IS THALASSEMIA A DYNAMICAL DISEASE?
Giani U.*, Filosa A.**
* Department of Communication Sciences.I Faculty of Medicine. University of Naples.
v. S. Pansini,S. 80131 Naples. Italy
**Department of Pediatrics. Cardarelli Hospital. Naples
Abstract
Recently the traditional view of " health " as " regularity" has been challenged, and normality
is coinceived as a sort of constrained randomness and pathology as a loss of the so called spectral
reserve.Dynamical diseases would be due to changes in the qualitative dynamics corresponding to
bifurcations in the non linear equations describing the system. In this respect, some hematological
diseases were modeled in terms of differential-delay equations by assuming a delayed regulation of
blood cells production, In the present paper the temporal evolution of the hemoglobin destruction rate
of 23 thalassemic children is analyzed. The results indicate that these models are to be partialy revised
and that Thalassemia can be conceived as a dynamical disease. A relation between the qualitative
dynamics of Hb rate of destruction and the clinical evolution is suggested.
1.0 INTRODUCTION
Reiman (1963) described a group of diseases whose symptoms recurred at
seven days interval which he called periodic diseases. In all of these diseases
oscillations appeared in physiological systems not normally characterized by
oscillations.
An extension of this concept was introduced by Mackey and Milton (1987)
who defined as dynamical diseases those diseases occurring in. an intact
physiological control systems " operating in a range of control parameters that leads
to abnormal dynamics".
These authors argued that the qualitative dynamics of a "healthy system " can
change due to changes in one or more parameters and that these changes can be
conceived as corresponding to bifurcations in the non linear equations describing the
system. So, the dynamical diseases may arise because of pathological alteration in
underlying control parameters.
From the phenomenological point of view, these diseases occur either when a
new periodism appears or when an old " normal" oscillatory pattern is disrupted.
However, Mackey and Milton did not give any interpretation of the concept of
“normality”: a “normal " or " healthy " system can be either periodical or aperiodical
and becomes " abnormal " whenever qualitative changes in its dynamics arise.
On the other hand Goldberg , West et al.(1985) argued that a number of
diseases processes and aging itself are characterized by a narrowing of the power
spectrum with a decrease of high frequency components and that this behavior
contrasts with the broad inverse power-law spectra seen in normal conditions.
They suggested (Golberg and West, 1987) that, although transitions to and from the
steady state and periodic behavior may be deleterious " chaos may be the normal state
of affairs ": the healthy status is a sort of organized variability or of constrained
randomness.
So, dynamical diseases are characterized by a loss of the physiological variability
and by the appearance of pathological usually low frequency periodicities(see also
Rensing, 1987).
From a more general and philosophical point of view, complexity may be the
salient feature shared by all" healthy " systems.
A full discussion of this problem can be found also in Pool (1989).
1.1 Dynamical diseases and blood cell regulation
Several periodical hematological diseases were observed e.g. autoimmune
hemolitic anemia (Mackey and Glass 1977), cyclical neutropenia (Mackey 1977),
aplastic anemia (Mackey and Milton 1987), cyclic trombocytopenia (Morley 1970).
It was also shown that oscillatory dynamics in hematological control systems arise
from the application of chemotherapic agents in normal dogs (Kennedy 1970 ) or of
marrow-seeking radioactive compounds in normal mice (Gumey, 1981).
These " diseases" are interesting from the system dynamics point of view
because they are susceptible of being modeled in terms of differential-delay equations.
For example, Mackey and Milton(1987) suggested a model of hematopoiesis based
on the hypothesis of a delayed feedback mechanism, and explained the 30-70 days
periodic oscillation of neutrophil counts in some form of mielogenic leukemias
by assuming that neutrophil production is a delayed decreasing function of
increasing neutrophil density. If time delay is of order of 5 days then the system is
stable. If the delay is increased the system becomes unstable and a cascade of
bifurcations occurs until chaos is observed at 20-days delay.
Murray (1989) recognized that these results are to be considered rather as a
“modelling exercise" ,and that their practical relevance is not definitely demonstrated.
