Because estrogen production and age are strong covariates, distinguishing their individual impact on hypothalamo-pituitary regulation of growth hormone (GH) output is difficult. In addition, at fixed elimination kinetics, systemic GH concentration patterns are controlled by three major signal types [GH-releasing hormone (GHRH), GH-releasing peptide (GHRP, ghrelin), and somatostatin (SS)] and by four dynamic mechanisms [the number, mass (size), and shape (waveform) of secretory bursts and basal (time invariant) GH secretion]. The present study introduces an investigative strategy comprising 1) imposition of an experimental estradiol clamp in pre- (PRE) and postmenopausal (POST) women; 2) stimulation of fasting GH secretion by each of GHRH, GHRP-2 (a ghrelin analog), and l-arginine (to putatively limit SSergic restraint); and 3) implementation of a flexible-waveform deconvolution model to estimate basal GH secretion simultaneously with the size and shape of secretory bursts, conditional on pulse number. The combined approach unveiled the following salient percent POST/PRE contrasts: 1) only 27% as much GH secreted in bursts during fasting (P < 0.001); 2) markedly attenuated burstlike GH secretion in response to bolus GHRP-2 (29%), bolus GHRH (30%), l-arginine (37%), constant GHRP-2 (38%), and constant GHRH (42%) (age contrasts, 0.0016 ≤ P ≤ 0.027); and 3) a 160% prolongation and 32% abbreviation of the time required to achieve maximal GH secretion after injection of l-arginine and bolus GHRP-2, respectively (both, P < 0.001). Accordingly, age selectively determines both the size (amount) and shape (waveform) of GH secretory bursts in healthy women independently of the short-term estrogen milieu.
- growth hormone-releasing hormone
growth hormone (GH) and sex-steroid concentrations decline together in aged mammals (26, 40). Although estrogen is a prominent positive determinant of GH secretion in humans, whether reduced GH output in aging is due to relative estrogen deficiency is difficult to parse for several reasons. First, GH secretion is correlated negatively with age and positively with estrogen availability, whereas age is related inversely to estrogen concentrations (38). Second, GH secretion is controlled via multiple peptidyl pathways that are both stimulatory and inhibitory (26, 33, 40). Finally, the dynamic mechanisms that govern plasma GH concentrations include, for any given distribution volume and elimination kinetics, both pulsatile (burstlike) and basal (time invariant) secretion (40).
Whereas the regulation of basal (<10% of total) GH secretion has not been well studied, pulsatile hormone release is controlled by three major factors, viz., the number, size, and shape of discrete secretory bursts (17). Plasma hormone concentrations are defined further by the diffusion, advection, and elimination of secreted molecules (16, 17). Because age and sex-steroid availability do not significantly influence the elimination kinetics or frequency of GH pulses (38, 40), the size and shape of secretory bursts constitute the primary determinants of GH secretion patterns. What remains unknown is how age and sex steroids individually modify these main physiological determinants. This basic question is significant, because GH patterns convey important regulatory information to target genes, such as those encoding transcripts for skeletal-muscle IGF-I, liver cytochrome P-450 enzymes, and EGF and LDL receptors (7, 22, 29, 40).
The present analyses test the hypothesis that age independently of short-term estrogen availability governs the size and shape of peptide-regulated GH secretory bursts in healthy humans. To this end, 19 normal women [10 pre- (PRE) and 9 postmenopausal (POST)] underwent a total of 95 individual infusion and blood-sampling sessions under a systemic estradiol (E2) clamp (38, 40). To investigate peptide-specific secretory dynamics, subjects received separate infusions of GH-releasing hormone (GHRH), GH-releasing peptide (GHRP)-2 (an analog of the GHRP ghrelin), and l-arginine [a putative inhibitor of somatostatin (SS) outflow]. GH secretory-burst size and shape were estimated simultaneously with basal hormone release conditional on a priori candidate pulse times using a recently validated statistically based deconvolution model (16, 17). The new deconvolution approach was designed to overcome an earlier impasse in accurately quantifying pulsatile and basal hormone secretion together (39). Thereby, we test the hypothesis that age stratum and secretagogue type jointly determine the size and shape of GH secretory bursts in an experimentally controlled estrogenic milieu.
