Fractal Properties of the Thyrotropic Feedback Control: Implications of a Nonlinear Model Compared with Empirical Data.

329
J. W. Dietrich1, A. Tesche2, C. R. Pickardt1 and U. Mitzdorf3
1 Medizinische Klinik, Klinikum
der LMU, CampusÊInnenstadt
Ziemssenstra§e 1, D-80336 MŸnchen,
F. R. Germany
dietrich@med.uni-muenchen.de
renatepickardt@aol.com
2 Chirurgische Klinik, Klinikum der
LMU, CampusÊInnenstadt
Nu§baumstra§e 20, D-80336
MŸnchen, F. R. Germany
Arthur.Tesche@lrz.uni-muenchen.de
3 Institut fŸr Medizinische
Psychologie
Ludwig-Maximilian University of
Munich
Goethestr. 31, D-80336 MŸnchen, F. R.
Germany
Abstract
More than 70 years after the discovery of the
pituitary-thyroid feedback control mechanism, a
classical endocrine regulation system, most of its
parameters have been identified. However, the
regulation of its central component in the pituitary
gland, probably responsible for pulsatile release of
thyrotropin (TSH), remains obscure. In order to
infer its structure from the systemÕs behaviour,
four different pituitary models were created and
compared regarding their fractal properties. Based
on non-competitive inhibition of TSH release by
thyroid hormones Ð a physiologically plausible
correlation Ð two of the models added stochastic
stimulation by central signals and two added an
additional intrapituitary feedback loop. The model
combining non-competitive inhibition with both
additional effects showed the same fractal
dimensions as real time series, while simpler
models exhibited significantly lower complexity in
the time series they yielded. The results suggest
that both effects play a role in the generation of
TSH pulses in the human pituitary.
1. Introduction
Pulsatile release of hormones is a common phenomenon
in endocrine systems. It is achieved by combining an
analog signal encoding with amplitude or frequency
modulation. Pulsatility contributes to reliable information
transfer. Furthermore, the oscillating hormone levels help
to prevent desensitisation of their target cells that would
be caused by down-regulation of specific receptors. The
pituitary-thyroid feedback control (thyrotropic feedback
control) involves two classes of hormones, the peptide
hormones TRH (thyroliberin) and TSH (thyrotropin) and
two thyroid hormones (T4 or thyroxine and T3 or
triiodothyronine). Although little is known about the
temporal patterns of TRH levels in the portal system of
the pituitary stalk, the release of TSH into blood plasma
is known to occur in a pulse-like manner. Being similar
to other signalling pathways based on peptide hormones
the thyrotropic information transfer occurs via amplitude
modulation of the TSH level. Faster oscillations with rate
of 5 to 20ÊperÊ24Êh are superimposed on a circadian
rhythm with maximum TSH levels early in the morning.
The mechanism causing the fast TSH oscillations is
unknown. The previously favored hypothesis assuming a
pulsatile input of TRH at the pituitary yielding
corresponding TSH pulses has been disproved by Samuels
et al. [1993], who showed that subjects receiving
constantly high doses of TRH still exhibited TSH levels
with significant pulses. Until now TSH patterns were
mainly classified by comparatively simple measures like
amplitude and rate. More sophisticated approaches using
methods from non-linear systems science to measure the
complexity of the signal patterns have been applied to
other endocrine control systems, e.g. to the release of
PTH in the calcium-phosphate homeostasis.
Thyroid diseases play a crucial role in both individual and
public health. The physiology of the thyrotropic feedback
control and the factors influencing its behaviour are only
partly understood. Previous models of the pituitary-
thyroid feedback control were implemented in a
Óbehaviourally isomorphicÓ way using different classes of
equations (linear, logarithmic, exponential or
polynomial), their parameters optimised to yield
behaviour resembling that of a living organism [Danziger
and Elmergreen, 1956; Roston, 1959; Norwich and
Reiter, 1965; DiStefano, 1969; Saratchandran et al., 1976;
Wilkin et al, 1977; Cohen, 1990 and Li et al., 1995].
Even though these models deliver possible ways in which
the system might be realized, this approach also exposes
the models to charges of being arbitrary.
Exhaustive investigations of the human pituitary in vivo
are practically impossible. Therefore the objective of this
study was to develop a parametrically isomorphic
simulative approach for gathering information about the
structure of pituitary information-processing from the
systemÕs behaviour.
2. Methods
2 .1 Th e m o de l
In order to elucidate the unknown mechanism of pulsating
TSH release, a number of physiologically consistent
models were created. Where empirically determined input-
output-relations were available, all models share the same
Fractal Properties of the Thyrotropic Feedback Control
Implications of a Nonlinear Model Compared with Empirical Data
CYBERNETICS AND SYSTEMS
2002

