modelling the feto-placental circulation: i. a distributed network predicting umbilical...

Ultras,~und in Med. & Bi~l. Vol. 18, Nos. 6/7, pp. 535-544. 1992 0301-5629/92 $5.00 + .00 Printed in lhe U.S.A. © 1992 Pergamon Press Ltd.

eOriginal Contribution

M O D E L L I N G T H E F E T O - P L A C E N T A L C I R C U L A T I O N : 1.

A D I S T R I B U T E D N E T W O R K P R E D I C T I N G U M B I L I C A L

H A E M O D Y N A M I C S T H R O U G H O U T P R E G N A N C Y

CATERINA GUIOT, t PIER GIORGIO PIANTA $ and TULLIA TODROS* t Dip. Anatomia e Fisiologia Umana dell'Universit~ di Torino, *Centro Studi Dinamica dei Fluidi CNR, and

*Ist. Ginecologia e Ostetricia dell'Universitfi di Torino, Torino, Italy

(Received 14 September 1991 ; in final form 26 February 1992 )

Abstract--The modifications of the Doppler flow velocity parameters occurring in the feto-placental circulation throughout pregnancy have been reproduced on the basis of a mathematical model. Some simple assumptions were made, such as the progressive development of a dichotomous villous vessel network and the increase of the perfusion pressure and of the umbilical arteries dimensions throughout pregnancy. Moreover, both the viscous and capacitive characteristics of the vascular bed were taken into consideration in order to predict the mean values of blood volume, flow and velocity and the pulsatility index. Their value is shown to depend on few parameters, and mainly on the cross-sectional area ratio between the vessels belonging to two succeeding generations.

Key Words: Mathematical model, Doppler ultrasound, Flow velocity waveforms, Pulsatility index, Haemodynam- ics, Fetal circulation, Placental blood flow.

INTRODUCTION

The study of Doppler Flow Velocity Waveforms (EVW's) of the umbilical artery is becoming a widely used technique in obstetrics, although there is still much debate about its role among the biophysical methods of fetal monitoring. Some indices [A/B ratio (Stuart et al. 1980), RI (Pourcelot 1974) and PI (Gosling and King 1975)] are used to describe the waveforms.

The physio-pathological phenomena underlying the modifications of the FVW's have not yet been completely elucidated. It is assumed that the main determinant in changing FVW's indices is the down- stream impedance of the vascular bed which, in the particular case of the umbilical circulation, is thought to be determined at the level of the arteries of the tertiary stem villi (Giles et al. 1985; Gudmundsson and Marsal 1988; Trudinger et al. 1985 ). Experimen- tal animal work supports this belief (Noordam et al. 1987).

However, it is well known that the shape of the Doppler FVW's also depends on the input pressure,

Address correspondence to: Caterina Guiot, Dip. Anatomia e Fisiologia Umana dell'Universith di Torino, Corso Raffaello 30, 1-10125 Torino, Italy.

535

cardiac contractility, distance of the sampling site from the heart, vessel wall compliance and reflection (Griffin et al. 1983). How do all these different vari- ables interplay in determining the actual shape of the waveform? A model approach, either based on animal experiments or on physical assumptions, would be valuable as a first step in answering this main ques- tion.

In this paper, we describe a mathematical model aimed at better understanding the physiological changes of the umbilical artery FVW's indices throughout pregnancy. The feto-placental circulation has already been approached by modelling techniques (Reuwer et al. 1986; Thompson and Stevens 1989) on the basis of "lumped" networks predicting the main haemodynamical features. These models mimic the placental vasculature at term only and totally dis- regard vessel branching [except for Thompson and Stevens (1989) who consider two orders of vessels]. A more realistic scheme, describing the feto-placental architecture and its modifications occurring throughout pregnancy, is proposed hereafter.

METHODS

In order to develop the model, some basic data were assumed from animal experiments, pathological

536 Ultrasound in Medicine and Biology Volume 18, Numbers 6/7, 1992

studies of the placental vascular bed and direct measurements obtained in the human fetus.

