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The Journal of Clinical Endocrinology & Metabolism Vol. 89, No. 11 5753-5761
Copyright © 2004 by The Endocrine Society

Erosion of Endogenous Testosterone-Driven Negative Feedback on Pulsatile Luteinizing Hormone Secretion in Healthy Aging Men

Johannes D. Veldhuis, Ali Iranmanesh and Daniel M. Keenan

Division of Endocrinology and Metabolism (J.D.V.), Department of Internal Medicine, Mayo Medical and Graduate Schools of Medicine, General Clinical Research Center, Mayo Clinic, Rochester, Minnesota 55905; Endocrine Service (A.I.), Medical Section, Salem Veterans Affairs Medical Center, Salem, Virginia 24153; and Department of Statistics (D.M.K.), University of Virginia, Charlottesville, Virginia 22908

Address all correspondence and requests for reprints to: Johannes D. Veldhuis, Division of Endocrinology and Metabolism, Department of Internal Medicine, Mayo Medical and Graduate Schools of Medicine, Mayo Clinic, 200 First Street Southwest, Rochester, Minnesota 55905. E-mail: veldhuis.johannes{at}mayo.edu.


   Abstract
Top
 Abstract
Introduction
 Subjects and Methods
 Results
 Discussion
 References
 
The present study tests the intuition that successful aging in men is marked by: 1) impaired feedforward by endogenous LH concentrations (con) of testosterone (Te) secretion (sec); and/or 2) attenuated feedback by unmanipulated Te con of LH sec. The goal was to assess both implicit linkages analytically without disrupting normal pathway coupling. This strategy required: 1) assay of paired LH and Te con sampled every 10 min for 24 h in 13 older (O) (ages 60–78 yr) and 13 young (Y) (ages 18–30 yr) men; 2) deconvolution-based estimation of LH and Te sec rates; 3) lag-specific cross-correlation analyses of the relationships between LH and Te con and sec; and 4) statistical contrasts by age stratum. Salient outcomes were: 1) O and Y men maintain comparable LH con drive of Te sec, viz maximal r = +0.51 and r = +0.52, respectively, at an optimal time lag of 50 min (both P < 0.001 against random LH and Te associations); 2) elderly subjects exhibit reduced Te con inhibition of LH sec [minimal r = –0.008 (O) vs. r = –0.10 (Y), P < 0.01 at a time lag of 40 min]; 3) mean (24-h) LH con do not differ by age; and 4) molar Te/sex hormone-binding globulin con are lower in the elderly than in Y individuals (P < 0.01).

In conclusion, noninvasive analyses predict that attenuation of endogenous Te feedback restraint on the hypothalamo-pituitary unit may be an early biological marker of adaptive changes in the GnRH-LH-Te ensemble axis in the healthy O male.


   Introduction
Top
 Abstract
 Introduction
Subjects and Methods
 Results
 Discussion
 References
 
TOTAL, BIOAVAILABLE, AND free testosterone (Te) concentrations (con) decline in many healthy older (O) men (1, 2, 3, 4, 5, 6). However, the primary mechanisms that subserve relative hypoandrogenemia in normal aging are not fully understood. Current concepts include relative failure of hypothalamic drive of GnRH outflow to anterior-pituitary gonadotrope cells, impaired stimulatory actions of endogenous LH pulses on Te biosynthesis and secretion (sec), and augmented or blunted Te-dependent negative feedback on the male hypothalamo-pituitary unit (1, 7). Which of these several mechanisms emerges in the early stages of healthful aging remains unclear. Two (nonexclusive) challenges inherent in assessing mechanistic issues are: 1) ensemble hypothalamo-pituitary-gonadal adaptations in successful aging are initially subtle and evolve slowly over time; and 2) pharmacological interventions used to study feedback mechanisms definitionally disrupt normal signal exchange within the axis (1).

Recent noninvasive approaches to appraise coordinate regulation of an interlinked feedback and feedforward control system include regularity statistics designed to detect subtle adaptations in feedback-sensitive control of the orderliness and synchrony of LH and Te release patterns (8). Such measures predict significantly impaired integrative control of LH and Te sec or Te and LH sec in healthy successfully aging, compared with young (Y), men (9, 10, 11). An emergent query then becomes whether inferred pathway disruption in the aging male is due to a reduction in: 1) pulsatile LH-dependent stimulation (feedforward) of Te sec; and/or 2) circulating Te-enforced inhibition (feedback) of LH sec; or 3) both.