From the experimental point of view the main difficulty is that the two
processes of destruction and production of blood cells are not directly observable.
In this respect Thalassemia( a genetic disease in which an anomalous and
ineffective hemoglobin is produced)can be considered as a sort of natural experimental
situation. In fact, thalassemic people are given transfusion therapy in order to
maintain the mean exhogenous hemoglobin level above 10 mg/dl. This Hb level is
thought to inhibit bone marrow autonomous blood production and to block
ineffective hematopoiesis so preventing progressive bone marrow expansion and
other dangerous side effects.So, the spontaneous blood production is blocked and
substituted by transfusions, and one can directly observe the time evolution of the
destruction rate of exhogenous Hb which in turn is a marker of red blood cells
consumption rate.
Figure 1 shows that this rate is not constant but varies almost unpredictably
both within and between subjects.
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Despite this, one can notice that physicians ,in order to scheduling the time of
the next transfusion, usually formulate a raw prediction of the rate of Hb destruction
in the days after the transfusion. So, at the n-th transfusion setting, they compute the
future daily consuption of Hb by taking the difference between the level reached after
the previous transfusion and the actual pretransfusional level and by dividing it by the
interval between the last and the actual transfusion. This figure is then projected into
the near future and the time of the next transfusion is scheduled so that the Hb level
will not, hopefully, fall below 10 mg/dl.
This procedure, which is well radicated in medical practice, may be a cue that
physicians, albeit unconsciuosly, perceive that at least a short term prediction is
sound. This means that they implicitly assume that the rate of Hb destruction is not
merely random.
So, in the present work the temporal evolution of Hb destruction rate of a set
of thalassemic children is analyzed in order to:
1, estabilish whether fluctuations in the rate of Hb destruction are to be considered as
random or as chaotic,
2. verify whether the Mackey's and Milton's hypothesis of delayed regulation can be
applied to red blood cells,
3. judge whether there is sufficient evidence for considering Thalassemia as a
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3. judge whether there is sufficient evidence for considering Thalassemia as a
dynamical disease,
4. suggest some relationships between typologies of temporal evolutions of Hb
destruction and the clinical evolution,
5. help physicians to rationalize their therapeutic behavior.
2. METHODS AND RESULTS
An analysis of the temporal series of Hb daily consumption of 23
thalassemic children was carried out. Table 1 summarizes the salient features of each
case.
‘CASE, SEX AGE RECORD TOTAL NUMBER] AVERAGE
(years) LENGHT OF INTERVAL
(days) TRANSFUSIONS| BETWEEN
|TRANSFUSIONS|
1 M 16.5 1813 116 15.62
2 FE 18.3 2828 172 16,44
3 F 15.7 2918 172, 16.96,
4 M 15.1 2933 163 17.99
5. M 14.0 3194 193 16.54
6 F 10.7 2046 108 18.94
7 M. 13.0 2937 183 16.04
8 E Tey 2068 104 19.88,
9 F 12.2 2056 140 14.68
10 M 17.0 2099 136, 15.43,
il M 12.0 2060, 102 20.19
12 M. 11.6 2037 1 22.38
13 F 142 2203 161 13.68
14 M 12.2 2114 149 14.18
45 = 12.5 1823 109 16.72,
16 F 12.6 1943 92 21.11
17 F Md 1996, us 17.35
18 F 10.0 1964 106 18.52
19 M 9.6 1861 108, 17.23
20 F 11.6 2101 98 21.43
2 F 13.2 2282 118 19.33
22 M 12.0 2162 124 17.43
23 M. 14.5 2199 135 16.28
Table 1
Since lags between transfusions were not fixed each time series was interpolated
by a cubic spline function (de Boor, 1978), and the Hb destruction rate at 10 days
intervals was, so, " estimated". Each series was tranformed into a zero mean time.
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series by subtracting the mean value .
