion, the steady-state serumconcentration increases, and from V=A/C it follows that A will not changesignificantly.Please note that the independence of V and CL is still true in the casewhere V and CL change as a result of the same mechanistic cause, as isprobably the case here. The inhibition of Na-K APTase may result in adecrease of V (as a result of reduced uptake of digoxin in various tissues)and CL (as a result of reduced uptake and excretion by the tubular cells).However, this does not imply that the decrease of V results in a decrease ofCL.> Sorry, Hans, but K and CL are in fact both equally useful and> interconvertible.Of course they are both useful and interconvertible. That's not thediscussion. The point is that, physiologically and mechanistically, K isdependent on CL, and not the other way around. That is not irrelevant, asshown in my comment to the digoxin example.> If you prefer Cl for some reason, that is your choice.In my contributions to the current and previous discussions in this groupyou may find the arguments for my preference for CL. Please read my twoquestion in the message to Luis Pereira, about extracorporeal circulationand tissue binding. I would be happy to get your view.> Read the refs please. They are good math, not arguments.Here we agree. The refs are mathematics. I prefer pharmacokinetics, based onphysiological arguments. If you prefer math ignoring physiologicy, ourdiscussion stops.best regards,Hans ProostDear Luis,Thank you for your extensive reply.> I'll just try once more to point out where the fallacy is, as explained in> so many textbook references in much more detail.Indeed, many textbooks are still presenting the basics of pharmacokineticswrongly, or at least inconsistently. Please see 'Clinical Pharmacokinetics(and Pharmacodynamics)' by Rowland and Tozer. This is state-of-the-artpharmacokinetics (and pharmacodynamics).> What I cannot explain are statements like "It is not a conclusion that CL> is the same ... k and k10 are dependent on one and the same CL".Clearance (total body clearance) is the primary parameter reflecting theelimination capacity of the body. In a one-compartment model, theelimination rate constant K equals CL/V; in multi-compartment models withelimination from the central compartment, the elimination rate constant k10equals CL/V1.> When mass transfer occurs by passive diffusion, according to Fick's first> law, the rate (mass per time) is proportional to the concentration> gradient in a perfectly homogeneous, or well stirred, system.This is a good starting point. Look at the word 'concentration'.> Then, according to the same first order (passive diffusion) assumption,> the rate is proportional to the driving force, i.e. the mass.The driving force is concentration, as was the starting point of yourreasoning. You cannot changes the rules during the game. The driving forcedoes not change by mathematical transformations.> So, if instead one decides to extend the homogeneity notion of a perfect> solution, to the biological system at hand, then the rate is proportional> to the concentration.Yes, I fully agree. Concentration is the driving force.> A key reminder here, should always be the true definition of concentration> and the fact that in heterogeneous systems, we may only interpret it in> light of an apparent (abstract, not biological) volume.The definition of plasma and blood concentration is quite clear. Why can wenot interpret a concentration without an apparent, abstract volume?> In regression terms, and depending on the irregularity of the parameter> space on which an optimization algorithm may try to minimize an objective> function, it does often help to try different parameterizations, since> they will generate different parameter space landscapes.This is irrelevant to this discussion.> What is often the case too, is the forgetfulness about the meaning of a> blood concentration and the price to pay for the conception of an apparent> volume.I don't understand how the meaning of a blood concentration can be a pointof discussion. And the 'conception of an apparent volume' is necessary torelate amount in the body to (blood or plasma) concentration, or to relateclearance to elimination rate constant, but not for the definition ofclearance.> So, to say that "the blood (or plasma) concentration of a drug ... passes> organs ... and metabolism and secretion ... are all dependent on the blood> (or plasma) concentration", is as obvious as stating that the sun goes> around the earth up in the sky. It's a correct phenomenological> observation which depending on the purpose intended may very well suffice,> as it suffices to follow the sun to count the days.This statement was meant to clarify why it is rational to say thatconcentration is the driving force for elimination, either by filtration,metabolism, diffusion or active transport. I cannot imagine any otherreasonable view.> It's just our choice and responsibility.As you explained above in detail, there is firm theory that explains thatthe rate of diffusion (as well as the rate of filtration and the rate ofmetabolism) is proportional to concentration. This is not a matter ofchoice.> Unlike Hans said, the concept of clearance was not a late discovery, but> it has been available all along since the 1930's (vide 'History of> Pharmacokinetics' by JG Wagner).Thank you for pointing to this.> It's a valuable concept bearing in mind that it's a mixed parameter> relating the rate and concentration, given that the latter refers to an> abstract conception of volume, rather than a physical one.This was perhaps the opinion in the 1930's, but in the 1970's this idea hasbeen refined by the physiological approach by Rowland and others. Onceagain, clearance (total body clearance) is a measure of the eliminationcapacity of the body. As stated above, we do not need an abstract conceptionof volume to define a concentration.> So, to say that "clearance is more "biologically relevant" than> elimination rate constant" depends on how relevant one wants to be, and> it's just a matter of opinion (don't tell me it's a club membership> preference because I just won't argue with that).I want to be very relevant, and for this reason I understand that theelimination rate constant is dependent on clearance, and for that reason Isay that clearance is 'more biologically relevant' than elimination rateconstant.> Hans' example of a low extraction drug is in fact self-explanatory. His> mistake starts in saying that "the elimination rate (amount/time) and the> elimination rate constant are the same".Sorry for be unclear. I meant that the VALUES of the elimination rate(amount/time) at a blood concentration of 10 mg/l are the same in case A andcase B, and that the VALUES of the elimination rate constant are the same incase A and case B.> Particularly with a low extraction ratio, one should always rationalize in> terms of the intrinsic clearance which is independent of blood flow> limitations.Yes indeed. Did you even think about the reason why the term 'intrinsicclearance' is used, and not 'intrinsic elimination rate'?> Only doing it ALL one may say that we'll start perhaps understanding> 'biological relevance'.Yes, but I missed clearance in your list. Actually, you gave a long essay onhepatic elimination, but I cannot find a single argument in favor of therate constant approach, and not a single argument contra the clearanceapproach.Unfortunately I did not find answers to my questions with respect toextracorporeal circulation and tissue binding. I would be happy to get yourreply.best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics, Toxicology and TargetingUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlands`
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