A stochastic differential equation model of diurnal cortisol
patterns
Emery N.
Brown1,
Patricia M.
Meehan1, and
Arthur P.
Dempster2
1 Neuroscience Statistics Research Laboratory, Department of
Anesthesia and Critical Care, Massachusetts General Hospital,
Division of Health Sciences and Technology, Harvard Medical
School/Massachusetts Institute of Technology, Boston 02114; and
2 Department of Statistics, Harvard University,
Cambridge, Massachusetts 02138
 |
ABSTRACT |
Circadian modulation of episodic bursts is recognized as the
normal physiological pattern of diurnal variation in plasma cortisol
levels. The primary physiological factors underlying these diurnal
patterns are the ultradian timing of secretory events, circadian
modulation of the amplitude of secretory events, infusion of the
hormone from the adrenal gland into the plasma, and clearance of the
hormone from the plasma by the liver. Each measured plasma cortisol
level has an error arising from the cortisol immunoassay. We
demonstrate that all of these three physiological principles can be
succinctly summarized in a single stochastic differential equation plus
measurement error model and show that physiologically consistent ranges
of the model parameters can be determined from published reports. We
summarize the model parameters in terms of the multivariate Gaussian
probability density and establish the plausibility of the model with a
series of simulation studies. Our framework makes possible a
sensitivity analysis in which all model parameters are allowed to vary
simultaneously. The model offers an approach for simultaneously
representing cortisol's ultradian, circadian, and kinetic properties.
Our modeling paradigm provides a framework for simulation studies and
data analysis that should be readily adaptable to the analysis of other
endocrine hormone systems.
circadian rhythms; deconvolution; kinetics; sensitivity analysis; ultradian rhythms
 |
INTRODUCTION |
CORTISOL IS A STEROID
HORMONE that in humans is primarily responsible for regulating
metabolism and the body's response to inflammation and stress. The
24-h plasma profile of cortisol is comprised of episodic release of
15-21 secretory events whose magnitudes vary in a regular diurnal
pattern (30, 52, 53, 56, 57). Plasma levels of the hormone
are lowest from 2000 to 0200, climb rapidly through the late night and
predawn hours, reach a maximum between 0800 and 1000, and decline
throughout the course of the day into the evening (56,
57). Findings in human studies suggest that the circadian
pacemaker governs a significant component of the diurnal variation in
plasma cortisol levels (13, 53). For this reason, cortisol
is often used along with core temperature and plasma melatonin levels
to study the properties of the human circadian system (5, 6, 13,
14).
An accurate description of cortisol's diurnal patterns requires a
representation that models the relation between the circadian and
noncircadian components underlying the diurnal variation. Biochemical
and physiological evidence from human investigations suggests that
diurnal variation in plasma cortisol levels is governed primarily by
the following three factors: 1) ultradian timing of the
cortisol secretory episodes (24, 30, 52, 53, 56, 57);
2) circadian control of cortisol secretory amplitudes
(24, 52, 53); and 3) the kinetics of cortisol
synthesis in the adrenal glands and infusion into (20, 22, 24,
53) and clearance from the plasma by the liver. There is also
measurement error variation as a result of the cortisol immunoassay
(30, 53).
Cortisol secretion is under the control of the
hypothalamic-pituitary-adrenal axis (Fig.
1). In the hypothalamus, communication between the suprachiasmatic nuclei, the site of the circadian pacemaker, and the paraventricular nuclei, the site of
corticotropin-releasing hormone (CRH), occurs most probably by direct
neural connections (40). CRH, secreted from the
paraventricular nuclei into the hypophyseal portal blood vessels, is
the principal hypothalamic factor responsible for inducing release of
ACTH from the anterior pituitary (41, 49). The frequency
of cortisol secretory events by the adrenal gland is tightly coupled to
the episodic release of ACTH from the anterior pituitary
(30). The amount of cortisol released in each secretory
episode appears to be regulated primarily by changes in the amplitude
rather than the frequency of the secretory episodes (53).
This amplitude modulation is believed to be controlled by the circadian
pacemaker through modulation of ACTH release (17, 52).

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Fig. 1.
Compartment representation of the 2-dimensional linear
stochastic differential equation of plasma cortisol levels. Arrows,
direction in which neural and humoral signals are propagated through
the hypothalamic-pituitary-adrenal (HPA) axis; +, excitatory
interactions; , inhibitory interactions. Direct neural connections
between the suprachiasmatic nucleus (SCN), site of the circadian
pacemaker, and the paraventricular nucleus (PVN) are most likely
responsible for circadian modulation of PVN release of
corticotropin-releasing hormone (CRH). CRH stimulates anterior
pituitary release of ACTH, which induces the adrenal gland to
synthesize and release cortisol. The adrenal compartment (Eq. 7) includes all of the HPA axis elements within the dotted line.
The plasma compartment (Eq. 8) is where the diurnal cortisol
rhythm is observed. The rate of cortisol movement from the adrenal
gland to the plasma compartment is governed by the infusion rate
constant ( I), whereas the clearance rate constant
( C) governs cortisol removal from the plasma compartment
by the liver. Cortisol exerts a negative feedback effect at the levels
of the hypothalamus and the anterior pituitary. This negative feedback
is not part of our current model. See Glossary for other
definitions.
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Because no cortisol is stored in the adrenal gland, initiation
of cortisol secretory episodes by ACTH is due to induction of de novo
cortisol synthesis from cholesterol by the G protein-coupled receptor-mediated increase in cholesterol desmolase activity and transcription of genes encoding the enzymes required to synthesize cortisol (57). After synthesis, cortisol diffuses into the
circulation where ~3-10% of the hormone is free and the
remainder is transported bound to either albumin or cortisol-binding
globulin (57). Cortisol is absorbed from the plasma in
various tissues where it executes its regulatory functions as a steroid
hormone. In the hypothalamus and the anterior pituitary, cortisol
inhibits, respectively, release of CRH and ACTH by negative feedback
(Fig. 1) (57). The hormone is cleared by the liver through
reduction of the A ring in the steroid backbone followed by conjugation
with glucuronic acid to form several water-soluble compounds that are
excreted in the urine (20). The kinetics of cortisol
infusion into and clearance from the plasma have been empirically
classified as first order (22, 53). The biochemistry of
cortisol suggests that a minimum of two compartments must be considered
to represent its diurnal pattern (Fig. 1).
Reported concentrations of plasma cortisol depend critically on the
reliability of the immunoassay used to measure it. The minimal
detectable concentration of the cortisol immunoassays has been reported
as 0.5 µg/dl with both intra- and interassay percent coefficients of
variation ranging from 5 to 10% (8, 30, 53). The
coefficients of variation change appreciably with the number of
replicates assayed per sample. Therefore, the immunoassay error
represents an important source of measurement variation and must be
included explicitly in a model of the hormone's diurnal variation
(5, 7).
A complete mathematical description of diurnal cortisol variation
should include all of the neural and humoral feedforward and feedback
linkages between the hypothalamus, anterior pituitary, and the adrenal
gland (40), as well as the effects of exogenous factors
such as sleep state, stress, and meals (28, 46).
Simultaneous measurement of the variables necessary to specify
correctly all components of such a model is not possible in humans and
is only possible to a limited extent in other species. Instead of
attempting to specify all of these relations, an alternative approach
is to use known physiology to define a minimal model of the features necessary to describe the observed diurnal patterns. If the
model is sufficiently parsimonious, then the parameters can be
estimated from the experimental data. If the minimal model can be
estimated for individual subjects, then this information can be used to define individual and population differences in normal and diseased conditions. The minimal model may also help define and set limits on
some of the structures in the more comprehensive model. Specifying a
minimal model of cortisol's diurnal variation is the approach we take
in this report.
Current data analysis methods and mathematical models of cortisol have
described some subsets of the three physiological factors underlying
cortisol's diurnal variation. None characterizes the relation among
the essential components in a single equation system. These models are
primarily deterministic in structure and do not take account of the
stochastic features of the hormone's diurnal pattern, which cannot be
attributed to measurement error from the immunoassay. We formulate a
minimal stochastic differential equation model based on these accepted
physiological properties of cortisol and use it to describe diurnal
variation in the hormone's plasma levels. We determine the model
parameters from previous studies of the hormone and show that the joint
physiological range assumed by these parameters may be succinctly
summarized by a multivariate Gaussian probability density. We establish
the plausibility of the model in a series of simulations and describe
the relation between our model and current quantitative methods used to
study diurnal cortisol patterns.
Glossary
µ(t) |
Average circadian amplitude function at time t
|
Aj |
Total amount of cortisol in a secretory event initiated at time
uj (µg/dl)
|
A |
Vector of secretory event amplitudes [A1,
A2,...,
AN(T)]T
|
cr |
Coefficient of the rth cosine harmonic of the mean circadian
amplitude function
|
dr |
Coefficient of the rth sine harmonic of the mean circadian
amplitude function
|
H1(t) |
Cortisol concentration in the adrenal glands at time t
|
H2(t) |
Cortisol concentration in the plasma space at time t
|
N(t) |
Number of secretory events occurring in an interval (0,t]
|
uj |
Time of the jth secretory event
|
u |
Vector of secretory event times
[u1,u2,...,uN(t)]
|
wk |
Waiting until the kth secretory event given the time of the
k 1st secretory event
|
yt |
Cortisol concentration measured in the plasma at time t
(µg/dl)
|
C |
Rate constant for clearance of cortisol from the plasma
(min 1)
|
I |
Rate constant for infusion of cortisol from the adrenal glands into the
plasma (min 1)
|
A |
Circadian amplitude function coefficient of variation
|
1 |
Location parameter of the gamma waiting time probability density
|
2 |
Scale parameter of the gamma waiting time probability density
|
µw |
Mean waiting time between secretory events
|
2µ(t) |
Variance of the circadian amplitude function at time t
|
2w |
Variance of the intersecretory event times
|
 |
Covariance matrix of the cortisol infusion and clearance rate constants
|
 |
Covariance matrix of the mean circadian function parameters
|
 |
Covariance matrix of intersecretory event waiting time parameters
|
|
Vector of model parameters:
= [ I, c, 1, 2,c0,c1,d1,c2,d2]
|
|
Vector of the mean circadian function parameters: = [c0,c1,d1,c2,d2]
|
 |
METHODS |
Model formulation.
