INTRODUCTION

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IN AGING PRIMATES, reproductive quiescence emerges in midlife in the female, and reproductive potential declines during the later lifetime in the male (1, 4, 5, 8, 26, 31). Clinical analyses in aging men have disclosed progressive age-related decrements in testosterone's bioavailability (so called andropause) and somatotropic-axis output (so called somatopause) (4, 5, 30, 35). The pathophysiological mechanisms underlying waning activity of the gonadotropic and somatotropic axes are not known but likely contribute to the frailty that accompanies aging (21-24).

Dynamic properties of the aging male reproductive axis show evident disruption before any reproductive loss occurs (36). For example, the coordinate release of luteinizing hormone (LH) and testostrone becomes markedly asynchronous in healthy older men despite normal (young adult) serum concentrations of LH and testostrone (12, 16, 27). The joint secretion patterns of ACTH and cortisol are likewise disturbed in aging humans despite normal daily secretion rates of both hormones (19). These data suggest the postulate that intra-axis feedback-dependent (network level) control of neurohormone secretion is impaired in the earlier stages of aging.

Deterioration of bihormonal secretory synchrony in a neuroendocrine ensemble (above) could denote regulatory defects in interglandular pathway communication (3-5, 18, 20, 32, 34, 36). However, interpreting or predicting a priori the precise site, nature, relevance, and/or magnitude of a postulated linkage defect in a complex interconnected axis (6, 7, 26) is difficult on intuitive grounds alone. Here, we implement an earlier computer-assisted biomathematical model (6, 7, 26) to test selected hypotheses for disrupted control of the GnRH-LH-testesterone feedback axis in the healthy aging male.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS COMPUTER-ASSISTED EXPERIMENTS APPLICATION TO CLINICAL DATA NEW HYPOTHESES SUGGESTED BY... DISCUSSION APPENDIX REFERENCES

Introduction

View larger version (29K): [in this window] [in a new window]   Fig. 1.   Schematized interactive model of the ensemble male gonadotropin-releasing hormone-luteinizing hormone-testicular (GnRH-LH-Te) axis, with feedforward (double arrows) and feedback (single arrows) as simulated here. Dashed lines denote hypotheses of aging-associated pathway disruption explored here (METHODS). sec, Secretion.

As formalized in detail in Refs. 6 and 7, we will denote the three primary hormone concentrations [GnRH (G), LH (L), and Te] in blood as: X(t) = [X_G(t), X_L(t), X_Te(t)] and their respective rates of secretion as: Z(t) = [Z_G(t), Z_L(t), Z_Te(t)]. Assuming simple conservation of mass, we begin with

(1)

The rate of hormone elimination is represented as proportional to its concentration [X(t)]. This has been experimentally observed by injecting purified hormone into the blood and monitoring its exponential disappearance rates in hormone-deficient volunteers (26).

An earlier more comprehensive formalism (6, 7) relates how each rate of secretion: Z_G(t), Z_L(t), Z_Te(t) depends on the history of relevant hormone concentrations within the interconnected feedback system {[X_G(s), X_L(s), X_Te(s)], s  t}.
We assume that GnRH feedforward on LH, LH feedforward on Te, and Te feedback on GnRH or LH are exerted via a time-delayed (time delay = [l₁, l₂]) and time-averaged effect of the relevant hormone concentration (6, 7). Thus, at time t and 0 < t  l₂  t  l₁ < , the feedback signal intensity (i.e., denotes a time average) is
We consider that feedback is exerted on the relevant control nodes (system components) in Fig. 1 via putatively monotonic dose-response logistic functions (denoted by H(·)'s); see Fig. 2. For the simplified male GnRH-LH-Te axis, we allow four major feedback/feedforward dose-response functions: H_1,2(·), which describes the GnRH pulse firing rate as a joint function of Te (feedback) and GnRH (autofeedback) concentration, H₃(·), which gives the rate of GnRH pulse-mass accumulation as a function of Te (feedback) concentration, H₄(·), which defines the rate of Te secretion as a function of LH (feedforward) concentration, and, H_5,6( · , · ) for the rate of accumulation of the LH pulse mass as a function of Te (feedback) and GnRH (feedforward); see Figs. 1 and 2. The subscripts correspond to feedback/feedforward relationships [(1)-(7)] given in below; the time delays for the jth relationship are denoted as (l_j,1, l_j,2).

