# PharmPK Discussion - Log-linear or linear trapezoid

PharmPK Discussion List Archive Index page
• On 3 Aug 2000 at 22:54:19, Korbtham Sathirakul (pyksk.-at-.mahidol.ac.th) sent the message
`Dear AllI am now suprising that there are still be quite a few people estimate theAUC by using the linear trepazoid. According to Gabrielsson textbook forPK and PD data analysis, their group strongly suggested the log/lineartrepazoid. I myself teach my students to do the log/linear trepazoid.However, i found that in pharm industry, quite a few of people still usedthe linear one. Could anybody please contribute to comment on using linearor log/linear trepazoid. Thank you very much.Best regardsAsoc. Prof. Dr. Korbtham SathirkaulFaculty of PharmacyMahidol UniversityBangkok, Thailand.`
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• On 4 Aug 2000 at 11:29:20, "Derrick J (Rick) Bates" (DJ.Rick.Bates.-at-.pnl.gov) sent the message
`	My own preference is to dispense with either of the two methods below	(which I label AUTs - Area Under the Trapezoids) that were originally	developed as a quick and dirty approximations to the real	AUC - Area under the Curve.	I prefer to use the real Area under the Curve from the fitted model.	For example, in the simple exponential model, this is easily estimated	as Co/ke, where Co is the estimated concentration at time 0	and ke is the estimated elimination coefficient.`
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• On 4 Aug 2000 at 14:29:51, Roger Jelliffe (jelliffe.-a-.usc.edu) sent the message
`Dear Dr. Sathirakul:         About AUC's. Why not make a model of the behavior of thedrug, and then simulate with the model and compute the AUC that way?You could use a Bayesian approach based first on a population modeland then on whatever serum data you obtain from an individualpatient. You would also capture the relationships between the doses,the concentrations found, and the AUC, and it would be easy tocompute the regimen to achieve a desired concentration for a desiredtime, or a desired profile until a desired AUC is reached? Anycomments, anyone?Roger JelliffeRoger W. Jelliffe, M.D. Professor of Medicine, USCUSC Laboratory of Applied Pharmacokinetics2250 Alcazar St, Los Angeles CA 90033, USAPhone (323)442-1300, fax (323)442-1302, email=  jelliffe.aaa.hsc.usc.eduOur web site=  http://www.usc.edu/hsc/lab_apk****`
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• On 4 Aug 2000 at 14:30:25, "Paul B. Laub" (plaub.-a-.incyte.com) sent the message
`Here are several useful papers on the AUC / noncompartmental analysis topic --Robert Purves"Optimum Numerical Integration Methods for Estimation of Area-Under-the-Curve(AUC) and Area-Under-the-Moment-Curve (AUMC)" in JPB 20:  211-226 (1992).KC Yeh and KC Kwan, "A Comparison of Numerical Integrating Algorithms byTrapezoidal, Lagrange, and Spline Approximation" Journal ofPharmacokinetics andBiopharmaceutics (JPB) 6:  79-98 (1978).ML Rocci, Jr and WJ Jusko, "LAGRAN program for area and moments inpharmacokinetic analysis" Computer Programs in Biomedicine 16:  203-216 (1983)P.B. Laub and J.M. Gallo"NCOMP - A Windows-based Computer Program forNoncompartmental Analysis of Pharmacokinetic Data"J. Pharm. Sci. 85:  393-395 (1996)Paul--                          -- 30 -- 30 -- 30 --Mr. Paul B. Laub      Expression Bioinformatics     (650) 845-5411 (voice)Incyte Genomics                                     (650) 855-0572 (fax)3160 Porter Dr.          Palo Alto, CA 94304        plaub.-a-.incyte.com`
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• On 6 Aug 2000 at 22:17:07, Stephen Day (shday.aaa.yahoo.com) sent the message
`I find this post needs some comment.>>  	My own preference is to dispense with either of the>  two methods below>  	(which I label AUTs - Area Under the Trapezoids)>  that were originally>  	developed as a quick and dirty approximations to>  the real>  	AUC - Area under the Curve.Really? This method is used in every other field ofscience to calculate the area under a curve. I canassure you it was not developed as a quick and dirtyapproximation - it was just good mathematics (andstill is).If the rate of change of concentration is first-orderwith respect to concentration, the Log-linear methodgives the *exact* area between two data points. Noquick and dirty here.>>  	I prefer to use the real Area under the Curve from>  the fitted model.Models are never perfect, therefore you will stillonly find an estimate of the "real" AUC in this way.Why rely on the accuracy of any model when you can useone of the trapezoid methods?=====Stephen DayMerck-Frosst Centre for Therapeutic ResearchKirkland, QC    CANADA`
