# PharmPK Discussion - Model equations

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• On 17 Jun 2000 at 08:34:41, ml11439.-at-.goodnet.com (Michael J. Leibold) sent the message
`The following message was posted to: PharmPKProfessor,     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 for eachoral 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/Tauand the time (t) would represent the total time of administration as ifit were a continuous infusion. So, FD/Tau could be substituted into theabove 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 amultiple dose function to represent the case of several doses, as themultiple dose condition is factored into the FD/Tau term and the time factor.The first order absorption model however, requires a multiple dose functionderived 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.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 1975 >Dear Mike, > >Last year, I followed your discussions on pharmacokinetic modelling and amwondering whether you can help me. I need the analytical equations for aparticular model. > >We conducted a comparative pharmacokinetic study on immediate and sustainedreleaase dosage forms of pentoxyfylline. This drug obeys the two compartmentmodel. Upon oral administration, it is absorbed and goes through the liverbefore entering the blood circulation. It has a high first pass effect. Infact, it is solely disposed through this organ(liver) with six metabolites.I was not interested in this metabolism, but followed plasma levels ofunchanged drug. What I need is, two compartment model drug with zero orderrelease and first order absorption model equations for blood level. Liveracts like the peripheral compartment into which absorption occurs. Theelimination is also through the liver(no unchanged drug, apparently) and notfrom the central compartment. But central compartment is the one sampled. > >Searching the literature did not yield the equations I needed. My Laplacetransformation knowledge is not enough for this pupose, unfortunately. Ifyou could be kind enough to derive these equations, I would be mostgrateful. 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. Ineed >the analytical equations for a particular model. >  >We conducted a comparative pharmacokineticstudy >on immediate and sustained releaase dosage forms of pentoxyfylline. Thisdrug >obeys the two compartment model. Upon oral administration, it is absorbedand >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 followedplasma >levels of unchanged drug. What I need is, two compartment model drug withzero >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 forthis >pupose, unfortunately. If you could be kind enough to derive theseequations, 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 >  > >href="mailto:ilbeyi.-at-.tr-net.net.tr">ilbeyi.aaa.tr-net.net.tr >  >`
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