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Inspection of multidimensional phase spaces with an application to the dynamics of hormonal systems
U. Brosa, H.-M. Harms, K. Prank, R.-D. Hesch
To cite this version:
U. Brosa, H.-M. Harms, K. Prank, R.-D. Hesch. Inspection of multidimensional phase spaces with an
application to the dynamics of hormonal systems. Journal de Physique II, EDP Sciences, 1991, 1 (3),
pp.273-278. �10.1051/jp2:1991168�. �jpa-00247516�
Classification
Physics
Absbncls 05.40 87.00 87.25Short Communication
Inspection of multidimensional phase spaces Wth
anapplication
to the dynamics of hormonal systems
/
U.
Brosa(I),
H.-M.Harms(2),
lLPrank(2)
and R.-D.Hesch(2)
(1)
FBPhysik, Philipps-Universit3t Marburg
and Brosa GmbH, Am Brocker lbr 4, D-3572Ambneburg, Germany
(2) Abteilung
KlinischeEndoknnologie
im Zentrum InnereMedizin,
Medizinische HochschuleHannover, Konstanty-Gutschcw-StraBe
8, D-3lK© Hannover 61,Germany
(Received
2Januafy
1991,accepted
mfinal fern
15Januafy 1991)
Abstract. We look
directly
into multidimensionalphase
spaces. This is useful if little is known about suitable observables andunderlying
laws. Thedynamical
system we examine is the humanbody,
in
particular
the secretion of theparathyroid
hormone(PTH).
tme series of PrH concentrations are transformed to multidimensional data sets. From theirrepresentations
inphase
space we derive asuitable observable: the average lifetime of a PrH fluctuation. It
provides
a dear-aut discrimination between health and two metabolic bonediseases,
viz. osteoporosis andhyperparathyroidism.
Thederivation is done step
by
step: First we consider multidimensionaldisplays
and observe that it is certain correlation function whichplays
an important role. Then asingle
number is taken from thatcorrelation
function,
and a threshold value issuggested.
Since Boltzmann's time the
phase space
has been a valuableconcept
to discussdynamical systems.
The state of any
system
can be describedby
asingle point
in that space, and its evolutionby
asingle uajectory.
Theadvantages
of thisconceptual simplici%/
have beenqualified by practical difficulties
with the visualization of
geometrical objects
inhigh-dimensional
spaces. So for along
timejust pictures
onplanes
weredrawn,
and henceonly
suchdynamical properties
assingular points
andlimiting cycles
weregenerally
understood. It was notsurprbing
that thediscovery
of thestrange
attractors coincided with the
widespread
use ofperspective pictures,
so-called 3Dgraphics. Using
modern
computer technology
we can look into more than three-dimensional spaces. Here we wfllonly present
anelementary technique, namely
theprojection
of ahigh-dimensional space
onto three-dimensionalsubspaces. Already
thb allowsinsights
which were never achieved before.The classical
approach
of statisticalphysics
tophase space
is thecomputation
ofaverages.
lftrhnces,
covariancematrices,
correlationfunctions, entropies
and all the fractal dimensions[ii represent
averages. Infact,
any observable can be construed as some mean value. Thisapproach, however,
is less fruitful if almostnothing
is known about the nature of theunderlying system.
Thenumber of observables is infinite. Which one
gives
the relevant information?274 JOURNAL DE PHYSIQUE II N°3
In such situations direct views into the
phase space
can behelpful.
One obtains at leastinklings
and can derive from these more
specific questions
or, in otherwords,
suitable observables.Hence,
our
approach
bpreparation
rather thanreplacement
of classical methods. For definiteness wepresent
anexample
fromendocrinology,
viz. the secretion of theparathyroid
hormone(PTH)
and its relation to
osteoporosis
andhyperparathyroidism.
Why
is PTHimportant?
In adult mammalian life the structure and function of boneundergoes
a
permanent plastic ad4ption,
called"remodelling".
