MODELING IN PHYSIOLOGY
A biomathematical model of time-delayed feedback in the
human male hypothalamic-pituitary-Leydig cell axis
Daniel M.
Keenan1 and
Johannes D.
Veldhuis2
1 Division of Statistics, Department of Mathematics,
University of Virginia, Charlottesville 22903; and
2 Division of Endocrinology, Health Sciences Center, and
National Science Foundation Center for Biological Timing, University of
Virginia, Charlottesville, Virginia 22908
 |
ABSTRACT |
We develop,
implement, and test a feedback and feedforward biomathematical
construct of the male hypothalamic [gonadotropin-releasing hormone
(GnRH)]-pituitary [luteinizing hormone (LH)]-gonadal [testosterone (Te)] axis. This stochastic differential equation formulation consists
of a nonstationary stochastic point process responsible for generating
episodic release of GnRH, which is modulated negatively by short-loop
(GnRH) and long-loop (Te) feedback. Pulsatile GnRH release in turn
drives bursts of LH secretion via an agonistic dose-response curve that
is partially damped by Te negative feedback. Circulating LH stimulates
(feedforward) Te synthesis and release by a second dose response. Te
acts via negative dose-responsive feedback on GnRH and LH output, thus
fulfilling conditions of a closed-loop control system. Four computer
simulations document expected feedback performance, as published
earlier for the human male GnRH-LH-Te axis. Six other simulations test
distinct within-model coupling mechanisms to link a circadian
modulatory input to a pulsatile control node so as to explicate the
known 24-h variations in Te and, to a lesser extent, LH. We conclude
that relevant dynamic function, internodal dose-dependent regulatory
connections, and within-system time-delayed coupling together provide a
biomathematical basis for a nonlinear feedback-feedforward control
model with combined pulsatile and circadian features that closely
emulate the measurable output activities of the male
hypothalamic-pituitary-Leydig cell axis.
neuroendocrine; biomathematics; stochastic
differential equations; reproductive axis
 |
INTRODUCTION |
THE MALE REPRODUCTIVE AXIS consists of three
physiologically distinct but interacting functional control nodes: a
gonadotropin-releasing hormone (GnRH) pulse generator, endowed by
hypothalamic neurons; luteinizing hormone (LH), produced in the
anterior pituitary gland; and testosterone (Te), secreted by Leydig
cells in the testes. In health, this multinodal feedback and
feedforward interactive system yields a pseudo-steady-state output of
pulsatile (episodic) neurohormone release that shows circadian
modulation, e.g., 24-h variations in serum Te concentrations and, to a
lesser extent, in LH (30). Dose-response relationships have been
largely defined for individual nodes acting in isolation, e.g., GnRH's
stimulation of LH secretion (1) and LH's dose-dependent stimulation of Te secretion (10). Similarly, earlier simulation modeling of neurohormone release has typically encompassed a single hormone, not
the entire interacting feedback system. A major limitation inherent in
thus isolating system components that are so highly coupled
physiologically via time-lagged feedforward (e.g., LH-Te) and feedback
(e.g., Te-LH) mechanisms is omission of the influences due to
communications among the component(s). Moreover, artificial isolation
of functional elements can make inference of system behavior more
difficult. Given these issues, we present an initial biomathematical
model of an entire interconnected three-nodal system, i.e., the (male)
hypothalamic (GnRH)-pituitary (LH)-gonadal (Te) axis, and test its
basal pulsatile output, modulated circadian responses, and predicted
performance in selected simulations and prior human experiments.
Glossary
i |
Elimination rate constant for hormone i
|
i |
Basal secretion rate of hormone i
|
t |
Discretization step size used in computer simulations
|
dS(t) |
Incremental secretion in interval
(t, t + dt) (from Ref. 16)
|
dWi(t) |
Stochastic noise term (via differential of Brownian motion) (see
section VIII)
|
dXi(t) |
Incremental change in concentration of hormone i at time
t
|
Zi(t)dt |
Incremental change in secretion of hormone i at time t
|
 |
Parameter that (probabilistically) controls interpulse lengths
|
GnRH |
Gonadotropin-releasing hormone
|
H( · ) |
Dose-response (interface) functions
|
IID |
Independent and identically distributed
|
(lj,1, lj,2) |
Time-delayed interval for jth feedback interaction
|
( · ) |
Cosine function specifying periodic (circadian) input
|
( · ) |
(Stochastic) pulse generator intensity function
|
LH |
Luteinizing hormone
|
Mji |
Pulse mass j for hormone i
|
p(t) |
Pulse generator (stochastic) intensity
|
i( · ) |
Pulse shape for hormone i
|
SDE |
Stochastic differential equation
|
T ij |
Pulse time j for hormone i
|
Te |
Testosterone
|
Wi( · ) |
ith Brownian motion process (one of "noise" processes)
(see section VIII)
|
Xi(t) |
Concentration of hormone i at time t
|
Yik |
Observed concentration of hormone i at time
tk
|
Zi(t) |
Secretory rate for hormone i at time t
|
 |
I. GENERAL METHODS |
A. Background Physiology
An important initial question in capturing physiological behavior of
the male hypothalamic-pituitary-gonadal axis in a biomathematical construct is: What is an appropriate level at which to formulate the
model? We have chosen to focus on the rate of change in blood hormone
concentration, since this is a principal measurement variable in health
and disease. Here, we do so in continuous time. By evaluating resultant
hormone concentrations one can test predictions of the biomathematical
formulation experimentally via available measurements, and by
structuring in continuous time one can compare data under different
sampling schemes.
Our thesis is that the complex dynamic of the male
hypothalamic-pituitary-gonadal feedback system occurs because the rate of secretion of any given hormone within the system (GnRH, LH, and Te)
depends on relevant time-delayed, nonlinear feedback and feedforward
signals derived from and acting on all or some of the components of the
system. We further assume that pertinent dose-responsive interfaces
(either inhibitory or stimulatory) connect hormone signal input to
nodal output; e.g., a GnRH concentration-LH secretion rate
dose-response relationship functionally links the hypothalamic GnRH
signal and the time-delayed pituitary LH secretory output (8, 9,
11-14, 25, 30, 35). Similarly, a dose-response relationship exists
for LH concentrations and Leydig cell Te secretion, as established by
in vitro and in vivo experiments (2, 7, 10, 30). By formulating a
physiologically linked network using the three primary and interacting
nodes of hypothalamus, pituitary gland, and testes (Fig.
1), we can examine basal hormone output, test system performance in specific computer simulations compared with
prior clinical experiments, evaluate relevant internodal linkages, and
later predict possible axis dysregulation.

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Fig. 1.
Schematic illustration of time-delayed negative feedback ( ) and
positive feedforward (+) within human male gonadotropin-releasing
hormone-luteinizing hormone-testosterone (GnRH-LH-Te) axis. Broad
arrows, feedforward (+) stimulus-secretion linkages; narrow arrows,
feedback ( ) inhibition. "H" functions are developed
further in section I and Fig. 2 and define dose-response relationships
at each feedback interface within axis (see section VIIIA2).
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|
1. Core equations.
Considering the foregoing background, we can relate the "true blood
hormone (GnRH, LH, or Te) concentrations" in vivo to corresponding secretion rates. To this end, let time 0 represent the onset of the observation period, XG(t),
XL(t), and
XTe(t) (t
0) the hormone concentration values, and ZG(t),
ZL(t), and
ZTe(t) (t
0) the corresponding hormone secretion rates for GnRH, LH, and Te,
respectively, over time. The core model of a dynamic GnRH-LH-Te
feedback system will then encompass the following general equations.
These equations state that, by definition, for each hormone signal
involved (GnRH, LH, or Te), the rate of change of its concentration in
blood is the difference between its rates of elimination and
production
|
(1)
|
where XG(0),
XL(0), and XTe(0) are
specified initial GnRH, LH, and Te concentrations and
G,
L, and
Te are the rates of elimination of
the respective hormones; the rates could be allowed to be concentration
dependent, as may be the case for higher levels of LH (32).
Concentration (and secretion rate) units are mass of neurohormone
(secreted) per unit distribution volume (per unit time).
