Study and Validation of a Model
of Fetoplacental Circulation
1.1. Reologia del sangue - Blood Rheology
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Il sangue è un tessuto fluido le cui cellule sono sospese in un
mezzo liquido detto plasma. Per questo motivo presenta caratteristiche
fluidodinamiche particolari che, data la loro influenza nelle trattazione
presente, vengono descritte nei paragrafi che seguono.
1.1. Blood Rheology .
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Blood is a fluid tissue, with solid cells in a liquid component: the
plasma. For this reason its fluidodinamic characteristics are peculiar.
Due to the importance of such characteristics in this study, they are described
in the following paragraphs.
1.1.1. Viscosity . 
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The blood viscosity is function of the protein concentration
[1], of the haematocrit (Ht), of the pH of
plasma [2][3][4], and also of the temperature
(this dependence is negligible in physiological condition).
Blood is considered as a Newtonian fluid for high values of the
gradient dv/dn (for the arterial flow) and not
Newtonian for low values, because in these circumstances the development
of groups of 7-10 erythrocytes (rouleaux) is common; in this last
case the viscosity is no longer constant. Starting from a mean value of
about 3,51 cP (with share rate = 100 sec.-1), the viscosity grows (for
share rates up to 0,1 sec.-1) until 57,09 cP [5].
Therefore the knowledge of the viscosity, including its variations,
is extremely important in the study of the blood rheology.
Among the main factors from which the blood viscosity depends, the
haematocrit is quite effective. The erythrocytes tend to reduce the speed
gradient dv/dn. This is the reason why the viscous term rises. Such increase
is reduced by the deformability of the erythrocytes (that reduces the effect
between the speeds of adjacent fluid threads) [4].
The formula of Bull describes this dependence:
µs = µp*(1+2,5*Ht)
considering:
µs = blood viscosity
µp = viscosity of plasma
Ht = haematocrit
The formula is not accurate for high values of haematocrit, for which
the relation is no longer linear [3].
Fig.2: Absolute viscosity versus the haematocrit in normal blood.
Indicatively, as it is function of several parameters, we can say that
for Ht=45% the viscosity of blood is:
at 20ºC µ=3,45 cP
at 37ºC µ=2,72 cP
and that is it decreases with the temperature, even if in the physiological
field of variation of the temperature it can be considered constant: the
relationship µplasma/µwater increases with the temperature
while µwater decreases, so, at least in first approximation, merging
the two tendencies, the viscosity of plasma does not change.
1.1.2. Microcirculation . 
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The above considerations have been reported considering
bigger diameters (at least 0,5 millimeter). Considering smaller vessels,
due to the presence of the erythrocytes, the viscosity decreases until
the minimal value are related to diameters of about 15 µm, under
which the viscous term increases again [6]. The
described phenomenon is called Fåhreus-Lindqvist effect (from the
name of the students who discovered it). It is due to to the tendency of
the erythrocytes to migrate towards the center of the vessel, concentrating
themselves along the axis; in this way they move along the vessel with
higher speed than plasma.
Fig.3: Viscosity of blood versus the diameter of the vessel (µm)
The speed difference indicates that the red globules pass more quickly
along a vessel than plasma and that the haematocrit in the vessel, calculated
in motion, will be lower than the one of the same blood in conditions
of null speed. This effect (auto-dilution), particularly meaningful for
arteries of small diameter, explain the described lessening of viscosity.
As a consequence of the lack of erythrocytes at the surface of the vessel,
a layer of plasma (between 1 and 3 µm) with lower viscosity is created.
Its thickness is function of the diameter of the vessel, the haematocrit,
etc.
We can analyze the influence of this layer on the viscosity of
blood inside the vessel.
We suppose to outline the flow of two not mixable fluids; plasma
flows close to the surface, with thickness s and viscosity µp. The
remaining blood, containing erythrocytes, with viscosity µc.
Fig.4: The
layer of plasma. 
The equation of the motion of the fluid in the vessel, in cylindrical
coordinates, in the mentioned hypothesis can be expressed as null for the
radial and tangential members of the speed, while the axial member is proportional
to the distance from the axis:
vr=0
v0=0
vz=vz(r)
The following two equations, valid respectively for blood and for plasma,
are soon obtained:
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vc finite with r=0
vp =0 with r=R
vc =vp with r=R-S
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Resolving the system the total capacity is obtained:
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where DP is the difference of pressure between
two sections of the vessel at distance L.
From the comparison with the formula of Poiseuille:
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we conclude that in the examined case we can introduce the equivalent
viscosity:
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Thus the viscosity is dependent by the geometry of the vessel and by
the thickness of the plasma layer: considering it constant versus R [4],
decreasing the diameter of the vessel, the equivalent viscosity decreases
too.
The described behavior is part of already explained phenomenon of Fåhreus-Lindqvist.
1.1.3. Differences between Maternal and Fetal Blood .
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The study of the rheology of fetal blood and its
comparison with the one of the adult are of extreme importance to the development
of the model. There are some differences on which it is worth to discuss.
The blood viscosity is slightly lower in the fetus (1,08 cP) that in
the adult (1,37 cP) due to smaller concentration of proteins. On the other
hand the fetal blood has values of haematocrite higher than that one of
the adult. Moreover its erythrocytes, equally deformable, have greater
dimensions; they cause a greater resistance to their passage through vessels
with diameter lower than 5 µm.
These two opposite tendencies in some way compensate, and their simultaneous
action makes the viscosity of the fetal blood similar to that one of the
adult.
The fetus has higher values of haematocrite than its mother. Abnormal
values of the haematocrite and/or of the viscosity are often index of dysfunctions
or pathologies: IUGR, pre-eclampsia etc.[7].
In the following part (cf.. par. 2.2.2) some
tables that resume the various characteristics of the fetal and maternal
blood [8] are reported.
Assuming the mentioned data and applying the formula of Bull, for the
fetal blood flowing along vessels with diameter > 300 µm we can consider:
µ = 1,08*(1+2,5*0,47) = 2,35 cP
For smaller vessels the relations on the equivalent viscosity explained
in the previous paragraph have to be considered.
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Last updated: October 1, 2003