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modeling in physiology A biomathematical model of time-delayed feedback in the human male hypothalamic-pituitary-Leydig cell axis DANIEL M. KEENAN1 AND JOHANNES D. VELDHUIS2 1Division of Statistics, Department of Mathematics, University of Virginia, Charlottesville 22903; and 2Division of Endocrinology, Health Sciences Center, and National Science Foundation Center for Biological Timing, University of Virginia, Charlottesville, Virginia 22908 Keenan, Daniel M., and Johannes D. Veldhuis. A GnRH’s stimulation of LH secretion (1) and LH’s dose- biomathematical model of time-delayed feedback in the hu- dependent stimulation of Te secretion (10). Similarly, man male hypothalamic-pituitary-Leydig cell axis. Am. J. earlier simulation modeling of neurohormone release Physiol. 275 (Endocrinol. Metab. 38): E157–E176, 1998.—We has typically encompassed a single hormone, not the develop, implement, and test a feedback and feedforward entire interacting feedback system. A major limitation biomathematical construct of the male hypothalamic [gonado- tropin-releasing hormone (GnRH)]-pituitary [luteinizing hor- inherent in thus isolating system components that are mone (LH)]-gonadal [testosterone (Te)] axis. This stochastic so highly coupled physiologically via time-lagged feed- differential equation formulation consists of a nonstationary forward (e.g., LH-Te) and feedback (e.g., Te-LH) mecha- stochastic point process responsible for generating episodic nisms is omission of the influences due to communica- release of GnRH, which is modulated negatively by short-loop tions among the component(s). Moreover, artificial (GnRH) and long-loop (Te) feedback. Pulsatile GnRH release isolation of functional elements can make inference of in turn drives bursts of LH secretion via an agonistic dose- system behavior more difficult. Given these issues, we response curve that is partially damped by Te negative present an initial biomathematical model of an entire feedback. Circulating LH stimulates (feedforward) Te synthe- interconnected three-nodal system, i.e., the (male) hypo- sis and release by a second dose response. Te acts via negative thalamic (GnRH)-pituitary (LH)-gonadal (Te) axis, and dose-responsive feedback on GnRH and LH output, thus fulfilling conditions of a closed-loop control system. Four test its basal pulsatile output, modulated circadian computer simulations document expected feedback perfor- responses, and predicted performance in selected simu- mance, as published earlier for the human male GnRH-LH-Te lations and prior human experiments. axis. Six other simulations test distinct within-model cou- pling mechanisms to link a circadian modulatory input to a Glossary pulsatile control node so as to explicate the known 24-h variations in Te and, to a lesser extent, LH. We conclude that ai Elimination rate constant for hormone i relevant dynamic function, internodal dose-dependent regula- bi Basal secretion rate of hormone i tory connections, and within-system time-delayed coupling Dt Discretization step size used in computer simula- together provide a biomathematical basis for a nonlinear tions feedback-feedforward control model with combined pulsatile dS(t) Incremental secretion in interval (t, t 1 dt) (from and circadian features that closely emulate the measurable Ref. 16) output activities of the male hypothalamic-pituitary-Leydig dWi(t) Stochastic noise term (via differential of Brown- cell axis. ian motion) (see section VIII) dXi(t) Incremental change in concentration of hormone i neuroendocrine; biomathematics; stochastic differential equa- at time t tions; reproductive axis Zi(t)dt Incremental change in secretion of hormone i at time t g Parameter that (probabilistically) controls inter- pulse lengths THE MALE REPRODUCTIVE AXIS consists of three physiologi- GnRH Gonadotropin-releasing hormone cally distinct but interacting functional control nodes: a H(·) Dose-response (interface) functions gonadotropin-releasing hormone (GnRH) pulse genera- IID Independent and identically distributed tor, endowed by hypothalamic neurons; luteinizing (lj,1, lj,2) Time-delayed interval for jth feedback interac- hormone (LH), produced in the anterior pituitary gland; tion and testosterone (Te), secreted by Leydig cells in the l(·) Cosine function specifying periodic (circadian) in- testes. In health, this multinodal feedback and feedfor- put ward interactive system yields a pseudo-steady-state L(·) (Stochastic) pulse generator intensity function LH Luteinizing hormone output of pulsatile (episodic) neurohormone release j Mi Pulse mass j for hormone i that shows circadian modulation, e.g., 24-h variations p(t) Pulse generator (stochastic) intensity in serum Te concentrations and, to a lesser extent, in ci(·) Pulse shape for hormone i LH (30). Dose-response relationships have been largely SDE Stochastic differential equation i defined for individual nodes acting in isolation, e.g., T j Pulse time j for hormone i 0193-1849/98 $5.00 Copyright ௠ 1998 the American Physiological Society E157 E158 PULSATILE GNRH-LH-TESTOSTERONE FEEDBACK AXIS Fig. 1. Schematic illustration of time- delayed negative feedback (2) and posi- tive feedforward (1) within human male gonadotropin-releasing hormone-lutein- izing hormone-testosterone (GnRH-LH- Te) axis. Broad arrows, feedforward (1) stimulus-secretion linkages; narrow ar- rows, feedback (2) inhibition. ‘‘H’’ func- tions are developed further in section I and Fig. 2 and define dose-response relationships at each feedback inter- face within axis (see section VIIIA2). Te Testosterone itary gland, and testes (Fig. 1), we can examine basal Wi(·) ith Brownian motion process (one of ‘‘noise’’ pro- hormone output, test system performance in specific cesses) (see section VIII) computer simulations compared with prior clinical Xi(t) Concentration of hormone i at time t Yi Observed concentration of hormone i at time t experiments, evaluate relevant internodal linkages, k k and later predict possible axis dysregulation. Zi(t) Secretory rate for hormone i at time t 1. Core equations. Considering the foregoing back- I. GENERAL METHODS ground, we can relate the ‘‘true blood hormone (GnRH, A. Background Physiology LH, or Te) concentrations’’ in vivo to corresponding secretion rates. To this end, let time 0 represent the An important initial question in capturing physiologi- onset of the observation period, XG(t), XL(t), and XTe (t ) cal behavior of the male hypothalamic-pituitary- (t $ 0) the hormone concentration values, and ZG(t), gonadal axis in a biomathematical construct is: What is ZL(t), and ZTe (t )(t$0) the corresponding hormone an appropriate level at which to formulate the model? secretion rates for GnRH, LH, and Te, respectively, over We have chosen to focus on the rate of change in blood time. The core model of a dynamic GnRH-LH-Te feed- hormone concentration, since this is a principal mea- back system will then encompass the following general surement variable in health and disease. Here, we do so equations. These equations state that, by definition, for in continuous time. By evaluating resultant hormone each hormone signal involved (GnRH, LH, or Te), the concentrations one can test predictions of the biomath- rate of change of its concentration in blood is the ematical formulation experimentally via available mea- difference between its rates of elimination and surements, and by structuring in continuous time one can compare data under different sampling schemes. production Our thesis is that the complex dynamic of the male dXG (t) hypothalamic-pituitary-gonadal feedback system oc- 52aGXG(t)1ZG(t) curs because the rate of secretion of any given hormone dt within the system (GnRH, LH, and Te) depends on dXL(t) relevant time-delayed, nonlinear feedback and feedfor- 52aLXL(t)1ZL(t)(t$0) (1) ward signals derived from and acting on all or some of dt the components of the system. We further assume that dXTe ( t ) pertinent dose-responsive interfaces (either inhibitory 52a X ( t ) 1 Z ( t ) or stimulatory) connect hormone signal input to nodal d t Te Te Te output; e.g., a GnRH concentration-LH secretion rate dose-response relationship functionally links the hypo- where XG(0), XL(0), and XTe (0) are specified initial thalamic GnRH signal and the time-delayed pituitary GnRH, LH, and Te concentrations and aG, aL, and aTe LH secretory output (8, 9, 11–14, 25, 30, 35). Similarly, are the rates of elimination of the respective hormones; a dose-response relationship exists for LH concentra- the rates could be allowed to be concentration depen- tions and Leydig cell Te secretion, as established by in dent, as may be the case for higher levels of LH (32). vitro and in vivo experiments (2, 7, 10, 30). By formulat- Concentration (and secretion rate) units are mass of ing a physiologically linked network using the three neurohormone (secreted) per unit distribution volume primary and interacting nodes of hypothalamus, pitu- (per unit time). PULSATILE GNRH-LH-TESTOSTERONE FEEDBACK AXIS E159 Next we incorporate feedback and feedforward rela- to 1.5-fold) physiological range. This is thought to tionships within the hypothalamic-pituitary-Leydig cell reflect homeostatic feedback control, which we embody axis by defining mathematically, via ‘‘H’’ functions, how in the coupled equations above (Eq. 1). More explicitly, each hormone secretion rate (at any instant in time t)of in young men, serum LH and Te concentrations are ZG(t), ZL(t), and ZTe (t ) depends, in a nonlinear manner, positively cross-correlated at a 120- to 50-min lag (Te on (all
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