View larger version (77K): [in this window] [in a new window]   Fig. 2.   Dose-response feedback and feedforward (interface) functions embedded in the male interlinked multinodal GnRH-LH-Te axis. H interface functions for experiments 0-2 (A) and experiments 3-5 (B). Each experimental condition (dashed curves) corresponds to 2 rows displaying the 4 H dose-response functions: e.g., experiment 0, the top 2 rows of A; experiment 5, the bottom 2 rows of B. The normal (experiment 0) H functions are presented for comparison in each experimental panel as continuous curves. The 4 H dose-response functions are Te and GnRH's bivariate inhibition of the pulsatile GnRH firing rate (by respectively heterologous and homologous negative feedback) [H1,2( · , · )], Te's inhibition of rate of accumulation of GnRH pulse mass [H3(·)], Te and GnRH's joint (respectively, negative and positive) control of the rate of accumulation of LH pulse mass [H5,6( · , · )], and LH's stimulation of the rate of Te secretion [H4(·)]. Experiments 0-5 are experiment 0, normal young male; experiment 1, heightened Te feedback on GnRH and LH; experiment 2, attentuated LH feedforward on Te (A); experiment 3, heightened GnRH feedforward on LH; experiment 4, reduced Te feedback, triggering accelerated GnRH pulse frequency, and decreased GnRH pulse mass; and, experiment 5 combined experiments 1 and 2 (B). secr, Secretion.

GnRH-LH-Te Feedback/Feedforward Interactions

Hypothalamus: Te and GnRH Feedback [interactions (1)-(3)] on GnRH. The blood Te concentration (ng/dl) is assumed to exert a negative time-delayed (nominally 80-90 min) feedback (11, 12), and GnRH concentration (pg/ml) a (auto) negative time-delayed (approximately 1-1.5 min) feedback (2), on the GnRH pulse firing rate (given in pulses/day) (see Figs. 1 and 2)

(2)

The Te concentration (ng/dl) is allowed to exert negative and time-delayed (e.g. approximately 80-90 min) (26) feedback on the rate of GnRH hormone synthesis (pg · ml¹ · h¹) (see Figs. 1 and 2)

(3)

Testes: LH feedforward [interaction (4)] on Te secretion. The blood LH concentration (IU/l) is viewed as producing a positive time-delayed (~20-30 min) (12, 26) feedforward action on Te rate of hormone synthesis (ng · dl¹ · h¹) (see Figs. 1 and 2)

(4)

Pituitary: GnRH feedforward and Te feedback [interactions (5)-(7)] on LH secretion. The portal GnRH concentration (pg/ml) is presented as a positive time-delayed (~0.5-1.5 min) feedforward signal driving the rate of pituitary LH hormone (mass) synthesis (IU · l¹ · h¹) (2) (see Figs. 1 and 2). Conversely, the plasma Te concentration (ng/dl) exerts a negative time-delayed (nominally 80-90 min) feedback on rate of LH hormone synthesis (IU · l¹ · h¹) (26).

(5)

where T, T, ... are the LH pulse times, and The function (·) is a sum of exponentials, which thus describes the expected cascade of cellular reactions (2nd and 3rd messengers, etc.) initiated by the GnRH signal acting on LH synthesis and storage for release (7).

(6)

Circadian Rhythm

(7)

Construct of the Male GnRH-LH-Te Axis

Pulse generator. The pulse times of GnRH: T, T, T, ... are the result of a (point) process with an intensity

(8)

where (t)dt is interpretable as the instantaneous probability of a pulse in the infinitesimal time increment (t, t + dt). The conditional density for T given T and (·) will be required to satisfy

(9)

resulting in GnRH pulse times T, T, ... , and LH pulse times (due to the refractory period r_L and time-delay _L):
and the associated counting processes: and This model for the pulse generator, based on assumed loosely coupled GnRH neuronal firing within an ensemble of excitable hypothalamic neurons, is: 1) dynamical, in the sense that the probabilistic structure of the pulse times depends on feedback within the system, and 2) flexible, wherein a Weibull renewal process (corresponding to a constant ) allows for variability of the interpulse interval lengths, independent of the mean, via the parameter   1 (6, 7).

