New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
A feedbackcontrolled ensemble model of the stressresponsive hypothalamopituitaryadrenal axis

Communicated by Wylie Vale, The Salk Institute for Biological Studies, La Jolla, CA (received for review June 27, 2000)
Abstract
The present work develops and implements a biomathematical statement of how reciprocal connectivity drives stressadaptive homeostasis in the corticotropic (hypothalamopituitaryadrenal) axis. In initial analyses with this interactive construct, we test six specific a priori hypotheses of mechanisms linking circadian (24h) rhythmicity to pulsatile secretory output. This formulation offers a dynamic framework for later statistical estimation of unobserved in vivo neurohormone secretion and withinaxis, doseresponsive interfaces in health and disease. Explication of the core dynamics of the stressresponsive corticotropic axis based on secure physiological precepts should help to unveil new biomedical hypotheses of stressorspecific system failure.
The stressresponsive hypothalamoadrenocorticotropic (ACTH)adrenal (cortisol) axis is critical in initiating lifesustaining adaptive reactions to internal (disease) and external (environmental) stressors. This neuroendocrine ensemble exhibits prominent timedependent dynamics reflected in vividly pulsatile (ultradian) and 24h rhythmic (circadian) output (1, 2). Episodic secretion is driven by hypothalamic neuronal pacemakers, which secrete the pituitary signaling peptides CRH (ACTHreleasing hormone) and AVP (arginine vasopressin) (3, 4). These agonists singly and synergistically stimulate ACTH synthesis and secretion (feedforward), which in turn promotes the timelagged and doseresponsive biosynthesis of cortisol. Cortisol feeds back to inhibit CRH/AVP and ACTH production via timedelayed concentrationdependent (integral) and rapid, ratesensitive (differential) mechanisms (5). These core physiological linkages mediate a homeostatic (servocontrol) system governed by nonlinear and timedelayed feedforward and feedback signal interchanges. We postulate that such interactive properties generate the observed complex dynamics of this dynamics.
A networklike notion of joint feedforward and feedback control of the ACTHadrenal axis was adumbrated by KellerWood and Yates nearly two decades ago (5), and reinforced subsequently by Liu et al.'s credible indirect estimates of in vivo CRH, ACTH, and cortisol halflives (6). Here, we extend these fundamental concepts to a multivalent, interactive, doseresponsive, and timedelayed biomathematical model that achieves coupling of circadian and pulsatile outputs. Thereby, we explore six a priori hypotheses of coupling mechanisms to link 24h periodic (circadian) rhythmicity to pulsatile (ultradian) secretion, and illustrate the utility of a networklike biostatistical formalism to assess dynamic interfaces within an integrative neuroendocrine system.
Methods
Based on a statistically validated interactive model for the feedback/feedforward control of the male reproductive axis (7), here we implement an extended formulation to encompass the unique interactions inherent in the HPA axis (see Appendix). The systemspecific details are highlighted below and illustrated schematically in Fig. 1.
CRH/AVP Pulse Generator.
We envision that hypothalamic pulse generators for CRH and AVP drive episodic ACTH release after a slight time delay and poststimulus refractory interval (ref. 8; see Appendix). For simplification, we consider CRH and AVP as a combined feedforward signal, wherein corticotropic synergy is achieved by modifying the joint CRH/AVP doseresponse curve (below).
Overview of Model.
Let [X_{C/V}(t), X_{A}(t), X_{C}(t)] designate the evolving hormone concentrations at time t and [Z_{C/V}(t), Z_{A}(t), Z_{C}(t)] denote the corresponding instantaneous hormone secretion rates for CRH/AVP (C/V), ACTH (A), and cortisol (C). Structurally, we also define instantaneous rates of hormone synthesis [S_{C/V}(t), S_{A}(t), S_{C}(t)] and of ACTH release from new synthesis [R_{A}(t)].
