# Multi-Compartment Pharmacokinetic Models

## Apparent Volumes of Distribution

The concentration of drug in the body is determined not only by the rate constant values but also by the apparent volume of distribution. In the case of the two compartment model a number of volume terms can been defined.

### V1

The apparent volume of the central compartment, V1 or Vc, can be calculated as:

Equation 19.5.1 Apparent Volume of Central Compartment

This parameter is important because it allows the calculation of the highest plasma concentration or Cp0 after an IV bolus administration. This concentration may result in transient toxicity. V1 can also be used in dose calculations.

### Varea ( = Vß)

Varea or Vß is defined as:

Equation 19.5.2 Apparent Volume, Varea

Because of the relationship with clearance and β and with V1 and kel this parameter is quite useful in dosing calculations. This parameter can be readily calculated via AUC and β values from the 'raw' data and is therefore commonly quoted.

### Vss

Vss, V steady state defined as:

Equation 19.5.3 Apparent Volume, Steady State

This term relates the total amount of drug in the body at 'steady state' with the concentration in plasma or blood

Figure 19.5.1 Plot of X1 (Plasma) and X2 (Tissue) Compartment Concentrations, Showing 'Steady State' with Both Lines Parallel

The relationship between volume terms is that:

Varea > Vss > V1

And for a one compartment model the values for all these parameters are equal.

### Example Calculation

As an example we can look at the data in the table below.

Table 19.5.1 Two Compartment Pharmacokinetics
Time (hr) Concentration (mg/L) Cplate (mg/L) Residual (mg/L)
0.5 20.6 8.8 11.8
1 13.4 7.8 5.6
2 7.3 6.1 1.2
3 5.0 4.7 0.3
4 3.7 3.7 -
6 2.2
8 1.4
10 0.82
12 0.50

Figure 19.5.2 Plot of Cp versus Time Illustrating the Method of Residuals

The first two columns are the time and plasma concentration which may be collected after IV bolus administration of 500 mg of drug. These data are plotted (n) in Figure 19.5.2 above. At longer times, after 4 hours, out to 12 hours the data appears to follow a straight line on semi- log graph paper. Since α > β this terminal line is described by B • e-β • t.

Following it back to t = 0 gives B = 10 mg/L. From the slope of the line β = 0.25 hr-1. Cplate values at early times are shown in column 3 and the residual in column 4. The residual values are plotted (o) also giving a value of A = 25 mg/L and α = 1.51 hr-1 Note that α/β = 6, thus these values should be fairly accurate.

B = 10 mg/L, β = (ln 10 - ln 0.5)/12 = 2.996/12 = 0.25 hr-1

A = 25 mg/L, α = (ln 25 - ln 0.27)/3 = 4.528/3 = 1.51 hr-1

Therefore Cp = 25 • e-1.51 • t + 10 • e-0.25 • t

We can now calculate the micro-constants.

The AUC by the trapezoidal rule + Cplast/β = 56.3 + 2.0 = 58.3 mg.hr.L-1, [Note the use of β] thus

Notice that Varea > Vss > V1 [34.3 > 26.7 > 14.3]

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