Analysis of ordinary differential equation models representing the dynamics of hormone axes

Abstract

The endocrine system plays a vital role in regulating physiological processes essential for maintaining overall health, i.e. metabolism, reproductive functions, and early brain development. Central to this system are hormonal axes in which the hypothalamus releases regulatory hormones that act on the pituitary gland. These interactions prompt the secretion of tropic hormones that stimulate specific target organs, creating a tightly regulated system governed by feedback mechanisms. Mathematical models provide a powerful framework for understanding these feedback systems and the effects of external stimuli. These models typically consist of nonlinear Ordinary Differential Equations (ODEs), with the number of equations varying based on the complexity of the system being represented. This work focuses on the mathematical analysis of models describing the Hypothalamus-Pituitary-Thyroid (HPT) and the Hypothalamus-Pituitary-Ovarian (HPO) axis. Selected models are calibrated against clinical hormone measurements to assess their alignment with biological observations. The results indicate that current mathematical models require further refinement. To evaluate the robustness and applicability of these models, comprehensive stability and sensitivity analyses are conducted. Sensitivity analysis, performed both locally and globally, quantifies the impact of parameter variations on model outputs, and enables the identification of critical parameters that govern system dynamics. Stability analysis of an HPT-axis model identified an asymptotically stable equilibrium point, which was found to be independent of the model parameters.