Feedback and feedforward between the hormones is incorporated by mathematical interface or doseresponse “H” functions. The latter designate (at any instant in time t) how the rate of synthesis of each hormone [e.g., S_{A}(t)] depends in a nonlinear manner on pertinent input by prior concentrations (delayed concentrationsensitive or integral feedback) or prior rates of change of concentrations (ratesensitive or differential feedback). We use logistic functions to approximate such doseresponsive behavior, assuming a onedimensional version given by: Integral feedback is implemented here for CRH/AVP's stimulation of ACTH, ACTH's stimulation of cortisol, and cortisol's inhibition of ACTH synthesis and CRH/AVP synthesis/secretion, whereas ratesensitive feedback is used to incorporate cortisol's inhibition of ACTH and CRH/AVP release (5).
Pituitary: Feedback and FeedforwardControlled Release of ACTH.
To model ACTH production, we assume that (i) the hypothalamopituitary portal blood CRH/AVP concentration (pg/ml) exerts a positive timedelayed (0.5–1.5 min before onset) feedforward effect, and the blood cortisol concentration (μg/dl) exerts a negative slow (timedelayed) (30–60 min) feedback effect, on the rate of ACTH synthesis (pg/ml per h; ref. 5); and (ii) the hypothalamopituitary portal blood CRH/AVP concentration (pg/ml) exerts a potentially steep timedelayed (0.5–1.5 min before onset) feedforward effect, and the rate of change of blood cortisol concentration exerts a rapid (timedelayed) (5–30 min) feedback action on pituitary ACTH release (pg/ml per h; Fig. 1).
The synergistic effect of CRH and AVP on ACTH synthesis is manifested as an elevation in the upper asymptote of the corresponding H (feedforward) doseresponse function. A rapid succession of CRH/AVP pulses outside the refractory window would thus exert both a combined and a supraadditive effect on ACTH synthesis [S_{A}(t)] and release [R_{A}(t)].
Adrenal Gland: Feedforward by ACTH.
We envision a nearly continuous basal rate of cortisol secretion. Blood ACTH concentrations (pg/ml) superimpose timedelayed (10–20 min) feedforward to elevate the rate of adrenal cortisol (C) synthesis and diffusive release (μg/dl per h; Fig. 1). Because ACTH evokes a complex facilitative cascade of second and more distal messengers in the adrenal zona fasciculata, we render such priming by way of a shortlived (5–10 min) leftshift of the ACTHcortisol doseresponse function H_{5} (increased adrenal sensitivity) with an elevation in its upper asymptote (rise in ACTH efficacy). The delayed sustained effect of ACTH is incorporated via multiplication of the ACTH input into H_{5} by a linear combination of exponential functions [Γ_{A}(⋅)], as described in ref. 7. Thereby, we emulate the adrenal secretory response observed experimentally.
Hypothalamus: Feedback on CRH/AVP.
The intervalaveraged blood cortisol concentration (μg/dl) exerts timedelayed (60–80 min) integral feedback, and the rate of change of blood cortisol concentration imposes rapid (5–30 min) ratesensitive feedback, on CRH/AVP synthesis/secretion (pg/ml per h; ref. 5, Fig. 1).
The foregoing primary connections do not exclude the existence of other withinaxis interactions: e.g., the blood cortisol concentration (μg/dl) might also exert slow (integral) negative feedback on basal ACTH release or the CRH/AVP pulsefiring rate(s) (see Discussion).
Parameters for the Population and Individual.
To represent the diversity among individuals, we allow for variations in in vivo hormone elimination rates, the degree of CRH/AVP synergism and ACTH priming of cortisol synthesis/secretion, the amplitude and phase of the circadian rhythm, and doseresponse parameters. Conversely, we consider structural mechanisms, such as the pulse shapes for CRH/AVP and ACTH secretion as populationally defined and relatively consistent among subjects.
Linking Circadian Rhythms to Pulsatile Output.
