Algorithm for application of Fourier analysis for biorhythmic baselines of pharmacodynamic indirect response models

The change of an indirect pharmacological response R(t) can be described by a periodic time-dependent production rate k(in)(t) and a first-order loss constant k(out). If k(in)(t) follows some biological rhythm (e.g., circadian), then the response R(t) also displays a periodic behavior. A new approach for describing the input function in indirect response models with biorhythmic baselines of physiologic substances is introduced. The present approach uses the baseline (placebo) response R(b)(t) to recover the equation for k(in)(t), Fourier analysis provides an approximate equation for R(b)(t) that consists of terms (usually two or three) of the Fourier series (harmonics) that contribute most to the overall sum. The model differential equation is solved backward for k(in)(t), yielding the equation involving R(b)(t). A computer program was developed to perform the square L²-norm approximation technique. Fourier analysis was also performed based on nonlinear regression. Cortisol suppression after inhalation of fluticasone propionate (FP) was modeled based on the inhibition of the secretion rate k(in)(t) using ADAPT II. The pharmacodynamic parameters k(out) and IC₅₀ were estimated from the, model equation with k(in)(t) derived by the new approach. The proposed method of describing the input function needs no assumption about the behavior of k(in)(t), is as efficient as methods used previously, and is more flexible in describing the baseline data than the nonlinear regression method.