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 kin(t) and a first-order loss constant kout. If kin(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 Rb(t) to recover the equation for kin(t). Fourier analysis provides an approximate equation for Rb(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 kin(t), yielding the equation involving Rb(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 kin(t) using ADAPT II. The pharmacodynamic parameters kout and IC50 were estimated from the model equation with kin(t) derived by the new approach. The proposed method of describing the input function needs no assumption about the behavior of kin(t), is as efficient as methods used previously, and is more flexible in describing the baseline data than the nonlinear regression method.