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Dear all,
I have constructed a PK/PD model similar to what follows:
dA1/dt= -A1*C*K12
dA2/dt= A1*C*K12 - A2*K20
I need to compare the time course of the system with respect to the
transfer from compartment 1 to 2 and the elimination from compartment
2. The main objective is to determine which process is faster: transfer
from A1 to A2, or elimination from A2. My problem is that K12 is a
second order rate constant, dependent on A1 and C, whereas K20 is a
first order rate constant, dependent on A2. Thus K12 has the units of
1/(C*time), while K20 has the unit of 1/time. Can anybody suggest a
general approach in comparing the two rates? References to published
material are highly welcome.
Regards,
Toufigh Gordi
[What is the differential equation for 'C'? Does C vary with time
within a particular experiment? - db]
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I would have thought that a simple and functionally-relevant way of
doing
this would be to see how your model behaves. I assume that the chemical
starts in compartment 1 - if not then what follows may not apply. If
your
model predicts only low levels in compartment 2 at any timepoint then
tranfer out of 2 is fast compared to transfer from 1 to 2. If chemical
builds up in compartment 2 then the opposite is true. You could do
this by
simulation, or as the model is so very simple you could analytically
derive
the maximum concentration in compartment 2 over time, given known
initial
conditions.
There is a danger in overinterpretting the parameters of such models and
forgetting that it is how the whole model and data behaves that is
important,
Kim
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Dear Dr. Gordi,
With respect to your question about the model:
> I have constructed a PK/PD model similar to what follows:
>
> dA1/dt= -A1*C*K12
> dA2/dt= A1*C*K12 - A2*K20
I have three comments:
1. I do not understand the rationale of this model. As was asked by
David Bourne: what is C?
It seems that your equations refer to an absorption model, in which the
rate of absorption is dependent also on the concentration (or whatever)
of something else?
2. The question how to compare the rate constants is simple. The
absorption rate constant (or 'drug entry constant') is C*K12, which has
the unit of reciprocal time.
3. The question why one would compare the rate constants is not clear
to me. Perhaps there may be a valid reason for this; if so, please
explain. If C*K12 can be regarded as an absorption rate constant, and
K20 as an elimination rate constant, a comparison of the numeric values
is meaningless. Both constants refer to completely different processes.
'C*K12' is the rate of removal of drug from comp. 1 divided by the
amount in comp. 1, and 'K20 is the rate of removal of drug from comp. 2
divided by the amount in comp. 2. Both rate constants are dependent on
the volumes of the corresponding compartments. What would we learn from
the knowledge which parameter, C*K12 or K20, is largest?
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.aaa.farm.rug.nl
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It seems that the problem is more complex than I thought. I had
originally given these equations:
"
dA1/dt= -A1*C*K12
dA2/dt= A1*C*K12 - A2*K20
"
Here comes additional information about the model. The full model
describes the effect of a drug on the number of organisms. C is
time-variable drug concentrations, which could be simplified as
following a mono-exponential elimination rate (although in reality it
could be much more complex, but still follow a mono-exponential decay).
A1 is the number of viable organisms, which due to a direct drug effect
are transferred to an "injured" form (A2). Thus, A2 is the number of
injured organisms, which are then cleared from the body. The full model
is again more complex than the one presented here, but I don't find it
of relevance to my aim, i.e. comparing the rate of transfer from A1 to
A2 and A2 out. The observed number of organisms are the sum of A1 and
A2 since there is no practical way of separating the viable and injured
organism. To complicate the situation even more, there is another
transfer to A2: from another viable, but not visible, organism
compartment, with another second-order rate constant.
During the model-building, I tried a model without any "injured"
compartment but the fits gets much better when it is added to the
system, which would imply the A2 to 0 is slower than A1 to A2. However,
I would like to have a more solid evidence of this by a possible
comparison of the two rates. So the question is: is there any way of
comparing a first-order (in this case K20) rate constant and a
second-order (K12) rate constant?
I think it is a good idea, as proposed by Kim Travis, to simulate the
model and see how the numbers change.
Thank you!
T. Gordi
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The following message was posted to: PharmPK At 02:15 AM 1/20/2004,
Hans Proost wrote:
"1. I do not understand the rationale of this model. As was asked by
David Bourne: what is C?
It seems that your equations refer to an absorption model, in which the
rate of absorption is dependent also on the concentration (or whatever)
of something else?"
I also wonder what C refers to in the equation.
For passive diffusion, which governs the absorption for most drugs, the
absorption rate is dependent on two concentrations - one on each side
of the membrane. Let's remember that the modern definition of
absorption is crossing the apical membrane into the enterocytes, not
reaching the portal vein or systemic circulation (this was emphasized
repeatedly at the Oral Drug Delivery Course in Vail, Colorado last
week).
For passive diffusion, Ficks' Law says that molecules cross a membrane
at a rate that is determined by the difference in concentration across
the membrane:
J = D * (C1 - C2)
2. The question how to compare the rate constants is simple. The
absorption rate constant (or 'drug entry constant') is C*K12, which has
the unit of reciprocal time.
The practice of assuming an absorption rate "constant" is a simplifying
assumption that, in our experience, rarely holds true. In fact, Ka is
never a constant, although in some situations, it can be treated as one
with acceptable error.
Consider the equation for Fick's Law above and what can cause the flux
(J) to change.
D: for absorption, this is the product of the local effective
permeability (Peff) and the surface/volume ratio. The local effective
permeability, Peff, changes with position in the gastrointestinal
tract. It is different in duodenum, jejunum, ileum, and colon for most
drugs. The differences are caused by different surface area, different
tight junction gap (for paracellular transported drugs), different pH
(ionization effects), and different transporter expression (for influx
or efflux). Sometimes the differences are small. Usually the colon Peff
is significantly different from the small intestine values.
C1 and C2: assume C1 is the lumen concentration, and C2 is the
concentration in the enterocytes. When C1 >> C2, you can treat C2 as
zero. But how long does that last? As the drug is absorbed, C1 drops
and C2 increases. When they become equal, absorption stops. When the
enterocyte concentration is higher than lumen, the flux is reversed,
and drug is exsorbed/secreted into the lumen (a phenomenon that has
been observed for iv doses). The old-fashioned approach of assuming an
absorption rate "constant" ignores this, and it ignores the fact that
the absorption rate steadily decreases as C1 decreases and C2
increases. In our experience, using a concentration gradient (Fick's
Law) approach provides far more accurate and consistent results -
enabling a single model to predict plasma concentration-time results
for different doses without a change in model parameters.
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (SIMU)
1220 W. Avenue J
Lancaster, CA 93534-2902
U.S.A.
http://www.simulations-plus.com
Phone: (661) 723-7723
FAX: (661) 723-5524
E-mail: walt.aaa.simulations-plus.com
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