1. Placental vascular structure The following is a schematic representation of

the placental morphology (Habashi et al. 1983; Kauf- mann 1982; Kaufmann et al. 1988; Lee and Yeh 1983 ). After the insertion of the umbilical arteries in the chorionic plate, a progressive dichotomous arterial branching occurs giving rise to 16-24 chorionic arteries. Further branching originates the feeding arteries of the placental villi, and the vascular architecture develops according to the progressive formation of primary, secondary and tertiary stem villi.

Such a structure was mimicked with a network of branching vessels of progressively reduced cross-sectional area and length, connecting the umbilical artery with the capillary bed, and its symmetric counter- part linking the capillaries with the umbilical vein (Fig. 1). The placental vasculature at term can be therefore represented as a dichotomous tree with 14- 15 generations of branches. The first generations of villous vessels probably start at around 8 weeks of gestational age and the rate of proliferation, always remarkable from that time on, is nevertheless known to slow down during the last trimester ( Reuwer et al. 1986 ). It is therefore very difficult to relate the development of each generation of vessels with the fetal gestational age. It seems reasonable, in any case, to assume that the mid-pregnancy vascular tree is already well developed (around the 8th generation).

The diameter of the umbilical artery is assumed to increase with gestational age. On the basis of direct 2D echographic measurements, we assumed two possible values for this diameter at term, namely 3 mm and 4 mm.

The ratio between the total cross-sectional area A of the vessels belonging to two succeeding generations, represented by the parameter k, is assumed to be ranging between 1 and 1.5 (see eqn A1 in the Ap- pendix). The cross-sectional area of any vessel at the nth generation can be related to that of the umbilical artery A0 (eqn A2). The ratio between the lengths of the branches of two succeeding generations has been assumed to be constant as well (eqns A3, A4, A5). The variation of A0 was taken into account by eqn (A9).

With the previous assumptions, the total length of the vasculature, its volume and the overall resistance can be predicted (eqns A6, A8) provided the Hagen-Poiseuille law (eqn A7) is assumed to hold. This hypothesis is supported by the absence of turbu- lence even in the largest feto-placental vessels [a Reynolds number between 130 and 1060 in the umbilical arteries is reported in Thompson and Stevens (1989)].

To account for the fact that the Doppler flow measurements in the umbilical artery of the human fetus are usually performed at some distance from the insertion of the chorionic plate, a piece of umbilical artery at the inlet of the system (and of umbilical vein at the outlet) has been considered. This varies its diameter and length throughout pregnancy: its cross-sectional area isA0, while its length is related to the diameter and is assumed to reach 10 cm or 20 cm at term.

2. Pressures Although data on the blood pressure measured at

the arterial and venous side of the placental circulation are not exhaustive, we assumed both the mean and the pulse arterial pressure to increase with gestational age (Dawes 1968; Johnson et al. 1991; Reuwer

ii

• - '5 " " i . ~

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,

............. ~ .......... I'?7ZS;--~-'J" . . . . . I--.-.-i .............. " '- .......... E:Z2";:--;-~ . . . .

" . . . . . . . . . U ' ; : : : - - : - - - - . : . . . . J

. . . . . . . . . . i . . . . . . . . . . ! : ; : ' . ' . " . ' : . ' ~ : Z ! - . - . ~ ' - " 'q. I . . . . . . . .

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: . . . . . . . . i : ' . ' ."" ."." Z . ' . - _ . ' ! - . . _ . . ~

Fig. 1. Schematic drawing of the placental network•

Modelling the feto-placental circulation: I e C. GUIOT et al. 537

et al. 1986). Since pressure oscillations are known to be almost completely damped along the vascular bed, the venous pressure is considered to be steady.

