## Abstract

The present study extends a recent composite model of in vivo interglandular signaling to assess the impact of age on *1*) nonequilibrium exchange among diffusible and protein-bound testosterone (Te); *2*) elimination of total and free Te; *3*) basal and pulsatile Te secretion (sec); *4*) the implicit feedforward function mediating luteinizing hormone (LH) concentration (con) drive of instantaneous Te sec; and *5*) possible stochastic variability of the predicted LH con-Te sec dose-response linkage. To this end, we measured LH and Te con every 10 min for 24 h in healthy young (*n* = 13) and older men (*n* = 13). Statistical comparisons of analytic estimates revealed that elderly subjects manifest *1*) reduced maximal burstlike LH-stimulated Te sec (impaired stimulus efficacy); *2*) depressed half-maximally LH-stimulated Te sec (lower Leydig-cell responsivity); *3*) decreased pulsatile and total Te sec; *4*) elevated basal Te sec; *5*) a prolonged half-life of total but not free Te con; and *6*) delayed time evolution of LH and Te sec bursts. In contradistinction, age did not influence estimated LH-pulse potency (ED_{50}), steroidogenic sensitivity (slope term), or stochastic variability of LH-Te coupling. On the basis of these data, we postulate that aging in the human male alters specific dose-response attributes linking LH con and Te sec and disrupts the time waveform of LH and Te sec bursts.

- reproduction
- human
- male
- testis
- androgen
- Leydig feedback
- luteinizing hormone linkage
- testosterone
- feedforward

aging in the male is marked by gradual depletion of systemic testosterone (Te) availability. The primary cause is unknown. Proposed mechanisms in the human include diminished hypothalamic gonadotropin-releasing hormone (GnRH) secretion (sec), impaired gonadotrope-cell function, and reduced Leydig-cell steroidogenesis (1, 26). In the last regard, injection of the luteinizing hormone (LH) surrogate, human chorionic gonadotropin (hCG), in older men fails to stimulate maximal young adultlike Te concentrations (con; 4, 9, 17, 21). In counterpoint, in a pilot analysis, midphysiological pulses of recombinant human LH increased total Te con comparably by age (6). Because continuous hCG exposure downregulates Leydig-cell Te sec (26), we appraised endogenous pulsatile LH con-dependent drive of Te sec in uninfused, unstimulated, and unblocked healthy young and older men using a novel analytic formalism (11).

## METHODOLOGY

### Clinical Protocol

Thirteen healthy young (ages 18-30 yr) and 13 older (ages 60-79 yr) men participated after providing written informed consent approved by the Institutional Review Board. Inclusion and exclusion criteria, screening biochemical data, diet, physical activity, and demographics have been described (19, 20, 28). Volunteers were admitted to a General Clinical Research Center for two nights and the intervening day. Ambulation was permitted but not vigorous exercise or daytime sleep. Three meals were provided daily. Room lights were put out by 2300. Alcohol, smoking, caffeine, and medications were disallowed in the study unit. Subjects underwent frequent (10 min) blood sampling for 24 h beginning at 0800 after overnight adaptation.

### Assays

LH con and total Te con were quantitated by high-specificity, two-site monoclonal, robotics-assisted immunoradiometric assay and solid-phase RIA, respectively (19, 20). All measurements (145 samples of both LH and Te per subject) were detectable (>3 SD above threshold); namely, for LH ≥0.2 IU/l (Second International Reference Preparation) and for Te ≥20 ng/dl. Samples from any one subject were assayed together. The within-assay median coefficient of variation (CV) was 4.7% (range 4.1 to 5.3%) for LH and 4.5% (3.8 to 5.1%) for Te. Between-assay variability averaged 6.1% (LH) and 5.8% (Te).

