## Abstract

The present study investigates the time-varying control of pituitary hormone secretion over the day and night (D/N). To this end, we implemented an analytical platform designed to reconstruct simultaneously *1*) basal (nonpulsatile) secretion, *2*) single or dual secretory-burst waveforms, *3*) random effects on burst amplitude, *4*) stochastic pulse-renewal properties, *5*) biexponential elimination kinetics, and *6*) experimental uncertainty. The statistical solution is conditioned on a priori pulse-onset times, which are estimated in the first stage. Primary data composed of thyrotropin (TSH) concentrations were monitored over 24 h in 27 healthy adults. According to statistical criteria, 21/27 profiles favored a dual compared with single secretory-burst waveform. An objectively defined waveform change point (D/N boundary) emerged at 2046 (±23 min), after which *1*) the mass of TSH released per burst increases by 2.1-fold (*P* < 0.001), *2*) TSH secretory-burst frequency rises by 1.2-fold (*P* < 0.001), *3*) the latency to maximal TSH secretion within a burst decreases by 67% (*P* < 0.001), *4*) variability in secretory-burst shape diminishes by 50% (*P* < 0.001), and *5*) basal TSH secretion declines by 17% (*P* < 0.002). In contrast, the regularity of successive burst times and the slow-phase half-life are stable. In conclusion, nycthemeral mechanisms govern TSH secretory-burst mass, frequency, waveform, and variability but not evidently TSH elimination kinetics or the pulse-timing process. Further studies will be required to assess the generality of the foregoing distinctive control mechanisms in other hypothalamo-pituitary axes.

- secretion
- thyrotrope
- hypothalamus
- pulsatility

valid appraisal of in vivo mechanisms that direct basal, pulsatile, and 24-h rhythmic modes of hypothalamo-pituitary hormone secretion is made difficult by the unique dynamics of interactive neurohormone systems. Analytical challenges are introduced by systemic feedback modulation; the prolonged half-life of sialylated pituitary glycoproteins; strong statistical correlations among unknown basal (time-invariant) hormone release, ligand-specific elimination rates, and secretory-burst number, location, shape, and amplitude; stochastic effects on sequential pulse timing and individual burst mass; and incompletely understood determinants of nycthemeral [day/night (D/N)] variations (2, 5, 14, 15, 22, 34, 37, 38). In the case of the hypothalamo-pituitary-thyroidal axis, thyrotropin (TSH) secretion responds concomitantly to feedback inhibition (e.g., by thyroxine, glucocorticoids, and dopamine) and feedforward stimulation [e.g., by TSH-releasing hormone (TRH)] (1, 15, 18, 19, 21-27, 29, 39, 40). The present study develops and implements an analytical formulism intended to reconstruct and quantify key properties of time-varying TSH secretion under multivalent signaling, as a paradigm of pituitary-hormone secretory control.

## METHODOLOGY

### Methods

*Clinical protocol.* Healthy adults (*N* = 27) provided written voluntary, informed consent approved by the Institutional Review Board of the Leiden University Medical Center. After overnight adaptation to the inpatient study unit, subjects underwent repetitive blood sampling (2.0 ml every 10 min) for 24 h beginning at 0800. Fourteen men and thirteen women participated with respective mean ages of 46 and 41 yr (ranges: male 29-65 yr and female 32-50 yr).

*Assay.* TSH concentrations were quantitated in each serum sample (*N* = 145/24 h) in duplicate in a precise and sensitive immunofluorometric assay (Wallac Oy, Turku, Finland). The detection limit is 0.03 mU/l, the interassay coefficient of variation (CV) is 5%, and the intra-assay CV is 4.4% in the concentration (mU/l) range studied here (20).

### Model of Diurnal TSH Secretion

*Overall structure.* The present study implements and adapts a recent Bayesian model of pulsatile neurohormone secretion, basal release, stochastic pulse times, flexible secretory-burst shape, random effects on burst mass, biexponential elimination kinetics, and combined experimental uncertainty in sample collection, processing, and assay (8, 10, 13). Reconstruction of the foregoing principal features is schematized in Fig. 1. In this analytical formalism, a priori estimates are made of pulse-onset times, *T*^{1}, *T*^{2},..., *T ^{n}*, by use of a previously constructed methodology (9). Then, as a second stage, analyses simultaneously quantify secretion and elimination parameters conditional on the prior set of putative pulse times (7). The present adaptation creates an additional allowance for two (independent) secretory-burst waveforms, which are demarcated over 24 h by way of a statistically identified change point (time).

