Mathematical modeling of circadian cortisol concentrations using indirect response models: comparison of several methods.

Six mathematical functions to describe the chronobiology of cortisol concentrations were assessed. Mean data from a dose-proportionality study of inhaled fluticasone propionate were fitted with an indirect response model using various biorhythmic functions (single cosine, dual ramps, dual zero-order, dual cosines, and Fourier series with 2 and n-harmonics) for production rate. Data with known parameters and random variation were also generated and fitted using the ADAPT II program. Fitted parameters, model estimation criteria, and runs tests were compared. Models with preassigned functions: the dual ramps, the dual zero-order and the dual cosines provide maximum and minimum times for cortisol release rate, were suitable for describing asymmetric circadian patterns and yielding IC50 values. Fourier analysis differs from the other methods in that it uses the placebo data to recover equations for cortisol secretion rate rather than by postulation. Nonlinear regression for Fourier analysis, instead of the L2-norm method, was useful to characterize the baseline cortisol data but was restricted to a maximum of two harmonics. Apart from the single cosine function, which predicts symmetrical cortisol concentrations, all methods were satisfactory in describing the baseline and suppressed cortisol concentrations. On the other hand, Fourier series with L2-norm produced the best unbiased estimate for baseline patterns. The Fourier method is flexible, accurate, and can be extended to other drug-induced changes in normal periodic rhythms.