Next we incorporate feedback and feedforward relationships within the
hypothalamic-pituitary-Leydig cell axis by defining mathematically, via
"H" functions, how each hormone secretion rate (at any
instant in time t) of ZG(t),
ZL(t), and
ZTe(t) depends, in a nonlinear manner,
on (all or some of) the prior ( · ) hormone concentrations
[XG( · ),
XL( · ), and
XTe( · )] over appropriately time-delayed intervals. In addition, we will allow for the stochastic variability, which occurs in several respects within the axis. Equation 1 will first be replaced by stochastic differential
equations (SDEs), Eqs. 6-10, which will allow for a
stochastic (feedback-driven) pulse generator. In section VIIIE,
a more extended stochastic formulation is presented, which, in addition
to the foregoing, attempts to describe further in vivo biological
variability. Experimental uncertainty also arises from sampling and
measurement errors due to sample collection and processing and
subsequent assay of the hormone concentration. For the LH axis, such
sampling-related technical variability has been estimated as
3-15% (29-36). Biological variability would influence the
time evolution of the actual unobserved "true concentration"
levels, whereas technical variability does not. Both factors affect any
individual "realized" continuous concentration series or its
discrete-time sampling.
Our feedback construct of the in vivo male GnRH-LH-Te axis thus
consists of a system of coupled SDEs, one for each of GnRH, LH, and Te.
The SDEs are of the most basic type, i.e., random ordinary differential
equations, in that the forcing function is now a random process; in
section VIII a more extended formulation is considered. The elimination
function acting on hormone concentrations will be assumed initially to
be exponential (31, 32). We will impose time-delayed, nonlinear
(dose-responsive) feedback by relevant hormone concentrations on future
secretion rates and on the intensity of the point process, which
describes the GnRH pulse generator's bursting activity (below). We
posit that these principal biological features produce the observed
oscillatory nature of such dynamic physiological systems.
 |
II. FEEDBACK BEHAVIOR ANTICIPATED IN THE MALE
GONADOTROPIN-RELEASING HORMONE-LUTEINIZING HORMONE-TESTOSTERONE AXIS |
Pulses of GnRH from the hypothalamus drive corresponding episodes of
pituitary LH secretion in a nearly uniformly 1:1 ratio (1, 8, 9, 14,
25, 27, 30, 35). Pulsatile LH concentrations bathing Leydig cells in
the testes in turn stimulate Te secretion in a dose-dependent manner
(7, 10). Thus serum Te and, to a lesser extent, serum LH concentrations
vary episodically over 24 h. In addition, available data are consistent
with circadian variations, with maximal hormone concentrations of LH
and Te occurring in the human during the later portion of nighttime
sleep (34). In general, the amplitude of LH pulses tend to vary
inversely with event frequency and to be maximal at night (2, 9, 24, 25, 33). Despite these variations, mean daily serum LH and Te
concentrations remain within a relatively narrow (0.5- to 1.5-fold) physiological range. This is thought to reflect homeostatic feedback control, which we embody in the coupled equations above (Eq. 1). More explicitly, in young men, serum LH and Te concentrations are positively cross-correlated at a +20- to 50-min lag (Te following LH) and negatively cross-correlated at a
80- to 100-min lag
(testosterone preceding LH) (34). The former reflects LH's
dose-dependent feedforward action on Leydig cell Te biosynthesis. The
latter presumably reflects Te's negative feedback on GnRH-LH secretion (30). Reduction or removal of Te's negative-feedback signal via an
androgen-receptor antagonist or an inhibitor of Leydig cell
steroidogenesis increases the frequency and amplitude of pulsatile LH
release as well as, in the former case, the mean serum Te concentration
(17, 29, 36). Conversely, continuous intravenous infusion of Te in
steroidogenesis-inhibited men suppresses pulsatile LH release by
reduced LH (and presumptively GnRH) pulse frequency with escape of
occasional higher-amplitude LH pulses (36). These published clinical
data provide a basis for testing the dynamics of our feedback model of
the GnRH-LH-Te axis (see below).
 |
III. SPECIFIC METHODS |
A. Constructing GnRH-LH-Te Secretion
The pituitary gland releases a small, approximately time-invariant
amount of LH into the blood, which is often referred to as constitutive
or basal secretion (1, 8, 25, 30), here designated as
. In addition,
regulated (nonbasal) LH secretion is driven by feedforward (GnRH) and
inhibited by feedback (Te) interactions within the GnRH-LH-Te axis. We
assume that this potentially regulated secretion of LH can occur in two
forms: pulsatile release and continuous release. To accommodate
pulsatile LH release, we assume a strict dependence of each LH pulse on
a GnRH pulse signal (8, 13, 25, 35). We designate activity of the
so-called GnRH pulse generator via a stochastic point process, which
has a probability of activating a GnRH (LH) pulse in the next time increment (t +
t) that varies on the basis of
time-delayed negative feedback by Te. Because the feedback is time
delayed, our point process will not be a Markov process (see section
VIII), and the distribution of waiting times will not be simply
exponential.
To represent explicitly the changing rates of pulsatile and continuous
secretion over time, we assume that GnRH of hypothalamic origin signals
the gonadotropic cell to increase its release and synthesis of LH
molecules. The GnRH stimulus acts over a finite time to increase the
rate of de novo LH biosynthesis while concurrently prompting the nearly
immediate release of (readily releasable) membrane-associated, storage
granule-encapsulated LH. Newly synthesized LH molecules are
encapsulated into such secretory granules, which diffuse out toward the
gonadotropic cell membrane as facilitated by cytoskeletal elements to
direct vectorial drift (30). When storage granules containing LH reach
the cell membrane, they can undergo secretory exocytosis (when the GnRH
signal is activated) or remain marginated (during secretory quiescence)
and thereby accumulate, awaiting the next GnRH signal to trigger
release.
For nonprotein hormones (unlike LH), there is no known granule
accumulation process but, rather, more nearly continuous release, as
presumed here for the steroid hormone Te (7, 10, 30). In contrast, the
foregoing granule storage-release process applies to LH and GnRH, for
which accumulated intracellular secretory granules become available for
rapid initial release at the onset of the next signal, resulting in
pulsatile release. The proximate signal for such a pulse of LH release
is GnRH, and for GnRH release it is a balance of stimulatory and
inhibitory neurotransmitter inputs.
In contrast to pulsatile LH release by gonadotropic cells, some LH
molecules are not encapsulated or are encapsulated but not hormonally
regulated and diffuse out toward and through the plasma membrane, thus
contributing the so-called basal secretion described above (30).
Because the pituitary gland contains an array of individual
gonadotropic cells, integrating LH secretion over all constituent cells
yields the overall LH secretion rate compounded from basal and
pulsatile release. Similar integration of secretion over the ensemble
of dispersed hypothalamic GnRH neurons is pertinent.
1. Continuous secretion.
For continuously released Te (30), the secretion rate at time t
will be assumed to depend on XL( · ),
the concentration levels of the input stimulus, LH, over some
time-delayed interval, e.g., from t
l2
to t
l1, as transduced via some
dose-response function H4( · )
|
(2)
|
where
Te is the Te basal release rate. Thus
we model Te release via the function
H4( · ) above; the precise assumed
form of H4( · ) is given in sections
IVA and VIII.
2. Pulsatile secretion.
Pulsatile secretion (of GnRH and LH) is marked by pulse times, when the
rate of hormonal secretion rapidly increases, followed by a possibly
not so rapid decrease; this secretory episode will be called a pulse
(30). Here we allow one principal stochastic pulse generator, i.e.,
hypothalamic GnRH, from which pituitary LH inherits its episodicity of
release (30). Thus, for the GnRH-LH-Te axis, there will be one set of
pulse times, T0, T1,
T2, ... , where without loss of generality we
will assume T0 = 0.
To accommodate GnRH and LH synthesis and accumulation in storage
granules, let Mj denote the amount (mass)
of hormone accumulated from the last pulse time
(Tj
1) to the present pulse time (Tj) this accumulation will be
available for release at time Tj. Then, to
define the time course of hormone release, let
( · ),
where
(s) = 0, s
0, called the pulse shape, be a
function normalized to integrate to 1, which represents the instantaneous rate of secretion in units of mass of hormone released per unit time and distribution volume.
Let us represent a pulse at time t, having started at pulse
time Tj, by a function
Mj
(t
Tj),
where Mj is the mass (or amplitude) of the
pulse. The overall rate of secretion at time t,
Zt), is then assumed to be given by the sum of basal
(
) and (summated) pulsatile secretion
|
(3)
|
where H( · ) designates feedback
control functions that regulate the mass of GnRH or LH accumulating.
[In section IVA these will be
H2( · ) in the case of GnRH and
H5,6( · ) in the case of LH.] This
is a plausible approximation of LH release in many physiological
contexts (1, 8, 9, 11-14, 19, 22, 24-27, 30-36). In
sections IV and VIII we explicitly expand on the form of the above
H( · ) control functions and the feedback
interactions.