*Secretion.* Each pulse time, *T ^{j}*, marks the onset of a secretory burst. Conceptually, a burst reflects abrupt exocytotic discharge of hormone-containing granules followed by less rapid secretion of newly synthesized molecules (9, 10). The mass of hormone secreted in any given burst is assumed to be the sum of a finite amount of minimally available stores, a linear function of hormone accumulation over the preceding interpulse interval, and stochastic variability in individual burst mass, defined algebraically as follows 1 where

*M*is the mass secreted; η

^{j}_{0}is basal cellular accumulation; η

_{1}is a linear coefficient operating over the preceding interburst interval,

*T*

^{j}-T^{j}^{−}

^{1}; and

*A*is random effects on burst mass (9, 10).

^{j}The mass contained in any given burst, *M ^{j}*, is released according to an adaptable (hormone-, subject-, and condition-specific) waveform. The waveform (evolution of the instantaneous secretion rate over time) is represented algebraically via a three-parameter generalized gamma function (8). The latter probability distribution encapsulates the unit-area normalized rate of secretion (in mass units) over time (in min) per unit distribution volume (in liters) by way of 2

The β parameters endow adaptive asymmetry by way of variability in the onset rate (β_{1}β_{3}), peakedness (β_{3}), and dissipation rate (β_{3}/β_{2}) of the secretion event (7). However, the gamma family is quite flexible, and the symmetrical waveforms (e.g., Gaussian) that have been used in other hormone models (32) can be well approximated by the three-parameter gamma family.

Pulsatile hormone release is reconstructed as the product of the mass of hormone secreted per burst and the normalized Ψ function. Thus the total secretion rate, Z, is the sum of time-invariant (basal) hormone release, β_{0}, and pulsatile secretion 3

*Dual-waveform model of burstlike pituitary-hormone secretion.* We test the hypothesis that there is an unknown transition time (change point) in the waveform of discrete secretory bursts. Objectively, the change point demarcates statistically independent putative day [ψ^{(D)}] and night [ψ^{(N)}] waveform functions defined by corresponding parameters β_{1} ^{(D)}, β_{2} ^{(D)}, and β_{3} ^{(D)} and β_{1} ^{(N)}, β_{2} ^{(N)}, and β_{3} ^{(N)}, respectively. According to this construction, there may or may not be a statistical requirement for representation of dual secretory-burst waveforms. The distinction is made on statistical grounds, wherein the one- and two-waveform model outcomes for each data set are compared via the Akaike Information Criterion (AIC). Specifically, suppose that there are two models, the first parameterized by *p* parameters and the second, a larger model that contains the first, parameterized by *p* + *m* parameters. The AIC criterion states that the second model is appropriate if twice the number of additional parameters, 2*m*, is less than twice the log value of the likelihood ratio test of the first to the second model. The factor of two is important only for ease of chi-square calculations and can be dropped in application of the criterion. Thus the measure penalizes enhancement of the regression fit (sum of squares of residuals) achieved solely by adding *m* new parameters. A lower AIC value thus favors a given model on the basis of a principle of assumed statistical parsimony.

*Biexponential kinetics of disappearance.* In the present formulation of hormone elimination, fast- and slow-rate constants primarily capture the respective effects of advection and diffusion (α_{1}) and irreversible metabolic loss (α_{2}) from the circulation (8). Earlier we showed that at any instant in time, *t*, the ligand concentration X(*t*), sampled at a given point in the circulation, *x*, can be described by (coupled) differential equations defining the overall elimination process in the foregoing terms (7). The analytical solution of this ensemble representation is given as 4 where *A* is the (amplitude) proportion of rapid to total elimination, Z(r) denotes the aggregate secretion rate (see below), and X(0) gives the starting hormone concentration (7, 10). The observed hormone concentration profile, Y_{i}, is then a discrete time sampling of *N* data points predicted by the foregoing continuous processes, as distorted by superimposed observational error, ϵ_{i} 5

*Stochastic burst times.* Pulse times are viewed as arising by way of a stochastic process definable jointly by mean burst frequency and interburst-interval regularity (7, 11). This notion is represented by a two-parameter Weibull probability distribution. In the latter renewal process, successive pulse times, *T ^{j}*, emerge statistically as the partial sums of incremental, independent, and identically distributed positive random variables,

*S*

_{i}6 The classical Poisson counting process is a single-parameter renewable process identified by the rate parameter, λ. In the Poisson model, random variables,