The current study implements a novel analytical platform designed to compare bidirectional in vivo signaling between LH and Te (feedforward) and Te and LH (feedback) noninvasively in successfully aging O and Y men. The outcome measure is the relative strength of the linkage between the physiological input (effector con) and the time-lagged physiological output (glandular sec) (Subjects and Methods). Thereby, we quantitate the impact of age stratum in healthy men on endogenous: 1) pulsatile LH con stimulation of Te sec; and 2) time-varying Te con inhibition of LH sec.


   Subjects and Methods
Top
 Abstract
 Introduction
 Subjects and Methods
Results
 Discussion
 References
 
Clinical protocol

Volunteers comprised healthy Y (n = 13; ages 18–30 yr) and O (n = 13; ages 60–78 yr) community-dwelling men with (mean ± SEM) body mass index values of 23 ± 1.4 and 25 ± 1.7 kg/m2, respectively. Inclusion criteria were: an unremarkable medical history, physical examination, and biochemical screening tests of hepatic, renal, metabolic, and hematologic function; total Te con more than 320 ng/dl (multiply by 0.0347 for nmol/liter); estradiol less than 45 pg/ml (multiply by 3.67 for pmol/liter); LH, 0.8–10 IU/liter; FSH, 4–15 IU/liter (both First International Reference Preparation); prolactin, 2–12 µg/liter; hematocrit more than 38%; and provision of voluntary informed consent approved by the local Institutional Review Board. Exclusion criteria were: hepatic, renal, hematologic, metabolic, or endocrine abnormalities on biochemical screening (above); a history of cryptorchidism, delayed puberty, or male infertility; steroid, alcohol, or drug abuse; exposure to any neuroactive or pychotropic medications within 5 biological half-lives; shift work or transmeridian travel (exceeding three time zones in the prior week); weight loss or gain (≥2 kg change in 6 wk); and failure to provide informed consent.

Subjects were admitted to the General Clinical Research Center for two nights and the intervening day. After overnight adaptation to a forearm iv catheter, subjects underwent repetitive blood sampling (2.0 ml) every 10 min for 24 h beginning at 0800 h. Three meals were provided daily. Participants were allowed to sit in a chair and ambulate to the lavatory but not to exercise, smoke, or drink caffeinated or alcoholic beverages. Lights were extinguished at 2300 h.

Laboratory analysis

LH and Te con (145 samples/subject) were quantitated in duplicate in a single run by immunoradiometric assay and solid-phase RIA, exactly as described (12, 13). Some data series were used earlier as controls in smaller interventional studies. None have been reported in the manner analyzed here. Within-assay coefficients of variation were 6.2 and 5.8%, and between-assay coefficients of variation were 8.3 and 7.9%, for LH and Te, respectively. No samples fell within fewer than 3 SDs of the assay threshold (LH, 0.3 IU/liter; and Te, 24 ng/dl) determined on hypopituitary serum.

Quantitation of hormone sec rates

LH and Te sec rates were estimated via a differential equation-based biexponential model of hormone sec and elimination (elim), as described recently mathematically (10, 14, 15). This methodology first estimates pulse-onset times of LH (and thereby implicitly Te) by an image boundary-detection technique (11). Based upon the resultant set of a priori values, coupled convolution equations are used for statistical estimation of all parameters of LH and Te sec and elim simultaneously in continuous time. Further analyses (below) used the estimated sample (discretized) LH and Te con (reconvolved) and sec (deconvolved) values (14, 15, 16, 17). Earlier experimental validation comprised direct sampling of all three of GnRH, LH, and Te con in the awake unrestrained ram and stallion every 5 min for 12 h and 4 h, respectively, and both LH and Te con in the human spermatic vein every 15 min for 17 h (18).

