PharmPK Discussion - Allometry

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• On 10 Feb 2004 at 14:11:26, "Jorge Duconge" (duconge.at.cieb.sld.cu) sent the message
`Dear colleagues,Can anyone of you help me understand how the AUC expression from theallometric approach (viz, AUC=1/a) is derived?. Unfortunatelly, Icannot follow the most frequently provided deduction on papers andtextbooks. Using a single bolus dose, mono-exponential model asexample:C = [Dose/Vd]* exp(-CL/Vd*t)but allometrically, Vd= c*BW^d and CL=a*BW^b, then substituing,C = [Dose/c*BW^d]* exp(- (a*BW^[b-d])/c * t)Finally, to obtain AUC we should integrate C over time (from 0 toinfinity) and here I have always found a sort of hocus-pocus as all thetexts I have read (not many!!) "skip" this part and yield a reducedexpression AUC = 1/a ("a" being the allometric coefficient forclearance). Assuming that both [dose/c*BW^d] and [(a*BW^[b-d])/c] areconstants, then after integration and cancel similar terms out of theexpression I obtained,AUC = [Dose/a*BW^b], which is much more reasonable to me as per AUC=Dose/CL and remembering that CL=a*BW^b.So, can anyone of you explain this to me?. What is exactly the step Ihave missed?. Thanks a lot in advance and best wishes,Jorge DucongeJorge Duconge, PhD. MSC.Center for Biological Evaluation & Research,Institute of Pharmacy and FoodsUniversity of Havana, CubaHavana 36 CP 13600.Tel. 537 271 9532Fax 537 33 6811e-mail: duconge.aaa.cieb.sld.cu`
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• On 11 Feb 2004 at 10:58:24, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message
`Jorge,I don't know of any common derivation in textbooks of an allometricexpression for AUC starting from a bolus input one compartment model.The derivation you finally use:CL=aBW^CAUC=Dose/CLseems the simplest way to do this.You only need to understand allometric predictions for fundamental PKparameters to be able to derive any other PK statistic dependent onthese parameters (such as deriving AUC from Dose/CL or Tmax fromln(Ka-CL/V)/(Ka-CL/V)).NickNick Holford, Dept Pharmacology & Clinical PharmacologyUniversity of Auckland, 85 Park Rd, Private Bag 92019, Auckland, NewZealandemail:n.holford.-at-.auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556http://www.health.auckland.ac.nz/pharmacology/staff/nholford/`
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• On 10 Feb 2004 at 22:30:00, jbyczkowski.-at-.netscape.net sent the message
`Nick Holford  wrote:"...Tmax from ln(Ka-CL/V)/(Ka-CL/V)...">>Nick:(Ka-CL/V)/(Ka-CL/V) = 1.therefore, ln(Ka-CL/V)/(Ka-CL/V) = 0.I do not think that this is what you meant ;-)Best wishes.          Janusz Z. Byczkowski, Ph.D.,D.Sc.,D.A.B.T.                          Consultant                      212 N. Central Ave.                      Fairborn, OH 45324                      voice (937)878-5531                   secure fax (702)446-9127                     e-mail januszb.aaa.AOL.com     homepage: http://members.aol.com/JanuszB/index.htmlJZB Consulting web site: http://members.aol.com/JanuszB/consult.htm[Probably t(max) = (ln(ka) - ln(CL/V))/(ka - CL/V) - db]`
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• On 11 Feb 2004 at 17:14:18, "Jorge Duconge" (duconge.-a-.cieb.sld.cu) sent the message
`Dear professor Nick,Thanks for your excellent remark. I understand your point. However, theallometric derivation for AUC starting from a bolus inputmono-exponentialmodel that I mentioned in my earlier mail was indeed takenfrom"Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts andApplications, second edition, 1997, by Johan Gabrielsson and DanielWeiner,on page 160, 3.8 Interspecies Scaling, 3.8.3 What is allometry?,wherein theauthors stated:"Substitution of the volume and clearance in a mono-exponential model byequations 3:247 (i.e., Cli = a BWi^b, BW means body weigth) and 3:249(i.e.,Vi = c BWi^d) givesC = Div/V exp^[-Cl/V]*t = D/[cBW^d] exp^[-(aBW^[b-d])/c *t](3:252)Integrating equation 