330
parametrically isomorphic basis. Modifications have been
implemented at the level of the pituitary gland, whose
dynamic properties are not yet well characterized.
The system was analysed in two ways: First, the
equations were solved analytically to obtain instant
solutions of the mean equifinal hormone levels. In a
second step, a computer simulation was generated to
obtain time series of the respective hormone levels. The
simulation program was developed in Pascal on an Apple
Macintosh workstation.
The model was based on two principal mechanisms of
biological information transfer, the Michaelis-Menten-Hill
kinetics and the ASIA element, supplemented by several
feedback loops for the binding of hormones to plasma
proteins.
Michaelis-Menten-Hill kinetics are known to determine
the behaviour of enzymatic conversion processes and
receptor-mediated signal transduction systems. The
subsystems respond with
ya = GxeD + xe
(1)
to an input signal xe, where G is the maximum possible
response of the transduction element and D is the input
signal yielding half of the maximum response G.
The temporal behaviour of the systemÔs variables was
modelled with ASIA elements (analog signal memory
with intrinsic adjustment) that essentially consists of a
variable stimulating its own degradation with

dy
dt = a x(t) – b y(t) (2)
in a first order linear feedback loop [Dietrich 2000]. Here y
denotes the output signal, a the input gain factor and b a
gain factor for output extinction. In the equifinal state the
subsystemÔs behaviour will converge to
y
¥
=
a x(t)
b (3)
with a first order time constant of

t 1 =
1
b
. (4)
Binding of thyroid hormones to plasma proteins was
simulated in a 0th order linear feedback loop according to
the mass action law with

HF = HT – K P HF
, (5)
where [HF] denotes the concentration of the free hormone,
[HT] the total hormone level, K a binding constant and
[P] the concentration of the respective plasma protein, e.g.
TBG. In equilibrium, a level of
HF =
HT
1 + K P (6)
will result.
For each level of signal transfer the respective equations
were mapped to values taken from empirical studies
(Tables 2 to 4).
Obviously, information processing at the pituitary level
occurs in a more complex way. Due to the paucity of
empirical input-output relations four distinct subsystems
were created differing in the temporal pattern of TRH
release into the hypothalamo-pituitary portal vessels as
well as the presence or absence of an ultra-short feedback
loop connecting level and release of pituitary TSH (Fig.
1). Common to all four models was the assumption of a
non-competitive inhibition of TSH release by receptor
bound triiodothyronine ([T3]R) in the form of

d[TSH]
dt =
a S GH [TRH]O
(DH + [TRH]O) (1 + LS[T3]R)
– b S [TSH]
.(7)
[TRH]O is the TRH-level in the pituitary stalk vessels;
see Tables 2 and 3 for an explanation of other symbols.
Further variants 3 and 4 of the pituitary model include an
ultra-short feedback mechanism of TSH in the pituitary
interstitium [TSH]z on its own release according to

d[TSH]
dt =
a S GH [TRH]O
(DH + [TRH]O) (1 + LS[T3]R) Z
– b S [TSH] (8)
with
Z = (1 + SS[TSH]zDS + [TSH]z)
. (9)
The pituitary models 1 and 3 assume a portal TRH level
that is Ð except for circadian variation Ð constant, whereas
versions 2 and 4 implement additional stochastic
variations of the TRH level in the hypothalamo-pituitary
vessels. To be congruent with observations made with
other peptide hormones the TRH-fluctuations were
simulated with a Gaussian random generator delivering a
log-normal distribution.
Table 1: Characteristics of the four variants of the pituitary
model (CV = circadian variation, LGN = log-normal Guassian
noise:
Version TRH level Ultra-Short Feedback
1 CV Omitted
2 CV and LGN Omitted
3 CV Present
4 CV and LGN Present