Agonist-driven (nonbasal) rates of synthesis [S(·)] for GnRH, LH, and Te.
(feedback from Te, plus allowable variability)
(feedforward from LH, plus allowable variability)
(feedforward/feedback from G and Te, plus allowable variability)
(accumulation of newly synthesized GnRH and LH in secretory granules, and where  represents the time rate of release of newly synthesized granules.)

Pulse masses M^j for GnRH and LH.
(normalized rates of secretion)
(fractional mass M remaining at time T)
(Available mass equals that not released from previous pulse plus newly synthesized granules.)

The male axis (GnRH, LH, Te) core equations. Combining the above (below, x₊ denotes max{x, 0})

(10)

where the W_i(·)'s are independent standard Brownian motions and _idW_i(t), subscripts i = G, L, Te, describe the in vivo biological variability due to the diffusive behavior (mixing, advection, turbulence, etc.) of the neurohormone within the blood. The values of _i, i = G, L, Te, were chosen so that the resulting coefficients of variation were approximately 6%. Te synthesis and secretion are assumed to be coupled (without an intervening response cascade) by simple diffusion of the synthesized steroid, because no granular storage form exists for Te (unlike LH) (26).

Allowance for variation among individuals. To allow for the variation among individuals, certain parameters may vary randomly about a population value. In particular, we allow the Te basal secretion _Te (Eq. 10) to be uniformly distributed between 1,000 and 1,500 ng · dl¹ · h¹, the circadian rhythm amplitude B₁ (Eq. 7) to be uniformly distributed between 0.2 and 0.5 with B₀ = 1, the pulse regularity parameter  (Eq. 9) to be uniformly distributed between 2 and 6, and the slope of the Te feedback on the instantaneous rate of GnRH pulsing (Eqs. 6, 8, and 9) to be uniformly distributed between 0.005 and 0.009 (6, 7, 26).

(11)

Overview of Approximate Entropy (and Cross-Approximate Entropy)

For a given data series Y_k, k = 1, ... , N, ApEn is calculated as follows: the (sample) standard deviation of Y, SD(Y), is calculated, and two values, m (pattern length) and r (discriminating threshold), are specified. The validated choices for m and r for neurohormone series of the length sampled here are m = 1, r = 0.2SD(Y). One then calculates

(12)

Removing trends before ApEn calculation. The formulations of ApEn and cross-ApEn apply to times series that are at least asymptotically stationary (14, 15, 17). One direct consequence of this assumption is an approximately constant marginal standard deviation SD(Y), which is used as part of the yardstick [r = 0.2SD(Y)] to monitor significant pattern differences. If a series is not stationary in its mean, then one would need to remove an estimated mean function, so that the resulting estimated standard deviation is not inflated. In the case of Te, which can exhibit prominent rhythmicity over 24 h (and to a lesser extent, LH), any major effects of a circadian rhythm in the concentration should thus be minimized before ApEn and cross-ApEn are computed. In Application to Clinical Data, ApEn and cross-ApEn are calculated for clinical data from older men and young men; Fig. 3 (left) displays a Te concentration (solid line) profile of a young male sampled every 10 min over 24 h (starting at 8 AM). The effect of the circadian rhythm on Te concentrations and its (sample) SD are apparent, which in part is a consequence of the nonstationarity in the mean.

View larger version (38K): [in this window] [in a new window]   Fig. 3.   Illustrative impact of detrending paradigms on a 24-h serum Te concentration profile sampled every 10 min in 1 young man. A: top: undetrended (original) Te time series with cosine fit (dashed line) or heat equation fit (dashed-dotted line); middle: cosine detrended; and bottom: heat equation detrended. B: corresponding Te periodograms for the original series (top) and the cosine (middle) and heat equation (bottom) detrended (dashed line). Each is superimposed on the original periodogram (solid line) to show that only low frequencies are removed.

