# PharmPK Discussion - Dose linearity

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• On 4 Jan 2006 at 11:58:35, "Shilpi Khan" (SKhan.-at-.cedracorp.com) sent the message
`The following message was posted to: PharmPKHappy new year to all,I have a question regarding dose linearity. In general, we examine, doselinearity using a power model,P = a * Dose^bwhere P represents the dependent variable (Cmax, AUClast, AUCinf) and, aand b are constants.  A value of b~1 indicates linearity.(To perform this simple regression in SAS, we use PROC REG on logtransformed data. The value of slope is 'b' and the exponentiation ofthe intercept is 'a')Do you have any comment or suggestion on this model?Thanks in advance,ShilpiShilpi Khan, M.S.Staff Scientist/BiostatisticianCEDRA CorporationAustin, TX 78754`
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• On 4 Jan 2006 at 13:30:11, dan combs (dan_combs.at.comcast.net) sent the message
`The following message was posted to: PharmPKShilpi:Here is a reference for the method you mentioned.  One can use SASProc Mixed or the WinNonlin Linear Mixed Effects modelling feature.Excel power curve plots give the same 'a' and 'b' parameter estimatesbut does not compute confidence intervals, which are needed for theassessment.BP Smith, FR Vandenhende, KA DeSante, NA Farid, PA Welch, JTCallaghan, ST Forgue. Confidence Interval Criteria for Assessment ofDose Proportionality. Pharmaceutical Research, Vol. 17, No. 10, 2000.Dan CombsCombs Consulting Service`
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• On 4 Jan 2006 at 17:26:34, Angusmdmclean.-at-.aol.com sent the message
`ShilpiYou regession model is as you indicate and I reproduce it below:P = a * Dose^b where P is a Pk  exposure parameter e.g. Cmax.My comment is as follows:Can you tell me what your null hypothesis (Ho) is and how are youevaluating the p value for this test.Angus McLean Ph.D,8125 Langport Terrace,Suite 100,Gaithersburg,MD 20877Tel 301-869-1009fax 301-869-5737`
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• On 5 Jan 2006 at 09:22:05, "Vikesh Kumar Shrivastav" (vikeshk.shrivastav.-a-.ranbaxy.com) sent the message
`The following message was posted to: PharmPKHi ShilpiYes the equation of power model for testing Dose Linearity isP = a * Dose^bwhere P represents the dependent variable (Cmax, AUClast, AUCinf)But i think apart from log-transformation data also needs to be dose-normalized either to Higher Dose or Lower Dose.On log transformation our equation becomes: Log(p) = a + b*log(dose)Our null hypothesis is H0: b = 1.We can use Proc Reg in SAS on the log-transformed data.You can add following statement in Proc reg inorder to test for b=1:    SLOPE : test Log_DOSE = 1;Hope this helps.....Vikesh S`
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• On 5 Jan 2006 at 09:12:12, "Henri Merdjan" (henri.-at-.novexel.com) sent the message
`The following message was posted to: PharmPKDear Shilpi,The suggested model makes perfect sense. However, I would like tochallenge the use of log-transformation. In my humble opinion, youmay do a better job by running non-linear regression on untransformeddata. In that case, linear regression on log-transformed data wouldstill be useful in providing initial parameter estimates.Hope this helps.HenriHenri MERDJAN, Pharm, AIHPHead of Drug Metabolism and PharmacokineticsNOVEXEL S.A.Parc Biocitech102 Route de NoisyF-93230 RomainvilleFranceTel     +33 (0)1 57 14 07 45Fax    +33 (0)1 48 46 39 26Web   www.novexel.com`
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• On 5 Jan 2006 at 18:57:26, =?ISO-8859-1?Q?J=FCrgen_Bulitta?= (bulitta.-at-.ibmp.osn.de) sent the message
`The following message was posted to: PharmPKDear Dr Merdjan, Dear All,I agree that the power model (AUC = a * Dose^b) is a reasonable wayto statistically test dose linearity. There might be some drawbacksfor Cmax, because the rate of absorption is sometimes slower athigher doses (e.g. if the drug has low solubility). In this casefitting the full population PK model will be superior to NCA in orderto better understand the PK data.It is important to adequately account for the between subjectvariability (BSV) in Cmax and AUC, irrespective if one uses linear orlog-scale to fit the data. I would assume a log-normal distributionfor Cmax and AUC. If one fits the power model on linear scale, onewould have to account for a proportional error structure (BSV)explicitly. Therefore, I would prefer ANCOVA on log-scale as a quickway to analyze dose linearity data, because an additive errorstructure on log-scale turns into a proportional error structure onlinear scale.Besides a criterion for statistical significance of dose linearity,an equivalence based criterion should also be considered. Even if adrug shows a nonlinear curve of AUC vs. dose, this may have only alimited effect on the pharmacodynamics at therapeutic doses. One wayto do so is to use equivalence statistics with dose-normalized AUC(and Cmax) in addition to fitting the power model.Hope this helpsBest regardsJuergen--Juergen Bulitta, M.Sc.Research ScientistIBMP - Institute for Biomedical and Pharmaceutical ResearchPaul-Ehrlich-Strasse 1990562 Nurnberg - HeroldsbergGermany`