The above core model addresses (ultradian) pulsatility and feedback/feedforward connections. To explore mechanisms that link the circadian rhythm to such shortterm secretory activities, we consider a 24h periodic internal neural clock (e.g., residing in the suprachiasmatic nucleus), the phase of which is set by relevant internal and environmental cues. Relevant circadian inputs might couple to the pulsatile network via 24h rhythmic control of: (model 1) the timedelayed negative feedback of cortisol concentrations on hypothalamic CRH and/or AVP synthesis/release (8); (model 2) the rapid ratesensitive negative feedback of cortisol on the CRH/AVPstimulated release of ACTH (5); (model 3) ACTH or cortisol's basal secretion rates; (model 4) the feedforward of CRH/AVP on the rate of accumulation of ACTH pulse mass (8, 9); (model 5) the sensitivity and/or maximum of CRH/AVP's doseresponsive stimulation of ACTH secretion, thereby encapsulating changing synergism between AVP and CRH (9, 10); and (model 6) the doseresponsive feedforward of ACTH on adrenal cortisol secretion (11)(see Fig. 1). Other hypotheses could include 24h variations in cortisol's rapid feedback on CRH/AVP release and/or its integral feedback on hypothalamic CRH/AVP pulse frequency or pituitary ACTH synthesis.
Results
The complex physiological output of the corticotropic axis is illustrated for three healthy men in Fig. 2 (Left column, Top to Bottom), which displays concurrent plasma ACTH and cortisol concentration profiles obtained by sampling blood at 7min intervals for 24 h. Fig. 2 (Right column, Top to Bottom) depicts computerassisted simulations using the circadian model of diurnally varying CRH/AVP synergy (model 5 above) to recapitulate both ultradian (pulsatile) and circadian (rhythmic) features.
To evaluate circadianpulsatile linkages (hypotheses 1–6 above) systematically, we shifted the sensitivity of each corresponding H (feedback or feedforward) doseresponse curve by smoothly varying the coefficient B in the relevant logistic (interface) functions. Fig. 3 summarizes modelspecific histogram predictions (each based on the same initial randomization seed) for 500 realizations. Asterisks in the subpanel for circadian model 5 mark the clinically observed values in six healthy young men. Circadian model 5 (24h varying CRH/AVP drive on ACTH secretion) and model 6 (a diurnal rhythm in ACTH's feedforward on cortisol production) each predict observed rhythmic properties of this axis (see Discussion).
Fig. 4 illustrates modelbased fitting of observed plasma ACTH concentration profiles in four young men. Table 1 summarizes predictions of ACTH kinetics and secretory dynamics for all six men so analyzed. Fig. 5 presents an illustrative computerassisted estimate of the unobserved CRH/AVP pulse signal based on analyzing simultaneously observed plasma ACTH and cortisol concentration time series in one healthy male.
Discussion
The present formalism explores the thesis that neuroendocrine ensembles operate homeostatically via organspecific and timedelayed doseresponsive facilitative or inhibitory interactions (7). To this end, we embody dynamics of the corticotropic axis via a biomathematical model, wherein relevant doseresponse interfaces serve to couple changing hypothalamicpituitary portal venous CRH/AVP concentrations to timedelayed stimulation of corticotrope ACTH biosynthesis and secretion. In turn, varying systemic blood ACTH concentrations drive nonlinear doseresponsive oscillations in cortisol secretion by steroidogenically responsive adrenal zonafasciculata cells. Biologically available cortisol feeds back on hypothalamic CRH/AVP and pituitary ACTH outputs by way of both delayed (timeintegrated) and rapid (ratesensitive) inhibitory mechanisms (see Methods). We incorporate these dynamic relationships in a core biostatistical construct of coupled stochastic differential equations along with biological variability. The resultant networklike formulation affords an objective, statistically valid, and conceptually tractable basis for predicting corticotropicaxis regulation (below).
