Modelling the feto-placental circulation: I. A distributed network predicting umbilical haemodynamics throughout pregnancy

Ultras,~undin Med. &Bi~l.Vol.18,Nos.6/7, pp. 535-544.1992

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eOriginal Contribution MODELLING THE FETO-PLACENTAL CIRCULATION: 1. A DISTRIBUTED NETWORK PREDICTING UMBILICAL HAEMODYNAMICS THROUGHOUT PREGNANCY 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, Haemodynamics, Fetal circulation, Placental blood flow.

INTRODUCTION

cardiac contractility, distance of the sampling site from the heart, vessel wall compliance and reflection (Griffin et al. 1983). How do all these different variables 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 question. 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 disregard 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.

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 downstream 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 ). Experimental 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,

METHODS Addresscorrespondenceto: Caterina Guiot, Dip. Anatomiae Fisiologia Umana dell'Universith di Torino, Corso Raffaello30, 1-10125 Torino, Italy.

In order to develop the model, some basic data were assumed from animal experiments, pathological 535

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

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 Appendix). 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 turbulence 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.

1. Placental vascular structure The following is a schematic representation of the placental morphology (Habashi et al. 1983; Kaufmann 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 counterpart linking the capillaries with the umbilical vein (Fig. 1). The placental vasculature at term can be therefore represented as a dichotomous tree with 1415 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.

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

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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 proportion 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 features 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 increase 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

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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 logarithmic 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 o f k when the umbilical artery tract

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DISCUSSION

is 20 c m at term, Figures lOa and lOb c o m p a r e the trend o f PI for k = 1.2 and 1.4 (c = 250 cm/s a n d L = 20 cm) in two different conditions: in Fig. lOa it is assumed PIP = 0.3 t h r o u g h o u t pregnancy, while in Fig. 10b it is a s s u m e d P~ = constant and having a value such that PIP = 0.3 when n = 8, but higher than 0.3 at the preceding generations and lower at the following ones.

S o m e model results are in full agreement with the experimental data, while others need further refinement.

1. Vessel dimensions T h e estimation o f the vessel dimensions gives as predicted diameter o f the 15th generation o f vessels a

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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

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2. Resistance and impedance of the vascular system

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.

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-

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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 substantially 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 K g - ' , 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 m L / m i n 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-

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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 abdominal 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 completely disregards blood inertia, since the contribution

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

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, a n d m a y p l a y a c o n s i d e r a b l e role. W a v e p r o p a g a t i o n effects have n o t b e e n a c c o u n t e d for, either. Both these p h e n o m e n a can be p r o p e r l y m o d e l e d o n l y with a fully n o n - l u m p e d m o d e l , i.e., by direct integration o f the n o n - l i n e a r f l u i d - d y n a m i c a l e q u a t i o n s . W o r k is in progress for i m p l e m e n t i n g these features in a continuous distributed parameter model.

6. Clinical implications T h e m a i n clinical i m p l i c a t i o n o f the p r e s e n t study is t h a t b o t h the b l o o d pressure in the u m b i l i c a l arteries a n d the characteristics o f the v a s c u l a r b e d e n t e r the d e f i n i t i o n o f P I ( e q n A 16), a n d so a b n o r m a l u m b i l i c a l PI values t h r o u g h o u t p r e g n a n c y c a n dep e n d b o t h on a defective d e v e l o p m e n t o f the fetal h e a r t a n d arterial system o r o n a n i m p a i r e d feto-placental vasculature. T h i s c o u l d also suggest s o m e specu l a t i o n s a b o u t the fact t h a t different values o f u m b i l i cal PI ( f r o m n o r m a l to e x t r e m e l y a b n o r m a l ) are f o u n d in I n t r a - U t e r i n e G r o w t h R e t a r d e d ( I U G R ) fetuses. Cases with a b n o r m a l values c a n be d u e to b o t h a higher resistance a n d a l o w e r b l o o d pressure with respect to the p h y s i o l o g i c a l values, while cases with n o r m a l or slightly a b n o r m a l v a l u e s c a n be d u e to a higher resistance c o m p e n s a t e d b y a higher b l o o d pressure. CONCLUSIONS O n the basis o f very s i m p l e h y p o t h e s e s o n the p l a c e n t a l c i r c u l a t o r y n e t w o r k a n d its h a e m o d y n a m ics, m a n y e x p e r i m e n t a l findings, b o t h f r o m clinical practice a n d f r o m a n i m a l m e a s u r e m e n t s , can be discussed a n d related to a few c o n t r o l l i n g p a r a m e t e r s . M a n y o f t h e m have b e e n scarcely investigated in the past, a n d a severe test o f the m o d e l p r e d i c t i o n s w o u l d be p r o b l e m a t i c at this stage. T h e m a i n result is t h a t a r e d u c t i o n o f the ratio k affects b o t h the v a s c u l a r resistance a n d i m p e d a n c e , a n d e v e n t u a l l y the pulsatility index, causing it to increase t h r o u g h o u t p r e g n a n c y . R e f i n e m e n t s o f the theory, i n c l u d i n g p r o p a g a t i o n p h e n o m e n a a n d inertial effects, are e x p e c t e d to i m p r o v e the m o d e l ' s accuracy a n d are r e q u i r e d 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 Adamson, S. L.; Langille, B. L. Effect of vasoactive agents and placental embolization on the vascular input impedance of the urn-

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544

Ultrasound in Medicine and Biology APPENDIX

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._~

Volume 18, Numbers 6/7, 1992 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:

(A1) C. = l~an/pc ~,

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)

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 Kirchhofflaws for the circuits. The input parameters V~ and I~ are related to their output counterparts Vo and Io by the transmission matrix B:

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

l~lli= B V° -to

(A3)

B(I, 1 ) = 1,

A direct proportional relationship between the vessel diameter and its length was assumed: 1, = M - 2 " ~a./Tr

B ( I , 2) - r.; B(2, 1) = iwc.;

(A4)

(we performed the calculation with M = 8). Then h can be expressed in terms o f 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:

(AI2)

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 o f 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:

L ( n ) = 2/o(1 - h"+~)/(l - h), V ( n ) = 2/oAo(1 - (hk)"+')/( l - hk).

n ~2N

(A6)

T(N) = [IB,.

(A14)

n=0

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 - ( 2 h / k 2 ) " + ' ) / ( l - 2 h / k 2)

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 ( A I 4 ) 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:

(AS)

Qo = (1°o - Pout,o)/R(N) Q, - (T(2, 2 ) / T ( I , 2))P, - (det T / T ( I , 2))Pout,, (AI5)

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:

can be written, being respectively Pout,oand 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

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

Pom,I = 0.

(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: Po(n)-Po(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 )

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.