For the sake of simplicity, the assumption is made of a simple pressure waveform, represented by a mean term Po and a sinusoidal term P~ (eqn A11 ). The mean value P0 is assumed to increase throughout pregnancy, according to eqn (A10), to the final value of 50 mmHg at term. The pulsatile term P1 is known to increase as well, in approximately the same propor- tion of the mean pressure, from mid-pregnancy until term (Johnson et al. 1991; Reuwer et al. 1986), but nothing is known about the first half of gestation. Therefore, the assumption that the pulsatility PIP of the arterial pressure (PIP = 2P,/Po) stays approximately constant throughout pregnancy is validated only for the last 5-7 generations. For calculations, we assumed PIP = 0.3, which was inferred from the available data previously quoted. The venous pressure was always assumed to be constant and equal to 5 mmHg.

3. Elastic properties of the ~vstem Taking elasticity into account, two main new fea-

tures can be introduced: (a) vessel expansion in order to store a variable volume of fluid depending on the applied pressure; and (b) pressure waveform propagation with a finite velocity c and partial reflection when sudden geometric and elastic properties variations occur.

In the present model, we disregarded the propagation effects. The vessel capacity was mimicked in the following way. Since a non-compliant vascular bed is basically equivalent to an electric resistive network consisting of a series of parallels of "generation" units, each element representing a vessel is assumed to be a pure resistor. Taking vessel compliance into account amounts to adding a capacitor parallel to the resistor.

A single vessel is therefore, in electrical terms, a "double bipole" connecting two voltage generators which stay for the input and output pressure across the vessel (Fig. 2 ).

Vessel compliance was related to the propagation velocity c by eqn (A 12). Although c probably varies from arteries to tertiary villous vessels, and its value changes during pregnancy as well, we assumed a constant reference value (250 cm/s or 350 cm/s).

4. Heart rate and blood viscosity The Heart Rate (HR) frequency is known to in-

crease in the fetus from about 125 to 175 beats per minute (bpm) between 7 and 9 weeks and decreases from about 175 to 145 bpm between 9 and 15 weeks, assuming later values ranging between 160 and 120

s ~ / N / N ~ :

A Vi "-' -C , , "" Vo

. [ Fig. 2. Electric network (double bipole) assumed to be the analog of each elementary vessel belonging to the placental

vascular tree.

bpm until term. For the sake of simplicity, we took its value to be constant and equal to 120 bpm. Similarly, because of the lack of data on fetal blood viscosity, a constant value of 0.033 poise was assumed for the kinematic viscosity t,.

RESULTS

The variation of a, vs. generations (in logarith- mic scale) assuming k = 1.2 is shown in Fig. 3, while Fig. 4 refers to vessel length variation between 0 and 15 generations, and Fig. 5 to the overall blood volume.

The behaviour of the total resistance R(n) vs. n for 3 different values of k, namely 1.2, 1.25 and 1.3, shows that R(n) generally decreases with the number of generations but, when k is small enough, it can eventually increase. This suggests the existence of a limiting value for k, which obviously depends on the assumption made with Ao: in our case, the limiting value is between 1.2 and 1.25 (Fig. 6).

If the mean value of the blood pressure across the placental vasculature is known, the mean blood flow and the mean blood velocity can be estimated as well. Figures 7 and 8 respectively show how the predicted mean blood flow and mean blood velocity change with generations and for different k values in the umbilical arteries. These figures have been obtained by adding to the circulatory placental system a portion (20 cm at term) of umbilical artery and vein respectively at the inlet and the outlet.

The pulsatility index (Pl), computed by means of eqns (A13-AI6), depends both on the vascular mechanical properties and the pressure conditions and mainly on PIP. To keep these two contributions distinct, we studied the quantity 0.3 PI/PIP, which corresponds to PI during the second half of pregnancy, when it is known that PIP is constant (Reuwer et al. 1986), and we assumed PIP = 0.3 (see 2. Pres- SlIfeS).

