Advances in mathematical modelling of the hypothalamic–pituitary–adrenal (HPA) axis dynamics and the neuroendocrine response to stress

Depression diagnostics using a nonlinear mathematical oscillatory model

BACKGROUND AND OBJECTIVES

It is known that long-term stress leads to trauma and very often to depression. Usually, the diagnosis of depression is dealt with by psychiatrists who, based on conversations and questions, diagnose the patient's illness and condition. Unfortunately, this diagnosis is not always reliable. To prevent the development of disease, it is necessary to detect illness in a timely manner. One of the indications of the possibility of the onset of disease is a disturbance in the level of hormones in the body, especially cortisol. The purpose of this study was to develop a mathematical model for cortisol variation resulting from stress which would be useful in making conclusions about depressive states.

METHODS

Rapid changes in cortisol concentration, according to ultradian rhythms, which are much faster than the daily circadian rhythm, is modelled as a truly nonlinear oscillator. The mathematical model contains two coupled first order differential equations. The stress is modeled as a pulsating action, described with a periodic trigonometric function, and cortisol production as a cubic nonlinear one. Three models for cortisol variation are considered: 1) the pure nonlinear model, 2) the periodically excited system, 3) and the chaotic system. The results from the study are supported with experimental measurements.

RESULTS

Without stress, cortisol variation is of an oscillatory type with a constant steady-state amplitude. Intensive stress causes a resonant phenomenon in cortisol oscillatory variation. The occasion is short and is usually without consequences. For long stress periods deterministic chaos occurs which permanently changes the levels of cortisol. This phenomenon is an indicator of depression. Results from the suggested models are compared with experimentally obtained ones and good quantitative agreement is obtained.

CONCLUSIONS

The nonlinear oscillator is a good model for indication of depression. The model provides not only general conclusions, but also individual ones, if personal characteristics are taken into consideration. Response of the model depends not only on the input data related to stress, but also on the system parameters that specify each individual. Findings obtained from this study have implications for the medical diagnosis and treatment of depression.

The Physiological and Pharmacological Significance of the Circadian Timing of the HPA Axis: A Mathematical Modeling Approach

As a potent endogenous regulator of homeostasis, the circadian time-keeping system synchronizes internal physiology to periodic changes in the external environment to enhance survival. Adapting endogenous rhythms to the external time is accomplished hierarchically with the central pacemaker located in the suprachiasmatic nucleus (SCN) signaling the hypothalamus-pituitary-adrenal (HPA) axis to release hormones, notably cortisol, which help maintain the body's circadian rhythm. Given the essential role of HPA-releasing hormones in regulating physiological functions, including immune response, cell cycle, and energy metabolism, their daily variation is critical for the proper function of the circadian timing system. In this review, we focus on cortisol and key fundamental properties of the HPA axis and highlight their importance in controlling circadian dynamics. We demonstrate how systems-driven, mathematical modeling of the HPA axis complements experimental findings, enhances our understanding of complex physiological systems, helps predict potential mechanisms of action, and elucidates the consequences of circadian disruption. Finally, we outline the implications of circadian regulation in the context of personalized chronotherapy. Focusing on the chrono-pharmacology of synthetic glucocorticoids, we review the challenges and opportunities associated with moving toward personalized therapies that capitalize on circadian rhythms.

The SCN-HPA-Periphery Circadian Timing System: Mathematical Modeling of Clock Synchronization and the Effects of Photoperiod on Jetlag Adaptation

Synchronizing the circadian timing system (CTS) to external light/dark cycles is crucial for homeostasis maintenance and environmental adaptation. The CTS is organized hierarchically, with the central pacemaker located in the suprachiasmatic nuclei (SCN) generating coherent oscillations that are entrained to light/dark cycles. These oscillations regulate the release of glucocorticoids by the hypothalamus-pituitary-adrenal (HPA) axis, which acts as a systemic entrainer of peripheral clocks throughout the body. The SCN adjusts its network plasticity in response to variations in photoperiod, leading to changes in the rhythmic release of glucocorticoids and ultimately impacting peripheral clocks. However, the effects of photoperiod-induced variations of glucocorticoids on the synchronization of peripheral clocks are not fully understood, and the interaction between jetlag adaption and photoperiod changes is unclear. This study presents a semi-mechanistic mathematical model to investigate how the CTS responds to changes in photoperiod. Specifically, the study focuses on the entrainment properties of a system composed of the SCN, HPA axis, and peripheral clocks. The results show that high-amplitude glucocorticoid rhythms lead to a more coherent phase distribution in the periphery. In addition, our study investigates the effect of photoperiod exposure on jetlag recovery time and phase shift, proposing different interventional strategies for eastward and westward jetlag. The findings suggest that decreasing photic exposure before jetlag during eastward traveling and after jetlag during westward traveling can accelerate jetlag readaptation. The study provides insights into the mechanisms of CTS organization and potential recovery strategies for transitions between time zones and lighting zones.

A mathematical representation of the reactive scope model

Researchers have long sought to understand and predict an animal's response to stressful stimuli. Since the introduction of the concept of homeostasis, a variety of model frameworks have been proposed to describe what is necessary for an animal to remain within this stable physiological state and the ramifications of leaving it. Romero et al. (Horm Behav 55(3):375-389, 2009) introduced the reactive scope model to provide a novel conceptual framework for the stress response that assumes an animal's ability to tolerate a stressful stimulus may degrade over time in response to the stimulus. We provide a mathematical formulation for the reactive scope model using a system of ordinary differential equations and show that this model is capable of recreating existing experimental data. We also provide an experimental method that may be used to verify the model as well as several potential additions to the model. If future experimentation provides the necessary data to estimate the model's parameters, the model presented here may be used to make quantitative predictions about physiological mediator levels during a stress response and predict the onset of homeostatic overload.

Modeling pulsativity in the hypothalamic-pituitary-adrenal hormonal axis

A new mathematical model for biological rhythms in the hypothalamic–pituitary–adrenal (HPA) axis is proposed. This model takes the form of a system of impulsive time-delay differential equations which include pulsatile release of adrenocorticotropin (ACTH) by the pituitary gland and a time delay for the release of glucocorticoid hormones by the adrenal gland. Numerical simulations demonstrate that the model’s response to periodic and circadian inputs from the hypothalamus are consistent with those generated by recent models which do not include a pulsatile pituitary. In contrast the oscillatory phenomena generated by the impulsive delay equation mode occur even if the time delay is zero. The observation that the time delay merely introduces a small phase shift suggesting that the effects of the adrenal gland are “downstream” to the origin of pulsativity. In addition, the model accounts for the occurrence of ultradian oscillations in an isolated pituitary gland. These observations suggest that principles of pulse modulated control, familiar to control engineers, may have an increasing role to play in understanding the HPA axis.

Integrating theoretical and empirical approaches for a robust understanding of endocrine flexibility

There is growing interest in studying hormones beyond single 'snapshot' measurements, as recognition that individual variation in the endocrine response to environmental change may underlie many rapid, coordinated phenotypic changes. Repeated measures of hormone levels in individuals provide additional insight into individual variation in endocrine flexibility - that is, how individuals modulate hormone levels in response to the environment. The ability to quickly and appropriately modify phenotype is predicted to be favored by selection, especially in unpredictable environments. The need for repeated samples from individuals can make empirical studies of endocrine flexibility logistically challenging, but methods based in mathematical modeling can provide insights that circumvent these challenges. Our Review introduces and defines endocrine flexibility, reviews existing studies, makes suggestions for future empirical work, and recommends mathematical modeling approaches to complement empirical work and significantly advance our understanding. Mathematical modeling is not yet widely employed in endocrinology, but can be used to identify innovative areas for future research and generate novel predictions for empirical testing.

Understanding virtual patients efficiently and rigorously by combining machine learning with dynamical modelling

Individual biological organisms are characterized by daunting heterogeneity, which precludes describing or understanding populations of ‘patients’ with a single mathematical model. Recently, the field of quantitative systems pharmacology (QSP) has adopted the notion of virtual patients (VPs) to cope with this challenge. A typical population of VPs represents the behavior of a heterogeneous patient population with a distribution of parameter values over a mathematical model of fixed structure. Though this notion of VPs is a powerful tool to describe patients’ heterogeneity, the analysis and understanding of these VPs present new challenges to systems pharmacologists. Here, using a model of the hypothalamic–pituitary–adrenal axis, we show that an integrated pipeline that combines machine learning (ML) and bifurcation analysis can be used to effectively and efficiently analyse the behaviors observed in populations of VPs. Compared with local sensitivity analyses, ML allows us to capture and analyse the contributions of simultaneous changes of multiple model parameters. Following up with bifurcation analysis, we are able to provide rigorous mechanistic insight regarding the influences of ML-identified parameters on the dynamical system’s behaviors. In this work, we illustrate the utility of this pipeline and suggest that its wider adoption will facilitate the use of VPs in the practice of systems pharmacology.

A Modeling and Machine Learning Pipeline to Rationally Design Treatments to Restore Neuroendocrine Disorders in Heterogeneous Individuals

Heterogeneity among individual patients presents a fundamental challenge to effective treatment, since a treatment protocol working for a portion of the population often fails in others. We hypothesize that a computational pipeline integrating mathematical modeling and machine learning could be used to address this fundamental challenge and facilitate the optimization of individualized treatment protocols. We tested our hypothesis with the neuroendocrine systems controlled by the hypothalamic–pituitary–adrenal (HPA) axis. With a synergistic combination of mathematical modeling and machine learning (ML), this integrated computational pipeline could indeed efficiently reveal optimal treatment targets that significantly contribute to the effective treatment of heterogeneous individuals. What is more, the integrated pipeline also suggested quantitative information on how these key targets should be perturbed. Based on such ML revealed hints, mathematical modeling could be used to rationally design novel protocols and test their performances. We believe that this integrated computational pipeline, properly applied in combination with other computational, experimental and clinical research tools, can be used to design novel and improved treatment against a broad range of complex diseases.

Circadian rhythms and the HPA axis: A systems view

The circadian timing system comprises a network of time-keeping clocks distributed across a living host whose responsibility is to allocate resources and distribute functions temporally to optimize fitness. The molecular structures generating these rhythms have evolved to accommodate the rotation of the earth in an attempt to primarily match the light/dark periods during the 24-hr day. To maintain synchrony of timing across and within tissues, information from the central clock, located in the suprachiasmatic nucleus, is conveyed using systemic signals. Leading among those signals are endocrine hormones, and while the hypothalamic-pituitary-adrenal axis through the release of glucocorticoids is a major pacesetter. Interestingly, the fundamental units at the molecular and physiological scales that generate local and systemic signals share critical structural properties. These properties enable time-keeping systems to generate rhythmic signals and allow them to adopt specific properties as they interact with each other and the external environment. The purpose of this review is to provide a broad overview of these structures, discuss their functional characteristics, and describe some of their fundamental properties as these related to health and disease. This article is categorized under: Immune System Diseases > Computational Models Immune System Diseases > Biomedical Engineering.

An integrate-and-fire model for pulsatility in the neuroendocrine system

A model for pulsatility in neuroendocrine regulation is proposed which combines Goodwin-type feedback control with impulsive input from neurons located in the hypothalamus. The impulsive neural input is modeled using an integrate-and-fire mechanism; namely, inputs are generated only when the membrane potential crosses a threshold, after which it is reset to baseline. The resultant model takes the form of a functional-differential equation with continuous and impulsive components. Despite the impulsive nature of the inputs, realistic hormone profiles are generated, including ultradian and circadian rhythms, pulsatile secretory patterns, and even chaotic dynamics.

Automatic control of Hypothalamus-Pituitary-Adrenal axis dynamics

BACKGROUND AND OBJECTIVE

In this study, a presentation is made for the automatic control of the hypothalamus-pituitary-adrenal axis which plays an important role in the immune stress responses and the circadian rhythms of mammalian organisms.

METHODS

Control approaches are implemented on a novel second order nonlinear system which accepts adrenocorticotropin hormone as an input and models the variation of plasma concentrations of adrenocorticotropin and cortisol respectively. The control methods are based on back-stepping and input-output feedback linearization techniques. The controllers adjust the adrenocorticotropin injection to maintain the daily rhythm of the cortisol concentration. In accordance with the periodicity of biological clock mechanism, we provide a sinusoidally varying cortisol reference to the controllers.

RESULTS

Numerical simulations are performed (on MATLAB) to demonstrate the closed loop performance of the controllers. Major concerns in the selection of the control gains are chattering and negative concentration in responses. The simulation results showed that one can successfully find gain levels which do not lead to those issues. However, the gains lie in different ranges for back-stepping and feedback linearization based controllers.

CONCLUSION

The results showed that, both back-stepping and feedback linearization based controllers fulfilled their duty of synchronization of the cortisol concentration to a reference daily periodic rhythm. In addition to that, the risk of negative valued adrenocorticotropin injection can be eliminated by properly choosing the controller gains.

Allostatic adaptation and personalized physiological trade-offs in the circadian regulation of the HPA axis: A mathematical modeling approach

The hypothalamic-pituitary-adrenal (HPA) axis orchestrates the physiological response to unpredictable acute stressors. Moreover, the HPA axis exhibits prominent circadian activity and synchronizes peripheral circadian clocks to daily environmental cycles, thereby promoting homeostasis. Persistent disruption of homeostatic glucocorticoid circadian rhythmicity due to chronic stress exposure is correlated with the incidence of various pathological conditions including depression, diabetes and cancer. Allostatic habituation of the HPA axis, such that glucocorticoid levels retain homeostatic levels upon chronic exposure to stress, can therefore confer fitness advantages by preventing the sustained dysregulation of glucocorticoid-responsive signaling pathways. However, such allostatic adaptation results in a physiological cost (allostatic load) that might impair the homeostatic stress-responsive and synchronizing functions of the HPA axis. We use mathematical modeling to characterize specific chronic stress-induced allostatic adaptations in the HPA network. We predict the existence of multiple individualized regulatory strategies enabling the maintenance of homeostatic glucocorticoid rhythms, while allowing for flexible HPA response characteristics. We show that this regulatory variability produces a trade-off between the stress-responsive and time-keeping properties of the HPA axis. Finally, allostatic regulatory adaptations are predicted to cause a time-of-day dependent sensitization of the acute stress response and impair the entrainability of the HPA axis.

What Is Stress? A Systems Perspective

The term "stress" is used to describe important phenomena at multiple levels of biological organization, but finding a general and rigorous definition of the concept has proven challenging. Current models in the behavioral literature emphasize the cognitive aspects of stress, which is said to occur when threats to the organism are perceived as uncontrollable and/or unpredictable. Here we adopt the perspective of systems biology and take a step toward a general definition of stress by unpacking the concept in light of control theory. Our goal is to clarify the concept so as to facilitate integrative research and formal analysis. We argue that stress occurs when a biological control system detects a failure to control a fitness-critical variable, which may be either internal or external to the organism. Biological control systems typically include both feedback (reactive, compensatory) and feedforward (predictive, anticipatory) components; their interplay accounts for the complex phenomenology of stress in living organisms. The simple and abstract definition we propose applies to animals, plants, and single cells, highlighting connections across levels of organization. In the final section of the paper we explore some extensions of our approach and suggest directions for future research. Specifically, we discuss the classic concepts of conditioning and hormesis and review relevant work on cellular stress responses; show how control theory suggests the existence of fundamental trade-offs in the design of stress responses; and point to potential insights into the effects of novel environmental conditions, including those resulting from anthropogenic change.