## Abstract

The mechanisms by which aging progressively depletes testosterone (Te) availability in the male are unknown. Accordingly, the objective was to estimate brain gonadotropin-releasing hormone (GnRH) outflow (release and action), which cannot be observed directly, on the basis of downstream effects on pituitary luteinizing hormone (LH) secretion. LH, in turn, feeds forward on (stimulates) gonadal Te secretion, which then feeds back on (inhibits) GnRH-driven LH secretion. LH and Te concentrations were measured repetitively (every 10 min) over 18 h during graded pharmacological blockade of endogenous GnRH outflow in 24 healthy 20- to 72-yr-old men. Data were analyzed using a new age-dependent regression model of GnRH-LH-Te interactions to estimate pulsatile LH secretion and elimination, GnRH outflow, LH feedforward, and Te feedback. By incorporating regression on age within the dose-response model, we show that aging erodes all three primary forward and reverse pathways linking the brain, pituitary gland, and testes. Aging is associated with concomitant deficits in GnRH → LH feedforward, LH → Te feedforward, and Te → GnRH/LH feedback. The analytical formalism should be generalizable to other ensemble regulatory systems, such as those that control growth, reproduction, stress adaptations, and glucose metabolism.

- gonadotropin
- men
- human
- testosterone
- testis
- model

aging, undernutrition, protracted illness, and environmental toxins reduce production of major anabolic hormones such as testosterone (Te) in men and animals (3, 7, 22). Chronically diminished androgen availability, in turn, predicts increased all-cause mortality, cardiovascular risk, visceral adiposity, long bone fractures, muscle wasting, cognitive impairment, and sexual dysfunction (20, 35). Nonetheless, dissection of the root cause(s) of Te deprivation has been thwarted by the strongly interlinked nature of brain-pituitary-gonadal regulation (27). In principle, such dynamics are controlled via repeated incremental adjustments in feedforward (stimulation) by hypothalamic gonadotropin-releasing hormone (GnRH) and pituitary luteinizing hormone (LH) and feedback (inhibition or repression) by gonadal Te (11, 13, 27) (Fig. 1). In particular, neuronal bursts of GnRH evoke pulses of LH from pituitary gonadotropes (30). LH pulses stimulate time-delayed Te secretion by Leydig cells, as verified by direct spermatic vein sampling in humans (11). Te undergoes diffusion, advection, elimination, and binding to two plasma proteins, sex hormone-binding globulin (SHBG) and albumin (11, 17). Te feeds back to inhibit GnRH outflow (secretion and action) (28, 31).

Quantification of ensemble GnRH-LH-Te dynamics remains difficult for several reasons. First, there are strong nonlinear dependencies among *1*) hypothalamic GnRH → LH feedforward, *2*) LH → Te feedforward, and *3*) Te → GnRH/LH feedback. Second, primary pathophysiology putatively induces secondary adaptations at other sites, thereby obscuring the nature of the original defect (11, 13, 17, 27, 30). Third, hypothalamically secreted GnRH cannot be measured accurately in the general circulation. Only LH and Te concentrations can be observed directly. Hence, a strategy is needed to reconstruct GnRH's stimulation of LH secretion and Te's inhibition of GnRH secretion and action. The present analyses introduce a new age-dependent regression structure embedded within the dynamic model of conjoint GnRH-LH-Te regulation.

## METHODS

We employ a dual strategem: *1*) formulation of a mathematical model of the entire system (observed and unobserved signals), which incorporates age as a dependent variable, and *2*) experimentation, in which graded amounts of a selective GnRH receptor antagonist are used to inhibit endogenous GnRH action competitively.