A total of 19 healthy PRE (N = 10) and POST (N = 9) women enrolled in and completed all five study sessions. Participants provided voluntary written informed consent, and the study was approved by the Mayo Institutional Review Board. The protocol was approved by the US Food and Drug Administration under an investigator-initiated new drug number. Exclusion criteria were recent transmeridian travel or night-shift work (within 7 days), significant weight change (>2 kg in 1 mo), body mass index (BMI) <19 or >29 kg/m2, acute or chronic organ system illness, anemia, psychiatric treatment, substance abuse, or failure to provide informed consent. Volunteers had no known or suspected cardiac, cerebral, or peripheral arterial or venous thromboembolic disease, breast cancer, or untreated gallstones. None was receiving neuroactive medications. Inclusion criteria were an unremarkable medical history and physical examination, as well as normal screening laboratory tests of hepatic, renal, endocrine, metabolic, and hematologic function.
The mean ± SD [range] age was 28 ± 3.2 [24–31] and 62 ± 9.3 [51–78] yr in PRE and POST volunteers, respectively. Corresponding BMIs were 26 ± 6.3 [19–29] and 25 ± 4.5 [20–29] kg/m2 (P = not significant). PRE women did not use oral contraceptives and had normal menarchal and menstrual histories and a negative pregnancy test. POST status was confirmed by concentrations of FSH > 50 IU/l, LH > 20 IU/l, and estradiol < 30 pg/ml (<81 pmol/l). After the personal physician's approval, POST volunteers (3 subjects) discontinued any sex hormone replacement at least 6 wk before the study.
The study was a parallel-cohort, repeated-measures, double-blind, prospectively randomized comparison of the effects of single secretagogues on the size and shape of GH secretory bursts during controlled E2 repletion in healthy POST versus PRE women. To achieve age-independent estrogen deprivation, the gonadotropin-relaxing hormone agonist, leuprolide acetate (3.75 mg depot im), was administered twice 3 wk apart (38, 40). Leuprolide was given to both POST and PRE subjects to obviate any unexpected confounding by the downregulation regimen. The first injection was given in young volunteers within 8 days of menstrual bleeding and within 48 h of a negative blood pregnancy test and in older women 6 or more wk after withdrawal of any estrogen supplements. Graded transdermal E2 repletion was accomplished on an outpatient basis, starting on the day of the second leuprolide injection (day 1). The E2 dose was changed every 4 days beginning at 0.05 mg/day followed by 0.10, 0.15, and 0.20 mg/day [Estraderm (Novartis)]. The highest E2 dose (0.2 mg/day) was administered for 10 days (days 14–23, inclusive). Infusion studies were performed during the last week of this 10-day window. The transdermal paradigm was designed to elevate serum E2 concentrations into the normal late follicular-phase range of 100–150 pg/ml (38, 40). On the last day of the study, oral micronized progesterone (100 mg nightly) was begun for 12 days, according to standards of good medical practice for women with an intact uterus.
Secretagogue infusions and sampling paradigm.
Each subject underwent five randomly ordered, double-blind infusion sessions on separate days. Volunteers received a standardized outpatient meal of 8 kcal/kg distributed as 20% protein, 50% carbohydrate, and 30% fat at 1800 the night before study and then remained fasting overnight and until the end of sampling. At 0700 the next morning, catheters were placed in contralateral forearm veins to allow blood sampling (1.5 ml) every 10 min for 6 h from 0800 to 1400. Concomitantly, saline (20 ml/h iv) was infused from 0800–1000 before the following secretagogue infusions: 1) GHRH continuously from 1000 to 1400 at a constant rate of 0.33 μg·kg−1·h−1, 2) GHRP-2 continuously from 1000 to 1400 at a constant rate of 0.33 μg·kg−1·h−1, 3) l-arginine 30 g (0.17 mol) continuously from 1000 to 1030, 4) GHRP-2 (0.33 μg/kg iv) bolus at 1030, and 5) GHRH (0.33 μg/kg iv) bolus at 1030.