This concernshematopoiesb (production
ofblood)
as well as the biomecllanical architecture of bone. PTH is the dominant hormone involved in theremodelling.
It worksby receptor-operated regulation
of geneexpression
in bone cells withsubsequent adaption
of the structural bone networkOsteopororis
is a disease where the three-dimensional architecture of bone isdestroyed
and its total mass decreases. Inosteoporotic subjects
thedynamic pattern
of PTH secretion bgrossly
altered. This leads to increased bone
resorption.
The boneplates (ostea)
becomeperforated, pomtic,
hence the name of the disease. At a certaindegree
theperforations
become irreversiblecausing
biomechanicalinstability
and bone fracture.Hyperparathyroidism
is a disease where theparathyroid glands
deliver more PTH than is needed. Thebone,
in contrast toosteoporosis,
exhibits ahigher connectivity
ofplates aid
anincreased bone mass.
We
prefer
thisexample
because it is not a schematic model(think
of all thespin systems)
where the mostimportant
observables and fundamental laws are known in advance. With PTH concen-trations,
there is atpresent
nosimple~criterion
to discriminate between health and diseases. Ourobjective
is to find such a criterion andto
discuss its relation tosystemic (affecting
theentire~bodily system) processes.
Within the static
concepts
ofendocrinology,
mean values of PTH concentrations were taken to indicate diseases. Yet this b notsuf%ciently
reliable. Since bone exhibitsprima facie
a chaos- alike structure, weexpect
arelationship
to thedynamic
hormonal modulation of this structure which shouldproceed
in a deterministicinformation-processing system.
lb say itcrudely:
Theremodelling
of bonerequires non-stationary dynamics
of PTH secretion. Thereforeexpressions
as "not
enough
PTH" or_ "too muchPTH",
which refer to the average PTHproduction,
serveonly
for first orientation.Healthy
PTH secretion is in fact adynamical process,
withlarge
andirregular
fluctuations[2]. Healthy
secretion has even features of a chaotic evolution as one can showby Grassberger-Procaccia analysis
andby Lyapunov exponents [3].
In
patients
withosteoporosis
we could associate the loss of bone mass with a bifurcation from the "normal"dynamic pattern
towards "low"dynamics
which resembles very much asteady
statemeaning
that cellularreceptors
arebiologically
at rest.Generally
fluctuations are small in os-teoporosis.
Therefore the nextattempt
to establish a suitable criterion would beby
vanances.Unfortunately
also thisattempt fails, first,
because sometimesosteoporotic
fluctuations can be aslarge
ashealthy
ones, andsecond,
because there w this other bonedisease, hyperparathyroidism.
In
patients
withhyperparathyroidism
we observelarge
fluctuations which are in most caseslarger
than in
healthy subjects.
We haveportrayed
this as"high" dynamics [4].
'Consideration
of variances favors anothermisjudgement, namely
withrespect
to theorigin
of the diseases. Based on varwnces one is inclined to think that a too-much is as harmful as a too-small. Yet we will see that
hyperparathyroidism
andosteoporosis
are malfunctions due toqualitatively
different causes.In the measurements blood was drawn from
patients every
two minutes. One measurementtypically
lasted 6 hours. From thesamples,
PTH concentrations are determinedby
suitable as- says. Details can be found in[2].
l~et us denote the time ofdrawing by (tn
0 n=
1, 2, )
andby c(tn)
thecorresponding
concentrations of PTH.Following
lhkens [5] we rearrange rite series(c(tn)
0 n =1, 2, )
togenerate points
in a nine-dimensional space:iC(ti)>
C(t2)>C(t3), C(t4), C(t5), C(t6), C(t7), C(t8), C(t9) iC(t2), C(t3), C(t4)> C(t5), C(t6), C(t7), C(t8)> C(t9), C(tic)i iC(t3), C(t4), C(t5), C(t6), C(t7), C(t8), c(t9), c(tic), c(tit)i
,
We call the first variable in these lines xi, the second z2, the third z3 and so forth. lb obtain a
la)
(b)
*
Fig.
