# 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 the
volume of distribution for aminoglycosides and vancomycin at our
institution.

The following formulas are being used

Vd = 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 min
extrapolated to the same time}

Would someone be able to comment on the appropriateness of each one.

Regards

Minyon 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 part
of the equation accounts for the fact that the Cp is obtain at t-T
after 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 drug
from the previous dose(s) that needs to be accounted for. The
concentration of drug remaining at the same time (end of infusion) is
estimated as Cmin*exp(-ke*(T+tx)) where Cmin is the trough
concentration and tx is the time between the time of trough
concentration 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 is
present prior to the dose (first dose) Cmin = 0 and the equation can
be 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)) in
place of (1-exp(-ke*(T-t))) - this equation would only be correct if
you define t as the time post infusion rather than the time from the
start 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 parenthesis
were misplaced. This equation is correct if indeed you extrapolate
Cmax and Cmin to the time in which the infusion ends.

Cmax = C(1)/exp(-ke*(t-T)) where C(1) = measured peak

Cmin = C(2)*exp(-ke*(tx+T)) where C(2) = measured trough

see above for definitions

I hope this helps
David 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 you
present are
probably different ways of parameterizing a one-compartment model.

More generally, one has the differential equations which describe the
behavior of the drug in question. This structural model may be small or
large, linear or nonlinear. Usually (not always) the differential equations
are written in terms of the amount of drug in the various compartments.

In addition, the model usually has output expressions. For example, the
measured serum concentration may be described as the amount of drug in the
serum compartment divided by the apparent volume of distribution of that
coompartment. Thus the Vd is a parameter in the equations describing the
behavior and the response of the drug to an input (a dosage regimen).

These are general ways of describing and modeling drug behavior. The
volume of distribution is simply one of the various parameters in whatever
structural model one uses. There is no single "correct" formula.

Best regards,

Roger Jelliffe
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USC Lab of Applied Pharmacokinetics
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