The foregoing peptide doses approximate 50% of maximal stimulation in POST women to mimic physiological rather than pharmacological actions, whereas the l-arginine dose is maximally effective as a positive control (38, 40).
Plasma GH concentrations were measured in duplicate by automated ultrasensitive double-monoclonal immunoenzymatic, magnetic particle-capture chemiluminescence assay using 22-kDa recombinant human GH as assay standard (Sanofi Diagnostics Pasteur Access, Chaska, MN). All samples (N = 185) from any given subject were analyzed together. Sensitivity was 0.010 μg/l (defined as 3 SDs above the zero-dose tube). No serum GH values fell below 0.020 μg/l. Interassay coefficients of variation (CVs) were 7.9% and 6.3%, respectively, at GH concentrations of 3.4 and 12.1 μg/l. Intraassay CVs were 4.9% at 1.12 μg/l and 4.5% at 20 μg/l. Cross-reactivity with GH-binding protein or 20 kDa GH is <5% (38, 40). Serum LH, FSH, testosterone, and estradiol concentrations were quantified by automated competitive chemiluminescent immunoassay (ACS Corning, Bayer, Tarrytown, NY), and total IGF-I, prolactin, and sex hormone-binding globulin (SHBG) concentrations by immunoradiomedic assay, as described earlier (38, 40).
Earlier deconvolution methods in some cases yield nonunique estimates of basal and pulsatile hormone secretion and elimination rates (39). To address this technical impasse, basal and pulsatile GH secretions were estimated simultaneously using a new maximum-likelihood deconvolution methodology discussed fully in appendix (16, 17). The basic assumptions are that 1) peaks in concentrations reflect the mass of hormone released in delimited secretory bursts, the waveform of which is defined by a three-parameter generalized γ-probability density; 2) combined diffusion, advection, and irreversible elimination can be represented via biexponential kinetics; and 3) parameter estimation is statistically conditioned on a priori estimates of pulse-onset times obtained by an incremental smoothing algorithm, as previously described (16, 17).
A modification of the general model was implemented wherein the principal analytical outcomes are cohort-defined estimates of basal and pulsatile GH secretion during saline infusion (in μg·l−1·h−1), the summed mass of GH secreted in bursts after stimulation with an individual secretagogue (in μg·l−1·h−1), and the reconstructed shape of GH secretory bursts, defined by the modal time in minutes to attain maximal secretion. Interpulse-interval times were modeled as a two-parameter Weibull probability density rather than a one-parameter Poisson process. The Weibull renewal process permits different degrees of variability of interpulse-interval times about the statistical mean, as required for physiological data (17). Unlike the Poisson distribution that defines interpulse variability as a CV of 100% (SD/mean × 100%), the Weibull density includes an additional term (γ), which allows lesser variability than 100% (γ > 1.0) at any given probabilistic mean frequency (λ).
An unpaired, two-tailed Student's t-test was utilized to compare experimentally independent measures. P < 0.05 was construed as statistically significant. Data are presented as means ± SE or the modes.
On the last day of the E2 clamp, PRE and POST women did not differ with respect to fasting serum concentrations of E2, SHBG, LH, IGFBP-1, or total testosterone (Table 1). In contrast, POST compared with PRE (POST/PRE × 100%) concentrations were 43% for IGF-I (P = 0.002), 70% for IGFBP-3 (P < 0.001), and 360% for FSH (P < 0.001).
Curves predicted by the deconvolution model are illustrated in two PRE and two POST subjects (median cohort outcomes) in Fig. 1. The fact that predicted curves are not readily distinguished from the measured GH concentration profile illustrates the fidelity of the model with the biology. Figure 2A depicts the 50 deconvolution-estimated GH secretory profiles in PRE women, and Fig. 2B gives the 45 corresponding GH secretory profiles in POST women. In both cohorts, bolus GHRP-2 infusion induced the most GH release, whereas constant GHRH infusion evoked the least.
