Abstract
We present a nonlinear randomeffects stochastic differential equation (SDE) model of combined basal and pulsatile hormone secretion with a seriesspecific hormone halflife and conditional pulse times. The construct uses a threeparameter pulse shape (generalized gamma function) to allow variably skewed secretory bursts superimposed on a finite basal hormone secretion rate. The analysis imbeds stochastic elements at three levels: a variable mass of hormone accumulation (of which the random effect is a part) during interpulse intervals, nonuniform secretion with hormone admixture into the circulation, and technical (sampling and assay) experimental uncertainty. We implement maximum likelihood estimates of secretory parameters (basal and pulsatile secretion and halflife) with asymptotic standard errors. The model applied to illustrative human luteinizing hormone (LH) time series suggests contrasts in basal LH secretion rates (e.g., greater in postmenopausal women than men) and LH secretory burst mass (e.g., higher in older women), but not LH burst frequency or distributional LH halflives (7–40 min). For validation, in two infused (human recombinant) LH profiles, we implement partially constrained mono and biexponential versions of the model with fixed (a priori assumed) versus variable LH basal secretion rates. We conclude that a statistically supported, nonlinear, random effects, SDEbased construct can evaluate jointly basal and pulsatile LH secretory rates and LH halflife in 24 h, episodically varying serum LH concentration profiles. This new reducedparameter analytic strategy should be useful to explore further the pathophysiological mechanisms of altered neurohormone secretion.
 pulse
 neurohormone
 pituitary gland
 model
 biomathematics
in the present paper , we implement a model for reconstructing from observed plasma hormone concentration time series the basal and pulsatile secretion rates of a protein hormone, such as luteinizing hormone (LH). Statistical methods for estimating the key parameters of the model are presented and applied illustratively to LH data for both the adult male and pre and postmenopausal female. Then, based on the estimated parameters, we estimate (reconstruct) the unobserved secretion rate time series.
Consider the “true concentration” in vivo of a given single hormone. Let t
_{0} (≥0) represent the beginning of the observation period. Let [X(t ),t ≥ t
_{0}] be the hormone concentration evolving over time and [Z(t ),t ≥ t
_{0}] the hormone secretion rate over time. The rate of change in the concentration [X(t )] is described by the differential equation
We have chosen to model and analyze the hormone concentration [X(t )] and the rate of secretion [Z(t )] in continuous time. If one did not model with this perspective, it would be difficult to compare usual experimental data observed under different sampling schemes. We assume that, for a given subject, the observations of blood hormone concentrations are made at times t_{k}
, k = 1, … , n during the period [L
_{1},L
_{2}], with t
_{0} =L
_{1}, t_{n}
=L
_{2}. What is then observed is Y_{k}
We postulate that an accurate description of hormone concentrations requires a stochastic formulation, which incorporates the various forms of uncertainty that occur within the different time and space scales. Here we allow for three such basic forms of variation.
First, at the cellular/glandular scale, there are variations in the instantaneous rates of synthesis of a given hormone within and among cells. For a hormone secreted in pulses, we distinguish between the instantaneous rate of synthesis (or intracellular production) and the instantaneous rate of secretion (which for a protein hormone is based on an accumulation and subsequent release of previously synthesized hormonecontaining granules). Conceptually, the instantaneous rate of synthesis represents an “idealized” (or expected) rate, whereas the “realized” rate for a population of molecules and cells should be a stable random variation about this expected rate. This dispersion of variability (noise) results in the inclusion of theA_{j} terms in Eq. 4. Such random variations could reflect nonuniform withincell and betweencell metabolic milieus (e.g., ATP energy stores), biochemical signals (e.g., first and second messengers), and cell biological functions (e.g., cytoskeletal apparatus state of polymerization, phosphorylation state of secretoryrelated proteins, etc.) (2, 5, 8, 9, 15, 18).
Second, at the level of glandular secretion of an array of hormone molecules into the circulatory system, there is nonuniform release topographically among the cells and subsequent mixing of the population of hormone molecules within the bloodstream, and this microscopic biological variability (noise) should impact the resulting concentration level. This noise is represented by the Brownian motion term ς_{w}dW(t ) in Eq.6 (1, 11, 16, 18).
