- On 10 Feb 2004 at 14:11:26, "Jorge Duconge" (duconge.at.cieb.sld.cu) sent the message
Dear colleagues,

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Can anyone of you help me understand how the AUC expression from the

allometric approach (viz, AUC=1/a) is derived?. Unfortunatelly, I

cannot follow the most frequently provided deduction on papers and

textbooks. Using a single bolus dose, mono-exponential model as

example:

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 to

infinity) and here I have always found a sort of hocus-pocus as all the

texts I have read (not many!!) "skip" this part and yield a reduced

expression AUC = 1/a ("a" being the allometric coefficient for

clearance). Assuming that both [dose/c*BW^d] and [(a*BW^[b-d])/c] are

constants, then after integration and cancel similar terms out of the

expression 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 I

have missed?. Thanks a lot in advance and best wishes,

Jorge Duconge

Jorge Duconge, PhD. MSC.

Center for Biological Evaluation & Research,

Institute of Pharmacy and Foods

University of Havana, Cuba

Havana 36 CP 13600.

Tel. 537 271 9532

Fax 537 33 6811

e-mail: duconge.aaa.cieb.sld.cu - On 11 Feb 2004 at 10:58:24, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message

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Jorge,

I don't know of any common derivation in textbooks of an allometric

expression for AUC starting from a bolus input one compartment model.

The derivation you finally use:

CL=aBW^C

AUC=Dose/CL

seems the simplest way to do this.

You only need to understand allometric predictions for fundamental PK

parameters to be able to derive any other PK statistic dependent on

these parameters (such as deriving AUC from Dose/CL or Tmax from

ln(Ka-CL/V)/(Ka-CL/V)).

Nick

Nick Holford, Dept Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New

Zealand

email:n.holford.-at-.auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556

http://www.health.auckland.ac.nz/pharmacology/staff/nholford/ - On 10 Feb 2004 at 22:30:00, jbyczkowski.-at-.netscape.net sent the message

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Nick Holfordwrote:

"...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.html

JZB Consulting web site: http://members.aol.com/JanuszB/consult.htm

[Probably t(max) = (ln(ka) - ln(CL/V))/(ka - CL/V) - db] - On 11 Feb 2004 at 17:14:18, "Jorge Duconge" (duconge.-a-.cieb.sld.cu) sent the message

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Dear professor Nick,

Thanks for your excellent remark. I understand your point. However, the

allometric derivation for AUC starting from a bolus input

mono-exponential

model that I mentioned in my earlier mail was indeed taken

from"Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and

Applications, second edition, 1997, by Johan Gabrielsson and Daniel

Weiner,

on page 160, 3.8 Interspecies Scaling, 3.8.3 What is allometry?,

wherein the

authors stated:

"Substitution of the volume and clearance in a mono-exponential model by

equations 3:247 (i.e., Cli = a BWi^b, BW means body weigth) and 3:249

(i.e.,

Vi = c BWi^d) gives

C = Div/V exp^[-Cl/V]*t = D/[cBW^d] exp^[-(aBW^[b-d])/c *t]

(3:252)

Integrating equation 3:252 yields

AUC 0 to infinity = D/[cBW^d]* Integral from 0 to infinity

{exp^[-(aBW^[b-d])/c *t] dt} (3: 253)

which after evaluation reduces to

AUC 0 to infinity = 1/a

(3:254)."

So, according to your viewpoint my concern for AUC derivation could be

solved understanding the empirically-deduced allometric equations for

fundamental PK parameters (viz, CL and V), and using these I could get

an

AUC expression equal to Dose/aBW^b (Dose/CL), but not any like AUC

=1/a, and

this is exactly the point I would like to raise. Many regards,

Jorge

Jorge Duconge, PhD. MSC.

Center for Biological Evaluation & Research,

Institute of Pharmacy and Foods

University of Havana, Cuba

Havana 36 CP 13600.

Tel. 537 271 9532

Fax 537 33 6811

e-mail: duconge.aaa.cieb.sld.cu - On 13 Feb 2004 at 16:26:30, "Hans Proost" (j.h.proost.aaa.farm.rug.nl) sent the message

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Dear Jorge Duconge,

You wrote, quoting "Pharmacokinetic and Pharmacodynamic Data Analysis:

Concepts and Applications, second edition, 1997, by Johan Gabrielsson

and

Daniel Weiner, on page 160, 3.8 Interspecies Scaling, 3.8.3 What is

allometry?:

"What is meant by a therapeutic availability can be corelated

very well with the availablity of drug or metabolite or its right

stereoisomeric form, in right concentration or quantity

for appropriate amount of time at the site of action(s).

"

Please note that the derivation of eq. 3:254 is wrong, as can be seen

easily

(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 first

edition of the book by Gabrielsson and Weiner. Does anyone know whether

the

above error is corrected in the third edition of 2002?

Best regards,

Hans Proost

Johannes H. Proost

Dept. of Pharmacokinetics and Drug Delivery

University Centre for Pharmacy

Antonius Deusinglaan 1

9713 AV Groningen, The Netherlands

tel. 31-50 363 3292

fax 31-50 363 3247

Email: j.h.proost.-at-.farm.rug.nl - On 16 Feb 2004 at 16:21:27, "Jorge Duconge" (duconge.aaa.cieb.sld.cu) sent the message

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Dear Hans,

Many thanks for your comments, but the problem seems to be more complex

than

merely a wrong deduction. Actually, while reading a seminal paper by

Harold Boxenbaum and

Richard D'Souza (Physiological Models, allometry, neoteny, space-time

and pharmacokinetics,

from Pharmacokinetics, Edited by A. Pecile and A. Rescigno, Plenum

Publishing

Co., NY, 1988, on page 206), I found the authors made mention of the

same statement (viz, The area

under the curve is 1/a), and once again considering an example in which

it is assumed a drug

eliminated from the body monoexponentially, but this time in order to

refer to the AUC of the

semilogarithmic syndesichron plot for antipyrine disposition in eleven

mammals, by using

C/[Dose/BW^d] versus t*BW^(b-d)*W^z coordinates (W means brain mass

[Kg] and z is its

corresponding 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 to

obtain superimposable curves with AUC equal to 1/a as can be figured

out from the last paragraph on this page. Can anyone confirm this

assumption?.

Best regards,

Jorge Duconge

Jorge Duconge, PhD. MSC.

Center for Biological Evaluation & Research,

Institute of Pharmacy and Foods

University of Havana, Cuba

Havana 36 CP 13600.

Tel. 537 271 9532

Fax 537 33 6811

e-mail: duconge.aaa.cieb.sld.cu - On 25 Feb 2004 at 00:30:40, "Harold Boxenbaum" (harold.at.arishel.com) sent the message
Hi,

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I believe I and Bob Ronfeld derived the requested relationship (or one

similar to it) in the following publication:

Boxenbaum, H. and R. Ronfeld. Interspecies pharmacokinetic scaling and

the Dedrick plots. Am. J. Physiol. 245: R768\0x201A\0xC4\0xEBR775 (1983).

Harold Boxenbaum, Ph.D.

Pharmaceutical Consultant

Arishel Inc.

14621 Settlers Landing Way

North Potomac, MD 20878-4305

(P) 301-424-2806

(F) 301-424-8563

Email: harold.at.arishel.com

Website: www.arishel.com

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