Periodic rhythmicity. If the effect of the circadian rhythm were exerted purely via a cosine function, then its removal by an estimated cosine function would probably suffice. We tested this notion via trigonometric regression at the first discrete Fourier frequency  = (2/n), which will provide the least-squares fitted cosine function, Acos(t) + . The periodogram I⁽ⁿ⁾(·), a normalized version of the squared modulus of the Fourier transform of a series {Y_k} provides the squared amplitude at this first frequency and all the other discrete Fourier frequencies

Figure 3 displays the fitted cosine function superimposed (interrupted line) on the Te concentrations (Fig. 3, top left). The cosine-detrended series is shown for comparison (Fig. 3, middle left). The periodogram of the original Te concentrations is compared on Fig. 3, top right. The sample SD of the cosine-detrended series is 65, which is a reduction from 79 of the original series. There appear to be residual nonstationarities after cosine detrending: see Figure 3, middle right, wherein the periodogram of the cosine-detrended series is superimposed (interrupted line) on the periodogram of the original series. One could attempt to remove these additional low frequencies more systematically, as suggested in the following subsection.

Mean function estimation via smoothing by the heat (or diffusion) equation. If modulation of the 24-h neurohormone profile should occur in a more complicated nonlinear manner, then the effects would probably be smeared over a low-frequency band. One way to remove a complicated nonlinear trend, without necessary knowledge of its form a priori, is by implementing the heat equation. In this strategy, the data serve as the initial conditions with Dirichlet, Neumann, or mixed boundary conditions. We have used Dirichlet conditions and subtracted the resulting smoothed version from the data Y_k, with (the mean) then added back. This procedure removes the low-frequency band and thus achieves a "straightening out," without making a priori assumptions about its form (see more detailed motivation in APPENDIX).

	COMPUTER-ASSISTED EXPERIMENTS

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On the basis of the mathematical construct of the GnRH-LH-Te axis given in Construct of the Male GnRH-LH-Te Axis (above and Refs. 6, 7, 26), we performed selected computer-assisted experiments, as described below. Each consisted of 500 simulations implemented in Matlab.

Normal young male (experiment 0). Published young-adult male 24-h serum LH and Te concentration time series (12, 18) are emulated realistically by the foregoing biomathematical construct (6, 7). The H dose-response functions for the normal young male are displayed in Fig. 2A. Figure 4 displays one realization from each of the six (numbers 0-5) experiments, with the randomness in each realization being initiated with the same seed. These profiles offer a visual impression of the dynamics produced by modifying a given dose-response H function. Figure 5 gives the resulting histograms of the 500 values of (heat equation detrended) LH ApEn, Te ApEn, and the cross-ApEn. And, Fig. 6 presents the histograms of the 500 LH and Te means, and the ratio of the 24-h amplitude/mean of the simulated Te profiles for each of the experiments. We obtained comparable histograms in the case where there was no cosine modulation of LH's feedforward on Te (i.e., B₁ = 0 in Eq. 7) (not shown). Thus comparable model-specific predictions emerge both before and after eliminating all cosine rhythmicity. Simulations of joint LH and Te time series yielded LH and Te ApEn and LH-Te cross-ApEn histograms with median values of 1.2, 1.8, and 2.1, respectively. These values in the young adult male served as baseline model parameters for further comparisons with simulated disturbances of H functions in various aging models (experiments 1-7, below).

View larger version (33K): [in this window] [in a new window]   Fig. 4.   Realizations of LH and Te from each of the 6 (0-5) computer-assisted experiments. Each row is a pair of plots that illustrates a single realization of (simulated) LH and Te concentrations for 2 computer-assisted experiments, as follows: top, experiment 0 (left), experiment 1 (right); second, experiment 2, experiment 3; third, experiment 4, experiment 5, using the same initial seed for generating randomness. Experiment 0: control (baseline); experiment 1: enhanced Te feedback on GnRH and LH; experiment 2: blunted LH feedforward on Te; Experiment 3: elevated GnRH drive of LH; experiment 4: reduced Te feedback on GnRH and LH; experiment 5: combined increased Te feedback and decreased LH feedforward (experiments 1 and 2).

