# PharmPK Discussion - Two Compartment Absorption Model 1st and 0 order Models

PharmPK Discussion List Archive Index page
• On 24 Nov 2001 at 19:49:17, "Mike Leibold" (m_leibold.at.hotmail.com) sent the message

The following message was posted to: PharmPK

Dr.Whelan,

The Xo term in the pharmacokinetic bolus equations is simply
the dose. In the first order absorption model, the dose appears
as D (with F equal to fraction absorbed). In the zero order
absorption model, the dose appears as Ko, representing the dose
absorbed over time(usually represented as the dosage interval in
the case of sustained release forms such as oral theophylline).

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)]

I hope this was what you were looking for!!

Mike Leibold, PharmD, RPh
ML11439.-a-.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
4) Godfrey, Keith, Compartment Models and Their Application, New York,