Given an observation interval (0,T], we define at time
t, H1(t), the concentration of
cortisol in the adrenal gland, and H2(t) the
concentration of cortisol in the plasma. We let
N(t) be the counting process that defines the
number of secretory events occurring in an interval (0,t)
for t
(0,T]. We let
uj, j = 1, ... , N(T) be the times in (0,T] at which
secretory episodes are initiated, and we let wk,
k = 1, ... , N(T) be the
waiting times between secretory events. It follows from the definitions
of uj and N(t) that
|
(1)
|
|
(2)
|
where dN(u) indicates a change in the
counting process at u. Heuristically speaking,
dN(u) is 1 if there is a secretory event at
u, and 0 otherwise. We assume the value for
wk is a renewal process defined as independent,
identically distributed gamma random variables whose probability
density has location and shape parameters
1 and
2, respectively (23). We use the gamma
probability density because it is a two-parameter model, which offers a
flexible description of both the mean and variance of the
intersecretory event interval. The expected value and variance
of wk are respectively
|
(3)
|
|
(4)
|
Let Aj, j = 1, ...,
N(T) be the amplitude or total amount of hormone
contained in the secretory event initiated at time
uj. We assume that the magnitude of the
Aj values is a random variable modulated in a
time-dependent manner by the circadian system. To represent the
stochastic nature of this modulation, we define the two-harmonic mean
circadian amplitude function µ(t) and variance function 
as
|
(5)
|
|
(6)
|
where
A > 0 is a coefficient of variation.
We assume that each At is a Gaussian random
variable with mean µ(t) and variance 
.
Let
I be the infusion constant governing the rate
at which cortisol enters the plasma from the adrenal gland and
C be the clearance parameter describing the rate at
which cortisol is cleared from the plasma by the liver. We assume the
kinetics of the infusion and clearance processes to be first order
(Fig. 1).
The rate of change in the concentration of adrenal cortisol is equal to
the rate of synthesis minus the rate of infusion from the adrenal gland
into the plasma. Similarly, the rate of change in the concentration of
plasma cortisol is equal to the rate of infusion of cortisol into the
plasma from the adrenal minus the rate of its clearance from the plasma
by the liver. Under the assumption of first-order kinetics for cortisol
infusion and clearance, the rates of change in cortisol concentration
in the adrenal gland and plasma may be written as
|
(7)
|
|
(8)
|
The solution to this equation system on the interval
(t0,t] is
|
(9)
|
where H(t) = [H1(t),H2(t)]',
G(t) = [At,0]'
and
|
(10)
|
where
=
I/(
I
C).
To complete the model description, we assume that during a time
interval of length T we collect n blood samples,
which are assayed for cortisol. We let
yt1, ... , ytn denote the
concentration of cortisol at times 0 < t1 < t2 <...<
tn
1 < tn
T and assume the
yti satisfy the equation
|
(11)
|
where
ti are independent, zero
mean Gaussian random variables with variance
2
ti defined
primarily by the immunoassay measurement error.
Taken together, Eqs. 9 and 11 describe a
two-dimensional state space model; the former is the state equation and
the latter is the observation equation. The state vector of this model
is H(t), and its components are the
concentrations of cortisol in the adrenal gland and in the plasma.
Equations 7 and 8 define how these
variables change over time. It is a stochastic differential equation
system because its forcing term
AtdN(t) is a random amount of cortisol input at a random time t. The solution to
Eqs. 7 and 8 is the stochastic convolution
integral in Eq. 9. This model also belongs to a class of
stochastic processes known as filtered point processes
(45) in which the counting process is defined implicitly
by the gamma probability model we assumed for the intersecretory event
times. The mark process, At, is a Gaussian
random variable whose mean (Eq. 5) and variance (Eq. 6) are periodic functions of time.
Characterizing the probability density of the model parameters.
We develop a two-step approach to simulating the model in Eqs.
1-11. We represent the joint probability density of the
observed plasma cortisol levels, the secretory amplitudes, the
secretory event times as
|
(12)
|
where y = [yt1,
... , ytn],
A = [A1, ... , AN(T)],
u = [u1, ... , uN(T)], and
= [
I,
C,
1,
2,c0,c1,d1,c2,d2]; f(y|A,u,
)
is the multivariate Gaussian probability density defined in Eq. 11; f(A|u,
) is the joint
probability density of the secretory event amplitudes whose mean and
variance follow implicitly from Eqs. 5 and 6; and
f(u|
) is the joint
probability density of the secretory event times defined implicitly by
the gamma probability density for the intersecretory event times given
in Eqs. 1-4.
If we assume that different individuals have different values of
,
then simulation of plasma cortisol data from Eq. 12 with
fixed is equivalent to studying variation in the hormone's plasma levels for an individual. Therefore, we consider the expanded model
|
(13)
|
where f(
) denotes the probability density
that summarizes the between-individual variation in
. We simulate
our cortisol model by first drawing
from f(
)
and then simulating y, A, and u given
from
f(y,A,u|
).
This two-step approach is equivalent to simulating between- and
within-individual variation in plasma cortisol patterns. The
between-individual variation is due to interindividual differences in
ultradian, circadian, and kinetic properties of cortisol and is
summarized by f(
). The
within-individual variation is summarized by
f(y,A,u|
).