#### Clinical experiments and overview of modeling.

Twenty-four healthy adults, ages 20–72 yr, provided voluntary written informed consent to participate in this study, which was approved by the Mayo Clinic Institutional Review Board. Ten young and 8 older subjects were evaluated earlier using a nonpopulational model (14), and 23 subjects were assessed using an approximate-entropy construct (19). Subjects were healthy community-dwelling men within 20% of ideal body weight with unremarkable medical inventories and physical examinations. Each subject underwent four separate randomly ordered, overnight sampling studies ≥10 days apart. Blood was withdrawn from a forearm intravenous catheter every 10 min for 18 h beginning at 1800, and LH and Te concentrations were assayed. At 2 h after the start of sampling, ganirelix acetate (GRX) was injected subcutaneously at 0 (saline), 0.1, 0.3, or 1.0 mg/m^{2}. GRX is a potent, selective long-acting competitive antagonist of the GnRH receptor (2). GRX, SHBG, and albumin concentrations were measured in a 2-h serum pool (16–18 h). Serum concentrations of LH, Te, GRX, SHBG, and albumin were determined as described previously (23, 29). Assay sensitivities were 0.010 IU/l for LH (First International Reference Preparation) and 8 ng/dl for total Te. No samples were undetectable. The general clinical paradigm of graded GnRH receptor antagonism was presented earlier and used for different purposes in a subset of the larger cohort studied here (14, 19, 21). One study involved in vivo validation (14), another showed model free estimation of the orderliness of LH secretion (19), and yet another assessed LH-Te feedforward via a linear estimation method (21).

Modeling consisted of the following sequential strategy: *1*) estimate multiple sets of potential LH pulse times by a validated diffusion-based incremental-smoothing methodology (10, 12), *2*) estimate the biexponential kinetics of LH elimination and the LH secretion function simultaneously by a conditional maximum-likelihood (pulse number-penalized) methodology using each 18-h LH concentration-time series (5, 13), *3*) estimate the free Te secretion rate via a logistic LH concentration → Te secretion feedforward model (17), and *4*) use the estimated LH secretion rates (from *step 2*) and the free Te concentration (from *step 3*) to reconstruct the virtual GnRH signal and GnRH → LH dose-response function, which could have produced the estimated LH secretion rates (5, 14). The new feature is the inclusion within the overall model of age-dependent regression of each key regulatory parameter of interest. The mathematical concepts are summarized in the appendix.

## RESULTS

On the control day, concentrations of total Te (408 ± 61 ng/dl, *n* = 24 subjects) and LH (4.1 ± 0.38 IU/l) were age invariant*.* Linear extrapolation to the age extrema of 20 and 80 yr yielded respective concentrations of SHBG of 19 and 42 nmol/l, bioavailable Te of 287 and 206 ng/dl, and free Te of 11.2 and 6.3 ng/dl and fraction of Te bound to SHBG of 35% and 56% (*P* ≤ 0.05 for each slope measure). By analysis of variance, GRX dose (*P* < 0.001), but not age (*P* = 0.44), determined LH concentrations (*P* < 0.001).

Figure 2 depicts the impact of age (independent variable) on the inhibitory relationship between measured GRX concentrations (independent variable) and calculated pulsatile LH secretion rates (dependent variable). The three-dimensional contour was determined by combined analyses of 10,368 individual-sample LH measurements and 96 individual-sample GRX measurements in the group of 24 men. Inhibitory sensitivity to GRX increased linearly and doubled over the regression-projected age span of 20–80 yr (*P* < 0.01). Inhibitory potency (1 half-maximally inhibitory GRX concentration) did not differ over the same age extrema (1.1 and 0.85 ng/ml, respectively).

Figure 3 illustrates 18-h LH and Te concentration and secretion time series in one 24-yr-old subject. The GnRH receptor antagonist reduced pulsatile LH concentrations, calculated Te secretion rates, and measured total Te concentrations. Analytically estimated multiple LH → Te feedforward dose-response functions at any given GRX dose illustrate inferred pulse-by-pulse variability in LH efficacy. The latter, as quantified by the coefficient of variation, decreased linearly from 40% to 22% between projected (model-extrapolated) ages of 20 and 80 yr (*P* < 0.01).

Four-parameter logistic regression analysis of LH → Te feedforward disclosed that age attenuates LH efficacy (*P* = 0.047; Table 1, Fig. 4). Over the model-extrapolated age range of 20–80 yr, LH efficacy declined by 49% (from 268 to 136 ng/dl per 8-h pulsatile Te secretion). Estimated testis sensitivity and LH EC_{50} (half-maximally effective LH concentration) did not differ significantly with age.

