- On 17 Jun 2000 at 08:34:41, ml11439.-at-.goodnet.com (Michael J. Leibold) sent the message

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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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