A standard approach for assessing the sensitivity of a model to the
choice of parameters is to compare the results of simulations from the
model as the parameters are changed one at a time. Along with
characterizing between-subject variation in plasma cortisol levels,
this two-step simulation approach suggested by Eq. 13 allows
all model parameters to be changed simultaneously.
We assume that f(
) is a multivariate Gaussian
probability density, which has the form
|
(14)
|
where
= [
1,
2],
= [c0,c1,d1,c2,d2],
and
= [
I,
C]. The probability
densities f(
), f(
), and
f(
) summarize, respectively, the uncertainty in
the ultradian, the circadian, and the kinetic parameters. We make the
Gaussian assumption because a Gaussian probability density is
completely defined by its mean vector and covariance matrix. We will
derive its mean vector and covariance matrix from published reports
that have quantitatively modeled diurnal cortisol patterns. We limited
our analysis to those investigations in which subjects were monitored
for at least 24 h and intersample intervals were 20 min or less.
We excluded the study of Mortola and colleagues (35),
which analyzed diurnal cortisol patterns in healthy female subjects,
because they did not report the phase of the menstrual cycle
during which the subjects were studied. It is now appreciated that the
character of women's circadian rhythms is different at different
phases of the menstrual cycle (42). We summarize below the
analysis used to develop each of these three components of
f(
).
Ultradian parameter probability density: f(
).
Linkowski and colleagues (30) reported the number of
cortisol secretory events in 24 h for each of seven subjects. We
computed for each subject the average intersecretory event interval by dividing 24 h by the number of secretory events. Veldhuis and colleagues (53) reported the individual average and SD of
the intersecretory event intervals for six subjects. Assuming equal weighting of each of these 13 subjects, we computed
µw and 
, the
estimated between-subject mean and variance, respectively. On the basis
of the assumption that the intersecretory event interval process is
described by a gamma probability density, the method of moments
estimate of the mean values of
1 and
2
can be computed as (23)
|
(15)
|
We derive the covariance matrix for
1 and
2 from the data presented by Veldhuis and colleagues,
since they provided estimates of µw and

for six individual subjects
(53). We used Eq. 15 to convert each subject's
estimate of µw and

into estimates of
1 and
2. Using the six pairs of
1 and
2 along with estimates of the means of
1
and
2 obtained by combining the data from Refs.
30 and 53, we computed the sample covariance
matrix using the standard formula (1). The estimates of
the mean vector and covariance matrix are
|
(16)
|
Hence, we take f(
) to be the bivariate
Gaussian probability density whose mean vector and covariance matrix
are given in Eq. 16.
Kinetic parameter probability density: f(
).
Hellman et al. (22), Weitzman et al. (56),
Linkowski et al. (30), and Veldhuis et al.
(53) reported individual clearance half-lives for two,
seven, seven, and six subjects, respectively. Jusko et al.
(24) reported the average half-life and its SD for seven
subjects. We combined the 22 subjects from the studies in Refs.
22, 57, 30, and 53
to compute the mean and SD of the clearance half-life. We converted
these mean and SD estimates into estimates of the mean and variance of
C using the delta method formula (38)
|
(17)
|
|
(18)
|
where
and s
are the
mean and variance of the half-life estimates, respectively. We also
converted the estimates of the average and SD of the half-life from the
seven subjects in Ref. 23 to mean and variance estimates
of
C using Eq. 16. We combined the estimates
of the mean of
C and the variance of
C
from the 22 subjects with the ones computed from the 7 subjects by
taking a weighted average. This yielded mean and variance estimates for
C of 0.645 min
1 and 0.015 min
2.
The analyses of Veldhuis et al. (53) provide information
on
I and its variability. These authors reported an
average infusion half-life of 16 min with a SD of 0.61 min in six
subjects. Using the delta method formula in Eqs. 17 and 18, we computed the mean and variance of
I to
be 2.71 min
1 and 0.074 min
2, respectively.
Although the mean estimate seemed reasonable, we suspected that this
variance estimate may understate the uncertainty in
I
because it was based on only six subjects. Therefore, because the
sample variance estimate is approximately distributed as a
2 random variable with five degrees of freedom, and the
99th percentile of this probability distribution is 0.669, we used this
value as the estimate of the variance of
I. Because we
identified no reports that jointly estimated
C and
I, we assumed the covariance between
C
and
I was zero and represented the joint probability density of these two parameters as the bivariate Gaussian density whose
mean vector and covariance matrix are, respectively
|
(19)
|
Circadian parameter probability density: f(
).
We estimated the probability density for the circadian parameters by
the analysis of plasma cortisol data collected from four healthy male
subjects studied on 24-40 h constant routines (13). We fit an approximate form of the model in Eqs. 9 and 11 to
each subject's plasma cortisol data by Bayesian Markov chain Monte Carlo methods (8, 9, 32). Our implementation of this
algorithm provided an approximate Gaussian probability density for the
circadian parameters for each subject. The average of the mean vectors
and covariance matrices across the four subjects was
|
(20)
|
where the lower triangle is omitted because of symmetry.
Therefore, we took f(
) to be the Gaussian
probability density whose mean vector and covariance matrix are defined
in Eq. 20. The value of
A, the
coefficient of variation, could not be determined from previous
studies. Therefore, we simulate our model for the following three
values of
A: 0.1, 0.3, and 0.5.