View larger version (53K): [in this window] [in a new window]   Fig. 5.   Histograms of the LH approximate entropy (ApEn), Te ApEn, and (LH-Te) cross-ApEn (after detrending by heat equation). Each row presents histograms of LH or Te ApEn and LH-Te cross-ApEn values derived from 500 simulations in the computer-assisted experiments 0-5 (METHODS and Fig. 4). The scales have been aligned to allow visual comparisons. Analyses apply cosine modulation of LH feedforward on Te. Arrows denote mean model predictions in the control ("young adult") setting. The numerical values in parentheses at the top of each column are the mean values of LH ApEn, Te ApEn, and (LH-Te) cross-ApEn for experiment 0 (normal). These values serve as baseline model parameters. Fdbk, Fdfwd, feedback and feedforward, respectively; CrApEn, cross-ApEn.

View larger version (54K): [in this window] [in a new window]   Fig. 6.   Histograms of model-predicted mean LH, mean Te, and ratios of Te amplitude to mean Te concentrations according to 5 specific hypotheses of aging (METHODS and Fig. 4). Each row depicts histograms of the individual LH means, Te means, and ratios of Te amplitude to Te mean. Data are based on 500 simulations of computer-assisted experiments 0-5. The scales have been aligned to enhance visual comparisons. Simulations embody cosine modulation of LH fdfwd on Te. Data are presented otherwise as described in the Fig. 5.

Heightened Te feedback in aging (experiment 1). As reported experimentally by Winters et al. (34) and Deslypere et al. (3) parenteral delivery of androgen seems to impose greater feedback inhibition of LH release in aging than in young men. To simulate the impact of this inference mathematically, we left-shifted the dose-response relationship of Te feedback on both GnRH and LH (Fig. 2A). The resultant histograms of uni- and bivariate ApEn of LH and Te concentrations and of the fractional (24 h) Te cosine amplitude are given in Fig. 6. Te ApEn rose, whereas Te concentration and Te cosine rhythmicity fell, consistent with clinical observations in older men. However, LH ApEn, LH-Te cross-ApEn, and the mean 24-h LH concentration fell, which would be inappropriate for known aging data (13).

Attenuated LH feedforward on Te (experiment 2). Human experiments using injected human chorionic gonadotropin as a surrogate for an LH stimulus have revealed a significant decline in maximally stimulated Te production in older men (26, 30, 31). Other clinical studies concur with this notion (12). To simulate this putative physiological decrease in testis responsiveness to LH, we reduced the maximum of the LH-Te dose-response curve, thus limiting LH actions on Leydig cells (Fig. 2A). This paradigm recapitulated all six observed changes in aging: higher uni- and bivariate ApEn for LH, Te, and LH-Te; reduced mean Te and elevated mean LH concentrations; and, lower 24-hour Te rhythmicity (Fig. 5 and 6).

Heightened GnRH feedforward on LH (experiment 3). A recent detailed intravenous GnRH dose-responsive study of LH release in older versus young men revealed greater maximal LH secretion after GnRH in older individuals (37). To represent this possible primary change in the aging male axis, the maximum of the GnRH-LH dose-response function was increased (Fig. 2B). This simulation mimicked expected age-related changes in LH-Te only by way of elevated 24-h serum LH concentrations (Fig. 6).

Accelerated GnRH pulse frequency and decreased GnRH pulse mass (experiment 4). Deconvolution analysis of intensively sampled (every 2.5 to 10 min) serum LH concentrations in young and older men has disclosed reduced LH secretory burst amplitude, possibly associated with a rise in LH pulse frequency (11, 26, 31). The latter conjecture of accelerated GnRH pulse frequency as a primary mechanism in aging was evaluated by augmenting the mean frequency (intensity) of the point process defining GnRH pulsatility (reduced mean waiting times) (Fig. 2B). All three ApEn measures increased (univariate LH and Te, and bivariate LH-Te), but serum LH and Te concentrations did not rise and fall, respectively, as recognized in older individuals.