B. Circadian Modulation
In addition to basal and pulsatile GnRH and LH release and continuous
Te secretion, there is a prominent circadian (24-h) rhythm in serum Te
concentrations. In men, serum Te concentrations exhibit an ~24-h
cycle, with maximal values at around 4-6 AM, dropping later in the
day and evening by 15-70% and then rising again in the night
(34). An important issue is how to appropriately incorporate this
"rhythm" in the GnRH-LH-Te axis in a manner consistent with
experimental data.
We previously formulated a model of the pulsatile secretion (not
concentration) of one hormone, without external feedback inputs, as
illustrated by episodic LH secretion (16). Pulsing was described by an
inhomogeneous Poisson process, for which the deterministic intensity
( · ) function was 24-h periodic, thus describing a
circadian rhythm. In the simulations, the
( · ) function
was assumed to consist of one harmonic, with amplitudes B0 and B1 and phase shift
1 being chosen so
( · ) would vary between a high of 1 at 4 AM and a low of 0.6 at 4 PM
(B0 = 0.85, B1 = 0.15,
1 =
4)
The mass of LH available for release within a pulse,
Mj, was assumed to be a linear function of
the preceding interpulse length: the longer the interpulse interval,
the greater the accumulation of LH-containing granules available for
release
where Mj represents the pulse
masses,
0 denotes a minimum amount of mass always
secreted per pulse, and
1 is a constant rate of hormone
accumulation. Incremental secretion, dS(t) (from the
pituitary gland of a horse), was evaluated previously (16). If
S(t) is the cumulative secretion up to and including time
t, then the incremental secretion between sample times
ti and
ti + 1[S(ti + 1)
S (ti)] can be obtained by integrating the
differential dS( · ); we previously (16) formulated
dS(t) as
The Z(t) in our present formulation
corresponds to dS(t)/dt in this earlier construction.
Thus the only feedback in the earlier formulation (16) was through the
length of the previous interpulse interval, but the instantaneous rate
of accumulation,
1, was constant and, hence, unaffected
by feedback; also the point process representing the pulse times was
modulated by a deterministic (periodic) intensity function
( · ), but not by feedback. Whereas this preliminary model is very reasonable and appears to fit certain secretory data
quite well, occasionally there may be very short interpulse intervals
followed by relatively large pulse masses, and vice versa (23). We
reasoned that this pulse-to-pulse variability might be explicable if
one models the entire system. In addition, by incorporating
physiologically pertinent feedback and feedforward connections, one
should derive more meaningful estimates of the true variability within
the intact system than by modeling release of a single hormone in
isolation (6).
 |
IV. COMPOSITE BIOMATHEMATICAL CONSTRUCT OF THE MALE
GONADOTROPIN-RELEASING HORMONE-LUTEINIZING HORMONE-TESTOSTERONE AXIS |
A. Dose-Response Linkages [H( · ) Functions]
The detailed mathematical motivations of the present formulation, i.e.,
elimination rates, feedback interactions, pulse shape, and pulse
generator, and time discretization of the output are given in section
VIII. Here we present the (time-delayed) feedback-feedforward dose-response functions that embody the interactions within and confer
nonlinear dynamics on the (male) pulsatile GnRH-LH-Te axis. In section
IVB we present six possible adaptations of this basic structure, each incorporating a circadian rhythm in a different mechanistic manner.
Experimental evidence suggests that internodal feedback (e.g., GnRH
feedforward on LH, LH feedforward on Te, Te feedback on GnRH or LH) is
exerted via a time-delayed time average of the effector concentration.
Thus, throughout the simulation, we will use the following notation,
where 0
t1
t2 <
, to
denote feedback signal intensity (i.e.,
denotes a time average)
Using empirically inferred dose-response (possibly
multivariate) logistic functions, which have been evaluated in in vitro and in vivo experiments in humans and animals (1, 30, 35), we then
define how such feedback interacts with the various
components
|
(4)
|
If Bi > 0, the feedback is
positive (i.e., feedforward effect); if Bi < 0, the feedback is negative. Accordingly, for the simplified male
GnRH-LH-Te axis, there are four major relevant feedback-feedforward
dose-response functions: H1( · ) for
the GnRH pulse firing rate as a function of Te concentration, H2( · ) for the rate of GnRH
pulse-mass accumulation as a function of Te concentration,
H4( · ) for the rate of Te secretion
as a function of LH concentration, and
H5,6( · , · ) for the
rate of LH pulse-mass accumulation as a function of Te and GnRH
concentrations. The subscripts correspond to feedback-feedforward
relationships (1-7 in section VIII). A function
H7( · ), not modeled by a logistic form, describes a refractory condition of the GnRH pulse generator as a
consequence of possible short-loop negative autofeedback of GnRH on its
own release (see section VIII); also, an optional interaction given by
H3( · ), not part of the basic
construct, allows for the Te basal secretion rate to vary with a 24-h
periodicity. Feedback connections are thus mediated via relevant
dose-response functions H1( · ),
H2( · ),
H4( · ), and
H5,6( · , · ), wherein the maxima, minima, slopes, and midpoints reflect the operating behavior of GnRH-LH, LH-Te, Te-GnRH, and Te-LH linkages. Also, the time
delays for the jth feedback interaction will be expressed throughout by lj,1 and
lj,2. For example, the negative feedback of
blood Te concentration (in ng/dl) on the rate of hypothalamic GnRH
pulse-mass accumulation (in
pg · ml
1 · h
1)
would be of the form
|
(5)
|
Also, until time t is above the maximum time delay,
the feedback will not be over the full time-delay interval, but rather only the amount of time since time 0 (as discussed in section VIII). Figure 1 displays the above feedback connections, and Figure 2 illustrates the corresponding
mathematical functions.

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Fig. 2.
Dose-response feedback and feedforward functions for male GnRH-LH-Te
axis. First (leftmost) vertical column displays, for a normal subject,
dose-response feedback and feedforward functions of Fig. 1: Te
inhibition of GnRH pulse firing rate
[H1( · )], Te inhibition of GnRH
pulse-mass accumulation rate
[H2( · )], LH stimulation of Te
secretion rate [H4( · )], and Te
and GnRH jointly acting on LH pulse-mass accumulation rate
[H5,6( · , · )].
Second column depicts H1( · ) and
H2( · ) for simulation 1 (consisting of flutamide treatment),
H4( · ) for simulation 2 (ketoconazole treatment), and
H5,6( · , · ) for
simulation 1. Third column presents assumed dependency of slope
(B1) as a function of Te in
H1( · ) for simulation 3 (ketoconazole treatment with Te infusion add back); 2 pulse shapes,
G( · ) (solid line) and
L( · ) (dashed line); 24-h periodic
modulating function ( · ) with its 12-h phase-shifted
version superimposed (dashed line); and assumed dependency of lower
asymptope (D) as a function of Te in
H5,6( · , · ) for
simulation 3. [Dose-response H( · )
functions are defined in section VIIIA2, and simulations are
summarized in sections VB1 and VB4.]
|
|
Figure 2 depicts the various dose-response feedback functions for the
simulation experiments; the nominal parameter values for the feedback
functions and elimination rates are given in section VIIIA
along with their motivation. In the sensitivity analysis (section
VC), these nominal values are doubled and halved to
demonstrate the dependency of the structural dynamics on the parameter
values. The pulse shapes
G( · ) and
L( · ) were taken to be generalized
densities as displayed in Fig. 2 (2nd row, 3rd column); the modulating
function
( · ) (and its 12-h phase shift) used in the
circadian mechanistic simulations is also displayed in Fig. 2 (3rd row,
3rd column). The variances for the combined technical and measurement
errors were such that the coefficient of variation was 6%. The
parameter
in the construction of the pulse generator (see Eq. 7) was taken to be 2 (see section VIIIB1).
B. Basic SDE Construct
Our formulation of the male GnRH-LH-Te feedback system is thus governed
by the following coupled SDEs. The pulse times of GnRH
(T1, T2, ...) are the
result of a point process with a so-called stochastic pulse intensity
|
(6)
|
where H7(t) is a refractory
condition whereby GnRH release may be inhibited by autofeedback (see
section VIII). The conditional density for
Tk given
Tk
1 and
( · )
will be required to satisfy
|
(7)
|
The secretory pulse masses for GnRH and LH, respectively,
are given by
|
(8)
|
|
(9)
|
The basic components of the male axis (GnRH, LH, and Te) are
thus
|
(10)
|
The SDEs in Eqs. 6-10 correspond to the core
construct (Eq. 1), now having incorporated the
feedback-feedforward relationships and a stochastic pulse generator
(feedback-driven) variability. Thus the "true" in vivo
concentration processes [(XG(t),
XL(t), and
XTe(t), t
0] are the
resulting solutions of the above system of equations. In section
VIIIE, a generalization of Eq. 10 is discussed with
additional stochastic biological variability in the feedback equations
per se.