Cross-correlation procedure

Auto- and cross-correlation analyses were applied to relevantly paired effector-response (input-output) time series, viz: 1) LH con and Te sec; 2) Te con and LH sec; and 3) LH sec and Te sec in both directions. The first two linkages define feedforward and feedback interactions, respectively. The cross-correlation coefficient, r, provides a measure of the relative strength of linear coupling between the two measures at any given time lag (delay in min separating the correlated values) (19, 20, 21). To avoid spurious cross-correlation estimates due to autocorrelation within individual LH and Te time series, the complete auto- and cross-covariance matrix was evaluated at each time lag for all four of LH/Te, LH/LH, Te/LH, and Te/Te relationships (21). To obviate inflation of the type I (false-positive) statistical error rate associated with assessing different time delays between LH and Te, statistical confidence intervals (CIs) for r were adjusted for lag time (Appendix).

Statistical contrasts

Statistical contrasts by age were made by comparing lag-adjusted analytical 95% CIs of the mean difference in cohort r values.


   Results
Top
 Abstract
 Introduction
 Subjects and Methods
 Results
Discussion
 References
 
Statistical comparisons revealed that (24-h) mean (and range) LH con and Te con did not differ between the healthy O and Y men studied here; viz, respectively, LH (IU/liter) 3.4 (range, 1.1–7.5) and 3.3 (range, 1.5–4.6), and total Te (ng/dl) 401 (range, 325–554) and 588 (range, 430–617). SHBG con (nmol/liter) values were significantly elevated in the O, compared with Y, cohort at 87 ± 5.3 (O) and 46 ± 4.7 (Y) (P < 0.01). Thus, the mean molar ratio of Te to SHBG con was reduced significantly in elderly participants (P < 0.01). Serum total IGF-I con values were also lower in O volunteers (P < 0.01), consistent with biological aging.

Figure 1Go presents paired LH con (A) and Te con (B) time series measured in blood sampled every 10 min for 24 h in four Y (left) and four O (right) men. Asterisks placed on the x-axis line mark the beginning of individual LH pulses. Visual inspection suggested lower-amplitude and more irregular LH pulses in the elderly cohort. Te con values in O men hovered in the range of 400 ng/dl (14 nmol/liter).



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  		FIG. 1. Measured (continuous lines) and analytically calculated (interrupted curves) con of LH (A) and total Te (B) in four Y (left) and four O (right) healthy men. Volunteers underwent blood sampling every 10 min for 24 h. Time zero minutes (x-axis) denotes 0800 h. Asterisks placed on the x-axis line of the LH plots identify a priori pulse-onset times determined by image-boundary detection (Subjects and Methods). Note unequal scaling of ordinate to facilitate visualization of pulsatile LH release (A).

 
Figure 2Go depicts deconvolution estimates of LH sec and Te sec over 24 h in two Y (top) and two O (bottom) subjects. Both LH and Te sec unfolded in punctuated bursts, which appeared to be coupled in time. By way of comparison, Fig. 3Go, A and B, illustrate paired LH con and Te sec profiles (feedforward) and, conversely, paired Te con and LH sec time series (feedback). These two directionally distinct relationships were used to test putatively physiological linkages, under the expectation that an effector hormone con acts upon glandular sec. The latter is distinct from conventional cross-correlation analysis, which links two con series.



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  		FIG. 2. Analytically reconstructed 24-h LH (continuous curves) and Te (interrupted lines) secretion (Secr) profiles illustrated in two Y (top) and two O (bottom) men. Data are presented otherwise as described in the Fig. 1Go legend.

 


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  		FIG. 3. A, Deconvolution-estimated 24-h LH con (continuous curves) and Te sec (interrupted lines) time series in two Y (top) and two O (bottom) men. B, Relationship between time-varying Te con (interrupted lines) and LH sec (continuous curves) monitored every 10 min over 24 h in two Y (top) and two O (bottom) men. See Fig. 1Go legend.

