Study and Validation of a Model
of Fetoplacental Circulation

3.2. Risultati del modello Results of the Model |
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Vengono ora presentati i risultati delle elaborazioni effettuate con il modello.
L'utilizzo di un software dedicato all'analisi di circuiti elettronici su un circuito idraulico, avente anche tutte le caratteristiche di variabilitā di un sistema biologico, comporta problemi che sono stati risolti con opportuni artifici.
Le spiegazioni dei paragrafi seguenti fanno riferimento alle figure 75 e 76.

3.2. Results of the Model |
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A software tool dedicated to electric networks is quite flexible, but requires to be adapted to a hydraulic system, using the methods described in the previous chapters.
The explanations of the following chapters refer to figures 75 and 76.
Due to the complexity of the net, the required PC platform is: 80386, 33MHz or superior, with coprocessor, and a plotter.

3.2.1. Physiological Situation |
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The diagrams obtained from the processing show the waves of pressure and flow in several points of the placental circulatory system. They do not refer to the situation in the remaining part of the fetal circulation as (cf. 3.1.10) it has been modeled with a functional block.
In order to make a direct comparison with the values of the medical literature the units of measure adopted are the following:
mmHg for the pressures, cc/min for the flows.
Refer to the appendix for the conversion into SI units.
Fig.78: Onde di portata in aorta fetale discendente (picco sistolico maggiore) e nelle arterie ombelicali, in cc/min.
Waveforms of the blood flow in fetal descendant aorta (green) and umbilical arteries (red), in cc/min.
The diagram of the flow in descendent aorta and in the umbilical arteries obtained with the model (figure 78) coincides with the waves obtained with the Doppler velocimetry.
For a quick comparison, figure 79 is a velocimetry of a normal fetus.
Fig.79: Velocimetria delle arterie ombelicali. Situazione fisiologica.
Velocimetry of the umbilical arteries. Physiological situation.
The wave visualized from the velocimetry can be turned over, due to the position of the probe versus the direction of the flow.
The 2 waves of figure 78 suggests some interesting observations; in particular that, qualitatively:
* the systolic peak is stronger in aorta, where it appears with a smaller delay versus its generation by the heart;
* the value of flow at end diastole is relatively higher in the umbilical arteries than in aorta;
* the pulsatility is smaller in the umbilical arteries than in aorta.
The model gave the following numerical values for the umbilical arteries:
.

systolic peak of flow cc/min | 651
end-diastole flow cc/min | 116
mean flow cc/min | 291
max pressure mmHg | 63.3
min pressure mmHg | 49.6

Fig.80: Curve di pressione (in mmHg) in vari punti della placenta.
Curves of pressure (in mmHg) in several points of the placenta.

3.2.2. Pathological Situations |
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We assumed that a pathology affecting a certain number of placental villi affects also the arteries that feed them and the veins that drain from them. Therefore the simulation of pathologies affecting the fluidodynamics of the placenta has been carried out interrupting the connections with a variable number of cotyledons.
In the simulation this means to exclude one or more branches of the circuit.
Consequently an increase of the total resistance, and a lessening of the capacitance is applied: the time constant t does not change and the shape of the wave of flow in the umbilical arteries remains almost unchanged, as verified in the real case.
Since the element of comparison of the curves obtained during the processing with the model consists in the echo-Doppler analysis of the umbilical arteries, the following diagrams and the numerical calculations refer mainly to such data.

In figure 81, the obtained curves have been analyzed simulating generalized pathologies affecting low percentages (lower than 15%) of villi: qualitatively the variations compared to the physiological situation are quite little, difficult to be noticed with the modern diagnostic systems.

Fig.81: Curve di portata (cc/min) con patologia del 14,28% dei villi.
Flow with 14.28% of pathology.

Increasing the pathological situation (pathology of 30% of the villi of cotyledons), in the model, some changes appear (see figure 82):
1, lessening of the systolic peak of flow in the umbilical arteries (about -3%),
2, lessening of the value of flow at end-diastole, more evident than in aorta (about -35%),
3, lessening of the medium flow (about -10%),
4, accentuation of the slope down in the umbilical arteries.

