Study and Validation of a Model
of Fetoplacental Circulation

3.1. Ipotesi del modello Hypothesis of the Model |
| Next §

Un modello matematico costituisce un'ipotesi semplificata, in termini quantitativi, sul funzionamento di un sistema. Il processo di modellazione comporta una serie di astrazioni che devono essere considerate nel trarre informazioni sul funzionamento del sistema basandosi sui dati forniti dal modello. Poiché l'intero sistema risulta eccessivamente complesso per essere descritto in modo completo, si prende spunto da modelli semplici che ne descrivono solo un particolare. Questi vengono successivamente verificati con osservazioni sperimentali e gradualmente migliorati, quindi possono essere connessi per ottenere il modello globale.
Dal punto di vista della simulazione al computer il sistema cardiovascolare può suddividersi in tre parti: il cuore (simulato mediante un'equazione che rappresenti la curva di portata del sangue), il sistema vascolare, i meccanismi di regolazione (includono fattori chimici e neuronici che regolano le variabili del sistema in funzione del loro valore).
E' stato considerato un modello a parametri concentrati. Si è ricorso ad un analogo elettrico, che è stato analizzato con il calcolatore mediante MICRO-CAP III.

3.1. Hypothesis of the Model |
| Next §

A model allows the quantitative analysis, based on simplified hypothesis, of a more complex system.
The process of modeling consists of several abstractions that must be considered while getting information from the model on how the system works.
The physiologic system is surely more complex and some of its elements cannot be considered by the mathematical analysis, where too heavy simplifications could move the results away from the reality.
Anyway, when the system is too complex to be analyzed in all its aspects, simpler models are used to start the analysis. Then, with further analysis and comparisons with the real case, these models are more and more improved, getting more complex.
In general, considering the cardiovascular system, it can be split into 3 main parts:

- Heart: its model is an equation of the blood flow: Q = Q(t);
- Vascular system: Q and P are functions of time and position;
- Tuning elements: they include chemical and nervous elements that tune the variables of the system. If they are not considered, the system is steady-state, characterized by steady cardiac frequency, mean flow and resistances.

We used an electric model (concentrate parameters), allowing a normal PC to analyze it using MICRO-CAP III (Spectrum Software).
As described by the following chapters, the hypothesis are:

- steady-state model:
- R and C are not functions of time
- steady blood flow per cycle
- concentrate parameters: each group of elements is a RC cell
- vessels characterized by linear elastic function
- fluid:
- viscous, not newtonian
- with negligible inertia
- flow:
- laminar
- with pulsations

3.1.1. System Modelling |
Prev. § | Next §

Figure 57 shows a diagram of the fetoplacental circulation. The relatively big dimensions of the brain and of the placenta, and the considerable length of the funiculus can be observed.

Fig.57: Il sistema circolatorio feto-placentare.
Diagram of the fetoplacentar circulation.

The irregular branching of the vascular tree, the not linear elasticity of vessels, the complex function of the heart flow and the not Newtonian behavior of blood do not facilitate a direct approach.
The placenta and the vessels of the funiculus will be considered with higher accuracy, as our study is finalized to the analysis of the behavior of this field. Reference is made to morphometric data available in literature and calculated from the experiments on placentas perfused with formaldehyde and glutaraldehyde. The rest of the system differs from the real morphology, for the following reasons:

- at the moment the literature is not able to supply data sufficient for an accurate model;
- for our aims it is necessary to guarantee to the placenta conditions close to the real ones;
- in this way it was possible to realize a model extremely elastic, allowing further developments of the single blocks.

In particular the heart has been represented like an element able to supply the real output curve of flow, without reference to the physical structure of the organ.
The following paragraphs describe the obtained relations for the analysis of single aspects of the system, that will be useful in order to define the elements of an equivalent electric net.

Fig.58: Schema a blocchi del modello.
Block-diagram of the model.

Fig.59: Schema della placenta.
Block diagram of the placenta.