Figure 9 represents the predicted 0.3 PI/PIP vs. n at different values ofk when the umbilical artery tract

538 Ultrasound in Medicine and Biology

a,~ ( c m 21

Volume 18, Numbers 6/7, 1992

0 1 0 - -

-1 '10--

-2 1 0 - -

-3 1 0 - -

-4 10 - -

[] []

[] n

D []

D []

D []

[]

D []

D D

lq

I I J 1 J I gene~auon 0 3 6 9 12 15

Fig. 3. Cross-sectional area of a villous vessel predicted by the model vs, generations when k = 1.2 (umbilical artery diameter at term = 4 ram).

is 20 cm at term, Figures lOa and lOb compare the t rend of PI for k = 1.2 and 1.4 (c = 250 cm/s and L = 20 cm) in two different condit ions: in Fig. lOa it is assumed PIP = 0.3 th roughou t pregnancy, while in Fig. 10b it is assumed P~ = constant and having a value such that PIP = 0.3 when n = 8, but higher than 0.3 at the preceding generat ions and lower at the following ones.

D I S C U S S I O N

Some model results are in full agreement with the experimental data, while others need further re- finement.

1. Vessel dimensions The est imation o f the vessel d imensions gives as

predicted diameter o f the 15th generat ion o f vessels a

[ . ( c m )

3 - -

2 - -

I -

[ ]

D

[3

[ ]

[ ] [ ]

I I I 0 3 6

[]

[] [] [] [] [] D j j ~ 9enera l ion 9 12 15

Fig. 4. Length of a villous vessel predicted by the model vs. generations when k = 1.2 (umbilical artery diameter at term = 4 ram).

Modelling the feto-placental circulation: 1 • C. GUIOT et al. 539

v<c 3)

60--

30--

D,,

l> •

. .., .. , . . . ~ ~ p., a a ~ " '-' ° ° °

J ~ ~ r , j ~ ~ y e n e r a . o n 0 3 6 9 12 15

Fig. 5. Total blood volume in the fetal placental vasculature vs, generations for different values o f k (square: k = 1.3; dot: k = 1.4; triangle: k = 1.5), umbilical artery diameter at term = 3 ram. (Since our model refers to one umbilical artery only, the predicted value is expected to be one half of the experimental one.)

value (80 microns when the umbilical artery diameter at term is equal to 4 ram, 60 microns when it is assumed to be 3 mm) slightly larger than the ones measured from vinilic casts (Kaufmann et al. 1988 ). The reasons for this difference may be: • the fact that we assumed the viscosity not to change

with the vessel diameter, thus disregarding impor-

tant effects related to the haematocrit reduction (Pries et al. 1990);

• the assumption of a simple relationship between the cross-sectional areas of succeeding generation vessels [biological systems are normally much more complicated, and a fractal relation would probably fit more properly the experimental data, as was

R(n ) ( g c m 4 s -1)

4 410 --

2.10 4 --

O D O O D O O D D O D ~o

° o I ~ O O 0 O 0 O

~' ' i ~ , ~ ~ g e n e r a t i o n

0 3 6 9 12 15

Fig. 6. Total fetal placental resistance vs. generations for different values of k (square: k = 1.2; dot: k = 1.25; triangle: k = 1.3), umbilical artery diameter at term = 3 ram.

540 Ultrasound in Medicine and Biology Volume 18, Numbers 6/7. 1992

Qo ( cm3~1 )

+

4-

"1" I>

4-- t- ~> •

+ •

2 - • D O O O O O O °n

LI i

, , , , , ger~erauorl

0 3 6 9 12 15

Fig. 7. Mean blood flow in the umbil ical artery vs. generations for different values o f k (square: k = 1.2; dot: k = 1.3; triangle: k = 1.4; cross: k = 1.5), umbi l ica l artery d i ame te r at t e rm = 3 m m . (Since our model refers to one

umbi l ica l artery only, the predicted value is expected to be one ha l f o f the venous b lood flow.)

shown for the bronchial tree (West et al. 1986) and for the pulmonary vasculature (Lefevre 1983)];

• the fact that the vessels we simulate are much more pressurized than those considered in casts, giving rise to a disagreement in size.