Analytical reconstruction of the underlying shape of individual secretory bursts disclosed comparable waveforms in fasting POST and PRE women during saline infusion (Fig. 3). The analytical mode of the waveform, defined by the time delay from secretory-burst onset to maximal secretion, was used to compare burst shapes (Table 2). In the unstimulated (saline infusion) state, modes were 21 and 23 min in POST and PRE subjects, respectively, which were not different. In contrast, POST women exhibited delayed peak GH secretory responses to l-arginine, and earlier peak GH secretory responses to bolus GHRP-2, compared with PRE women (both, P < 0.001). Relative precision of the modal estimate (defined practically here as SE/mode × 100%) ranged from 1.7% to 7.4%, except in the case of PRE women after bolus GHRH (108%) and POST women during continuous GHRH (256%). The two circumstances reflected inexplicably high prestimulus GH concentrations in several women.
The amount of GH secreted in bursts (mass released per unit distribution volume per unit time) was reduced in POST women to only 27% than that in PRE subjects during saline infusion (P < 0.0001) (Fig. 4). In contrast, estimated basal (nonpulsatile) GH secretion did not differ by age cohort, viz., POST 0.077 ± 0.028 and PRE 0.144 ± 0.082 μg·l−1·h−1 (P = 0.44). Percent basal of total GH secretion tended to be higher in POST than in PRE individuals (4.5% vs. 2.2%; P = 0.053), because total GH secretion was markedly reduced in POST subjects (Fig. 4). Expressed as percentages, POST/PRE stimulated pulsatile GH secretion values were bolus GHRP-2 (29%), bolus GHRH (30%), l-arginine (37%), constant GHRP-2 (38%), and constant GHRH (42%) (0.0016 ≤ P ≤ 0.027) (Fig. 5).
Table 3 shows that the interpulse interval (proportionate to the reciprocal of GH pulse frequency) was not influenced by age. In addition, γ (a measure of interpulse-interval variability) was only minimally, albeit significantly (P = 0.015), increased in POST compared with PRE women. This difference signifies reduced pulsing variability in the older than in the young cohort.
The present combined experimental and analytical paradigm discloses 1) marked reductions in the estimated size of both endogenously maintained and exogenously stimulated GH secretory bursts and 2) prominent differences in the reconstructed shape of secretagogue-induced GH secretory bursts in healthy fasting POST compared with PRE women studied in a controlled E2-replete milieu. Since the experimental goal of imposing comparable E2 milieus in the two study cohorts was attained, these observations indicate that age, independently of short-term estrogen availability, strongly determines both the amount and waveform of pulsatile GH secretion but not pulse frequency. In the first regard, members of all three major classes of GH secretagogues were 238% to 345% more effective in PRE than POST individuals, whereas E2 concentrations differed by only 23%. In the second regard, the analytically reconstructed time course of burstlike GH secretion was age dependent such that the time delays to attain maximal GH release after the onset of a burst differed by absolute standard-deviate (z) scores of 6.7 for l-arginine and 7.6 for bolus GHRP-2 stimulation. An interesting contrast was observed in the GH pulsing mechanism, wherein POST women manifested lesser interpulse-interval variability than PRE individuals. The age-related distinction in GH regularity mimics the LH pulse-regeneration difference reported in older and young men (18). The age-associated contrasts were selective, given that mean GH pulse frequency did not differ by menopausal age or secretagogue type.
