# One Compartment IV Bolus

## Apparent volume of distribution, V

Figure 4.6.1 Concentration as a function of Time

We can use Equation 4.6.1 to calculate the plasma concentration at any time when we know kel and Cp0. However, usually we don't know Cp0 ahead of time, but we do know the dose. A dose in mass units, maybe in mg. To calculate Cp0 we need to know the volume that the drug is distributed into. That is, the apparent volume of the mixing container, the body. This apparent volume of distribution is not a physiological volume. It won't be lower than blood or plasma volume but for some drugs it can be much larger than body volume. It is a mathematical 'fudge' factor relating the amount of drug in the body and the concentration of drug in the measured compartment, usually plasma.

## Defining Equations:

Equation 4.6.2 Definition for Apparent Volume of Distribution

Equation 4.6.3 Relationship between Amount and Concentration

Immediately after the intravenous dose is administered the amount of drug in the body is the IV dose. Thus:

Equation 4.6.4 Volume calculated from Dose and Cp0

or

Equation 4.6.5 Initial Concentration calculated from Dose and V

Combining Equation 4.6.4 and Equation 4.6.1 we are able to derive an equation for drug concentration as a function of time given values of Dose, V, and kel.

Equation 4.6.6 Concentration as a function of Time

The one compartment model assumption is that there is a rapid equilibration in drug concentrations throughout the body, however, this does not mean that the concentration is the same throughout the body. This is illustrated in Figure 4.6.1. In the first beaker the concentration throughout the beaker is the same and the apparent volume of distribution is the same as the size of the beaker. In the second beaker after a rapid equilibrium, distribution between the solution (representing plasma) and the charcoal (representing various tissues of the body) may be complete. However, drug concentrations within the beaker (representing the patient) are not uniform. Much of the drug is held with the charcoal leaving much smaller concentrations in the solution. After measuring the drug concentration in the solution the apparent volume of the patient is much larger, the apparent volume of distribution is much larger.

Figure 4.6.1 Apparent Volume of Distribution

## Units:

The units for the apparent volume of distribution are volume units. Most commonly V is expressed in liters, L. On occasion the value for the apparent volume of distribution will be normalized for the weight of the subject and expressed as a percentage or more usually in liters/kilogram, L/Kg.

## Determining Values of V:

The usual method of calculating the apparent volume of distribution of the one compartment model is to extrapolate concentration versus time data back to the y-axis origin. See Figure 4.5.1 for an example. This gives an estimate of Cp0. When the IV bolus dose is know the apparent volume of distribution can be calculated from Equation 4.6.3, above.
Table 4.6.1 Example values for apparent volume of distribution (Gibaldi, 1984)

 Drug V (L/Kg) V (L, 70 kg) Sulfisoxazole 0.16 11.2 Phenytoin 0.63 44.1 Phenobarbital 0.55 38.5 Diazepam 2.4 168 Digoxin 7 490

Note, the last figure in this table, for digoxin, is much larger than body volume. This drug must be extensively distributed into tissues, leaving low concentrations in the plasma, thus the body as a whole appears to have a very large volume of distribution. Remember, this is not a physiological volume.

The line in the figure below, Figure 4.6.2, was calculated with a dose of 450 mg, apparent volume of distribution of 15 L and elimination rate constant of 0.20 hr-1. Calculate the curve with other parameter values using the interactive graph.

Figure 4.6.2. Concentration versus time

Click on the figure to view the interactive graph

References
• Gibaldi, M. 1984 "Biopharmaceutics and Clinical Pharmacokinetics", 3rd ed., Lea & Febiger, Chapter 12, page 214
• Activated carbon at Wikipedia
• Volume has been discussed on the PharmPK listserv