### Model of LH Feedforward Drive of Te Secretion

In overview, the analysis reconstructs how pulsatile LH con drives Te sec after a time delay (13-16). The implicit in vivo dose-response relationship is represented algebraically by a monotonic four-parameter logistic function (11, 14). The latter is motivated theoretically and empirically (7). Thereby, we estimate response sensitivity (maximal slope), agonist potency (half-maximally stimulatory concentration, ED_{50}) and stimulus efficacy (asymptotically maximal sec; see appendix; Ref. 10). The shape (time evolution) of unobserved LH sec bursts is represented via a three-parameter generalized Gamma probability density. LH sec burst mass (integral of sec event) is rendered as the sum of basal accumulation, a weak linear function of the preceding interpulse length and a random effect on amplitude (12). Two novel features are implemented. First, the LH-induced Te sec burst is modeled as the output of a simultaneously estimated four-parameter logistic dose-response function. Specifically, the reconstructed (1-min discretized) LH con pulse serves as the time-varying input and the instantaneous Te sec rate the output. Second, the model incorporates statistical allowance for possible stochastic dose-response adaptations in either pulse-by-pulse agonist efficacy or response sensitivity, but not both simultaneously (appendix). The maximum-likelihood estimation (MLE) proceeds via simultaneous statistical estimation of all sec, elimination, dose-response, and stochastic parameters conditional on prior estimation of pulse-onset times (11, 12): Fig. 1.

*Statistical contrasts by age.* Sec and kinetic estimates were compared between cohorts by the rank-sum test or unpaired Student's *t-*test under a one-tailed contrast based on a postulated decrease in Te output in aging. Outcomes from the two stochastic models were compared within a cohort by cognate paired tests. *P* < 0.05 was construed as significant. Data are given as the mean ± SE, median (range), or CV.

## RESULTS

Figure 2 depicts observed (measured) and analytically estimated (fit) LH and total Te con sampled every 10 min for 24 h in 4 young and 4 older men. Mean LH con (IU/l, based on 145 measurements per subject) was 3.3 ± 0.3 (3.1) young and 3.4 ± 0.49 (2.8) older [*P* = not significant (NS)], and total Te con (ng/dl) 488 ± 26.4 (513) young and 401 ± 38.7 (371) older (*P* = 0.019).

Figure 3 illustrates time-varying total (measured and calculated), protein-bound, and free Te con and Te sec with reconstructed LH con-Te sec dose-response functions in one young and older subject. Calculated time and dose-response profiles are given under the two stochastic models of allowable random pulse-by-pulse variability in agonist efficacy or response sensitivity (METHODOLOGY).

Figure 4 summarizes estimated in vivo LH con-Te sec dose-response properties in young and older cohorts. Analyses distinguish between possible stochastic variability in efficacy and sensitivity. However, age-related contrasts obtained by the two models were comparable within 3%; namely, *1*) 41-44% reduction in LH efficacy in older compared with young; *2*) no age differences in Te sec sensitivity (slope) and LH potency (ED_{50}); and *3*) 43-45% reduction in Te sec responsivity (half-maximal Te sec response, Te ED_{50}).

Figure 5 presents analytically reconstructed four-parameter logistic functions for LH con drive of Te sec for the two stochastic allowances. On the basis of 24-h mean LH con values (marked on *x*-axes), LH feedforward operates within the mid-to-upper one-third of the physiological LH con range.

The degree of inferred stochastic variability in dose-response parameters within a pulse train was comparable by age and stochastic model (data given as median cohort CV for young and older): *1*) LH efficacy, 19% and 22%; and Te ED_{50} 19% and 22% (variable-efficacy model) and *2*) testis sensitivity, 28% and 21%; and LH ED_{50}, 24% and 25% (variable-sensitivity model).

Figure 6 summarizes statistical estimates of half-lives of free and total Te con in the stochastic sensitivity construct. Half-times of rapid and delayed phases of free Te disappearance were ∼0.82 and 2.7 min, respectively. The rapid phase predicts diffusion and advection, and the slower phase primarily elimination. Values did not differ by type of stochastic model or age stratum. The monoexponential half-life of total [free plus sex-hormone binding globulin (SHBG) and albumin bound] Te was relatively prolonged at 68 ± 5.4 (64) in older compared with 50 ± 5.6 (45) min in young (*P* < 0.025) in the sensitivity construct. There was a similar trend (*P* < 0.10) in the efficacy model (median 57 and 45 min in older and young men, respectively).