 
Cross-correlation analysis was applied to paired con and sec series. The quantitative endpoint is the cross-correlation coefficient (r value) linking LH and Te or Te and LH at any given time lag (minutes separating correlated values). Figure 4Go gives cohort mean (and 95% CI) cross-correlation r values (y-axis) as a function of time lag (x-axis) for the feedforward relationships defined between: 1) LH sec and Te sec (left); and 2) LH con and Te sec (right) in Y (top) and O (middle) men. In this representation, feedforward is defined by positive r values unfolding within a time-lag window of –150 to –10 min (left side of each plot). Age comparisons are made statistically via the mean (and lag time-adjusted 95% CI) difference in r between the O and Y cohort at each LH-Te time lag (bottom). Review of the data presented in Fig. 4Go (left) shows that: 1) LH sec rises 30 min before Te sec increases in both age groups; and 2) LH sec and Te sec are equally coupled in the two cohorts, as defined statistically by mean maximal individual r values of 0.41 (Y) and 0.42 (O) [P < 0.001 vs. chance associations; P = nonsignificant (NS) by age]. Key outcomes inferred from Fig. 4Go (right) are that: 1) LH con increase 50 min before Te sec in both age groups; and 2) increments in LH con and Te sec are highly correlated with r values of +0.52 (Y) and +0.51 (O) (P < 0.001 vs. chance associations; P = NS by age).



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  		FIG. 4. Cross-correlation coefficients (r, y-axis) relating LH con to Te sec (left side) as well as LH con to Te con (right side) in 13 Y (top) and 13 O (middle) men. Positive r values signify feedforward (and negative r, feedback) at indicated time lags (x-axis, 10-min increments) between the paired measures. Time lags of –150 to –10 min inclusive (left side of each panel) denote that changes in LH precede changes in Te (and, conversely for positive time lags). The NS mean difference in r values by age (O minus Y subjects) denotes no detectable feedforward contrast between the two cohorts (bottom). Each numerical value (and the flanking arrows) gives the optimal time lag of significant mean r (and the range of time lags at which mean r is significant). Error bars signify ±2 lag-penalized SDs (wherein one-tailed P < 0.05 and P < 0.01 correspond to 1.64 and 2.33 SDs, respectively).

 
Figure 5Go presents cross-correlation r values for the feedback pairs Te con and LH sec (left column) and Te con and LH con (right column). Conventional cross-correlation analysis of Te con and LH con is shown for comparison with earlier approaches. Cross-correlation analysis of Te con and LH con predicted a feedback time delay of +90 to +120 min in Y men and large 95% CI of r values in both age groups. In contrast, evaluating the negative-feedback relationship between Te con and LH sec revealed that rising Te con precedes falling LH sec by 40 min in Y men with corresponding mean minimal r = –0.10 (P < 0.01 vs. chance associations). In the same correlation model, O men exhibited no detectable negative-feedback relationship (r = –0.008, P = NS vs. random coupling). These outcomes in O and Y men differ significantly (P < 0.025).



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  		FIG. 5. Estimated negative feedback by Te con of LH sec (left side) and by Te con of LH con (right side) in healthy Y (top) and O (middle) men. Negative cross-correlation r values (y-axis) denote Te inhibition of LH output. Positive time lags (10–150 min) indicate that increased Te con precedes reduced LH release by the corresponding interval. The significantly negative algebraic difference (O minus Y) in mean r values (bottom) signifies feedback failure in elderly subjects. Data are presented as outlined otherwise in the Fig. 4Go legend.

 
To verify attenuation of negative feedback in the elderly cohort, we applied cross-correlation analysis to paired LH sec and Te sec time series (Fig. 4Go, left). In this assessment, O men evinced detectable negative feedback (r = –0.14, P < 0.01 vs. random associations), which was significantly less than that estimated in Y volunteers (r = –0.33, P < 0.001 vs. chance associations) (P < 0.025, by age, at a feedback time lag of 20 min).

Figure 6Go highlights the dispersion of individual feedforward/feedback r values and time lags in all 26 subjects for each of the four cross-correlation models; i.e. relationships defined by LH con/Te con (top), LH con/Te sec (upper middle), LH sec/Te con (lower middle), and LH sec/Te sec (bottom). Inspection of these scatterplots establishes a range of interindividual biological variability, which would not otherwise be evident from mean correlation values (above). For example, occasional O men maintain relatively strong LH-Te or Te-LH correlations, and conversely for Y men.



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  		FIG. 6. Interindividual variability in cross-correlation r values (y-axis) associated with LH-Te feedforward (left half of panels with x-axis lag times –150 to –10 min) and Te-LH feedback (right half of panels with lag times +10 to +150 min). Data are the lag time (minutes) and the maximal absolute value of r in each of 26 healthy men. Relationships between paired LH and Te are shown for (top-to-bottom): LH con/Te con; LH con/Te sec; LH sec/Te con; and LH sec/Te sec. Symbols distinguish Y vs. O subjects and the direction of signal coupling (feedforward vs. feedback) as follows: Y and O feedforward, x and o; and Y and O feedback, * and +, respectively.

 

   Discussion
Top
 Abstract
 Introduction
 Subjects and Methods
 Results
 Discussion
References
 
The present noninvasive paradigm unveils marked attenuation of endogenous Te con-dependent feedback inhibition of pulsatile LH sec in a cohort of successfully aging O, compared with healthy Y, men. In contradistinction, the same methodology forecasts statistically comparable strength of unmanipulated LH con-dependent feedforward stimulation of Te sec. According to this formalism, feedback and feedforward time delays are independent of age stratum. Therefore, we postulate that an early mechanism of gonadal-axis disruption in the healthy O male entails reduced negative feedback by prevailing Te availability.

The primary finding is that, in the absence of experimental intervention, elderly, compared with Y, men manifest quantifiably reduced feedback restraint of LH sec by total Te con, while maintaining analytically similar submaximal feedforward drive of Te sec by pulsatile LH con. Both outcomes depend upon mathematical estimation of paired LH and Te con and sec rates via a coupled differential equation-based deconvolution model validated independently in three mammalian species (14, 15, 18, 22). At present, we are unaware of any comparable statistical appraisal of bidirectional LH and Te feedforward and feedback control from which to draw complementary inferences.

By way of qualification, first, quantitatively reduced negative-feedback signaling by endogenous total Te con in O subjects would not exclude significant inhibition of LH sec by higher con of exogenously delivered Te. Analogously, preserved steroidogenic responsiveness to midphysiological (submaximally effective) LH con in elderly volunteers would not exclude diminished testis responses to pharmacological stimulation by hCG. Second, given the relatively small size of the cohorts evaluated here (n = 13 Y, n = 13 O men), further investigations will be needed to establish the populational generality of our inferences. And, third, aging of the male gonadal axis putatively gives rise to a graded spectrum of mechanistic adaptations, which are modified by comorbidity (7). For example, the present focus on successfully aging men detects impairment of inhibitory Te-LH signaling without any change in the mean (24-h) LH con, but a significant reduction in the molar Te/SHBG con ratio. In other contexts of aging, LH con may be increased (in healthy individuals or subjects with primary gonadal failure) or decreased (in a subset of apparently healthy men or O patients with comorbidity affecting hypothalamo-pituitary function) (1, 7, 23, 24). The precise relationship among such mechanistic categories has not been elucidated.

In earlier clinical studies, pharmacological supplementation with Te or 5 {alpha}-dihydro Te by iv, transdermal, or im routes variously suppressed LH con equivalently, more or less in O than Y men (7, 25, 26, 27, 28, 29). The basis for such disparate reports is not clear. From a clinical vantage, pertinent factors might include: differences in subject selection and cohort size; type, dose, route, and duration of exogenous Te or 5 {alpha}-dihydro Te administration; duration and frequency of blood sampling; specificity of the LH assay; and/or timing and method of analyzing LH release. From an experimental perspective, earlier approaches of monitoring the fall in single-sample or mean LH con induced by exogenous Te or its reduced derivative differ from appraising endogenous time-varying Te-dependent inhibition of pulsatile LH sec based on repetitive sampling over 24 h. Implementation of a noninvasive analytical formalism in the current investigation was intended to more faithfully reflect individual biological differences in: 1) pulsatile LH and Te con and sec; and 2) undisturbed bidirectional signaling between the hypothalamo-pituitary unit and the testis. Both objectives are relevant, inasmuch as, in Y men, both Te con and the time pattern Te delivery into the bloodstream determine feedback suppression of LH sec (7, 14, 15, 16). For example, short-term pharmacological depletion of Te con followed by continuous iv or sc addback of a midphysiological amount of Te reduces 24-h mean LH con and sec more than pulsatile iv infusion of the same total dose of Te (30, 31, 32). Assuming that both the amount and time course of Te availability determine negative-feedback signaling, then attenuated total Te con-dependent inhibition of LH sec in elderly men may reflect: 1) reduction in free and/or bioavailable (non-SHBG-bound) Te con; and/or 2) alteration in the con pattern of secreted Te, as inferred by statistical regularity analyses (Introduction). In the first regard, the SHBG con was higher and the molar ratio of Te/SHBG con significantly lower in O volunteers. Lower Te bioavailability would not require or exclude concomitantly impaired target-tissue sensitivity to any given Te con (33, 34). In principle, therefore, inferred erosion of negative feedback by total Te con in O individuals could result from diminished Te bioavailability, disruption of the Y adult-like regularity of Te con patterns in the blood, and/or diminished hypothalamo-pituitary responses to Te. Controlled Te infusion paradigms, combined with novel techniques to estimate pulsatile sec of free and albumin-bound Te noninvasively, will be required to distinguish among these mechanistic considerations (18, 35).