Diminished pulsatile GH secretion appears to characterize both aged and hypogonadal individuals (26, 40). The present experimental design demonstrates that reduced pulsatile GH secretion in older women is due to diminutive GH secretory-burst size rather than to fewer pulses and that smaller secretory bursts are not attributable to short-term differences in systemic concentrations of E2, testosterone, or SHBG (Table 1). In particular, both endogenously maintained GH secretion and GH secretory responses to nearly physiological amounts of GHRH and GHRP-2 (a ghrelin-receptor agonist) were reduced in POST compared with PRE individuals. Stimulation with a pharmacological dose of l-arginine corroborated decreased burstlike GH secretion in POST subjects. The last outcome is important because l-arginine is believed to elicit GH secretion by restricting hypothalamic SS outflow and disinhibiting GHRH and ghrelin drive (1, 12, 36, 37). Accordingly, the accompanying findings point to regulatory deficits in all three major peptidyl pathways that converge on GH secretion in aging women. Limited studies in the monkey and human are consistent with but do not directly prove this unifying postulate (2, 25, 27, 30). Other laboratory data in rodents suggest that aging can alter hypothalamo-pituitary expression of GHRH, SS, and cognate receptors (8, 9, 21, 24, 42).
GHRPs, such as GHRP-2 and ghrelin, are unique in their multifaceted capabilities to stimulate somatotropes directly in vitro, synergize with a maximally effective dose of GHRH in vivo, release GHRH from the arcuate nucleus into hypothalamo-pituitary portal blood, and oppose certain central neural actions of SS (albeit not the release of SS into portal blood) (reviewed in Refs. 3, 10, 19, 40, and 41). Genetic models in the mouse, GHRP-receptor antagonist studies in the rat, and rare GHRP-receptor mutations in the human together support a role for ghrelin in maintaining GH secretion, body composition, and somatic growth especially in the female (28, 33, 43). Thus one may hypothesize that POST women with reduced responses to a ghrelin-receptor agonist, as observed here, respond less well than PRE individuals to 1) GHRH, because injected GHRH should synergize with endogenous ghrelin, and 2) l-arginine, given that this amino acid is thought to mimic SS withdrawal by evoking reboundlike GHRH and thereby GH release (10, 14, 33, 36, 43).
The physiological mechanisms that supervise basal (time invariant) GH secretion remain poorly understood. In the mouse, deletion of the SS receptor subtype 1 (SSTR1) gene elevates basal GH release in vitro (20). However, in the human, SSTR3 and SSTR5 may be more important mediators of somatotrope inhibition (32). In addition, IGF-I can exert repressive effects on both the hypothalamus and pituitary gland (13, 31), whereas E2 can stimulate GH synthesis by ectopic pituitary tissue in vivo and pituitary cells in vitro (4, 5, 34). Whether such mechanisms modulate basal GH secretion in the aging human or animal is not yet established.
GH secretory bursts can be monitored directly in pituitary-venous blood in some larger animals (reviewed in Ref. 40). Although invasive studies are not possible in humans, recently validated analytical methods allow one to reconstruct time-varying secretion rates using serial plasma hormone concentrations and thereby estimate both the size and shape of discrete secretory events (16, 17). Analyses of GH secretory-burst shape delineated considerable asymmetry of the release process in healthy young and older adults (Fig. 3). In particular, under baseline conditions, estimated instantaneous GH secretion rates within any given delimited burst increased to a maximum within 22 ± 2 min (the waveform mode) and then declined gradually over the next 50 min. An asymmetric time course was also predicted recently for TSH, LH, and ACTH (17, 18). A remarkable observation was that l-arginine infusion abbreviated and prolonged the mode in PRE and POST women, respectively, resulting in a prominent (10.2 min) difference by age. Inasmuch as burstlike GH secretion is mediated via exocytosis of GH-containing granules (6), the delay in the timing of maximal GH release in older individuals could signify reduced drive to the exocytotic process or impaired mechanics of exocytosis. Because POST women attained maximal GH secretion significantly (6.3 min) earlier than PRE women given a bolus of GHRP-2, we infer that the basic exocytotic mechanism is intact. Therefore, the secretory delay in aging subjects more likely reflects an unknown defect in the hypothalamo-pituitary pathway of l-arginine action, viz., altered signaling by secondary mediators such as GHRH, SS, or nitric oxide (15, 23, 35, 37). Since BMI was similar in the two cohorts, we cannot attribute the age-related effect readily to this physical feature. In consideration of the experimental paradigm used, the unexplained defect in POST women is independent of short-term estrogen availability, albeit not necessarily independent of estrogen action (40, 42). A speculative explanation for more rapid GH release after bolus GHRP-2 infusion in POST than in PRE women is greater immediately releasable exocytotic GH stores (6). In principle, the latter could be associated with increased baseline SSergic outflow in older individuals, which is opposed by the ghrelin analog (26, 38, 40). Other possible mechanisms are less evident, given that the waveform contrast was not observed after l-arginine, GHRH (bolus or continuous), or continuous GHRP infusions.