Calculated total (basal plus pulsatile) Te sec was 42% lower in older men than young (mean of the 2 stochastic models, *P* < 0.025): Fig. 7. Lower total Te sec in older men was due to a commensurate (55%) reduction in pulsatile Te sec (*P* < 0.01). Basal Te sec, albeit a minor component of total Te sec, was 2.5-fold higher in older than young men (*P* < 0.05). Thus the percentage contribution of basal to total Te sec was elevated at 28 ± 3.9 (23) in older vs. 7.7 ± 2.3 (6.8) in young men (*P* = 0.02). Figure 8*A* depicts analytically reconstructed individual LH and Te sec-burst waveforms (time-evolution of sec rates within a pulse). Figure 8*B* shows that the time required to complete the second (postmaximum) half of LH and attain the maximum of Te sec bursts was significantly extended in older compared with young men.

## DISCUSSION

According to the present analytic formalism, in vivo LH-driven pulsatile and total daily Te sec rates are reduced by ∼50% in older compared with young men. In contrast, albeit a minor (<7%) component, time-invariant (basal) Te sec is elevated in the elderly individuals. This prominent contrast emerged in the face of equivalent (24-h mean) LH con and an 18% reduction in total Te con in the older male group. The current noninvasive analytic estimates are consistent with extrapolations from radiolabeled steroid infusion data obtained in a small number of young volunteers (24). An earlier nonparametric deconvolution model using literature-derived Te kinetics also forecast attenuation of pulsatile but not basal Te sec in aging men (18). The basis for decreased pulsatile and increased basal Te sec in the aging male has not been elucidated. A plausible hypothesis is that the 45-50% reduction in LH sec-burst mass reported in elderly men impairs LH drive of pulsatile Te sec (15, 18, 20, 28). Physiological stimuli of basal Te sec are not established, but might include interpulse or 24-h mean LH con. In this regard, interpulse LH con is elevated, whereas 24-h rhythmic Te con is blunted with age (26).

Analytic reconstruction of the endogenous LH con-Te sec dose-response function predicted a 44% reduction in LH efficacy (maximal rate of burstlike Te sec) in older men. The latter estimate matches the degree of decline in total Te sec (above). The age distinction was selective, because calculated LH potency (ED_{50}) and Leydig-cell sensitivity (slope of dose response) did not differ by age. Whether the projected reduction in LH efficacy is reversible is unknown. However, pulsatile subcutaneous infusion of a fixed dose of LH and continuous incubation of Leydig cells with LH fails to normalize reduced Te sec in the aged male Brown-Norway rat (2, 8).

The current biomathematical construction incorporates potential asymmetry of burst shape by way of a (non-Gaussian) three-parameter generalized Gamma density. The Gamma density defines a probabilistic sec-burst waveform (unit-area normalized time evolution of instantaneous sec). Thereby, we identified less rapid completion of the second half of LH sec bursts in elderly individuals, despite normal initial unfolding of sec rates. Because maximal LH sec developed within 18-20 min, we hypothesize that release of stored LH sec granules may proceed nearly normally in aging individuals. On the other hand, to the degree that continuing LH sec requires de novo glycoprotein synthesis and packaging, aging may impair GnRH-enhanced transcription and/or translation of LH β-sub-unit and vectorial processing for secretion (24-26).

Estimation of Te sec-burst shape necessitated propagation of analytically reconstructed pulsatile LH con series through the simultaneously estimated four-parameter LH-Te dose-response function (APPENDIX). This approach predicted that Te sec bursts evolve to a maximum within 35-40 min and wane over the subsequent 80-120 min. Asymmetric Te sec events are inferable by the more rapid onset than offset of *1*) endogenous LH-stimulated Te sec in the human spermatic vein and *2*) agonist-induced Te sec by in vitro perifused rodent Leydig cells (3, 5, 29). Statistical contrasts by age disclosed that older men manifest a prolonged latency to achieve peak Te sec rates in bursts. The precise steroidogenic mechanisms that mediate this alteration are not established.