Third, at the level of the sample removal, processing, and assay from the human subject, there are additional contributions to experimental uncertainty (e.g., withinassay measurement errors). This is represented by the e_{i} terms in Eq. 8.Experimentally, the error in the automated measurements of protein hormones is estimated to have a typical coefficient of variation of 3–6% (5, 6, 16).
In Ref. 8, the pulsatile secretion (not concentration) of one hormone was modeled and applied to pituitary LH secretion data from a horse. This paper builds on that initial model in three ways. First, it extends the biomathematical construct and analysis to that of hormone concentration by including the modeling of hormone elimination; second, and as importantly, we here allow for greater flexibility in the modeling of the variable (LH) pulse masses via random effects, without introducing an inordinate number of parameters; and, third, we implement a maximum likelihood estimation (MLE) strategy with the calculation of appropriate secretoryparameter statistical confidence intervals. In Refs. 7 and 8 the pulse masses were assumed to be a linear function of the preceding interpulse lengths: the longer the interpulse interval, the greater the accumulation of LH mass. The former model is reasonable and appears to fit certain data well. However, such a model is rather rigid if a small interpulse interval is followed by a large mass (or vice versa), as we illustrate here. Failure to allow for more flexibility in possible burst mass values could produce spurious estimates of the true secretory variability (discussed below).
METHODS
Modeling Hormone Elimination and Secretion
Elimination.
The elimination rate constant α of a biological molecule from a particular (single compartment) sampling space is related to its halflife (t _{1/2} ) by: exp (−αt _{1/2} ) = ½. For the components of the human male and female reproductive axes, we note that direct quantification of biexponential LH disappearance rates in hypopituitary men injected with highly purified human pituitary LH yielded a mean rapid (first) component LH halflife of ∼18 min, a slow (second) component halflife of ∼90 min, and an average fractional contribution of the former to total decay amplitude of 0.63 (20). Thus a monoexponential approximation of such kinetics would suggest a halflife within this broad range and its experimental uncertainties. Although the rapid and slower algebraic phases of hormone elimination do not necessarily correspond to definable anatomic compartments, the more rapid phase of hormone disappearance is thought to reflect largely hormone distribution within the vascular space after abrupt secretion or infusion, whereas the slower component may result from irreversible metabolic removal of the hormone from blood. Indeed, rate constants for the latter correlate with the sialic acid content of human LH infused into hypophysectomized rats, consistent with a view of a sialoreceptormediated uptake and removal of LH by metabolically relevant tissues (2). In accordance with this concept, primarily a distributional phase halflife may be evident during spontaneous LH secretion pulses, whereas irreversible metabolic removal would proceed more slowly. We discuss below that with rapidly recurring LH secretory pulses, the slower putative removal process may be less evident in the plasma LH concentration profile and could mimic a low rate of basal (nonpulsatile) LH secretion between peaks. This point will be illustrated further in a paradigm of infused LH pulses.
Pulsatile secretion.
We have viewed the secretion model as arising in two stages. The first stage concerns the mechanism that governs the time occurrences of the bursts (or pulses); the second stage concerns the resulting shapes and masses of the bursts (at the various pulse times).
PULSE TIMES.
In the present work, we are not so concerned with estimating the probabilistic structure of the pulse times. For example, we will condition on the observed (estimated) pulse times. Various authors have developed methods for estimating the pulsing mechanism (3, 5, 6, 16,17). The method for pulse detection applied here is presented in Ref.8. Elsewhere, in Ref. 9, models of varying degree of complexity are presented for the pulse generator; there we indicate that the most general formulation allows for the probability of a pulse in the next time increment (t, t + dt ) to depend on timedelayed nonlinear feedback by some or all of the hormones of the system (axis). The present implementation will assume conditional independently estimated pulse times, based on which secretory pulse measures will be reconstructed.
PULSE SHAPE.
