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:

..

vc finite with r=0
vp =0 with r=R
vc =vp with r=R-S

..

Resolving the system the total capacity is obtained:
..

where DP is the difference of pressure between two sections of the vessel at distance L.
From the comparison with the formula of Poiseuille:

..

we conclude that in the examined case we can introduce the equivalent viscosity:

..

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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