caa a22 4500 44533004X CHVBK 20180317142734.0 cr unu---uuuuu 170323e20111001xx s 000 0 eng 10.1007/s00285-010-0384-2 doi (NATIONALLICENCE)springer-10.1007/s00285-010-0384-2 The minimal model of the hypothalamic-pituitary-adrenal axis [Elektronische Daten] [Frank Vinther, Morten Andersen, Johnny Ottesen] This paper concerns ODE modeling of the hypothalamic-pituitary- adrenal axis (HPA axis) using an analytical and numerical approach, combined with biological knowledge regarding physiological mechanisms and parameters. The three hormones, CRH, ACTH, and cortisol, which interact in the HPA axis are modeled as a system of three coupled, nonlinear differential equations. Experimental data shows the circadian as well as the ultradian rhythm. This paper focuses on the ultradian rhythm. The ultradian rhythm can mathematically be explained by oscillating solutions. Oscillating solutions to an ODE emerges from an unstable fixed point with complex eigenvalues with a positive real parts and a non-zero imaginary parts. The first part of the paper describes the general considerations to be obeyed for a mathematical model of the HPA axis. In this paper we only include the most widely accepted mechanisms that influence the dynamics of the HPA axis, i.e. a negative feedback from cortisol on CRH and ACTH. Therefore we term our model the minimal model. The minimal model, encompasses a wide class of different realizations, obeying only a few physiologically reasonable demands. The results include the existence of a trapping region guaranteeing that concentrations do not become negative or tend to infinity. Furthermore, this treatment guarantees the existence of a unique fixed point. A change in local stability of the fixed point, from stable to unstable, implies a Hopf bifurcation; thereby, oscillating solutions may emerge from the model. Sufficient criteria for local stability of the fixed point, and an easily applicable sufficient criteria guaranteeing global stability of the fixed point, is formulated. If the latter is fulfilled, ultradian rhythm is an impossible outcome of the minimal model and all realizations thereof. The second part of the paper concerns a specific realization of the minimal model in which feedback functions are built explicitly using receptor dynamics. Using physiologically reasonable parameter values, along with the results of the general case, it is demonstrated that un-physiological values of the parameters are needed in order to achieve local instability of the fixed point. Small changes in physiologically relevant parameters cause the system to be globally stable using the analytical criteria. All simulations show a globally stable fixed point, ruling out periodic solutions even when an investigation of the ‘worst case parameters' is performed. Springer-Verlag, 2010 HPA axis nationallicence CRH nationallicence Cortisol nationallicence ACTH nationallicence Dynamical system nationallicence Mathematical modeling nationallicence Vinther Frank Roskilde University, NSM Building 27, 4000, Roskilde, Denmark aut Andersen Morten Roskilde University, NSM Building 27, 4000, Roskilde, Denmark aut Ottesen Johnny Roskilde University, NSM Building 27, 4000, Roskilde, Denmark aut Journal of Mathematical Biology Springer-Verlag 63/4(2011-10-01), 663-690 0303-6812 63:4<663 2011 63 285 https://doi.org/10.1007/s00285-010-0384-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00285-010-0384-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Vinther Frank Roskilde University, NSM Building 27, 4000, Roskilde, Denmark aut NATIONALLICENCE 700 1- Andersen Morten Roskilde University, NSM Building 27, 4000, Roskilde, Denmark aut NATIONALLICENCE 700 1- Ottesen Johnny Roskilde University, NSM Building 27, 4000, Roskilde, Denmark aut NATIONALLICENCE 773 0- Journal of Mathematical Biology Springer-Verlag 63/4(2011-10-01), 663-690 0303-6812 63:4<663 2011 63 285 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer