Impulsive model of endocrine regulation with a local continuous feedback

Whereas development of mathematical models describing the endocrine system as a whole remains a challenging problem, visible progress has been demonstrated in modeling its subsystems, or axes. Models of hormonal axes portray only the most essential interactions between the hormones and can be described by low-order systems of differential equations. This paper analyzes the properties of a novel model of a hypothalamic-pituitary axis, portraying the interactions in a chain of a release hormone (secreted by the hypothalamus), a tropic hormone (produced by the pituitary gland) and an effector hormone (secreted by a target gland). This model, unlike previously published ones, captures two prominent features of neurohormonal regulation systems, namely, the pulsatile (episodic) production of the release hormone and a complex non-cyclic feedback mechanism that maintains the involved hormone concentrations within certain biological limits. At the same time, the discussed model is analytically tractable; in particular, the existence of a so-called 1-cycle featured by a single pulse over one period is proven mathematically

Impulsive Goodwin’s Oscillator Model of Endocrine Regulation: Local Feedback Leads to Multistability

The impulsive Goodwin’s oscillator (IGO) is a hybrid model that captures complex dynamics arising in continuous systems controlled by pulse-modulated (event-based) feedback. Being conceived to describe pulsatile endocrine regulation, it has also found applications in e.g. pharmacokinetics. The original version of the IGO assumes the continuous part of the model to be a chain of first-order blocks. This paper explores the nonlinear phenomena arising due to the introduction of a local continuous feedback as suggested by the endocrine applications. The effects caused by a nonlinear feedback law parameterized by a Hill function are compared to those arising due to a simpler and previously treated case of affine feedback law. The hybrid dynamics of the IGO are reduced to a (discrete) Poincaré map governing the propagation of the model’s continuous states through the firing instants of the impulsive feedback. Bifurcation analysis of the map reveals in particular that both the local Hill function and affine feedback can lead to multistability, which phenomenon has not been observed in the usual IGO model

Robust One-Step Estimation of Impulsive Time Series

The paper deals with the estimation of a signal model in the form of the output of a continuous linear time-invariant system driven by a sequence of instantaneous impulses, i.e. an impulsive time series. This modeling concept arises in, e.g., endocrinology when episodic hormone secretion events and elimination rates are simultaneously estimated from sampled hormone concentration measurements. The pulsatile secretion is modeled with a train of Dirac impulses constituting the input to a linear plant, which represents stimulated hormone secretion and elimination. A previously developed one-step estimation algorithm effectively resolves the trade-off between data fit and impulsive input sparsity. The present work improves the algorithm so that it requires less manual tuning and produces more accurate results through the use of an information criterion. It is also extended to handle outliers and unknown basal levels that are commonly recognized issues in biomedical data. The algorithm performance is evaluated both theoretically and experimentally on synthetic and clinical data.Comment: 26 pages, 11 figure

DynPeak : An algorithm for pulse detection and frequency analysis in hormonal time series

The endocrine control of the reproductive function is often studied from the analysis of luteinizing hormone (LH) pulsatile secretion by the pituitary gland. Whereas measurements in the cavernous sinus cumulate anatomical and technical difficulties, LH levels can be easily assessed from jugular blood. However, plasma levels result from a convolution process due to clearance effects when LH enters the general circulation. Simultaneous measurements comparing LH levels in the cavernous sinus and jugular blood have revealed clear differences in the pulse shape, the amplitude and the baseline. Besides, experimental sampling occurs at a relatively low frequency (typically every 10 min) with respect to LH highest frequency release (one pulse per hour) and the resulting LH measurements are noised by both experimental and assay errors. As a result, the pattern of plasma LH may be not so clearly pulsatile. Yet, reliable information on the InterPulse Intervals (IPI) is a prerequisite to study precisely the steroid feedback exerted on the pituitary level. Hence, there is a real need for robust IPI detection algorithms. In this article, we present an algorithm for the monitoring of LH pulse frequency, basing ourselves both on the available endocrinological knowledge on LH pulse (shape and duration with respect to the frequency regime) and synthetic LH data generated by a simple model. We make use of synthetic data to make clear some basic notions underlying our algorithmic choices. We focus on explaining how the process of sampling affects drastically the original pattern of secretion, and especially the amplitude of the detectable pulses. We then describe the algorithm in details and perform it on different sets of both synthetic and experimental LH time series. We further comment on how to diagnose possible outliers from the series of IPIs which is the main output of the algorithm.Comment: Nombre de pages : 35 ; Nombre de figures : 16 ; Nombre de tableaux :

Follicle-Stimulating Hormone Receptor: Advances and Remaining Challenges

International audienc

A mathematical model for LH release in response to continuous and pulsatile exposure of gonadotrophs to GnRH

In a previous study, a model was developed to investigate the release of luteinizing hormone (LH) from pituitary cells in response to a short pulse of gonadotropin-releasing hormone (GnRH). The model included: binding of GnRH to its receptor (R), dimerization and internalization of the hormone receptor complex, interaction with a G protein, production of inositol 1,4,5-trisphosphate (IP(3)), release of calcium from the endoplasmic reticulum (ER), entrance of calcium into the cytosol via voltage gated membrane channels, pumping of calcium out of the cytosol via membrane and ER pumps, and release of LH. The extended model, presented in this paper, also includes the following physiologically important phenomena: desensitization of calcium channels; internalization of the dimerized receptors and recycling of some of the internalized receptors; an increase in G(q )concentration near the plasma membrane in response to receptor dimerization; and basal rates of synthesis and degradation of the receptors. With suitable choices of the parameters, good agreement with a variety of experimental data of the LH release pattern in response to pulses of various durations, repetition rates, and concentrations of GnRH were obtained. The mathematical model allows us to assess the effects of internalization and desensitization on the shapes and time courses of LH response curves

Multiscale mathematical modeling of the hypothalamo-pituitary-gonadal axis

International audienceAlthough the fields of systems and integrative biology are in full expansion, few teams are involved worldwide into the study of reproductive function from the mathematical modeling viewpoint. This may be due to the fact that the reproductive function is not compulsory for individual organism survival, even if it is for species survival. Alternatively, the complexity of reproductive physiology may be discouraging. Indeed, the hypothalamo-pituitary-gonadal (HPG) axis involves not only several organs and tissues, but also intricate time (from the neuronal millisecond timescale to circannual rhythmicity) and space (from molecules to organs) scales. Yet, mathematical modeling, and especially multiscale modeling, can renew our approaches of the molecular, cellular and physiological processes underlying the control of reproductive functions. In turn, the remarkable dynamic features exhibited by the HPG axis raise intriguing and challenging questions to modelers and applied mathematicians. In this article, we draw a panoramic review of some mathematical models designed in the framework of the female HPG, with a special focus on the gonadal and central control of follicular development. On the gonadal side, the modeling of follicular development calls to the generic formalism of structured cell populations, that allows one to make mechanistic links between the control of cell fate (proliferation, differentiation or apoptosis) and that of the follicle fate (ovulation or degeneration) or to investigate how the functional interactions between the oocyte and its surrounding cells shape the follicle morphogenesis. On the central, mainly hypothalamic side, models based on dynamical systems with multiple timescales allow one to represent within a single framework both the pulsatile and surge patterns of the neurohormone GnRH (gonadotropin-releasing hormone). Beyond their interest in basic research investigations, mathematical models can also be at the source of useful tools to study the encoding and decoding of the (neuro-)hormonal signals at play within the HPG axis and detect complex, possibly hidden rhythms, in experimental time series