What we then actually observe is a discrete-time sampling of these with
attendant technical/measurement error due to sample collection,
processing, and assaying
In the simulations, the
i values are
independent and identically distributed (IID) normal, mean zero and
with variances such that the coefficients of variation for
YGk, YLk, and
YTek are 6% to fall within an
anticipated operating range of 3-15% for typical available assay
coefficients of variation (29-36).
C. Mechanisms for Incorporating a Circadian Rhythm
We can include a circadian rhythm acting deterministically via several
possible mechanisms, in which there is deterministic circadian input to
the GnRH-LH-Te feedback axis at any of one or more nodes or control
functions: 1) Te (negative) feedback on GnRH burst frequency,
2) Te (negative) feedback on GnRH burst mass, 3) basal
Te secretion rate, 4) LH (positive) feedforward on Leydig cell
Te secretion, 5) Te (negative) feedback on LH secretion mass,
6) GnRH (positive) feedforward on LH secretion, and 7)
GnRH autonegative feedback.
Section VIII gives a further description of the first six possible
models for the circadian rhythms in Te and LH. We test (see section V)
the predictions of these six putative circadian coupling schemes and
show that some, but not all, predict physiological variations in 24-h
serum Te (more than LH) concentrations. Mechanism 7 (GnRH
autofeedback) is less likely, we speculate, which would entail
day-night variations in GnRH feedback inhibition of its own secretion.
 |
V. COMPUTER-ASSISTED SIMULATIONS OF THE MALE GONADOTROPIN-RELEASING
HORMONE-LUTEINIZING HORMONE-TESTOSTERONE AXIS |
Simulations from the above discretization of the biomathematical
formulation of the GnRH-LH-Te time-delayed feedback axis were
implemented (using Matlab). The sampling interval was 30 s, and the
simulations were extended for 48 h. The second 24 h were recorded, thus
removing any startup effects. The six circadian mechanistic linkages
were simulated, as were data from four previously published clinical
experiments (see section VB, 1-4). For each simulation,
500 realizations were accomplished. We display the first realization
along with the average of the 500 realizations superimposed in Figs.
3-6,
which depict pulsatile LH, Te, and GnRH concentrations over a 24-h
period for the six circadian mechanisms and simulations of earlier
clinical experiments 1-4. For circadian mechanism
4, the secretion rates of LH, Te, and GnRH are also displayed (Fig.
3, bottom).

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Fig. 3.
Feedback modulation of GnRH pulse firing rates and pulsatile-circadian
coupling mechanism 4. Top: 9 panels in the 3 horizontal rows
illustrate feedback modulation of GnRH pulse firing rates for 6 circadian mechanisms (1-3 in row 1 and
4-6 in row 2; see sections IVC and
VIIIC) and for feedback simulations 1-3 (row
3; see sections VB1 and VB4). Bottom:
mechanism 4, i.e., 24-h modulation of feedforward action of LH
on rate of Te secretion. Simulations of concentration levels
(left) and secretion rates (right) for LH
(top), Te (middle), and GnRH (bottom) are shown
over 24 h, discretized to every 30 s, sampled every 10 min. Each plot
consists of 1 realization with average of 500 realizations
superimposed.
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Fig. 4.
Pulsatile-circadian coupling mechanisms 1, 2,
3, and 5. A: mechanism 1, i.e.,
periodic 24-h modulation of negative-feedback actions of Te on GnRH
pulse generator firing rate. B: mechanism 2, i.e.,
similar modulation of negative-feedback effects of Te on GnRH secretion
rate (mass). C: mechanism 3, i.e., diurnal modulation
of basal secretion rate of Te. D: mechanism 5, i.e.,
nyctohemeral modulation of negative feedback of Te on rate of LH
secretion. Simulations of concentration levels for GnRH (top),
LH (middle), and Te (bottom) over 24 h, discretized to
every 30 s, sampled every 10 min are shown. Each plot consists of 1 realization with average of 500 realizations superimposed.
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Fig. 5.
Pulsatile-circadian coupling mechanism 6 and feedback
simulations of clinical experiments 1-3. A:
mechanism 6, i.e., 24-h rhythmic modulation of stimulatory
actions of GnRH on rate (mass) of LH secretion. B:
simulation 1, i.e., decreased Te negative feedback on LH and
GnRH as imposed experimentally via flutamide, an antiandrogen.
C: simulation 2, i.e., nearly complete (90%)
withdrawal of Te negative feedback achieved via ketoconazole, which
blocks testicular steroidogenesis. D: stimulation 3, i.e., nearly complete (90%) withdrawal of Te negative feedback, for 48 h, along with a total daily dose of 8 mg of Te infused continuously
over 24 h. Simulations of concentration levels for GnRH (top),
LH (middle), and Te (bottom), over 24 h, discretized to
every 30 s, sampled every 10 min are shown. Each plot consists of 1 realization with average of 500 realizations superimposed.
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Fig. 6.
Feedback simulation 4 (45-, 60-, 90-, and 120-min GnRH pulses).
In simulation 4, GnRH pulse generator is "clamped" to
simulate fixed periodic injections of GnRH every 45 min (A),
60 min (B), 90 min (C), and 120 min (D).
Simulations of concentration levels for GnRH (top), LH
(middle), and Te (bottom), over 24 h, discretized to
every 30 s, sampled every 10 min are shown. Each plot consists of 1 realization with average of 500 realizations superimposed.
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A. Implementation of Possible Circadian Rhythm Mechanisms
As illustrated in Figs. 3, 4A, and 5A, individual and
mean (of 500) realizations of pulsatile LH, Te, and GnRH output show a
range of 24-h periodic rhythms that emulate circadian variations in
these neurohormone concentrations. Figure 3, top displays
individual and mean (of 500) realizations of the (instantaneous) GnRH
pulse firing rate (no./day) for each of the six possible circadian
rhythm models and for simulations of clinical experiments
1-3 described below (see sections VB and
VIIIC); the GnRH pulse generator was "clamped" in
feedback simulation 4, and hence there is no entry for it in
Fig. 3, top.
In the first linkage mechanism considered between pulsatile and
circadian rhythms, we test the proposition that Te negative feedback on
GnRH secretory burst frequency is modulated by a circadian input, e.g.,
with greater feedback (inhibition) at night. These simulations elicited
relatively little 24-h variation in serum LH and Te concentration
profiles (Fig. 4A, top and middle). Simulations of a second mechanism of coupling a circadian oscillator (e.g., speculatively via suprachiasmatic nucleus input) to GnRH neuronal secretory burst mass/rate also yielded relatively small 24-h rhythms for all three primary components of the male axis: GnRH, LH, and Te
concentrations (Fig. 4B). In contrast, clinical observations suggest greater Te than LH circadian rhythmicity (34). Third, simulating a construct of 24-h rhythmicity of basal Te secretion (unmodulated by LH) clearly predicts (Fig. 4C, middle)
rhythmic serum Te variations over 24 h with lesser changes in GnRH and LH. Fourth, proposed circadian modulation of LH's stimulation of Te
secretion, via progressive shifting of the LH-Te dose-response curve
sensitivity across 24 h, yields strong (Fig. 3, bottom) day-night variations in LH and Te (but not GnRH) concentrations. This
also is consistent with known physiology. Fifth, 24-h modulation of
Te's feedback on LH secretory mass yields predictions with predominant
rhythms in LH greater than Te concentrations (Fig. 4D).
Finally, periodic variation over 24 h of GnRH's feedforward (stimulatory) action on LH release evokes relatively little diurnal rhythmicity in Te concentrations (Fig. 5A). Thus,
qualitatively, varying basal Te secretory rates, Te's feedback on LH
secretion, or LH's feedforward action on Te via a deterministic
periodic (24-h) input will emulate the known (human) physiology of
greater Te than LH circadian variation.
B. Simulations of Earlier Clinical Feedback Experiments
To explore further the performance of this multinodal SDE feedback
model of the GnRH-LH-Te axis, the following four computer-assisted simulations, corresponding to known prior clinical experiments (as
referenced below), were performed. The incorporation of the circadian
rhythm for these experiments was via mechanism 4, i.e., the
24-h periodic modulation of LH feedforward on Te secretion rate
(discussed in section IVC).