Joint probability density of the model parameters: f(
).
Because no single study provided information about all of the
parameters in our model, the covariances between the kinetic, ultradian, and circadian parameters could not be estimated. Therefore, we assumed these covariances to be zero. Combining Eqs. 16,
19, and 20, it follows that the joint probability
density of the parameters is the multivariate Gaussian density whose
mean and covariance matrix are
|
(21)
|
Immunoassay error.
As stated in the INTRODUCTION the intra-assay coefficients
of variation for the cortisol immunoassay measurement error range between 0.05 and 0.10 when multiple replicates of the blood sample are
assayed and may be as high as 0.15 when a single replicate is assayed
(7, 8). In either case, the probability density of the
immunoassay error can be reasonably well approximated by a Gaussian
density (5, 7). Under the assumption that each blood
specimen is assayed in duplicate, we take the coefficient of variation
of the errors in Eq. 11 to be 0.07.
Simulation algorithm of diurnal cortisol patterns.
Given measurement times 0 < t1 < t2 <...< tn
1 < tn
T, the simulation algorithm
for the cortisol model defined in Eqs. 1-9 proceeds
through the following seven steps
|
(22)
|
All simulations were carried out using the Splus programming language.
 |
RESULTS |
Figure 2 gives a graphic
illustration of the steps in the simulation algorithm given in
Eq. 22. In this simulation and all subsequent ones, we
assume that the initial cortisol measurement is made at midnight and
therefore that the initial concentrations of cortisol in the adrenal
gland and the plasma compartments are zero. An illustration of the
secretory event times and amounts of cortisol produced in the adrenal
compartment as generated by steps 1-4 of the
algorithm are shown in Fig. 2A. The circadian modulation of
the secretory process is shown in the variation of the amplitudes of
the secretory events. Ultradian variation in secretory event timing is
less pronounced because the values of the mean and variance of the
intersecretory event times in this example are 72 and 9 min,
respectively.

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Fig. 2.
Graphic summary of a 24-h realization of the cortisol simulation
algorithm in Eq. 22 for a given value of . A:
times and amounts of cortisol produced in the adrenal compartment in
the 24-h period (steps 2-4 of the algorithm).
B: time course of the impulse response function of a
secretory event initiated at time 0 as defined by Eq. 9. C: evaluation of Eq. 9 to obtain the true
plasma cortisol levels for the secretory episodes in A
(step 5). D: observed plasma cortisol level is
composed of the true plasma cortisol in C plus the error due
to the cortisol immunoassay (steps 6 and
7).
|
|
Step 5 of the algorithm integrates the stochastic
differential model in Eqs. 7 and 8 to obtain the
solution in Eq. 9 (Fig. 2C). The impulse response
function for this system, as shown in Fig. 2B, is derived
directly from Eq. 9. That is, for a given secretory event,
i.e., a uk and an Ak, the
infusion and clearance parameters govern the fraction of
Ak that appears in the plasma at a given time
after initiation of the secretory event. Within ~30 to 40 min is when
the largest fraction of the Ak enters the plasma. By 6 h, all of the cortisol in this secretory event has reached the plasma. In step 6 of the algorithm, Gaussian
noise based on the properties of the cortisol immunoassay is added to the true plasma cortisol levels. The coefficient of variation is set at
0.07, making the variance proportional to the square of the true plasma
cortisol level. Finally, in step 7, the immunoassay noise is
added to the simulated true plasma level (Fig. 2D). The difference between C and D in Fig. 2, shows the
effect of the immunoassay error on the measured plasma cortisol levels.
To simulate the cortisol model, we chose the observation interval
T to be 48 h, the time between measurements to be 10 min, and
A to be 0.1 (Fig.
3), 0.3 (Fig.
4), and 0.5 (Fig.
5). As stated above, all simulations
start at midnight when both the initial adrenal and plasma cortisol
concentrations are assumed to be zero. [Initial condition estimates
for times other than midnight can be obtained by beginning at midnight
(zero initial condition) and simulating the model over several days.
The value of the cortisol series taken at the time of interest several
days into the simulation should be a reasonable initial guess.] We drew different valves of
from f(
), and for
each value of
A, we simulated six 48-h
cortisol series. Figures 3, 4, and 5 show the same value of
. That
is, for example, Figs. 3A, 4A, and 5A were all simulated with the same random draw of
. Figures 3-5
differ only in their choice of
A. Therefore,
Figs. 3-5 each represent 48 h of plasma cortisol levels for
the same individual with three different amplitude coefficients of
variation. Different panels in Figs. 3-5 represent different
individuals. Together, the 18 panels in Figs. 3-5 represent both
between- and within-subject variation in plasma cortisol diurnal
patterns.

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Fig. 3.
Forty-eight-hour simulations of the population variation in
cortisol plasma levels with chosen randomly from
f( ) in A-F and
A = 0.1. Each panel (A-F) is
simulated with a different value of chosen randomly from
f( ). Corresponding panels in Figs. 4, 5, and 6
were simulated with the same values of but a different value of
A.
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Fig. 4.
