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**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 Cp^{0}. However, usually we don't know Cp^{0} ahead of time, but we do know the **dose**. A dose in **mass units**, maybe in mg. To calculate Cp^{0} 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.

**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 Cp ^{0}**

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**

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

**Figure 4.6.2. Concentration versus time**

Click on the figure to view the interactive graph

- 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

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