1. Simulating decreased Te negative feedback on LH and GnRH.
The negative-feedback actions of Te (presumptively on the masses of
GnRH and LH secreted and on the GnRH pulse generator frequency) have
been antagonized pharmacologically in clinical experiments via the
administration of a nonsteroidal antiandrogen, flutamide, which
inhibits Te's binding to the androgen receptor (17, 29). This
pharmacological treatment would in effect shift the Te
negative-feedback dose-response curves in our biomathematical construct
to the right. Thus to simulate flutamide's actions, we have introduced
an 80% shift (by way of an evident decrease) in feedback sensitivity of GnRH and LH pulse mass and GnRH pulse frequency to Te feedback inhibition (Figs. 1 and 2). As shown in Figs. 5B and 3
(top, 3rd row, 1st column), this simulation disclosed increased
serum Te concentrations, with accelerated LH secretory burst frequency and amplified LH secretory burst mass, as reported earlier in young men
(17, 29).
2. Simulating nearly complete (90%) withdrawal of Te negative
feedback.
A low and constant blood concentration of Te has been induced
clinically by administration of the drug ketoconazole, which blocks the
ability of the testis to synthesize Te (36). To achieve this state in
our SDE feedback construct, we fixed Te secretion at 20 ng · dl
1 · h
1
(vs. 1,350 ng · dl
1 · h
1 in
nominally normal conditions). The corresponding simulations correctly
predicted (Fig. 5C and Fig. 3, 3rd row, 2nd column) the
clinical observations reported earlier in young men treated with this
inhibitor (36), i.e., an increase in the frequency and mass of LH
secreted per burst.
3. Simulating nearly complete (90%) withdrawal of endogenous Te
secretion followed by continuous Te (add-back) infusions.
In six men treated earlier with the drug ketoconazole for 48 h (see
section VB2) to block Te biosynthesis and then intravenously infused with 8 mg of Te continuously over hours 24-48,
serum Te concentrations rose on average from 45 to 1,000 ng/dl over the latter interval (36). In this published clinical experiment, virtually
all the Te in the system arises from the infusion. Our feedback
simulations predicted (Fig. 5D, and Fig. 3, top, 3rd row, 3rd column) the clinically evident pulsatile LH profiles in these
young men (36). Here, we assumed that the gradual increase in LH
secretion over hours 0-24 of ketoconazole treatment and, hence, Te withdrawal (but before Te add back) tended to reduce the
negative-feedback minima for Te on LH secretory burst mass and GnRH
pulse frequency, as shown in Fig. 2 (top right and bottom right, respectively).
4. Simulating GnRH pulse generator as "clamped" at various
frequencies by fixed periodic injections of GnRH.
Simulations consisted of fixed GnRH injections every 45, 60, 90, or 120 min, and the output profiles of serum LH and Te concentrations were
recorded (Fig. 6A). Here, GnRH (exogenous) is no longer
regulated negatively by itself or by Te negative feedback. This
paradigm evinced the key previously reported features of an inverse
relationship between LH pulse frequency and burst mass and of a
plateauing rise in mean LH concentrations at higher LH pulse
frequencies (LH secretion ultimately is inhibited by rising Te negative
feedback; Fig. 6), as reported earlier independently in in vivo
experiments in the sheep and human (9, 25).
C. Sensitivity (Analysis) of GnRH-LH-Te Output to Changes in
Dose-Response Parameter Values
To ascertain the dependency of the structural dynamics of our SDE
feedback model on the parameters of the dose-response functions, we
varied their respective values in two respects (cf. Fig. 2): 1)
the maximum and minimum of the dose-response functions were simultaneously increased, then simultaneously decreased, and 2) the midpoint of the dose-response function was shifted to the left,
then to the right.
In particular, for feedback-feedforward interactions 1,
2, and 4, each of their corresponding dose-response
functions is parameterized by values A, B1,
C, and D; in the case of interactions 5 and
6, the dose-response function was parameterized by A,
B1, B2, C, and
D. The maximum and the minimum were modified by multiplying C and D first by 0.5, then by 2. In the case of
interactions 1, 2, and 4, the midpoint was
modified by multiplying B1 first by 0.5, then by 2;
in the case of interactions 5 and 6, we modified B1 by 0.5 and 2 and B2 by 0.5 and 2. The resulting modified (vs. unmodified) dose-response functions
for interactions 1, 2, and 4 are displayed in
the first two rows of Fig. 7 and those for interactions 5 and 6 in the third row. Corresponding
realizations from our model are shown in Fig.
8, i.e., the resulting concentrations of LH
and Te over time.

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Fig. 7.
Sensitivity analysis: modifications of dose-response functions.
Dose-response functions (see Fig. 1) corresponding to
feedback-feedforward interactions 1, 2, and 4 (as defined in section VIIIA2) are parameterized by values
A, B1, C, and D and
interactions 5 and 6 by A,
B1, B2, C, and
D. For each dose-response function, 2 variations were
considered: 1) maximum and minimum of dose-response functions
were simultaneously increased, then simultaneously decreased, and
2) midpoint of dose-response function was shifted to left, then
to right. Maximum and minimum were modified by multiplying C
and D first by 0.5 and then by 2. In case of dose-response
(interface) functions enumerated in section VIIIA2 for
interactions 1, 2, and 4, midpoint was modified
by multiplying B1 first by 0.5 and then by 2; in
case of interface function for interactions 5 and 6,
B1 was modified by 0.5 and 2 and
B2 by 0.5 and 2. Resulting modified
H( · ) or dose-response functions for
interactions 1, 2, and 4 are displayed in
rows 1 and 2 and those for interactions 5 and
6 in row 3. [H( · ) functions
are identified by way of relevant interfaces in GnRH-LH-Te axis in Fig.
1.] Within each display row from left to right,
dose-response plots modified by 0.5, original plot, and plot modified
by 2 are shown. For example, row 1, column 1 shows results of
modifications of GnRH firing rate as a function of Te feedback
[H1( · ) function]: original
function (dotted line), maximum and minimum multiplied by 0.5 (solid
line), and maximum and minimum multiplied by 2 (dashed line). (In
corresponding location in Fig. 8 are 2 plots, one each for realized LH
and Te concentrations; within each are a solid-line plot corresponding
to 0.5-fold and a dashed-line plot corresponding to 2-fold modification
of maximum and minimum values of Te feedback on GnRH pulse firing rate
dose response.)
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Fig. 8.
Sensitivity analysis. Simulation realizations of SDE model
corresponding to each dose-response
[H1( · ),
... , H5,6( · )] function
modification given in Fig. 7. For each simulation, resulting
concentrations of LH and Te are shown. For example, in row 1, column 1 (LH) and column 2 (Te), results of modifications
of Te feedback on GnRH firing rate
[H1( · ) function] are shown:
maximum and minimum multiplied by 0.5 (solid line) and maximum and
minimum multiplied by 2 (dashed line). Subplots correspond to those in
comparable locations in Fig. 7 (in vertical pairs of LH and Te
subplots) from left to right: row 1, H1( · ) and
H2( · ); row 2, H2( · ) and
H4( · ); row 3, H5,6( · ).
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VI. EXPLORABLE ISSUES, GIVEN A MATHEMATICALLY FORMULATED MULTINODAL
FEEDBACK-CONTROL CONSTRUCT |
Specific queries in GnRH-LH-Te axis pathophysiology may be addressed,
we believe, using a formalized (SDE) construct of neuroendocrine feedback, such as presented here. A foremost aim is to aid in the
design of experiments that are likely to help discriminate between
alternative plausible explications of pathophysiology; e.g., what
circadian coupling mechanisms will link 24-h variations in a
deterministic input to ultradian (short-term pulsatile) neurohormone release? In this regard, our six possible models of circadian control
of GnRH-LH-Te coupling suggest specific relevant experiments, such as
the evaluation of diurnal changes in pituitary responsiveness to GnRH
(14), in Leydig cell sensitivity to LH, and/or in GnRH's susceptibility to Te's negative feedback. Further relevant questions are as follows: How responsive are circadian variations in serum Te
(and LH) concentrations to small changes in and/or different deterministic (24-h) input schemes (above)? What degree of enhanced circadian regulation is achieved when two or more (rather than a
single) coupling mechanisms are implemented in concert? Which deterministic circadian linkage may be susceptible to disruption in a
(particular) pathological state? Which linkages are most stable to
variations in, e.g., feedback-feedforward signal strength or
variability in the biological signals? Which circadian-coupling mechanism(s) emulate alterations in, e.g., puberty, aging, the menstrual cycle, and menopause?
Other physiological questions become relevant given a mathematically
formulated GnRH-LH-Te feedback system. For example, how important are
feedforward vs. feedback interfaces in defining the quantifiable
regularity or orderliness of neurohormone release? Application of
approximate entropy measures (20, 21) would be one useful tool in
evaluating the following questions: How do changes in the coupling
strengths (dose-response functions), intensities, or complexity of
feedback and feedforward connections influence the orderliness of
neurohormone release? What artifacts in pulsatile secretion or entropy
estimates could be introduced by imposing strong circadian variations
on the ultradian rhythms?