Figure 5 illustrates estimates of three-dimensional GnRH/LH/Te feedforward-feedback relationships in six subjects. Each surface plot represents model-predicted pulsatile LH secretion rates as a function of “reconstructed” virtual GnRH outflow (arbitrary units) and free Te concentrations (feedback signal). Age was a significant negative determinant of maximal virtual GnRH outflow (efficacy; *P* = 0.017; Table 1). Regression on age of all 24 individual exponential Te-feedback coefficients [(log IU·l^{−1}·min^{−1})·(ng·dl^{−1})^{−1}, defining the magnitude of free Te's inhibition of GnRH outflow], yielded a slope of −0.0006 ± 0.0002 (*P* = 0.012), predicting a 51% decrease across the model-extrapolated age extrema of 20 and 80 yr (Table 1). These outcomes indicate that age blunts maximal virtual GnRH drive and free Te's negative feedback on GnRH outflow-dependent LH secretion. Table 2 summarizes all parameters estimated for the entire cohort of 24 men, along with parameter SE.

Age affected placebo-associated, but not GRX-modified, GnRH pulse frequency, which was taken as the LH pulse frequency. For example, in relation to age, administration of GRX (1.0 mg/m^{2}) increased GnRH/LH pulse number (per 24 h) to 29 ± 0.89 from the placebo value of 20.4 ± 1.4 in older men and to 28 ± 1.4 from the placebo value of 16.7 ± 0.67 in young men (*P* < 0.01 for age contrast after placebo, but *P* > 0.10 for age contrast after GRX). Age did not affect estimates of *1*) the waveform mode of the virtual GnRH signal, *2*) the rapid and slow half-lives of elimination of LH (means of 13 and 97 min, respectively), and *3*) half-lives of free Te (0.86 and 4.9 min) or slow half-life of total Te (57 min) in this group of 24 men.

## DISCUSSION

Healthy aging in male mammals, including humans, results in gradual Te depletion and impoverished anabolic drive to bone, muscle, brain, and sexual organs (1, 20). The fundamental basis of Te deprivation in this setting remains unknown. A major obstacle to establishing pathophysiology is that GnRH and free Te concentrations are not observed and can be estimated only via a mathematical model of the entire axis (5, 13, 14). Here, using data from all subjects considered together, we extend the ensemble construct to include regression of key dose-response parameters on age. This new analytical framework predicted that age significantly attenuates *1*) virtual GnRH's outflow (secretion to and action on the pituitary gland), *2*) endogenous LH's maximal stimulation of gonadal Te production (feedforward efficacy), and *3*) Te's negative feedback on GnRH outflow-driven LH secretion. In contradistinction, age did not affect LH potency, testis sensitivity, basal LH secretion, the estimated waveform of virtual GnRH secretory bursts, Te's feedback on GnRH/LH pulse frequency, or the elimination kinetics of LH and Te. Accordingly, a population-based age-dependent regression model of interlinked control among GnRH, LH, and Te unveils multiple regulatory deficits in the aging male gonadal axis.

A salient inference was that virtual GnRH outflow (secretion and action) declines by ∼50% over the model-extrapolated age range of 20–80 yr. The decrease, estimated analytically, is consistent with the age-associated increase in LH's inhibitory susceptibility to graded incremental exposure to a specific competitive GnRH receptor antagonist (Fig. 2). The concept stated intuitively is that greater inhibition of LH pulse size by a fixed amount of a competitive GnRH antagonist denotes less endogenous GnRH drive, with the assumption that pituitary responses to GnRH are preserved with age, as strongly implied by existing data (2, 15, 29, 32, 36). The analytical outcome in the human also is consistent with evidence of altered GnRH synaptology (24), impaired release of GnRH from cultured hypothalamic fragments (8), preserved GnRH action in vitro and in vivo (9, 31), and decreased spontaneous and castration-induced pulsatile LH secretion (26) in the aged male rat. Proof of our inference would require direct measurement of GnRH secretion in (human) hypothalamopituitary portal blood.