The current statistical representation allows for (but does not require) stochastic fluctuations in either of two fundamental dose-response properties, efficacy or sensitivity. This novel formulation unmasked significant individual variability of pulse-by-pulse LH efficacy and Leydig-cell sensitivity, namely, ∼25% (expressed as a CV). The latter median estimate exceeds intra-assay CVs of LH and Te by greater than threefold (*P* < 0.001). Inferred dose-response variability was independent of age stratum or stochastic model type. The physiological basis, albeit unknown, might reflect *1*) fluctuations in antecedent LH pulse amplitude, duration, interval, or interpeak values; *2*) inconsistent pulse-by-pulse LH bioactivity; *3*) nonuniform intragonadal delivery of LH; *4*) asynchronous neurogenic, paracrine, and/or autocrine inputs to Leydig cells; and *4*) variable steroidogenic desensitization and resensitization. The foregoing considerations are important by way of suggesting testable hypotheses.

### Perspectives

This work highlights a fourfold strategy to quantitate jointly deterministic (dose responsive) and stochastic (random) mechanisms that link neuroendocrine signals in vivo; namely: coupled differential equations to encapsulate time-dependent adaptations; nonequilibrium exchange of free ligand among bound and free compartments and irreversible elimination; analytic reconstruction of implicit dose-response attributes; and stochastic allowance for unexplained biological variability arising in key dose-response measures. To estimate interglandular signaling properties reliably in an intact host requires primary experimental validation and independent statistical verification. The first was accomplished for LH-Te by extended (4-12 h) and frequent (5 min) direct sampling of all three of GnRH, LH, and Te con in the conscious, unrestrained, and uninfused stallion and ram and simultaneous monitoring (every 20 min for 17 h) of LH and Te con in the human spermatic vein (10, 27). Statistical verification was achieved by formal mathematical proof of maximum-likelihood asymptotic parameter estimates (12). From a broader perspective, this general investigative strategy should have utility in dissecting multisignal regulation in other interlinked endocrine and nonendocrine biological systems.

## APPENDIX

### Statistical Model of LH Feedforward on Te Secretion

*Pulse times.* LH pulse-onset times are predicted by a previously constructed pulse-detection methodology, and then treated as fixed, T^{1},T^{2},... T^{m} (13).

*LH sec rate and con.* The shape of LH sec bursts is represented by a unit area-normalized three-parameter (generalized Gamma probability density) waveform, ψ_{L}. The waveform comprises instantaneous sec rates evolving over time and is superimposed on a time-invariant basal sec rate (β_{L}). Sec burst mass arises from intraglandular basal hormonal accumulation (η_{0,L}), increases as a linear function of interpulse interval length and exhibits random effects on amplitude . The total LH sec rate, Z_{L}(*t*), is the sum of basal sec and the product of burst mass and waveform 1

For fast and slow rates of elimination given by and , we established that the time-varying LH con for a starting con of X_{L}(0) is 2

Sample hormone con with experimental uncertainty, ϵ_{L}(i), is then defined by Y_{L,i} = X_{L}(*t*_{i})+ ϵ_{L}(i), i = 1,..., *n*. Deconvolution analysis proceeds by MLE of all sec and elimination parameters simultaneously (12): Fig. 1*A*.