Maximal cross-correlation r values of LH con-dependent drive of Te sec were statistically indistinguishable in the 13 Y and 13 O individuals evaluated here. Feedforward time delays were also comparable by age. Analysis of paired LH sec and Te sec time series corroborated age-invariance of the strength and timing of unmanipulated LH-Te feedforward coupling. Collectively, these data indicate that submaximal stimulation of total Te sec by endogenous LH con may be preserved when very healthy O male volunteers are studied.

In pharmacological paradigms, hCG injection increases total Te con less in O than Y men (36, 37, 38, 39). HCG administration provides a complementary test of Leydig-cell steroidogenesis by assessing maximal rather than submaximal Te sec responses. In particular, hCG differs fundamentally from LH in several respects: 1) hCG binds to the LH/hCG receptor nearly irreversibly; 2) at the doses typically employed, hCG has at least 100-fold greater lutropic effect than an endogenous LH pulse; and 3) hCG injection significantly down-regulates Te sec by the testis (1, 7, 23, 33, 34, 35, 36, 37, 38, 39). Supraphysiological LH drive, induced by doubling LH con during a 14-d iv infusion of GnRH pulses every 90 min, also fails to increase Te con maximally in O compared with Y men (13). And, LH/hCG efficacy is reduced in the aged male rodent as assessed by ex vivo testis perfusion and in vitro Leydig-cell culture (40, 41). The present analysis is unique by way of examining Te sec stimulated by endogenous, physiological, nonmaximal, pulsatile LH con. In this circumstance, we hypothesize that LH potency (half-maximally stimulatory LH con) and/or testis sensitivity (slope of implicit LH con-Te sec response function) may be preserved in O men. In contradistinction, our data do not directly address endogenous LH efficacy (maximal stimulatory response). Recent implementation of a new nonlinear technical platform for in vivo dose-response estimation, in fact, predicts that endogenous LH efficacy (extrapolated asymptotically) declines significantly in successfully aging men (35). Thus, viewed statistically, the present linear cross-correlation strategy provides strongly complementary mechanistic insights into submaximal LH-Te feedforward activity and unveils, for the first time, significant loss of endogenous Te feedback on LH sec in healthy aging men.

In summary, the current study combines intensive and extended blood sampling, deconvolution-based reconstruction of LH and Te con and sec, and time-lagged (feedback and feedforward) cross-correlation analysis of effector-response coupling strength in 13 O and 13 Y healthy men. Thereby, we estimate feedforward and feedback actions of LH and Te without experimental manipulation of the gonadal axis. This noninvasive analytical approach discloses: 1) undetectable Te con-dependent feedback inhibition of LH sec, and significantly reduced negative coupling between LH sec and Te sec in O men; 2) comparable LH con-dependent feedforward stimulation of Te sec, and preserved positive coupling between LH sec and Te sec in elderly individuals; and 3) similar 24-h mean LH but a lower ratio of Te/SHBG con in O volunteers. These collective outcomes allow the hypothesis that aging-related erosion of negative feedback by Te con of hypothalamo-pituitary drive of LH sec reflects biologically reduced availability, altered sec patterns, and/or impaired action of Te on the hypothalamo-pituitary unit.