3.1.2. Elasticity of the Vessels |
Prev. § | Next §

Due to the characteristic of elasticity of vessels calculated a relationship between blood volume and pressure.
Using a spherical geometry neglects the variation of pressure along the axis of the single section of a vessel represented by the elastic element. The used symbols represent the following quantity:
P = internal pressure
f = force for unit of length on the membrane
V0 = volume at P=0
r0 = beam at P=0.
Imposing the equilibrium to the translation of an element of section we obtain: 2prf = Ppr2, P = 2f/r
Supposing that the elasticity is linear, f = k(r-r0), P = 2k(1-r/r0) and as the result is: .
In particular the gauge of the vessel versus the difference between internal and external pressure is: .
The elasticity constant k depends from the Young constant E of the tissue of the vessel, and from the ratio of the vessel gauge and thickness.

3.1.3. Resistances |
Prev. § | Next §

The law of Hagen-Poiseuille regards the laminar flow of a Newtonian liquid in rigid cylindrical pipes: it can be applied to the circulation if blood and the vascular system obeys to such theoretical principles. The principle of Newton is not directly applicable to blood: the anomalies of the viscous behavior are due to the presence of the cells suspended in the plasma. The importance of this phenomena is however much limited in vessels having diameter bigger than 0.5 millimeters if the speed of the flow is within the physiological values. Therefore the fluid is Newtonian at least in the greater vessels, the flow is laminar, but it is pulsating in all the arterial tree. The consequent accelerations and decelerations of the blood induce variations of kinetic energy not considered in the equation of Poiseuille. Moreover we must consider that the beam of vessels is function of the blood pressure.
The resistance R of a vase section is defined as the difference between the pressure to the section of input and the one at the output, divided by the flow Q: DP/Q
Assuming the flow as a current, the pressure as a voltage and the electrical charge as a volume, we immediately find, for a circuit hydraulic, the validity of the laws of Kirchhoff and the fundamental theorems of the electrical nets (table 27).

hydraulic q. |  | electric q. |
P |  | V |
Q |  | I |
V | = Q×t | Q | = I×t
t |  | t |
C | = Q×t/P | C | = I×t/V
R | = P/Q | R | =V/I
L | = P×t/Q | L | = V×t/I

Tab.27: Corrispondenze tra grandezze idrauliche ed elettriche.
Relationship between hydraulic and electrict quantities.

Replacing the law of Poiseuille in the definition of hydraulic resistance, the relation is obtained: it is function of the gauge of the vessel, according to the obtained relation of the previous paragraph. The complete formula is: .
The tolerances due to the variability of physiological data are higher than the difference (2k-P)4-(2k-Pm)4, since the elastic constants of placental vessels are much greater than the maximum excursions of the pressure. Therefore the value (2k-P)4 is practically constant during the cardiac cycle and equal to (2k-Pm)4, being Pm the mean pressure in the cycle.
These considerations allow to represent the hydraulic resistances in the model by means of normal electrical resistances, of constant value, independent by the pressure drop.

3.1.4. Hydraulic Capacitances |
Prev. § | Next §

The variation of volume of vessels due to varying of the pressure during the cardiac cycle in the model is simulated by means of hydraulic accumulators.
The fundamental law of the capacitance is: C = dQ/dV
being Q the charge and V the applied voltage.
For analogy in the hydraulic case we obtain: C = dV/dP
representing with V the volume of the vessel section and with P the difference of pressure between input and output. Since V=lpr2, replacing the relation seen for r(P) and deriving versus P the result is .
The value (2k-P)3 introduces negligible variations regarding (2k-Pm)3, similarly to the case of resistances: in the model we used normal capacitors of fixed value computed considering the values of Pm.

3.1.5. Inertia |
Prev. § | Next §

The pulsating flow induces accelerations and decelerations, therefore the inertial members of the motion of the blood should be considered (they are represented by inductors in analogous electrical nets).
L = DP/(dF/dt) = DP/aA
where a is the acceleration, A the area of the section. Considering that F = m*a and PA = m*a, L = m/A2 = rl/pr2.
Replacing the expression of r(P) we obtain the complete formula, where we can still replace P, variable in the time, with Pm.
The values for the placental vessels, for the funiculus and for the aorta appear negligible in the range of the frequency that characterizes the wave of the cardiac flow, as other authors wrote [42][43]. Therefore inductors do not appear in our model.