*S*

_{i}, have an exponential distribution with a probabilistic average interval length (waiting time) of 1/λ. However, the latter formulation does not accommodate possible biological dissociation between mean event frequency and interevent variability, inasmuch as the means ± SD of the set of interpulse-interval lengths are equal definitionally. Thus, in the Poisson model, the CV (CV = mean/SD × 100%) of interburst waiting times is fixed at 100%. The latter diverges from estimated values of 15-40% for many hormones (4, 28, 30, 31). The Weibull probability density adds the parameter γ, which uncouples mean burst frequency from statistical regularity of interpulse-interval lengths. In the Weibull construct, the conditional expectation for

*T*given

^{k}*T*

_{k}

_{−}

_{1}is 7 where λ denotes the mean burst frequency (events observed/unit time), γ is the regularity of interpulse-time delays, and

*s*is a dummy variable on time (8, 11). The mean, variance, and CV of the Weibull renewal process are defined by 8 where Γ(·) is the classical mathematical Gamma function (Γ and lowercase parameter γ are unrelated). Accordingly, in the Weibull density, the CV of interpulse (waiting) times depends only on γ (and not frequency λ). The Poisson process is a derivative function, wherein γ = 1. In the Weibull expansion, γ > 1 signifies more regular (less variable) interpulse-interval lengths.

*Data presentation.* Scatterplots are presented to highlight the dispersion of values in the cohort of 27 subjects. Values in the text are cited as means ± SE, with median in parentheses. D/N parameter comparisons (dual- vs. single-burst waveform models) are made by a two-tailed, paired *t*-statistic. *P* < 0.05 was used to assign statistical significance.

## RESULTS

Figure 2 illustrates application of single and dual secretory-burst waveform models to quantify secretion and elimination kinetics underlying 10-min observed 24-h serum TSH concentration profiles in one subject. The plots highlight estimated pulse-time locations (asteriks placed on *x*-axis), measured and model-projected TSH concentration profiles (*top*), reconstructed TSH secretion rates, the waveform change point (time; *middle*), and TSH secretory-burst shape (*bottom*).

Figure 3 depicts observed and analytically reconstructed serial TSH concentrations (*top* of each diptych) and calculated TSH secretion rates (*bottom* of each) under a postulated dual secretory-burst model in six normal adults. In the group of 27 subjects, the statistically defined change point (see *Methods*) occurred at *clock time 2046* (±23 min) (2052).

Individual estimates of probabilistic TSH secretory-pulse frequency and D/N waveform-change point times are presented in Fig. 4. Daily TSH secretory burst number averaged 17.8 (±0.40) (18). The latter estimate (calculated a priori) is independent of subsequent choice of burst-waveform model (see *Methods*). Approximately 48% (±1.8%) (47%) of TSH pulses emerged within the 766 ± 23 (772)-min daytime interval preceding the statistically identified change point. The 24-h normalized event frequency was 15 ± 0.44 (15) (based on a daytime projection) and 18 ± 0.57 (18) (nighttime projection) (*P* < 0.001).

As highlighted further in Fig. 4 (*top*), representation of 24-h TSH time series under a single- vs. dual-waveform model did not influence calculated biexponential TSH kinetics; namely, rapid-phase half-lives (in min) were 9.6 ± 1.6 (6.9) and 7.9 ± 0.60 (7.0), and slow-phase half-lives were 72 ± 7.3 (72) and 90 ± 11 (70) in the single- and dual-burst formulation, respectively. On statistical grounds, rapid-phase disappearance half-times are not determinable definitively for values less than the following: (ln 2) × sampling interval (here, 10 min; see Ref. 8). Therefore, the above rapid-phase kinetic estimates denote maximally realizable values.

Projected total daily TSH secretion rates (in mU · l^{-1} · 24 h^{−}^{1}) were comparable by model form; i.e., 60 ± 7.7 (50) and 56 ± 7.0 (50) in the single- and double-waveform constructs, respectively (Fig. 4, *bottom*). Pulsatile TSH secretion rates (summed daily pulse mass, in mU · l^{-1} · 24 h^{−}^{1}) were also independent of waveform representation; i.e., 40 ± 5.8 (26) (single-burst type) and 25 ± 5.2 (14) [two-burst type; *P* = not significant (NS)]. Basal secretion rates (in mU · l^{-1} · 24 η^{-1}) were 21 ± 4.2 (16) (one-burst construct) and 25 ± 5.2 (14) (two-burst construct; *P* = NS), thus contributing approximately one-third of total TSH output.