Fig.82: Curve di portata (cc/min) con patologia del 28,56% dei villi.
Flow with 28.56% of pathology.

Simulating pathologies that affect 60% or more of the villi, the model shows stronger changes compared to the physiological situation (see figure 83):

Fig.83: Curve di portata (cc/min) con patologia del 57,16% dei villi.
Flow with 57.16% of pathology.

1, halving of the capacity at end-diastole in the umbilical arteries,
2, appearance of retrograde flows in aorta,
3, further lessening of the medium flow (about - 20%).
The diagrams show that the placenta, also without regulation systems, can sustain light anomalous situations that are commonly found at the term of gestation. In particular we noticed that the placenta, an extremely primitive organ but however over-dimensioned, at least from the haemodynamic point of view, compensates pathologies that affect up to 15-20% of the villi.

Fig.84: Curve di portata (cc/min) con patologia del 71,44% dei villi.
Flow with 71.44% of pathology.

Figure 85 allows to compare the results of the model.

Fig.85: Confronto delle curve di portata (cc/min) al variare del grado di patologia.
La curva superiore č fisiologica.
Comparison of the waves of flow at different ratios of pathology.
The upper wave is physiological

An interesting aspect of the waves of flow in the umbilical arteries obtained from the simulation is maintenance of their shape: the pulsatility does not change appreciably, and only in case of important pathologies (affecting at least 35% of cotyledons) there is a sort of rounding towards the bottom of the diastolic arc.

Fig.86: Confronto delle curve di pressione (mmHg) nelle arterie ombelicali al variare del grado di patologia.
Comparison of the waves of pressure in umbilical arteries at different ratios of pathology.

The waves of pressure calculated by the model show an increment already with pathologies of 20-30% of vessels. Such changes affects the blood circulation: in a real situation, characterized by a system of regulation of the arterial pressure (still not introduced into our model), the cardiac capacity would decrease, with consequent remarkable worsening of the blood oxygenation. It is not yet possible to carry out clinical verifications on this particular effect, since the measurement of the arterial pressure of a fetus is strongly invasive.

Differently from the results obtained by the models previously developed, also in case of pathologic situation the waveforms of flow match with the velocimetries (supplied by dr Enrico Ferrazzi of the San Paolo hospital of Milan), as evidenced by the figures 87 and 88.

Fig.87: Flussimetria delle arterie ombelicali. Infarto del 5% del parenchima.
Velocimetry of umbilical arteries. Placenta affected by infarctuation of 5% of parenchima.

Comparing the first velocimetry (the placenta, dissected after the childbirth, had infarct of 5% of the tissue dedicated to the exchange of gases) with that physiological one there are not strong differences. The second one refers to a pathological situation more serious (infarct of 12% of the parenchyma), and the peak at end-diastole is almost not detectable.

Fig.88: Flussimetria delle arterie ombelicali. Infarto del 12% del parenchima.
Velocimetry of umbilical arteries. Placenta affected by infarctuation of 12% of parenchima.

The following tables 33, 34 and 35 summarize the data of the simulation.

Pathology % | max cc/min | var. % | min cc/min | var. % | mean cc/min | var. %
14.28 | 639 | -1.84 | 108 | -6.09 | 280 | -3.78
28.56 | 634 | -2.61 | 100 | -13.04 | 268 | -7.90
57.16 | 608 | -6.61 | 73 | -36.52 | 237 | -18.56
71.44 | 574 | -11.83 | 48 | -58.26 | 196 | -32.65
85.72 | 517 | -20.58 | -1 | -100.87 | 133 | -54.30

Tab.33: Confronto di dati relativi alla portata istantanea nelle arterie ombelicali in situazioni di patologia rispetto ai valori fisiologici.
Comparison between flow in umbilical arteries in situations of pathology and the physiological values.