3.1.6. Model of the Vessel |
Prev. § | Next §

All the sections of vessels are characterized by a resistance, calculated considering the dimensions measured or given by the literature, or, as far as the system "periphery", obtained from the ratios of pressure_drop/flow.
The capacitances have been considered only for the vessels interested by pulsating flow, therefore the stiffness of the venous sections, even if characterized by a greater capacity than the arteries [44], are not considered.
The hydraulic accumulators must simulate the internal and external pressures.
We used the following subdivision:

- Arteries of the placenta close to the blood of the intervillous space: the capacitor is connected between the output of the resistance and a point characterized by the steady pressure [14] of the intervillous space. The configuration is like a low-pass filter.
- Arteries of the placenta close to the amniotic fluid: the capacitor are inserted as in the above case, as the pressure of the amniotic fluid is equal to the pressure of the maternal blood in the intervillous space [14].

Fig.60: Collegamento degli elementi capacitivi.
Connection of the capacitors.

- Arteries belonging to the functional modeling: since reference is not made to the physiological situation of these blocks, the capacitors of the network are connected to ground.

The value of the steady pressure applied in this way to the capacitors does not influence the behavior of the model, but it is here just in compliance with the physiological situation.

3.1.7. Placental Vessels |
Prev. § | Next §

The solutions exposed in the 5 previous paragraphs must be applied to the blocks of the placenta introduced at the beginning (figure 59) on the base of the classification of vessels described in the part relative to the physiology.
The vascular net has been translated into a tidy diagram that can represent the peculiar characteristics. We explain the arterial branches, as the venous part is approximately symmetrical. For each of the segments the methods that allow to calculate the values to assign to the members of the model are explained. The employed relations and the numerical calculations are better explained in appendix.

- Chorionic arteries (figure 61):
  Fig.61: Arterie corioniche Corionic arteries

  The sections A represents the continuation of the umbilical arteries in the point of insertion of the funiculus in the placenta. They split 3 times according to Arts [25]. The total section is constant along the distributions.
  The volumes that characterize the vessels have been estimated considering the constants of elasticity k already determined in a study previously carried out near our laboratory [45].

- Arteries of the cotyledons
  We considered a placenta with 50 cotyledons (medium value according to Walsh and Lind [46]), that constitute units in parallel, as shown in figure 62:

  Fig.62: Cotiledone fetale, parte arteriosa
  fetal cotyledon, arterial part

  For the calculation of the resistances it was necessary to obtain the values of gauge and length of each section. The data available in literature [20] quantify only the following values:
  1, volume occupied totally by trunci, rami and ramuli;
  2. lengths of the ramuli;
  3. percentage of lumina in the samples of tissue of trunci, rami and ramuli.
  On the base of these values we estimated first the number of elements in parallel for every section, considering that:
  1. rami and ramuli split by 2
  2. the trunci are as much as the cotyledons
  3. from a truncus approximately eight rami of the first order are generated
  4. the rami of the tenth order are approximately six millions: they originate 44 to 48 millions terminal villi.
  We noticed an increment in the lumina: the area is is 0.19 cm2 at the level of chorionic vessels, 5 cm2 (weighed average) in rami, 62 cm2 in the ramuli, 70 cm2 in the mature interm. villi. In order to quantify the radius of each section of vessels pertaining to rami and ramuli we assumed, separately for rami and ramuli, a linear increment of such section that could allow the connection with the known values of adjacent structures respecting the averaged values mentioned.

  Fig.63: Linearizzazione dell'incremento di sezione totale di passaggio nei vasi placentari.
  Linearization of the section of placental vessels.

  The obtained values are used for the calculation of the resistances, considering Pm = 40 Torr in all the vessels.

- Villi
  From each ramulus of the tenth order 2 mature interm. villi are departed, each of them gives origin to 4 terminal villi in series. 10% of the length of the terminal villi are constituted by the sinusoids, sections characterized by a bigger diameter, that facilitates the transfer of substances between fetal and maternal blood (cf.. par. 1.2.3.).
  The resistance values have been estimated still considering Pm = 40 Torr.

At this point we calculated the total resistance of the placenta with 2 different methods:
1. by means of the sum of the resistances of the single sections
2. dividing the difference of physiological pressure between input of the umbilical arteries and umbilical vein by the blood flow.
The results are:

1,09985*109
1,0669*109

The 2 values are obtained using 2 totally independent ways. They are based on the hypothesis applied to available data the first one, and directly on the base of sure calculations the second one. Their matching, in our opinion, confirms the correctness of the hypotheses and of the methods.

