# PharmPK Discussion - Equation for 3 compartment modelling

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• On 15 Oct 2001 at 11:01:00, "Laszlo Hollos" (laci.-a-.hollos.net) sent the message
`The following message was posted to: PharmPKHello!I would like to draw a plot graph showing the blood concentration of a  drug after a short infusion. It's pharmacokinetic is based on a 3  compartment model. I have the following equation to calculate the  rate of infusion to maintain a certain level. k12, k21, k13, k31, k10  are the coefficients between compartments, Vc is the volume of  central comaprtment, C is the target concentration, R is the rate of  infusion, t is the time.R= C*Vc(k10+[k12*e^(-k21*t)]+[k13*e^(-k31*t)]). If it is, C is be  equal R/Vc(xxxx), but this does not explain what happens after a  short infusion, because as soon as R goes zero, the C will be zero,  which is incorrect. Can anybody help me? Please do not suggest  softwares, because I know a few, but I would like to understand how  it mathematically works.Thank you, Laszlo`
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• On 15 Oct 2001 at 23:39:01, David_Bourne (david.at.boomer.org) sent the message
`[Two replies - db]From: "Rob Ariano" Date: Mon, 15 Oct 2001 15:32:32 -0500To: david.at.boomer.orgSubject: Re: PharmPK Equation for 3 compartment modellingThe following message was posted to: PharmPKWhere have you found this equation describing the serum-conc. timeprofile for an infused drug in a 3-cmpt model?  Notice for t=zero,you actually achieve a value for C other than zero; which isimpossible in the absence of a load!Regards,Robert Ariano, Pharm.D.,BCPSClinical Pharmacist Critical CareSt.Boniface General Hospital;  &Associate Professor of Pharmacy,& Medicine, University of Manitoba,204-237-2050 Phone204-237-2165 FAXrariano.at.sbgh.mb.cawww.sbgh.mb.ca---From: "Mike Makoid" Date: Mon, 15 Oct 2001 16:08:37 -0500To: david.-a-.boomer.orgSubject: RE: PharmPK Equation for 3 compartment modelingThe following message was posted to: PharmPK>I would like to draw a plot graph showing the blood concentration of a>   drug after a short infusion. It's pharmacokinetic is based on a 3>   compartment model. I have the following equation to calculate the>   rate of infusion to maintain a certain level. k12, k21, k13, k31, k10>   are the coefficients between compartments, Vc is the volume of>   central comaprtment, C is the target concentration, R is the rate of>   infusion, t is the time.>R= C*Vc(k10+[k12*e^(-k21*t)]+[k13*e^(-k31*t)]). If it is, C is be>   equal R/Vc(xxxx), but this does not explain what happens after a>   short infusion, because as soon as R goes zero, the C will be zero,This is an incorrect assumption on your part.  What happens after an IVBolus to the same model? If you can answer that question, you shouldunderstand.  Realize the body treats the drug the same way as an IVbolus after the infusion is stopped.  After the infusion is terminated,the body doesn't care how the drug got, or how long it took.  It justwants to get rid of it.  So at the end of the infusion, you have a newquestion - the IV bolus question BUT with different initial conditions =not all of the drug is in the central compartment at time right afterthe infusion termination, some is in each of the compartments andexcreted.  Treat it as Two questions, before and after termination. mmMichael Makoid, Ph.D.Professor and ChairDepartment of Pharmacy SciencesSchool of Pharmacy and Allied Health ProfessionsCreighton University2500 California PlazaOmaha, NE 68178Voice 402 280 2952Fax   402 280 1883Cell  402 250 4618`
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• On 17 Oct 2001 at 10:54:38, David Bourne (david.-a-.boomer.org) sent the message
`[Two replies - db]From: "Edward Dennis Bashaw" Date: Mon, 15 Oct 2001 21:57:52 -0400To: david.aaa.boomer.orgSubject: Re: PharmPK Equation for 3 compartment modellingThe following message was posted to: PharmPKLaszlo,You are being too literal in your interpretation of time.  In your equationat time =0 C=0, this is the initial condition prior to the infusion.  Oncethe infusion begins the clock starts.  Your error is that you are countingDOWN time instead of it proceeding in a  forward manner.  At the end of a 5min infusion the time is not "0" but "5".  Also you are confusing rate ofinfusion with dose.  Rate is the measure of transfer of drug or amount intothe system.  Rate in this equation is not an end unto itself as you have ithere, you need to re-consider the left side of the equation as yoursimplification of "R" is the main problem here.  Personally, I wouldrecommend you consult a copy of Shargel & Yu's (or other PK) textbook for amore detailed explanation, using the two compartment model as a start andthen expanding it to fit your needs by adding in the appropriate exponents.Dennis Bashaw, Pharm.D.Team Leader, PharmacokineticsUS Food and Drug Administration---From: "Hans Proost" Date: Wed, 17 Oct 2001 10:09:43 METTo: david.aaa.boomer.orgSubject: Re: PharmPK Equation for 3 compartment modellingThe following message was posted to: PharmPKDear Dr. Hollos,As pointed out by others, a steady-state plasma concentration willbe obtained only if one starts with a bolus dose equal to Css.Vc(i.e. a loading dose), followed by a continuously changing infusionrate R according to your equation, which is known as BET (Bolus-Elimination-Transfer) (Lauven PM, Der Anesthesist 1982;31:15-20,in German):R= Css*Vc(k10+[k12*e^(-k21*t)]+[k13*e^(-k31*t)])During the infusion the plasma concentration will be exactlyconstant (provided that your patient behaves according to thismodel).After stopping the infusion, the plasma concentration will decreasein a similar way as after a single bolus, following a three-exponential profile. The rate constants will be equal to that afterbolus injection, but the intercept will not! The intercepts aredependent on the duration of the infusion (of course, asymptoticallyapproaching a constant value after long infusion).These intercepts can be obtained from convolution of drug input I(bolus dose and BET infusion) and the 'unit impulse response' UIR(plasma concentration profile after bolus administration of a unitdose):C(t) = integral(0-t) [ I(tau) . UIR(t-tau) d tau ]This equation for the plasma concentration profile C(t) results inrather complicated derivations (taking into account the bolus doseand the infusion from time zero to T_inf), which need quite a lot ofpaper to write down. But it certainly works.Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax  31-50 363 3247Email: j.h.proost.-at-.farm.rug.nl`
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• On 28 Oct 2001 at 14:56:24, "Laszlo Hollos" (laci.-at-.hollos.net) sent the message
`The following message was posted to: PharmPKDear AllI would like to say thank to all who responded to my original e-mail  regarding the mathematical modelling of three compartment  pharmacokinetics of iv administered drugs. The original equation  describing the serum level time profile was published in an  anesthesiology journal in 1998, which explained how the propofol  target-controlled infusion (Diprifusor) worked. Now I understand  from your letter, that it is not simple enough to be described by only  one simple equation.Thanks again, Laszlo`
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