# PharmPK Discussion - Meaning of PK curve as a probability curve

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• On 7 Feb 2013 at 10:16:05, Bernard Murray (Bernard.Murray.-a-.gilead.com) sent the message
`Hello there,I have been mentoring one of my colleagues on the principles ofnoncompartmental analysis.  They have training in statistics so, forbetter or worse, I skipped the usual MRT = AUMC/AUC approach and insteadshowed them that the concentration-time curve could be converted to aprobability curve by dividing each concentration by the AUC and thenreplotting.  They were then comfortable that classical statisticalmoment theory could be applied to calculate the mean of the distribution(MRT) straight from the raw first moment curve.During the conversation, one of their questions made me hesitate, so Iwanted to check with you to make sure I wasn't misleading them."What is the meaning of the probability curve, calculated from the PKcurve?"My answer was that it was that each point represented the probability offinding drug in plasma at that particular time.Does that sound right?As statisticians they were disappointed that we didn't have much use forhigher statistical moments (for variance, skewness and kurtosis) in PK,but when I pointed out the potential errors resulting in multiplying a24-hr concentration by 24^4, and the huge area extrapolations that wouldlikely be needed for the higher moments, they could see the practicallimitations.All the very best,BernardBernard Murray, Ph.D.Senior Research Scientist, Drug MetabolismGilead Sciences[Reminds me an applied math course I helped to teach many years ago. Inone example themathematics instructor turned the PK equation into stochastic events - db]`
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• On 7 Feb 2013 at 18:12:01, Roger Jelliffe (jelliffe.at.usc.edu) sent the message
`The following message was posted to: PharmPKDear Bernard:  I don't understand. What good does noncompartmental modeling do foryou that real compartmental modeling does not do better, especiallynonparametric population modeling? How do you plan to use these models?Howdo you use them, for example, to develop dosage regimens of drugs formaximally precise dosage regimens as with nonparametric models andmultiplemodel dosage design? Can you help me?Some references1.   Jelliffe R, Schumitzky A, Bayard D, Milman M, Van Guilder M, WangX,Jiang F, Barbaut X, and Maire P: Model-Based, Goal-Oriented,IndividualizedDrug Therapy: Linkage of Population Modeling, New "Multiple Model"DosageDesign, Bayesian Feedback, and Individualized Target Goals. Clin.Pharmacokinet. 34: 57-77, 1998.2.  Jelliffe R: Goal-Oriented, Model-Based Drug Regimens: SettingIndividualized Goals for each Patient. Therap. Drug Monit. 22: 325-329,2000.3.  Jelliffe R, Bayard D, Milman M, Van Guilder M, and Schumitzky A:Achieving Target Goals most Precisely using Nonparametric CompartmentalModels and "Multiple Model" Design of Dosage Regimens. Therap. DrugMonit.22: 346-353, 2000.4.  Leary, R., Jelliffe R., Schumitzky, A., and Van Guilder, M Anadaptive grid non-parametric approach to pharmacokinetic anddynamic(PK/PD)population models, 14-th IEEE Symposium on Computer Based MedicalSystems,389-394, 2001.5.  Jelliffe R: Estimation of Creatinine Clearance in Patients withUnstable Renal Function, without a Urine Specimen. Am. J. Nephrology,22:320-324, 2002.6.  Bayard D, and Jelliffe R: A Bayesian Approach to Tracking Patientshaving Changing Pharmacokinetic Parameters. J. Pharmacokin. Pharmacodyn.31(1): 75-107, 2004.7.  Bustad A, Terziivanov D, Leary R, Port R, Schumitzky A, and JelliffeR: Parametric and Nonparametric Population Methods: Their ComparativePerformance in Analysing a Clinical Data Set and Two Monte CarloSimulationStudies. Clin. Pharmacokinet., 45: 365-383, 2006.8.  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 ofGentamicinand Vancomycin Data Collected From Patients Undergoing CardiothoracicSurgery. Ther. Drug Monit. 30:67-74, 2008.9.  Jelliffe R, Schumitzky A, Bayard D, Leary R,  Botnen A,  Van GuilderM, Bustad A, and Neely M: Human Genetic variation, PopulationPharmacokinetic - Dynamic Models, Bayesian feedback control, andMaximallyprecise Individualized drug dosage regimens. Current PharmacogenomicsandPersonalized Medicine, 7: 249-262, 2009.10.  Neely M, van Guilder M, Yamada W, Schumitzky A, and Jelliffe R:Accurate Detection of Outliers and Subpopulations with Pmetrics, aNonparametric and Parametric Pharmacometric Modeling and SimulationPackagefor R. Therap. Drug Monit. 34: 467-476, 2012.11.  Tatarinova T, Neely M, Bartroff J, van Guilder M, Walter Yamada W,Bayard D, Jelliffe R, Leary R, Chubatiuk A, and Schumitzky A: TwoGeneralMethods for Population Pharmacokinetic Modeling: Non-Parametric AdaptiveGrid and Non-Parametric Bayesian. J. Pharmacokin. Pharmacodyn, in press.Very best regards,Roger JelliffeRoger W. Jelliffe, M.D., F.C.P., F.A.A.P.S.Professor of Medicine,Founder and Co-Director, Laboratory of Applied Pharmacokineticswww.lapk.orgUSC Keck School of Medicine2250 Alcazar St, Room 134-BLos Angeles CA 90033`
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• On 8 Feb 2013 at 03:46:06, "Wang, Yaning" (Yaning.Wang.-at-.fda.hhs.gov) sent the message