To define secretory pulse shape over time, given any particular pulse time, a function ψ( ⋅ ) will be specified. This denotes the normalized rate of secretion per unit mass of hormone per unit distribution volume per unit time. We have used a generalized gamma family of densities (i.e., normalized to integrate to 1) to model the pulse
PULSE MASS.
The basal or nonpulsatile rate of production of hormone will be represented by a constant β_{0}. By M^{j} we denote the amount of mass accumulation of hormone from the last pulse time (T _{j − 1} ) to the present pulse time (T_{j} ); this accumulation will begin to be released at time T_{j} . Let ψ( ⋅ ) be
the pulse shape, defined above, which represents the instaneous rate of secretion per unit mass per unit distributional volume. We will represent a pulse at time t, having started at pulse timeT, by a function M × ψ(t − T ), ψ(s) = 0, s ≤ 0, whereM is the mass of the pulse. We will assume that eachjth pulse mass is given by
MODELING HORMONE CONCENTRATION PROFILES
Given the foregoing physiological motivation, we thus propose as a biomathematical formulation of the rate of secretionZ( ⋅ ) and the concentration levelX( ⋅ ) of pulsatile hormone secretion superimposed on a finite basal (time invariant) hormone secretion rate, β_{0}, the following
To summarize, we formulated a construct to represent the intermittently observed hormone concentrations, starting at the level of the (unobserved) continuous rate of pulsatile secretion and allowing for random variation not only in the pulse times, but also via three other sources of stochastic variation (as reviewed in the introduction):1) anticipated biological variation in the hormone mass accumulated between pulse times, which is reflected by the inclusion of the A_{j}
terms in Eq. 4
, whereA_{j}
terms are IID normal (0,
RECONSTRUCTING THE PULSATILE HORMONE SECRETION RATE
Consider a discretetime sampling of X( ⋅ )
In Ref. 24, the asymptotics for the MLE of θ
Also, we are interested in the actual values of random effectsA, which embody biological variations in pulse mass (Eq.4 ). Because the random effects A_{j} are not observable, we will useE(A_{j} ‖Y _{1},Y _{2}, … , Y_{n} ); in Ref.24 a precise formula for calculating these predicted values is presented. Also, to calculate the asymptotic standard deviations of the MLEs, one can calculate the information matrixI_{n} (θ̂_{n} ) on the basis of our model. The explicit expressions for the information matrix are given in Ref. 24.
On the basis of the parameter estimateθ̂_{n} and the predicted random effects
To calculate the total daily LH secretion, we integrate the reconstructed LH secretion rate, Ŝ, from 0 to 1,440 min; because the normalized secretion rate ψ( ⋅ ) integrates to one, we can use the following very accurate approximation
Also, consider the (particular) likelihood equation with respect to η_{0} (evaluated at the MLEθ̂_{n} )
We can use this approximation to calculate desired standard errors for daily LH basal and daily pulsatile LH secretion
Also, to obtain the standard error for the halflife (t _{½} ), from the standard error for the elimination rate [α = log(2)/t _{½} )], we use the δmethod (a firstorder approximation).
ILLUSTRATIVE APPLICATIONS
Adult Male and Female 24h Serum LH Concentration Profiles
We here use previously published 24h serum LH concentration time series, in which blood was sampled at 10min intervals and the subsequent sera were submitted to LH immunoradiometric assay (IRMA) in young men, young women at three stages of the menstrual cycle, and in estrogenwithdrawn postmenopausal women (5, 14, 15, 22). Illustrative fitted profiles with calculated LH secretion rates are shown in Fig1.
Table 1 gives the LH secretion statistics for the three illustrative LH data groupings: young males, premenopausal females, and postmenopausal females. SDs are given in parentheses. Table 2 shows the individual parameter estimates in two of the subjects. We also estimate parameter SD values for each of the 10 key parameters in the model.