Another explorable issue is the seemingly more complex and biologically
plastic multivalent feedback and feedforward control system of the
female reproductive axis. Important questions in the female axis are as
follows: How can longer-term cyclical (e.g., monthly) variations in
gonadotropin output originate (e.g., by threshold, switch, or trending
mechanisms) from short-term diurnal deterministic and ultradian
(stochastic) elements? Whereby is overall day-to-day output stability
ensured in such a formulation? How is the monthly female preovulatory
surgelike increase in gonadotropin secretion generated or disrupted
while still allowing preserved ultradian rhythms? What are the earliest
predictable changes in system performance with female reproductive
aging? How is the feedback altered plausibly in pathophysiology, e.g.,
the LH hypersecretory states of polycystic ovaries and testicular
feminization?
A well-defined and relevantly constructed neuroendocrine feedback model
of the GnRH-LH-Te axis should also eventually allow prediction of the
activity and regulation of unobserved functional nodes. For example,
computer-assisted simulations might help predict when measurements of
LH and Te release would allow accurate prediction of (unobserved) GnRH
output or GnRH-LH dose-response properties. This is significant, since
GnRH cannot be measured in hypothalamic-pituitary blood in the human.
What minimal experimental data would be most useful in making valid
predictions of unobservable dose-response functions (e.g., GnRH
feedforward on LH)? What blood sampling scheme(s) will ensure good
statistical power for defining altered GnRH feedback or feedforward
properties? Will computer-based simulations help guide the design of in
vivo animal experimentation or clinical study? For example, one might
ask, What predictions arise from the hypothesis that the GnRH pulse
generator is resistant to Te's negative feedback in patients with
inborn androgen-receptor mutations? Are these predictions consistent
with clinical pathophysiology? What experiments could be done next to
further elucidate underlying mechanisms inferred from model
simulations?
Generalization of these SDE feedback concepts to other neuroendocrine
axes, e.g., the growth hormone-releasing hormone-somatostatin-growth hormone-insulin-like growth factor I axis and the arginine
vasopressin-corticotropin-releasing hormone-ACTH-cortisol axis, is
plausible. This is practicable by modifying the core SDEs in accordance
with, e.g., physiologically relevant internodal connections, particular
dose-response functions, pertinent elimination rate constants, and
relevant time delays. Indeed, a general SDE feedback construct may also
apply on a microscopic scale to autocrine and paracrine feedback
regulatory systems at the cellular level within an organ or gland and
may possibly be relevant to modeling intracellular biochemical
regulatory pathways. Generalization will also likely help clarify other
basic issues in physiological control systems: What factors govern
normal physiological variability? What is the impact of nonmonotonic
dose-response/control functions and/or variably weighted or
stochastically varying control functions on short-term and long-term
physiological outputs? How sensitive is long-term system behavior to
initial conditions, the magnitude and nature of the stochastic
contributions, the choice and dispersion of time delays, the
introduction of queues, thresholds, switches, or long-term trends, and
the complexity of negative- and positive-feedback connections? Is
additional stochastic variability in the feedback equations (see
section VIII) required to model pathophysiology and, if so, under what conditions? Refinement of explicit SDEs and other formulations of
physiological control systems should help address these and other
issues.
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VII. DISCUSSION AND SUMMARY |
The salient and novel features of the current biomathematical
multinodal SDE feedback control formulation of the male
hypothalamic-pituitary-testicular axis include 1)
physiologically dictated regulatory nodes (GnRH-LH-Te) that are linked
relevantly, 2) internodal interfaces defined by appropriate
feedforward/agonistic (GnRH-LH and LH-Te) and feedback/antagonistic (Te-GnRH/-LH) dose-response functions, 3) time delays in
feedback and feedforward interactions among input concentrations and
GnRH, LH, and Te secretion output, with pertinent individual hormone elimination rates, 4) stochastic elements embodied at
potentially three levels, i.e., in the pulse generator waiting times,
the system of differential equations itself (see section VIII:
biological variability), and the sample observations (technical noise),
thus allowing for uncertainties in the model/construct as well as the measurements, 5) continuous functions to allow arbitrary
discretization to any coarse or fine time structure, 6)
subsidiary linkages to incorporate plausible mechanisms of coupling
between a deterministic periodic oscillator (e.g., circadian input) and
the pulsatile GnRH-LH-Te feedback system, and 7) comparison of
biomathematical simulations with several specific earlier published
clinical experiments to assess computer-assisted predictions vs.
published LH-Te responses to defined physiological perturbations. These
features should offer a more plausible, realistic, and physiologically
pertinent formulation of combined feedback and feedforward dynamic
regulation of the male hypothalamic-pituitary-testicular axis and, by
extension, other physiological neuroendocrine systems.
Our SDE biomathematical construct of the GnRH-LH-Te axis showed
stability over multiple realizations, with prolonged realizations, and
at high-intensity discretization (e.g., every 30 s). Mechanistic analyses revealed that, in principle, periodic input onto, or in
control of, each node and/or dose-response function in this feedback control system could participate in generating or maintaining a circadian pattern of LH and Te release. More realistic were models
with a periodic oscillator acting to modulate over 24 h the LH-Te
feedforward dose-response curve or Te's negative feedback effects on
LH secretory burst mass or basal Te secretion rates. For these
linkages, circadian variations emerged in serum Te (>LH > GnRH) concentrations. Diurnal variations in the efficacy or potency of Te's
feedback inhibition of GnRH pulse frequency or GnRH pulse mass provide
additional possible circadian-pulsatile linkage models. Perhaps most
plausible a priori would be circadian variations in GnRH pulse
frequency and/or mass and/or in GnRH's sensitivity to
feedback inhibition by Te (long-loop negative feedback), since a
circadian oscillator mechanism resides in the suprachiasmatic nucleus
with connectivity to the (mediobasal) hypothalamus where GnRH neurons
originate (30). However, these mechanistic paradigms did not seem to
predict strongly 24-h variations in serum Te > LH concentrations. The
clinical observation that fixed pulses of GnRH at 90-min intervals
result in relatively little day-night variation in serum LH
concentrations in prepubertal boys, but significant (30%) variations
in serum Te levels in the same individuals (13) would be consistent
with a notion of 24-h variations in basal Te release or in LH-Te
feedforward coupling. Both of these mechanisms yielded appropriate
circadian variations in Te > LH concentrations in our simulations.
Thus model simulations suggest corresponding animal experiments to help
in ultimately establishing which mechanism(s) is most appropriate.
Another circadian mechanism explored here require(s) diurnal
differences in LH secretory responses to GnRH stimulation (not
generally established in the human to our knowledge; see above). The
trivial consideration that neurohormone (GnRH, LH, or Te) elimination
rates or distribution spaces vary substantially over 24 h also is not
documented to our knowledge. Thus the present work suggests selected
relevant experiments to help clarify basic physiological principles of
(deterministic) circadian-(stochastic) pulsatile system coupling within
the male reproductive axis.
Simulations with the SDE construct embodying time-delayed
dose-responsive positive and negative feedback within the male
GnRH-LH-Te axis showed appropriate reactivity to selectively altered Te
milieus and variable Te feedback potency. The computer-assisted
simulations yielded qualitative inferences similar to those published
independently in human and animal experiments. For example, a so-called
castration response to Te withdrawal is recognized in vivo whether
after administration of an androgen-receptor antagonist (17, 29) or a
Te biosynthesis inhibitor (36). Our simulations also predicted augmented LH secretory burst frequency and mass in this context. Moreover, clamping GnRH pulse intervals at 120, 90, 60, or
45 min yielded higher but plateau levels of LH and Te concentrations, akin to in vivo responses in endogenous GnRH-deficient patients treated
at fixed but escalating GnRH injection frequencies (25). Consequently,
the coupled SDE simulator reacts appropriately to selected
perturbations in the GnRH-LH-Te feedback and/or feedforward signals.
Application of the present simulator model could forseeably identify
new and relevant hypotheses of altered GnRH-LH-Te network regulation
underlying the depressed circadian rhythmicity in serum Te
concentrations in healthy aging men (4) and the greater serial
disorderliness of the LH (and Te) release process also recognized in
older men (21). In addition, whereas LH pulse times have traditionally
been modeled as essentially a renewal process [typically inferred from
studies in normal men (5, 30)], which precludes evident feedback
structure in the pulsing mechanism, the existence of strongly negative
autocorrelation in successive LH interpulse interval values in
midluteal phase women (24) suggests feedback, which could be explored
in a suitable mathematical formulation of multivalent time-delayed
feedback activity within the female GnRH-LH-progesterone/estradiol
axis.