Fig.64: Tratto arterioso dei villi.
Arterial branch of villi.

In order to perform further verifications with the values in literature we calculated the total lengths of vessels pertaining to the single mature interm. villi and terminal villi, and the sum of all the villi: the first sum of lengths, equal to approximately 60 km, matches with the calculations performed by Jackson [17]; the second sum, equal to 85,3 km, confirms the calculations of Knopp [47] and Kaufmann [20].

3.1.8. The Ventricle |
Prev. § | Next §

The block "heart" includes both the left and right side, also if they work in parallel. Swanson e Clark obtained [48] the equation of the cardiac flow during the systole: , considering:
q(t) flow
Q peak value
t time 0<t<TS
TS systole duration.
The following times are known from the fetal ECG:
RR 429 ms
TS 164 ms
SD 13.8 ms
RR is the duration of a cardiac cycle, between 2 consecutive R waves, and SD is the standard deviation. The frequency is 140 b/minute (see figure 65). V is the mean velocity in aorta and D is the vessel diameter. The measurements of these 2 parameters are synchronized with the ECG.
From the diameter we can compute the vessel area; multiplying it by the velocity, the instantaneous flow is obtained.

Fig.65: Flow and diameter of the aorta.

The 2 curves are slightly shifted at their beginning and at their peaks. Lingman suggested that this delay is due to the limited time resolution of the instrument, but probably the beginning of the expansion of the vessel and the velocity are simultaneous.
Anyway the peak of velocity happens before the peak of diameter.
The fetal ventricle is therefore represented by a current generator, whose equation (periodic, with period RR) is the positive arc of a Swanson-Clark wave, synchronized with 0 and TS.
The function of the cardiac flow (see figure 66) is composed by a big number of harmonics as its derivate has discontinuities. In order to obtain such equation for the whole RR interval, we computed the Fourier series of an opportune substitutive equation.

Fig.66: Cardiac flow according to Swanson and Clark

We considered only the systolic diagram, and the values of RR and TS, as a different diagram during the diastolic interval of the wave can be easily zeroed by MICRO-CAP III using a boolean function [49].
In this way the parts with null value of the flow have been substituted by arcs of sinusoid, in order to avoid discontinuities (angles) of the whole function:

Fig.67: Swanson-Clark wave modified with sinusoids

The 2 sinusoids of figure 67, necessary to consider the interval [-T/2;T/2] have the following equations:

as 0.53=2(RR-Ts).
This new wave has been analyzed with the Fourier series. We considered only the first 12 coefficients, as the others are lower than 10-3. The coefficients are listed by Table 28, the new wave compared with the Swanson-Clark one is shown in figure 68.

Calculation of the Fourier series.

a0 | -0.0690316 | b3 | 0.0145081
a1 | 0.0848862 | a4 | -0.00961521
b1 | 0.182702 | b4 | 0.00700865
a2 | -0.00581456 | a5 | -0.00744054
b2 | 0.0316751 | a6 | -0.00327702
a3 | -0.00889979 | a7 | -0.00214784

Tab.28: Coefficients of the Fourier series

Fig.68: Swanson-Clark wave compared to the wave obtained with the Fourier series

The differences are lower than the tolerances of the original equation, so our approximation is considered good enough. The 2 waves have the same shape, and the difference of their areas, equal to the volume of blood ejected during a cycle, in our approximation is 0,142%.
The negative part is 'cut' by MICRO-CAP III defining the 'user function' of the current generator in the following way:
Fx = Fcycle*(F+ ³ 0)
where Fcycle is the Fourier series described, and F+ is a periodic function, with T=RR, positive only in the systolic phase of Fcycle. The value of (F+ ³ 0) is 1 when F+ is not negative, otherwise it is 0.
F+ is: .

Fig.69: The cardiac flow according to our model.

3.1.9.The Atrium |
Prev. § | Next §

The model of the ventricle is not able to represent the whole cardiac behavior: A flow generator between the arterial and the venous systems should generate the same pulsatility, in both the systems. In order to avoid this mistake we have to insert the components able to simulate the atrial function.
The analysis of a model of fetal atrium based on real data is not a target of this thesis. Also in this case we used a functional model: to dump the pulsatility of the venous system we forced a steady pressure at the input of the cardiac generator (in our electric net, a DC voltage generator is used).