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• On 6 Aug 2000 at 22:17:25, "Lakritz, Jeffrey" (LakritzJ.at.missouri.edu) sent the message
`Dear Drs. Jelliffe, Sathirakul, et al,	While we are on the subject of AUC's.  I analyzed some data on a 3rdgeneration cephalosporin after IV, IM and subcutaneous administration inanimals.  The AUC after IM and sub Q came out significantly higher thanafter IV, so bioavailability is >100%.  I determined the AUC using thetrapezoidal approximations.  Since I am going to have to go back and try todetermine how this happened, could anyone suggest why?  I suppose, thatafter IV the drug is more rapidly cleared via the kidney than after IM/SQ.Any thoughts for a beginner?Jeff Lakritz DVM, PhD.`
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• On 7 Aug 2000 at 11:25:47, David_Bourne (david.aaa.boomer.org) sent the message
`[A few replies from over the weekend - db]Sender: PharmPK.aaa.boomer.orgReply-To: Nick Holford MIME-Version: 1.0From: Nick Holford Date:  Sat, 05 Aug 2000 08:46:49 +1200To: david.-at-.boomer.orgSubject: Re: PharmPK Re: Log-linear or linear trapezoidThe following message was posted to: PharmPK"Derrick J (Rick) Bates (by way of David_Bourne)" wrote:>          My own preference is to dispense with either of the two methods below>          (which I label AUTs - Area Under the Trapezoids) that were originally>          developed as a quick and dirty approximations to the real>          AUC - Area under the Curve.The "AUT"s (0-inf or over SS Dosing Interval) are themselves only ofany real PK value as initial estimates of Clearance/F.>          I prefer to use the real Area under the Curve from the fitted model.>          For example, in the simple exponential model, this is>easily estimated>          as Co/ke, where Co is the estimated concentration at time 0>          and ke is the estimated elimination coefficient.---More generally (any number of compartments) "real" AUC can becomputed from Dose*F/CL. Any (linear) PK model should beparameterizable in terms of CL/F so you only need to know oneparameter (CL/F) (rather than two, C0,ke, as Derrick suggests).But if I have an estimate of CL/F why would I have anypharmacokinetic reason to bother calculating AUC? Perhaps the onlyreason is to satisfy regulatory bioequivalance guidelines -- but inthis context any pharmacokinetic science (e.g. compartmental models)is typically too hard for the regulators and one must resort toempirical AUC, Tmax, Cmax statistics developed at the Walt DisneySchool of Pharmacokinetics.AUC (derived from Dose*F/CL) has some use as an empirical "exposure"quantity in pharmacodynamics. But because information about time isnecessarily lost by integration of C wrt t learning about scheduledependence is very hard. I believe the AUC approach topharmacodynamics was also developed at the Walt Disney School ofPharmacokinetics. Many graduates of WDSP seem to be working in theanti-cancer field which is perhaps a reflection of why it isdifficult to find examples that demonstrate that AUC is any betterthan dose as a predictor of drug response in this area.--Nick Holford, Dept Pharmacology & Clinical PharmacologyUniversity of Auckland, Private Bag 92019, Auckland, New Zealandemail:n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm---Sender: PharmPK.aaa.boomer.orgReply-To: "J.G. Wright" MIME-Version: 1.0From: "J.G. Wright" Date:  Mon, 7 Aug 2000 12:07:42 +0100 (GMT)To: david.at.boomer.orgSubject: Re: PharmPK Re: Log-linear or linear trapezoidThe following message was posted to: PharmPKDear Stephen,Let say I give a bolus dose, and then take a couple of samples.  In allbut a few cases (eg enterohepatic recycling), concentration will decreasewith time. So if I put myfirst point at time zero, concentration zero, I will miss a big chunkof the AUC and obtaincorrespondingly biased estimates of clearances.  