Dear colleagues,Back to the Top
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
Back to the Top
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/
Back to the Top
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
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JZB Consulting web site: http://members.aol.com/JanuszB/consult.htm
[Probably t(max) = (ln(ka) - ln(CL/V))/(ka - CL/V) - db]
Back to the Top
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
Back to the Top
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
Back to the Top
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
Hi,Back to the Top
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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