In summary, a multivalent, physiologically structured SDE model
embodying relevant time-delayed and dose-dependent agonistic and
antagonistic feedback and feedforward connections among principal regulatory nodes exhibits pulsatile neurohormone output that closely emulates that of the normal male GnRH-LH-Te axis. This biomathematical construct with its relevant stochastic components shows appropriate dynamic behavior after defined manipulations of selected feedback (e.g., Te) or feedforward (e.g., GnRH) signals. The SDE model also
predicts several possible specific mechanisms by which circadian input
may be coupled to a pulsatile axis. Extension of this formulation to
the female reproductive axis and to nonreproductive feedback control
systems should be possible. Moreover, its embellishment, via
complementary features (e.g., in queuing theory) and as additional knowledge emerges regarding other facets of the male axis, will likely
help stimulate new insights into the complex nonlinear dynamics
underlying highly interactive and integrated neuroendocrine axes.
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VIII. APPENDIX |
A. Elimination Rates and Feedback Interactions
1. Elimination rates.
The elimination rate
of a biological molecule from a particular
sampling space is related to its half-life (t1/2)
as follows: exp (
t1/2) = 1/2. Here,
we assume the following 1) GnRH has a nominal
t1/2 of 1
3 min (8, 25, 30); thus
G = 0.23-0.69 min
1; 2) LH
has a nominal t1/2 of 50
80 min (30,
32); thus
L = 0.0087-0.014 min
1;
3) Te has an approximate t1/2 of 15 min
(19, 30); thus
Te = 0.046 min
1.
2. Feedback interactions.
For the intact male GnRH-LH-Te axis, we allow for the following
possible interactions and nominal time-delayed intervals over which the
feedback occurs [these interactions are denoted by corresponding H( · ) functions:
H1( · ), ... , H7( · )]:
1) the blood Te concentration (ng/dl) exerts a negative
time-delayed (25-60 min) feedback on GnRH pulse firing rate (no.
of pulses/h) (22, 26, 28); 2) the blood Te concentration
(ng/dl) exerts a negative time-delayed (25-60 min) feedback on the
rate of GnRH pulse-mass accumulation (pg · ml
1 · h
1);
3) the basal Te secretion rate varies with a periodicity of 24 h; 4) circulating LH concentration (IU/l) exerts a positive time-delayed (20-30 min) feedforward action on Te secretion rate (ng · dl
1 · h
1)
(10, 19, 34); 5) the blood Te concentration (ng/dl) exerts a
negative time-delayed (25-60 min) feedback on rate of pituitary LH
mass accumulation
(IU · l
1 · h
1)
(11, 13, 27); 6) hypothalamic-pituitary portal blood GnRH concentration (pg/ml) exerts a positive time-delayed (0.5-1.5 min)
feedforward effect on the rate of LH mass accumulation
(IU · l
1 · h
1)
(1, 9, 35); and 7) the hypothalamic GnRH pulse generator because of autonegative feedback exhibits a refractory period of
~1-3 min after each pulse time (30).
The formulation of interaction 7, a refractory condition,
denoted by H7( · ), will not be via a
dose-response function; this is discussed in section VIIIB1;
also, interaction 3, allowing Te basal secretion rate to vary
with a 24-h periodicity, will not be part of the basic GnRH-LH-Te axis
but represents one possible mechanism for incorporating a circadian
rhythm (see Eqs. 16 and 20). Accordingly, for the
simplified male axis, there are four principal feedback dose-response
functions: H1( · ) for Te feedback on
the GnRH pulse firing rate (interaction 1),
H2( · ) for Te feedback on the rate
of GnRH pulse-mass accumulation (interaction 2),
H4( · ) for the rate of LH-driven Te
secretion (interaction 4), and
H5,6( · , · ) for Te
feedback and GnRH feedforward on the rate of LH pulse-mass accumulation
(interactions 5 and 6). We then use empirical
dose-response logistic functions to define how such feedback
interacts with the various axis components
The
values of A, B1, C, and D
and A, B1, B2,
C, and D used in the computer experiments were
empirically determined on the basis of presumed normative physiology in
the healthy man (2, 4, 5, 13, 14, 17, 19, 21, 25, 29-36); the
values were 2.06,
0.005, 28, and 3, for
H1( · ), 3.57,
0.008, 60, and 1 for H2( · ),
2.76, 0.8, 900, and
10 for H4( · ), and 2,
0.0074, 0.35, 5, and 1 for
H5,6( · , · ).
Sensitivity analysis (Figs. 7 and 8) was used to explore the
physiologically relevant limits and impact of varying absolute
parameter values on key measures of GnRH, LH, and Te output. The time
delays for the above jth feedback interaction will be expressed
throughout by lj,1 and
lj,2. So far, in vivo lag times are not
well established, but they have been estimated by cross-correlation
analysis or by catheterization (1, 2, 7-11, 14, 19, 21, 25, 30,
34).
We integrate various processes over time-delayed intervals.
Consequently, until time t is above the maximum time delay, the feedback will not originate over the full time-delay interval but,
rather, only from the amount of time since time 0; to
accommodate this we could use the notation s
0 to define
the maximum of s and zero. For example, the feedback of Te
concentration (in ng/dl) on the rate of GnRH pulse-mass accumulation
(in
pg · ml
1 · h
1)
would be of the form
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(12)
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However,
for simplicity we have not used this notation, but such limits
(s
0) have been analytically and computationally implemented.
B. Pulse Generator and Pulse Shape
1. Pulse generator.
Experimentally, the pulse times of LH closely mimic those of GnRH but
are slightly time delayed (1, 8, 13, 14, 25, 27, 35). Hence, we have
defined the male GnRH-LH-Te axis as having one principal pulse
generator, i.e., with output (hypothalamic) in the form of GnRH, with
pulse times T0, T1,
T2, ... . Although in young men there are
often ~15-30 GnRH pulses in a 24-h period (30), in the present
new model, unlike previous constructs (6, 16), the time structure of
the pulses (and their amplitudes) is dynamic, being determined by
time-delayed feedback from Te and GnRH itself. The latter feedback
introduces a brief refractory state.
To formulate the refractory condition of interaction 7 mathematically, let N(t) denote the number of
pulses up to and including time t, for then
TN(t) is the time of the last pulse. Let
be a small positive value. The time delays for
interaction 7 are l7,1 = 0 and
l7,2 = 1 min. We can define the refractory condition as H7(t)
Thus,
at a time t during the interval
(Tj, Tj + l
7,2) the value of H7(t)
will be the small value
; at other times the value of
H7(t) will be 1. (In the simulations of
section V,
was taken to be 0.)
In the present model the evolving concentrations of Te and GnRH exert
feedback on the GnRH pulse generator through a pulse intensity
function. For a stationary or nonstationary Poisson process, the
intensity function
is a fixed deterministic function; it is not
using any feedback. In addition to including feedback, one would like
to be able to control (in a probabilistic sense) the pattern of
interpulse lengths. The Weibull distribution has an additional
parameter for that purpose, call it
, and as
increases it forces
a more regular pattern (the Poisson is the special case
= 1). In
our formulation the function
is now a stochastic process (because
of the feedback), not a deterministic function. The present formulation
of the stochastic pulse generator allows for a modulation of the
instantaneous rate of pulsing by time-delayed feedback, in addition to
control over the regularity of interpulse length. Just as the
deterministic intensity function in the nonstationary Poisson can be
viewed as a deterministic time transformation of the stationary
Poisson, our model can be viewed as a stochastic time transformation of
a Weibull renewal process. In the simulations (section V) the value of
was taken to be 2, chosen to produce more regularity than the
Poisson but still allowing a fair amount of flexibility.
More precisely, there is a "pulse generator" intensity
(t), for which
(t)dt describes the
probability of firing in the infinitesimal time increment
(t, t + dt)
|
(13)
|
In
Fig. 3 (1st row, 1st column), we show the feedback function
H1( · ) of Te on frequency output of
the GnRH pulse generator. The conditional density for
Tk given
Tk
1, Tk
2, ... ,
T0 and
( · ) will then be required
to satisfy
|
(14)
|
We allow the probabilistic structure of the pulse times to
depend on time-delayed feedback and flexibility in the variability of
the interpulse lengths when the mean is (roughly) known. As a
consequence of the time delays, the Tk values
cannot satisfy a Markov property (at least not in a finite-dimensional
sense). However, conditional on the pulse intensity
( · ), the Tk values will, in fact,
be Markov; hence, our pulse generator construct incorporates
time-delayed feedback while at the same time allowing for relative
simplicity. Also, the parameter
1 allows for flexibility in the
variability of the interpulse lengths when the mean is (roughly) known.