If you back-extrapolatein some way, then you are modelling and you may a well do it in astatistical manner, acknowledging uncertainty.Join-the-dots pharmacokinetics (whatever good mathematics it calls upon)ignores two fundamental factors - the dosing history and the existence oferror.  You say that the loglinear trapezoidal gives an exact area betweentwo points - this would only be true if the samples are error-less.  Theyare measured with error and it probably will  not be the same at bothpoints.There is a common misconception that noncompartmental methods aremodel-independent.  This is not the case, they are simply based onassumptions which are somewhat physiologically strange (straight linesjoin the dots,on some scale) and hence unstated.  As the use ofAUC to calculate clearance depends on many of the assumptions required fora simple compartmental model, speed and perhap simplicity are all that isgained.In a pharmacological context, there is a lot to understand.  Join-the-dotspharmacokinetics doesn't acknowledge its assumptions and furthermoreprovides no diagnostics for when they are violated. A method cannot bedivorced from its context - trapezoidal AUCs are quick and very dirty.  On the other hand,timing error won't make much difference...Regards,James Wright---Sender: PharmPK.-a-.boomer.orgReply-To: "Bhatti, Masood" MIME-Version: 1.0From: "Bhatti, Masood" Date:  Mon, 7 Aug 2000 09:22:52 -0400To: david.-a-.boomer.orgSubject: RE: PharmPK Re: Log-linear or linear trapezoidThe following message was posted to: PharmPKDear Jeff,Is it possible that your drug is undergoing first pass lung metabolism afterI.V. administration?Masood BhattiSection LeaderPurdue Pharma---Sender: PharmPK.-a-.boomer.orgReply-To: zhao wang Mime-Version: 1.0From: zhao wang Date:  Mon, 07 Aug 2000 09:05:01 -0500To: david.at.boomer.orgSubject: Re: PharmPK Re: Log-linear or linear trapezoidThe following message was posted to: PharmPKI think:1) IV does not make kidney clearance "more rapidly"and this the oneof the assumptions that is made to estimate BA;2) Try normalize the dose by body weight for each subject being studied;3) Try use the same subject for different administration;4) Simultaneously IV, oral and sub Q modeling to estimate BA will bemore accuracy but you need a software like SAAM II.5) If each study (different dose administration ) is from differentsubject, there will be some variation of Vd and Cle from eitherinter-or intra individuals and therefore,  if the variation of BAestimation is within this range, it should be acceptable, 105% of BAdoes not mean BA > 100%, but the variation of estimation.Just some thoughts.Zhao Wang, M.D.Northwestern University Medical SchoolAnesthesia Research---Sender: PharmPK.aaa.boomer.orgReply-To: "David S. Farrier" Mime-Version: 1.0From: "David S. Farrier" Date:  Mon, 07 Aug 2000 10:35:49 -0400To: david.-at-.boomer.orgSubject: Re: PharmPK Re: Log-linear or linear trapezoidThe following message was posted to: PharmPKA group from Phoenix International's PK Department presented a poster in1996 at the AAPS annual meeting. They ran simulations comparing the resultsof calculating AUC using trapezoid, log-trapezod, and spline smoothingmethods. Their conclusion was: given the normal variability of plasma leveldata, the log-trapezoid and spline methods did not give a statisticallybetter estimate of the AUC than the standard trapezoid rule.We built and tested a module for PK Solutions that produced a side-by-sidecomparison of AUC values (partials and cumulative) calculated by both thestandard trapezoid and log-trapezoid methods. The overall results werestatistically so similar, except with rare and unlikely plasma profiles,that we decided to only use the standard trapezoid method for our software.Not that log-trapezoid is any more difficult or sophisticated, but why makethings more esoteric when the simpler approach is adequate and justified.DavidDavid S. Farrier, Ph.D.Summit Research Services1374 Hillcrest DriveAshland, OH 44805 USATel/Fax:  (419)-289-9207Email:     DFarrier.aaa.SummitPK.comWeb Site: www.SummitPK.com[I seem to remember a paper by Kwan (some time back) that came to asimilar conclusion...I had a graduate student do some (limited)simulations once and we came to same conclusion. However, Purvespaper and suggestion convinced to add a one after the othercomparison in Boomer output. Linear trapezoid and Purves' linear /log-linear method - db]`