Pulsatile LH secretion was modeled as the triple consequence of *1*) competitive antagonism of GnRH action by a GnRH receptor inhibitor, *2*) nonlinear GnRH-LH feedforward (stimulatory) dose-response properties, and *3*) exponential noncompetitive feedback (inhibition) by free (protein-unbound) Te concentrations. The free Te concentration represents a plausible physiological signal, given its strong correlation with clinical features of anabolism and empirical evidence that free steroids can inhibit hypothalamopituitary drive (33). A key outcome of this analysis was that age significantly attenuates free Te's inhibition of GnRH outflow. In a subanalysis, age equally muted feedback by bioavailable and total Te concentrations (14). These outcomes were selective, because age did not significantly influence the feedback effect of Te concentrations on LH secretory-burst frequency. Although hypothalamopituitary androgen and estrogen receptor density may decline in aging animals (25), the biochemical basis of specific impairment of Te's negative feedback on GnRH outflow, but not GnRH/LH pulse frequency, is not known.

Trophic hormones such as LH direct target tissue responses via implicitly asymptotic dose-response functions. Analytical reconstruction of the nonlinear function transducing pulsatile LH's stimulation of Te secretion indicated that age diminishes (by 49%) LH efficacy (Fig. 4). Reduced testis responsivity to extrapolated maximal LH drive would be consistent with previous pharmacological data obtained using high doses of human chorionic gonadotropin (33). Taken together, noninvasive model-based dose-response estimation thus identifies impaired endogenous GnRH drive, decreased Te feedback on GnRH outflow (possibly compensatory to low GnRH output), and reduced LH efficacy in stimulating Te secretion.

### Perspectives and Significance

A generalizable age-dependent regression model of the male gonadal axis provides an analytical formalism to quantify joint feedback and feedforward adaptations associated with aging in the human. Extensions of this basic construct to other interlinked physiological systems should have utility in probing age-dependent mechanisms of multipathway failure more generally. Moreover, proteomics research has unveiled a large repertoire of putative and confirmed physiological regulators. Given that direct measurements of multiple nonsystemic (locally active) mediators can be difficult in the intact animal and human, the concept of analytically estimating unobserved signals within a regulatory ensemble may have utility in investigating other networks. One approach to calibrating reconstructed virtual signals may be injection of a measurable homologous signal.

## GRANTS

This work was supported in part by National Center for Research Resources Grant M01 RR-00585 to the Mayo Clinic, National Institutes of Health Grants RO1 AG-023133, R01 DK-060717, AG-031763, and K01 AG-019164, and National Science Foundation Award DMS-0107680.

## ACKNOWLEDGMENTS

We thank Pamela Roebuck and Paul Takahashi for assistance with protocol implementation and Donna Scott and Ashley Bryant for manuscript and graphics support.

