Nonlinear perpendicular least-squares regression in pharmacodynamics.

Currently available software for nonlinear regression does not account for errors in both the independent and the dependent variables. In pharmacodynamics, measurement errors are involved in the drug concentrations as well as in the effects. Instead of minimizing the sum of squared vertical errors (OLS), a Fortran program was written to find the closest distance from a measured data point to the tangent line of an estimated nonlinear curve and to minimize the sum of squared perpendicular distances (PLS). A Monte Carlo simulation was conducted with the sigmoidal Emax model to compare the OLS and PLS methods. The area between the true pharmacodynamic relationship and the fitted curve was compared as a measure of goodness of fit. The PLS demonstrated an improvement over the OLS by 20.8% with small differences in the parameter estimates when the random noise level had a standard deviation of five for both concentration and effect. Consideration of errors in both concentrations and effects with the PLS could lead to a more rational estimation of pharmacodynamic parameters.