An alternative method for the estimation of the terminal slope when a few data points are available

Phase plane plots are graphical expressions for differential equations plotting the state derivative dc/dt versus the state c. Using these plots, we developed a novel method for the estimation of the terminal slope from time-concentration data. The values of the derivatives used for the construction of the phase plane plots were calculated by two different methods of numerical differentiation. The first method (D1) is based on the classical calculation of slope of the line connecting two successive data points. The alternative method (D2) relies on an initial second-order polynomial interpolation utilizing three successive data points followed by the calculation of the derivative at each one of the concentration values. A forced-through-zero linear regression of the phase plane plot data is used to derive an estimate for the slope. For comparative purposes, the standard approach based on the semilogarithmic plot was also applied. For a hypothetical drug absorbed by first-order process into a one-compartment model, simulated time-concentration data disturbed by a Gaussian zero mean random error with various coefficients of variation were generated. Various sampling schedules, with two, three, four, or five data points, were utilized for the estimation of the terminal slope. Performances of the proposed methods on simulated data were expressed by means of root-mean-squared error, bias, and standard deviation. In all cases, D2 was superior to D1. The D2 method outperforms the standard method in that it furnishes estimates closer to the real values in all cases when two data points and in most cases when three data points were used. All methods behave similarly when four or five data points were used.