*LH-Te agonist-response interface function.* Time-averaged LH con stimulates time-delayed Te sec via a four-parameter logistic dose-response function. We include statistical allowance for possible within-subject, pulse-by-pulse random variability in the asymptotic maximum (agonist efficacy) or response slope (target-gland sensitivity). The theoretic foundation for cooperative and saturable dose-responsiveness was highlighted earlier (7). In this dual (deterministic and stochastic) formulation, Te sec is defined as for *t* such that T^{j} ≤ *t* < T^{j+1}, j = 1,... .m, wherein Z_{Te}(*t*), *t* ≥ 0, is instantaneous Te sec, β_{Te} is the lower asymptote, η_{0,Te} is the potency (ED_{50}), η_{1,Te} is sensitivity (slope), and η_{2,Te} is efficacy (maximal Te sec) within the j^{th} pulse. Random effects, , operate on either efficacy or sensitivity, but not both: Fig. 1*B*. F_{L}(*t*) is the time-integrated and delayed LH con feedforward signal.

*Fate of secreted Te.* At each time *t*, free Te entering the blood may remain unbound [free (F)] or associate reversibly with SHBG (S) or albumin (A): Fig. 1*C*. Let denote the three primary Te compartments, which sum to the total Te con, X_{Te}(*t*). The partitioning of total Te among S, A, and free fractions is governed by rates of association and dissociation and con of (occupied and unoccupied) S and A, as follows

Equilibrium dissociation constants estimated in Ref. 22 are M, M with binding capacities of S = 49 ± 2.3 × 10^{-9} M (young men) and 89 ± 5.8 × 10^{-9} M (older men), and A = 6.8 ± 0.1 × 10^{-4} M (both age groups). The forward rate constant is the product of the association rate constant and con of unoccupied protein binding sites: and . Thereby, we estimate equilibrium fractions of free, SHBG-, and albumin-bound testosterone of 2, 43, and 55%. Joint Te kinetics, including exchange with SHBG and albumin, are defined by the solutions, j = 1,2 4 where B^{(j)} is above matrix; δ(*t*) = [Z_{Te}(*t*),0,0]′; and are the rapid- and slow-phase free Te elimination rates; a_{Te} and 1 - a_{Te} are fractional elimination amplitudes; and total Te con is 5

*Estimation of LH feedforward signal and Te sec response.* Continuous model functions are discretized to a 1-min scale to estimate LH feedforward (con, IU/l) and Te sec (ng·dl^{-1}·min^{-1}) (11, 12, 15, 16). The half-time of Te's association with (e.g., 0.07 s) and dissociation from (e.g., 0.32 s for albumin and 3.3 s for SHBG) plasma proteins is rapid compared with the sampling interval (e.g., 10 min). Thus *equations 4* and *5* can be solved recursively in time (*t*_{i}) 6

Sample Te con with experimental uncertainty, ϵ_{Te}(i), is

The resultant parameter set is θ = [α^{(1)}, α^{(2)}, β, η_{0}, η_{1}, η_{2}, γ_{A}, γ_{ϵ}, X(0)], where the subscript Te is left implicit, and where γ_{A} is the SD of efficacy or sensitivity random effects []. In the stochastic efficacy model, the likelihood function is Gaussian and therefore analytically obtainable. In the sensitivity model, random effects contribute nonlinearly. Thus in the first stage of estimation, random effects are held fixed, and a conditional Gaussian likelihood model is maximized. And, in the second stage, parameters are held fixed and Gaussian random effects are calculated via an Expectation-Maximization algorithm (21a). The iterative solution yields the MLE of θ. Estimates of sec rates are conditional expectations evaluated at the MLE, q̂ 7

Predicted (fitted) LH and Te con profiles, Ŷ _{k,i}, i=1,..., *n*; k=L, Te, are convolution integrals of the sec and elimination functions: Figs. 2 and 3.

## GRANTS

Support was provided by Grants K01-AG-19164 and R01-AG-23133 from the National Institutes of Health (Bethesda, MD); DMS-0107680, a National Science Foundation Interdisciplinary Grant in the Mathematical Sciences (Washington, DC); and M01-RR-00585 from the National Center for Research Resources (Rockville, MD) to the Mayo Clinic and Foundation General Clinical Research Center.

## Acknowledgments

We thank K. Bradford for excellent assistance in text preparation, statistical presentation, and graphical illustrations.

## Footnotes

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

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- Copyright © 2004 the American Physiological Society