3.1.10. The Peripheral Net |
Prev. § | Next §

The block named "periferia" in figure 58 is all that part of the fetoplacental circulation not yet described: all the districts of the body, including the lungs. As we did not obtain reliable clinic data, we used a functional model, without reference to the reality of the vascular net, but only considering the whole fluido-dynamic situation: 60% of the whole cardiac flow passes through the placenta, so the resistance of the peripheral net is 6/4 of the placental resistance, including the funicular vessels (see 1.2.1.).
This model allows the calculation of the changes of the blood flow ratio between the placenta and the rest of the system caused by placental pathologies without feedback systems.
In order to create this functional model, at the beginning we started from the placental model. In particular we analyzed the waves of the flow in abdominal aorta and funicular arteries obtained through the model: they are quite similar to the available velocimetries (see figure 70).

Fig.70: Blood flow in abdominal aorta and umbilical arteries (cc/min), in normal condition, obtained by the placental model of figure 77

We tried to simulate the peripheral net by a resistance and a capacitor in parallel to the placental model: it should allow to obtain the known waveforms. Anyway this simple solution gave wrong results:

Fig.71: Blood flow in abdominal aorta and umbilical arteries (cc/min) after connecting one RC cell.
The ratio between max and min peak is clearly wrong.

Other than the mismatch of the peak values, there is a backward aortic flow, due to the different behavior of the branches in parallel versus the pressure. The RC circuit of the peripheral net caused a different frequency response of the system, compromising the results.
For this reason we started considering a vascular net similar to the placental system. The values of resistance of each part are proportional to the relevant placental values, always considering the ratio 6/4. Moreover we inserted 2 RC cells, called "primi vasi (first vessels)", correspondent to the aorta, with the same t=R*C.

Fig.72: Diagram of the partition of the blood flow in the fetoplacental circulation

The obtained results, considering the 2 parameters of comparison, match exactly the ones obtained by the stand-alone placental model, shown in figure 70.
Thus we considered this simulation able to satisfy our requirements.

Fig.73: Comparison between aortic flow (upper curve) and umbilical flow.
Foreground: curves obtained with the whole peripheral net.
Background: curves obtained with one RC cell.

3.1.11. Values of the Variables |
Prev. § | Next §

The following tables summarize the values applied to the components.
Data on placental morphology have been collected mainly from Kaufmann [20], and Arts [25], as already mentioned (cf. 1.2.). The peak value of the systholic flow was calculated by several researchers [11][12][50]. The values calculated by Lingman and Marsal (peak flow 65 cc/kg/min at 40 weeks, versus 139 cc/kg/min at 28 weeks on a population of only 21 cases) were not considered as mismatching with all the other authors, that performed their studies later, and using more accurate equipment.

|  |  |  | Vasi vessels |  |  | |¬ | Stem | villi |  | ®
|  | Arterie | |¬ | corionici |  | ®| |  | |¬ | rami |  | ®|
|  | funicolo | I | II | III | IV | trunci | I | II | III | IV
N |  | 2 | 2 | 5 | 12 | 25 | 50 | 400 | 800 | 1600 | 3200
r° | m m | 1750 | 1750 | 1100 | 700 | 500 | 325 | 275 | 240 | 200 | 180
l | mm | 520 | 30 | 30 | 30 | 30 | 5 | 10 | 20 | 5 | 5
S | cm2 | 0.19 | 0.19 | 0.19 | 0.19 | 0.19 | 0.17 | 1.04 | 2.75 | 4.5 | 6.2
R | Nsm-5x106 | 166 | 9.6 | 24.5 | 62.3 | 114.9 | 50.62 | 24.57 | 42.37 | 10.89 | 8.26
C | m5N-1x10-12 | 170 | 23.2 | 23.2 | 23.2 | 23.2 | 0.6 | 7.4 | 22.4 | 7.7 | 12.3
L | kg× m-4 | 28 | 3.3 | 3.34 | 3.44 | 3.24 | 0.64 | 0.2 | 0.28 | 0.04 | 0.04
k | Torr | 1320 | 280 | 280 | 280 | 280 | |¬ |  | 1450 |  | ®
D P | Torr | 6.225 | 7.92 | 7.92 | 7.92 | 7.92 | |¬ |  | 6.12 |  | ®