Pathology % | P max mmHg | var % | P min mmHg | var %
14.28 | 66.4 | +4.90 | 51.9 | +4.64
28.56 | 69.5 | +9.79 | 54.6 | +10.08
57.16 | 80.1 | +26.54 | 64.5 | +30.04
71.44 | 89.7 | +41.71 | 73.4 | +47.98
85.72 | 104.6 | +65.24 | 87.8 | +77.02

Tab.34: Confronto di dati relativi alla pressione nelle arterie ombelicali in situazioni di patologia rispetto ai valori fisiologici.
Comparison between pressure in umbilical arteries in situations of pathology and the physiological values.

Pathology % | systolic peak cc/min | var % | end-dyastole flow cc/min | var %
14.28 | 925 | -0.64 | 55 | -8.33
28.56 | 915 | -1.72 | 39 | -35.00
57.16 | 900 | -3.33 | 8 | -86.67
71.44 | 869 | -6.66 | -23 | -138.3
85.72 | 807 | -13.32 | -96 | -260.0

Tab.35: Confronto di dati relativi alla portata in aorta fetale in situazioni di patologia rispetto ai valori fisiologici.
Comparison between flow in fetal aorta in situations of pathology and the physiological values.

Generally adimensional indices are used by physicians (cf. 1.3.3). The advantage is the possibility to quantify in simple way morphologic data of the diagrams, obtaining values independent from the insonation angle.
The diagrams of figures 89, 90 and 91, obtained by the model, show the how such indices and other values of the curves of flow and pressure change with the level of pathology.

Fig.89:A) Andamenti normalizzati dei picchi di portata nelle arterie ombelicali.
B) Andamenti della portata media e della differenza tra massimo e minimo (cc/min).
A) normalized curves of the systolic peak in umbilical arteries.
B) curves of mean flow and differences between min and max values, versus the ratio of pathology.
Fig.90: A e B) Valori dell'onda di pressione nelle arterie ombelicali (mmHg).
Wave of pressure in umbilical arteries (mmHg).
Fig.91: Indici PI, RI e A/B in funzione del grado di patologia.
PI, RI and A/B versus the ratio of pathology.

The model, also under this aspect, supplies results matching with they data in literature [34], even if on the physiological values of such indices there is not a full agreement between the researchers. Commonly the following limits are accepted [34]:
PI 0.65 to 2.8 (Reuwer)
RI 0.6 to 0.9 (Arbeille)
A/B 2.1 to 6.4 (Stuart).
Other authors think that the physiological situation is characterized by a lower pulsatility index (PI < 2).
The values obtained with the simulation are summarized in table 36.

Pathology % | PI | A/B | RI
0 | 1.86 | 5.66 | 0.823
14.28 | 1.90 | 5.91 | 0.831
28.56 | 1.99 | 6.34 | 0.842
57.16 | 2.26 | 8.32 | 0.880
71.44 | 2.68 | 11.96 | 0.916
85.72 | 3.89 | -- | 1.002

Tab.36: Confronto degli indici adimensionali PI, A/B e RI riferiti alla portata nelle arterie ombelicali in situazione fisiologica e patologica.
Comparison of PI, A/B and RI in physiological and pathological situations.

In the model there is an increment of all the indices increasing of the percentage of interrupted rami, as in the real case.
We carried out further calculations, changing the resistances of the chorionic arteries or of the stem-villi only, in order to execute a comparison with a previous simulation [45] that considered this type of localized pathology, but the results are not significative. The diagram of figure 92 allows to compare the variations of the flow in the umbilical arteries due to a doubling of the mentioned resistances. The differences are small. The increase of the resistances of vessels of the chorionic plate cause greater variations respect to a similar increase of the resistances in the stem-villi. Since the minimum (flow at end-diastole) does not change, the index PI is almost unchanged. The increase of 70% of the resistance of the terminal villi does not produce meaningful variations.