2. Resistance and impedance of the vascular system The prediction of a "critical value" for k between

two different trends, namely a situation in which the vascular resistance decreases with the number of generations and an opposite situation in which the resis-

U o ( c m ~1)

0-

40-

I

I

f

0

÷

4. c> t> •

r7 O 0

÷ ÷

i>

÷ ÷

t>

I>

÷

÷

t> t>

O O O O O o

÷ 4-

I>

o m

, , , , , gene ra l i on 3 6 9 12 15

Fig. 8. M e a n b lood velocity in the umbi l ica l ar tery vs. genera t ions for different values of k (square: k = 1.2; dot: k = 1.3; triangle: k = 1.4; cross: k = 1.5), umbil ical artery d i ame te r at t e rm = 3 m m .

Modelling the feto-placenta| circulation: 1 • C. GUIOT et al. 541

0.3 PI/PIP

3 6

2 -

I l - I T

E] El

Q

[]

[]

13

El

El

[3

I I I I I I generation 0 3 6 9 12 15

Fig. 9. PI/PIP vs. generations for different values ofk (square: k = 1.2; dot: k = 1.3; triangle: k = 1.4; cross: k = 1.5), umbilical artery diameter at term = 4 ram, c = 250 cm/s, umbilical artery length = 20 cm.

tance increases, is in full agreement with previous findings (Gudmundsson and Marsal 1988; Stuart et al. 1980). Both in experimental models and in clinical practice, in fact, normal pregnancies show a progressive fall in the placental resistance, while complicated pregnancies are commonly related to non-decreasing resistance (Dawes 1968; Giles et al. 1985).

The absolute value predicted for the total resistance when the umbilical artery diameter at term is 3 mm is comparable with the values ranging between 10 9 and 4 10 9 N m -5 s, quoted in the literature (Thompson and Stevens 1989) on the basis of experimental data. Larger diameters produce a lower total resistance.

Only one estimate of the vascular impedance is available for comparison. Since in our model only the mean value and the amplitude of the sinusoidal term of the pressure and the flow are taken into account (their actual waveform is considered unknown), we can compute the modulus and the phase of the system impedance for the 0th and the first harmonics only. The predicted values, nevertheless, show the same behaviour of those computed from the experimental haemodynamical data on the fetal sheep (Adamson and Langille 1991 ).

3. Blood volume, mean flow and velocity The blood volume predicted in Fig. 5 varies sub-

stantially with k, and represents approximately half the quantity of fetal blood filling the placenta, since only one umbilical artery was taken into account. It is

very difficult to get experimental estimations for this parameter: as far as we know, in only one previous study (Yao et al. 1969) a value of about 125 mL at term is quoted, which is comparable with that predicted by our model when k ranges between 1.4 and 1.5 and the umbilical artery diameter at term is 3 mm. In the other case (diameter equal to 4 ram), a reasonable value of the total blood volume can be already obtained at k -- 1.4.

Figures 6 and 7 represent the mean values of the blood flow and velocity, respectively. Their values depend crucially on that of the total vascular resistance (eqn A15). The mean blood flow predicted by the model pertains to a single umbilical artery, and should therefore be compared with one half of that measured in the umbilical vein. The values reported in the literature (Dawes 1968) for the fetal sheep at term range between 150 and 200 mL min J Kg- ' , while for the human fetus lower values, between 100 and 120 mL min -~ Kg -~, are quoted (Eik-nes et al. 1980). Our results are in better agreement with the data coming from the animal models, since a value of about 300 mL/min is predicted when k = 1.4 and the umbilical artery diameter at term is 3 mm. The mean blood velocity predicted by the model is slightly larger than that measured by pulsed Doppler in the human fetal aorta (Tonge and Wladimiroff 1986).

4. Pulsatility index As far as the prediction of PI is concerned, the

main limits are due to ignorance about: (a) the pulsa-


Pt

3 - -

2 - -

1 - -

0 Q

[ ] • • •

Q []

Q • •

[]

[]

I I I 1 I 0 3 6 9 12

[]

[]

l g e n e r a t i o n 15

(a)

4 - -

2 - R ~ ~ O O O O U O O U O O D O

• • • 0 0 0 0

r ~ I 1 z l generalion 0 3 6 9 12 15

(b)

Fig. 10. PI vs. generations For different values o fk (square: k = 1.2; dot: k = 1.4), umbil ical artery diameter at term = 4 ram, c = 250 cm/s. (a) PIP is kept constant at 0.3; (b) PIP decreases (see text).

tility of the arterial pressure in the umbilical cord, which strongly affects the final results; and (b) the elasticproperties of the vessels, possibly variable with their dimensions and hystological characteristics.