Infused LH Pulse Profiles
To evaluate our modelbased estimates of LH pulse mass in a defined experimental context, we infused five different doses of human recombinant LH (Serono Laboratories, Norwell, MA) intravenously as 1min (7.5, 15, or 30 IU) or 8min (50 or 75 IU) squarewave pulses in two leuprolide [gonadotropin releasing hormone (GnRH) agonist, TAPS Pharmaceutical]suppressed healthy young men (T. Mulligan and J.D. Veldhuis, unpublished observations). Infusions were administered every 2 h for four to eight consecutive injections of the same dose. Volunteers were sampled every 10 min for later assay of serum LH concentrations by IRMA (First International Reference Preparation) with the first blood sample withdrawn immediately before the first LH injection.
Five models of combined basal and pulsatile LH release (infusion) were evaluated, and the known mass of LH injected (IU/volunteer) per pulse was regressed against the calculated mass of LH “secreted” per burst (IU/l of distribution volume) (Fig.2 A ). The inverse of the slope of this line approximates the LH distribution volume, which was measured earlier (20). In the first two models, the LH halflife was computationally estimated as a single exponential disappearance function for each LH dose infused, with or without constrained low basal rates of (residual) endogenous LH secretion. The latter reflected incomplete suppression by leuprolide and was calculated from the first measured (preinjection) basal serum LH concentration in each series. Compared with an expected LH distribution volume of 3.6–6.4 liters (75–86 kg subjects, with projected 4.5–8% of body weight as LH distribution volume), these two models overpredicted this value at 9.81 (variable basal) and 10.0 liters (constrained basal). Two other models, one with and the other without a constrained low basal rate of endogenous LH secretion, assumed known a priori measured biexponential LH disappearance kinetics, namely, a rapid and slower mean LH halflife of 18 and 90 min, respectively, with 37% of the total decay amplitude attributable to the slower component (20). Both models predicted mean LH distribution volumes similar to expectation, namely 3.96 (variable basal) and 3.60 liters (constrained basal). A fifth model also allowed the foregoing twocomponent LH kinetics, but assumed zero residual (basal) LH secretion and yielded an estimated LH distribution volume of 3.76 liters. Examples of the modelbased fits to the serum LH concentration profiles, as well as the reconstructed LH secretion rates, after the 7.5 (low) and 50 IU (higher) dose LH infusions for three models are shown in Fig. 2 A. The three models illustrated are zero basal, constrained basal, and variable basal (endogenous) LH secretion.
DISCUSSION
Here, we develop, implement, and illustrate a stochastic differential equation (SDE) biomathematical formulation of combined basal and pulsatile LH secretion in concert with a subjectspecific LH halflife and conditional pulse times to quantitatively analyze (24 h) serum LH concentration time series, e.g., as obtained earlier in healthy young men and premenopausal as well as postmenopausal women (5, 1416). This new construct of hormone secretion and removal allows for variably shaped [e.g., skewed or asymmetric (1, 11)] LH secretion pulses superimposed upon a timeinvariant basal (LH) secretion rate. In addition, secreted LH molecules are admixed in the bloodstream and subjected to a monoexponential (or higher order) elimination function (20, 21). The smaller number of essential parameters defining overall LH secretion and removal in this model [namely 10 parameters per 144sample data set according to the present notion, versus 20–30 parameters by modelspecific deconvolution analysis (18)] allows us to explore the otherwise difficult issue of joint estimation of basal and pulsatile hormone secretion rates assessed concurrently with halflife (19). Moreover, the statistical basis for the present MLE of basal and pulsatile LH secretion and LH halflife permits the construction of statistical confidence intervals for each of these parameters, as well as a reconstruction of the randomeffects term defining stochastic variability in anticipated LH secretory burst mass.
