8.1. Introduction

When a tablet of drug is taken orally, in general, it reaches the stomach and begins to disintegrate and is absorbed (A). When dissolved into solution in the stomach acid, the drug is passed on to the small intestine (Rowland and Tozer, 1980). At this point, some of the drug will pass through and be eliminated (E) from the body. Some will be metabolized (M) into a different substance in the intestine, and some drug will be picked up by the walls of the intestine and distributed (D) into the body. This last bit of drug substance passes through the liver first, where it is also often metabolized (M). The drug substance that remains then passes through the liver and reaches the bloodstream, where it is circulated throughout the body.

Pharmacokinetics (PK) is the study of absorption, distribution, metabolism, and excretion (ADME) properties (Atkinson et al., 2001). Following oral administration, the drug is held to undergo these four stages prior to being completely eliminated from the body. PK is also the study of what the body does to a drug, (as opposed to what a drug does to the body).

Measurement is central to PK, and the most common method is to measure by means of blood sampling how much drug substance has been put into the body (i.e., dose) relative to how much drug reached the systemic circulation. For bioequivalence trials (U.S. Food and Drug Administration (FDA), 2003), at least 12 samples are collected over time following dosing with an additional sample collected prior to dosing.

As the drug is absorbed and distributed, the plasma concentration rises and reaches a maximum (called the C_max or maximum concentration). Plasma levels then decline until the body completely eliminates the drug from the body. The overall exposure to drug is measured by computing the area under the plasma concentration curve (AUC). AUC is derived in general by computing the area under each time interval and adding up all the areas. Other common summary measures are:

• T_max (time of maximum concentration),

• T_1/2 (half-life of drug substance)

More details of techniques used in the derivation of AUC may be found in Yeh and Kwan (1978). C_max and T_max are derived by inspection of the data, and T_1/2 is estimated by linear regression in accordance with the description of Rowland and Tozer (1980).

In this chapter, we will concentrate on the modeling of AUC and C_max using PROC MIXED (note that, in general, the methods described are applicable to other summary measures). It is acknowledged that AUC and C_max are log-normally distributed (Crow and Shimizu, 1988) for the purposes of this discussion in accordance with the findings of Westlake (1986), Lacey (1995), Lacey et al. (1997), and Julious and Debarnot (2000). Thus AUC and C_max data are analyzed under ln (natural-logarithmic) transformation (Box and Cox, 1964).

Analysis of ln-transformed AUC and C_max data in clinical pharmacology follows the general principles of analysis of repeated-measures, cross-over data using mixed effect linear models (Jones and Kenward, 2003; Wellek, 2003; Senn, 2002; Chow and Liu, 2000; Vonesh and Chinchilli, 1997; Milliken and Johnson, 1992). To illustrate this assumption, consider a study with n subjects and p periods where an observation y (either AUC or C_max, etc.) is observed for each subject in each period. Y may be expressed as a pn-dimensional response vector. Then, in matrix notation, this model can be expressed as

Y = Xβ + Z₁u + Z₂e,

where Xβ is the usual design and fixed effects matrix, u is multivariate normal with expectation 0 and variance-covariance matrix Ω, i.e.,

u ~ M V N (0, Ω),

and

e ~ M V N (0, Λ),

and u is independent of e. Subjects are assumed to be independent, and Z₁ and Z₂ are design matrices used to construct the variance-covariance structure as appropriate to the study design and desired variance components. The RANDOM and REPEATED statements in PROC MIXED are included as appropriate to calculate the desired variance-covariance structure.

Restricted maximum likelihood modeling (REML, Patterson, 1950; Patterson and Thompson, 1971) is applied to calculate unbiased variance estimates and the degrees of freedom (Kenward and Roger, 1997) are used to calculate appropriate degrees of freedom for any estimates or contrasts of interest.

Nonparametric statistical analysis of cross-over data will not be discussed further in this chapter but can be accomplished using SAS. Readers interested in such techniques should see Jones and Kenward (2003), Wellek (2003), Chow and Liu (2000) and Hauschke et al. (1990) for more details.

Other topics of general interest to readers of this chapter include Population Pharmacokinetics (FDA Guidance, 1999) and Exposure-Response Modeling (FDA Guidance, 2003). SAS is not generally used to support such nonlinear mixed-effect analyses, but is used in data management support (which will not be discussed here). It is theoretically possible to do such nonlinear mixed-effects modeling using the NLMIXED procedure, and we refer readers interested in such analyses to Atkinson et al. (2001), Sheiner et al. (1989), Sheiner (1997), Sheiner and Steimer (2000), and Machado et al. (1999) for more information on these topics.

To save space, some SAS code has been shortened and some output is not shown. The complete SAS code and data sets used in this book are available on the Web site at http://support.sas.com/publishing/bbu/companion_site/60622.html