The minimal model of the hypothalamic–pituitary–adrenal axis

Abstract

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.

Hypothalamic-pituitary-adrenal axis dysregulation and cortisol activity in obesity: A systematic review

“…Borrowing from the field of systems engineering, robustness theory describes how architectural features of a system (in this case, negative cortisol feedback loops, adrenal/non-adrenal interactions, and 11␤-HSD and other enzymatic interactions) give rise to system functioning. Dynamic systems modeling has been used successfully within the context of HPA function (Aschbacher et al, 2012;Sriram et al, 2012;Vinther et al, 2011), as well as in predicting metabolic health (Aschbacher et al, 2014). This approach to modeling HPA activity in obesity may be more appropriate and is a promising avenue for future research.…”

Transcriptional implications of ultradian glucocorticoid secretion in homeostasis and in the acute stress response

“…These prior studies consider the origin of ultradian rhythms as due to either discrete bursting (21,22) or an unstable fixed point driving oscillatory dynamics (2,20,24,28,40,50). The latter class of models rely on either unphysical parameter values or time delays to induce oscillations, as illustrated by Vinther et al (49).…”

“…been widely used in HPA axis models and glucocorticoid pharmacodynamic models (2,20,30,40,49,50,56), which allows for simpler models that do not explicitly account for cortisol-binding proteins.…”

“…In contrast to ultradian rhythms, which are largely assumed to arise due to dynamics internal to the HPA axis, circadian rhythms are typically viewed as a centrally mediated external signal that drives 24 h variability in glucocorticoid secretion (38). For this reason, prior models that have considered this issue typically drive a model parameter by an imposed sinusoid with a 24 h period (2,20,24,49,50). This has previously been applied to the HPA axis model in Eq.…”

Stability analysis of a hypothalamic-pituitary-adrenal axis model with inclusion of glucocorticoid receptor and memory

“…During the past decade, the mathematical modelling of the HPA axis has been intensively studied [1,2,3,4,5,6,7,8,9,10,11,12]. In 2005, Savic and Jelic [10] presented one of the mathematical models of HPA axis described by a system with three nonlinear differential equations with one delay.…”