Multiple solutions, illegal parameter values, local minima of the sum of squares, and anomalous parameter estimates in least-squares fitting of the two-compartment pharmacokinetic model with absorption

Abstract

When the two-compartment model with absorption is fitted to data by nonlinear least squares, in general six different outcomes can be obtained, arising from permutation of the three exponential rate constants. The existence of multiple solutions in this sense is analogous to the flip-flop phenomenon in the one-compartment model. It is possible for parameter estimates to be inconsistent with the underlying physical model. Methods for recognizing such illegal estimates are described. Other common difficulties are that estimated values for two of the rate constants are almost identical with very large standard deviations, or that the parameter estimation algorithm converges poorly. Such unwanted outcomes usually signal a local (false) minimum of the sum of squares. They can be recognized from the ratio of largest to smallest singular value of the Jacobian matrix, and are, in principle, avoidable by starting the estimation algorithm with different initial values. There also exists a class of data sets for which all outcomes of fitting the usual equations are anomalous. A better fit to these data sets (smaller sum of squares) is obtained if two of the relevant rate constants are allowed to take complex conjugate values. Such data sets have usually been described as having “equal rate constants.” A special form of the model equation is available for parameter estimation in this case. Precautions relating to its use are discussed.