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• On 8 Aug 2000 at 10:29:56, ml11439.aaa.goodnet.com sent the message
`Dr.Lakritz,     The reason why bioavailability(F)= AUCoral/AUCiv is thatthe first order absorption equation Cp= [KaFd/Vd(Ka-Ke)][e-Ket- e-Kat]when integrated from infinity->0 reduces to:             AUCoral= FD/VdKe     When compared to AUCiv= D/VdKe             AUCoral= (F)*AUCiv     Since an F value greater than one is not possible, the possibilityof a decreased Ke when given by the IV route is one of the most likelypossibilities. At any rate, some factor is affecting the determinationof the Ke value when given by different routes.                         Mike Leibold, PharmD, RPh                         ML11439.aaa.goodnet.com`
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• On 8 Aug 2000 at 11:07:27, "stefans" (stefans.-at-.moag.gov.il) sent the message
`Dear Friends,I've been following your interesting discussion concerning the AUCdetermination. I'd like to add one more aspect to the discussionthat, in my understanding, nobody has addressed yet.The analytical error at the end of a concentration vs. time curveincreases (especially when the concentrations are analyzed to thelimit of the method). There appears to be a possibility that onlydata points that fall on the "positive" side of the LOQ will berecorded while those on the "negative" side would fall in thecategory of non-quantifiable. This seems to be a possibilityespecially with shallow curves.My question is: What is the effect of the analytical error at orclose to the LOQ on the determination of AUC by curve fittingespecially when weighting is used?With best regaards,Stefan Soback, DVM, PhDKimron Veterinary InstituteBeit DaganIsrael`
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• On 8 Aug 2000 at 21:20:09, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message
`by way of David_Bourne wrote:>>       The reason why bioavailability(F)= AUCoral/AUCiv is that>  the first order absorption equation Cp= [KaFd/Vd(Ka-Ke)][e-Ket- e-Kat]>  when integrated from infinity->0 reduces to:>>               AUCoral= FD/VdKe>       Since an F value greater than one is not possible, the possibility>  of a decreased Ke when given by the IV route is one of the most likely>  possibilities. At any rate, some factor is affecting the determination>  of the Ke value when given by different routes.Algebra   CL=Vd x KeBiology   Ke=CL / Vd From a biological perspective clearance and volume of distribution arequite distinct. Ignoring random variation one can consider them to be"constant". The elimination rate "constant" is only a constant if CL andVd do not change. A change in either CL or Vd will alter Ke.However, ONLY a change in CL will change AUC. A change in Vd cannotchange AUC (for a drug wihe linear kinetics integrated from 0-inf for asingle dose or over a dosing interval at steady state).The asssumption of linear kinetics is easily violated when oral dosesare given. Rate dependent extent of absorption can occur easily becauseportal vein concs can be high enough to cause the mixed orderelimination during first pass. Inspection of plasma conc profiles mayonly support first order elimination.A higher AUCoral than AUCiv can occur if the oral absorption rate israpid. This can lead to erroneous conclusions. One of the most infamousexamples is the widely cited but erroneous assertion that the stomach isa major source of first pass metabolism of ethanol. Frezza M. di PadovaC. Pozzato G. Terpin M. Baraona E. Lieber CS. High blood alcohol levelsin women. The role of decreased gastric alcohol dehydrogenase activityand first-pass metabolism [published errata appear in N Engl J Med 1990May 24;322(21):1540 and 1990 Aug 23;323(8):553]  New England Journal ofMedicine. 322(2):95-9, 1990 Jan 11. These authors completely ignored themixed order elimination of ethanol and interpreted the ratio ofAUCoral/AUViv as a measure of extent of absorption. The NEJM is not anauthoratative source for PK issues!--Nick Holford, Division of Pharmacology & Clinical PharmacologyUniversity of Auckland, Private Bag 92019, 85 Park Road, Auckland, NZemail: n.holford.at.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm`