The effect of the length of the umbilical artery is similar to that observed in vivo where the PI is higher when the measurements are performed at the abdomi- nal insertion of the umbilical cord rather than at the placental insertion. However, the main determinant

of the PI is the k value which affects its magnitude and behaviour at varying n: in general, small values of k generate an increase of PI with the number of generations while a large enough k can be responsible for its decrease, at least for the last generations.

5. Final considerations Finally, it has to be stressed that our model com-

pletely disregards blood inertia, since the contribution

Modelling the feto-placental circulation: 1 • C. GUIOT el al. 543

of the kinetic term to the total pressure has not been considered. This term, on the contrary, is probably quite large at least at the inlet and at the outlet of the system, and m a y p lay a cons iderab le role. Wave p rop- agat ion effects have no t been accoun ted for, ei ther . Both these p h e n o m e n a can be p roper ly m o d e l e d on ly with a fully n o n - l u m p e d mode l , i.e., by direct integra- t ion o f the non - l i nea r f lu id -dynamica l equa t ions . W o r k is in progress for i m p l e m e n t i n g these features in a c o n t i n u o u s d i s t r ibu ted p a r a m e t e r model .

6. Clinical implications The ma in cl inical imp l i ca t ion o f the present

s tudy is tha t bo th the b l o o d pressure in the umbi l i ca l ar ter ies and the character is t ics o f the vascular bed en te r the def in i t ion o f P I ( eqn A 16), a n d so a b n o r m a l umbi l i ca l PI values t h r o u g h o u t p regnancy can depend bo th on a defect ive d e v e l o p m e n t o f the fetal hear t and ar ter ia l system or on an i m p a i r e d feto-placenta l vascula ture . This cou ld also suggest some spec- u la t ions abou t the fact tha t different values o f umbi l i - cal PI ( f rom n o r m a l to ex t r eme ly a b n o r m a l ) are found in In t r a -Ute r ine G r o w t h R e t a r d e d ( I U G R ) fe- tuses. Cases with a b n o r m a l values can be due to bo th a h igher resis tance and a lower b lood pressure with respect to the phys io logica l values, while cases with n o r m a l or sl ightly a b n o r m a l va lues can be due to a higher resis tance c o m p e n s a t e d by a h igher b lood pressure.

C O N C L U S I O N S

On the basis o f very s imple hypotheses on the p lacenta l c i rcu la to ry ne twork and its h a e m o d y n a m - ics, m a n y expe r imen ta l findings, bo th f rom cl inical pract ice and f rom an ima l measu remen t s , can be dis- cussed and re la ted to a few cont ro l l ing parameters . M a n y o f t h e m have been scarcely inves t iga ted in the past, and a severe test o f the m o d e l p red ic t ions would be p r o b l e m a t i c at this stage.

T h e ma in result is tha t a r educ t ion o f the ra t io k affects bo th the vascular resis tance and impedance , a n d even tua l ly the pulsa t i l i ty index, causing it to increase t h roughou t pregnancy . Ref inemen t s o f the theory, inc lud ing p r o p a g a t i o n p h e n o m e n a and iner- t ial effects, are expected to i m p r o v e the mode l ' s accu- racy and are requ i red to m a k e extensive use o f it.

Acknowledgment--This research has been supported by the Italian National Research Council (CNR), targeted project: "Prevention and Control of Disease Factors," subproject SP7 no. 9100093PF41.

REFERENCES

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bilico-placental circulation. Eighteenth International Meeting of The Society for the Study of Fetal Physiology, De Eemhof, The Netherlands: 1991.

Dawes, G. S. Fetal and neonatal physiology. Chicago: Year Book Medical Publ.; 1968,

Desoer, C. A.: Kuh, S. Basic circuit theory. New York: McGraw- Hill Inc.: 1969.