Caveats include the relatively small number of subjects studied (N = 19), the somewhat short baseline sampling interval (10 h), and the large variance of occasional GH secretory-burst modes. Further studies will be needed to assess the impact of more prolonged E2 clamps on GH secretion as well as IGF-I and FSH concentrations and to quantify dose-responsive actions of GHRH and GHRP in various fixed steroidal milieus. The experimental E2 paradigm implemented here is not intended for clinical application. Although recent dose-response analyses indicate that short-term E2 supplementation can potentiate stimulation by GHRH, GHRP-2, and ghrelin and attenuate inhibition by SS in POST women (38, 40), whether age per se modulates these estrogenic effects is unknown.
In summary, POST compared with PRE women studied in an experimentally controlled estrogenic milieu exhibit prominent attenuation of the size (but not number) of endogenous and exogenously driven GH secretory bursts and marked secretagogue-selective differences in the time delay to maximal GH secretion after burst onset. A parsimonious interpretation of these outcomes is that factors associated with aging 1) attenuate the hypothalamo-pituitary effects of GHRP/ghrelin and GHRH; 2) alter the waveform of GH secretory bursts without disrupting the basic exocytotic process; and 3) impair amino acid-induced GH secretion, which is putatively mediated by SS withdrawal and reboundlike GHRH release. More generally, the present studies illustrate a conjoint strategy of clamping systemic sex-steroid availability and applying variable-waveform deconvolution analysis to dissect physiological regulation of a dynamic endocrine axis.
APPENDIX: VARIABLE WAVEFORM DECONVOLUTION ANALYSIS
From a technical perspective, there are five interventional assignments involving both pre- and postmenopausal women. The following model applies to each of the two groups. Each subject, j = 1, 2, …, was sampled every 10 min for 6 h under each of the five conditions. The five infusion types are here denoted as k = 1, 2, 3, 4, 5. At a given time, t, the GH secretion rate (unobserved) and GH concentration (measured) in subject j for condition k are designated by Z (t) and X(t), respectively. The group basal (nonpulsatile) GH secretion rate is given by γ, with a random effect [Rj(k)] allowing for variation for each subject and intervention day: γ + Rj(k). Burstlike hormone secretion, before and following secretagogue injection at time T*, is described by two terms: 1) the waveform or instantaneous (unit-area normalized) rate of secretion over time, ψ(·); and 2) the mass (M) of GH released per unit distribution volume in the burst (in μg/l) (17). The interventional secretagogue was administered at time T* = 2 h. A preinjection (baseline) waveform is defined [ψ(0)], as well as waveforms for the k = 1, 2, 3, 4, 5 interventions. These waveform functions (burst shapes) are defined by the generalized Gamma probability density: (1) The three β parameters of the Gamma distribution permit variable asymmetry or Gaussian-like symmetry of secretory-burst shape.