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• On 8 Aug 2000 at 21:21:20, Nick Holford (n.holford.-at-.auckland.ac.nz) sent the message
`"stefans (by way of David_Bourne)" wrote:>>  The analytical error at the end of a concentration vs. time curve>  increases (especially when the concentrations are analyzed to the>  limit of the method). There appears to be a possibility that only>  data points that fall on the "positive" side of the LOQ will be>  recorded while those on the "negative" side would fall in the>  category of non-quantifiable. This seems to be a possibility>  especially with shallow curves.>>  My question is: What is the effect of the analytical error at or>  close to the LOQ on the determination of AUC by curve fitting>  especially when weighting is used?The role of analytical error (and other sources of random variation) canbe recognized by using a model based approach and bypassing the WaltDisney AUC step all together.--Nick Holford, Division of Pharmacology & Clinical PharmacologyUniversity of Auckland, Private Bag 92019, 85 Park Road, Auckland, NZemail: n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm`
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• On 9 Aug 2000 at 10:07:07, prashant bodhe (prashnvb.-at-.dr.com) sent the message
`Dear MembersMy opinion on the issue is as follows1. In case of biological research field especially like PK, we can closelyapproximate the facts and not determine them exactly. And I feel itimpossible and un-necessary as well to have 100% accuracy. (one of thefactors is cost involved).With the accuracy and precision levels currently prevailing, the purpose ofstudying the PK of drugs is generally seved.2. The impact of the values close to LOQ on AUC can be ignored. Because thecontribution of the corresponding AUC(Tm to Tn) to the total AUC isgenerally small.3. In very few cases it may not be so. Then the analytical method needs'upgradation'`
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• On 9 Aug 2000 at 10:08:12, "David S. Farrier" (DFarrier.aaa.SummitPK.com) sent the message
`>The role of analytical error (and other sources of random variation) can>be recognized by using a model based approach and bypassing the Walt>Disney AUC step all together.>>-->Nick Holford, Division of Pharmacology & Clinical PharmacologyTrue Nick, except FDA only accepts the Disney-like, model-independentapproach for bioequivalence studies.David S. Farrier, Ph.D.Summit Research Services1374 Hillcrest DriveAshland, OH 44805 USATel/Fax:  (419)-289-9207Email:     DFarrier.-a-.SummitPK.comWeb Site: www.SummitPK.com`
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• On 10 Aug 2000 at 23:27:09, Roger Jelliffe (jelliffe.-a-.usc.edu) sent the message
`Dear Members:         About accuracy and precision. They ARE important. At leastfor potentially toxic drugs, we are not playing with data, we aremodeling it so we can act on it OPTIMALLY, that is, to develop dosageregimens to achieve desired target goals with maximal precision.         The issue of LOQ is also important here. I hate to belaborthis point, but when we have no other info about the specimen exceptthe measured value itself, then there most certainly IS a LOQ.However, when we do most PK work, that is not the case. We know, withreasonable precision, when the doses were given and when the sampleswere obtained. So we know the drug is really present. Even simplelinear models show us that the last molecule is never excreted. So,instead of having to ask, as we must in toxicological work, if thedrug is PRESENT OR NOT, and having therefore to develop a LOQ, weknow the drug is present. The question being asked is not the same asin toxicology. It is instead - HOW MUCH drug is present?         