- On 6 Aug 1998 at 09:41:27, "Avent, Minyon" (minyona.aaa.baylordallas.edu) sent the message

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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.

Thank you for your assistance

Regards

Minyon Avent - On 7 Aug 1998 at 11:18:32, "David Nix" (nix.-at-.Pharmacy.Arizona.EDU) sent the message

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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 - On 21 Aug 1998 at 14:27:17, Roger Jelliffe (jelliffe.-a-.hsc.usc.edu) sent the message

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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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