The high correlations among basal and pulsatile rates of LH secretion and concurrent halflife of elimination of LH in earlier highly parameterized convolution models of neurohormone release and removal pose a formidable challenge to reliable estimation of these interdependent parameters by parametric approaches (19). The current implementation of a more parsimonious model of LH secretion and removal, including relevant stochastic contributions at several levels and a restricted resultant parameter set, begins to address some of these constraints, and might thus ultimately allow resolution not only of pulse number and mass, basal hormone release, and hormone halflife, but also of asymmetric LH secretorypulse shape. Our use of the generalized gamma function (with 3 parameters) to mathematically depict a potentially asymmetric LH secretory burst shape seems to be suitable for evaluating the hormone secretion rate contour over time within a pulse (see preliminary analysis in Table 2 and direct sampling data in Refs. 1, 11). Such estimates should be technically accomplishable in peripheral blood with greater accuracy under conditions of sufficiently rapid blood sampling and adequately specific and reproducible hormone assays. For example, a recent clinical study sampled blood every 2.5 min in young and older men throughout the night, allowing a high density of serum LH measurements over time (13). Our formalization of pulse shape via a simple threeparameter gamma function, with a term to define the steepness of upstroke, another to depict the rapidity of declining secretion after its maximum, and a third to impart peak sharpness quantification should be useful in eventually appraising secretory event variations in different pathophysiological states. Without highly specific and precise assay methods, however, neither this nor other models would likely allow accurate discrimination of (LH) secretory pulse shape. Moreover, when pulse shape is reconstructed, we recommend comparisons of analytic results with direct secretory gland sampling, e.g., as carried out in the intact horse (1), ovariectomized sheep (11), or human (17).
By applying the present SDE nonlinear randomeffects model to physiological serum LH concentration time series in men and young and older women, we could illustrate possible contrasts among healthy individuals with respect to LH secretory burst frequency and mass and basal LH secretion rate but not apparent (distributional) LH halflife. In particular, the postmenopausal woman exhibited a higher rate of calculated basal LH secretion than the young men, without any major disparity in (distributional) LH halflife. In addition, LH secretory burst mass was considerably (4fold) higher in the older woman than that in young men. In contrast, in the young woman in the midluteal phase of the menstrual cycle, LH pulse frequency was low with an interpulse interval of ∼2.5–3 h. In the luteal phase, the apparent basal LH secretion rate was intermediate and contributed ∼50% of total daily LH secretion. Thus the postmenopausal woman and young men may represent extremes in basal LH secretion rates and LH secretory burst mass. The healthy young woman in the early follicular phase of the menstrual cycle seems to represent nearmaximal pulsatile LH secretion (74 ± 6%) and the lowest basal LH secretion rate. During the late follicular phase, an accelerated LH pulse frequency, increased mass per burst, and augmented basal LH release rate are suggested here. We emphasize that considerable further analyses in a larger group of men and women will be important to definitively test the generality of these illustrative inferences. The basis for our estimates must be distinguished from that of earlier deconvolution estimates of LH secretion (15) in that here we implement an SDE model with random effects, jointly estimate combined pulsatile and nonpulsatile (basal) LH release, condition our analysis of secretion rates on independently estimated pulse times, and apply maximum likelihood statistical estimation (MLE).
Our estimates of LH halflife during pulsatile LH release under physiological conditions in the human male and female are consistent with independent calculations of LH distribution rates (rapidphase elimination) after the bolus injection of highly purified human pituitary LH in hypopituitary men (20). Namely, such earlier experiments showed an initial distributional phase LH halflife of disappearance from plasma of ∼18 min, a delayed (slow, second) component halflife of 90 min, and a fractional contribution of the rapid component to the total amplitude of LH decay of 0.63 (20). Here we find a range of apparent (rapid) halflives of endogenous pulsatile LH decay toward basal of 7.4–27 min, thus suggesting that in vivo human pituitary LH secretion occurs with sufficient frequency that steady state is rarely attained in plasma and that the rapid phase of LH distribution tends to predominate. Alternatively, we point out that these data either argue against a combined model of basal (nonpulsatile) and pulsatile LH secretion or suggest a biexponential LH decay structure in these physiological states. Data by way of validation (Fig. 2 A ) suggest the latter (biexponential) interpretation.
