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

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• On 24 Nov 2001 at 19:49:17, "Mike Leibold" (m_leibold.at.hotmail.com) sent the message
`The following message was posted to: PharmPKDr.Whelan,   The Xo term in the pharmacokinetic bolus equations is simplythe dose.  In the first order absorption model, the dose appearsas D (with F equal to fraction absorbed). In the zero orderabsorption model, the dose appears as Ko, representing the doseabsorbed over time(usually represented as the dosage interval inthe case of sustained release forms such as oral theophylline).   The two compartment model can be adapted to oral absorption bymodifying 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 releasemedications can be modeled as above with a smaller Ka relativeto the immediate release form, or can be modeled as a zero orderprocess (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 isthe input function: KaFDe-kat or Ko.    The matrix respresentation of these systems of differentialequations is useful for solving the Laplace transformed systemsfor 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 compartmentyields 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 andthen dividing by Vc yields equations for the plasma concentration foreach 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.comReferences1) Gibaldi, M., Perrier, D., Pharmacokinetics, New York, Marcel Dekker   19752) Gibaldi, M., Perrier, D., Pharmacokinetics 2nd ed, New York, Marcel Dekker  19823) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug   Intelligence Publications 19754) Godfrey, Keith, Compartment Models and Their Application, New York,   Academic Press 1983`
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