To examine the putative neurointegrative mechanisms subserving commingled circadian and ultradian rhythmicity, we first simulated the outcomes of six plausible circadianpulsatile coupling hypotheses based on earlier studies of the CRH/AVPACTHcortisol feedback axis (1, 2, 5, 6, 8–10). Computerassisted experiments predicted that 24h variability in the coupling strength of pituitary ACTH's drive of adrenal cortisol secretion (circadian model 6) can generate diurnal rhythmicity of both cortisol and ACTH release (Fig. 3). CRH/AVP's joint stimulation of ACTH synthesis and secretion (circadian model 5) also engendered nyctohemeral variations in both cortisol and ACTH output. The latter model would accord with preserved circadian ACTH rhythmicity in severely cortisoldeficient (Addisonian) patients (12), and in transgenic CRH knockout mice administered a constant exogenous CRH stimulus (presumptively accompanied by diurnally variable endogenous AVP release) (10). Unvarying CRH stimulation in healthy humans or patients with postoperative Cushing's Disease also sustains cortisol rhythmicity, as plausibly mediated via circadian models 5 or 6. In contrast, four other postulated circadianultradian linkage mechanisms (e.g., 24h rhythmic changes in cortisol's rapid or delayed negative feedback on CRH/AVP or ACTH production) failed to capture expected nyctohemeral ACTH and cortisol changes. Because the foregoing analyses evaluated single linkage mechanisms only, further study of joint coupling models and their presumptive pathophysiological disruptions will also be important hereafter.
The current statistically founded networklike ensemble (Fig. 1) recreates stable and pulsatile and circadian patterns of ACTH and cortisol secretion (Fig. 2). The resultant output also exhibits subordinate (higher frequency) variations, which emerge from the complex dynamics of nonlinear and timevarying feedback and feedforward signaling among regulatory nodes, as predicted from simpler reductionistic mathematical models (13).
The present biomathematical construct also allows estimation of ACTH and cortisol secretion rates and their respective in vivo kinetics, conditional on the inferred pulse times (Fig. 4, Table 1). To extend this notion, we illustrate the combined assessment of ACTH and cortisol secretory behavior and CRH/AVP pulse times based on simultaneous measurements of two of the three signals (Fig. 5). Thereby, one could begin to estimate in vivo doseresponse interface functions, while accounting correctly for fullsystem interactions and variability (7).
The complexity of corticotropicaxis control includes putative gender differences in the ACTHcortisol doseresponse relationship, hypothalamic CRH/AVP rhythmicity, and cortisol feedback sensitivity (14). A formalized combined feedback and feedforward model should offer the basis for further exploration of the networklevel implications of such sex differences. Analogously, semiquantitative modeling should aid in appraising the causes and consequences of disruption of selected neuroendocrine doseresponse functions in infancy, aging, stress, and disease by complementing clinical intuition. Indeed, intuitive perspectives of dynamic axis behavior are difficult to validate or refute otherwise, given the multivalent, timelagged, nonlinear doseresponsive and integral and ratesensitive feedback properties of this homeostatic system. Such statistical analyses will pose new analytical challenges and may require novel experimental data (5–7, 15).
