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The following message was posted to: PharmPK
"Hans Proost (by way of David Bourne)" wrote:
> You proposed to use the following equations:
> > V(blood)*C(blood)'=Q*C(liver)-Q*C(blood)-k12*C(blood)+k21*C(tissue)
> > V(tissue)*C(tissue)'=k12*C(blood)-k21*C(tissue).
> These equations are wrong, as can be seen from the dimensions:
> k12 and k21 cannot be rate contants with dimension 1/time.
> Written in this form, k12 and k21 are intercompartmental
> In classical compartmental modeling, differential equations are
> written in terms of amounts, and in that case rate constants are
I agree with your concern that the symbols used for the equations are
misleading but the equations appear to be correct otherwise. 'k12'
does indeed have dimensions of volume/time and would be more
appropriately written as a clearance e.g. 'CL12'. However, there is
no need to insist on using rate constants.
IMHO rate constants are obsolete artefacts and modern understanding
of pharmacokinetics is based on clearance (David -- are you reading
[Yes - I'll try to answer within your message even though it is
probably confusing - quicker than waiting for the original to be sent
out ;-) - db]
Your teaching web site http://www.boomer.org/c/p1/ would be wonderful
[thank you :-) - db]
if you used clearance instead of rate constants e.g.
http://www.boomer.org/c/p1/Ch04/Ch0403.html Equation IV-1 implies
rate of elimination is dC/dt
[I'm trying to get the students started with the basics of first
order kinetics - seemed a reasonable way to start - db]
when I think it is more sensible to think of elimination rate as
dA/dt because this immediately leads to prediction of the maintenance
dose rate from the steady state elimination rate).
[I don't get to multiple doses for many chapters later - BTW how do
you calculate Cpmin and Cpmax using clearance? - db]
When solving differential equations it makes no difference from a
strictly mathematical perspective whether you use rate constants or
clearances (provided the dimensions are correct).
[You are right about the differential equations. Actually I have no
problem using rate constants with dC/dt or dA/dt, especially for
empirically based compartmental models. I use dA/dt and C with PBPK
using blood flow == clearance does that count? - db]
The important reason to parameterise in terms of clearance and
volume comes when applying pharmacokinetics in a population context.
Covariates usually influence clearance and volume independently.
Clearance and volume are reasonable population constants whose group
values can then be sensibly adjusted in light of covariate
information such as weight, renal function, other drugs, etc.
[You can do the same thing with rate constants if you have an
Changes in clearance can be readily applied to other models e.g.
with a different number of compartments, but this won't work with rate
[I don't understand this - db]
Nick Holford, Divn Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand
email:n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
[I'm sure you know a lot of the early PK work (preceding my
involvement ;-) was done with physical pharmacy people familiar with
chemical kinetics. Thus the use of dC/dt = -k*Cp. Even today it seems
easier to get students to basic dosage regimen development and
graphical parameter estimation using compartmental models. You are
right you can do some of this with clearance approaches, especially
maintenance infusions and steady state concentrations. But how do you
calculate how long it takes to get to therapeutic levels? I know you
can reparameterize using CL = V*k or k = CL/V. I still think the
choice is a matter of what type of model you are using.
For compartmental models - empirically defined you can easily use kel
and V to calculate dosing regimens, parameters from data, effect of
renal functions, effect of protein binding, non-linear elimination
The non-compartmental, 'model independent', moment analysis method
have the advantage of being easy to automate and give a number of
useful parameters. However, the 'finer' detail available with more
'complete' modeling can be lost.
For even more detail, physiologically based pk models are available
including blood flow and clearance terms. These models require more
data and extensive modeling.
Many experienced pharmacokinetic analyst is able to work with any of
these model types. For beginners needing some detail the
compartmental approach is attractive.
I've revisited a text or two today and thought of some simulations
for my graduate student to try since I found you message in my inbox.
I do need to revisit clearance...maybe I can add some non
compartmental material earlier in class. Actually I find I'm teaching
two new courses in our new PharmD program combining the first course
material referenced above with computer modeling and equation
derivation (http://www.boomer.org/c/p2/ - still just pdf from my
powerpoint presentations). I may need to include more clearance
emphasis is this material. Our clinical pk course is taught as an
elective by another faculty member. - db]
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The following message was posted to: PharmPK
Dear Dr. Holford,
Thank you with your reply on my comments. My concern was
related both to the use of the symbol k12 (suggesting the meaning
of a rate constant) and the use of the word microconstant, also
suggesting a rate constant.
I apologize to Dr. Nagaraja that my statement was as imprecise as
In addition you wrote:
> IMHO rate constants are obsolete artefacts and modern
> understanding of pharmacokinetics is based on clearance
I fully agree, as I do with your excellent explanation about the
difference. I am happy with your ongoing efforts to clarify the
difference between PK and mathematics.
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
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