I. Time series of PTH secretion under osteoporosis(small
clouds in the lowerleft)
and m ahealthy
person
(big structures) Subsequent
points of the series are connectedby
lines. Part(a)
holds if thedelay
between points is just I time unit(2
mm).
Part(b)
descnbes the behavior for adelay
of4 time units. Between thebeginning
and end of every axis PTH concentrations vary from 25 to 125 ngA. This is afigure
forpeople
seeing such data for the first time.Generally,
data fromosteoporotic
andhealthy subjects
are less wellseparated. Anyway,
theshape
differences discussed in the text seemalways
to be present.276 JOURNAL DE PHYSIQUE II N°3
representation
of thesubspace (zi,
z2,z3),
we select the first three columns of the above scheme anddbplay
them in a Cartesian coordinatesystem.
Anexample
b shown inFig.
la.We may choose any combination of three columns. Most
important
for thepresent purpose
aresynopses
like(xi,
z3,z5), (xi,
z4,z7)
and(xi,
z5,z9)
sincethey
reflect the correlations in the series.Consider,
forexample, Fig.la.
Itdisplays
the short-time correlations. The smallspherical
cloud stems from
osteoporosis,
while the club-like cloudbelongs
to health. The centroids of these cloudscorrespond
to the meanvalues,
their extensions to the varhnces. Moresignificant
are theirshapes: Sphericity
of a cloud tells us that the data are almost uncorrelated. The stretchedshape
ofa club
signalizes
finite correlation. We may check thisby Fig.
lb. The data areplotted
here in thesubspace (zi,
z5, z9)
to indicate medium-time correlations. We see that the small cloud b almost the same as inFig.
la.Clearly,
zero correlations can't be made smaller. Thebig club, however,
became broader. This
corresponds
to apartial
loss of correlations which occurs because thereis,
more
delay
between the variables xi, z5 and z9 than between xi, z~ and z3.We may summarize the
interpretation
ofFig.I
as follows: PTH secretion inosteoporotic
sub-jects
resembles anoisy
fixedpoint.
Infact, comparison
of the samesamples using
different assays has shown that mostosteoporotic
fluctuations are artefacts causedby
the limited resolution of the immunoradiometric detectors.Healthy
PTHsecretion, ho~vever,
occurs in definedpulses.
Atypical
lifetime of such apulse
is 5 timeunits,
I.e. 101ilinutes.In
Fig.2
datagenerated by
a personsuffering
fromhyperparathyroidism
arejuxtaposed
to thehealthy
behavior. That mean value and variance of PTH concentrations arebigger
inhyper- parathyroidism
than inhealth,
is often the case, however notalways.
More reliable areagain
theshapes
of the clouds: Health maintains someregulation,
whereas fluctuations inhyperparathy-
roidism seem to
bi
uncorrelated.Hence we propose the familiar correlation function
~
((c(tn+>)-
<c(tn) >)(c(tn)-
<c(tn) >))
>
I(c(tn)-
<c(tn) >)2)
a% a suitable observable.
Averages
are taken over n. For data fromhealthy subjects
andpatients
with
hyperparathyroidism
pi isplotted
inFig.3.
We see that thehyperparathyroitic
correlation functiondecays significantly
faster than the function of health. l~et us define the average lifetime of a fluctuationby
that timetj
at which pi takes the value 0.5. FromFig.3
we can derive tllat theaverage lifetime of a
healthy
fluctuation is about 10 minutes whereas that ofhyperparathyroidism
b at most 4 min. The average lifetime under
osteoporosis
b even shorter.Thus we have reduced the
complex
data fromdynamic
hormone secretion to astraightforward yes/no
decision. We can describe therecipe
for the decbion between health and disease as follows:Measure the average lifetime of the PTH fluctuations. If this time1s
longer
than 7minutes,
theperson
does not suffer fromosteoporosis
orhyperparathyroidism. Among
all data sets studiedby
us, no
exception
from this rule was found.The criterion
appears simple enough.