Tab.29/A: Dati dei vasi della placenta. Continua ®

| ¬ |  |  |  | stem | villi |  |  |  | ®| | |¬ | villi | ®|
| |¬ |  |  |  | ramuli |  |  |  |  | ®| | interm. |  |
| I | II | III | IV | V | VI | VII | VIII | IX | X | maturi | term. | sinus.
Nx1000 | 11.75 | 23.5 | 47 | 94 | 187 | 375 | 750 | 1500 | 3000 | 6000 | 12000 | 45000 | --
r° | 150 | 140 | 120 | 98 | 77 | 59 | 44 | 33 | 25 | 18 | 14 | 7.2 | 25
l | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 0.45 | 1 | 0.1
S | 9.4 | 15.7 | 22.1 | 28.3 | 34.6 | 40.9 | 47.2 | 53.5 | 59.8 | 66.1 | 70 | 30 | --
R | 1.01 | 0.664 | 0.609 | 0.676 | 0.870 | 1.22 | 1.89 | 2.85 | 4.09 | 7.05 | 3.70 | 350.5 | 0.372
C | 7 | 12.3 | 17.8 | 24 | 29.4 | 34.7 | 38.7 | 43.4 | 50.3 | 52 | 20 | 250 | --
L | .0014 | .0008 | .0006 | .0004 | .0003 | .0003 | .0003 | .0002 | .0002 | .0002 | .00006 | .002 | .00002
k | ¬ |  |  |  |  | 1450 |  |  |  | ®| | 1850 | 1850 | 1850
D P | ¬ |  |  |  |  | 6.12 |  |  |  | ®| | 0.142 | 13.4 | 0.014

¬ Segue: Tab.29/B: Dati dei vasi della placenta.
¬ Data of placental vessels

R | N× s× m-5 | 1.09985× 10 9
L | kg× m-4 | 14.5
D P | Torr | 41.09

Tab.30: Dati sulla placenta completa.
Values introduced into our model.

| R x106 | D P | L | C x10-12 | t
Aorta | 7.17 | 0.36 | 4.96 | 324.8 | 2.3 ms
Periferia | 2050 | -- | 21.75 | -- | --
Vena omb+dotto+vena cava | 346.7 | 13 | -- | -- | --

Tab.31: Valori per il dimensionamento del modello.
Values introduced into our model.

Cuore Hearth | Q m3× s-1 | 36.12× 10-6
P vena cava | Torr | 4
P amniotica | Torr | 12

Tab.32: Valori per il dimensionamento del modello.
Values introduced into our model.
.

3.1.12. Cardiac Generator |
Prev. § | Next §

MICRO-CAP III allows defining only arbitrary wave voltage generators.
As a current generator was needed, we used a voltage driven current generator, as shown in figure 74. A linear function, with the parameter Q (fetal peak flow), is the correspondent relation. The voltage generator is split into 4 generators F1, F2, F3 e F4, in series, each one containing 3 of the 12 coefficient of the Fourier series previously described, as the complete function was too long to be used by the program. The zeroing of the sum during the diastolic intervals has been obtained zeroing all the function (see fig. 74 and 75).

Fig.74: Parametri del generatore di corrente.
Parameters of the current generator.

Fig.75: Parametri dei generatori di funzione.
Parameters of the function generators.

3.1.13. The Model |
Prev. § | Bottom

Figure 76 shows the equivalent electric net, implemented using MICRO-CAP III.
All the components representing the described blocks have been inserted.
The placenta has been split into 3 branches, connected in parallel, each one including a different percentage of vessels of the cotyledons (14.28%; 28.56% and 57.16%). This trick makes easier to study pathological situations that can be analyzed supposing to clamp several cotyledons, in steps of 14.28%.

Fig.76: Circuito elettrico analogo del modello.
Equivalent electric diagram of the model.

Fig.77: Schema elettrico analogo della placenta.
Equivalent electric diagram limited to the placental net.

Ho me | Abstract | Index | Download | Links | Contact us | 3.2.

best viewed with res. 800 x 600.
Last updated: October 1, 2003