Now comes the question of weighting the data optimally. Mostpeople agree that weighting data by its Fisher information isappropriate - the reciprocal of the variance of the data point. Itworks quite well. The point is that when you determine the assayerror and express it as a polynomial function of the concentration,that important relationship continues over the entire range of theassay, down to and including the blank, if you set it up correctly.This point is discussed in more detail in an article in Therap DrugMonit 15:380-393, 1993, especially the section on Evaluating theCredibility of Population Parameter Values and Serum Level Data, pp.386-391.Best regards,Roger JelliffeRoger W. Jelliffe, M.D. Professor of Medicine, USCUSC Laboratory of Applied Pharmacokinetics2250 Alcazar St, Los Angeles CA 90033, USAPhone (323)442-1300, fax (323)442-1302, email=  jelliffe.-a-.hsc.usc.eduOur web site=  http://www.usc.edu/hsc/lab_apk*******`
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• On 11 Aug 2000 at 13:04:24, "Derrick J (Rick) Bates" (DJ.Rick.Bates.aaa.pnl.gov) sent the message
`>  I've been following your interesting discussion concerning the AUC>  determination. I'd like to add one more aspect to the discussion>  that, in my understanding, nobody has addressed yet.>>  The analytical error at the end of a concentration vs. time curve>  increases (especially when the concentrations are analyzed to the>  limit of the method).	First, a digression from your main point below.	Well, it is true that the analytical error/precision at the end of	a concentration vs. time curve typically increases in a RELATIVE	sense, i.e, the analytical error relative to the result increases.	It is also true that the analytical error/precision at the end of	a concentration vs. time curve generally decreases in an ABSOLUTE	sense, i.e. the analytical error/precision itself gets smaller and	smaller towards the end of the concentration vs. time curve	(basically the low end of the calibration curve).>  There appears to be a possibility that only>  data points that fall on the "positive" side of the LOQ will be>  recorded while those on the "negative" side would fall in the>  category of non-quantifiable. This seems to be a possibility>  especially with shallow curves.>	Happens all the time in the data I see.	But this is a topic that I will avoid getting mired in here.	Oops, now a second digression just occurred to me.	Another reason I prefer to use AUCs (Area Under Curve)	instead of AUTs	(Area under Trapezoids)	is because of the ability to generate uncertainty estimates	(measures of quality) for AUCs whereas I seem to have to accept	AUTs with no concept of uncertainty or quality measure for them.>  My question is: What is the effect of the analytical error at or>  close to the LOQ on the determination of AUC by curve fitting>  especially when weighting is used?>	Since the model fitting programs I am aware of try to minimize	the ABSOLUTE errors instead of the RELATIVE errors then	analytical error at the end of the curve should have minimal	effect on the determination of the AUC.	Having said that, it must of course be caveated when weighting	is used.  The end effect of weighting is minimize what become	"relative" errors.	My personal experience has been that LOQ and weighting seem to	have minimal effect on the AUC estimates except in those cases	where the terminal elimination becomes very flat and gets	extrapolated to infinity. It is this extrapolation to infinity	that is the source of the problem.  The same problem would exist	for the AUT if it was extrapolated to infinity by the terminal	elimination estimate.`
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• On 16 Aug 2000 at 16:07:03, Roger Jelliffe (jelliffe.at.usc.edu) sent the message
`Dear Derrick:         Good for you! You are right - the absolute error, not therelative error, is the important thing. The SD and the variance, notthe CV%, are what is important in determining the credibility of adata point by its Fisher informationBest regards,Roger JelliffeRoger W. Jelliffe, M.D. Professor of Medicine, USCUSC Laboratory of Applied Pharmacokinetics2250 Alcazar St, Los Angeles CA 90033, USAPhone (323)442-1300, fax (323)442-1302, email=  jelliffe.aaa.hsc.usc.eduOur web site=  http://www.usc.edu/hsc/lab_apk*****`
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