# PharmPK Discussion - 3-compartmental model

PharmPK Discussion List Archive Index page
• On 10 May 2000 at 22:31:47, Chantal Le guellec (leguellec.aaa.med.univ-tours.fr) sent the message
`Dear all,Could someone indicate me equations describing plasma concentration of adrug following 3-exponential pharmacokinetics and administered asIV-infusion, expressed as a function of intercompartmental constants andvolumes of each compartment.We were requested by anesthesiologists ofour institution who wish to implement such a model forcomputer-controlled infusion of anesthetic drugs.Many thanks for your help.Chantal Le GuellecLaboratoire de pharmacologie et toxicologieCHU de Tours, France`
Back to the Top

• On 11 May 2000 at 20:56:10, David_Bourne (david.at.boomer.org) sent the message
`[Two replies - db]Date: Thu, 11 May 2000 00:48:47 -0700 (MST)X-Sender: ml11439.aaa.pop.goodnet.comTo: PharmPK.-at-.boomer.orgFrom: ml11439.-at-.goodnet.com (Michael J. Leibold)Subject: Re: PharmPK 3-compartmental modelChantal,     This is actually an answer from a previous discussion, butalso applies to your question:     Triexponential refers to the presence of three slopes in thelog plasma concentration versus time curve.     Cp = Ae-at + Be-bt + Ge-gt     In linear compartmental pharmacokinetics this is interpreted as athree-compartment model:                          Xo                           \                     k12           k13              Cpt2<------->Cpt1<------->Cpt3                     k21    \      k31                             \->k10     This is a linear mammillary model with elimination from a centralcompartment wich usually represents the plasma compartment. The systemis described by linear first order differential equations describingthe change in compartmental amounts of drug with time:     dX1/dt= -(k13+k10+k12)X1    +  k21X2        +k31X3     dX2/dt=       k12X1         -  k21X2     dX3/dt=       k13X1                         -k31X3     This system of differential equations can be solved by Laplacetransforms and matrix algebra. The matix representation of the Laplacetransformed system of differential equations is:                    [SI-A][Xs]= [Us]     Where the Laplace transform of the system of differential equationsabove is equal to the Laplace transformed vector of dose input, forexample: (Ko/s)(1-e-Ts).     [(s+k13+k10+k12)      -k21         -k31][X1s]     [(Ko/s)(1-e-Ts)]     [    -k12            (s+k21)         0 ][X2s]   = [      0       ]     [    -k13              0        (s+k31)][X3s]     [      0       ]     This can be solved by matrix algebra to yield Laplace transformedquantities of each compartment, most important of which is the centralcompartment quantity (X1s):     X1s= [(Ko/s)(1-e-Ts)(s+k21)(s+k31)]/[(s+a)(s+b)(s+g)]     The inverse Laplace transform of the above results in an equationdescribing the amount in the central compartment as function of time,and dividing this by the Vc (volume of the central compartment) resultsin the equation for the plasma concentration over time:    Cp= Ko(k21-a)(k31-a)(1-e-aT)(e-at')/[a(b-a)(g-a)Vc]  +        Ko(k21-b)(k31-b)(1-e-bT)(e-bt')/[b(a-b)(g-b)Vc]  +        Ko(k21-g)(k31-g)(1-e-gT)(e-gt')/[g(a-g)(b-g)Vc]     The above equation describes the triexponential plasma concentrationcurve of a drug being administered by a an intermittent infusion, whereT= infusion time and t'= time after infusion.     The multiple dose form of the above equation is:Cp= Ko(k21-a)(k31-a)(1-e-aT)(1-e-naTau)(e-at')/[a(b-a)(g-a)Vc(1-e-aTau)]  +     Ko(k21-b)(k31-b)(1-e-bT)(1-e-nbTau)(e-bt')/[b(a-b)(g-b)Vc(1-e-bTau)]  +     Ko(k21-g)(k31-g)(1-e-gT)(1-e-ngTau)(e-gt')/[g(a-g)(b-g)Vc(1-e-gTau)]      Examples of drugs which exhibit triexpoential plasma concentrationcurves are vancomycin and aminoglycosides. However, this is difficult todetect as the curves really appear two compartment. Vancomycin has aninitial rapid distribution phase (T1/2~=7min) which can be detected withnumerous plasma samples and careful analysis. Aminglycosides have a longwashout phase due to renal tissue binding and release which can be detectedwith plasma samples taken long after the drug has been administered todetect a slow decline (T1/2~=100-200 hours) in plasma concentrations as drugis released from renal tissue.                    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) Schumacher, G.E., Therapeutic Drug Monitoring, Norwalk, Appleton&    Lange 19954) Evans, W.E., Schentag, J.J., Jusko, W.J., Applied Pharmacokinetics 3rd ed,    Vancouver, Applied Therapeutics 19925) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug    Intelligence Publications 1975---Reply-To: "David Foster" From: "David Foster" To: Subject: Re:  PharmPK 3-compartmental modelDate: Thu, 11 May 2000 07:54:18 -0700X-Priority: 3Are you interested in fitting a sum of three exponential to the data andusing this to estimate plasma concentration, or fitting a three compartmentmodel to the data?  If you are using sums of exponentials and fitting thisto concentration data, the equation is:c(t) = A0 - A1*exp(-a1*t) - A2*exp(-a2*t) - A3*exp(-a3*t)where A0 - A1 - A2 - A3 = 0 (assuming there is no drug in the system at timezero).There are occasions, even when using a constant infusion directly intoplasma, wherethis constraint may have to be relaxed.  In addition, there are occasionswhen a slightdelay is required.  IN this case, t in the equation for c(t) is replaced byc - tlag where tlag is the time of the delay.If you are using a three compartment model, you just need to structure yourmodel according to how you want the compartments hooked together, thensimulate the experiment on the model.  Software such as SAAM II can doeither of these quite easily.`
Back to the Top

• On 14 May 2000 at 22:29:16, exfamadu.-at-.savba.sk sent the message
`Dear Dr. Le GuellecI  recommend to use a model describing the behavior of the anestheticdrug during and after the end of a short- or long-time  IV-infusion (1),instead of a model describing only a post infusion part of theconcentration-time profile of the drug. In general, such a model can be ofany order (lower or higher than a 3rd-order model). This model allows toutilize the information  about the behavior of the drug contained in themeasurements over the whole time interval from the beginning of theinfusion to the end of the sampling and that even in the earliest phaserepresenting intravascular mixing. Moreover this model and a computer-controlled pump allow to adjust  an adequate flow of the drug into thepatient body.1. Durisova M., et al., Meth. Find. Exp. Clin. Pharmacol., 1998, 20, 217.With best regards,Maria DurisovaDipl. Engineer Maria Durisova D.Sc.Senior Research WorkerScientific SecretaryInstitute of Experimental PharmacologySlovak Academy of SciencesSK-842 16 BratislavaSlovak Republichttp://nic.savba.sk/sav/inst/exfa/advanced.htm`
Back to the Top

Want to post a follow-up message on this topic? If this link does not work with your browser send a follow-up message to PharmPK@boomer.org with "3-compartmental model" as the subject

Copyright 1995-2010 David W. A. Bourne (david@boomer.org)