Estimating Integrals Using Quadrature Methods with an Application in Pharmacokinetics

The estimation of integrals using numerical quadrature is common in many biological studies. For instance, in biopharmaceutical research the area under curves is a useful quantity in deriving pharmacokinetic parameters and in providing a surrogate measure of the total dose of a compound at a particular site. In this paper, statistical issues as separate from numerical issues are considered in choosing a quadrature rule. The class of Newton-Côtes numerical quadrature procedures is examined from the perspective of minimizing mean squared error (MSE). The MSE are examined for a variety of functions commonly encountered in pharmacokinetics. It is seen that the simplest Newton-Côtes procedure, the trapezoidal rule, frequently provides minimum MSE for a variety of concentration-time shapes and under a variety of response variance conditions. A biopharmaceutical example is presented to illustrate these considerations.