Checking out an example from drug research

One common nonlinear regression problem arises in drug development research. As soon as scientists

start testing a promising new compound, they want to determine some of its basic pharmacokinetic

(PK) properties. PK properties describe how the drug is absorbed, distributed, modified, and

eliminated by the body. Typically, the earliest Phase I clinical trials attempt to obtain basic PK data as

a secondary objective of the trial, while later-phase trials may be designed specifically to characterize

the PKs of the drug accurately and in great detail.

Raw PK data often consist of the concentration level of the drug in the participant’s blood at various

times after a dose of the drug is administered. Consider a Phase I trial, in which 10,000 micrograms

(μg) of a new drug is given as a single bolus, which is a rapid injection into a vein, in each

participant. Blood samples are drawn at predetermined times after dosing and are analyzed for drug

concentrations. Hypothetical data from one participant are shown in Table 19-2 and graphed in Figure

19-6. The drug concentration in the blood is expressed in units of μg per deciliter (

).

Remember, a deciliter is one-tenth of a liter.

Several basic PK parameters, such as maximum concentration, time of maximum concentration, area

under the curve (AUC), are usually calculated directly from the concentration-versus-time data,

without having to fit any curve to the points. But two important parameters are usually obtained from a

regression analysis:

The volume of distribution (

): This is the effective volume of fluid or tissue through which the

drug is distributed in the body. This effective volume could be equal to the blood volume, but

could be greater if the drug also spreads through fatty tissue or other parts of the body. If you know

the dose of the drug infused (Dose), and you know the blood plasma concentration at the moment of

infusion (

), you can calculate the volume of distribution as

. But you can’t

directly measure

. By the time the drug has distributed evenly throughout the bloodstream, some

of it has already been eliminated from the body. So

has to be estimated by extrapolating the

measured concentrations backward in time to the moment of infusion (

).

The elimination half-life (λ): The time it takes for half of the drug in the body to be eliminated.

TABLE 19-2 Blood Drug Concentration versus Time for One Participant

Time after Dosing (In Hours) Drug Concentration in Blood (μg/dL)

0.25

57.4

0.5

54.0

1

44.8

1.5

52.7

2

43.6

3

40.1

4

27.9

6

20.6

8

15.0

12

10.0