Technical appendix

Secretion and kinetic estimation. The sample set comprises 145 paired LH and Te con measurements in each of 13 Y and 13 O men (Subjects and Methods). As a first stage in the estimation of sec and elim parameters, underlying LH pulse-onset times, TL1, TL2,... , TLm, are predicted by a previously described method (22). Then, conditional on these pulse times, all parameters defining LH and Te sec, ZL and ZTe, and biexponential elim are estimated simultaneously by a maximum-likelihood procedure (see Refs.17, 18, 19). The elim function has a fast, {alpha}(1), and slow, {alpha}(2), phase of elim and a fixed (populationally defined) proportion of rapid to total elim. In this construction, the rapid rate constant reflects primarily advection and diffusion of secreted molecules in the bloodstream, and the slow rate constant incorporates irreversible metabolic clearance (18, 19). The model allows valid estimation of an unknown admixture of basal (ßL, ßTe) and pulsatile (burst-like) LH and Te sec. Basal is defined as time-invariant release. Pulsatile sec arises via a flexible sec-burst shape (waveform), which is represented statistically by a three-parameter generalized Gamma probability density. The latter function relates the instantaneous sec rate within a burst to time:

{psi} is normalized to integrate to unity, which renders the waveform estimate analytically independent of sec-burst mass. Pulsatile LH sec is the product of the normalized waveform and the three-part sum of ongoing accumulation of releasable hormone ({eta}0,L), a weak correlation ({eta}1,L) effect of the length of the prior interpulse interval (TLj – TLj–1) and sec-burst mass (ALj). Total LH sec is the sum of basal and pulsatile sec:

The secretion rate of Te (ZTe) is described as a logistic function, driven by the LH feedforward signal (18). Convolution of the sec and elim functions gives rise to the time-varying hormone con(X), as formulated earlier (14, 15, 16, 17, 18, 35). Observed LH and Te con time series are then a discrete time sampling of the foregoing underlying continuous processes plus observational error:

Figures 1–3GoGoGo illustrate statistical estimates of LH and Te sec rates, ^L,i, ^Te,i, and analytically reconstructed LH and Te con profiles by this methodology.

Cross-correlation analyses. To account for short-term trends or epochs in hormone release, a 2-h time-moving average is removed from con profiles before cross-correlation analysis. The resulting stationarized LH and Te con series are designated by L,i and Te,i. By construction, multivariate (stationary) time-series methods can be applied to the discrete time-dependent process defined by the interlinked set of con and sec estimates. To this end, let {rho}Lc,Tes (·) denote the feedforward cross-correlation function linking LH con (c) and Te sec (s), and let {rho}Tes,Ls (·) define the corresponding negative-feedback relationship. The estimate of {rho}Lc,Tes (h) for lag h (≥0) is given by:


{jcem-5753}

Based upon estimation of both autocorrelation functions (LH/LH and Te/Te) and both cross-correlation functions (LH/Te, Te/LH), the multivariate Bartlett’s formula (40) is applied to calculate asymptotic variances for each time lag h and k:

The square root of each variance estimate forms the basis for computing the corresponding SD of lag-specific r values shown in Figs. 4Go and 5Go.


   Acknowledgments
 
We thank Kandace Bradford, Kimberly Coulter, and Kris Nunez for excellent assistance in text presentation and graphical illustrations.


   Footnotes
 
This work was supported by Grants K01 AG19164 and AG23133 from the National Institutes of Health (Bethesda, MD); DMS-0107680 Interdisciplinary Grant in the Mathematical Sciences from the National Sciences Foundation (Washington, D.C.); and M01 RR00585 to the General Clinical Research Center of the Mayo Clinic and Foundation from the National Center for Research Resources (Rockville, MD).

Abbreviations: CI, Confidence interval; con, concentration(s); elim, elimination; NS, nonsignificant; O, older; sec, secretion; Te, testosterone; Y, young.

Received February 27, 2004.

Accepted July 30, 2004.


   References
Top
 Abstract
 Introduction
 Subjects and Methods
 Results
 Discussion
 References
 

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