Eik-nes, S. H.: Brubakk, A. O.; Ulstein, N. K. Measurement of human fetal blood flow. Br. Med. J. 280:283-287; 1980.

Giles, W. B.: Trudinger, B. J.; Baird, P. J. Fetal umbilical artery flow velocity waveforms and placental resistance: Pathological correlation. Br. J. Obstet. Gynaecol. 92:31-38: 1985.

Gosling, R. G.; King, D. H. Ultrasound angiology. In: Marcus, A. W.: Adamson, L. S., eds. Arteries and veins. Edinburgh: Churchill Livingstone; 1975:61-98.

Griffin, D.: Cohen-Overbeek, T.: Campbell, S, Fetal and uteroplacental blood flow. Clin. Obstet. Gynecol. 10:565-602: 1983.

Gudmundsson, S.: Marsal, K. Umbilical artery and uteroplacental blood flow velocity waveforms in normal pregnancy--A cross- sectional study. Acta Obstet. Gynecol. Scand. 67:347-354; 1988.

Habashi, S.; Burton, G. J.; Steven, D. H. Morphological study of the fetal vasculature of the human term placenta: Scanning elec- tron microscopy of corrosion casts. Placenta 4:41-56: 1983.

Johnson, P.: Allan, L. D.; Tynan, M.: Maxwell, D. J. Fetal cardio- vascular pressures: Direct measurement, interpretation and clinical application. Fourth International Conference on Fetal and Neonatal Physiological Measurements, Noordwenhorst, The Netherlands, May 12-15: 1991,

Kaufmann, P. Development and differentiation of the human placental villous tree. Bibl. Anat. 22:29-39; 1982.

Kaufmann. P.: Luckhardt, M.: Leiser, R. Three-dimensional representation of the fetal vessel system in the human placenta. Tro- phoblast Res. 3:113-137; 1988.

Lee, M. L. L.: Yeh, M. N. Fetal circulation of the placenta: A com- parative study of human and baboon placenta by scanning elec- tron microscopy of vascular casts. Placenta 4:515-526:1983.

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Noordam, N. J,: Wladimiroff, L. W.: Lotgering, F. K.: Strujk, P. C.; Tonge, N. H. Fetal blood flow velocity wavelbrms in relation to changing peripheral vascular resistance. Early Hum. Dev. 15:119-127; 1987.

Pourcelot. L. Applications cliniques de l'examen Doppler transcu- tane. In: Peronneau. P., ed. Velocimetric ultrasonor Doppler. 1974:213-240.

Pries, A. R.: Secomb, T. W.; Gaehtgens, P.: Gross, F. Blood flow in microvascular networks. Circ. Res. 67:826-834: 1990.

Reuwer, P. J. H. M.: Nuyen, W, C.; Beijer, H. J. M.; Heethaar, R. M.: Haspels, A. A.: Bruinse, H. W. Feto-placental circulatory competence. Eur. J. Obstet. Gynecol. Reprod. Biol. 21:15-26; 1986.

Stuart, B.: Drumm, J.; Fitzgerard, D. E.; Duigan, N. M. Fetal blood velocity waveforms in normal pregnancy. Br. J. Obstet. Gynae- col. 87:780-785: 1980.

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Trudinger, B. J.: Giles, W. B.; Cook, C. M.; Bombardieri, J.; Col- lins, L. Fetal umbilical artery flow velocity waveforms and placental resistance: Clinical significance. Br. J. Obstet. Gynaecol. 92:23-30: 1985.