Figure 5 (mode, mean, and SD) compares analytically estimated properties of TSH secretory bursts under the formalism of one or two distinct waveforms. In technical terminology, the ψ function defines the (unit-area normalized) time evolution of instantaneous secretion rates within any given burst (see Fig. 1*B*). The dual-waveform (2-ψ) formulation predicts significant D/N distinctions, namely, before and after the objective burst-shape change point (see *Methods*; Fig. 5*B*). Table 1 summarizes quantitative contrasts in the mode, mean, and SD of the predicted time latency (in min) to maximal secretion attained in the normalized secretory burst. An important inference is that the modal and mean times to peak TSH secretion are abbreviated markedly following (compared with before) the D/N waveform boundary; i.e., respective (D − N) difference values were 102 ± 15 (93) (modal time delay) and 112 ± 11 (107) min (mean time delay; both *P* < 0.001). Comparison of the interindividual variability (SD) of the mean time latency with maximal TSH release revealed greater uniformity at night, i.e., difference of intersubject SDs between day and night: −43 ± 14 (−17) min (*P* < 0.001). The latter estimate signifies less between-person variability in TSH burst shape at night than during the day. The dual-waveform representations in Fig. 6 illustrate *1*) the D/N distinction in burst-shape variability and *2*) the nighttime shift toward more rapid TSH release within bursts. Predictions of a single-burst formulation are given by way of comparison. According to the AIC statistical penalty term (see *Methods*), a dual secretory-burst model represents diurnal and nocturnal TSH release more aptly (lower absolute AIC value) than a single-pulse construct in 21 of 27 individuals (Fig. 6, *right*).

Figure 5 (mass and pulsatile) presents the calculated mass of TSH secreted per burst (in mU/l). Data are shown for the interval before vs. after the objective diurnal/nocturnal change point time in each subject. Cohort mean values are summarized in Table 1. On the basis of the statistical change point location inferred in the 2-ψ model, we made D/N comparisons in both models. The latter revealed consistent nighttime amplification of burst mass, namely, by 1.8-fold (1 ψ) and 2.1-fold (2 ψ) (*P* < 0.001 vs. daytime). Pulsatile TSH secretion (in mU · l^{-1} · interval^{−}^{1}) rose commensurately, i.e., by 1.7-fold (1 burst) and 2.0-fold (2 bursts) (*P* < 0.001).

The regularity of the TSH pulse-renewal process (defined by analysis of the set of waiting times between consecutive bursts) was appraised statistically by γ of the Weibull renewal process (see *Methods*). Individual reconstructions of probability distributions and regularity (γ) of interpulse-interval lengths are shown in Fig. 7, and cohort mean values are given in Table 1. Statistical appraisal demonstrated stable interpulse-interval regularity (γ) over 24 h and elevated TSH pulse number (λ of Weibull process) after the waveform change point. Because the CV of interpulse-interval lengths (stochastic variability) is determined by γ independently of probabilistic mean event frequency (see *Methods*), these data indicate that the reciprocal property, regularity, of the TSH pulse-renewal process is stable nycthemerally.

## DISCUSSION

The present composite formulation of deterministic and stochastic control of combined basal and pulsatile TSH secretion over 24 h introduces the concept of diurnal variation in secretory-burst waveform. Compared with a single-burst construct, dual-waveform representation of pulsatile TSH secretion was justified statistically in 21 of the 27 individuals. Implementation of the latter analytical construct disclosed that mechanisms driving the nocturnal elevation of plasma TSH concentrations include 2.1-fold amplification of TSH secretory-burst mass (in mU/l), 1.2-fold acceleration of probabilistic mean secretory-burst frequency (in events/h), and 67% abbreviation of the time latency (in min) to maximal TSH secretion within a burst. These day/night adaptations are specific, inasmuch as the quantifiable regularity of the stochastic pulse-renewal process and the in vivo elimination half-life of TSH did not vary over 24 h (Fig. 8). Whether the foregoing array of adaptive mechanisms is exclusive to physiological regulation of the TSH axis is not known. Generality of the current model form will be particularly relevant to explore in neuroendocrine systems that are marked by prominent diurnal and/or circadian rhythmicity, such as the corticotropic, somatotropic, lactotropic, and gonadotropic axes (4, 28, 30, 31).