See however [2] and thepapers
cited there to understand that itsdiscovery
wasby
no meansstraightforward.
Now that we found a suitable observable to discriminate between health and
disease,
we are on safergrounds
withhypotheses regarding
the mechanisms of PTH modulation.In
healthy conditions,
PTHproduction
and secretion is modulatedby complex
metabolicpro-
cesses which take
place,
at leastpartly,
far away from theparathyroid glands.
We know this be-cause the average lifetime of PTH fluctuations
(10 min)
isbigger
than the recurrence time ofblood circulation
(2-4 min).
la)
16)
Fig.
Z Similar tofigure
I, but here PrH secretion inhyperparathyroidism
iscompared
withhealthy
be- havior. The data from thehealthy subject
are the same as infigure
I. Due to the different scalethey
appearthis time in the lower left corner; in this
figure
PTH concentrations vary from 125 to 913 ngA. The differencein sizes is not
always
asspectacular
as shown here.Only
the correlations seem to begeneral
properties.1.oo
Pj
o 75
0 50
0 25 ,,
'~-~_~l'
I'0 00
0 2 4 6 8
Fig.
3. Correlations pi ofhealthy
PTH secretion(solid line),
secretion inhyperparathyroidism (dashed)
and osteoporosis
(dotted)
as functions of timeii.
One time unit has 2 minutes. Thisfigure
was obtainedby
averaging over data sets described in [2]. The ind1vldual correlation functions are quite s1mllar to their average representatives Please notice- The difference between health and disease is much more conspicuousin
figure
2.278 JOURNAL DE PHYSIQUE II N°3
In
osteoporosis,
theparathyroid glands
haveapparently
lost theirability
to answer external modulation.They just constantly produce
some PTH(which might
correlate withnon-dynamic
basal
secretion).
Finally
inhyperparathyroidism
thePTH-producing
cells still canrespond
tomodulation,
butthey
are modulated on a different levelby
different substances. These substances must be pro- duced at aplace
not far away from theparathyroid glands.
One of themprobably
ischromogranin
A
[6],
a direct inhibitor which is stored in the samesecretory granules
as PTH.Chromogranin
Amight play
the role of a emergency brakebeing
much faster than the normalmodulation, acting
however
only
when a veryhigh
level of PTH concentration istransgressed.
In summary, conventional
approaches
of statistics have failed toanalyse
data on PTH pro- duction[2]. They
did notpermit systemic interpretations beyond
theconcept
of static reference ranges of variables determined incirculating
bloodf.
Our newapproach
ofrepresenting
data in multidimensionalphase
spaces offers a newconcept
topathophysiology.
We can characterizethe
dynamics
ofapparent
wellbeing
and canportray
the evolution of diseases[~.
Forphysics
wefind that vbualization of
objects
inhigh-dimensional
spaces eases the very firststeps
of research.References
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H-G-,
Deterministic Chaos, VCH(Weinheim)
1988.[2] HARMS,
H.-M., KAPrAINAU.,
KOLPMANNW-R.,
BRABAW G. and HESCHR.-D.,L
ChmEndocnno&~gy
Metaboksm 69(1989)
843 and the references therein.[3] PRANK K., HARMS
H.-M.,
KAYSER Ch, BRABANTG.,
OLSEN L.E and HESCHR.-D.,
Thedynamic
code: information transfer in hormonal systems in:Proceedings
of the Advanced NATO ResearchWorkshop
onComplex Dynamics
andBiological
Evolution(Hindsgavl
in Denmark1990)
Plenum Press(New York,
inpress)
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T,
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SCHULZ W, ZIEGLER R. and BRESSEL M., Kkn. Wochenschr 65(1987)
643[5j TAKENS F, Lecmre Notes m Math 898
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FAWIOTrOB-H-,
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