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A P P E N D I X

The ratio between the total cross-sectional area A of the vessels belonging to two succeeding generations (n - 1 )tfi and nth is given by:

k = A./A._~ (A1)

and the cross-sectional area a. of the single vessel belonging to the nth generation is related to the umbilical artery cross-sectional area A o as:

an - (k /2)"Ao. (A2)

A similar law can reasonably hold for the vessels length/, and a new parameter h is introduced as

h - [~/ln , (A3)

A direct proportional relationship between the vessel diameter and its length was assumed:

1, = M - 2 " ~a./Tr (A4)

(we performed the calculation with M = 8). Then h can be expressed in terms of k as:

h = k~72. (AS)

When the geometry of the system is known, it is possible to express the total length L of the vascular network and the total blood volume Vat any given generation n:

L(n ) = 2/o(1 - h"+~)/(l - h),

V(n) = 2/oAo(1 - (hk)"+') / ( l - hk). (A6)

The viscous resistance r is expressed, for each single vessel o f nth generation, by means of the Hagen-Poiseuille law (p is the blood density, equal to 1 g c m 3):

G = 8rrpvl./a:. (A7)

and the total resistance is:

R(n) = 2r0(l - (2h /k2 )"+ ' ) / ( l - 2h / k 2) (AS)

where R (n) is an increasing function of n as long as ro is assumed to be constant. Since the umbilical arteries diameter increases during pregnancy, however, ro and R (n) can decrease.

The law relating the umbilical artery cross-sectional area measured when the system has reached the nth generation with its value at term (n = N) is:

Ao(n) = Ao(N)( (n + 1 ) / ( N + 1)) 2/3 , (A9)

Inserting (A9) in (A2) and then (A2) in (A7) and at last (A7) in (A8), the total vascular resistance can be evaluated.

The mean arterial pressure P0 increases until the term value at n = N as:

P o ( n ) - P o ( N ) ( ( n + I ) / ( N + 1)) 2/s (AI0)

and the feeding pressure P is assumed to be sinusoidal (plus an offset term):

P ~ Po + Ptexp(i~°t) (AI 1 )

where w = 27rHR, Pressure and flow at the inlet can be connected to their counterparts at the outlet via a transmission matrix B (Desoer and Kuh 1969), whose elements are linear combinations of the vessel resistance r. , its capacity (7. and the circular frequency w.

The vessel belonging to the nth generation will be therefore characterized by its viscous resistance (A7) and its capacity:

C. = l~an/pc ~, (AI2)

c being the wave propagation velocity of the pressure pulse. The "transmission matrix" description of a "double bipole"

electric network like the one described in Fig. 2 expresses the Kirch- hofflaws for the circuits. The input parameters V~ and I~ are related to their output counterparts Vo and Io by the transmission matrix B:

l~lli= B V° - to

B(I , 1 )= 1,

B( I , 2) - r.;

B(2, 1) = iwc.;

B(2, 2) = 1 + io~rnc.. (AI3)

Since a parallel of q equal networks has the same electric properties of a single network whose resistance is 1/q times the original value and whose capacitance is q times the original capacitance, the branched structure is equivalent to a series disposition of networks with proper values of resistance and capacitance.

The branched structure can therefore be mimicked with a series of parallels o f equal elementary "'rnc,'" networks. To each generation pertains a proper transmission matrix, and the series arrangement amounts to multiplying all the matrices up to twice the maximum number of generations N considered:

n ~ 2 N

T(N) = [ I B , . (A14) n = 0

If a piece of umbilical artery and vein is added to the system, the corresponding matrices Ba and By are to be multiplied respectively at the beginning and the end of the expression (AI4) to get T(N) .

By denoting as Qo and Q~ respectively the mean and the pulsating values of the flow, which can be written in the same form of (AI l ) apart from a phase term, the final expressions:

Qo = (1°o - Pout,o)/R(N)

Q, - (T(2, 2 ) / T ( I , 2))P, - (det T / T ( I , 2))Pout,, (AI5)

can be written, being respectively Pout,o and Pout,~ the mean and the pulsating components of the outlet pressure,

In what follows, we will assume that the pressure oscillation is completely attenuated by the system, so that

Pom, I = 0.

We finally get the pulsatility index PI as

PI = 2Q~/Qo - (T(2, 2 ) R ( N ) / T ( 1, 2))PIP (A16)

where

PIP = 2Pl/Po

which clearly shows evidence of the fact that PI depends both on the vascular properties and on the inlet pressure waveform.

modelling the feto-placental circulation: i. a distributed network predicting umbilical...

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