Type of pathology | Syst. peak cc/min | var. % | end-diast. cc/min | mean flow cc/min | var % | PI
none | 651 |  | 115 | 288 |  | 1.86
arterie cotiledoni | 638 | -2.0 | 115 | 278 | -3.5 | 1.88
arterie corioniche | 617 | -5.2 | 115 | 267 | -7.3 | 1.88

Tab.37: Confronto di dati di portata nelle arterie ombelicali con patologie caratterizzate dal raddoppio delle resistenze delle sole arterie dei cotiledoni o delle arterie corioniche.
Comparison of data about flow in umbilical arteries with pathology causing doubled resistances of chorionic arteries or arteries of the cotyledons.
Fig.92: Portata (cc/min) nelle arterie ombelicali.
Situazione fisiologica e con raddoppio delle resistenze dei soli stem-villi o delle sole arterie corioniche.
Flow (cc/min) in umbilical arteries.
Physiologic case and doubling the resistances of stem-villi or chorionic arteries.

Still considering figure 85, the reduction of delay of the peak of systolic flow in the umbilical arteries versus its generation by the cardiac contraction is an element of interest. It is resulting by the model but still not confirmed. The diagram of figure 93 shows this result.

Fig.93: Ritardo (ms) del picco sistolico di portata nelle arterie ombelicali in funzione del grado di patologia dei villi.
Delay (ms) of the systolic peak of flow in umbilical arteries versus the ratio of infarctuation of villi

The phenomenon is not depending on the particular values assigned to geometries of vessels, but it is due to the complex structure of the hydraulic network, constituted by a long series of filter elements. We also tried to determine which the importance of a variation of the diameters of the vessels of the chorionic plate, since the values used in the model, obtained from the literature [20][25], could still not be confirmed by the analyses done on the placentas perfused with formalin.
For this reason we carried out a simulation (Monte Carlo analysis) introducing random variations of the resistances of the chorionic arteries and the trunci, within a tolerance of 70%: the figure 94, evidencing that the situation is substantially unchanged, confirms the results of the model.

Fig.94: Portata in arterie ombelicali. Curve generate mediante Analisi Monte Carlo con MICRO-CAP III: simulazione effettuata con variazione casuale delle resistenze delle arterie corioniche e dei trunci entro una tolleranza del 70%.
Blood flow in umbilical arteries (MonteCarlo simulation) applying 70% tolerance to values of resistances of chorionic arteries and trunci.

Therefore we think that it would be opportune to execute a clinical verification of this phenomenon through a synchronization of a echo-Doppler examination of the umbilical arteries with a fetal electrocardiogram (FECG): the instrumentation generally available in a hospital should allow to perform this examination in a non-invasive way.
In case the data on the shortening of the systolic delay was confirmed, we would probably obtain an instrument more reliable than the adimensional indices, finally able to perform early diagnoses. This type of analysis is still independent from the angle of insonation Doppler, and is easy to perform. Moreover the reduction of the systolic delay in function of the level of pathology of the placental villi is appreciable also for low percentages (see table 38).

Rate of pathology % | Delay ms | Variation %
0 | 33.5 | --
14.28 | 31.5 | -6
28.56 | 29.5 | -12
42.84 | 27 | -19.5
57.16 | 24 | -28.5
71.44 | 20.5 | -39
85.72 | 15.5 | -54

Tab.38: Ritardo del picco sistolico di portata nelle arterie ombelicali in funzione del grado di patologia dei villi.
Delay of the systolic peak of flow in umbilical arteries versus the ratio of infarctuation of villi.

The developed model did not introduce any feedback circuits. In the real situation the system fetus-placenta has several systems of regulation, in particular:
1, The placental vessels change their geometry and resistances in function of the partial pressures of gases;
2, The fetal nervous system is able to change the cardiac flow in function of the arterial pressure;
3, It is proved that the blood viscosity is increased by placental pathologies.
These elements, not completely known, can affect the pathological situation. The increased blood viscosity and the regulation of the arterial pressure cause a bigger change of the parameters.

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Last updated: October 1, 2003