331
+
[TRH]O
LS
DH
[T3]z
GH
6
[T3]R
DR
GR
1
+ +
+
[IBS]
[T3]N K31
dTSH*
dt
+
[TRH]O
LS
DH
[T3]z
a S2
w + b S2
GH
[TSH]z 6
[T3]R
DR
DS
GR
SS
1
1
5
+
+
+
+ +
[IBS]
[T3]N K31
dTSH*
dt
Pituitary versions 1 and 2
Pituitary versions 3 and 4
Figure 1: Different versions of the pituitary model
Table 2: Numerical implementation of the Michaelis-Menten-
Hill kinetics. Empirically determined values from D'Angelo
et al. [1976], Dumont and Vassart [1995], Greenspan [1997],
Lazar and Chin [1990], Okuno et al., [1979], van Doorn and
van der Heide [1985], Visser et al. [1983] and from own data.
Symbol Explanation Value
GH Secretion capacity of the pituitary 817 mU/s
DH Damping constant (EC50) of TRH at
the pituitary
47 nmol/l
GT Secretion capacity of thyroid gland 3.4 pmol/s
DT Damping constant (EC50) of TSH at
the thyroid gland
2.75 mU/l
GD1 Maximum activity of type I
deiodinase
28 nmol/s
KM1 Dissociation constant of 5Ô-
deiodinase I
500 nmol/l
GD2 Maximum activity of type II
deiodinase
4.3 fmol/s
KM2 Dissociation constant of 5Ô-
deiodinase II
1 nmol/l
GR Maximum gain of TR b receptors 1 mol/s
DR EC50 for central T3 100 pmol/l
SS Brake constant of TSH ultra-short-
feedback
100 l/mU
DS EC50 for TSH at the pituitary 50 mU/l
Table 3: parameterization of ASIA elements. Empirical values
from [Benvenga and Robbins, 1998; Duntas et al., 1990;
Greenspan 1997; Gru§endorf 1988; Odell et al., 1967;
Oppenheimer et al. 1967]
a R Dilution factor for peripheral TRH 0.4 l-1
b R Clearance exponent for TRH 2.3 e-3 s-1
a S Dilution factor for TSH 0.4 l-1
b S Clearance exponent for TSH 2.3 e-4 s-1
a T Dilution factor for T4 0.1 l-1
b T Clearance exponent for T4 1.1 e-6 s-1
a 31 Dilution factor for peripheral T3 2.6 e-2 l-1
b 31 Clearance exponent for T3P 8 e-6 s-1
a 32 Dilution factor for central T3 1.3 e5 l-1
b 32 Clearance exponent for central T3 8.3 e-4 s-1
a S2 Dilution factor for pituitary TSH 2.6 e5 l-1
b S2 Clearance exponent for central TSH 140 s-1
Table 4: Dissociation constants of hormone binding. Values
from Li et al. [1995].
K30 Dissociation constant T3-TBG 2 e9 l/mol
K31 Dissociation constant T3-IBS 2 e9 l/mol
K41 Dissociation constant T4-TBG 2 e10 l/mol
K42 Dissociation constant T4-TBPA 2 e8 l/mol
2 .2 Fra cta l di m e n s i o n i n g
In order to compare simulated to real time series,
recordings made from volunteers by Brabant et al. [1990],
Greenspan et al. [1986] and Samuels et al. [1990] were
digitized and processed by two programs for calculating
the fractal dimensions of the signal patterns.
The first measure of complexity used was the fractal
capacity dimension (D0). This approach covers the
graphical representation of the time series with squares of
successively varied border-length s using the mesh-
counting theorem. For each length it counts the number
N(s) of squares covering the curve. With
D0 = lim
s fi 0
log N(s)
log 1s (10)
the capacity dimension can be calculated and compared for
real and simulated time series.
By means of a second approach to determine the dataÕs
complexity the so-called correlation dimension D2 [Loistl
and Betz 1996] was calculated.
After embedding the time series x1, x2, x3 É xN into the
m-dimensional vector
À i = xi, xi + p, xi + 2p,…, xi + m – 1 p (11)
the local density
ni e = 1N u0( e – À j – À i )Sj = 1
N
(12)

