# PharmPK Discussion - Volume of Distribution

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• On 6 Aug 1998 at 09:41:27, "Avent, Minyon" (minyona.aaa.baylordallas.edu) sent the message
`We have a number of different formulas that are being used to calculate thevolume of distribution for aminoglycosides and vancomycin at ourinstitution.The following formulas are being usedVd = Dose (1 - e -ke( tinf) ) / Cpk*ke*tinf(1 - e-ke(t - tinf) )Vd = Dose (1 - e -ke( tinf) ) / Cpk*ke*tinf(1 - e-ke(t ) )Vd = Ko (1 - e -ke( tinf)   / ke * ( Cmax - Cmin)  {with Cmax and C minextrapolated to the same time}Would someone be able to comment on the appropriateness of each one.Thank you for your assistanceRegardsMinyon Avent`
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• On 7 Aug 1998 at 11:18:32, "David Nix" (nix.-at-.Pharmacy.Arizona.EDU) sent the message
`In typical computure language (see definitions below):Vd = Dose(1-exp(-ke*tinf)) / (Cpk*ke*tinf)*(1 - exp(-ke(t-tinf)))this equation is useful only following the first dose - the last partof the equation accounts for the fact that the Cp is obtain at t-Tafter the infusion stops.For second dose and beyond:C(t) = ko * (1-exp(-ke*T)) / (ke*V)*exp(-ke*t-T)      single dose equation for concentration at any time post      infusion       NOTE:  ko=Dose/T,  T=time of infusion,      exp - exponential function for variable enclose in ( ), ke      is the elimination rate constant, and t is the elapsed      time from the start of the infusion.      When one has a trough-peak drug concentrations, Vd can be      determined assuming superpostion.  To simplify the equation      determine the peak concentation at the end of the infusion (Cp)      - this is done by estimating the Cp given the measure peak      concentration and ke.Cp = ko*(1-exp(-ke*T)) / (ke*V)   -- there will be remaining drugfrom the previous dose(s) that needs to be accounted for.  Theconcentration of drug remaining at the same time (end of infusion) isestimated as Cmin*exp(-ke*(T+tx))  where Cmin is the troughconcentration and tx is the time between the time of troughconcentration and the beginning of the next drug infusion.       Cp = ko*(1-exp(-ke*T)) / (ke*V) + Cmin*exp(-ke*(T+tx))rearragement provides:       Cp-Cmin*exp(-ke*(T+tx)) = ko*(1-exp(-ke*T)) / (ke*V)       V = ko*(1-exp(-ke*T)) / (ke* (Cp-Cmin*exp(-ke*(T+tx))))I think this is the equation that you should be using - If no drug ispresent prior to the dose (first dose) Cmin = 0 and the equation canbe simplified.> Vd = Dose (1 - e -ke( tinf) ) / Cpk*ke*tinf(1 - e-ke(t ) )For this equation the last part is substituted with (1-exp(-ke*t)) inplace of (1-exp(-ke*(T-t)))  - this equation would only be correct ifyou define t as the time post infusion rather than the time from thestart of infusion.>> Vd = Ko (1 - e -ke( tinf)   / ke * ( Cmax - Cmin)  {with Cmax and C min> extrapolated to the same time}Vd = Ko*(1-exp(-ke*T))  /  (ke * ( Cmax - Cmin))  some parenthesiswere misplaced.  This equation is correct if indeed you extrapolateCmax and Cmin to the time in which the infusion ends.Cmax = C(1)/exp(-ke*(t-T))      where C(1) = measured peakCmin = C(2)*exp(-ke*(tx+T))   where C(2) = measured troughsee above for definitionsI hope this helpsDavid Nix`
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• On 21 Aug 1998 at 14:27:17, Roger Jelliffe (jelliffe.-a-.hsc.usc.edu) sent the message
`Dear Minyon:	What's with the formulas? It appears that most of the ones youpresent areprobably different ways of parameterizing a one-compartment model.	More generally, one has the differential equations which describe thebehavior of the drug in question. This structural model may be small orlarge, linear or nonlinear. Usually (not always) the differential equationsare written in terms of the amount of drug in the various compartments.	In addition, the model usually has output expressions. For example, themeasured serum concentration may be described as the amount of drug in theserum compartment divided by the apparent volume of distribution of thatcoompartment. Thus the Vd is a parameter in the equations describing thebehavior and the response of the drug to an input (a dosage regimen).	These are general ways of describing and modeling drug behavior. Thevolume of distribution is simply one of the various parameters in whateverstructural model one uses. There is no single "correct" formula.Best regards,Roger Jelliffe************************************************Roger W. Jelliffe, M.D.USC Lab of Applied PharmacokineticsCSC 134-B, 2250 Alcazar St, Los Angeles CA 90033**Note our new area codes below, since 6/15/98!**Phone **(NOTE NEW AREA CODE AND PREFIX)** (323)442-1300, Fax (323)442-1302email=jelliffe.at.hsc.usc.edu************************************************You might also look at our Web page for announcements ofnew software and upcoming workshops and events. It ishttp://www.usc.edu/hsc/lab_apk/************************************************`
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