## Footnotes

Glossary

*a*_{L}and (1 −*a*_{L})- Fast and slow fractions of biexponential LH elimination
*a*_{Te}and (1 −*a*_{Te})- Fast and slow fractions of free Te elimination
*A*_{G}^{k,j}- Random-effect component of
*j*th GnRH pulse mass for*series k* *A*_{L}^{k,j}- Random-effect component of the
*j*th LH pulse mass for*series k* - ED
_{50} - ED
_{50}coefficient for LH secretion as a dose-response function of GnRH and free Te *f*_{1}- Coefficient for free Te inhibition of LH efficacy on GnRH
- F
_{L}^{(k)}(*t*) - LH feedforward signal (on Te secretion rate)
- G
^{(k)}(*t*) - Unobserved GnRH signal, acting on the pituitary, at
*time t* - Ĝ
_{i}^{(k)}(*i*= 1,…,*n*) - Reconstruction of unobserved GnRH signal, based on estimated LH secretion rate and estimated free Te concentrations
- [GRX
^{(k)}] and*K*_{d}^{GRX} - GRX concentration (
*k*= 0, 1, 2, 3) and GRX dissociation constant - IPI
- Interpulse interval (min)
*k*- Index for
*k*= 0, 1, 2, 3 (i.e., 4 levels of GRX for each subject) *K*_{a}^{S}- Association constant for Te binding to SHBG (2 binding sites)
- MLE
- Maximum-likelihood expectation
*n*^{A}*K*_{a}^{A}- (Apparent) association constant for Te binding to albumin (multiple binding sites)
*t*- Time (min)
*T*^{k,1},*T*^{k,2},…,*T*^{k,m}- LH pulse times (same as GnRH pulse times) for
*k*= 0, 1, 2, 3 - [totalTe], [TeS], [TeA], and [FrTe]
- Concentrations at
*time t*(left implicit) of total Te, Te bound to SHBG (S), Te bound to albumin (A), and free (Fr) Te *X*_{L}^{(k)}(*t*)- LH concentration at
*time t*for*series k* *X*_{Te}^{(k)}(*t*) = [*X*_{FrTe}^{(k)}(*t*),*X*_{TeS}^{(k)}(*t*),*X*_{TeA}^{(k)}(*t*)]- 3 components of Te concentration for
*series k* *X*_{Te}^{(k)}(*t*)- True (i.e., without measurement error) Te concentration at
*time t*for*series k* *X̂*_{FrTe,i}^{(k)}- Estimated free Te concentration at
*time t*_{i} *Y*_{Te,i}^{(k)}=*X*_{Te}^{(k)}(*t*_{i}) + ε_{Te,i}^{(k)}(1 = 1,…,*n*)- Observed total Te concentrations
*Z*_{L}^{(k)}(*t*)- LH secretion rate (mass·unit volume
^{−1}·unit time^{−1}) *Z*_{Te}^{(k)}(*t*)- Te secretion rate (mass·unit volume
^{−1}·unit time^{−1}) *Ẑ*_{L,i}^{(k)}(*i*= 1,…,*n*)- Estimated LH secretion rate, reconstructed as the conditional expectation of the unobserved LH secretion rate (at the MLE), given the observed LH concentrations
*Ẑ*_{Te,i}^{(k)}(*i*= 1,…,*n*)- Estimated Te secretion rate, reconstructed as the conditional expectation of the unobserved Te secretion rate (at the MLE), given the observed Te concentrations
- α
_{L}^{1}and α_{L}^{2} - Fast and slow LH elimination rates (min
^{−1}) - α
_{Te}^{(1)}and α_{Te}^{(2)} - Fast and slow free Te elimination rates (min
^{−1}) - β
_{L}^{(k)} - Basal LH secretion rate (mass·unit volume
^{−1}·unit time^{−1}) - β
_{Te} - Basal Te secretion rate (mass·unit volume
^{−1}·unit time^{−1}) - ε
_{Te,i}^{(k)}(1 = 1,…,*n*) - Measurement error at
*time t*_{i}for total Te concentration - η
_{0,L}^{(k)}and η_{1,L}^{(k)} - Parameters for mean pulse mass, i.e., coefficients for a linear function of the preceding IPI [e.g., the
*j*th IPI (*T*^{k,j}−*T*^{k,j−1})] - η
_{0,Te}, η_{1,Te}, and η_{2,Te} - Potency, sensitivity, and efficacy parameters of the logistic dose-response function for Te secretion, driven by the LH feedforward signal
- η
_{0,G}and η_{1,G} - Parameters for mean GnRH pulse mass, i.e., coefficients for a linear function of the preceding IPI [
*e.*g., the*j*th IPI (*T*^{k,j}−*T*^{k,j−1})] - κ
_{1}^{S}, κ_{−1}^{S}, κ_{1}^{A}, and κ_{−1}^{A} - On- and off-binding rate coefficients for free Te to SHBG (S) and albumin (A)
- κ˜
_{1}^{S}and κ˜_{1}^{A} - Forward binding rates for free Te to SHBG (S) and albumin (A)
- ψ
_{L}(*t*−*T*^{k,j}) - 3-Parameter γ (probability density) function representing normalized pulse mass secretion at
*time t*for pulse beginning at*time T*_{L}^{k,j} - ψ
_{G}(*t*−*T*^{k,j}) - 3-Parameter γ (probability density) function representing normalized GnRH pulse mass secretion at
*time t*for pulse beginning at*time T*^{k,j} - λ̂
^{(k)} - Estimated frequency for IPI length under the Weibull model of IPI

- Copyright © 2009 American Physiological Society

## APPENDIX

### Biomathematical Model of GnRH, LH, and Te Dynamics

The age-dependent regression model adds a coefficient for an age effect on any one physiological parameter in the model and then uses data from all subjects combined to solve the regression function, along with the physiological parameter set. The parameters of the overall model are summarized below.

