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The following message was posted to: PharmPK
Dear all,
I was wondering if someone out there could help me to understand what is
the best way to write differential equations: should I use mass terms or
concentration terms?
For a onecompartment model, one can start with either mass or
concentration and will get the same equation using V1 as the converting
factor, since there's only one volume in the model.
A1=total amount, C1=conc, V1=apparent volume of the compartment
Start with mass:
(1) dA1/dt =  k10 * A1
(2) dA1/dt / V1 = k10 * A1 / V1
(3) dC1/dt = k10 * C1
Start with concentration:
(4) dC1/dt = k10 * C1
However, when we talk about a twocompartment model, since there are two
different volume terms involved, V1 and V2, the differential equations
are no longer interchangeable (Note the difference between eqn (5) and
(10), or eqn (7) and (8)):
A1=total amount in compartment 1, C1=conc in compartment 1, V1=apparent
volume of compartment 1
A2=total amount in compartment 2, C2=conc in compartment 2, V2=apparent
volume of compartment 2
k12=rate constant from compartment 1 to 2, k21=rate constant from
compartment 2 to 1
Start with mass:
(5) dA1/dt = k12 * A1 + k21 * A2
(6) dA1/dt / V1 = k12 * A1 / V1 + k21 * A2 / V1
(7) dC1/dt = k12 * C1 + k21 * A2 / V1
Start with concentration:
(8) dC1/dt = k12 * C1 + k21 * C2
(9) dC1/dt * V1 = k12 * C1 * V1 + k21 * C2 * V1
(10) dA1/dt = k12 * A1 + k21 * C2 * V1
I've seen usage of eqn (5)(7) in many publications, but I've also seen
eqn (8) in some textbooks as well. I'm a little confused regarding which
one is valid, and why.
Then, if we move on to a 2ndorder process, let's say that 1+2=3, and
everything happens within one compartment:
********* ********* kon *********
* A1,C1 * + * A2,C2 * > * A3,C3 *
********* ********* < *********
koff
V=apparent volume of the compartment
kon=association rate constant, koff=dissociation rate constant
Start with mass:
(11) dA1/dt = kon * A1 * A2 + koff * A3
(12) dA1/dt / V = kon * A1 * A2 / V + koff * A3 / V
(13) dC1/dt = kon * A1 * C2 + koff * C3
(14) Or: dC1/dt = kon * C1 * A2 + koff * C3
Start with concentration:
(15) dC1/dt = kon * C1 * C2 + koff * C3
(16) dC1/dt * V = kon * C1 * C2 * V + koff * C3 * V
(17) dA1/dt = kon * A1 * C2 + koff * A3
(18) Or: dA1/dt = kon * C1 * A2 + koff * A3
In this case, depending on whether we use mass or concentration to start
with, we get different equations again. (Note the difference between
eqn(11) and (17)/(18), or eqn(13)/(14) and (15). It seems that
eqn(15)(18) are the ones used in publications, which makes sense in
chemical kinetics where concentrations are used in writing rate
equations.
I guess when we talk about mass transfer from one compartment to
another, we should use mass terms, but when there's formation of new
entities (reactive system), concentration terms should be used. But
again, I did see eqn(8) in some textbook (Shargel and Yu, Applied
Biopharmaceutics and Pharmacokinetics), and still wondering if this is a
valid expression, and how do we explain the difference.
I may have just trapped myself in a loop and couldn't get out. If
someone is willing to spend some time explaining all this to me, that
will be very much appreciated!!
Thanks,
yinuo
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The following message was posted to: PharmPK
Pang Yinuo,
It's rare to have closed 2compartment noneliminating PK equations in
textbooks and in the literature.
Chemical kinetics receives a slightly different treatment than
pharmacokinetics, because with the former, you may be dealing with a
closed
reaction chamber, where the volume of a "second compartment" or reactive
element may be shared. With PK however, your kinetic processes may
describe
membrane transfer or better yet, tissue binding, where the volume is not
shared by the mass. Right away, you will recognize that to describe the
"concentration of drug bound to tissue" is not meaningful, and
consequently
are often termed "apparent" volumes for this reason (although I don't
like
that term personally).
So, volumes are not input variables in PK models, but rather are
parameterized wherever possible.
The correct approach is to directly derive your models in terms of your
system under observation.
dA1/dt = k12 * A1 + k21 * A2
dC1/dt = k12 * A1/V1 + k21 * A2 / V2
are both correct, ...for a closed system.
Hope it helps.

Shawn D. Spencer, Ph.D., R.Ph.
Assistant Professor of Biopharmaceutics
Florida A&M College of Pharmacy
Tallahassee, FL 32307
shawn.spencer.at.famu.edu
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Yinuo
The book (Shargel and Yu, Applied Biopharmaceutics and Pharmacokinetics)
contains number of misprints and errors. The 2compartment model
equations (4.4)(4.5) that is presented there (chapter 4) do not satisfy
mass balance equations, and should not be used. So, your system (1)(3)
is correct, (5)(7) is correct, (8)(10) is not correct, (11)(14) is
not correct (rate of drugtarget complex production is proportional to
concentrations of components, not to their amounts), (15)(18) is
correct if you assume equal volumes for A1, A2 and A3
This is not the only case when books and published journal articles
contain errors or misprints, so it is always helpful to check
assumptions behind equations and the underlying biology.
Thanks
Leonid

Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
email: LGibiansky at quantpharm.com
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