Combined increased Te feedback (experiment 1) and decreased LH feedforward (experiments 2 and 5). The results for experiment 5 (combined experiments 1-2; see Fig. 2B) are shown in Figs. 5 and 6. This model predicted only that ApEn of LH would rise and that concentrations of Te would decline, as recognized in aging men.

	APPLICATION TO CLINICAL DATA

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We next evaluated the impact of detrending of the LH and Te time series using clinical data from healthy older and young men (described in Refs. 11, 12, 18). As shown in Fig. 7, both the cosine and heat equation detrending procedures lowered the SDs of the LH and Te time series, thus reducing r [e.g., 0.2SD(Y)] in Eq. 12 and increasing ApEn, preserving an age contrast. The latter was also prominent in analysis of the untransformed (nondetrended) data (Fig. 7).

View larger version (18K): [in this window] [in a new window]   Fig. 7.   Application of statistical regularity calculations to clinical data in young (n = 11) and older (n = 13) men. ApEn (1,20%) and cross-ApEn (X-ApEn) were used as regularity statistics to monitor subpattern reproducibility of LH and Te output in young and older men, in whom blood was sampled every 10 min for 24 h for later assay of serum LH and Te concentrations. ApEn and X-ApEn for LH and/or Te were calculated without detrending ("untransformed") and after cosine versus heat-equation detrending (METHODS). P values are estimated via Student's t-test (unpaired, 2-tailed, unequal variances). Higher ApEn denotes greater relative pattern irregularity.

	NEW HYPOTHESES SUGGESTED BY COMPUTER-ASSISTED SIMULATIONS

TOP ABSTRACT INTRODUCTION METHODS COMPUTER-ASSISTED EXPERIMENTS APPLICATION TO CLINICAL DATA NEW HYPOTHESES SUGGESTED BY... DISCUSSION APPENDIX REFERENCES

Earlier clinical experiments in young men have disclosed that LH ApEn rises consistently on pharmacological withdrawal of Te negative feedback, e.g., via treatment with the antiandrogenic drug, flutamide (25), or the steroidogenesis inhibitor, ketoconazole (29). Consequently, we also tested the network-predicted response to simulated attenuation of androgen negative feedback on GnRH and LH (experiment 6). Such a postulated age-related reduction in Te negative feedback (right shifted response curves) yielded an anomalous rise in serum Te concentrations, which is not evident in older individuals, but predicted appropriate increases in ApEn of LH and Te individually and jointly, as well as a rise in LH concentrations (Figs. 8, A and B).

View larger version (95K): [in this window] [in a new window]   Fig. 8.   Computer-assisted experiments 6 and 7. A (facing page): the H dose-response functions for experiment 6 (isolated damping of Te feedback on GnRH and LH secretion) are displayed in the top 2 rows, and those of experiment 7 (dual muting of Te feedback and reduced LH feedforward on Te secretion) in the middle 2 rows in the same format as Fig. 2. The normal (experiment 0) H functions are presented for comparison in each experimental panel as continuous curves. The bottom 2 rows portray illustrative predicted single realizations of 24-h blood LH and Te concentration time series in experiment 6 (left) and experiment 7 (right). B: corresponding paired histograms for 500 realizations of experiment 6 and experiment 7 (as depicted in Figs. 5 and 6): top 2 rows are the histograms of the LH ApEn, Te ApEn, and (LH-Te) cross-ApEn; bottom 2 rows are the histograms of model-predicted mean LH, mean Te, and ratios of Te amplitude to mean Te concentrations. Arrows denote mean values in control ("young adult") simulations.

