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Professor,
The two compartment model can be adapted to oral absorption by
modifying the input function of the differential equation.
dX1/dt= -(k10+k12)X1 + (k21)X2 + KaFDe-Kat
dX2/dt= (k12)X1 - (k21)X2
In the above case, the oral absorption is modeled as a first-
order process KaFDe-Kat, with Ka= first oral absorption constant,
F= bioavailability, and D= the dose. Oral sustained release
medications can be modeled as above with a smaller Ka relative
to the immediate release form, or can be modeled as a zero order
process (or infusion model):
dX1/dt= -(k10+k12)X1 + (k21)X2 + Ko
dX2/dt= (k12)X1 - (k21)X2
As you can see, the difference in the two systems above is
the input function: KaFDe-kat or Ko.
The matrix respresentation of these systems of differential
equations is useful for solving the Laplace transformed systems
for the amount in the plasma compartment:
[SI-A][Xs]= [Us]
A) First order oral absorption:
[(s+k10+k12) -k21 ][X1s]= [KaFD/(s+Ka)]
[ -k12 (s+k21) ][X2s] [ 0 ]
B) Zero order oral absorption:
[(s+k12+k12) -k21 ][X1s]= [Ko(1-e-Ts)/s]
[ -k12 (s+k21)][X2s] [ 0 ]
Solving the above matrices for the amount in the central compartment
yields the following Lapace transforms:
A) First order oral absorption:
X1s= [(KaFD)(s+k21)]/[(s+a)(s+b)(s+Ka)]
B) Zero order absorption:
X1s= (Ko)(1-e-Ts)((s+k21)/[(s)(s+a)(s+b)]
Taing the inverse Laplace transforms of the above equations and
then dividing by Vc yields equations for the plasma concentration for each
oral absorption model:
A) First order absorption model:
Cp= (KaFD)(k21-a)e-at/[(Vc)(b-a)(Ka-a)] +
(KaFD)(K21-b)e-bt/[(Vc)(a-b)(Ka-b)] +
(KaFD)(k21-Ka)e-Kat/[(Vc)(a-Ka)(b-Ka)]
b) Zero order absorption model:
Cp= Ko(k21-a)(1-e-at)/[(Vc)(a)(b-a))] +
Ko(k21-b)(1-e-bt)/[(Vc)(b)(a-b)]
In the zero oder absorption model, the term Ko would represent FD/Tau
and the time (t) would represent the total time of administration as if
it were a continuous infusion. So, FD/Tau could be substituted into the
above equation to give the following equation:
Cp= (FD/Tau)(k21-a)(1-e-at)/[(Vc)(a)(b-a)] +
(FD/Tau)(k21-b)(1-e-bt)/[(Vc)(b)(a-b)]
Since this is a continuous infusion model, it would not require a
multiple dose function to represent the case of several doses, as the
multiple dose condition is factored into the FD/Tau term and the time factor.
The first order absorption model however, requires a multiple dose function
derived from the superposition priciple:
A) First order absorption model multiple dose equation:
Cp= (KaFD)(k21-a)(1-e-naTau)e-at/[(Vc)(b-a)(Ka-a)(1-e-aTau)] +
(KaFD)(K21-b)(1-e-nbTau)e-bt/[(Vc)(a-b)(Ka-b)(1-e-bTau)] +
(KaFD)(k21-Ka)(1-e-nKaTau)e-Kat/[(Vc)(a-Ka)(b-Ka)(1-e-KaTau)]
I hope this was of some Help!!
Mike Leibold, PharmD, RPh
ML11439.aaa.goodnet.com
References
1) Gibaldi, M., Perrier, D., Pharmacokinetics, New York, Marcel Dekker
1975
2) Gibaldi, M., Perrier, D., Pharmacokinetics 2nd ed, New York, Marcel Dekker
1982
3) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug
Intelligence Publications 1975
>Dear Mike,
>
>Last year, I followed your discussions on pharmacokinetic modelling and am
wondering whether you can help me. I need the analytical equations for a
particular model.
>
>We conducted a comparative pharmacokinetic study on immediate and sustained
releaase dosage forms of pentoxyfylline. This drug obeys the two compartment
model. Upon oral administration, it is absorbed and goes through the liver
before entering the blood circulation. It has a high first pass effect. In
fact, it is solely disposed through this organ(liver) with six metabolites.
I was not interested in this metabolism, but followed plasma levels of
unchanged drug. What I need is, two compartment model drug with zero order
release and first order absorption model equations for blood level. Liver
acts like the peripheral compartment into which absorption occurs. The
elimination is also through the liver(no unchanged drug, apparently) and not
from the central compartment. But central compartment is the one sampled.
>
>Searching the literature did not yield the equations I needed. My Laplace
transformation knowledge is not enough for this pupose, unfortunately. If
you could be kind enough to derive these equations, I would be most
grateful. Your effort would no doubt be cited in any paper I produce.
>
>Thanking you for your kind cooperation, I remain,
>
>Prof.Dr.Ilbeyi Agabeyoglu
>Dept.Pharmaceutical Technology,
>Faculty of Pharmacy,
>Gazi University,
>Ankara,
>Turkey
>
>ilbeyi.-at-.tr-net.net.tr
>
>
>
>
>
>
>
>
>Dear Mike,
>
>Last year, I followed your discussions on
>pharmacokinetic modelling and am wondering whether you can help me. I
need
>the analytical equations for a particular model.
>
>We conducted a comparative pharmacokinetic
study
>on immediate and sustained releaase dosage forms of pentoxyfylline. This
drug
>obeys the two compartment model. Upon oral administration, it is absorbed
and
>goes through the liver before entering the blood circulation. It has a high
>first pass effect. In fact, it is solely disposed through this organ(liver)
with
>six metabolites. I was not interested in this metabolism, but followed
plasma
>levels of unchanged drug. What I need is, two compartment model drug with
zero
>order release and first order absorption model equations for blood level.
Liver
>acts like the peripheral compartment into which absorption occurs. The
>elimination is also through the liver(no unchanged drug, apparently) and not
>from the central compartment. But central compartment is the one
>sampled.
>
>Searching the literature did not yield the
>equations I needed. My Laplace transformation knowledge is not enough for
this
>pupose, unfortunately. If you could be kind enough to derive these
equations, I
>would be most grateful. Your effort would no doubt be cited in any paper I
>produce.
>
>Thanking you for your kind cooperation, I
>remain,
>
>Prof.Dr.Ilbeyi Agabeyoglu
>Dept.Pharmaceutical Technology,
>Faculty of Pharmacy,
>Gazi University,
>Ankara,
>Turkey
>
>
>
>
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Copyright 1995-2010 David W. A. Bourne (david@boomer.org)