Forty-eight-hour simulations of the population variation in
cortisol plasma levels with chosen randomly in each panel from
f( ) and A = 0.3. Each
panel (A-F) is simulated with a different value of chosen randomly from f( ). Corresponding panels
in Figs. 3 and 5 were simulated with the same values of but a
different value of A.
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Fig. 5.
Forty-eight-hour simulations of the population variation in plasma
cortisol levels with f( ) chosen randomly in each
panel from f( ) and A = 0.5. Each panel (A-F) is simulated with a different value of
chosen randomly from f( ). Corresponding
panels in Figs. 3 and 4 were simulated with the same values of but
a different value of A.
|
|
As expected, the variability in the plasma cortisol levels increases as
A increases from 0.1 (Fig. 3) to 0.3 (Fig. 4)
to 0.5 (Fig. 5). Between-day variation within subjects can be
appreciated by comparing within a panel the first 24 h with the
second 24 h. For example, in Fig. 4, A and
F, the plasma levels in the first 24 h are higher than
those in the second, whereas in Fig. 3A the plasma levels in
the second 24 h are higher. For
A = 0.1 (Fig. 3), there is less between-day variation in the plasma cortisol
levels. The simulated data from each value of
A show a good resemblance to the diurnal
patterns seen in actual cortisol series (see Fig. 8).
The simulated circadian amplitude function for the panels in Figs. 3 to
5 is shown in Fig. 6, along with the
locations and amplitudes of the secretory events. The amplitude
functions have the same asymmetric bimodal shape due to the
two-harmonic model for this process defined in Eqs. 5 and 6.
The largest mode of the circadian function occurs between 0600 and
0900, and the second smaller mode occurs between 2100 and 2200. The
amplitude is near zero around midnight for nearly all of the series. A
close comparison of the largest mode of the circadian amplitude
function with the time of maximum plasma level in the early morning
(Figs. 3-5) shows that the latter tends to lag behind the former
by ~45-60 min. This lag is due to the infusion and clearance
processes, i.e., the impulse response function (Fig. 2B).
Because there is infusion of cortisol from the adrenal gland into the
plasma, all of the hormone cannot enter the plasma instantaneously.
Moreover, concomitant with its entry into the plasma, the cortisol is
being metabolized in the liver. The 45- to 60-min lag is consistent
with the time for a secretory event to reach its maximum effect on the
plasma as shown in Fig. 2B. Although each simulated
circadian function has the same general shape and time course, there is
significant between-subject variation in its structure. Between days,
for a given subject, the average amplitude function is the same because it is periodic.

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Fig. 6.
Simulated circadian amplitude functions and pulse locations of the
plasma cortisol series in Figs. 3-5. Each panel (A-F)
is simulated with a different value of chosen randomly from
f( ). A-F of the simulated secretory
event amplitudes and pulse locations correspond to A-F for
the cortisol series in Figs. 3-5. Plasma cortisol concentrations
were converted to amounts of cortisol using the volumes of distribution
estimates reported in Ref. 53 and normal total daily
cortisol production reported in Ref. 19.
|
|
To appreciate better within-subject variation in plasma cortisol
levels, we simulated an additional set of six cortisol series with
fixed at
0 and
A = 0.3 (Fig.
7). That is, for all six draws,
is
set equal to
0 in step 1 of the simulation algorithm (Eq. 22). These simulated series show that a given
subject can have appreciable day-to-day variation in plasma cortisol
levels.

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Fig. 7.
Forty-eight-hour simulations of the within-individual variation in
plasma cortisol levels obtained by simulating the model with fixed
at 0, the mean of f( ), and
A = 0.3. Each panel (A-F)
represents a different simulation of the cortisol model using = 0.
|
|
As a qualitative assessment of the ability of our model to
reproduce the diurnal patterns in actual cortisol series, we plot in
Fig. 8 cortisol data from eight healthy
male subjects measured on a 24-h constant-routine protocol (6,
13). The data were collected as part of the study conducted by
Dr. Charles A. Czeisler to measure the ability of bright light to shift
the phase of the human circadian pacemaker at a fixed phase of the
circadian cycle (3). In this study, five groups of either
seven or eight subjects received a 5-h light treatment of either 0.03, 10, 180, 1,260, or 9,500 lux. The subjects whose cortisol data are
shown in Fig. 8 were in the 9,500-lux group. These cortisol
measurements were made at 20-min intervals during the baseline constant
(routines the subjects underwent before receiving light therapy). These cortisol series are representative of the data from the 39 subjects studied under this protocol.

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Fig. 8.
Actual cortisol series collected from 8 healthy male
subjects on a 24-h constant routine. The cortisol samples were taken at
20-min intervals. Each panel (A-H) indicates a different
subject.
|
|
These experimental cortisol plasma levels are consistent with the
asymmetric circadian amplitude modulation defined in Eqs. 5 and 6 and shown in Fig. 6. The rising phase of the cortisol cycles requires 6 h or more, and the decline phase is
~10-12 h. Three different patterns are evident both between and
within subjects. These are fine pulsatile activity (Fig. 8,
C and F), extended periods of short or low
pulsatile activity (Fig. 8, B, D, F,
and H), and periods of apparently large pulses (Fig.