The foregoing analysis was also combined with experiment 2 (reduced LH feedforward on Te) to test the hypothesis of dually impaired Te feedback and LH feedforward [see Fig. 8, A and B (experiment 7)]. This bipartite postulate forecasted increased LH and/or Te ApEn and cross-ApEn values and reduced Te and elevated LH concentrations, as reported in older men (see introduction).

	DISCUSSION

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To test specific hypotheses of interglandular pathway disruption within the aging male GnRH-LH-Te axis, the present experiments implement an earlier stochastic differential equation-based biomathematical model that embodies the dynamic time-delayed and dose-responsive feedback and feedforward linkages known within this interactive system (6, 7, 26). We here apply this more formal computer-assisted construct to explore selected hypotheses of altered regulatory behavior due to aging. Several primary hypotheses for more disorderly LH and Te secretion patterns in healthy older men originated from published clinical studies (1, 3, 8, 11, 12, 16, 18, 20, 25, 26, 31, 32, 34). In addition, new (secondary) hypotheses emerged on the basis of initial predictions of the ensemble model. Available clinical literature points to multiple plausible pathway (interglandular) or nodal (glandular)-level disturbances in the aging male; e.g., hypothalamic GnRH release, reduced GnRH feedforward on LH secretion, impaired LH feedforward on Te production, and altered Te feedback on GnRH and LH output. However, most earlier experimental investigations of necessity have addressed individual nodal alterations in isolation; i.e., by analyzing GnRH's dose-responsive stimulation of pituitary LH release (37), hCG's feedforward drive of gonadal Te secretion (31), and Te's feedback restraint of GnRH/LH production (22, 24). The present networklike analyses show that single-node studies cannot readily distinguish between partial failure of a control node and disruption of internodal signaling (16). Indeed, a crucial prediction of the present computer-assisted experiments is that disturbing the function of any one regulatory locus disrupts the networklike behavior of this axis, due to the strong feedback and feedforward dependencies that interlink GnRH, LH, and Te secretion (6, 7, 26). Because dynamic interfaces in neuroendocrine axes are multiple, nonlinear, time-lagged, and stochastically variable, a priori intuitive predictions of the impact of nodal disruption and/or pathway failure on overall system performance are very difficult. Indeed, the current formalism highlights the unexpected implications of some earlier intuitions (below).

As a sensitive new statistical tool to quantify the network-level consequences of selected disruption of any given regulatory site and/or its interconnections, we applied the univariate and bivariate (joint) regularity measures, ApEn and cross-ApEn (12, 14-18, 27). This family of measures appears to capture among the earliest quantifiable changes in the pattern orderliness of reproductive hormone secretion in the healthy aging male (16) and female (18). As complementary outcome measures, we also used mean and 24-h rhythmic LH and Te output to monitor aging hypothesis-specific predictions (26). Among the five primary (single node) hypotheses explored, we observed that only partial failure of LH's feedforward drive of Te secretion forecasts the principal changes in LH and Te reported in aging men, namely: 1) elevated LH concentrations, 2) more disorderly LH and Te release, considered both individually and jointly, 3) reduced Te concentrations, and 4) blunted 24-hr rhythmicity of Te secretion. Accordingly, from a feedback/feedforward model perspective, muted feedforward drive by LH of Te secretion appears to be necessary and sufficient to predict the multifold GnRH-LH-Te disturbances documented consistently in clinical experiments (26).

In complementary computer-assisted simulations, we imposed accentuated versus reduced Te negative feedback on GnRH-LH release. These putative pathway defects recapitulated some, but not all, the expected age-related changes in LH and Te secretion recognized in the aging male. In particular, model-based enhancement of Te's negative feedback on GnRH and LH secretion, as suggested by reports of heightened suppression of LH production by pharmacological delivery of androgens in older men (3, 33, 34), forecasted a rise in Te ApEn and fall in Te concentrations, but not an increase in LH ApEn and LH-Te cross-ApEn, as required by clinical data (16). Conversely, computer-simulated attenuation of Te's negative feedback on GnRH and LH secretion, as intimated in other clinical studies (26), predicted higher LH concentrations and LH and Te ApEn and LH-Te cross-ApEn, but an anomalous rise in serum Te concentrations. Thus the foregoing computer-assisted hypothesis testing argues against excessive or deficient Te feedback on GnRH/LH as an exclusive mechanism driving the constellation of reported neuroendocrine changes in the human male reproductive axis in aging.