332
as relative number of neighbour points of an attractor
point À i whose distance is smaller than e could be
calculated with the heaviside function

u0(x) = 0, x < 01, x > 0
. (13)
Subsequently by averaging over several reference points
the correlation integral
C e = 1
M 2
u0( e – À j – À i )S
i, j = 1
i „ j
M
(14)
as the number of correlated vectors normalized over the
number of possible vector pairs M2 could be calculated.
For each embedding dimension, formally similar to the
definition of the capacity dimension, a specific local
correlation dimension D2 can be obtained from
D2 = lim
e fi 0
log C( e )
log e
. (15)
The first maximum of the local correlation dimensions
D2, arranged by increasing embedding dimension m, was
regarded as a global correlation dimension of the time
series.
Eight real time series adopted from Brabant et al. [1990],
Greenspan et al. [1986] and Samuels et al. [1990] were
respectively compared with 8 time series generated from
each pituitary model.
The capacity-dimension was calculated with the Fractal
Dimension Calculator 1.5 by Paul Bourke (Astrophysics
and Supercomputing Centre, Swimburne University of
Technology, Hawthorn, Melbourne, Australia, available
via http://astronomy.swin.edu.au/pbourke/fractals/fracdim/).
Correlation and embedding dimensions were calculated
with the application C(D2) (J.ÊW. Dietrich, University of
Munich, Germany, available from
http://link.medinn.med.uni-muenchen.de/cybermed/
nonlin/cd2/).
3. Results
All four models showed the same results for equifinal
hormone levels as those obtained by analytically solving
the model equations for TSH, FT4 and FT3. The equation
system is solved via a cubic equation as casus
irreducibilis with three mathematically real solutions.
Only one of these solutions is realizable in a biological
context, the other two results being negative (Table 5).
Table 5: Solutions for the equilibrium levels of the simulated
hormone levels (see text for explanation):
Parameter Solution 1 Solution 2 Solution 3
TSH 1.8 mU/l -1.2 mU/l -1.2 mU/l
FT4 1.4 ng/dl -2.4 ng/dl -2.4 ng/dl
FT3 3.5 pg/ml -6.0 pg/ml -6.0 pg/ml
Obviously, the simulated parameters are located within
known reference regions for healthy individuals.
The differences between the four variants are disclosed in
the comparison of the time series delivered by the
computer simulations (Fig 2).
Version 1
Version 2
Version 3
Version 4
Figure 2: The behaviour of the four versions of the pituitary
model (time series over 24 hours simulated time).
Fractal properties are implied by the four variants as
shown in Table 6 and Fig. 3.
Table 6: Fractal properties of the models and empirical time
series (D0: Capacity Dimension, D2: Correlation Dimension,
m: Embedding Dimension, **: p<0,001, t-test for
independent samples, comparison of empirical and simulated
time series):
Mean
Dimensions
D0 D2 m
Empirical 1.20 1.75 19.63
Model 1 0.96** 0.76** 1.00**
Model 2 1.04** 0.77** 1.00**
Model 3 0.99** 0.74** 1.13**
Model 4 1.18 1.91 20.14
As revealed by all dimension measures, the pituitary
models 1, 2 and 3 exhibit significantly lower complexity
than real time series, whereas the fractal behaviour of
model 4 is comparable to that of empirical data (except for
a smaller variance of simulated time-series for the capacity
dimension).