Fawa La
v V V yy V V
f V yey V
V V f° v
N A
Vv V V
wh LMA ak Lr
LA JING TL abl a
Figure 3: Power spectra
ltl a
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SHY:
CASE AR() ARQ) CONSTANT RESIDUAL | CORRELATION
STANDARD | DIMENSION
DEVIATION.
1 0189 =.00086, 04823 2.57,
E 0116 ~,00024 03819 2.55
3 OTL -.00003, 03395 2.20
4 30434 =.00003 03115 2.21
5 08448, =.00129 04432 2.72
6 36683 00207 04504 0.91
7 =20618 00670 0374, 2.76
8 4363 00040 0024 1.03
9 =.15533 00064 0576 2.91
10 0587 00067 0333, 2.75,
in 4622 00016 0160 211
12 6675 00320 0124 17
13 21148, 00170 0474 2.77
14 0976 ,00006 0380 2.85
15 12767, ,00004 0370 171
16 5830, .00017 0131 2.24
17 4575, ,00008 0355 2.15
18, 2151 00156 0317 2.26
19 0143 00145 0495 1.74
20 0582 (00005, 0381 2.27
21 29416 00014 0316 2.13
2 -.0564 ‘00107 0473 2.03
23 22813 00065, 0335 2.55
Table 2
3. DISCUSSION
The main result of this work is that the temporal evolution of the destruction
rate of Hb in thalassemic people has a narrow-band power spectrum with a maximum
peak corresponding to a characteristic oscillation with a period of about fourty days.
This is at variance with what one would expect if only unpredictable random
factors such as, for example, intercurrent deseases, stressfull events, fever,
exercises and so forth affected Hb consumption.
The analysis of correlation dimensions seems to indicate that the time
evolutions are in reality chaotic. But, table 3 shows that they can also be modeled as
second order stochastic autoregressive processes.
We are not able at the moment to disambiguate among these two competing
hypotheses since it is known that stochastic processes different from white noise can
show a correlation dimension lower than the embedding one.
However, the problem arises of whether the observed quasi-periodic
fluctuations are physiological findings which are normally obscured by the
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mechanism of Hb production, which is blocked in transfused thalassemic people, or
are they a peculiar feature of blood consumption in thalassemic people.
At the present state of the analysis we can only be conservative and merely say
that the observed narrow-band spectra are consistent with the hypothesis that
Talassemia is a dynamical disease, and that other analyses are needed in order to
exclude that also the temporal evolution of Hb destruction rate in normal people is
quasi-periodical.
From the system dynamics point of view, the narrow-band power spectrum is
probably a cue of the existence of a control mechanism of the number of circulating
ted blood cells which is more complex than the one postulated by Mackey and
Milton. In fact, Machey's and Glass's model is based upon the assumption of a
delayed blood cell production with a constant rate of cell destruction, whereas our
data indicate that the rate of Hb destruction itself is quasi-periodical. So, other more
complex models are needed.
Another possible hypothesis is that the oscillations are caused by the
transfusions themselves which in fact substitute the normal blood production
mechanism, the main difference being that this artificial blood production is not
continuous.
Against this conjecture one can argue that the rate of Hb destruction is not
significantly crosscorrelated at any time displacement with Hb level attained at the
previous transfusion since this level, due to the therapeutical objectives, is kept
practically constant.
However, one cannot exclude that the artificial “discretization" of blood
production may induce oscillations in the Hb desctruction rate.
Nor oscillations can be thought as generated by a sort of physician dependent
feedback since physicians can control only the timing of transfusions but not the rate
of hemoglobin destruction.
Moreover, from table 2 one can notice that all of the second order
autoregressive coefficients are significantly negative, whereas the first order ones can
be zero, slightly positive or slightly negative. This leads to different types of power
spectra. Whether or not these different typologies correspond to different clinical
pictures and/or to different genetic structures is the object of an ongoing analysis.
Finally, from the pragmatical point of view, one must notice that
autocorrelograms indicate that the physicians' rule of thumb of projecting the past rate
of Hb destruction into the near future must be partially revised because of the inverse
relationship at 20 days lag and of the different values of the second order
autoregressive coefficient.
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