It was the inflexibility of the stationary and nonstationary Poisson
processes that suggested the above generalization. If
( · ) were a deterministic function and, hence, feedback
is absent, then the pulse generator reduces to a Markov process, including the stationary and nonstationary Poisson processes as special
cases.
2. Pulse shape.
The functions
G( · ) and
L( · ), resulting from an application of
Eq. 3 to GnRH and LH, respectively, are the rates of secretion given as mass of hormone released per unit of time and distribution volume. We have used a generalized
family of densities (i.e., normalized to integrate to 1) to define the pulses and accommodate a
spectrum of varying asymmetrical shapes
|
(15)
|
where
1 > 1,
2 > 0, and
3 > 0 are parameters that model the secretory burst shape. The
and
the Weibull families are included in this construction. Appropriate
choices of
values allow on the one extreme a nearly Gaussian
(symmetrical) secretory burst and, on the other extreme, a host of
variably rightward-skewed representations of secretory bursts. Such
asymmetry of secretion events is sometimes evident in in vitro and in
vivo experiments (1, 8, 18). The maximum (rate of secretion) of
( · ) occurs at
(r) =
2(
1
1/
3)1/
3.
The swiftness of the upstroke is controlled by
1 and
3, which thus provides some measure of the amount of
immediately releasable granule-contained GnRH or LH (30). The inclusion
of the parameter
3 allows for variably peaked events
that are not easily accommodated by
1 and
2 alone. The rate of decline in secretory rate after the
maximum (e.g., when hormone-containing granules are progressively depleted) is controlled by
2 and
3.
Figure 3 (2nd row, 3rd column) displays the
G( · ) and
L( · ) used in the simulations of
section V.
C. Incorporating a Circadian Rhythm
Given the above basic construct, we can consider the inclusion of a
circadian rhythm acting deterministically via several possible
mechanisms. The effects of the 24-h rhythm are observed primarily in
the cyclical pattern of serum Te concentrations over the day and, to a
lesser extent, in LH. An unresolved physiological issue is how the
circadian rhythm might interface with various pulsatile and feedback
loci within the GnRH-LH-Te axis. A general form of any (24-h) rhythmic
function, composed of m harmonics, is simply
|
(16)
|
where
appropriate values of the amplitudes
B0, ... , Bm and
the (possible) phases
1, ... ,
m will ensure the positivity of
( · ); in the simulations of section V, one harmonic
(m = 1) was used, with B0 = 0.85, B1 = 0.15, and
1 chosen so that the
maximum of
( · ) occurred at an appropriate time (e.g.,
4 AM).
Within the foregoing structure as developed for the GnRH-LH-Te axis,
six theoretically possible mechanisms for the circadian rhythms in Te
and LH are 1) circadian modulation of the negative-feedback actions of Te on the GnRH pulse generator firing rate, whereby
|
(17)
|
is
replaced by
|
(18)
|
2)
periodic modulation of the negative-feedback effects of Te on the GnRH
secretion rate (mass), such
that
|
(19)
|
3) diurnal modulation of a basal secretion rate of
Te itself,
|
(20)
|
4)
24-h modulation of the feedforward action of LH on the rate of
secretion of Te,
where
|
(21)
|
5) nyctohemeral modulation of the negative feedback
of Te on the rate of secretion of LH, whereby
|
(22)
|
is
replaced by
|
(23)
|
and/or
6) 24-h rhythmic modulation of the stimulatory actions of GnRH
on the rate (mass) of secretion of LH, in which
|
(24)
|
is
replaced by
|
(25)
|
Practically,
the impact of
(t) on loci 1-6 above is
accomplished by shifting the sensitivity of the corresponding
dose-response curve, i.e., by smoothly varying B in the
logistic function (Eq. 4). Not included is a seventh
formulation of GnRH autonegative feedback with circadian variation.
D. Time Discretization of the Male GnRH-LH-Te Axis for the Purpose
of Simulation
The discretization of the above system of SDEs, by use of an Euler
scheme, results in the SDEs that follow. Let
t = ti + 1
ti be the scale of the discretization
(e.g.,
t = 30 s); this is ordinarily smaller than the
actual sampling increment (e.g., sampling every 10 min). Here, we
take the sampling increment to be the same as the scale of
discretization,
t = 30 s. A sequence of IID uniform
(0, 1) random variables is created
(Uk, k
1), which will be used
in the construction of the pulse times.
Having constructed the processes up to time
tk with
Tj
1 being the last pulse time,
we construct Tj by solving the
(discretization of the integral) equation, with
Tj being the first
tk satisfying
because
the resulting conditional density of Tj
being
p[ · |Tj
1,
( · )], where
Having
constructed Tj, one can then calculate the
jth pulse masses
In
the above, the circadian mechanism is mechanism 4 (24-h
modulation of LH feedforward on the Te secretion rate).
Finally, what we observe is the above with "experimental" errors
due to sample collecting, processing, and assaying
|
(26)
|
where
the
i values are IID normal mean zero, and the
variances are such that the coefficient of variation for
YGk, YLk, and
YTek is 6%.
E. Stochastic Elements
The hypothalamic-pituitary-Leydig cell axis is a highly coupled system
with variability within its dynamics. Such variability arises not only
via internodal interactions but also at the levels of 1) the
stochastic GnRH pulse generator, which is further perturbed by
feedback, and 2) and multicellular hormone (GnRH, LH, or Te) secretion and subsequent mixing within the blood. In addition, random
experimental variations arise in the course of sample withdrawal, processing, and analytic assay. The motivation of our extension from
the deterministic differential equations (Eq. 1) to the SDEs (Eqs. 6-10) is to allow for precisely such aggregate
within-system variability in a continuous time formulation. Using the
probabilistic concept of Brownian motion, we can formulate the
component of biological variability resulting from instantaneous
secretion arising from nonuniformly secreting cells that are variously
arrayed about capillaries and the subsequent mixing of secreted
neurohormone molecules within the turbulent bloodstream (30); the use
of Brownian motion to describe such variation in diffusive behavior of
fluids (e.g., mixing and turbulence) can be mathematically justified
because of the cumulative effects of a large number of interactions
(e.g., cellular and molecular), each occurring on a very small time
scale. The biological variables, such as molecular admixture in the
bloodstream, intra- and intercellular nonuniformities in hormone
synthesis and secretion rate, local intraglandular variations in
microvascular perfusion nonuniform cellular exposure to paracrine and
autocrine regulatory molecules (distinct from GnRH, LH, and Te), and
unequal secretory cell energy stores (30), can be recognized in
aggregate by inclusion of an additional stochastic (Brownian)
contribution in the SDEs (see below).
The above additional stochastic elements in the feedback system per se
can be incorporated into the core equations (Eqs. 6-10) by
replacing Eq. 10 with
|
(27)
|
The SDEs above correspond to the core construct (Eq. 1), now having incorporated the feedback-feedforward relationships
and stochastic (biological) variability. Thus the true in vivo
concentration processes, XG(t),
XL(t), and
XTe(t) (t
0), are the
resulting solutions of the above system of equations. The exact
magnitude of this biological feedback system variation is not known but might vary in health and disease.
F. Mathematical Basis for SDE Feedback Time-Delay Construct of the
GnRH-LH-Te Axis
The mathematical basis for the above formulation (Eqs. 6,
9, and 27), as well as that of Eqs. 6-10,
is currently being studied. There is a proper probabilistic framework
in which such a system, with all the imposed time-delayed,
dose-responsive feedback interactions, stochastic elements, etc., is
achievable, with the realizations from the resulting processes,
XG(t),
XL(t), and
XTe(t) (t
0), being
continuous real functions. Such mathematical verification shows that
the various structures (e.g., feedback and pulsing), which are
"local" in nature, produce the proper long-term stable behavior.
The present model, in which time-delayed feedback from the
concentration processes, and not just the history of the prior pulse
times themselves, modulates the pulse generator, is novel to stochastic
modeling in general. For general references on point processes and
SDEs, see Refs. 3 and 23.
 |
ACKNOWLEDGEMENTS |
D. M. Keenan was supported by Office of Naval Research Contract
00014-90-J-1007. J. D. Veldhuis was supported in part by the National
Science Foundation Center for Biological Timing, the National
Institutes of Health General Clinical Research Center, NIH Research
Career Development Award 1K04-HD-00634 and P-30 Reproduction Research
Center Grant HD-28934 from the National Institute for Child Health and
Human Development.
 |
FOOTNOTES |
Address for reprint requests: J. D. Veldhuis, Div. of Endocrinology,
Dept. of Internal Medicine, Box 202, University of Virginia Health
Sciences Center, Charlottesville, VA 22908.
Received 11 March 1997; accepted in final form 18 March 1998.
 |
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