8A). Several of the series appear slightly smoother than the
simulated series in Figs. 3-5. This may in part be due to the
difference in sampling intervals, which is every 20 min for the actual
cortisol series and every 10 min in our simulated series. Overall, the
qualitative agreement between the model and actual cortisol series is good.
 |
DISCUSSION |
Development of methods for the analysis of endocrine hormone data
is an active area of research. These procedures fall typically into one
of four categories: pulse-finding algorithms, deconvolution procedures,
Fourier and harmonic regression methods, and approximate entropy
(ApEn). Pulse-finding algorithms use local statistical criteria such as
Bonferroni bounds, t-tests, or a specific multiple of the immunoassay coefficient of variation to identify in hormone data
series the times of secretory events (11, 33, 34, 36, 43, 51,
54). Deconvolution procedures use deterministic, usually linear
time-invariant convolution integrals to represent the relation between
inputs, secretory event times and amplitudes, and observed output
hormone levels (55). The deconvolution analyses use
convolution integral models to determine the secretory event times and
secretory amplitudes from the observed hormone levels. If the
hormone's diurnal pattern has an important circadian component, this
is generally extracted in a separate analysis using harmonic regression
methods (44, 46, 50, 53). ApEn is a technique that may be
used to assess regularity and complexity of any biological series
(36). Differences in regularity and complexity detected by
ApEn can help characterize differences in physiological and pathological states. With current methods, separate analyses are required to completely analyze circadian, ultradian, and kinetic components of a hormone data series.
The first step toward obviating multiple analyses is to devise a single
equation system that captures the principal physiological components in
the experimental data. We studied cortisol because much is known about
the physiology of its diurnal patterns. Our work builds on the
two-compartment deterministic differential equation model of
cortisol's diurnal patterns proposed by Jusko et al.
(24). The model by Jusko et al. represents circadian input
as a simple sine wave and the kinetic process as first order. Secretory
events are defined as local percentage changes in plasma hormone
levels, and the model is fit to experimental data by using nonlinear
least squares analysis. The model of Jusko et al. is, to our knowledge,
the first attempt at a single equation system description of the
ultradian, circadian, and kinetic properties of cortisol's diurnal
patterns. This model does not consider immunoassay uncertainty.
To improve upon this model, we represented the physiological properties
summarized by Jusko et al. as a stochastic rather than a deterministic
model. We modeled the ultradian process after a gamma renewal process
because this probability model is widely used to describe waiting time
phenomena (16, 23). From Eqs. 3, 4, 15,
and 16, the average intersecretory event time is ~83
min with a SD of 11 min. Hence, the probability of a first secretory event followed within 60 min or less by a second one is small to
negligible. The low probability of rapid-succession secretory events
may reflect a refractory period of cortisol synthesis induced by
negative feedback at the hypothalamus, anterior pituitary, and possibly
the adrenal gland. Moreover, the ultradian and circadian processes may
also interact in that the degree of feedback inhibition may depend on
the amount of hormone released in a secretory episode. For these
reasons, the assumptions of independence of the intersecretory event
times (renewal process) and of independence between the circadian and
ultradian processes are only plausible first approximations to be
modified in future versions of the model.
Formulation of the ultradian input to our model is based on the
reported high concomitance between ACTH and cortisol secretory episodes
and the fact that the adrenal gland maintains no stores of cortisol
(29-31, 57). Because these observations suggest that the ultradian process must have a substantial effect on the initiation of cortisol synthesis, the ultradian process defines in our model the
times at which the secretory events begin. This representation differs
from the model of Jusko et al. in which the ultradian process modulates
hormone release. It differs also from the convolution integral model of
Veldhuis and Johnson (55) in which the ultradian process
identifies the time at which one-half the amount of hormone in a given
secretory episode has been released. Whereas the ultradian process may
also affect adrenal cortisol release, we have modeled infusion as a
first-order kinetic process independent of the ultradian modulation.
The representation of our model's circadian component is based on the
finding that the circadian system modulates primarily the amplitude
rather than the frequency of the secretory events (53).
The shape of the mean circadian waveform deduced by applying a
preliminary version of the current model presented (8, 9) to the seven cortisol series in Ref. 56 closely resembles
those previously derived using Fourier methods (30, 50)
and those shown in Fig. 6. A second harmonic added to the circadian
amplitude function accounts simply for the asymmetry in the underlying
circadian process. Our decision to model the amplitude process at a
given time as a Gaussian random variable is arbitrary. We also studied a frequency-modulation model of the circadian input. None of its simulated outputs resembled actual cortisol data.
Because neither the form nor the magnitude of the amplitude variance
could be deduced from previous reports, we represented this parameter
as a function of the circadian amplitude. We obtained an apparently
realistic data series with coefficients of variation in the range from
0.1 to 0.5. This variability most probably reflects variability in both
the ACTH stimuli delivered to the adrenal gland and in the cortisol
synthetic response to the ACTH stimuli. Although there is evidence to
suggest that cortisol clearance from the plasma may have some degree of
diurnal modulation (15), we have found that the hormone's
kinetic parameters are well described as a time-invariant first-order
kinetic process.
The second step in defining a single equation system for the analysis
of cortisol's diurnal patterns is to show that model simulations
reliably reproduce experimental data. Our approach builds on the
endocrine hormone simulation model of Guardabasso et al.