`The following message was posted to: PharmPKHi, Bernard:AUC normalized concentration curve (Ct/AUC) can be described as follows(assuming F=1 for simplicity):Ct/AUC=Ct/(Dose/CL)=Ct/(Dose/(V*K))=(Ct*V/Dose)*KCt*V/Dose is the fraction of drug molecules remaining (alive) in thebody at time t. This is the same as survival function in statistics,S(t), proportion of people still alive at time t. K is approximately thefraction of drug (relative to the drug amount at time t) eliminated fromthe body within a small time interval (I like this way of interpreting Kbecause K can always be converted to a <1 number by scaling time to avery small unit, such as min, sec, to make K intuitively meaningful.E.g. K=1/hr can be expressed as 0.0167/min, meaning approximately 1.67%of remaining drug molecules is eliminated within 1 min). In survivalstatistics, K is the same as hazard or conditional failure rate (h), thefraction of people (relative to the people alive at time t) dying withina small time interval. Then (Ct*V/Dose)*K is the fraction of drugmolecules (relative to the total dose) eliminated at time t (within asmall time interval around time t). To make it more consistent with thestatistical description, (Ct*V/Dose)*K is the fraction of drug molecules(relative to the total dose) that has a residence time (survival in thebody) of t because this fraction of drug molecules are not eliminated(die) until time t. This is the same as the density function orunconditional failure rate, f(t), in survival statistics. And it is wellknown f(t)=S(t)*h. So at each t, Ct/AUC is just like a histogram summary(expressed as proportion) of drug molecules with different residencetimes. E.g. at Ct/AUC=0.1 at 2 hours means that 10% of drug moleculessurvived up to 2 hours (residence time=2). Ct/AUC=0.2 at 10 hours meansthat 20% of drug molecules survived up to 10 hours (residence time=10).Therefore, the raw Ct/AUC can be used to calculate the mean residencetime. I hope this will help your colleagues to understand this PKconcept better. There is a close link between PK and survivalstatistics. PK is basically a description of the drug moleculessurviving in the body.David:Any first order process can be described as a random walk process. Therate constant can be interpreted as the probability of moving to anotherstate (compartment) and 1-rate constant is the probability of staying atthe current state (compartment). Of course, the rate constant needs tobe converted to a <1 number as described above to make thisinterpretation meaningful.ThanksYaningYaning Wang, Ph.D.Associate Director for ScienceDivision of PharmacometricsOffice of Clinical PharmacologyOffice of Translational ScienceCenter for Drug Evaluation and ResearchU.S. Food and Drug Administration"The contents of this message are mine personally and do not necessarilyreflect any position of the Government or the Food and DrugAdministration."`
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• On 8 Feb 2013 at 09:46:45, Bernard Murray (Bernard.Murray.-at-.gilead.com) sent the message
`The following message was posted to: PharmPKHello Roger,I understand your concern about inappropriate use of NCA, but in thiscase it is being applied to nonclinical studies, either to earlydiscovery pharmacokinetics or to toxicokinetics.  The aim is only tohave simple, objective measures of exposure (AUC, Cmax etc.) andpersistence (MRT).  My colleague's question was largely triggered bycuriosity as to how MRT calculations were performed.  There are somenice papers on the use of higher statistical moments in pharmacokinetics(e.g. Weiss & Pang, J Pharmacokin Biopharm, 20: 253-278 [1992]) but, forreasons I mentioned before, they are not routinely applicable.I can assure you that we switch to compartmental modeling fordevelopment candidate molecules, or for those used in nonclinicalpharmacodynamic models.  It is unfortunate that we don't have time todevelop and validate more sophisticated models for some of our more"interesting" compounds.Thank you very much for the literature review.  I'll pass that to mycolleague and I am sure that there will be future mentoring sessions onthat topic.All the very best,BernardBernard Murray, Ph.D.Senior Research Scientist, Drug MetabolismGilead Sciences`
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• On 12 Feb 2013 at 18:12:14, (Angusmdmclean.at.aol.com) sent the message
`Yaning: Thank you for your informative  and thought provoking commentaryconcerning the significance of  the AUC normalized concentration curve Ct/AUC.Could I ask you to enlarge upon it a little to include the significanceofCmax/AUC inf quotient.ThanksAngus McLean Ph.D.`
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• On 14 Feb 2013 at 03:45:12, "Wang, Yaning" (Yaning.Wang.at.fda.hhs.gov) sent the message
`The following message was posted to: PharmPKAngus:Following "Ct/AUC is just like a histogram summary (expressed as proportion) ofdrug molecules with different residence times", Cmax(observed)/AUC is the peakof this histogram, representing the largest fraction of drug molecules with aspecific residence time: Tmax (observed).ThanksYaningYaning Wang, Ph.D.Associate Director for ScienceDivision of PharmacometricsOffice of Clinical PharmacologyOffice of Translational ScienceCenter for Drug Evaluation and ResearchU.S. Food and Drug Administration`
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• On 14 Feb 2013 at 08:51:32, (Angusmdmclean.-a-.aol.com) sent the message
`Yaning:  As you are aware Ct/AUC has the dimension of time.  Thereforecould I say that Cmax(observed)/AUC is the peak of the Ct/AUC histogramand  represents  the time of occurrence  of the largest fraction of drugmolecules with a specific residence time (Tmax (observed).Do you agree with this way of putting it?Angus McLean`
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