The present analytical formulation is distinctive by way of reconstructing 1) a common baseline (unstimulated) Gamma function for the cohort of young and another for the cohort of older volunteers, as well as each of the five interventions, k; and 2) a cohort-specific mean amount of GH secreted at baseline, M(0), as well as after each secretagogue intervention, M(k). For subject j, the m [=m(j,k)] pulse times for intervention k are denoted as Tj,l(k), l = 1,…, m(j,k). The mass secreted by subject j at pulse time Tj,l(k) is then M(0) plus a random variation, Aj,l(0), if the pulse is prestimulus, or M(k) plus a random variation, A, k = 1, 2, 3, 4, 5, if it is poststimulus. The pulse times for each profile were determined by a recently published pulse detection method. Trends are first removed, and the data are normalized to [0, 1], so that the algorithmic parameters do not depend on scale (16). The method then uses a nonlinear diffusion equation, with the diffusion coefficient inversely related to the rate of increase. Thus the putative pulse times are identified as points of rapid increase that are not easily smoothed away. The algorithm is run for a specified amount of algorithmic time, and the estimated pulse times are determined. The total (basal and pulsatile) GH secretion rate (in μg·l−1·min−1) in subject j under condition k (k = 1, 2, 3, 4, 5) is (2) and the predicted GH concentration is (3) (basal + prestimulus pulsatile + poststimulus pulsatile components), where a is the proportion of rapid to total elimination, α1 and α2 are rate constants of rapid and slow elimination, and X(0) is the starting hormone concentration (17). Here, α1 is fixed at 3.5 min and α2 at 20.8 min as reported for endogenous GH (11).
The model is represented fully by the set of parameters defined by (4) Measured GH concentrations, Y, are considered a discrete time sampling of the foregoing continuous processes, as distorted by observational error, εi: Y = X(ti) + ε, i = 1,…,n, k = 1, 2, 3, 4, 5.
We assume that the random effects for basal [Rj(k)] pulse masses [Aj,l(k)] and the observational errors (ε) are independent identically distributed Gaussian random variables, with mean zero and standard deviations, σR(0), σA(0), σA(k), σε(k), k = 1, 2, 3, 4, 5.
Because the preinjection parameters θ(0) describe the preinjection secretion for each subject under each of the five interventions, all of the parameters must be estimated simultaneously using all of the data. Using the above models and assumptions, a Gaussian likelihood can be written (16). Let l denote the log likelihood.
The discretized secretion rate, Z=Z(ti), i=1,…,n, is estimated by the conditional expectation evaluated at the maximum likelihood estimate, θ̂: (5) The reconstruction of the unobserved secretion rates involves statistical estimation of each subject's random effects contributing to GH secretory-burst mass (e.g., subject j, intervention k): Eθ̂(A, i = 1,…, n, r = 1, 2, 3, 4, 5), as well as the random effect for basal Eθ̂(R|Y, i = 1,…, n, r = 1, 2, 3, 4, 5).
Variances and covariances estimates of maximum-likelihood estimation parameter estimates θ̂ are obtained explicitly from the inverse of the estimated information matrix: ∑̂ = −(∂2l/∂θ∂∂′)−1, evaluated at the maximum likelihood estimate, θ̂.
Thereby, statistical confidence intervals are calculated directly for basal secretion γ̂ and waveform parameters, , β̂, and β̂, k = 0, 1, 2, 3, 4, 5. The statistical mode (most commonly represented value) of the time delay to attain the maximal GH secretion rate within a burst is given as (for k = 0, 1, 2, 3, 4, 5) h(β̂, β̂, β̂3(k)) = β̂2(k)[β̂1(k) − (1/β̂3(k))]1/β̂3(k). Variance of this value is computed by the multivariate δ method as )(∂h/∂β evaluated at (β̂ β̂, β̂), where σ̂i,j is the (i,j) element of .
This study was supported in part via the General Clinical Research Center Grant MO1-RR-00585 (to the Mayo Clinic and Foundation from the National Center for Research Resources, Rockville, MD) and National Institutes of Health (Bethesda, MD) Grants R01-AG-019695, AG-29362-01, and R21-DK-072095.
We thank Heidi Doe and Kay Nevinger for excellent support of manuscript preparation; Ashley Bryant for data analysis and graphics; the Mayo Immunochemical Laboratory for assay assistance; and the Mayo research nursing staff for implementing the protocol.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
- Copyright © 2007 the American Physiological Society