3:252 yieldsAUC 0 to infinity = D/[cBW^d]* Integral from 0 to infinity{exp^[-(aBW^[b-d])/c *t] dt}                      (3: 253)which after evaluation reduces toAUC 0 to infinity =  1/a(3:254)."So, according to your viewpoint my concern for AUC derivation could besolved understanding the empirically-deduced allometric equations forfundamental PK parameters (viz, CL and V), and using these I could getanAUC expression equal to Dose/aBW^b (Dose/CL), but not any like AUC=1/a, andthis is exactly the point I would like to raise. Many regards,JorgeJorge Duconge, PhD. MSC.Center for Biological Evaluation & Research,Institute of Pharmacy and FoodsUniversity of Havana, CubaHavana 36 CP 13600.Tel. 537 271 9532Fax 537 33 6811e-mail: duconge.aaa.cieb.sld.cu`
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• On 13 Feb 2004 at 16:26:30, "Hans Proost" (j.h.proost.aaa.farm.rug.nl) sent the message
`Dear Jorge Duconge,You wrote, quoting "Pharmacokinetic and Pharmacodynamic Data Analysis:Concepts and Applications, second edition, 1997, by Johan GabrielssonandDaniel Weiner, on page 160, 3.8 Interspecies Scaling, 3.8.3 What isallometry?:"What is meant by a therapeutic availability can be corelatedvery well with the availablity of drug or metabolite or its rightstereoisomeric form, in right concentration or quantityfor appropriate amount of time at the site of action(s)."Please note that the derivation of eq. 3:254 is wrong, as can be seeneasily(AUC independent of Dose, and of BW?!).The evaluation of eq. 3:253 results in your equation:AUC  0 to infinity = Dose/aBW^b (Dose/CL)Several years ago I made comments on the numerous errors in the firstedition of the book by Gabrielsson and Weiner. Does anyone know whethertheabove error is corrected in the third edition of 2002?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-.farm.rug.nl`
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• On 16 Feb 2004 at 16:21:27, "Jorge Duconge" (duconge.aaa.cieb.sld.cu) sent the message
`Dear Hans,Many thanks for your comments, but the problem seems to be more complexthanmerely a wrong deduction. Actually, while reading a seminal paper byHarold Boxenbaum andRichard D'Souza (Physiological Models, allometry, neoteny, space-timeand pharmacokinetics,from Pharmacokinetics, Edited by A. Pecile and A. Rescigno, PlenumPublishingCo., NY, 1988, on page 206), I found the authors made mention of thesame statement (viz, The areaunder the curve is 1/a), and once again considering an example in whichit is assumed a drugeliminated from the body monoexponentially, but this time in order torefer to the AUC of thesemilogarithmic syndesichron plot for antipyrine disposition in elevenmammals, by usingC/[Dose/BW^d] versus t*BW^(b-d)*W^z  coordinates (W means brain mass[Kg] and z is itscorresponding allometric exponent from the empiric power function CL =a*BW^b*W^z). Perhaps,Gabrielsson and Weiner were searching for a similar goal in order toobtain superimposable curves with AUC equal to 1/a as can be figuredout from the last paragraph on this page. Can anyone confirm thisassumption?.Best regards,Jorge DucongeJorge Duconge, PhD. MSC.Center for Biological Evaluation & Research,Institute of Pharmacy and FoodsUniversity of Havana, CubaHavana 36 CP 13600.Tel. 537 271 9532Fax 537 33 6811e-mail: duconge.aaa.cieb.sld.cu`
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• On 25 Feb 2004 at 00:30:40, "Harold Boxenbaum" (harold.at.arishel.com) sent the message
`Hi,I believe I and Bob Ronfeld derived the requested relationship (or onesimilar to it) in the following publication:Boxenbaum, H. and R. Ronfeld. Interspecies pharmacokinetic scaling andthe Dedrick plots. Am. J. Physiol. 245: R768\0x201A\0xC4\0xEBR775 (1983).Harold Boxenbaum, Ph.D.Pharmaceutical ConsultantArishel Inc.14621 Settlers Landing WayNorth Potomac, MD 20878-4305(P) 301-424-2806(F) 301-424-8563Email: harold.at.arishel.comWebsite: www.arishel.com`
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