Gaussian quadrature as a numerical integration method for estimating area under the curve

Area of Science:

• Pharmacokinetics and Pharmacodynamics
• Numerical Analysis
• Biostatistics

Background:

• Traditional methods for estimating area under the curve (AUC) over infinite time intervals often rely on piecewise interpolation and nonlinear regression extrapolation.
• This extrapolation can introduce theoretical inconsistencies with the initial data integration, potentially affecting the accuracy and optimal sampling strategy.
• Existing numerical integration techniques may have restrictions that limit their applicability to real-world data.

Purpose of the Study:

• To present a novel numerical integration method for estimating the area under the curve (AUC) over an infinite time interval.
• To overcome the limitations and theoretical inconsistencies associated with traditional extrapolation techniques in AUC estimation.
• To provide a more robust and theoretically consistent approach for AUC calculation in pharmacokinetic studies.

Main Methods:

• The proposed method utilizes Gauss-Laguerre quadrature, directly estimating AUC over the infinite interval without conventional extrapolation.
• Sampling points are strategically placed near the zeros of Laguerre polynomials for direct AUC estimation.
• A simple error management strategy is employed to address restrictions of the original Gaussian quadrature, ensuring practical precision.

Main Results:

• The Gauss-Laguerre quadrature method provides AUC estimates over infinite time intervals without the need for traditional extrapolation.
• This method demonstrates theoretical consistency between integration and estimation, unlike traditional schemes.
• Numerical simulations show competitive or superior performance in terms of bias and variance compared to trapezoidal, log-trapezoidal, Lagrange, and parabolas-through-the-origin methods.

Conclusions:

• The Gauss-Laguerre quadrature method offers a theoretically sound and practically precise approach for estimating AUC over infinite time intervals.
• This method eliminates the inconsistencies inherent in traditional extrapolation techniques, improving reliability.
• The proposed sampling design is advantageous as it requires no specific prior information, unlike previous variance-minimizing strategies.