The area under the concentration-time curve (AUC) is a simple but very important measure in pharmacokinetics. The estimate of AUC may be obtained as an immediate result of nonlinear regression analysis, in which a set of observed concentrations are fitted to a certain pharmacokinetic function by estimating its parameters. However, the role of AUC is more significant in cases where the estimate is obtained by direct integration of data with the spirit of noncompartmental pharmacokinetic

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... 1) Despite the importance of AUC in such a situation, there has only been a single class of methods for estimating AUC: piecewise interpolatory methods. This class of methods includes trapezoidal/log-trapezoidal rules, spline fitting, and the so-called Lagrange method, as well as hybrids of each of them. There have already been several reports regarding comparative performances of these methods. 2-6) Among the class of methods, Purves reported that "parabolas-through-the-origin then logtrapezoidal rule" (PTTO) performed best, 4) whereas Jawień pointed out the possible drawback of Purves' method in theoretical situations. 6) It remains controversial as to which method is optimal in actual settings. In the piecewise interpolatory methods, an interpolating polynomial is constructed for each sub-interval specified by two adjacent data points. Then, the approximate value of AUC is given as a sum of the partial areas bounded by the polynomials. Accuracy of approximation depends on the locations of the sampling points, and several criteria were proposed as part of an optimal sampling strategy. 7, 8) In general, as is the case with parameter estimation, [9] [10] [11] [12] these criteria were driven so as to minimize the variance of AUC estimates. However, these optimizing strategies seem to be useful in restricted settings, because these strategies require prior knowledge concerning both pharmacokinetic and variance models. Another difficulty involved in the piecewise interpolatory 70 This paper presents a numerical integration method for estimating the area under the curve (AUC) over the infinite time interval. This method is based on the Gauss-Laguerre quadrature and produces AUC estimates over the infinite time interval without extrapolation in a usual sense. By contrast, in traditional schemes, piecewise interpolation is used to obtain the area up to the final sampling point, and the remaining portion is extrapolated using nonlinear regression. In this case, there is no theoretical consistency between the quadrature and extrapolation. The inconsistency may cause certain problems. For example, the optimal sampling criterion for the former is not necessarily optimal for the latter. Such inconsistency does not arise in the method of this work. The sampling points are placed near the zeros of Laguerre polynomials so as to directly estimate the AUC over the infinite time interval. The sampling design requires no particular prior information. This is also advantageous over the previous strategy, which worked by minimizing the variance of estimated AUC under the assumptions of particular pharmacokinetic and variance functions. The original Gaussian quadrature is believed to be inappropriate for numerical integration of data because of several restrictions. In this paper, it is shown that, using a simple strategy for managing errors due to these restrictions, the method produces an estimate of AUC with practically sufficient precision. The efficacy of this method is finally shown by numerical simulations in which the bias and variance of its estimate were compared with those of the previous methods such as the trapezoidal, log-trapezoidal, Lagrange, and parabolas-through-the-origin methods.