Earlier deconvolution analyses assuming purely pulsatile LH secretion (zero basal) predict an LH halflife range of 60–130 minutes in human subjects (5, 15, 16, 21). Such values approximate the directly measured second (slow) component of LH removal after bolus intravenous injection of highly purified human pituitaryderived LH in LHdeficient men (90 min) (20) or the halflives of the metabolic removal of LH during (12, 20, 23) or after (10) steadystate LH infusions in the human (80–130 min). Accordingly, the shorter halflives of LH disappearance estimated in our combined pulsatilebasal secretion model with a nonlinear randomeffects SDE analysis reflect either primarily distributional (rapid phase) kinetics of LH and/or suggest an overestimation of the basal LH secretory rate due to (e.g., in postmenopausal women) rapidly successive LH pulse times with incomplete LH removal before the onset of the next secretory event and a coarse sampling rate (relative to pulsing rate). These issues are overcome largely via a biexponential decay structure (see below).
We explored the foregoing kinetic considerations further in experiments using bolusinjected human recombinant LH in five young men pretreated with the GnRH agonist leuprolide to produce a reversible LH deficiency state (see Fig. 2). In this clinical paradigm of experimentally reduced rates of basal LH release and known fixed exogenous LH injection doses given intravenously at 2h intervals, a singleexponential formulation of LH decay would yield overestimates of LH distribution volume (bottom 2 curves in Fig. 2 A ), underestimates of LH halflife, and overestimates of basal LH secretion rate. Accordingly, we caution that misapplication of a combined basal and pulsatile hormone secretion model with monoexponential kinetics to a known (virtually) purely pulsatile hormone release context may predict unduly short hormone halflives and excessive basal and total hormone secretion rates. A similar inference is made using bolus testosterone injections after pharmacological testosterone depletion by ketoconazole administration in men (not shown).
Using the current SDE construct, we have not attempted to fit the human LH data to a variable twocomponent halflife model of elimination, which would increase the parameter set. In principle, adding a second (slower) component of LH removal, possibly reflecting irreversible tissue uptake and degradation of LH (see introduction), would reduce the estimates of basal secretion presented here, while tending to increase the apparent mass of hormone secreted within bursts. Indeed, our experiments using varying doses of infused LH corroborate this prediction (see top 3 curves in Fig. 2 A ). Thus further evaluation of biexponential kinetic models of hormone disappearance will be quite relevant and instructive. In addition, we would note that independent prior knowledge of anticipated or known concurrent basal rates of (nonpulsatile) hormone release, whether zero or otherwise, would be helpful when appraising complex pulsatile hormone secretion profiles quantitatively.
Our formalization of LH secretory pulse mass includes a residual mass of LH accumulated since the last GnRHstimulated discharge or pulse and a constant that relates the rate of pulsemass accumulation to the prior interpulse interval length (respectively, η_{0}, η_{1} ). We have attempted here (Table 2) preliminarily to estimate these two conceptually distinct contributions to the mass of any given burst of secreted LH. When estimating secretion from highintensity sampling data with infrequent pulse events and/or from larger groups of LH time series these parameters may be estimable with greater accuracy. We urge the conduct of appropriate corresponding in vivo experiments to eventually establish the reliability of pulse composition estimates under various conditions.
The present implementation of an SDE model of pulsatile LH secretion superimposed on basal release with a random effect contributing to individual pulse mass depends on assumed (conditional) pulse times. We here use one particular methodology of objectively estimating possible LH pulse times and recognize that a variety of validated tools exist for this purpose (e.g., reviewed in Refs 5, 6, 16). The mechanisms that generate variable pulse times within the GnRHLHgonadal axis, and within other neuroendocrine axes, have been considered elsewhere by various authors (e.g., Refs 3, 4, 8).
Our notion of episodic LH secretion could likely be generalized or extended to certain other hormones, particularly protein hormones encapsulated in secretory granules, the release of which is triggered by an agonist hormone, e.g., corticotropin releasing hormone stimulating ACTH release, growth hormone (GH) releasing hormone stimulating GH secretion, GnRH stimulating folliclestimulating hormone release, etc. In addition, as suggested by animal experiments, our model makes allowance for some determinable finite basal hormone (LH) secretion rate, e.g., in the ewe in the gonadectomized state (11), in the ovaryintact horse (1), and in the human as inferred by inferior petrosal vein sampling of LH (17). Further studies will be required to quantify the extent of and variability in basal LH (and other hormone) secretion (as well as in pulse shape) as inferred by modeldependent statistical estimation (as illustrated here in Table 2) and validated by direct catheterization studies in experimental animals and human volunteers (e.g., undergoing sampling for independent clinical indications).