##### Model of LH pulse times, secretion, and concentrations.

For each of the four GRX regimens (*k* = 0, 1, 2, 3), LH and Te concentrations were sampled every 10 min for 18 h, and GRX was administered after 120 min. GRX, SHBG (S), and albumin (A) concentrations were also measured in a 2-h serum pool (16–18 h). This protocol was described by Keenan et al. (14) in a subset of the larger group studied here.

One must first estimate the LH pulse times on the basis of observed LH concentration data. A recently published pulse detection method was employed (12). It utilizes a nonlinear diffusion equation, with a diffusion coefficient inversely related to the rate of increase, identifying putative pulse times as points of increase, which are not easily smoothed away (Fig. A1). The pulse times were denoted *T*^{k,1}, *T*^{k,2},…, *T*^{k,m}, where the number of pulses (*m*) depends on *k*. LH secretion rates
*A*) that allows for biological variability in individual burst mass. An instantaneous normalized rate of secretion (ψ_{L}) is assumed to be a three-parameter generalized γ (probability density) function, with the concentrations given as
*m*, a function of *n*). The asymptotic consistency and normality of the MLE was established previously (18). Then LH secretion rates were estimated as conditional expectations evaluated at the MLE:
*A*_{L}, which could be more explicitly modeled as GnRH/Te → LH drive (4, 34).

##### Binding of Te to carrier proteins.

_{1}) were then calculated using subject- and strata (*k*)-specific SHBG and albumin concentrations and a literature-based estimate of free Te: 0.4 nM (28).

##### Model of (LH-driven) Te secretion and concentrations.

The LH feedforward signal,
*A*) are allowed on the efficacy parameter to permit possible pulse-by-pulse variations in stimulus efficacy. The logistic function parameters do not depend on GRX stratum (*k*). Let

The asymptotic consistency and normality of the MLEs were established previously (18). Te secretion rates were reconstructed as conditional expectations evaluated at the MLE:

##### Estimation of gonadotrope sensitivity to GRX inhibition.

According to classical concepts of a competitive ligand-receptor interaction, the magnitude of an observed biological response is determined jointly by concentration(s) of the agonist and any competing antagonist and properties of the receptor-effector response pathway (15). These minimal assumptions are satisfied, because *1*) GnRH is the exclusive or predominant physiological agonist of the cognate human pituitary receptor (*K*_{d} = 0.85 nM) and *2*) GRX (*K*_{d} = 0.69 nM) acts strictly competitively in vitro and in vivo (2, 6, 29).

##### Model of the GnRH feedforward signal on LH.

Because the elimination of GnRH occurs so rapidly, its concentration at the pituitary will be a “scaled” version of its secretion rate. Thus the following would approximate the unobserved GnRH signal
_{G} is a three-parameter generalized γ (normalized probability density) function and the amplitude is a linear function of the prior IPI plus a random effect
^{(k)}(*t*) (see above), a plausible representation of free Te, acting as a noncompetitive inhibitor, is

##### Free Te negative feedback on the LH (and thereby GnRH) pulse timing.

A Weibull renewal process (16) was fitted from the optimal LH pulse-time sets, with frequency allowed to depend on subject, strata (*k*), and mean free Te:
*k*).

##### Regression on age.

Parameter regressions on age are illustrated for the three significant regressions (all slopes negative) in Fig. A6 and for three nonsignificant regressions in Fig. A7 (see supplemental information in the online version of this article for individual parameter estimates and their correlations).

##### Standard errors of the model (n = 24 subjects) and of individual parameters in each subject.

The foregoing error terms can be estimated analytically as the inverse of the Fisher information matrix obtained via the likelihood method presented here. Table 2 gives SEs of model parameter estimates for the cohort (*n* = 24 men; see supplemental information for individual parameter SEs based on 432 observations per subject).