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• On 5 Jan 2006 at 13:13:11, Angusmdmclean.-at-.aol.com sent the message
`Henri:What you suggest to evaluate nonlinear regression on untransformeddata makes sense but exactly how would you introduce the nullhypotheses and obtain a p factor.  Exactly what computer program doyou have in mind.can you comment?Angus McLean Ph.D,8125 Langport Terrace,Suite 100,Gaithersburg,MD 20877tel 301-869-1009fax 301-869-5737`
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• On 11 Jan 2006 at 10:25:23, "Henri Merdjan" (henri.-at-.novexel.com) sent the message
`The following message was posted to: PharmPKDear Angus,I guess a sensible way of approaching the problem would be to statethe null hypothesis as:  " exponent=1 ". I would expect a nonlinearregression piece of software to provide confidence intervals aroundparameters' estimates (by the way, I am not recommending any specificsoft). Accepting/rejecting H0 is then straightforward. Just answerthe question: does the CI include 1? So far, this is theory.Real life may be a bit different. In my experience, the majordrawback of the hypothesis testing approach is that an exponent maybe statistically different from, however "close" to unity. Let me gothrough a numerical example. Let's imagine the exponent is 1.08 witha 95%CI of 1.03 to 1.13. In strict statistical terms, this is asignificant deviation from linearity. However, what is its practicalrelevance? Such a power model would suggest that each time you doublethe dose, the PK parameter of interest (eg Cmax or AUC) will increaseby a 2.1-fold factor instead of 2. Not a big deal!In summary, you may find some interest in setting acceptance limitsfor the exponent rather than, or in addition to, testing somehypothesis.Hope this helps,HenriHenri MERDJAN, Pharm, AIHPHead of Drug Metabolism and PharmacokineticsNOVEXEL S.A.FranceTel     +33 (0)1 57 14 07 45Fax    +33 (0)1 48 46 39 26Web   www.novexel.com`
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• On 12 Jan 2006 at 11:21:55, "J.H.Proost" (J.H.Proost.at.med.umcg.nl) sent the message
`The following message was posted to: PharmPKDear Henri,I fully agree with your statement about the relevance of confidenceintervals.But is real life, there may be also a different problem. What shouldone conclude if the exponent differs from 1 to a clinically relevantextent, but with a confidence interval including 1? E.g. exponent 1.3and confidence interval 0.9 to 1.7. What is the conclusion now?Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax 31-50 363 3247Email: j.h.proost.at.rug.nl`
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• On 12 Jan 2006 at 14:04:20, Angusmdmclean.-at-.aol.com sent the message
`Johannes:  what a physician want to know is if you double the dosewould you get double the maximum concentration (Cmax)? Given theadverse event profile of the drug on a case by case basis there maybe concern about the probability of the drug (on dose doubling)producing disproportionately higher adverse events on account ofhigher Cmax values (than anticipated from linearity).  Perhaps incases, where there is uncertainty about proportionalityconsiderations, then dose would be more carefully titrated withclinical response in mind to a patient over a period of time withintermediate dose strengths.Best Regards,Angus McLean Ph.D,8125 Langport Terrace,Suite 100,Gaithersburg,MD 20877tel 301-869-1009fax 301-869-5737`
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• On 13 Jan 2006 at 14:04:19, "Henri Merdjan" (henri.aaa.novexel.com) sent the message
`The following message was posted to: PharmPKDear Johannes,This is a fair point. The question you are asking illustrates thelevel of uncertainty possibly associated with a parameter estimate.In this particular example, the CI is so "wide" that you cannotreally decide whether the exponent value is actually closer to unity,or to the central estimate of 1.3, or to some even higher value. Insuch a scenario, I would simply recommend a careful dose escalation.Best regards,Henri`
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