Unlike the foregoing hypothesis of an isolated reduction in Te's negative feedback on GnRH-LH output in the aging male, a bipartite hypothesis, which combined the latter defect with impaired LH feedforward drive of Te secretion, predicted the expected (older male) neuroendocrine phenotype of elevated ApEn of LH and Te, higher cross-ApEn of LH-Te, increased LH and reduced Te secretion, and blunted 24-h rhythmic Te production. Accordingly, these biostatistical experiments allow for, but do not prove, a more complex gonadal-axis pathophysiology in older men, consisting of dual defects that involve both positive and negative control; i.e.: 1) an impairment of LH's feedforward on Te, and 2) a reduction in Te's feedback on GnRH and LH. To affirm or refute this bipartite prediction will require further clinical and laboratory experiments.

Further analyses were carried out to evaluate the network implications of heightened endogenous GnRH feedforward on LH in the aging male. This clinical hypothesis reflects the empirical finding that graded GnRH injections stimulate greater LH secretion in older men than young men (37). However, computer-simulated feedforward enhancement of GnRH on LH, although emulating increased LH output, failed to drive elevated ApEn or cross-ApEn of LH and Te or lower Te concentrations. According to these data, we posit that the reported heightening of GnRH potency and/or efficacy in older men may be a consequence (rather than a cause) of diminished Te feedback. Indeed, reduced Te feedback is inferrable from the well established 30-50% fall in bioavailable (nonsex hormone binding globulin associated) testosterone in older individuals (9, 10, 13, 26, 31, 33).

Accelerated LH pulse frequency and reciprocally reduced LH pulse amplitude have been reported in some aging men (3, 11, 26, 28, 31). We reasoned that both neuroendocrine features might be secondary consequences of restricted Te availability and hence diminished negative feedback on GnRH/LH. In particular, on the basis of the present biomathematical systems model, restraining Te negative feedback (by right shifting the corresponding negative dose response functions of Te on GnRH and LH secretion) projected an increase in GnRH/LH pulse frequency and a reciprocal decrease in LH peak amplitude. The latter inverse relationship is also consistent with reportedly damped gonadotrope responsiveness to higher GnRH pulse frequencies in experimental animals (2). Thus the foregoing model-predicted considerations highlight the thesis that examining any one control node (e.g., here GnRH's feedforward action on LH) in isolation may not illuminate full network operation (see introduction).

The bipartite postulate of age-related defects in LH drive of Te secretion along with augmented Te feedback on GnRH and LH output nullified three of the age-appropriate outcomes otherwise predicted by the single (uncombined) model of decreased LH feedforward on Te. In contrast, combining impaired LH stimulation of Te secretion with reduced Te feedback on GnRH/LH predicted relevant age-related entropic, mean and 24-h rhythmic changes in LH and Te output. These feedback distinctions, if validated independently by further in vivo physiological experiments, illustrate the challenge of intuitively based predictions and the utility of more formalized biomathematical tools to explore combinatorial defects within an axis.

The present study did not attempt to appraise all possible paired (or multiply combined) mechanisms of pathway disruption in the aging male. Thus other plurinodal or multipathway models might also explicate reported disturbances in the older male reproductive axis. Further studies will be needed to explore this interesting issue. In addition, it will be important to investigate further model sensitivity to different renditions of circadian linkages (e.g. imparting diurnal periodicity to GnRH pulse generation and/or GnRH's action on gonadotropes), of the interface functions for LH-Te and Te-GnRH-LH, of time-lagged and time-averaged inputs, and stochastic variability embodied in feedback and/or feedforward interfaces. The current model-based perspective on hypothesis testing will be important to explore further in relation to the more complex menstrual and aging-related dynamics of the female reproductive axis, as well as the integrative control of the thyrotropic, lactotropic, somatotropic, and corticotropic axes.

Perspectives