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

Six mathematical functions to describe the chronobiology of cortisol concen trations were assessed. Mean data from a dose-proportionality study of inha led fluticasone propionate were fitted with an indirect response model usin g various biorhythmic functions (single cosine, dual ramps, dual zero-order , dual cosines, and Fourier series with 2 and n-harmonics) for production r ate. Data with known parameters and random variation were also generated an d fitted using the ADAPT II program. Fitted parameters, model estimation cr iteria, and runs tests were compared. Models with preassigned functions: th e dual ramps, the dual zero-order and the dual cosines provide maximum and minimum times for cortisol release rate, were suitable for describing asymm etric circadian patterns and yielding IC50 values. Fourier analysis differs fi om the other methods in that it uses the placebo data to recover equati ons for cortisol secretion rate rather than by postulation. Nonlinear regre ssion for Fourier analysis, instead of the L-2-norm method, was useful to c haracterize the baseline cortisol data but was restricted to a maximum of t wo harmonics. Apart from the single cosine function, which predicts symmetr ical cortisol concentrations, all methods were satisfactory in describing t he baseline and suppressed cortisol concentrations. On the other hand, Four ier series with L-2-norm produced the best unbiased estimate for baseline p atterns. The Fourier method is flexible, accurate, and can be extended to o ther drug-induced changes ill normal periodic rhythms.