333
, 2 5
, 5
, 7 5
1
1 ,25
1 , 5
1 ,75
2
2 ,25
2 , 5
2 ,75
Correlation dimension
Pituitary model 4
Pituitary model 3
Pituitary model 2
Pituitary model 1
Empirical
0
5
1 0
1 5
2 0
2 5
3 0
m
Pituitary model 4
Pituitary model 3
Pituitary model 2
Pituitary model 1
Empirical
, 9
, 9 5
1
1 ,05
1 , 1
1 ,15
1 , 2
1 ,25
1 , 3
1 ,35
1 , 4
Capacity dimension
Pituitary model 4
Pituitary model 3
Pituitary model 2
Pituitary model 1
Empirical
Figure 3: Fractal Dimensions of the four pituitary models
compared with empirical data.
4. Discussion
The existence of a feedback loop interconnecting pituitary
and thyroid gland has been known for decades [Aron
1929, Crew 1930]. Nevertheless, the algorithms of
information processing inside the pituitary remained
unclear.
The inhibiting effect of TSH on its own release has been
observed in animals [Kakita and Odell 1986]. There is
controversy whether or not this kind of ÓUltra Short
FeedbackÓ exists in humans, although Prummel et al.
[1997] found TSH receptors in human pituitary tissue.
Like other peptide hormones, TSH is secreted
episodically. As TRH pulses have been ruled out as the
cause of these rhythms, the location and nature of the
pulse generator remain obscure.
Apart from the mechanisms occuring in the central organs,
the physiology of thyroid regulation seems to be well
characterized. Therefore, it was possible to develop a
model relying predominantly on empirical input-output
relations (again except from pituitary regulation). This
parametrically isomorphic model could then be used to
test different variants for the as-yet unknown information
processing within the pituitary gland.
In the form of a Michaelis-Menten-Hill kinetic with one
additional non-competitive inhibitory site, the first
pituitary model was physiologically plausible but
relatively simple. Alternative models added stochastic
stimulation by TRH and an ultra-short feedback loop
inhibiting the TSH release by interstitial thyrotropin in
the pituitary.
While all variants yielded the same equilibrium hormone
levels, their behaviour over time was different. All
versions failed to reach the complexity of real time series
Ð with the exception of the fourth model.
It may appear to be trivial to observe that the rising
complexity within an information processing structure
parallels the increasing complexity of the time series.
Nevertheless, the fact that simulated time series from
model 4 show nearly identical dimensions as real time
series, supports the hypothesis that they may be caused by
isomorphic processes. Certainly, the systemÕs behaviour
in vivo will be influenced by additional factors, an
assumption that is supported, e.g., by the larger standard
deviation of natural capacity and correlation dimensions as
compared with simulated data.
The results of the simulations suggest that regulation of
thyroid activity might be more complex than simple non-
competitive inhibition of the TRH mediated activation of
TSH release. Together with the identification of TSH
receptors in the pituitary tissue [Prummel et al. 1997] our
results support the hypothesis that an ultra-short feedback
loop in the pituitary gland may also play a role in human
physiology.
When inspecting the simulated time series (Fig. 2) it is
striking to note that the amplitudes of the TSH pulses
increase when the circadian rhythms yield a maximum of
the basal TSH activity. This observation can also be made
with empirical data. However, this effect has hitherto
rarely been described, e.g. in Adriaanse et al. [1993].
+
[TRH]O
LS
DH
t 1s
t 0s
dTSH*
dt
[TSH]
GT
DT
[Thy] [I-]
KT KI
t 1T t 0T
a T
w + b T
Thyroid
Organism
Environment
Pituitary
[T4]
[TBG] [TBPA]
K41
[FT4]
KM2
[FT4]
GD2
[T3]z
GD1
t 13p
t 03p
KM1
K41
[T3]p
[FT3]P1
2
K42
3
4
a S2
w + b S2
GH
[TSH]z 6
[T3]R
DR
DS
GR
SS
1
1
5
+
+
+
+
a S
w + b S
+
a 31
w + b 31
dT3P*
dt
+
+
dT4*
dt
+
+
dT3Z*
dt
a 32
w +b 32
t 13Z
t 03Z
+
[IBS]
[T3]N K31
Figure 4: Information processing structure of the overall
system according to version 4 of the pituitary model.
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