In summary, we identify, implement, illustrate, and discuss an SDE model of combined basal and pulsatile LH secretion and removal, with estimation of basal and pulsatile LH release as well as (distributional) LH halflife concurrently in healthy young men and premenopausal and postmenopausal women. These analyses suggest a spectrum of inferred physiological LH secretory partitioning between basal and pulsatile release with contrasts among different subject groups. Such a nonlinear randomeffects model should thus allow reconstruction of 24h serum LH concentration profiles in individuals in both health and disease. The present work also suggests possible later application of this technical strategy to the investigation of other neuroendocrine pathophysiologies, especially when independent information is available to define the structure of system behavior (e.g., biexponential halflives, anticipated relative partitioning of secretion into basal and pulsatile components).
Perspectives
An enlightened appraisal of neurohormone secretory systems should be based, in our view, on attempts to model from first principles of biology, namely, from the level of molecular synthesis, hormone secretion by individual cells, integration across a somewhat functionally heterogeneous cell population, nonuniform admixture of secreted molecules within the bloodstream or other fluids, timedelayed delivery of hormone to target tissues, variable but irreversible metabolic removal, recirculation of hormone, etc. As introduced here, stochastic uncertainty exits not only by way of the foregoing (nonuniform hormone synthesis, secretion, and representation across the cell population, with admixture in blood), but also at the level of blood or tissuefluid sample withdrawal, processing, and assay. In addition, biological variation not explained by the algebraic representation of the model is implicitly a stochastic contribution (i.e., stochastic elements operate in the apparent reconstruction of the biological processes themselves). Although a conditional determinant here of subsequently calculated secretory features, pulse times from neuroendocrine episodic secretory units may behave as point processes or renewal processes with finite or no memory for previous interpulse intervals and in this respect are of stochastic nature. We believe further that the amount of variability inferred within a neuroendocrine system is underestimated by examining any one output alone or by considering any one nodal function in isolation. Thus we propose that the full feedback system with its relevant physiological connections, including appropriate feedforward and feedback response interfaces, endows nonlinear features as well as appropriate variability both over time and by way of amplitude and feedback control. Accordingly, the entire system should ideally be modeled from first principles, embedding stochastic elements as appropriate, and with full feedback and feedforward connectivity pertinently interfaced by doseresponse curves. Finally, given available biomedical knowledge as “prior information,” the concept of Bayesian renditions of neuroendocrine analyses will be timely. Moreover, it will be important to develop efficient implementation of maximum likelihood methods for obtaining true multiparameter estimates, including estimates using alternative secretory pulse numbers and locations with refitting for a global optimization of true system behavior, given a model basis in first principles, stochastic elements, feedback control, and Bayesian prior knowledge. Under such circumstances, we forecast the utility to neuroendocrine physiologists of more comprehensive methodologies to delineate secretory rhythms, quantitate unobserved secretory outputs within the system, generate applicability to other neuroendocrine dynamics, and thereby assist in evaluating the impact of age, gender, and selected pathophysiologies on hormone regulation.
Acknowledgments
Support for this work was provided by the National Science Foundation Center for Biological Timing, National Institute of Child Health and Human Development Grant RCDA1K04HD00634, National Institutes of Health Reproduction Research Center Grant P30HD28934, and National Institute on Aging Grant R01AG14799.
Footnotes

Address for reprint requests: J. D. Veldhuis, Box 202, Endocrinology; Health Sciences Center and NSF Center for Biological Timing, University of Virginia, Charlottesville, VA 22908.

Present address for R. Yang: Pharmaceutical Research Associates, Charlottesville, VA 22903.

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 Copyright © 1998 the American Physiological Society