(21). These authors report an autoregressive moving
average process in which coefficients are nonlinear functions of the
decay parameters. Secretory event times are generated from a discrete Poisson approximation, the amplitudes are sampled from either a uniform
or truncated Gaussian probability density, and the observational errors
are Gaussian and depend on the immunoassay uncertainty. Our model makes
use of two extensions these authors suggested in their discussion: use
of a more general renewal process model to describe intersecretory
event intervals and a time-dependent circadian modulation. The
simulation algorithm proposed by these authors can be greatly
simplified and implemented in continuous rather than discrete time by
reexpressing their model as in Eqs. 1-11.
Three other stochastic simulation models relate to our work. Straume et
al. (47) reported a model for simulating hormone patterns
in which intersecretory event times are based on a logistic model,
secretory event amplitudes depend logarithmically on secretory event
times, and secretion is modeled as a Gaussian-shaped rate process.
Diurnal timing is either a square-wave (day-night) or sinusoidal
(circadian) process, hormone elimination is either a mono- or
biexponential process, and measurement uncertainty is due to assay
variability. The authors do not relate the several model components in
a single equation system; however, they do show that the model
successfully simulates plasma growth hormone levels. Keenan
(25) reports a model in which the true hormone level is a
one-dimensional stochastic convolution of secretory events from a
periodic, inhomogeneous Poisson process and a time-invariant gamma
distribution. Integrated Brownian motion noise added to the convolution
integral output defines the measured hormone level. Model fitting to
simulated data is by nonlinear least squares. This model differs from
ours in that the Poisson process governs secretory event times
(frequency modulation), whereas the gamma distribution subsumes both
the secretory event amplitudes and the kinetic processes in our model.
Keenan and Veldhuis (26) report a stochastic differential
equation model to simulate the hypothalamic-pituitary-gonadal axis.
Unlike our minimal feedforward cortisol model, this model specifies all
of the feedforward and feedback neural and humoral linkages in the
hypothalamic-pituitary-gonadal axis, including ultradian, circadian,
kinetic, and time-delay parameters. This model has the potential to
give a nearly complete description of the dynamics of the
hypothalamic-pituitary-gonadal system under normal and pathological
conditions. However, it requires specification of relations among
variables and postulating values of several parameters that are not
known or directly measurable. This equation system would require
simplification for use in data analysis. None of the other stochastic
models of endocrine hormones exploits physiological properties in its
design (4, 18, 37).
By summarizing in a probability density the values of the model
parameters from previous reports, we characterize their joint physiological range. The model may be used to simulate formally within-
and between-subject variation and to perform a sensitivity analysis in
which all parameters vary simultaneously. We propose this approach to
assess the model's sensitivity to its parameter values as a more
appealing alternative to that of varying individual parameters one at a
time. Simulated series generated with our model may be used to test
current pulse-finding, deconvolution, Fourier, and ApEn methods.
The final step in developing a single system for the analysis of
cortisol's diurnal patterns is to establish that the model can be fit
to an experimental cortisol time series. In a report currently under
preparation, we present a set of methods to fit our cortisol model to
experimental data using Bayesian Markov Chain Monte Carlo methods. The
joint parameter probability density in Eq. 21 can serve as a
prior probability density for the Bayesian analysis. Preliminary work
applying these methods has been reported previously (8, 9,
32). In addition to including feedback interactions, other
extensions of our cortisol model that we will consider are the
effect of factors exogenous to the hypothalamus-pituitary-adrenal axis,
such sleep states and meals (28, 46), and simultaneous modeling of cortisol and ACTH.
In summary, we have developed a model of cortisol's diurnal patterns
that unifies in a minimal physiologically based stochastic differential
equation the principal elements underlying current pulse-finding,
deconvolution, and Fourier data analysis methods. Our modeling paradigm
provides a framework for both simulation studies and data analysis that
should be readily adaptable to the analysis of other endocrine hormone
systems. Analysis of physiological systems with stochastic models may
help alleviate some presently unexplained discrepancies between
experimental data and model descriptions derived from deterministic models.
 |
ACKNOWLEDGEMENTS |
We thank three anonymous referees whose suggestions helped improve
the content and presentation in this work; Dr. C. A. Czeisler for
use of the cortisol data in Fig. 8; R. Barbieri, Y. Choe, and L. M. Frank for assistance with the data analysis and graphics; and B. Marshall for assistance with manuscript preparation.
 |
FOOTNOTES |
This work was supported by Robert Wood Johnson Foundation Grants 19122 and 23397; National Aeronautics and Space Administration (NASA) Grant
NAGW 4061 and NASA Cooperative Agreement NCC 9-58 with the
National Space and Biomedical Research Institute; National Institutes
of Health Grants 1-R01-GM-53559, 1-P01-AG-09975, 1-R01-AG-06072, 1-R01-MH-45130, and M01-RR-02635; an American Dissertation Fellowship from the American Association of University Women; and Army Research Office Grant DAAL03-91-6-0089.
Address for reprint requests and other correspondence: E. N. Brown, Dept. of Anesthesia and Critical Care, Massachusetts General Hospital, 55 Fruit St., Boston, MA 02114 (E-mail: brown{at}SRLB.mgh.harvard.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 21 December 1999; accepted in final form 5 October 2000.
 |
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