280 PART 5 Looking for Relationships with Correlation and Regression

may not converge to an answer, depending on the complexity of the function

you’re fitting and how close your initial estimates are to the truth. And in addition

to attending to these unique issues, analysts running a nonlinear regression face

all the other complications of multivariate regression, such as collinearity, as

described in Chapter 17.

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 deter-

mine some of its basic pharmacokinetic (PK) properties. PK properties describe

how the drug is absorbed, distributed, modified, and eliminated by the body. Typ-

ically, the earliest Phase I clinical trials attempt to obtain basic PK data as a sec-

ondary 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 partici-

pant’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 concentra-

tions. 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 ( g dL

/

). 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 (V}d^{): 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 (C 0), you can calculate the volume of distribution as V

Dose C

d

/

0^.

But you can’t directly measure C 0. By the time the drug has distributed evenly

throughout the bloodstream, some of it has already been eliminated from the

body. So C 0 has to be estimated by extrapolating the measured concentrations

backward in time to the moment of infusion (Time 0).»

» ^{The elimination half-life (λ): The time it takes for half of the drug in the body}

to be eliminated.