Whereas the current biostatistical construct of CRH/AVPACTHadrenal dynamics incorporates certain core feedback interactions as presently understood, further hypothalamic and extrahypothalamic regulatory inputs will also be important to consider later; e.g., independent CRH and AVP feedforward signaling, autofeedback by CRH and/or AVP, corticolimbic feedback inputs, and intrapituitary or intraadrenal paracrine regulation (3, 4, 8, 9–12, 16). A practicable and valid model structure is essential in considering such enhancements. Second, certain stressors applied during the neonatal period influence responsivity of the adult CRH/AVPACTHadrenal axis; e.g., maternal separation during a critical interval in infancy strongly modulates later stress reactivity in the adult (17). A suitable basic model formulation should aid in the later exploration of mechanisms underlying such longerterm neuroregulatory adaptations. Third, the corticotropic axis often exhibits different homeostatic adjustments to an acute versus chronic stressor (18, 19). This plasticity could point to timedependent “resetting” of selected physiological feedforward or feedback parameters. A relevant interactive model should find utility in examining such hypotheses. Fourth, neuroendocrine axes that regulate stress, growth, reproduction, and metabolism typically interact; e.g., AVP and CRH also negatively regulate the reproductiveaxis neuronal pulse generator or somatotropin secretion (20). Such betweenaxis linkages likely facilitate organismic adaptations to environmental stressors, but their mechanistic coupling has been difficult to formalize. Fifth, certain stress contexts unmask thresholdlike responses of the ACTHadrenal axis, whereas others unveil gradual neurointegrative changes (5, 8). Understanding putative “jump” mechanisms (rather than continuous doseresponsiveness) is stymied currently by limited quantitative undergirdings. And, lastly (ultra), shortloop feedback interactions are implicit in many biological systems, including the corticotropic axis; e.g., ACTH and/or betaendorphin can inhibit, whereas enkephalins can stimulate, CRH secretion (21). A more comprehensive biomathematical formalism should aid in examining the physiological implications of such shortloop regulatory effects.
In summary, the present analyses implement and explore the dynamics of an interactive (networklike) biomathematical formulation of the complex but autonomously regulated CRH/AVPACTHadrenal axis, as inferred clinically and experimentally from studies of single components in isolation. This new formalism embodies expected withinaxis physiological linkages via timedelayed, nonlinear, doseresponsive, ratesensitive, and integral feedforward and feedback controls. The ensemble features generate realistic pulsatile, 24h rhythmic and subordinate (patternsensitive) modes of ACTH, cortisol, and CRH/AVP secretion, and allow computerassisted predictions and hypothesis testing. The foregoing biostatistical encapsulation predicts that certain putative mechanisms of ultradiancircadian coupling are more likely than others to generate the jointly 24h rhythmic release of ACTH and cortisol observed in vivo. Accordingly, biostatistical tools of this evolving genre should help fuel novel insights into the adaptive physiology and pathophysiology of the CRH/AVPACTHcortisol axis and other complex homeostatic neuroendocrine systems.
Acknowledgments
Support for this work was provided by the University of Virginia Center for Biomathematical Technology, National Institutes of Health General Clinical Research Center Grant M01 RR00847, National Institute on Aging Grant R01 AG14799, and the Specialized Cooperative Center for Reproduction Research (U54 National Institute of Child Health and Human Development Grant HD28934).
Appendix
We assume that CRH/AVP signaling dictates the pulse times for ACTH after a finite time delay τ_{A}, reflecting hypothalamopituitary portal blood transit, and a poststimulus refractory interval, r_{A}, when further CRH/AVP inputs are ignored. Thus, there will be two corresponding sets of pulse times: T, T, T, … and T, T, T, … , where T = [Min_{j} {TT ≥ T + r_{A}}] + τ_{A}, with T ≤ 0, T = T + τ_{A}. Let N(t) denote the counting process associated with the ACTH pulse times. Here, we view the pulse times as a Weibull renewal process (15), where λ is a rate parameter (number of pulses/day) parameter and γ controls the regularity of interpulse interval lengths. Then, the conditional probability densities for T given T are given by: We denote a