Evaluation of methacholine dose-response curves by linear and exponential mathematical models: Goodness-of-fit and validity of extrapolation

Several models have been proposed to analyse dose-response curves recorded in bronchoprovocation challenge tests. The aims of the present work were: 1) to investigate which model (linear vs exponential) and which minimization method (trials and errors vs Levenberg-Marquardt) gives better results in terms of data interpolation (goodnesss-of-fit); and 2) to verify the validity of extrapolation by comparing forced expiratory volume in one second (FEV₁) observed after 4 mg methacholine with values extrapolated after truncation of the curves at 2 mg. For these purposes, methacholine dose-response curves were obtained in 832 subjects from a random population sample, as part of the European Community Respiratory Health Survey (ECRHS) in Italy. Methacholine was inhaled up to a maximum dose of 6 mg by dosimeter technique. The coefficient of determination (r²) was significantly higher with the exponential model (0.81 ± 0.22; mean ± SD) than with the linear model (0.69 ± 0.27). With both models, extrapolated values were usually lower than observed values. As a consequence, a 20% fall in FEV₁ with respect to postsaline FEV₁ was observed in only 24% and 21% of the tests, where a 20% fall had been predicted, respectively, according to the linear and the exponential model. In conclusion, exponential models are better than linear models with respect to data interpolation of methacholine dose-response curves. However, they are worse with respect to extrapolation to higher doses. With any model, extrapolation of dose-response curves by one doubling-dose should be avoided.