The precise hypothalamo-pituitary mechanisms that orchestrate recurrent burstlike secretion of TSH in the day and night are unknown. Hypothalamic signals that impact thyrotropes include, nonexclusively, TRH (agonistic) and somatostatin and dopamine (antagonistic) (1, 5, 16, 19, 24, 26, 39). Which, if any, of the foregoing effectors expressly determines the number and variability of discrete TSH secretory events is not established. In addition, the degree to which systemic feedback by triiodothyronine, thyroxine, and/or cortisol modulates the timing of discrete TSH secretory bursts is not clear. The present data indicate that, whatever the fundamental TSH pulse-generating process, relevant mechanisms accelerate TSH secretory-burst frequency at night without detectably altering intrinsic regularity of the stochastic pulse-renewal process.

Analytical reconstruction of TSH secretory-burst waveform (normalized rate of secretion over time) predicted significant abbreviation of the latency to maximal secretion at nighttime (Fig. 8). This novel adaptation could reflect nocturnally enhanced accumulation of readily releasable (exocytotically available) TSH stores, heightened thyrotrope sensitivity to agonistic inputs, and/or attenuated thyrotrope responsivity to antagonistic signals (1, 2, 14, 16, 19, 24-26, 37, 39, 40). For example, independent observations suggest that the hypothalamic outflow of somatostatin and dopamine may decrease over the nighttime interval, which decline may facilitate the sleep-related and late-day augmentation of GH and prolactin secretion, respectively (4, 22).

From a statistical vantage, a succession of discrete neurohormone secretory episodes can be viewed as random about a probabilistic mean waiting time (reciprocal of frequency) (7, 8). The dispersion of interevent delays measures the variability of the stochastic pulse-renewal process, here represented statistically by a Weibull distribution model. The latter formulation allows one to discriminate potentially separate biological control of burst frequency and interpulse regularity. For example, recent application of the Weibull analytical model unmasked paradoxically increased regularity of gonadotropin-releasing hormone (GnRH)/LH pulse regeneration in aging men and postmenopausal women compared with young counterparts (6, 11). Greater interpulse waiting-time regularity in the older male and female occurred in the face of elevated and unchanged daily LH pulse frequency, respectively. The precise neuroadaptive bases for physiological dissociation of stochastic interpulse-interval variability and pulse frequency in the thyrotropic and gonadotropic axes will be important to explore further.

Viewed mathematically, a stochastic pulse-renewal process denotes independently and identically distributed random, positive incremental waiting times. As one approach to validating this assumption in the case of TSH pulse regeneration, we applied the model-free and scale-invariant approximate entropy (ApEn) statistic. ApEn quantifies the order-dependent subpattern consistency (degree of process randomness) in numerical series (17, 35, 36). In a pulse-renewal process, the relative reproducibility of event recurrence times is definable by ApEn analysis of the observed sequence of successive interpulse-interval lengths. This application to TSH interburst waiting-time series revealed that the mean normalized ratio of ApEn of the original ordered series to ApEn of 1,000 randomly shuffled versions of the same series does not differ significantly from unity (not shown). The latter outcome is consistent with a presumptive stochastic pulse-renewal mechanism. ApEn analysis of sequential interburst-interval lengths in ACTH and LH time series has also predicted random pulse-timing properties in these systems (3, 7, 11, 33).

### Perspectives

The precise nycthemeral determinants of secretory-burst mass, frequency, and waveform are not established. Depending on the neuroendocrine axis, relevant diurnally varying internal inputs could include central neural-, activity/rest-, and sleep/wake-related signals, whereas extrinsic factors often include fasting, food intake, external temperature, and light exposure. The present general analytical formulation should facilitate dissection of relevant underlying mechanisms mediating neuroendocrine adaptations. Further insights should be gained by extending statistical estimation procedures to include feedback and feedforward dose-responsive interface properties that control secretory-burst evolution, as foreshadowed in recent single-waveform analyses of the gonadotropic and corticotropic axes (7, 12).

## DISCLOSURES

This work was supported in part by National Institute on Aging Grant K01-AG-19164 and National Institute of Diabetes and Digestive and Kidney Diseases Grant R01-DK-60717 (Bethesda, MD), National Center for Research Resources (Rockville, MD) General Clinical Research Center Grant RR-M01-00585 to the Mayo Clinic and Foundation, and National Science Foundation Interdisciplinary Grant in the Mathematical Sciences DMS-0107680.

## Acknowledgments

We thank K. Bradford for expert editorial and graphical aid.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 2003 the American Physiological Society