timeaveraged feedback signal at time t with time delay (l_{1},l_{2}) by: where Y(r) is either a hormone concentration or its rate of change at time r. In what follows, the subscripted numerics 1–7 for the interface (H) functions denote corresponding feedback/feedforward interactions (see Fig. 1): viz., ACTH synthesis (subscript 1,2) and release (subscript 3,4) are each joint functions of timedelayed CRH/AVP feedforward and slow and rapid cortisol feedback signals, respectively. CRH/AVP synthesis is analogously controlled jointly by respectively rapid and slow cortisol feedback (subscript 6,7). In refs. 7 and 15, we show that the mathematical effect of cascading targettissue reactions to a signal input is the multiplication of the initial feedback/feedforward signal by a linear combination of exponential functions, denoted by Γ_{C/V}(⋅) and Γ_{A}(⋅), which allows ongoing glandular responses after the signal is withdrawn. Let ψ_{A}(·) and ψ_{C/V}(·) represent the normalized rates of secretion per unit mass per unit distribution volume per unit time; these rates are presently modeled as 3parameter generalized gamma densities (7, 15). Accordingly, synthesis (S), release (R), accumulation (A), and fractional mass remaining for later secretion (Ψ) are given as: Based on the above constructions, the corresponding interactively controlled rates of secretion are given as: Secreted molecules undergo combined diffusion and advection in the bloodstream at very rapid rates (short halflife component, α_{1}) and are removed more slowly but irreversibly (long halflife component, α_{2}). If V_{i} is the assumed distribution volume for hormone i (i = C/V, A, C), we here approximate V_{i} in the human as: 1/2 ml for each of CRH and AVP, 3.5–5 liters for ACTH, and 7–8 liters for cortisol. If incremental secretion V_{i}Z_{i}(t)dt enters two (statistical) compartments with respective distributional volumes of V and V (V_{i} = V + V and a and proportional contents a = 1 − a, then VZ(t)dt = aV_{i}Z_{i}(t)dt, (j = 1,2, i = C/V, A, C). Here, we approximate a = 0.33, a = 0.67, i = C/V, A, C (1, 3, 16–18). The solution of the above (assuming VX(0) = aX_{i}(0), with X_{i}(0) being specified (initial condition) for i = C/V, A, C) is: which takes the form of a biexponential elimination rate. In the context of the above formulation, we have explicitly modeled the secretion rates Z_{i}(⋅), i = C/V, A, C, based on known physiological structure (Fig. 1). One can allow for additional biological variability as, e.g., because of within and amongcell heterogeneities in the instantaneous rate of production of each hormone, as well as turbulent admixing and diffusion of hormone molecules in the blood, by including relevant terms for such variabilities (7).
What one then observes is a discretetime sampling of these processes, plus joint uncertainty because of blood withdrawal, sample processing, and hormone measurement errors, ɛ_{i}(k):
Footnotes
Abbreviations
 ACTH,
 adrenocorticotropic hormone;
 CRH,
 ACTHreleasing hormone;
 AVP,
 arginine vasopressin
 Received June 27, 2000.
 Accepted December 28, 2000.
 Copyright © 2001, The National Academy of Sciences
References
 ↵
 ↵
 ↵
 Rivier J,
 Spiess J,
 Vale W W
 ↵
 Vale W W,
 Spiess J,
 Rivier C,
 Reivier J
 ↵
 ↵
 Liu B,
 Zhao Z,
 Chen L
 ↵
 Keenan D,
 Sun W,
 Veldhuis J D
 ↵
 ↵
 De Bold D R,
 Jackson R V,
 Sheldon W R Jr,
 Island D P,
 Orth D N
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 Keenan D M,
 Veldhuis J D
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
Citation Manager Formats
More Articles of This Classification
Biological Sciences
Medical Sciences
Related Content
 No related articles found.
Cited by...
 Rapid Glucocorticoid ReceptorMediated Inhibition of HypothalamicPituitaryAdrenal Ultradian Activity in Healthy Males
 Secretagogues govern GH secretoryburst waveform and mass in healthy eugonadal and shortterm hypogonadal men
 Hypocortisolemic clamp unmasks jointly feedforward and feedbackdependent control of overnight ACTH secretion
 Administration of recombinant human GHRH1,44amide for 3 months reduces abdominal visceral fat mass and increases physical performance measures in postmenopausal women
 Testosterone supplementation in healthy older men drives GH and IGFI secretion without potentiating peptidyl secretagogue efficacy
 Reconstruction of in vivo timeevolving neuroendocrine doseresponse properties unveils admixed deterministic and stochastic elements
 PGlycoprotein Protects Leukemia Cells Against CaspaseDependent, but not CaspaseIndependent, Cell Death